18736823 quantum world of nuclear physics

The Quantum Worldof Nuclear Physics

SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS

Founder and Editor: Ardeshir GuranCo-Editors: M. Cloud & W. B. Zimmerman

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Series A Volume 17

Founder & Editor: Ardeshir Guran

Co-Editors: M. Cloud & W. B. Zimmerman

The Quantum Worldof Nuclear Physics

Yuri A. BerezhnoyKharkov National University, Ukraine

\ ^ World ScientificN E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G - S H A N G H A I • H O N G K O N G • T A I P E I • C H E N N A I

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THE QUANTUM WORLD OF NUCLEAR PHYSICS

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Preface

Quantum physics, which governs the motions and interactions of micro-objects, is a firmly established part of our lives. It has provided workingformulae for the design of nuclear reactors, electronic devices, and super-conductive magnets, and has helped us understand processes occurring insolids, liquids, gases — even the stars. Many technologies are based directlyon quantum laws, which differ in principle from the vastly more familiarlaws of classical physics.

Quantum mechanics was originally formulated to explain the structureand properties of atoms. Its further development showed that it can de-scribe a huge variety of physical phenomena. It has served as a basis forthe creation of atomic and nuclear physics, elementary particle physics,and solid-state physics. It underpins the operation of semiconductors andlasers, of nuclear reactors and weapons, etc. These directions within sci-ence and technology are fascinating, and the many engineering and militaryapplications are of great importance in today's world.

Various aspects of quantum mechanics, describing the motion and struc-ture of molecules, atoms, atomic nuclei, and elementary particles, as well asthe structure of substances, do receive attention in high school physics text-books. The coverage given is, however, cursory at best; necessary conceptsare typically treated in a way that is both incomplete and inconsistent, andthe unfortunate reader is left without a solid understanding of this essentialbranch of physics.

The creation and development of quantum mechanics have also led todeep alterations in certain philosophical views regarding the world in whichwe live. This alone warrants an integrated and sufficiently simple presenta-tion of those quantum concepts that every educated person should know,regardless of his or her profession.

V

vi The Quantum World of Nuclear Physics

Quantum physics has shown that the main laws of Nature have a statis-tical rather than a dynamical character. This means that various physicalprocesses follow probabilistic laws, and the strict determinism of classicalmechanics is only revealed as a limiting case of this probabilistic descrip-tion. Moreover, the probabilistic behavior is characteristic of not only largeensembles but of individual objects (molecules, atoms, atomic nuclei, andelementary particles) as well.

Quantum mechanics may seem inaccessible to many people. The mainbarrier, however, is a simple hesitancy to abandon the habitual notionsof classical physics that are subject to constant reinforcement during oureveryday experience. Indeed modern physics, by its very character, lendsitself to understanding by anyone willing to devote sufficient time to itsstudy. The present book was written to facilitate this. No attempt hasbeen made to expound quantum mechanics to its fullest extent; rather, thepresentation is restricted to main ideas, concepts, and applications to thetheory of atoms and subatomic structures.

Despite the rather complicated mathematics that governs its behavior,the quantum world is a truly fascinating place to visit. The author invitesanyone with an interest in modern science to "take the plunge" into thismicroscopic universe of atoms and nuclei. The book will have fulfilled itsmission if it manages to spark some interest in atomic and nuclear physics.

The author is indebted to Michael Cloud and William B.J. Zimmermanfor the great work they have done editing the book, and to V.V. Pilipenkoand Yu. P. Stepanovsky for reading the manuscript and offering useful com-ments. He is also grateful to Jaime Humberto Lozano Parada and V. Yu.Korda for production assistance.

Yu. A. Berezhnoy

Contents

Preface v

1. Quantum Mechanics 1

1.1 Why Two Types of Mechanics? 11.2 Main Ideas and Principles of Quantum Mechanics 81.3 Measuring the Physical Characteristics of Microobjects . . . 171.4 Structure of Atoms 221.5 Structure of Matter 31

2. Fundamental Interactions 41

2.1 Gravitational Interaction 412.2 Electromagnetic Interaction 512.3 Weak Interaction 532.4 Non-Conservation of Parity in Weak Interaction 552.5 Strong Interaction 57

3. Structure of Atomic Nuclei 67

3.1 Composition and Properties of Nuclei 673.2 Shell Model of Nuclei 763.3 Collective Motions of Nucleons in Nuclei 813.4 Superfluidity of Nuclear Matter 86

4. Radioactivity of Atomic Nuclei 89

4.1 The Law of Radioactive Decay 894.2 Alpha-Decay 984.3 Beta-Decay 102

vii

viii The Quantum World of Nuclear Physics

4.4 Gamma-Radiation of Nuclei 1064.5 Exotic Types of Radioactivity 1074.6 Application of Radioactive Isotopes 109

5. Nuclear Reactions 115

5.1 Conservation Laws in Nuclear Reactions 1155.2 Nuclear Reaction Mechanisms 1195.3 Nuclear Optics 1275.4 Accelerators 1385.5 Detectors of Particles 145

6. Fission of Atomic Nuclei 147

6.1 Nuclear Fission Mechanism 1476.2 Chain Fission Reactions 1556.3 Nuclear Reactors 1596.4 Man-Made Synthesized Elements 162

7. Nuclear Astrophysics and Controlled Nuclear Fusion 167

7.1 Expanding Universe 1677.2 Creation of Atomic Nuclei 1717.3 Evolution of Stars 1767.4 Controlled Nuclear Fusion 181

Index 187

Chapter 1

Quantum Mechanics

1.1 Why Two Types of Mechanics?

We live in a complicated world. Our sense organs provide a steady flow ofinformation regarding the numerous phenomena that surround us. Powerfultechnological inventions also extend the reach of the human senses, givingus access to information more exact and complete than would otherwise beavailable.

The world of our sense perception is the macroscopic one. Here phys-ical phenomena are described by classical physics, which includes classicalmechanics, continuum mechanics (hydrodynamics and the theory of elastic-ity), thermodynamics, and electrodynamics. Because classical physics dealswith phenomena in which microscopic structure plays no significant role, itcannot yield a comprehensive theory of the structure of real substances.

The laws of classical physics govern the motions of objects whose lineardimensions are sufficiently large:

Rc\ > 10"6 m, say.

Nothing more powerful than an optical microscope will be needed to observesuch objects. Classical mechanics, in particular, describes the motions ofplanets, comets, stars, and galaxies.

But there exists another world, inaccessible to direct observationthrough our sense organs. This is the amazing world of micro-objects,in which physical phenomena are subject to the laws of quantum mechan-ics. The dimensions of molecules, atoms, atomic nuclei, and elementaryparticles are very small and could be characterized as

-Rqu < 1(T8 m.

l

2 The Quantum World of Nuclear Physics

Thus we have two distinct physical theories. One describes macroscopicphenomena, the other microscopic phenomena. Why do two types of me-chanics exist? The answer is far from simple. Let us pursue the questionin more detail.

Until the end of the 19th century, practically all physical phenomenawere described using classical mechanics. This subject was originally ex-pounded by Sir Isaac Newton in his "Philosophiae naturalis principia math-ematica", a monumental work published in 1687. The appearance of eachnew experimental fact had required only modification to some old equation,or perhaps the introduction of a new one, but had not cast doubt on clas-sical physics itself. Around the turn of the 20th century, however, existingapproaches based on classical physics failed to describe new experimentaldata on atomic and subatomic phenomena. Two of these were black bodyradiation and the photoelectric effect.

In physics, the term "absolutely black body" is used for an object thatabsorbs light (i.e., electromagnetic radiation) but does not reflect it. Amodel of a radiating absolutely black body is a closed box with impenetra-ble walls, in which a tiny hole is made so that we can observe the radiation.In 1860, Gustav Kirchhoff established that the intensity of black body radi-ation depends on temperature and frequency only, and not on the substancefrom which the walls are made.

In 1896 Wilhelm Wien found that the radiation energy per unit volumeand unit frequency (the density of black body radiation) decreases accordingto an exponential law

/9(w,T)~w3exp(-aa;/r),

where u> is the frequency, T is the temperature, and o is a constant. Wien'slaw holds for large frequencies (aui/T 3> 1). In 1900 Lord Rayleigh dis-covered that at low frequencies {aui/T <C 1) the radiation density has theform

p(u,T) ~w2T.

It was obvious that this latter relation, known as the Rayleigh-Jeans law,cannot hold for large frequencies; indeed it permits an absurd result latertermed the "ultraviolet catastrophe" — an unlimited increase in the radia-tion density p(u>,T). The behavior of the radiation density in the interme-diate range of frequencies was still an open question.

At the end of the 19th century the German physicist Max Planck inves-

Quantum Mechanics 3

tigated the dependence of the electromagnetic radiation intensity of a blackbody on frequency. Naturally, Planck was an adherent of classical physicsbecause there was no other science at that time. However, in 1900 he estab-lished that the available experimental data could be explained only underthe assumption that black body radiation is emitted in discrete portions.The energy of such a portion is given by the formula E = hu), where

h = A = 1.05457266 • 10~34 J • s. (1.1)

The constant h (here the bar reminds us of the division by 2TT) is calledPlanck's constant, and the energy portions are called quanta (from theLatin word "quantum", meaning "how much"). Planck's constant has thedimension of action (or of angular momentum); i.e., h is a quantum ofaction.

This result was really overwhelming, as a discrete character for energyis impossible in classical physics. Planck's law meant a qualitatively newapproach which required new physical and conceptual levels of understand-ing. This was the first "brick" to be laid in the foundation of an emergingphysical theory: quantum mechanics. For his work on the discovery ofenergy quanta, Planck was awarded the Nobel Prize in Physics for 1918.

P

/21

I

0 co

Fig. 1.1 Blackbody radiation density p versus frequency u>.

On the assumption that black body radiation is emitted in quanta,Planck obtained a formula for the radiation density, p(ui,T), which agreeswith the known limiting cases for small and large frequencies and also standsin good agreement with the experimental data over the whole frequency


range (Fig. 1.1):

P^T) = ^{etlr_iy (1-2)where k is Boltzmann's constant.

If the frequency is small (huj <C kT), then from (1.2) we can obtain theRayleigh-Jeans law

p(w,T)=u>2kT/(Tr2c3).

For large frequencies (fko 3> kT) Planck's formula (1.2) turns into Wien'slaw

p(u,T) = (w3h/TT2c3)exp{-fru/kT)

See Fig. 1.1.The next major step was taken by Albert Einstein in 1905. He in-

troduced the concept of a quantum of light and established that the pho-toelectric effect (the emission of electrons from a substance by light, aphenomenon discovered by Hertz in 1887) obeys the following law:

Em = hw - A, (1.3)

where Em is the maximum energy of an emitted electron, HUJ is the energyof the absorbed quantum of light, and A is the photoelectric work function(i.e., the energy required for emission of electrons from the substance).Einstein received the Nobel Prize in Physics for this discovery in 1921.Note that the term photon for a quantum of electromagnetic radiationwas introduced by J. Lewis in 1926.l Today the photon is regarded as anelementary particle with zero rest mass and spin 1, possessing energy fkoand momentum fuv/c.

In 1911 Ernst Rutherford discovered the atomic nucleus, whose linearsize turned out to be four orders of magnitude smaller than that of the atom.It became clear that an atom is a system in which electrons somehow movearound a very small nucleus, and the planetary model of the atom arose.There was, however, an important contradiction with classical physics. Anelectron moving along a curved trajectory through an electric field must

1 Lewis regarded the quantum of light as an indivisible atom, an assumption whichdid not stand the test of time. Nonetheless, his term "photon" has become a standardterm in physics and has even entered the popular lexicon. What would Star Trek bewithout photon torpedoes?

Quantum Mechanics 5

emit electromagnetic waves. The resulting loss of energy should be accom-panied by a decay in the electron's orbit, this process continuing until theelectron finally falls into the nucleus and the atom collapses. According toclassical physics then, any atom must cease to exist after some finite timeinterval. But it was already known that atoms are actually stable and haveno such constraint on their lifetimes.

This contradiction was resolved by Niels Bohr in 1913 in his famouswork "About the structure of atoms and molecules". Bohr theorized thatan electron in an atom can remain for an infinitely long time only in certainstates characterized by discrete values of the total energy. So the idea ofdiscreteness, already known for electron energy, was used to explain atomicstructure. Bohr obtained a correct formula for the electron energies in thehydrogen atom (the energy spectrum for this atom). For this purpose heused the equality of the centrifugal and electric forces acting on a chargedparticle (an electron) moving along a circular orbit in a Coulomb field:

^ = 4- (1-4)

Here e is the electronic charge (traditionally, in formula (1.4) the CGSEsystem of units is used), m is the electron mass, and rn and vn are the orbitalradius and velocity values. It is assumed that the mass of the nucleus isinfinitely large in comparison to that of the electron. The number n {mainquantum number} takes positive integer values n = 1, 2, 3 , . . . and serves toindex the possible states of the atom (the electron orbits).

Furthermore, Bohr supposed that the angular momentum of the electroncan take only discrete values:

mvnrn — nh. (1-5)

While the relation (1.4) is usually referred to as "classical," the relation(1.5) is referred to as "quantum mechanical" as it involves the quantizationof angular momentum (the requirement that angular momentum values bediscrete). However, in today's established theory of quantum mechanics,the angular momentum cannot be written as a product of the momentummvn and distance rn. So the left-hand side of (1.5) is a purely classicalexpression for the moment of momentum.

From (1.4) and (1.5) one can readily obtain the quantities rn and vn:

h2 2 e2 , .me2 hn


The total energy En of an electron in the hydrogen atom is a sum ofthe kinetic energy mv^/2 and the potential energy —e2/rn. Therefore, wefind that

_ me4

^—wj- (L7)

The negative sign signifies that the electron exists in a bound state. Thefirst five energy levels in the spectrum of the hydrogen atom are shown inFig. 1.2, where one can see the levels getting closer as the binding energy ofan electron decreases. Later, the energy expression (1.7) was obtained onthe basis of quantum mechanics via solution of the Schrodinger equation.

S = n=4n=2

n=1

Fig. 1.2 The first five levels of the energy spectrum of the hydrogen atom.

While deriving (1.7) Bohr postulated the concept of permissible orbitsof the electron, having radii rn, as well as the possibility of emission andabsorption of a certain portion (quantum) of electromagnetic radiation bythe electron during its transition between orbits. This is the essential con-tent of Bohr's two famous postulates. It is obvious that the energy fiu> ofsuch a quantum must equal the difference of the electron energies for theseorbits. If the electron passes from an orbit with energy En to one withenergy Es (we assume that n > s), it emits a quantum (a photon) withenergy

tlaJ = En-Es. (1.8)

Formula (1.8) is the Bohr frequency relation. Substituting the expres-

Quantum Mechanics 7

sion for the energy (1.7) into it, we obtain

» = £(?-;?)• (")By fixing the number s we obtain a series of frequencies (the Lyman

series for s = 1, the Balmer series for s = 2, the Paschen series for s = 3)within which the number n defines a certain line. For each series we mustkeep n > s. Bohr noted that from (1.9) it is impossible to obtain otherseries of frequencies that were ascribed to hydrogen at the time. He wasright to point out that these series belong to helium.

Bohr's frequency relation (1.8) or (1.9) was important because the radi-ation frequency u> did not coincide with that of the electron orbit. For thisreason the result differed in principle from, for example, those of Nicolson(who assumed these two frequencies must be equal). For his investigationof the structure of atoms and the radiation emanating from them, Bohrwas awarded the Nobel Prize in Physics for 1922.

Let us emphasize that in the works of Planck, Einstein, and Bohr, whichformed the basis for the development of quantum mechanics, the corre-sponding final formulae were correct, but the methods by which they werefound are subject to criticism. In fact, these scientific giants used theirpowerful intuition to guess the necessary results and causes of the corre-sponding physical phenomena. However, these results cannot be obtainedin the framework of classical physics as it existed at that time. Quantummechanics, as elaborated later, made it possible to derive these formulae bymeans of quite different and strict methods. In other words, the methodol-ogy for obtaining quantum results used by Planck, Einstein, and Bohr wasnot correct, and is now of mainly historical interest. Let us also note thatthe energy formula (1.7) obtained by Bohr is valid only for the hydrogenatom, but for more complex atoms Bohr's approach does not hold. More-over, the approach is valid only for circular orbits, and cannot be appliedto elliptical ones in this form.

The explanation of radiation from an absolutely black body, the photo-electric effect, and the structure of the hydrogen atom required the intro-duction of new concepts that turned out to be incompatible with classicalphysics. This quandary was the impetus for the creation of a new theory:quantum mechanics. So it is important to ascertain whether there existgeneral properties of our world which require the existence of two types ofmechanics.

In physics, there is a relation between the fundamental symmetry prop-


erties of space and time and the laws of conservation of certain physicalquantities. For example, the equivalence of all moments of time (temporalhomogeneity) leads to the law of energy conservation for a closed (isolated)system. Similarly, the equivalence of all points of space (spatial homogene-ity) leads to the law of conservation of momentum, and the equivalence ofall directions in space (the isotropy of space) leads to the law of conser-vation of angular momentum. These symmetry properties are geometricalsymmetries since they are not connected with concrete types of interaction.

Space possesses one more geometrical symmetry, namely that of simi-larity (scale invariance), which is connected with changes in spatial scale.However, the laws of nature turn out not to be invariant with respect toa similarity transformation (scale change). In reality then, the principleof similarity is not valid. This can be formulated as follows: the absolutesize of a body does matter; two systems that are geometrically similar butdifferent in scale are fundamentally different in mechanical behavior.

The failure of the similarity transformation was known to Galileo. Heunderstood that if the sizes of animals and men were to be essentiallyincreased, a significant increase in the firmness of their bones would becomenecessary; otherwise the bodies of giants would collapse under their ownweight. Another example is a model of a building (e.g., a house) made ofmatchsticks. If such a model were scaled up to realistic dimensions, it wouldcollapse similarly. So the principle of similarity is not valid at macroscopicscales.

The existence of the smallest natural building blocks — molecules andatoms — which have finite sizes, the existence of the elementary electriccharge, and the limiting speed of signal propagation (the speed of light),all imply a failure of the similarity principle and non-invariance of naturallaws with respect to scale transformations. Hence follows the existence oftwo types of mechanics: (1) the classical version, valid at distances largecompared with the linear sizes of molecules and atoms, and (2) the quantumversion, whose laws describe physical processes at distances comparablewith the linear sizes of molecules and atoms or smaller.

1.2 Main Ideas and Principles of Quantum Mechanics

In 1923 the French physicist Louis de Broglie suggested that a materialparticle having nonzero mass also possesses wave properties uniquely relatedto its mass and energy. He ascribed a wavelength to a free particle, which

Quantum Mechanics 9

is related to its momentum p by the formula

The quantity X (or A) has been called the de Broglie wavelength for theparticle. The concept of the wave nature of particles with nonzero restmass was developed by de Broglie in a series of works in 1924. He receivedthe Nobel Prize in Physics for the year 1929.

No one in France could grasp the profundity of de Broglie's idea. Hisdoctoral committee (consisting of the physicists Perrin and Langevin, themathematician Cartan, and the crystallographer Mauguin) may not haveeven conferred his degree. But Langevin sent the thesis to Einstein forevaluation, and the answer was "He has lifted an edge of the great curtain."De Broglie's formula (1.10) deserves a place alongside Planck's formulaE = hu and Einstein's formula E — me2.

With de Broglie's discovery it became clear that material particles pos-sess physical properties similar to those of quanta of electromagnetic radi-ation (photons): they have both wave properties and particle properties.The discovery of the wave nature of microparticles was a brilliant guesswhich led to a true upheaval in our understanding of the physics of mi-crophenomena.

In 1927 the Americans Davisson and Germer and, independently, theEnglishman G. Thompson, discovered electron diffraction by crystals ex-perimentally. This was a remarkable corroboration of the predicted waveproperties of material particles. Davisson and Thompson received the No-bel Prize in Physics for 1937.

We should stress that wave properties are inherent even in a single ma-terial microparticle. If one passes an electron beam of very low intensitythrough a crystal, so that individual electrons fly through the crystal in-dependently of one another, then with sufficiently long exposure the samediffraction pattern will be observed as for a beam of high intensity. Since anindividual electron causes the blackening of only one grain of the photoemul-sion on the screen, electron diffraction means that one can only indicate aprobability that an electron will reach some point of the screen. But it isimpossible to calculate its trajectory.

Thus, in 1924 two basic ideas of quantum mechanics were alreadyknown. The first was that of quantization: the possibility that a physi-cal quantity will take a discrete series of values under certain conditions.The second was the wave-particle duality of quantum objects, i.e., that any

(10)


quantum object (photon, electron, atom, molecule, etc.) is both a corpus-cle (particle) and a wave at the same time. Under different conditions theparticulate or wave properties of a certain object can be manifested to agreater or lesser extent.

Both quantum ideas were in absolute contradiction with classicalphysics, in which physical quantities take only continuous values. More-over, in classical physics a particle and a wave are incompatible, whereas inquantum mechanics a particle and a wave cannot be separated. We stressthat the wave-particle duality of a microobject should be understood asits potential ability to manifest its particulate or wave properties, depend-ing on the conditions under which it is observed. Particles and waves aredifferent forms of the same physical reality.

In 1925 the German physicists Heisenberg, Born, and Jordan elaboratedmatrix mechanics, which was the first variant of quantum mechanics. In1926 the Austrian physicist Erwin Schrodinger developed wave mechanics,based on an equation later named after him. To describe a microobjectstate, Schrodinger introduced the wave function (the i[>-function). In 1926-1927 the Englishman Paul Dirac made a great contribution to the elab-oration of mathematical techniques for quantum mechanics, and in 1927proposed the method of secondary quantization. For the creation of quan-tum mechanics, Werner Heisenberg was awarded the Nobel Prize in Physicsfor 1932. Schrodinger and Dirac were awarded the Nobel Prize in Physicsfor 1933 for the discovery of new productive forms of atomic theory.

In 1926 Born gave a statistical interpretation of the wave function V>(r)as a probability amplitude such that the quantity |i/>(r)|2d3r yields theprobability that the particle resides in volume d3r in the vicinity of thepoint r. We stress that the wave function can be a complex quantity. It canalso depend on time. For his fundamental research in quantum mechanics,especially for his statistical interpretation of V'(i'), Born was awarded theNobel Prize in Physics for 1954.

Although the mathematical body of quantum mechanics was mainlyelaborated in 1925-1926, it remained unclear why Heisenberg's matrix me-chanics and Schrodinger's wave mechanics yield the same results. Develop-ment of mathematical techniques has made it clear that these are simplytwo equivalent variants of the same science, subsequently called quantummechanics. The probabilistic character of the laws of quantum mechanicsis due to the intrinsic randomness of the behavior of microobjects. Quan-tum mechanics can only predict the probability that a physical quantitywill take a given value. For this reason, the probability concept plays a

Quantum Mechanics 11

fundamental role in quantum mechanics. However, probability in quantummechanics differs substantially from this notion elsewhere in physics.

The wave function is a probability amplitude and, generally speaking,can be a complex quantity. As distinct from other parts of physics, inquantum mechanics the wave functions (the probability amplitudes) areadded to each other, but not the probabilities themselves. In this way,interference terms arise in the total probability.

Let us illustrate the interference of probability amplitudes as an exampleof the superposition principle, which plays an important role in quantummechanics. In the simplest variant it can be formulated as follows. If agiven physical system can be in a state described by a wave function ip\and the same system can be in another state described by a wave functionfa, then it can also be in the state described by the function

^ = 01^1+^2^2, (1-11)

where oi and a<i are constants (generally speaking, complex) satisfying thecondition |ai|2 + |a2|2 = 1.

Let a measurement, carried out with the system under considerationin the state described by wave function fa, yield result 1, and a similarmeasurement, carried out with the system being in the state described bywave function fa, yield result 2. Then the same measurement, carriedout with the same system being in the state described by the wave function(1.11), will yield result 1 with probability |ai|2 and result 2 with probability

M2.The superposition principle leads to an important consequence: the

equations of quantum mechanics must be linear. Indeed, the Schrodingerequation, which is the basic equation of nonrelativistic quantum mechanics,is linear.

If the number of states in which the system can exist is greater thantwo, then the superposition principle appears as follows:

^ = X)an^n, (1.12)n

where the sum is extended over all possible states of the system. Thenumber of system states can be infinite. Then the sum in (1.12) will containan infinite number of addends.

The probability that the system is in the state described by the wave


function (1.11) is defined by the expression

w = h/f = M V i l 2 + M2|V>2|2 + (aiaSVi^a + a\a^l^)- (1.13)

The probability w consists of three terms. The first two are the probabili-ties of finding the system in the states 1 and 2, respectively, while the thirdrepresents an interference of the probability amplitudes. The occurrence ofinterference terms is characteristic of quantum mechanics. It is connectedwith the intrinsic wave properties of microobjects. The interference existingin quantum mechanics is the main conceptual difference between the classi-cal and quantum descriptions of objects and processes. It can be illustratedby considering a particle motion along different paths from point 1 to point2. In classical physics the probabilities of the particle motion along eachpath are summed. In quantum mechanics the amplitudes associated withthese paths are summed, and then the squared modulus of the resultingamplitude is the probability of the particle having passed from point 1 topoint 2.

In 1927 Heisenberg established the uncertainty relation. Let us con-sider its simplest form. If we introduce the root-mean-square deviations(dispersions) for the coordinate and momentum

<(Az)2} = (x2) - (x)\ <(APl)2> = {pi) - <Px)2, (1.14)

then it turns out that their product for any microobject cannot be smallerthan the value fh/2:

((Ax)2)((APx)2) > ^ . (1.15)

The angle brackets in (1.14) and (1.15) denote the averaging of quantitiesin the microobject state under consideration.

Formula (1.15) is the uncertainty relation for the position and momen-tum. It means that the more precisely we determine the position of amicroobject, the greater the indeterminacy of its momentum componentalong the same axis will be, and vice versa.

The uncertainty relation (1.15) shows that in quantum physics the si-multaneous use of the notions of position and momentum is nonsense, al-though the very notions of position and momentum do have physical mean-ing. Uncertainty relations similar to (1.15) also exist for other pairs ofquantities which are called conjugate pairs (for example, the energy of amicroobject and the time of its interaction with a measuring device, the


projection of the angular momentum onto the axis of quantization and theazimuthal angle, etc.).

Physical laws have, as a rule, the form of certain exclusions (restrictions)imposed upon physical quantities under consideration. However, only threeexlusions hold throughout physics.2 The first is the uncertainty relation,which forbids the product of the dispersions of two conjugate physical quan-tities to be less than fr?/4. The second is the impossibility of existence ofperpetual motion of the first and second kinds. The third does not allow aparticle with nonzero rest mass to increase its velocity from a value v < cto a value v > c (the impossibility of crossing the light cone).

The universal character of the uncertainty relation means that it isalso valid in classical physics. However, owing to the smallness of thequantity h, it is never manifested in classical physics because in this casethe uncertainties (dispersions) ((Ax)2) and ((Apx)2) are so small that theycannot be noticed. Let us emphasize that the existence of the uncertaintyrelation is a general law of quantum mechanics which is not connected witha method or accuracy of measurement of the corresponding quantities.

The uncertainty relation (1.15) leads to an important consequence.Since it is impossible to determine simultaneously a position of a microob-ject and the corresponding component of momentum with arbitrary accu-racy, then this implies the absence of any trajectory of the microobject,which is defined in classical physics as a function p(r). In other words, ifwe know the particle position at a given moment of time, we cannot know itat any subsequent moment. Rather, we can only determine the probabilityof finding the particle at a point in space at any subsequent moment.

The absence of trajectories of microobjects turns out to be of greatimportance for systems of identical particles. If at a given moment of timewe number identical particles, then at any following moment we cannotindicate where each of the numbered particles is.

This property of systems of identical particles can be formulated in thefollowing way: the physical properties of a quantum system of identicalparticles are invariant with respect to the permutation of any pair of par-ticles in this system. In particular, the probability defined by the squaredmodulus of the wave function of a system of identical particles, \4>\2, isinvariant with respect to such a permutation. However, it is clear that inthis case the wave function ip itself may be not invariant with respect tothe permutation of a pair of identical particles.

2 It is worth noting that the familiar conservation laws are not among these so-calledglobal exclusions.


There are two possibilities: the wave function is symmetrical and doesnot change under the permutation of a pair of identical particles, or it isantisymmetrical and does change sign. It is proved in quantum mechanicsthat, if the wave function of a system of identical particles is symmetrical(antisymmetrical) at a given moment of time, then it will be symmetrical(antisymmetrical) for all time.

Thus, a general problem arises to determine which systems of identicalparticles are described by symmetrical wave functions and which are de-scribed by antisymmetrical ones. It turns out that the symmetry propertiesof a wave function are completely denned by the spin (intrinsic angular mo-mentum) of particles which, like the orbital moment, is measured in unitsof Planck's constant h. If the spin of a particle is an integer (possibly zero),then the wave function is symmetrical, and if the spin is half-integral, thenthe wave function is antisymmetrical.

Spin is an inherent characteristic of a particle, like its mass and charge.It is a purely quantum characteristic, having no analogue in classicalphysics. In quantum mechanics the orbital moment of a particle, whichis related to motion in space, takes only discrete values that are multiplesof h. For this reason, in quantum mechanics it is convenient to measureangular momentum in units of Planck's constant h. The spins of differentparticles can have (in units of h) both integral and half-integral values. Thespins of electrons, protons, neutrons, and neutrinos are equal to 1/2, thespins of 7r-mesons are equal to zero, and the spin of a photon equals 1.

If identical particles possess an integral spin, then they obey the Bose-Einstein statistics and are called bosons. If identical particles have a half-integral spin, then they obey the Fermi-Dirac statistics and are calledfermions. A system of identical bosons is described by a symmetrical wavefunction, and a system of identical fermions by an antisymmetrical wavefunction.

In 1924-1925 Wolfgang Pauli investigated the properties of systems ofidentical particles, and established what was later to be called the Pauliexclusion principle:

Two identical fermions cannot be in the same state.

For this discovery Pauli was awarded the Nobel Prize in Physics for 1945.Although Pauli provided a proof of the exclusion principle, the latter hadbeen used intuitively by Bohr in 1921-1922 when he classified atoms on thebasis of their electron structure.

The validity of the exclusion principle can be easily understood as fol-


lows. Let us assume that two identical fermions are in the same state. Thentheir permutation does not change the wave function of the system, since inthis case the system turns into itself by virtue of the indistinguishability ofthe identical particles. On the other hand, the wave function of the systemof two identical fermions must be antisymmetrical, i.e., it must change itssign under their permutation. For this reason, the wave function in thecase under consideration equals zero. In other words, the probability oftwo identical fermions being in the same state is equal to zero.

If one considers a system of identical particles at low temperature, thenit turns out that a system of identical bosons behaves in a different wayas compared with a system of identical fermions. At absolute zero all theidentical bosons would be in their lowest energy state. In this case, Bosecondensation arises.

In the case of a system of identical fermions, no more than one particlecan be in each state, even at absolute zero, owing to Pauli's exclusionprinciple. For this reason, a certain distribution of fermions over energiesarises. Since atoms of different elements have different chemical properties,and since electrons in metals, even near absolute zero, have large energies,it is obvious that various physical characteristics of systems of identicalbosons and fermions near absolute zero must be different.

Let us consider another interesting quantum effect. The solution ofthe Schrodinger equation for the potential of a harmonic oscillator showsthat a quantum oscillator with frequency u> has the lowest energy EQ =tvuj/2, which is called its zero-point vibrational energy. In other words, thequantum oscillator differs from the classical one in the fact that it does notrest in its lowest energy state, but performs these zero-point vibrations withenergy E0- In quantum mechanics, it can be shown that the energy EQ isthe minimum energy of the oscillator which is permitted by the uncertaintyrelation.

This amazing property of quantum oscillators has been verified exper-imentally. In particular, it means that, when approaching absolute zero,atoms situated at points of the crystal lattice of a solid perform zero-pointvibrations. One can investigate the scattering of electromagnetic radiation(photons) on atoms of the lattice. From the character of this scattering attemperatures close to absolute zero one can determine whether atoms atthe points of the lattice rest or vibrate. These experiments have uniquelyshown that atoms at the lattice points perform the zero-point vibrationswith the energy Eo, because the zero-point vibrations substantially changethe character of the photon scattering.


Zero-point vibrations are also performed by any physical field even inthe absence of the particles that are quanta of this field. For example, zero-point vibrations of an electromagnetic field lead to a small change of theenergy levels of atoms as compared to those calculated without taking theminto account. This phenomenon was for first time observed experimentallyby the American physicist Willis Eugene Lamb in 1947. It has been calledthe Lamb shift, and has been explained by calculations carried out on thebasis of quantum electrodynamics. For this discovery Lamb was awardedthe Nobel Prize in Physics for 1955.

The concept of a field plays a very important role in modern quantumphysics. A field can exist even in the case when the corresponding particlesare absent. This state of a field is called vacuum. However, there arezero-point vibrations that occur in a vacuum, which sometimes leads to aspontaneous creation of a pair consisting of particle and antiparticle. Thispair is subsequently annihilated. For example, in an electromagnetic fieldelectron-positron pairs can be spontaneously created and annihilated. It isfor this reason that in quantum theory the distinction between a particleand field disappears.

The successes of classical physics had led to a profound confidence in thepossibility of unambiguous predictions of various physical events. The mostprominent representative of this conception of necessity was the famousFrench scientist Pierre-Simon Laplace. In his words,

"An intellect, which could know all the forces in the natureand the relative situation of the things composing it ata given instant and which could be powerful enough tosubject them to an analysis, would be able to comprehendthe motions of the greatest bodies in the Universe and ofa smallest atom by a single formula, and nothing wouldremain unknown, both the past and the future would beopen to him."

So Laplace believed that a knowledge of the positions and velocities ofall particles of a physical system made it possible to predict its behavior.This statement expresses the "Laplace determinism" that underlies classicalphysics.

In quantum mechanics the situation is different: it is impossible to knowthe positions and velocities of particles simultaneously, by virtue of theuncertainty principle. One can only define the wave function at an initialmoment of time, which is the most complete description of the state of a


physical system. In quantum mechanics, which uses the wave function, onlya probabilistic description of events is possible. Laplace's classical causalitydoes not hold for microobjects, whose behavior is inherently random.

The probabilistic character of the quantum description of the behaviorof microobjects is also rooted in the interaction of a microobject with therest of the world. It is just the attempt to isolate a microobject or somephysical system from the rest of the world that leads to the randomness intheir behavior. In the words of Heisenberg,

"It is necessary to draw one's attention to the fact that asystem which should be considered according to the meth-ods of quantum mechanics actually is a part of much largersystem and, ultimately, of the whole world. It is in an in-teraction with this large system, and we have also to sayin addition that microscopic properties of the large systemare, at least to a great extent, unknown".

That is why, by virtue of the probabilistic character of the quantum laws,the physical processes of microobjects are characterized by a probabilisticform of causality, of which Laplace's determinism is only a limiting case.

However, one should not think that quantum mechanics is not a deter-ministic theory. It is deterministic in the sense that it defines the law of thewave function variation with time. Elements of unpredictability and ran-domness arise only with attempts to interpret a microobject on the basisof classical conceptions about its positions and velocities.

1.3 Measuring the Physical Characteristics of Microobjects

An important criterion for the validity of a physical theory is agreementbetween calculated predictions and experimental data. Another is the abil-ity to predict results of future measurements from those already available.Hence the measurement of physical characteristics of microobjects is anissue of great importance in quantum mechanics.

We are macroscopic objects living in the macroscopic world. This ne-cessitates that any device (detector) used for quantum measurements alsobe a macroscopic object, so that we can perceive the information it obtains.Such a device must also, as a macroscopic system, exist in a rather unstablestate that can be easily changed under the influence of a microobject. Forexample, in a bubble chamber a superheated transparent liquid instantly


boils when a charged particle flies through it (Sect. 5.5). The resulting bub-bles of vapor allow us to see the particle trajectory. Here the thickness ofthe trajectory is large in comparison to the atomic scale, so the uncertaintyrelation is not violated. By such a rough approximation a microobject canbe considered classically.

The detector necessarily and significantly changes the state of the mi-croobject with which it interacts. In classical physics it is assumed thatthe influence of a measurement device on an object can be made arbitrarilysmall. During a quantum measurement, one cannot in principle neglectthe interaction between the detector and the microobject. Consequently,the uncontrollable character of the interaction between detector and mi-croobject leads to the necessity of the probabilistic description of quantumprocesses, since the measurement destroys the initial quantum state of themicroobject in an unpredictable way.

The process of a quantum measurement is irreversible: as a result ofthe measurement, the wave function of the microobject changes abruptly;i.e., a reduction or "collapse" of the wave function occurs. Reconstructionof the initial state of the microobject is absolutely impossible after themeasurement process has occurred. Thus the irreversibility of the processof measurement plays a fundamental role in quantum physics.

By virtue of this irreversibility, an irreproducibility of a single measure-ment arises. Since in each act of measuring the interaction between thedetector and microobject occurs in a different way, the measured resultswill be different. Only a sufficiently large number of measurements willgive a certain stable pattern of the distribution of results. This pattern canalso be obtained in another series including a sufficiently large number ofmeasurements.

The state of a microobject is not defined before a measurement. Aseries of measurements performed with the same detector over identical, asone would think, microobjects yields a set of different results. If a beam ofelectrons passes through a slit in a screen, then different electrons will reachdifferent points of a photoplate and a certain diffraction pattern will appear.In this case one can only determine the probability that the electrons arriveat different points of the photoplate. In other words, a certain statisticaldistribution of electrons on the photoplate arises which is not chaotic. Thetask of quantum mechanics is to determine the probability distribution forvarious physical quantities characterizing microobjects.

Let us now direct the electron beam onto a screen with two slits A andB (Fig. 1.3). If A is open and B is closed, then on the screen-detector we


Aw

....!?w ..... x

Fig. 1.3 Distribution of electrons passing through a screen with two slits.

observe the electron distribution I\ (x) that corresponds to a wave functionipi. If only B is open, the electron behavior will be described by a wavefunction ip2 and we will observe the electron distribution h{x). If A and Bare open, then the wave function of the electrons is ip = V>i + 4>2- In thiscase each electron, owing to its wave nature, will pass through both slitssimultaneously. The probability density of this process is denned by theexpression

w = |^|2 = i^i2 + ^ 2 | 2 + (1/^2 + v.1^5). (1.I6)

The third term in (1.16) describes the interference of waves passing throughthe two slits simultaneously. This case corresponds to the electron distri-bution I{x).

If the electrons are allowed to pass through the slits opened in turn,then the probability distribution of this process will be defined by a sum ofthe probabilities of electron passage through each slit separately:

w = \ipi\2 + \ip2\2- (1-17)

In this experiment the interference disappears; i.e., controlling the electronpassage through a certain slit in the screen destroys the interference. Inother words, quantum measurement destroys interference and we observe


the electron distribution Iz{x).Quantum measurements are characterized by the union of a measuring

device and the microobject over which the measurement is performed. It isthis inseparable union of a macroscopic (classical) device and the microob-ject under analysis that leads to their uncontrollable interaction changingthe microobject state.

Quantum mechanics is a probabilistic theory, which makes it principallydifferent from classical physics. However, it turns into classical mechanicsin the limiting case when Planck's constant h becomes negligible. Formally,the transition to classical mechanics is performed when one lets this quan-tity decrease to zero: h —» 0. Quantum mechanics is grounded in, andtherefore irrevocably linked to, classical mechanics.

The limit h —> 0 can be understood as follows. If n > 1, then, accordingto (1.5), the angular momentum of an electron in an atom becomes verylarge relative to Planck's constant: mvnrn 3> h. In other words, in the casen > 1 the constant h can be neglected and the discreteness of the angu-lar momentum disappears. Thus, quantum mechanics turns into classicalmechanics when quantum numbers are large — this is the correspondenceprinciple initially formulated by Bohr.

In particular, this means that for large quantum numbers the frequencyof radiation emitted by an atom at the transition from one state to anotherasymptotically coincides with the frequency predicted by the classical the-ory. If the atom passes from an excited state with energy En+i to a statewith energy En, then the radiation frequency equals uin = (En+\ — En)/h,where En and En+\ are defined by (1.7). Then

_ roe4 [ ( n \2

For n » 1 we get 1 — (n/n + I)2 « 2/n. Thus, we find that

_ me4 _ vn

h3n3 rn'

where the electron velocity vn and the "orbital radius" rn are defined bythe formulae (1.6).

In classical mechanics the frequency w of the electron revolution withvelocity v along the orbit of radius r is equal t o u = v/r. We can see thatfor n » 1 the quantum result coincides with this. Moreover, the distancebetween neighboring energy levels of the hydrogen atom tends to zero for


large quantum numbers (n —• oo). In this limiting case, the discreteness ofthe energy spectrum becomes less significant and the atomic system behaveslike a classical one.

Because of its probabilistic approach to the explanation of micropro-cesses, quantum mechanics was rejected by some scientists during its devel-opment. One of the opponents of the quantum theory was Einstein, whohelped sow the seeds for its creation. He proposed various arguments toprove that quantum mechanics is not valid. However, Bohr, also a giantof Physics, parried with cogent counterarguments in support of quantummechanics.

In 1935, a paper by Einstein, Podolsky, and Rosen appeared, in whichthey put forward their famous paradox. According to their initials, it be-came known as the EPR-paradox. We will not expound the essence of thisor certain other paradoxes that were put forward during the development ofquantum mechanics in order to prove its incompleteness. We shall only notethat the outstanding intellect of Niels Bohr resolved all these paradoxes.

Discussions regarding the completeness of the quantum-mechanical de-scription of microobjects have led to the conjecture that the uncertaintyin the behavior of a quantum object is explained by the existence of some"hidden" parameters, about which observers know nothing. Just the pres-ence of these hidden parameters could lead to the probabilistic behaviorof microobjects and to the uncertainty of the results of measurements. Itfollowed from this approach that knowledge of the hidden parameters couldallow one to predict exactly the microobject behavior, i.e., that the deter-minism of classical physics would triumph.

The first proof of the nonexistence of hidden parameters was given byJ. von Neumann. However, a formulation of the proposition was requiredwhich could experimentally corroborate the absence of hidden parameters.Finally, in 1965, Bell proposed a statement (the Bell theorem) that made itpossible to ascertain experimentally the distinction between the predictionsof quantum mechanics and the theory of hidden parameters.

Experiments based on the Bell theorem were carried out by Clause andFreedman at the University of California in 1972, as well as by Aspect,Dalibard, and Roger at the Paris Institute of Optics in 1982. These andother experiments have proved the validity of quantum mechanics and thefailure of the hidden parameter theory. Undoubtedly, experimental studiesin this direction will continue. However, the theory of hidden parameters,at least in its present form, does not agree with experimental data.


1.4 Structure of Atoms

Bohr's 1913-1922 works, which investigated the structure of atoms, gavecorrect results which were, in fact, guessed, since the methods used toobtain them were incorrect. Later, these results were obtained on the basisof quantum mechanics.

Bohr's formulas (1.6) and (1.7) are of value because they allow us topredict a characteristic linear size for the hydrogen atom, as well as a char-acteristic velocity and energy for the electron in the atom. For n = 1 (theground state of hydrogen) we have

n = - ^ w 0.53 • 1CT10 m,me1

e2Vl = — « 2.2 • 106 m/s,

4TflP

£?i = - 2 ^ - « - 1 3 . 6 eV. (1.18)

The quantity r\ has been called the radius of the first Bohr orbit. It showsthat the characteristic linear size of the atom, by order of magnitude, isRat ~ 10~10 m. Using the constants m, e, H, one can compose the quantityrat = ?l3/(me4) ss 2.5 • 10~17 s, which is a characteristic atomic time.

To understand to what extent the smallest particles of substance aresmall, let us consider a continuous chain composed of atoms lined up sideby side. If every inhabitant of our planet had put one atom into this chain,its length would be equal to several centimeters.

By virtue of the uncertainty relation (1.15), a microobject describedby quantum mechanics has no trajectory. This means that there exist noelectron orbits in atoms, only probabilities of different distances betweenthe electron and nucleus. For example, for the ground state of the hydrogenatom (n = 1) this probability equals

wi(r) = Cr2exp (~) , (1.19)

where C is a constant.The probability (1.19) has a maximum at r = r\, i.e., the radius of

the first Bohr orbit is the most probable distance between the electron andnucleus in the ground state of the hydrogen atom. We stress that in realitythe electron does not revolve around the atomic nucleus like a planet movingaround the Sun. It simply exists in the atom and can be found at any point


of the atom with a strictly definite probability. According to quantummechanics, a hydrogen atom in the ground state has spherical symmetry,whereas in classical mechanics a system consisting of two charged particlescannot be in a spherically symmetrical state.

We note that the initial approach of Bohr, which is based on classicalmechanics, gave the correct formula (1.7) for the energy of an electron ina hydrogen atom, but it led to an incorrect value of the electron orbitalmoment I. Indeed, for n = 1 (the ground state of the hydrogen atom),formula (1.5) yields the value I = 1 (in units of H); according to quantummechanics, it should be I = 0.

The structure of atoms is mainly denned by the electric interaction ofatomic electrons with the nucleus and between the electrons themselves. Ifthere are two or more electrons in the atom, then this is already a complexsystem of mutually interacting particles. However, it turns out that it isa good approximation for a complex atom to consider that each electronmoves in an effective central field which is created by the nucleus and otherelectrons. Such a field is called the self-consistent field.

A state of an electron in a central field (in a spherical atom) is char-acterized by the main quantum number n, the orbital moment I, and theprojections of the orbital moment m and of the spin a on the quantizationaxis. The electron spin projection can take two values: a = ±^. The elec-tron orbital moment projection (the magnetic quantum number) can take21 + 1 values, since it takes integer values with the limits —l<m<l. Theelectron orbital moment can take the values I = 0,1,2,..., and the mainquantum number n = I + 1; I + 2; I + 3; . . . .

According to the Pauli principle, only one electron can be in the statewith quantum numbers n,l,m, a. All of the atomic electrons with the samequantum numbers n and I are said to be in the same electron shell of theatom. Such electrons possess the same energy and are said to be equivalent.An electron shell can contain 2(21 + 1) equivalent electrons having differentvalues of m and a. All of the 2(2Z + 1) atomic electrons having the same nand I values form a perfectly occupied (closed) shell.

Table 1.1 Electron states.

1 0 1 2 3 4 5 6s p d f g h i

2(21+1) 2 6 10 14 18 22 26


To denote the states of electrons with definite values of orbital mo-ment, the literal notation presented in Table 1.1 is used. The lower line ofTable 1.1 contains the total number of electrons in a closed shell with thegiven values of quantum numbers n and I. The state of the atomic electronwith definite values of n and I is denoted by a figure indicating the value ofn and a letter indicating the value of /. Sometimes, one refers to electronswith main quantum numbers n = 1; 2; 3; 4; 5; 6; . . . , as being in the K-,L-, M-, N-, O-, P~, Q-, ..., shells, respectively.

/ / / / / / /1s 2s 3s 4s 5s 6s 7s

/ / / / / / / /2p 3p 4p 5p 6p 7p

/ ' / / / / /3d 4d 5d 6d 7d

4f 5f 6f 7f/ / / /

Fig. 1.4 The order in which atomic shells are filled is shown by arrows.

The order of filling of the atomic shells is shown by the arrows in Fig. 1.4.However, one should remember that there are some exceptions to this rule.The distribution of electrons in shells is called the electron configuration ofthe atom. Let us consider this for some particular atoms.

A hydrogen atom contains one electron in the state Is. In a helium atomthere are two equivalent electrons in the same state, which differ by thevalues of their spin projections on the quantization axis. For this reason,an atom of helium has electron configuration Is2, where the superscriptdenotes the number of equivalent electrons in this state.

A lithium atom has the electron configuration Is22s, while a berylliumatom has the configuration Is22s2. In the boron atom the first electronwith I = 1 (the p-electron) appears. The electron configuration of thisatom has the form Is22s22p. Further, the filling of the 2p-shell continues:a carbon atom has the electron configuration Is22s22p2, a nitrogen atomhas the configuration Is22s22p3, and so on. An atom of neon, which is aninert gas, has all shells filled: Is22s22p6.

In atoms of sodium and magnesium, the 3s-shell is filled. This isalso true for the 3j>-shells in atoms of the elements from aluminum toargon. In atoms of potassium and calcium the 4s-shell is filled, and


in atoms of the elements from scandium to zinc the 3<i-shell is filled.Here, the first violations of the successive filling of the electron statesin the shells appear: a chromium atom has the electron configurationIs22s22p63s23p64s3d5 instead of ...4s23d4, and a copper atom has theconfiguration Is22s22p63s23p64s3d10 instead of ... 4s23d9.

Atoms of the elements in which the 3d-shell is filled (from scandium tozinc, inclusive) form the ferrum series. Atoms of the elements in which the4d-shell is filled (from yttrium to cadmium, inclusive) form the palladiumseries. Atoms of the elements in which the 5rf-shell is filled (from lanthanumto mercury, inclusive, except for lanthanoids) form the platinum series.

The first /-electron appears in the atom of cerium. The elements withthe atomic number Z in the range 58 < Z < 71 (from cerium to lutetium,inclusive), in which the 4/-shell is filled, are called the lanthanoids or rare-earth elements. The elements with 90 < Z < 103 (from thorium to lawren-cium, inclusive), in which the 5/-shell is filled, are called the actinoids. Inthe inert gases helium (Z = 2), neon (Z = 10), argon [Z = 18), krypton(Z = 36), xenon (Z = 54), and radon (Z = 86), all shells are completelyfilled.

Calculations show that the average distances from the nucleus to theelectrons in the d- and, especially, /-shells are substantially smaller thanthose to the electrons in the s- and p-shells. The chemical properties ofelements are mainly defined by the electrons situated in the outer domainsof atoms where the s- and p-electrons can most probably be found. Forthis reason the elements of the ferrum series, in which the 3d-shells arefilled, have fairly similar chemical properties. Still more similar propertiesare possessed by the lanthanoids, in which the 4/-shell is filled. The sameholds for the palladium and platinum series, and for the actinoids. Thus,the periodicity of chemical properties of elements discovered by Mendeleevmeans, from the quantum-mechanical point of view, the replicating of thestructure of the electron shells being filled up.

The elements in whose atoms the s- and p-shells are filled are called theelements of basic groups, and the elements in whose atoms the d- and /-shells are filled are called the elements of transition groups. There are alsosome exceptions here. For instance, copper and zinc belong to the elementsof transition groups, but not to the basic ones.

Let us briefly touch upon the notion of valency of elements from thepoint of view of quantum mechanics. This notion can be used basically insimple compounds. Valency defines the ability of atoms to join with eachother. In 1927, Heitler and London related this ability to the value of total


spin of an atom. This was the beginning of quantum chemistry. Accordingto their investigations, the valency is the doubled spin of an atom joining acompound. The same atom can manifest different valencies depending onthe state in which it enters into a compound. Atoms enter into compoundsin such a way that their spins should mutually compensate. Let us considerthe valencies of elements of the basic groups.

(1) Atoms of elements of the first group (Li, Na, K, Pb, Cs, and Fr, beingalkaline metals) have spin 1/2 in the ground state and manifest thevalency 1. Their first excited state lies far from the ground state. Forthis reason they usually enter into compounds in the ground states.

(2) Atoms of elements of the second group (Be, Mg, Ca, Sr, Ba, and Ra,being the alkaline-earth metals) have zero spin in the ground state anddo not enter into compounds in this state. However, they have anexcited state whose last shell configuration is sp (instead of s2) withspin 1 and this is energetically close to the ground state. So atoms ofthese elements can enter into compounds in these excited states andmanifest the valency 2, in this case.

(3) Atoms of elements of the third group (B, Al, Ga, In, and Tl) haveelectron configuration of the last shell s2p and the spin 1/2 in theground state. Close to the ground state they have an excited state withthe configuration of the last shell sp2 and the spin 3/2. For this reason,the elements of this group manifest valencies 1 and 3. Exceptions arethe first two elements B and Al, which are always trivalent.

(4) Atoms of elements of the fourth group (C, Si, Ge, Sn, and Pb) have theelectron configuration of the last shell s2p2 with spin 1 in the groundstate, and close to it there is an excited state of configuration sp3 withspin 2. Hence they manifest valencies 2 and 4. The first two elementsof this group, C and Si, mainly manifest valency 4 (an exception is, forinstance, carbon monoxide CO).

(5) Atoms of elements of the fifth group (N, P, As, Sb, and Bi) in theground state have the configuration of the last shell s2p3 with spin 3/2.The closest excited state has configuration sp3s', where s' denotes theelectron state which has main quantum number being greater by onethan in the state s. The spin of this excited state is equal to 5/2. Forthis reason, the elements of the fifth group manifest valencies 3 and 5.

(6) Atoms of elements of the sixth group (O, S, Se, Te, and Po) in theground state have the configuration of the last shell s2p4 with spin1. Besides, they have excited states with the configurations s2p3s' and


sp3s'p' with spins 2 and 3, respectively. For this reason, the elements ofthe sixth group manifest valencies 2, 4, and 6. An exception is oxygen,which is always bivalent because its atoms enter into compounds onlyin the ground state.

(7) Atoms of the seventh group (F, Cl, Br, I, and At, being the halogens)in the ground state have the configuration of the last shell s2p5 withspin 1/2. They can also enter into compounds in the excited stateswith configurations s2p4s', s2p3s'p', sp3s'p'2, having spins 3/2, 5/2,7/2, respectively. For this reason, they can manifest valencies 1, 3, 5,7. An exception is fluorine, which is always univalent.

(8) Atoms of elements of the eighth group (He, Ne, Ar, Kr, Xe, and Rn,being the inert gases) in the ground state have perfectly filled shellswith zero spin. For this reason, they are practically always inert. Somecan enter into compounds; this stems from the transition of electronsfrom the outer filled shell to the states of the unfilled d- and /-shell,which are close in energy.

Quantum mechanics also makes it possible to explain the valencies ofelements of the transition groups. In atoms of these elements the fillingof the d- and /-shells occurs. Electrons in the d- and /-shells of atomsare situated deeper than the outer s- and p-electrons. For this reason,the interaction of these atoms with other atoms and molecules is usuallyweaker than for the atoms of basic group elements. In particular, amongthe compounds of transition group elements, molecules with nonzero spinsare often encountered. Atoms of these elements can manifest both even andodd valencies, which are defined both by the interaction of outer electronsand by the possibility to enter into compounds in excited states when thedeep-lying electrons pass over into the s- and p-shells.

When atoms join a molecule, the electron densities in the filled shellschange only slightly. However, in this case the electron densities in theunfilled shells change substantially. From this point of view, there are twoextreme cases. The first is the heteropolar or ionic bond, in which all valenceelectrons pass over from some atoms to others and the molecules consistof charged ions with perfectly filled shells. For example, in the moleculeNaCl, the sodium atom gives its s-electron to the chlorine atom, and thepositively charged ion Na+ together with the negatively charged ion Cl~form a molecule of sodium salt. Another extreme case is the homopolaror covalent bond, in which atoms remain neutral and the valence electronsbecome "collective". Examples occur in the molecules H2, CI2, and so on.


Naturally, there also exist intermediate types of bond which are partiallyionic and partially covalent in character.

Let us now consider some aspects of the atom structure. The wavefunction of an electron in a central field, i.e., in a spherical atom, can bewritten in the form

il>nlm{r,0,<P) = Rnl(r)Ylm(6,<p), (1.20)

where Rni(r) is the radial wave function, Yim{9,(p) is the spherical har-monic function whose definition can be found in mathematical referencebooks, and r, 9, tp are the spherical coordinates of the electron. We presentthe mathematical expressions for the first several normalized spherical har-monics:

Ym{0,<p) =-^=, Yw(9,<p)=iJ~cos9,V 4 T T V 4TT

Y1,±1(0,<p) = Ti\[fe±i'<'sme,V 47T

Y2,±1{0,<p)=±yJ^e±i*sm20,

Y2,±2(9,v) = -J]^e±2i*Sm2e.The spherical function Yjm (9, <p) does not depend on the form of inter-

action between the electron and the self-consistent field of the atom. Allthe information about this interaction is contained in the radial functionRni(r). The probability density of the electron being found in a state withthe quantum numbers n, I, m in the point having the coordinates r, 9, if isequal to

u w M . v ) = Wmm(r,9,<p)\2 = \Rni(r)\2\Yim{0,ip)\2. (1.21)

The squared modulus of the spherical function, \Yim(9, (p)\2, is a stan-dard mathematical expression that contains no information about interac-tion of the electron with the atom field. Unfortunately, in some textbooksthis quantity is called the "electron cloud" and pupils learn the pictures


representing \Yim(O, </?)|2- The notion of "electron cloud" is an unsuccessfulattempt to build up a visual model of a microobject (the atomic electron).

It is conventional to apply the term "cloud" to a dense mass of smallfloating particles (e.g., ordinary clouds of atmospheric water vapor con-densed high in the sky). For this reason, the notion of "electron cloud" forone electron in an atom has neither physical nor semantic meaning. Westress that the quantity \Yim(9, ip)\2 is only a part of the probability (1.21),which contains no information about the interaction of the electron withanother part of the atom. It is the same for all spherical atoms.

The state of an atomic electron with definite values of the quantumnumbers n,l,m is not observable, because the electron energy cannot de-pend on the orbital moment projection m; indeed, otherwise a rotation ofthe coordinate system could change the orbital moment projection valueand, together with it, the electron energy. The observable electron statesin an atom are those with definite values of n and I, whose energies are Eni.The wave function of such a state is defined by the expression

i

ipni{r,O,ip) = Rnl{r) J2 CmYim(e,y), (1.22)m=-l

where the cm are some constant (generally speaking, complex) coefficients.The quantity |cm|2 defines the probability of the electron being found in anunobservable state with the definite value of m.

Since the electron energy Eni in the state n, I does not depend on thequantum number m, it is usually said that this state is degenerate in m.The multiplicity of this degeneracy is equal to 21 + 1. Degeneracy of thestate with the energy Eni with respect to the electron spin projection ais equal to 2 since <r can only take two values ±1/2. For this reason, thedegeneracy multiplicity of the state with the energy Eni is equal to 2(21 + 1).This is just the maximum number of equivalent electrons in the shell withgiven values of n and I, which are indistinguishable.

The probability density for the electron in an observable state withquantum numbers n, I and energy Eni to be at a point with coordinatesr,6,ip is equal to

i 2

wnl(r,e,<p) = Wnl(r,e,tp)\2 = \Rnl(r)\2 £cmYi m(0,cp) . (1.23)m=-l

The angular dependence of the probability wni(r, #,</?) is the squared mod-


ulus of a superposition of spherical functions, but not of a single sphericalfunction. The radial portion of probability \Rni(r)\2 contains all the infor-mation about interaction of the electron with the field of the atom and de-fines, in particular, the average distance between the electron and nucleus.If the quantity r2\Rni(r)\2 has one maximum, then this state correspondsto a circular orbit, as a classical analog. If it has two or more maxima, thenthe classical analogs for these cases are elliptic orbits.

The electron energy in a hydrogen atom (1.7), as distinct from the elec-tron energies in other atoms, depends only on the main quantum numbern and not on the orbital moment I of the electron. Thus, an electron statein a hydrogen atom, having the energy En\, is degenerate in both m andI. For this reason, the wave function of an observable electron state in ahydrogen atom with energy En\ has the form

n - l I

iPn(r,0,cp) = Yl J2 cimRni{r)Ylm(e,v), (1.24)1=0 m=-l

where cim are some constant (generally speaking, complex) coefficients.The quantity |Qm|2 defines the probability that the electron is in an unob-servable state with definite values of the quantum numbers I and m.

The probability that the electron in a hydrogen atom, having energyEn, is at the point r,0,tp is equal to

n-l I 2

wn(r,e,cp) = \*ljn(r,e,v)\2= Y, Y, clmRni{r)Ylm{6,v) (1-25)/=0 m=-l

It is evident that, for the electron in a hydrogen atom which is in an observ-able state with the definite energy value En, it is impossible to separatea factor depending on the angles only. In other words, for a state of ahydrogen atom with the definite energy value En, the probability that theelectron is located at the point r, 9, if is not separable into radial and an-gular factors.

Thus an "electron cloud" is not a probability distribution describing thepresence of an atomic electron at certain points in space for an observedstate with definite energy, and does not correspond to modern conceptsabout atomic structure. This is simply an unsuccessful attempt to create avisual model of a microobject, which is impossible in principle. In serioustextbooks on quantum mechanics, the notion of an "electron cloud" is,naturally, not used.


1.5 Structure of Matter

An explanation of the structure and various properties of solids, liquids, andgases turned out to be possible on the basis of quantum mechanics. Fullconsideration of these questions requires substantial analysis using compli-cated mathematical techniques that lie beyond the scope of this book. Forthis reason, we shall restrict ourselves only to the consideration of someimportant properties of the condensed media, which are connected withtheir microscopic structure.

Many solids are crystals. A crystal is a form of substance in whicha regular arrangement of atoms having a three-dimensional periodicity isobserved. A crystal structure is defined by the geometry of the crystallattice in whose points atoms, ions, and molecules are situated. In biolog-ical crystals, the lattice points are occupied by proteins, nucleic acids, oreven viruses. Other solids are amorphous. These are characterized by anisotropy in their properties, i.e., by an absence of any strict periodicity inthe arrangement of their atoms, ions, molecules, or groups thereof.

Crystalline substances can be divided into three main groups with re-spect to their ability to conduct electric current: metals (conductors), di-electrics, and semiconductors. Metals have electrical conductivity in therange 108-106 (fim)"1. The corresponding range for dielectrics is 10~"8-10"10 (ffrn)"1. Semiconductors occupy an intermediate position betweenmetals and dielectrics in terms of conductivity. We should note that theproperties of substances with respect to the conduction of electric currentcan be substantially modified by external conditions such as temperatureand pressure.

Metals are widespread in Nature. It is enough to notice that undernormal conditions 83 of the first 106 elements are metals. Metals exhibit apractically free conduction of electric current. Their conductivity decreasesas temperature increases.

A metal is a crystal whose lattice points contain ions, because someatomic electrons (the valence electrons) break away from their atoms. Col-lectively, these free electrons can be considered as a gas within the metal.At the metal's boundary, there exists a potential barrier that prevents elec-trons from flying away from the metal.

If one applies a high-intensity electric field £ ~ 108 V/m to a metal-lic cathode, a cold emission current results. This is a purely quantummacroscopic phenomenon, which is connected with the sub-barrier passageof electrons known as the tunnel effect (Fig. 1.5). As a result, electrons


u(x) barrier withoutelectric field

U° * \

metal vacuum u^eSx

x

Fig. 1.5 The tunnel effect.

fly away from the metal and a current is observed. The dependence of thiscurrent /(£) on the electric field intensity £ is denned by an expression thatcan be obtained only on the basis of quantum theory:

I(£)=Ioexp(-£o/£), (1.26)

where IQ and £Q are constants.

E>c <D

o c //////7T

15

free atom crystal

Fig. 1.6 Energy levels of electrons within a crystal.

Since electrons are fermions, they obey the Pauli principle. For thisreason, electrons that have broken away from atoms and are practicallyfree over the volume of the crystal occupy different energy levels (Fig. 1.6).Distinct from a free atom, in a crystal the permitted values of electronenergies form bands as shown in Fig. 1.6. These bands are separated byforbidden regions. Each band contains a large number of energy levels thatcan be occupied by electrons. Widths of the bands increase as the electronbinding energy decreases, i.e., the deepest atomic energy levels correspond


to very narrow bands in the crystal. Let us note that in Fig. 1.6 the simplestcase is shown when the bands do not overlap. In reality, band overlappingis possible.

Now consider the mechanism of electric current flow through a crystal.In an unexcited crystal all free electrons tend to occupy the lowest permittedenergy levels. By the Pauli principle, electrons occupy all levels up to somehighest occupied level, whose energy is called the Fermi energy Ef. Fig.1.7 shows two cases of the highest occupied level situation in a crystal.From this viewpoint only two bands are of interest: the upper one, which isthe conduction band, and the lower one — the valence band. Energy levelsin all the bands lying below are completely occupied, and those in all thehigher-energy bands are empty.

iHM MMg <jj /AWWA? /////////

(D XXXXXXXXX TXTTAXTTTT

Fig. 1.7 The topmost energy level bands in a crystal.

If the lowest empty level is situated far from the upper edge of the band(Fig. 1.7a), then under the action of an electric field applied to the crystal,the electrons situated near the empty levels increase their energy and passover to the nearby empty levels. Their movement creates an electric currentin the crystal.

Such crystals are metals, and the electrons creating the electric currentin them are called conduction electrons. Any metal is characterized by acertain electrical resistance. It is caused by scattering of the conductionelectrons on thermal vibrations of the ions, forming the crystal lattice, andon structural inhomogeneities of the lattice (the impurity atoms and latticedefects). The resistance of a metal decreases with decreasing temperatureand increases with increasing temperature.

If the band is completely occupied (Fig. 1.7b), then the first empty levellying in the conduction band is separated from the last occupied level bya forbidden energy band of width e. The quantity e is usually equal toseveral eV. For example, for diamond it equals e = 5.5 eV. In this case,


the voltage applied to the crystal cannot transfer electrons from the valenceband to the conduction band, i.e., such a crystal is a dielectric and does notconduct electric current. Only a very high voltage (the material's breakdownvoltage) can transfer electrons to the conduction band.

If a substance has a narrow forbidden region (s < 1 eV), then ther-mal motion can cause a small number of electrons in the crystal to passfrom valence band to conduction band. The positions left empty in thevalence band are called holes. These behave like positively charged parti-cles with a mass equal to the electron mass and with the charge +e. Thenumber of electrons having sufficient energy for the transition from valenceband to conduction band is proportional to exp(—e/kT) for a wide rangeof temperatures T. It is obvious that the conductivity of the material willhave a similar temperature dependence. Such substances are called semi-conductors. Their electrical conductivity increases with temperature (incontrast to the behavior of metals as mentioned above), because of the in-crease in vibrational amplitude of the ions forming the crystal lattice. Thismechanism for conductivity is characteristic of pure semiconductors. Weshould note that at sufficiently high temperatures some dielectrics becomesemiconductors.

In a semiconductor, electrons are constantly making the transition fromvalence band to conduction band. The reverse transition occurs as well; herea conduction electron and a corresponding hole disappear as mobile chargecarriers. This process is called recombination. At any given temperaturean equilibrium arises between the numbers of electron transitions to andfrom the conduction band. In other words, the number of electrons in theconduction band and the number of holes in the valence band is conserved atany given temperature. Note that in semiconductors, electrical conductivityis due to both the motion of negatively charged electrons and the motionof holes in the opposite direction.

Some substances, with regard to their electrical properties, occupy anintermediate position between metals and semiconductors. They have acomparatively small number of electrons in the conduction band even atvery low temperatures, i.e., they can conduct only small currents. Suchsubstances are called the semimetals (examples include Bi, Sb, and As).

For practical aims one usually employs not pure semiconductors but de-liberately impure ones. They contain a small amount of admixtures whichsubstantially change their physical properties. The impurities are of twotypes: donors and acceptors. A donor impurity will donate surplus elec-trons to a semiconductor and create electron conductivity — the result is


an n-type semiconductor. An acceptor impurity will capture some electronsfrom the valence band and create hole conductivity, resulting in a p-typesemiconductor.

a b

gf /////A , '//////Si5 r _,J03 =5 y y fc iy V X X X X

Fig. 1.8 Creation of conductivity in doped semiconductors.

Fig. 1.8 shows how conductivity is created in impure or doped semicon-ductors. In the donor semiconductor (Fig. 1.8a) the electron to be releasedis on the energy level of a doped atom situated close to the conductionband. For this reason, even at a low temperature, such an electron canpass to one of the lowest empty levels of the conductivity band. Typicalexamples of donor semiconductors are Ge and Si containing sparse num-bers of P, As, or Sb atoms that have replaced atoms of the main elementat some lattice points. The ionization energy of the doping atoms is verysmall: e' « 0.01 eV for germanium and 0.04 eV for silicon. For this rea-son, even at 77 K nearly all doping atoms are ionized, and in this case thenumber of conduction electrons is determined by the donor concentration.

In an acceptor semiconductor (Fig. 1.8b) the doping atom has an emptyenergy level close to the valence band. The capture of an electron from thevalence band on this level of the acceptor leads to the rise of a hole in thevalence band and to the creation of hole conductivity. Typical examples ofacceptor semiconductors are Ge and Si containing sparse numbers of B, Al,or Ga atoms.

It is possible to grow a semiconductor (e.g., germanium) crystal con-sisting of two individually doped regions: one n-type and the other p-type.Such a pn crystal, if it has a narrow boundary region between the n- and p-regions, can be used as a rectifier {diode). If positive and negative voltagesare applied to the p- and n-regions, respectively, electrons will pass from then-region to the p-region and holes will move in the opposite direction. Soa continuous current will flow through the crystal. For an applied voltageof opposite polarity, the current will be practically nil.


More complicated devices can be constructed from semiconductor crys-tals of the npn-type. Here a narrow p-type layer, a few hundred-thousandths of a meter wide, is placed between two n-type regions. If,for example, a positive voltage is applied to the left end of such a semi-conductor and a negative voltage to its right end, then current flows. Theleft portion (emitter) is similar to a triode filament — it emits electrons —while the middle (base) and right (collector) portions are analogous to thetriode grid and anode, respectively. This npn-semiconductor (transistor)can function as an amplifier, increasing the voltage and power of a signaldelivered to its emitter. A pnp-semiconductor in which hole conductiontakes place will operate analogously.

Let us consider the magnetic properties of solids. Those substances thatcan be magnetized in an external magnetic field are said to be magnetic.Inside a magnetic substance during magnetization, an intrinsic magneticfield arises. There exist weak-magnetic substances: diamagnets and para-magnets. Diamagnetic substances include He, Ar, Au, Zn, Cu, Ag, Hg,water, and glass. Paramagnetic substances include O, Al, Pt, the alkalimetals, the alkaline-earth metals, and the rare-earth elements.

In the absence of an external magnetic field, the atoms and molecules ofdiamagnets do not possess any intrinsic magnetic moments. If a diamagnetis placed in a magnetic field, then in each atom or molecule an additionalcurrent is introduced which creates a magnetic moment. According toLenz's law, the current induced by an external magnetic field is always di-rected in such a way that it decreases the field inside the substance. For thisreason, the induced magnetic moments are aligned opposite the directionof the external magnetic field, and the diamagnet is magnetized. When theexternal magnetic field is removed, the diamagnet is demagnetized.

The atoms and molecules of a paramagnet have some intrinsic magneticmoments in the absence of an external magnetic field. In this case, however,random thermal motion leads to a chaotic orientation of their magneticmoments; i.e., the paramagnet does not possess an intrinsic magnetic fieldwhen an external magnetic field is absent. In an external magnetic field,the magnetic moments of atoms and molecules align along the field, andthe paramagnet is magnetized.

As already noted, when an external magnetic field is absent, the mag-netic moments of atoms and molecules in a paramagnet are oriented chaot-ically. However, there exist substances which have an ordered structure inabsence of an external magnetic field (Fig. 1.9). If in such a structure themean magnetic moments of atoms are oriented in the same direction (Fig.


t t t t t t t t t t tttUMtM

\/\A/\/\/

tmtmut

- \ / - \ / - \ / -

t + t + t * t * t * t

t t i t t t i t t t iFig. 1.9 Orientation of magnetic moments in structured materials.

1.9a) so that a macroscopic volume magnetization arises, then the materialis said to be ferromagnetic. But if the ordering is such that the overall mag-netic moment equals zero (the magnetic moments of neighboring atoms aredirected oppositely) and the macroscopic magnetization is absent, it is saidto be antiferromagnetic (Fig. 1.9b). Finally, if the magnetic moments canhave the opposite direction of ordering (Fig. 1.9c) but the magnetization isdifferent from zero, the material is ferrimagnetic. The critical temperatureabove which the magnetic ordering disappears is called the Curie temper-ature or Curie point Tc in the ferro- and ferrimagnetic cases, and the Neeltemperature or Neel point Tn in the antiferromagnetic case (after the Frenchphysicist Neel who, independently of Landau, explained the phenomenonof antiferromagnetism).

Ferrimagnetism can be considered as the most general case of magneticordering. From this point of view, ferromagnetism and antiferromagnetismare particular cases of ferrimagnetism. Examples of ferromagnetics are Fe,Co, and Ni; antiferromagnetics include MgO, FeO, and NiO, while ferri-magnetics include Fe3O4, CoFe2O4, and NiFe2O4. Note that the criticaltemperature values for various magnetic substances lie in a very wide range.


For instance, the ferromagnetics Dy, Ni, and Fe have Curie temperaturesof 37 K, 627 K, and 1043 K, respectively. Above the critical temperature,ferromagnetics and antiferromagnetics become paramagnets.

At temperatures T < Tc, a ferromagnetic material consists of domains.A domain is a small volume of substance having linear sizes on the order of10~4-10~5 m, inside which all magnetic moments of atoms are co-oriented,i.e., the domain has maximum possible magnetization. In the absence ofan external magnetic field, the magnetic moments of domains inside a fer-romagnetic sample are oriented chaotically so that the overall magneticmoment of sample is zero. Under the action of an external magnetic field,however, the magnetic moments of the domains align and the sample asa whole becomes magnetized. When the magnetic field is removed, someof the magnetic moments remain in alignment. For this reason, a residualmagnetism can arise and permanent magnets can be created.

P /

0 Tcr T

Fig. 1.10 Superconductivity in mercury.

Now let us consider phenomena that occur at very low temperatures.These are the superconductivity of some metals (e.g., Pb, Ta, Sn, Al, andNb) and alloys (e.g., NbsSn), and the superfluidity of liquid helium. Su-perconductivity was discovered by the Dutch physicist Kamerlingh-Onnesin 1911, when he investigated the electrical resistance of mercury at lowtemperatures. It turned out that at the critical temperature Tcr =4.15K, the resistance of mercury fell abruptly to zero (Fig. 1.10). Further, su-perconductivity of other substances was discovered. Each of them has itsown critical temperature TCT for transition into the superconducting state.For his investigation of the properties of matter at low temperatures, whichled to the production of liquid helium, Kamerlingh-Onnes was awarded the


Nobel Prize in Physics for 1913.If an electric current is induced in a superconducting metal ring, it will

circulate in this ring without any damping, because its flow through thesuperconductor is not accompanied by any heat evolution. This propertyof semiconductors is used for the construction of superconducting magnetsand other devices. It has also been established that a weak magnetic fielddoes not penetrate a superconductor. The expulsion of magnetic field froma superconductor (the Meisner effect) means that in an external magneticfield a superconductor is a perfect diamagnet.

The nature of superconductivity was explained by the American physi-cists Bardeen, Cooper, and Schrieffer in 1957. They were awarded theNobel Prize in Physics for 1972 as a result. According to their theory, twoconduction electrons in the crystal lattice of a metal create a bound state,the Cooper pair, which has zero total moment (spin). Let us consider themechanism of creation of the Cooper pair in more detail.

Two electrons experience Coulomb repulsion. However, an electroncan excite or absorb a quantum of the lattice vibrations, which is calleda phonon. If an electron excites a phonon and another one absorbs it, thensuch an exchange of a phonon leads to an attraction arising between theelectrons; moreover, the attractive force in a superconductor turns out to begreater than the force of Coulomb repulsion. It is just this attractive force,arising from the presence of the crystal lattice, that leads to the pairing ofelectrons. Such a bound electron pair behaves like a particle. For this rea-son it is called a quasiparticle, which is a boson. At temperatures close toabsolute zero, these quasiparticles experience Bose condensation. The Bosecondensate, consisting of the electron pairs, moves freely (experiencing noresistance) through the crystal, which leads to superconductivity.

The states of electrons in a superconductor are continuously changingand, for this reason, the compositions of electron pairs are continuouslychanging as well. Since the linear size of an electron pair is on the order of10~6 m, this is an example of the long-range coupling of particles.

At T = 0 K, all conduction electrons are paired. If T ^ 0, the probabil-ity of breaking an electron pair differs from zero. For this reason, at T ^ 0the unpaired electrons form a normal electron "liquid" in the crystal, andthe paired ones form a superconducting "liquid". Above T = Tcr the quasi-particles (Cooper pairs) become completely broken, and superconductivitydisappears.

Superfluidity is observed in liquid helium, which leaks through verynarrow channels and capillaries without friction. This phenomenon was


discovered by the Soviet physicist Kapitsa in 1938. He established that attemperatures below T\ = 2.17 K (the lambda point), liquid 4He becomessuperfluid. Kapitsa received the Nobel Prize in Physics for 1978 for thisdiscovery and for his fundamental research on low temperature physics.

For temperatures T > T\, liquid 4He is called He I; for T < T\, it iscalled He II. At the temperature T = T\ in liquid 4He, a phase transitionof the second kind occurs, i.e., in this case its internal structure abruptlychanges. Note that phase transitions of the second kind are undergoneby ferromagnetics and antiferromagnetics at the Curie point, as well as bysuperconductors at the critical point.

An explanation for superfluidity of liquid He II was given by Landau in1941. According to his theory, there are two types of elementary excitations(quasiparticles) in He II: phonons and rotons. At finite temperatures, apart of He II behaves like a normal viscous liquid and another part behaveslike a superfluid which possesses no viscosity. There is no friction whenthese liquids move through each other. The Soviet physicist Landau wasawarded the Nobel Prize in Physics for 1962 for his pioneering theoriesregarding condensed matter, especially liquid helium.

Consideration of the normal and superfluid components in He II is aconvenient way to describe phenomena occurring in a quantum Bose liquid.This does not mean that the liquid is actually separated into two fractions.In reality, in He II two types of motion can occur simultaneously: one ofthem identical to a viscous liquid, and the other corresponding to a super-fluid liquid. Both types of motion occur without transferring momentumfrom one to the other.

In 1972-1974, superfluidity of 3He was discovered, which is observedat temperatures of several mK. In this case two atoms of 3He, which arefermions, create a pair which is a quasiparticle (boson) and the Bose con-densation of these quasiparticles occurs, which leads to superfluidity ofliquid 3He.

Finally, we will note that quantum mechanics has made it possible toascertain the physical laws of various processes and phenomena observedin condensed media, which in principle cannot be explained on the basisof classical physics. Studies of the microscopic structure of solids allowedpeople to create devices and mechanisms widely used in modern technology.

Chapter 2

Fundamental Interactions

2.1 Gravitational Interaction

Physics is concerned with matter: its structure and motion. The motionof matter is due to certain forces acting between bodies. The motion ofgalaxies and stars, of planets and comets, of electrons in TV sets and atoms,of nucleons and quarks in atomic nuclei, radioactive decays of atomic nucleiand elementary particles as well as all various processes in the Universeare caused by interactions between different physical objects. So it is notsurprising that some of the most important questions in physics pertain tothe study of these interactions.

Over two millennia ago, the Greek philosopher Aristotle theorized thatall substance in the Universe consisted of four elements — earth, air, fire,and water — and that these were subject to the action of two forces. Thefirst was the force of gravity, which attracted earth and water downwards.The second was a "force of lightness", which served to attract fire and airupwards. Thus, Aristotle divided all of Nature into substances and forces.This approach has persisted in physics until the present day. Now, thereare four known types of interactions: gravitational, electromagnetic, andthe strong and weak nuclear forces. Let us consider each in turn.

The gravitational interaction, by intensity, is the weakest of all the in-teractions known to us. The gravitational forces have a universal character.This means that all matter is subject to them; this is what the law of univer-sal gravitation states. The range of gravitational forces is infinite. Gravita-tional forces are attractive. Gravitational interaction is mainly manifestedbetween macroscopic bodies; it determines the motions of various cosmicobjects: galaxies, stars, planets, etc. In the world of elementary particles,gravitational interaction is not directly apparent because of the very small

41


masses of the particles.Since the gravitational attraction of any two bodies is proportional to

their masses, the notion of mass, which is one of the most important char-acteristics of a physical object, plays a fundamental role not only in thetheory of gravitation, but in physics in general. The notion of mass hasbecome more complicated after the creation of relativity theory. For thisreason we shall consider it in detail.1

First, we shall consider the notion of mass in nonrelativistic (Newtonian)mechanics, which describes the motion of physical objects whose velocitiesare small relative to that of light. In this case, the momentum p of aphysical body is related to its velocity v by the formula

p = mv, (2.1)

where the coefficient of proportionality m characterizing properties of thebody is called the mass. The kinetic energy of the body is

If a force F acts on the body, its momentum changes with time accordingto the law

Since the acceleration a of the body is defined as the time derivative of itsvelocity,

differentiation of (2.1) with respect to time leads to Newton's second law

F = ma. (2.5)

The mass m entering into (2.1) and (2.5) is called the inertia! mass. Westress that in nonrelativistic mechanics, (2.1) and (2.5) are equivalent. Inthese equalities, the mass plays the role of a coefficient of proportionalitybetween the velocity and momentum, or between the acceleration and force.

Now consider the gravitational attraction of bodies. The law of uni-versal gravitation, discovered by Newton, states that the potential energy

^.B. Okun. The Concept of Mass. Uspekhi Fiz. Nauk, 1989, v. 158, N 3, p. 511.

(2.2)

(2.3)

(2.4)

Fundamental Interactions 43

of attraction of two bodies having masses M and m is determined by theformula

Ug(r) = - G ^ , (2.6)

where the constant G = 6.67- 10~u N-m2/kg2 and the negative sign signifiesattraction. Differentiating equation (2.6) with respect to the vector r andtaking into account the relation Fg = —dUg(r)/dr, we find the gravitationalforce

F. = -<££ , . (2.7)

Equations (2.7) and (2.6) are equivalent, and the mass that appears ineach is known as gravitational mass. It follows from (2.5) and (2.7) that theacceleration of a body moving in a gravitational field (e.g., a body fallingin the gravitational field of the Earth) does not depend on its mass. If Mis the mass of the Earth, we obtain

g = ^ = -<4r. (2.8)m r6

Since M « 6 • 1024 kg and the radius of the Earth RE « 6.4 • 106 m, we cansubstitute RE for r in (2.8) and find that g sa 9.8 m/s2.

Galileo was the first to establish the universality of free-fall acceleration.Afterwards, the independence of the quantity g on the mass and substanceof a falling body was confirmed by accurate experiments. The universalityof g leads to the equality of the inertial and gravitational masses. In otherwords, the masses m of the same body, which enter into formulae (2.1), (2.5)and (2.6), (2.7), are the same. The equality of inertial and gravitationalmasses constitutes the principle of equivalence that plays an importantrole in relativity theory. According to this principle, when inside a systemthat moves with acceleration, one has no way to distinguish the acceleratedmotion from gravitation.

In 1905 Einstein developed relativistic mechanics, which is also knownas special relativity. This allows one to describe the motion of objects atvelocities close to the speed of light c. The theory is based on the exper-imental fact that there exists a limiting speed of propagation for physicalsignals in Nature: c fa 3 • 108 m/s. General relativity, created by Einsteinin 1915, accounts for gravitation. Many phenomena observed in the Uni-verse and caused by the gravitational interaction of various cosmic objects


can be explained only on the basis of this theory. The special and generalrelativity theories are classical (non-quantum) theories.

In relativistic mechanics, the relation between momentum and velocityof a freely moving body is defined by the formula

P = §v, (2.9)

and the relation between energy and momentum is defined by

E2=p2c2+m2c4. (2.10)

The mass m and velocity v in (2.9) and (2.10) are the same quantities thatenter into the formulae (2.1)-(2.7) of nonrelativistic mechanics. It followsfrom (2.9) and (2.10) that in relativistic mechanics the energy of a bodywith m ^ 0 does not become zero even when its velocity and momentumare equal to zero: v = p = 0. In other words, the body's rest energy EQ,according to relativity theory, is given by

Eo = me2. (2.11)

Relation (2.11) is the basis of conventional and atomic energetics, aswell as of conventional and atomic military techniques. It was not knownin nonrelativistic mechanics. It follows from (2.9) that E = pc if v =c. Substituting this into equation (2.10), we conclude that the mass of aparticle moving with the speed of light is equal to zero. Conversely, if aparticle has zero mass, it always moves with the speed of light. There isno reference frame for the massless particle in which it would rest; i.e., inany frame of reference such a particle moves with the speed of light. Forthis reason the photon, which is a massless particle, is doomed to be an"eternal wanderer", flying with speed c until it ceases to exist as a resultof some electromagnetic process.

It is convenient to express the energy and momentum of a particle withm ^ 0 through mass and velocity. For this purpose, we substitute (2.9)into (2.10):

Wl-J)=m2c4. (2.12)

Denoting 7 = 1/^1 - v2/c2, we write (2.12) in the form

E = mc2~f. (2.13)


Substituting (2.13) into (2.9), we find that

p = mvy. (2.14)

The kinetic energy of a body in relativistic mechanics is

Ek = E-E0=mc2{-1-l). (2.15)

If u < c, expressions (2.14) and (2.15) turn into formulae (2.1) and(2.2) of nonrelativistic mechanics, respectively (for v -C c we have 7 «1 + v2/2c2 + •••). This limiting case confirms the statement that the bodymass m in nonrelativistic and relativistic mechanics is the same quantity.Now consider the relation between the force and acceleration in relativis-tic mechanics. For this purpose, let us differentiate equation (2.14) withrespect to time. Then, taking account of definitions (2.3) and (2.4), weobtain

F = m a 7 + ^ ( a - v ) 7 3 . (2.16)&•

It can be seen from (2.16) that generally, in relativity theory, the forcehas two different components: one directed along the acceleration and onedirected along the velocity. This makes formula (2.16) differ in principlefrom formula (2.5) of nonrelativistic mechanics. Equation (2.16) correctlydescribes relativistic particle motion. It has been corroborated by numerousexperiments. In particular, it formed the basis for the design of elementaryparticle accelerators that have operated around the world for years.

Multiplying equation (2.16) by the velocity v, we find

a " V = m 7 ( f + W ) = ^ (2'17)Substituting (2.17) into (2.16), we have

F - ( F - v ) ^ = m a 7 . (2.18)

If the force is perpendicular to the velocity, F i v , then we obtain

F = ma7. (2.19)

However, if the force is parallel to the velocity, F || v, we find

F = ma7 3. (2.20)

We see from (2.18)-(2.20) that in special relativity theory, the ratio ofthe different forces to the acceleration is substantially dependent on the


mutual orientation of the force and velocity, that is, in relativistic mechanicsthe mass cannot be defined as the ratio of force to acceleration. In thenonrelativistic limiting case (v <C c, 7 « 1), formulae (2.18)-(2.20) turninto expression (2.5).

Now, consider gravitational attraction in general relativity theory. Inthis theory, the gravitational field is determined by the energy-momentumtensor of the body. This complex quantity has ten independent compo-nents. In the simplest case of the interaction of two bodies, one having avery large mass M (for example, the Earth) and the other having a verysmall mass m (for example, an atom or electron), it can be shown that thegravitational force acting upon the particle of small mass m is

GME I"/ v2\ (r-v)vl

In the nonrelativistic limit u « c , the expression in square brackets reducesto r and E/c2 « m, i.e., we obtain (2.7). However, for v « c the quantitythat could play the role of the gravitational mass of the relativistic particledepends not only on the particle energy, but also on the mutual orientationof the vectors r and v. If r || v, this quantity is equal to E/c2, but ifr _L v, it is (E/c2)(l + v2/c2). For a photon we have m = 0 and v — c, i.e.,when r || v, its "gravitational mass" equals E/c2, and for r J. v it equals2E/c2. Hence, one can conclude that there is no sense in speaking aboutthe gravitational mass of a photon or any other relativistic particle, sincethis quantity is different for objects moving at different angles with respectto the direction of the gravitational force.

The mass of a body changes along with its internal energy. In particular,it changes during heating or cooling, as well as during changing the state ofaggregation of the constitution of the body. For example, during the heatingof iron by 200°C, the relative increase of its mass is Am/m = 10~12, whilefor the melting of ice into water we have Am/m = 3.7 • 10~12. This changeof mass is very small and cannot be detected in experiments.

Owing to the energy conservation law in chemical and nuclear reac-tions, the rest energies of bodies partially or completely transform into thekinetic energy of the reaction products if the overall mass of the particlesentering into the reaction exceeds the overall mass of the reaction products.For example, during methane combustion the following chemical reactionoccurs:

CH4 + 2O2 —> CO2 + 2H2O. (2.22)

(2.21)


As a result, the energy s = 35.6 MJ/m3 is released. Since the density ofmethane is p = 0.89 kg/m3, we find that Am/m = e/pc2 = 4 • 10~10.

The annihilation of an electron and a positron usually leads to the cre-ation of two photons, and the whole rest mass of the initial particles trans-forms into the kinetic energy of the photons created. On the Sun and somestars, due to thermonuclear reactions, a fusion of four protons occurs thatleads to the creation of an a-particle, two positrons, and two neutrinos:

4p—> a + 2e+ + 2ve. (2.23)

The mass of four protons is 3755.08 MeV, the mass of a-particle is 3728.35MeV, and the mass of two positrons is 1.02 MeV. In this process, the relativedecrease of the mass is considerable: Am/m — 0.66 • 10~2.

Thus, the relative change of mass Am/m in nuclear reactions is 7-8 orders higher than in chemical reactions. However, the mechanism ofenergy release in these reactions is the same: the rest energy of particlestransforms into their kinetic energy (i.e., heat).

Consider an atom of hydrogen, which consists of a proton and an elec-tron. Experiments show that the mass of the hydrogen atom is less thanthe sum of the masses of a proton and an electron by several hundred-thousandths of the electron mass. The explanation of this fact is connectedwith the circumstance that the energy of electric attraction of the protonand electron, binding these particles into the hydrogen atom, is considerablerelative to the electron rest energy.

More interesting examples are atomic nuclei. The simplest compositenucleus, the deuteron, consists of a neutron and a proton. The experimen-tally measured mass of deuteron is approximately 0.1% less than the sumof the masses of a neutron and a proton. The nucleus of uranium 238Ucontains 92 protons and 146 neutrons. The measured mass of this nucleusis approximately 5.2% less than the sum of the masses of the neutrons andprotons entering into its composition. The examples presented show thatmass is not an additive quantity; the mass of a composite object is notequal to the sum of the masses of the bodies from which it is composed.

Now, let us discuss the role of the notion of mass in nonrelativistic me-chanics and in relativity theory. The mass of an isolated physical object isconserved, and does not change under the transition from one frame of ref-erence to another. In relativity theory, the mass determines the rest energyof the body. This property of the mass was not known in nonrelativisticmechanics. At the same time, in relativistic theory the mass of a physical


system is not a measure of the quantity of substance.In relativistic mechanics, there is no principal difference between sub-

stance (the objects with m =/= 0 — neutrons, electrons, atoms, molecules,etc.) and radiation (photons, which have m = 0). It is possible that pho-tons are not the only particles having zero mass. Neutrinos and some otherparticles are difficult to observe experimentally because of their weak in-teraction with matter, but they are supposed to have zero mass as well.In relativistic theory, the mass of a system of particles (molecules, atoms,nuclei, etc.) does not coincide with the sum of the masses of the particlesthat compose it.

In relativity theory, force has components directed along the accelerationand velocity. Therefore, the mass cannot be defined as the ratio of force toacceleration, that is, in this case the mass is not a "measure of inertia" ofthe moving body. We should stress that relativistic mechanics is a physicaltheory in a four-dimensional pseudo-Euclidean space (Minkowski space).For this reason, the three-dimensional laws of nonrelativistic mechanics, towhich Newton's laws belong, are not realized in relativity theory in principle(aside from the limiting case v <C c, when relativistic mechanics turns intononrelativistic mechanics).

The mass of a body in relativity theory does not determine its interac-tion with a gravitational field. The gravitational force depends on the anglebetween the vectors r and v. Therefore in relativistic mechanics the massof a body cannot be defined as the ratio of its energy to the square of thespeed of light, i.e., the notions of gravitational and electromagnetic massesdo not exist. Furthermore, the notion of mass depending on the velocity ofthe body is spurious as well. Actually, the gravitational force acting upona horizontally flying photon is two times larger than the force acting upona vertically flying photon.

From relativity theory an important conclusion follows, which has agreat physical and methodological significance: any mass corresponds tosome energy, but not conversely. In other words, there is not a completeequivalence between mass and energy. The formula m = E/c2 which is,unfortunately, present in many textbooks, is based on the wrong assump-tion that the nonrelativistic relation (2.1) holds in relativistic mechanics.This is the relation (2.1) from which formula (2.5) follows. If we treat thequantity m = mo7 as a mass in it (here mo is the rest mass of the body),then we obtain

(lfl)F = mQ—1. (2.24)

at


Considering the force F to be constant, from (2.24) we determine the timeinterval t0 during which the body will reach the speed of light c:

j *!_=F_Jdtm (2.25)

Performing the integration in (2.25), we find

*o = ^f. (2.26)

This erroneous result arises from the use of (2.5) instead of (2.16), whichalso contains the term with 73. It follows from (2.16) that in reality a bodywith mo 7 0 can never reach the speed of light.

A dogmatic acceptance of the wrong formula m = mo7, according towhich the body mass increases with velocity, can turn out to be a seri-ous obstacle for students who wish to acquire a deeper understanding ofrelativity theory in the future.

Special relativity theory permits, in principle, the existence of particlesthat travel faster than the speed of light. These hypothetical particles,called tachyons (from the Greek word "tachis", meaning "fast"), must becreated with super-light-speed velocities. Their velocities can never ap-proach or drop below the speed of light if their masses are not equal tozero. However, such particles have never been observed.

It follows from general relativity theory that space becomes curved un-der the action of masses of bodies. We can use the following analogy forthis phenomenon. Imagine a sheet of thin rubber stretched between pegsin such a way that this sheet is flat. If we put a billiard ball upon thisrubber, then it will create a pit, that is, a deformation of the rubber willoccur and distortion of its surface will arise. Adding more balls will lead toan increase of the surface deformation and curvature. An analogous phe-nomenon is observed in space, if some matter is placed in it. However, theaverage density of matter in the Universe is small, p = 10~27 kg/m3, sospace curvature is also very small. Nevertheless, there exist effects causedby this space curvature, which can be observed experimentally.

The closest planet to the Sun is Mercury, which experiences thestrongest gravitational force from our star. Its elliptic orbit is ratherstretched out. According to general relativity, the major axis of the el-liptic orbit of Mercury should turn around the Sun by approximately onedegree each ten thousand years. This effect is called the turning of the


perihelion (the closest point to the Sun of the orbit) of Mercury. It wasobserved experimentally even before general relativity was put forth. Sub-sequently, even smaller changes in the orbits of the planets were measured.All these phenomena agree with the predictions of general relativity.

According to general relativity, a ray of light in a gravitational fieldfollows a curved, not straight, trajectory. This effect can be observed inthe following way. The light from stars passing near the Sun is deflectedfrom a straight line path. One can photograph the sky during a totalsolar eclipse and then, after a long time period (say, half a year), takeanother photograph of that part of the night sky which was close to theSolar disk during the eclipse. Since during this time the Sun will havemoved into another region of the sky, the light from the stars we are in-terested in will not be influenced by the gravitational field of the Sun. Acomparison of the photographs allows one to determine the deflection oflight in the gravitational field of our star. Such observations were first car-ried out in 1919. They coincided with the predictions of general relativitytheory.

For the same reason, the wavelength of a radio signal transmitted fromthe Earth into space will be getting longer. In other words, for an observerin space the electromagnetic oscillations in the radiating antenna occurmore slowly than for an observer on the Earth. For this reason, it turns outthat time on the Earth's surface passes more slowly than it does at distantlocations. The difference is approximately one second per fifty years, butmodern atomic clocks can detect it.

On the surface of the Sun the effect of time dilation is a thousandfoldgreater; it would be even greater on the surface of a neutron star. Thus,on the surface of a massive body, time passes more slowly than it does inspace. On Earth, time passes more quickly on mountain peaks than it doesin valleys. This effect has been observed experimentally by means of atomicclocks. The results of these measurements have exactly coincided with thepredictions of general relativity.

General relativity is a classical theory. A quantum theory of gravitationhas not yet been created. However, it is clear from certain physical rea-sons that in such a theory a massless particle having spin 2 should figure,which is called the graviton (a quantum of the gravitational field). Thegravitational force acting between two particles should be described in thequantum theory as an exchange of a graviton (or gravitons) between the in-teracting particles. The real gravitons must propagate in space in the formof waves, which are called gravitational waves in classical physics. However,


these waves are very weak, and it has not been possible to register themexperimentally.

2.2 Electromagnetic Interaction

Electromagnetic forces act between bodies having electric charges. Electro-magnetic interactions are also experienced by photons, which do not carryelectric charge, and by neutral particles possessing magnetic moments (e.g.,neutrons).

In Nature, there are two kinds of electric charges: positive and negative.Between two positive or two negative charges there exist repulsive forces,and between positive and negative charges an attractive force acts.

The electric force of the interaction between two charges is denned byCoulomb's law, which is similar in form to Newton's law of universal grav-itation (2.7):

Fe = k^-v. (2.27)

Here e\ and e^ are the charges, and k is a constant whose value dependson the system of units used to measure force. The lack of a negative signin (2.27) indicates that like charges repel and opposite charges attract,because an attractive force, as in the case of the gravitational interaction,is assumed to be negative.

The SI unit of electric charge is the Coulomb (C). In order that theforce may be expressed in Newtons, the constant in (2.27) should be equalto k = 8.987 • 109 N-m2/C2. There also exists a rarely used unit of electriccharge called the electrostatic unit, defined in such a way that the constantwould be k = 1. In this case, the force is measured in dynes (1 dyne = 1g-cm2/s2), and the separation r in cm.

Great progress in understanding electromagnetic interaction was madeby the outstanding English physicist James Clerk Maxwell. It was Maxwellwho introduced the notions of the electric and magnetic field intensities.The introduction of the field concept marked the rejection of the old "action-at-a-distance" concept — the idea of force as a direct action of one bodyupon another body located some distance away. The electric field can beconsidered as an external factor that does not depend on the properties ofthe given charged body, and that exerts a direct action upon it. The electricfield is determined by the disposition of all charges in space. Analogously,the magnetic field is determined by all currents (moving electric charges)in space.


The field concept plays a fundamental role in modern physics. Besideselectric and magnetic fields, which can be considered as a unified electro-magnetic field, there exist other fields that constitute objective realities inthe Universe. In other words, matter can exist in two forms: as particles,and as fields.

Electromagnetism was discovered by the Danish physicist Hans Chris-tian Oersted in 1820. This discovery had great repercussions on the devel-opment of science and technology (electric motors and generators, electriclighting, radio and television, information technology, particle accelerators,etc.). Later, Maxwell showed that the electromagnetic force is a sum of theelectric and magnetic forces. At small velocities v -C c the magnetic forceis small, but at large velocities it is of the same order of magnitude as theelectric force. Thus, the scale that determines unification of the electricand magnetic forces is the scale of the velocities of charged particles.

The greatest achievement of electromagnetism was the creation of thetheory of the electromagnetic field by Maxwell in 1860-1865, which hasbeen formulated as a set of equations named after him. Maxwell's equationsunified the electric and magnetic fields into a single electromagnetic field.

Maxwell established that visible light consists of electromagnetic waveswith frequencies between 4 • 1014 and 7.5 • 1014 s"1. Since the frequency vis related to the wavelength A by the formula A = c/v, the wavelengths ofvisible light range from 7.5 • 10~7 to 4 • 10~7 m. Later it was shown thatultraviolet, infrared, and Roentgen rays, as well as gamma radiation andradio waves, are also electromagnetic waves.

In large bodies (stars and planets), the amounts of positive and negativecharges are almost the same, i.e., the electromagnetic forces between theseobjects are small. By contrast, electromagnetic interaction plays a definitiverole in the world of atoms and molecules. Atoms, molecules, and solids owetheir existence to electromagnetic forces. Electromagnetic interaction isalso intrinsic to the chemical and Van der Waals forces that act betweenatoms at large distances, i.e., at distances large relative to the linear sizesof atoms. The range of action of electromagnetic interaction is infinite.

Electromagnetic processes in the microworld are described by quantumelectrodynamics. It allows us to explain numerous processes of an electro-magnetic nature, in which elementary particles, atomic nuclei, atoms, andmolecules take part. In quantum electrodynamics, the interaction of twocharged particles is considered as an exchange of one or several photonswhich are quanta of the electromagnetic field. These photons are emit-ted by one charged particle and absorbed by another. Since these photons


cannot be registered experimentally, they are called virtual photons. Thephoton possesses, as we already stated, zero mass and spin 1. Note that allexchange particles (the quanta of different fields) always have integer spins,i.e., they are bosons.

2.3 Weak Interaction

A comparatively small number of fundamental particles among the severalhundred currently known elementary particles are now considered as struc-tureless. Leptons (from the Greek word "leptos", meaning "light") belongto this class. Now we know three pairs of leptons. These pairs are usuallycalled the three generations of leptons. The first generation consists of theelectron e~ and the electron neutrino ve. The second generation consistsof the muon /J,~ and the muonic neutrino v^. The third generation consistsof the tau-lepton T~~ and the neutrino vT corresponding to it.

The masses of all neutrinos are apparently zero, or are so small that ithas not been possible to measure them. The masses of other leptons are asfollows: me = 0.511 MeV « 9.1 • 1CT31 kg, m^ = 106 MeV w 1.89 • lO""28

kg, and mT = 1784 MeV « 3.18 • 10~27 kg. All leptons have spin 1/2. Themuon and tau-lepton are unstable. Their half-lives are £1/2(AO = 2-2 • 10~6

sand £1 / 2 (T) =3.15- 10"13 s.How do the neutrino and antineutrino differ? Since in the theory it is

usually assumed that these particles have zero rest mass, they must natu-rally move with the speed of light. In this case, it is possible to characterizea particle by the projection of its spin upon the direction of its momentum.This quantity is called helicity. The spin of a neutrino is directed againstits momentum (left-handed helicity), and the spin of an antineutrino is di-rected along its momentum (right-handed helicity). Since these particlesmove at the speed of light, their helicities are conserved in any frame ofreference.

Leptons take part in weak interactions. In such a process, the numbersof leptons before and after the reaction are the same. Strictly speaking,the difference in the total number of leptons and the total number of an-tileptons is conserved. Moreover, the numbers of leptons of each generationare conserved separately. The conservation of the lepton number in weakinteraction processes is called the leptonic charge conservation law. If oneascribes the leptonic charge +1 to any lepton and —1 to any antilepton,then the summary leptonic charge before and after any reaction is conserved


(all other particles have zero leptonic charge). An example of a leptonicprocess is muon decay: /z~ —> e~ + ve + v^ (here the overbar denotes anantiparticle).

Hadrons also undergo weak interactions, however their main type ofinteraction is the strong one. For example, a free neutron decays accordingto the scheme n - t p + e" + ve. This process is the /3-decay of a neutronand is the basis of the /3-decay of atomic nuclei. The elementary reactionof the capture of an atomic electron by a neutron of the nucleus (usuallyfrom the if-shell), p + e~ —> n + ve, is a basis of the /('-capture which is avariety of the /3-decay. The pion (7r-meson) decays creating a muon and anantineutrino, n~ —> pT + PM.

The particles taking part in a weak interaction exchange intermedi-ate bosons. These possess, as the photon does, spin 1. There are threeintermediate bosons: the charged W+- and W^-bosons, and the neutralZ°-boson. Exchange particles responsible for the weak interaction are veryheavy. Their masses are mw = 80.22 ± 0.26 GeV « 1.43 • 10~25 kg andmz = 91.127 ± 0.007 GeV « 1.62 • 10~25 kg. Because of this, the rangeof action of the weak forces is very small, r\y = fr/m\yc « 10~18 m. Theweak forces are short-range ones.

In the theory of weak interaction, the process of neutron decay is con-sidered as two successive steps. First, the neutron emits a W~-boson andtransforms into a proton: n —> p + W~. Then the W~-boson decays intoan electron and an antineutrino in a time r « 10~26 s: W~ —> e~ + ve.Analogously, the /3-decay of nuclei should be considered as a two-stage pro-cess in which, first, an intermediate boson is emitted which further decaysinto leptons.

Intermediate bosons, predicted theoretically, were discovered in 1983.The Nobel Prize in Physics for 1984 was awarded to those who made decisivecontributions to the large project that led to the discovery of the W- andZ-bosons. The engineer Van der Meer created the particle storage ring usedin the experiment, and the physicist Rubbia led the team of 150 scientists.

Similar to Maxwell's unification of electricity and magnetism, a theoryhas been developed that has unified the electromagnetic and weak inter-actions. For the creation of this theory, the Nobel Prize in Physics for1979 was awarded to Glashow, Weinberg, and Salam. In the theory, theexistence of the W±- and Z°-bosons has been predicted, and it has beenestablished that three intermediate bosons and the photon must behavesimilarly at energies much greater than 100 GeV, i.e., they are just like


the same particle in this case. Nonetheless, at low energies these particlesessentially differ.

2.4 Non-Conservation of Parity in Weak Interaction

Along with the properties of homogeneity and isotropy of space that lead tothe laws of conservation of energy and momentum for an isolated physicalsystem, there is one more symmetry property of space — the invariance ofsystem properties with respect to mirror reflection, i.e., the simultaneouschange in the signs of all spatial coordinates.

In classical mechanics all processes are invariant with respect to theinversion transformation, since in this case the equations of motion do notchange. In quantum mechanics this invariance is not always present. Beforeconsidering the invariance with respect to the mirror inversion in quantumprocesses, let us consider the left-right (mirror) symmetry in Nature.

When a man looks in the mirror his face seems to be symmetrical.Leaves of trees, at first sight, also seem to be symmetrical. In fact, afterscrutinizing these objects it turns out that small deviations from symmetryare always present. People and animals are nonsymmetrical creatures, sincethe heart and some unpaired organs (for example, the liver) are situated offto one side. It is surprising that Nature has not formed both possible typesof people: those with hearts on the left side and those with hearts on theright side. All people (except for some anomalous cases) have their heartson the left side. Thus, biology departs from complete bilateral symmetry.

When studying microobjects, we also encounter asymmetry. The ab-sence of bilateral symmetry in some organic compounds of biological struc-tures was noticed by Paster as early as 1848. For example, DNA (deoxiri-bonucleic acid) molecules created by an organism look like right-handedscrews. This rule is the same both for DNA created by humans and othermammals, and for DNA created by bacteria. The corresponding "left-hand-screw" DNA molecules, which are the mirror reflections of the "right-hand-screw" molecules, are never created by living organisms.

How do organisms know that they only have to synthesize right-handedDNA molecules? Undoubtedly, this program is written in the genetic codesof all living organisms. But why did Nature choose just right-hand screwsand not permit left- and right-handed DNA molecules with equal probabil-ities? It is clear that in the process of evolution of life on Earth, somethinghad to occur which selected the right-handed sense of rotation in the struc-ture of DNA molecules in biological systems. This phenomenon remains


unexplained.Now let us consider quantum objects. We have seen that the neu-

trino always has left-handed helicity, and that right-handed helicity canbe possessed only by the antineutrino. Thus, in the world of leptons, thepreference for the left-handed screw is also observed — moreover, it existsfor all three types of neutrinos.

In quantum mechanics the wave function for some cases (e.g., for a par-ticle moving in a central field and having a definite value of the orbitalmoment) can remain unchanged or can change its sign under an inversion.Then the particle or quantum system state described by such a wave func-tion is said to have a definite parity. If the wave function does not changeunder inversion, the parity is positive — otherwise, it is negative. Parity isa conserved quantity in electromagnetic and strong interactions. However,it turns out that parity is not conserved in weak interactions.

Parity non-conservation was discovered in the decay of /('-mesons, whichdecay as particles with positive parity in some cases and negative parityin other cases. The American physicists Dao Lee and Chen Ning Yangestablished in 1956 that the proof of parity conservation only exists forelectromagnetic and strong interactions, but in weak interactions parity isnot conserved. For their fundamental investigations of parity laws, Lee andYang were awarded the Nobel Prize in Physics for 1957. In 1957, Wu withher collaborators carried out a special experiment which confirmed paritynon-conservation in weak interactions.

The idea of Wu's experiment centered on the following. The /3-decay ofcobalt nuclei was studied:

60Co—> eoNi + e-+ve. (2.28)

In this experiment, cobalt nuclei were polarized by means of a very strongmagnetic field, H ~ 107 A/m, in such a way that their spins were ori-ented along a certain direction. The radioactive substance was cooled toultralow temperatures T « 0.01 K in order to decrease the role of the ther-mal kinetic motion of the nuclei, which could destroy their polarization. Toreach such ultralow temperatures, the method of adiabatic demagnetizing ofparamagnet salts was used. In this case, the substance cooling is explainedby the existence of the magnetocaloric effect. This occurs if heat inflowfrom outside is absent, then the work of the paramagnet demagnetizing isdone solely due to its internal energy (the thermal kinetic energy). In thiscase, the preliminary magnetizing of the paramagnet should be performedisothermally. For technical reasons it was not easy to carry out this exper-


iment at that time. Nevertheless, it was accomplished. Electron emissionwas mainly observed in the direction opposite to the spin orientation of thenuclei, indicating the absence of mirror symmetry in the /3-decay process.In other words, during the /3-decay of nuclei due to weak interaction, parityis not conserved.

Numerous experiments carried out afterwards have convincingly con-firmed parity non-conservation in the /3-decays of different nuclei, i.e., theviolation of the parity conservation law in weak interactions. Returningto neutrinos, we should note that these particles are always polarized sincethey always have left-handed helicity (their spins are always directed againsttheir momenta). Thus, these leptons manifest parity non-conservation dur-ing the weak interactions. However, it remains unclear why the weak in-teraction prefers just left-handed helicity.

2.5 Strong Interaction

The existence of atomic nuclei was discovered by Rutherford in 1911, and heintroduced the term "nucleus" in 1912. However, an understanding of thestructure of atomic nuclei became possible only after the discovery of theneutron by Rutherford's disciple Chadwick in 1932, who observed neutroncreation during the irradiation of beryllium by a-particles:

a+ 9Be —• 12C + n. (2.29)

For this discovery, Chadwick was awarded the Nobel Prize in Physics for1935.

At first, physicists supposed that the neutron is a composite particleconsisting of a proton and an electron. However, in 1934 Chadwick andGoldhaber discovered that the neutron mass is larger than the sum of themasses of the proton and electron (now, it is known that the neutron massis equal to 1.00083 of the sum of the masses of the proton and electron). Forthis reason, the neutron was acknowledged as a new elementary particle.

After the discovery of neutrons, it became clear that atomic nuclei con-sisted of nucleons, i.e., of protons and neutrons. For the first time thisnotion regarding nuclei was developed by Heisenberg and, independently,by Ivanenko. The interaction between nucleons is called the nuclear inter-action. For a long time it was presumed to be fundamental. The intensityof this interaction surpasses by far that of all other types of interaction.

In order to compare the intensities of different interactions, let us con-


sider two protons separated by a distance r m h/(mpc) « 2 • 10"16 m, wheremp is the proton mass. We shall take the protons as point particles. Inthis case, the ratio of their energies due to the electromagnetic and strong(nuclear) interactions is a quantity of order 10~2, the ratio of their energiesdue to the weak and strong interactions is of order 10~5, and the ratio oftheir energies due to the gravitational and strong interactions is of order10"38.

The quantum theory of any interaction between two particles is basedupon the exchange of some other particles. A theory of nuclear forceswas first developed by the Japanese physicist Yukawa in 1935. Accordingto this theory, the nuclear interaction between nucleons occurs because ofthe exchange of 7r-mesons. There are three such particles: two charged n-mesons, TT+ and TT~, and one neutral 7r-meson, TT°. The spins of n-mesonsare equal to zero. The masses of the charged pions (7r-mesons) are mn+ =mff- = 139.57 MeV « 2.48 • 1(T28 kg, and the mass of the neutral pionequals m / = 134.96 MeV sa 2.40 • 10~28 kg, i.e., the masses of pionsare about 270-280 times greater than the electron mass. The pions werefound experimentally in 1947. These particles are unstable. They decayaccording to the following schemes: TT+ —> fj,+ + v^ (fi+ is an antiparticlewith respect to /x~), n~ —> ji~ + PM, TT° —> 7 + 7 (here 7 denotes aphoton). The half-decay periods of the charged pions are 2.6 • 10~8 s, andthat of the neutral pion is 0.83 • 10~16 s. For his prediction of the existenceof mesons on the basis of theoretical work on nuclear forces, Yukawa wasawarded the Nobel Prize in Physics for 1949.

V(r) 1

0 — \ 1 ' ' 1 .

\O5 to 1 . 5 ^ 7

Fig. 2.1 Nuclear interaction potential.

It is now known that the nuclear forces of the interaction between twonucleons cannot be explained on the basis of pion exchange only. In reality,the nuclear interaction potential is rather complicated (Figure 2.1). Thenuclear forces of attraction between two nucleons at sufficiently large dis-


tances, r « 2 • 10~15 m, are due to the exchange of a ir-meson. At smallerdistances, 0.4-10"15 m < r < 2-10~15 m, the nucleons apparently exchangetwo pions and, maybe, 77- or if-mesons which are, like pions, spinless par-ticles (the mass of the neutral 77-meson is 549 MeV, that of the iC+-mesonis 494 MeV, and that of the X0-meson is 498 MeV). At small distancesr < 0.4 • 10~15 m, where the interaction between two nucleons has thecharacter of repulsion, they exchange vector mesons (the p+-, p~-, and p°-mesons have masses of 769 MeV « 1.4 • 10~27 kg, and the w°-mesons havemasses of 783 MeV « 1.4 • 10"27 kg) having spin 1. Since the exchangeof a 7T-meson explains the nuclear forces at large distances, the range ofinfluence of these forces can be determined as rn = h/m^c. Substitutingthe pion mass into this formula, we find rn ss 1.4 • 10~15 m. Thus, nuclearforces are short-range, which is similar to the weak interaction forces.

The nuclear forces differ in essential ways from other known forces.These forces are mainly attractive, but at very small distances, r < 0.4 •10~15 m, they become repulsive. The nuclear forces have a finite range ofinfluence: at distances r > 2 • 10~15 m they are not practically observed.The nuclear forces possess the saturation property. For example, a nucleondoes not interact with all other nucleons in the same nucleus, but only withseveral neighbors. The nuclear forces depend on the mutual orientation ofspins of the interacting particles. They are noncentral forces and depend onthe value of the total spin of the system of interacting particles. The nuclearforces do not depend on the electric charges of the interacting particles, i.e.,the nuclear forces acting between two protons or between two neutrons arethe same as the forces between a proton and a neutron.

Nucleons are actually composite particles consisting of quarks, i.e., nu-clear forces are secondary, and the strong interaction is defined by the forcesbetween quarks. The particles undergoing the strong interaction are calledthe hadrons (from the Greek word "hadros", meaning "strong"). The con-cept of the quark structure of hadrons was independently formulated byGell-Mann and Zweig in 1964. For his works on the classification of ele-mentary particles and their interactions, Gell-Mann was awarded the NobelPrize in Physics for 1969. Gell-Mann took the particle name "quark" fromthe 1939 novel Finnegans Wake by Irish writer James Joyce. This noveldoes not translate well into other languages because it toys with seman-tics and puns. Joyce's novel describes the life of Mr. Finn, who sometimesassumes an appearance of Mr. Mark. Three quarks are children of Mr.Finn, which often appear in the role of Mr. Finn (Mr. Mark) himself. Fromhere, an association with elementary particle physics has arisen, since the


nucleons and some heavier particles consist of three quarks. The particlesconsisting of three quarks are called baryons (from the Greek word "barys",meaning "heavy"). The particles consisting of two quarks (strictly speak-ing, of a quark and an antiquark) are called mesons (from the Greek word"mesos", meaning "medium" or "intermediate"). We may also note thatthe word "quark" in the German means "curds".

Let us consider the quark structure of some baryons in more detail.Protons have mass mp = 938.3 MeV w 1.67 • 10"27 kg and spin 1/2. Theneutron mass is equal to mn = 939.6 MeV sa 1.67 • 10~27 kg and its spin is1/2. The masses of these particles differ only by 0.14% of the proton mass.Their main difference is that the proton has a positive elementary chargewhile the neutron is electrically neutral. To explain the structure of nucle-ons, it is sufficient to assume the existence of two types of quarks. Theseare denoted by the letters u and d (for "up" and "down", respectively).The up- and down-quarks form the first generation of quarks. The quarkspossess spin 1/2, so that it is possible to construct particles with differentspins out of them.

The most amazing property of quarks is their fractional electric charge.The quark u has the charge 2e/3, and the quark d has the charge —e/3.Quarks are the only particles known to have fractional charges. Now, letus consider how it is possible to build up nucleons out of the quarks u andd. The proton consists of two quarks u and one quark d, and the neutronconsists of one quark u and two quarks d. In order that the Pauli principlenot be violated, the spins of the two quarks u that enter into a proton andof the two quarks d that enter into a neutron should be directed oppositely,i.e., every such pair of identical quarks inside a nucleon has zero spin, andthe spin of the third quark defines the spin of the nucleon.

An essential difficulty has arisen in the attempt to explain the structureof short-lived A-particles on the basis of the quark model. The A-particleshave masses mA = 1232 MeV « 2.19 • lO"27 kg and spins 3/2. Four A-particles are known, which differ by their charges: A + + , A+, A0, and A~.To construct the particle A+ + , which has charge 2e, it is necessary to takethree quarks u with parallel spins in order that their total spin would equal3/2. However, in this configuration the Pauli principle, which prohibits anytwo identical fermions from being in the same state, would be violated.

This difficulty can be removed if one supposes that quarks are charac-terized by an additional quantum number — the color — which can takethree different values. It is assumed that any quark is always in one ofthree color states: blue, green, or red. For this reason, the particle A + +


consists of three quarks u — one blue, one green, and one red. Of course,the term "color" should not be understood literally; it is merely a labelapplied to a certain quantum state of a quark. The structure of hadrons isthen determined by the following rule: the quarks entering into a hadronmust have colors such that the hadron is a "white" object. Here, the opti-cal rule is taken into consideration: by mixing equal amounts of the actualcolors blue, green, and red, we get the color white.

After the creation of the quark model, numerous searches for free quarksin Nature and in laboratory settings were carried out. However, there hasbeen a singular failure to observe free quarks. Quarks can only be seeninside hadrons. For this, it is necessary to investigate the internal structureof hadrons (e.g., protons) using particles of very high energies. Since elec-trons take part only in electromagnetic interaction (their weak interactionpractically plays no role against the background of the more intense elec-tromagnetic interaction), from the character of their scattering on protonsone can learn how electric charge is distributed inside a proton. It hasbeen found from experiments that during interaction with protons, elec-trons are very often scattered to large angles, as if they collided with someelectrically charged point objects when flying through a proton. Moreover,from the character of electron scattering, the electric charges of these pointobjects inside protons have been successfully determined. Their chargesare equal to 2e/3 and —e/3, i.e., they coincide with the charges of quarks.For a homogeneous electric charge distribution inside protons, the electronscattering character would be quite different. Thus, quarks inside a protonhave been successfully observed.

The physical theory describing the strong interaction of quarks has beencalled quantum chromodynamics (from the Greek word "chromatos", mean-ing "color") because quarks are colored. It turns out that the interactions ofquarks possess interesting peculiarities. At small distances, i.e., at distancesmuch smaller than the linear size of the nucleon, r^ ~ 10~15 m, quarksbehave practically like free particles. This phenomenon has been calledasymptotic freedom. If the distance between two quarks increases, reachingthe order of rjy, then the force of attraction of the quarks rapidly increases.In other words, it is impossible to tear quarks away from each other. Forthis reason, quarks are "locked" inside hadrons. This phenomenon is calledconfinement. It follows from quantum chromodynamics that a colored ob-ject (e.g., a quark) in a free state cannot be observed. Observable objectsare only white objects, i.e., hadrons.

During interaction, quarks exchange particles called gluons (from the


word "glue", since they glue together quarks inside hadrons). There existeight different gluons. A gluon possesses spin 1 and its mass and electriccharge are both zero. Gluons, like quarks, are colored objects. However,unlike a quark, each gluon bears two colors. For this reason, interactingquarks change their colors during gluon exchange. Since a gluon is a coloredobject, it, like a quark, cannot be observed as a free particle. However,several gluons can form a "white" object that can be observed. Such gluonicformations are called glueballs. Glueballs created in various processes donot live long, and decay into hadrons.

Besides baryons, which are the hadrons that consist of three quarks,there also exist mesons — hadrons consisting of a quark and an antiquark.Let us consider the simplest family of these particles — pions — whichincludes three particles: TT+, TT~ , and n°. The meson TT+ consists of thequark u and the antiquark d (the charge of an antiquark is opposite tothat of a quark). Moreover, the 7r+-meson contains the combination ofthe blue quark u and the antiquark d, whose color is complementary toblue, i.e., together they yield white. This combination is added to the sameamounts of the configurations of the green quark u and the antiquark d,whose color is complementary to green, as well as of the red quark u and theantiquark d, whose color is complementary to red. The meson TT~ containsthe combinations of corresponding colored quarks d and antiquarks u, andthe meson TT° contains the combinations of u and u as well as of d andd. The spins of quarks and antiquarks that form the "white" pairs inside7r-mesons are directed oppositely. For this reason, 7r-mesons have zero spins.

There are also hadrons called vector mesons. Belonging to this class,in particular, are three p-mesons — p+, p~, and p° — whose spins equalunity. The /9-mesons are structurally similar to the 7r-mesons, but the spinsof the quarks and antiquarks entering them are parallel in each "white"pair. This configuration of two quarks turns out to be "heavier" than theconfiguration with oppositely directed spins. For this reason, the massesof p-mesons are much greater than those of 7r-mesons. There are otherknown mesons that contain quarks and antiquarks, both of the first andother generations.

Investigations of various processes observed at high energies have shownthat, besides the first generation of quarks, there also exist two other gener-ations. The second generation includes the charm quarks c and the strangequarks s. These carry electric charges 2e/3 and — e/3, respectively. Hadronscontaining c-quarks are called charmed particles; those containing s-quarksare called strange particles.


Table 2.1 Symmetry of first generation leptons and quarks.

Electric charge (e) - 1 - | - | 0 + | + § + 1

leptons e i/e

antiquarks u dquarks d u

antileptons Pe e

The third generation includes the top quarks t and bottom quarks b.Note that these designations parallel those of the first generation quarks uand d. The quark t carries charge 2e/3, while the quark b carries charge—e/3. So there are six known quarks; moreover, it has been establishedthat there are no other quarks. One can notice a certain symmetry betweenquarks and leptons, which also have three generations. These generationsof particles are usually displayed as

C-) (?) C-) CD 0 0- ^In Table 2.1, the observed symmetry of the leptons and quarks of the firstgeneration is shown.

All matter in the Universe consists only of first generation particles: thequarks u and d enter into nucleons forming atomic nuclei, and electrons,together with the nuclei, form atoms. While a supposition arises that itmight be possible to build up matter using particles of the second and thirdgenerations, this has not yet been discovered in Nature. Hadrons containingquarks of the second and third generations have been obtained only underlaboratory conditions (they have not been observed in cosmic rays).

At very high energies, a collision of hadrons can create a quark-gluonplasma, i.e., the structure of hadrons is demolished and matter consistingof quarks and gluons is created. Such matter has interesting properties,and its investigation makes possible the extension of our knowledge aboutquarks and gluons.

Let us summarize. It follows from what has been written above, thatthe interaction of two quantum objects is realized by an exchange of certainparticles with integer spins. For the electromagnetic interaction these arephotons; for the weak interaction they are the intermediate bosons W+,W~, and Z°; for the strong interaction they are gluons (eight of them);for the gravitational interaction they are gravitons that have not yet beenobserved experimentally. These particles, which are quanta of the corre-


sponding fields, have been called the gauge bosons. The spin of a particlewhich transfers interaction is of great importance: if its spin is even or zero,then two identical particles attract each other, but if its spin is odd, theymutually repel.

The particles transferring interaction are bosons and, for this reason,they do not obey the Pauli principle. Consequently there is no restrictionon the number of bosons that can be exchanged by the interacting particlesin one act of the interaction, i.e., the force of interaction can be large.The particles transferring interaction are virtual particles and cannot beregistered directly. However, the virtual particles do exist because onecan measure magnitudes of the effects connected with the interaction ofmaterial particles. Under certain conditions, the gauge bosons also exist asreal particles, i.e., in this case they can be registered by special detectors.

So far, the question remains open whether leptons and quarks are true"elementary" particles or whether they consist of some other particles. Toanswer this question, it is necessary to "probe" their structure at distancesmuch smaller than 10~18 m. To this end, particle accelerators must beconstructed that could attain energies at least as high as 104 GeV. Sofar, such energies have not been reached, and the answer to the questionregarding the structure of leptons and quarks is not known.

In conclusion, let us again touch upon the nature of nuclear forces. First,let us consider the interaction of atoms at large distances. Atoms are elec-trically neutral objects. For this reason, the usual electric forces do not actbetween atoms at distances large compared to their linear sizes. However,atoms have finite sizes, and the electric charges in them are distributedin a certain way. Therefore, atoms can have, generally speaking, nonzerodipole, quadrupole, octupole, and other electric multipole moments, whichare caused by their spatial distribution of charges. The presence of multi-pole moments leads to the fact that, as atoms approach, forces start to act.These have been named Van der Waals forces after the Dutch physicistwho was first to study them in the last century. They are attractive, andtheir dependence on distance between atoms is denned by the formula

Fw = -%, (2.31)r1

where C is a positive constant.The nuclear forces between hadrons are similar to the Van der Waals

forces in atomic physics. These forces decrease very rapidly with an increasein the distance between hadrons. Thus, nuclear forces are not fundamental;


they are a consequence of the superstrong quark-quark forces, which are thefundamental chromodynamical forces. It is these latter forces, caused byan exchange of gluons, that determine the strong interaction.

Chapter 3

Structure of Atomic Nuclei

3.1 Composition and Properties of Nuclei

In 1909 Geiger and Marsden, disciples of Rutherford, found that the mostprobable scattering angle of the 5.5 MeV.a-particles emitted by radioactivebismuth 214Bi (called RaC at that time), and passing through gold foil4 • 10~7 m thick, was 0.87°. However, approximately one a-particle in20,000 was scattered at an angle exceeding 90°, i.e., backward.

Rutherford's genius helped him understand this deviation of a smallnumber of cc-particles through large angles. He argued that a positivelycharged a-particle with sufficiently high energy could be backscattered bya collision with something extremely small, heavy, and charged within theatom. He introduced the notion of "atomic nucleus" by analogy with thecell nucleus in biology.

Rutherford made a simple calculation and found that the linear size ofnucleus was at least thousand times smaller than that of the atom. Theformula he obtained (which was later named after him) made it possible todescribe the scattering of one charged point particle by another. Rutherfordwas extremely lucky because his formula, obtained on the basis of classicalnotions, turned out to be correct in quantum mechanics, too, which wasunderstood much later. Therefore, in 1911 Rutherford drew the conclusionregarding the existence of an atomic nucleus.

An atomic nucleus possesses fascinating properties. It contains approx-imately 99.97% of the atomic mass, but occupies a volume ten thousandbillion times smaller than that of an atom. This means that the atom, asis the case with all matter, consists mostly of empty space. In other words,the density of nuclear matter is ten thousand billion times greater than thedensity of Earth matter. The carrying capacity of a single truck is sufficient

67


to transport just 2 • 10~13 m3 worth of nuclear matter.In 1911, Geiger and Marsden began a research program with a series

of precise measurements concerning the fraction of a-particles scattered atdifferent angles by atoms. In 1913, they reported that the experimentaldata obtained were in good agreement with Rutherford's formula. Thiswas the final confirmation of Rutherford's discovery of the atomic nucleus.

The simplest atomic nucleus — that of the hydrogen atom — is a proton.However, two protons cannot form a bound state. In order to "glue" protonsto form atomic nuclei, other particles should be added. After the discoveryof a neutron by Chadwick in 1932, Heisenberg, and independently Ivanenko,proposed a realistic model of the atomic nucleus. According to that model,the atomic nucleus consists of protons and neutrons and, evidently, is apositively charged composite particle. The number Z of protons in a nucleuscoincides with the atomic number, i.e., the number of electrons in a neutralatom. The sum of the number Z of protons and the number N of neutronsin a nucleus is called the mass number A = Z + N.

Before 1932, some scientists believed that atomic nuclei consisted of pro-tons and electrons. Then, the nucleus charge Z would be equal to the differ-ence between the number of protons and the number of electrons. However,that assumption was incorrect. First, an electron cannot be confined in aregion of space occupied by a nucleus because, according to the Heisenberguncertainty principle, such an electron would have so much kinetic energythat it could not remain inside the nucleus. Indeed, Ap ~ h/ Ax, where Apis the uncertainty in the electron's momentum. The uncertainty in the elec-tron's coordinate is defined by the linear size of the nucleus: Arc ~ 10~15

m. Since the velocity of an electron inside the nucleus must be very high —on the order of c — its kinetic energy is of order AE ~ cAp ~ ch/Ax ~ 100MeV. We see that AE is much greater than the experimentally measuredbinding energy of one particle inside a nucleus, which is of order 10 MeV.So an electron cannot be confined within a nucleus.

Other experimental facts are consistent with the result that electronscannot reside in nuclei. For instance, according to the electron-protonmodel, the 6Li nucleus should contain 6 protons and 3 electrons. Sinceproton and neutron spins are equal to 1/2 (in h units), the spin of 6Li(total spin of 6 protons and 3 electrons) will be a half-integer. However,experiments show that the 6Li nucleus possesses integer spin. The sameholds for the nuclei of 2H, 10B, and 14N.

Nuclei that contain Z protons can have different numbers of neutrons.Nuclei having the same numbers of protons Z but different N (or A) are

Structure of Atomic Nuclei 69

called isotopes. This nomenclature for varieties of the same element wasgiven by the British physicist Soddy in 1910, because in the table of chem-ical elements they occupy the same place (from the Greek "isos", meaning"same", and "topos", meaning "place"). For instance, hydrogen has 3 iso-topes: light hydrogen (protium) 1H, deuterium 2H, and tritium 3H. Xenon(Xe) has the largest number of isotopes: 28 (118 < A < 145). For hiscontribution to our knowledge of the chemistry of radioactive substances,and his investigations into the origin and nature of isotopes, Soddy wasawarded the Nobel Prize in Chemistry in 1921. Another British physicist,Aston, was awarded the Nobel Prize in Chemistry in 1922 for his discov-ery of isotopes in a large number of nonradioactive elements, and for hisenunciation of the whole-number rule.

Nuclei having the same A but different Z and N are called isobarsor isobaric nuclei (from the Greek "baros", meaning "weight"). Nucleiwith the same N but different Z (or A) are called isotones (from the Greek"tonos", meaning "strain"). A given nucleus with definite A and Z is calleda nuclide.

The mass of a nucleus depends on the total number of neutrons andprotons it contains. Provided the nucleons in a nucleus do not interact witheach other, making the nucleus a "gas" of free nucleons, its mass would bem(Z, A) = Zmp + (A — Z)mn, where mp and mn are the proton and neutronmasses. However, allowing for the interaction between nucleons, the massof a nucleus is smaller:

m(Z, A) = Zmp + (A - Z)mn - AM, (3.1)

where AM is called the mass defect. A stable nucleus has AM > 0.According to the theory of relativity, the nucleus rest energy EA =

m(Z,A) • c2. The product of the nucleus mass defect and the squaredvelocity of light is the nucleus binding energy: W(Z, A) = AM • c2.

Nucleus masses are measured in atomic mass units (a.m.u., or daltons).This unit is chosen to make the mass of the 12C carbon nucleus preciselyequal to 12 a.m.u.

Denoting u = 1 a.m.u., we introduce the magnitude

Am = m(Z, A) - Au. (3.2)

This quantity, called the mass excess, can be positive or negative. Conver-sion from a.m.u. to the units of energy, MeV, in which nuclear masses can


also be measured, is accomplished by the formula

u = 931.5016 MeV. (3.3)

Nuclear mass could also be expressed in kg through the conversion formula

u = 1.66 • 10"27 kg. (3.4)

Nuclear masses are measured experimentally with special devices calledmass spectrometers. They can also be defined from measurements of theenergies of different nuclear reactions, from the energies of a- and /3-decays,and from radiospectroscopic measurements of the frequencies of transitionsbetween rotational levels of molecules. The latter measurements are ableto define ratios of nuclear masses with high precision of the order of 10~5 —-icr6.

There are 287 nuclei in Nature. Most of these (168) are even-even nucleihaving even numbers of protons and neutrons. There are only 4 stable odd-odd nuclei: \R, |Li, 5°B, and y4N. There are 5 radioactive odd-odd nuclei:19K, 2°V> sfLa, ^6Lu, and | fTa. Elements that have Z = 43,61,85,and 87, and Z > 93, are obtained artificially because they do not occurterrestrially. There are no stable nuclei with A = 5,8,147, or A > 210.

Tin has the largest number of stable isotopes: 10. Light nuclei (up to20Ca) have approximately equal numbers of protons and neutrons. Further,with an increase in Z the number of neutrons in natural nuclei begins toexceed the number of protons. Thus, the heaviest natural nuclei have morethan 1.5 times as many neutrons as protons.

Nuclear binding energy can be denned on the basis of a hydrodynamicalmodel, where a nucleus is treated as a drop of an incompressible chargedliquid. Neglecting surface energy and the Coulomb interaction of protons,the nuclear binding energy would be proportional to the number A of nu-cleons in a nucleus (volume energy). The surface energy is proportional tothe area of the nuclear surface, i.e., to the square of the nuclear radius R,and decreases the binding energy.

Since the volume of a nucleus treated as a drop of an incompressibleliquid is proportional to the number of nucleons A, the radius of a nucleus isproportional to A1/3: R = r0A1/3, where r0 can be considered as a constantvalue. The surface energy is proportional to A2/3. The electrostatic energyof repulsion of protons in a nucleus is proportional to the squared chargenumber Z2 and inversely proportional to the nuclear radius, i.e., to A"1/3.This also decreases the binding energy. Therefore, considering volume,


surface, and Coulomb energies in a hydrodynamical (drop) model, we areled to the following formula for the nucleus binding energy:

W{Z,A)=aA-f3A2/3 --yZ2A~1/3, (3.5)

where a, /?, and 7 are constants. The quantity f3 is connected to the coef-ficient of surface tension a of a drop of nuclear "liquid" by the correlation/3 = 47rrQ<7, while the constant 7 is denned by the squared elementarycharge e: 7 = 3e2/(5ro).

Formula (3.5) for the nuclear binding energy can be improved by theaddition of two terms. Following the hypothesis of charge independenceof nuclear forces, symmetry between protons and neutrons in a nucleusshould exist. Thus the corresponding additional term (energy of symmetry)should depend only on the difference N — Z — A — 2Z and should be aneven function of it, i.e., the energy of symmetry will be proportional to(A — 2Z)2. This additional term decreases the nuclear binding energy.

Detailed analysis of the binding energies of nuclei shows that nucleonsof the same sort in a nucleus prefer to join in pairs. Such "pairing" ofnucleons leads to an increase in the nuclear binding energy. A correspondingadditional term (the energy of "pairing") is denned by 5A~3/4, where

{ \d\ for even-even nuclei,0 for nuclei with odd A, (3.6)

— \S\ for odd-odd nuclei.So the final formula for nuclear binding energy is

W(Z, A)=aA- $A2'3 - 1Z2A~1'3 - C ^ " ^ + 5A~3/A. (3.7)

This formula, known as Weizsdcker's semi-empirical formula for the bind-ing energy of the nucleus, was established in 1935 by the German physicistvon Weizsacker. The quantities a, j3, 7, £, and 5 in (3.7) are derived througha comparison of calculations with experimental data. One widespread (butnot unique) set of values is a — 15.75 MeV, /3 = 17.8 MeV, 7 = 0.71 MeV,C = 23.7 MeV, and |<5| = 34 MeV.

Fig. 3.1 demonstrates the dependence of binding energy per nucleonW(Z, A)/A (MeV) as a function of the nucleus mass number A. It canbe seen that the ratio W(Z,A)/A quickly increases in the region of thelightest nuclei, reaches a maximum at A = 56 (where W(Z, A)/A « 8.8MeV), and then decreases smoothly to about 7.5 MeV for heavy nuclei.The approximate constancy of the binding energy per nucleon (except for


3 ,

I / ^ ^

I i i i i i ^

80 160 A

Fig. 3.1 Dependence of the binding energy per nucleon in a nucleus as a functionof the mass number of the nucleus.

the lightest nuclei) is connected with the property of saturation of nuclearforces: each nucleon in a nucleus effectively interacts only with its nearestneighbors and not with every other nucleon.

Knowing the binding energy W(Z, A) of a nucleus, we can find its massby the formula

m(Z, A) = Zmp + (A- Z)mn (3.8)

a A P ,.2/3 7 ry<2 , 1/3 C {A-2Z)2 5 , _ 3 / 4

- -rA + ^A2/3 + -^Z2A~1/3 + \ - —- =A 3 /4.

Formula (3.8) can help us derive the correlation between A and Z for theisotopes of nuclei being stable against /3-decay. Under /3-decay the numberA is constant but Z changes. Assuming that (3.8) defines the dependenceof nuclear mass on its charge number Z for a constant mass number A, it ispossible to find the nucleus (isotope) with minimum mass (minimum restenergy) that will be stable against /3-decay. For this purpose, the derivativedm(Z, A)/dZ with constant A should be found and equated to zero. As aresult we obtain

Z = 1.98 + 0.015A2/3' ( 3 ' 9 )

The majority of nuclei have a spherical form. Some are deformed, i.e.,their equilibrium form is not spherical. But this nonsphericity is small.Thus, in the first approximation these nuclei can be considered as spherical.Since the nucleus volume is proportional to the number of nucleons A, thenucleus radius is proportional to A1/3: R — TQA1^. Values of ro found


by different experimental methods lie in the interval 1.1 • 10~15 m < r0 <1.5-10-15 m.

One of the simplest methods of measuring the linear sizes of nuclei isthe elastic scattering of fast neutrons by nuclei. This method is generallysuitable for medium and heavy nuclei whose sizes substantially exceed thewavelength of scattered neutrons strongly absorbed by nuclei. In this casethe neutron scattering is similar to the optical diffraction of light on a blacksphere (Fraunhofer diffraction). The extremum positions in the angulardistribution of scattered neutrons are completely denned by the nuclearradius. Experiments on the scattering of fast neutrons give ro = (1.3 —1.4) • 1(T15 m.

The distribution of charge (protons) in a nucleus is characterized by thescattering of electrons by nuclei because electrons only weakly interact withneutrons. Such experiments give values of nuclear (charge) radii smallerthan those found from the analysis of neutron scattering: ro = (1.2 — 1.3) •1CT15 m.

The radii of a-radioactive nuclei can be derived from observing an a-decay. The half-life of a-radioactive nuclei depends on the width of thepotential well the a-particle is in. Thus, the measured half-life gives thewell width and the nuclear radius as r0 = (1.45 - 1.50) • 10~15 m.

Analysis of 7-spectra of muon atoms is an important method of findingnuclear sizes. Provided the negatively charged muon is close enough to theatomic nucleus, it can be captured by the latter and move in the electricfield of the nucleus just like an electron. Such a system where one of theelectrons is replaced by a muon is similar to a usual atom and is calleda muon atom. Since the muon mass is more than 200 times greater thanthe electron mass (m^ « 207me), the mean distance between a muon andthe nucleus in an atom is approximately 200 times smaller than the meandistance between electron and the nucleus in an atom, i.e., unlike an electronthe muon is situated very close to the nucleus. The change in energy stateof a muon in an atom is accompanied by emission of a photon, the energyof which depends substantially on the nuclear radius. Analysis of 7-spectraof muon atoms give values of nuclear radii r0 — (1.1 - 1.2) • 10~15 m.

Studies of nuclear binding energies with the help of Weizsacker's for-mula, which contains the nuclear radius, give ro = (1.2 — 1.3) • 10~15 m.Experiments on the scattering of protons, a-particles, and different pro-jectiles by nuclei give nuclear radii in the region ro = (1.2 — 1.5) • 10~15

m.An atomic nucleus should not be considered as a sphere with a sharp


P(r)P(O)

1 ^ ^

0.5- Q 1—\

0 r

Fig. 3.2 Nuclear density p(r) as a function of the distance from the center ofnucleus r.

boundary because in the comparatively thin surface region the density ofnuclear matter changes from a value close to that in the nucleus center tozero. The nuclear density as a function of distance from the nucleus centeris well approximated by

P(r) = Vc)' (3-10)1 + exp i-^-i

where po is a normalization constant, C is the half-density radius, andd is the nuclear surface diffuseness value (Fig. 3.2). The density p(r) ischaracterized by the root-mean-square radius

( r 2 )= Id3rr2p(r). (3.11)

One can introduce the radius R of an equivalent homogeneous densitydistribution. This radius corresponds to the density p(r) at d = 0 and afixed atomic mass A. The radius R is related to (r2) by the formula

^ 2 = (r2). (3-12)

In this case the quantity R is exactly the nuclear radius. Note that nuclearmatter in the nucleus center has an enormous density of 2.2 • 1017 kg/m3.In order to imagine how large this is, we can note that one cubic millimeterof nuclear substance would have a mass of 220000 tons!

Like electrons, protons and neutrons have their own angular momenta— spins that equal 1/2 in h units. A nucleus consists of many protons andneutrons. Spins of separate nucleons of nuclei add according to the rule


for vector addition. Furthermore, nucleons inside a nucleus are in motionand have their own orbital moments. The latter are added with nucleonspins and give the total angular momentum of a nucleon at rest — its spin.The spin of a nucleus (in h units) can take integer and half-integer values:I = 0,1/2,1,3/2,.... Nuclei with odd A have half-integer spins, whilenuclei with even A have integer spins.

Ground state spins of stable even-even nuclei are equal to zero. Groundstate spins of stable natural nuclei with odd A do not exceed 9/2. Thatmeans that nuclear spin is small compared to the sum of the absolute valuesof the spins and orbital momenta of all nucleons forming a nucleus. Henceit follows that the majority of nucleons in a nucleus move so that their spinsand orbital moments compensate each other. Note that in Nature, thereare nuclei 50V with the ground state spin / = 6, 138La with / = 5, 176Luwith 1 = 7, and 180Ta with 1 = 8. However, all these nuclei are radioactive(although their half-lives are very large).

A deuteron is the simplest composite atomic nucleus, consisting of aneutron and a proton. A deuteron's spin is equal to unity, i.e., the spinsof nucleons forming a deuteron are always codirected. In Nature there areno bound states of a neutron and proton having zero spin. A deuteron'sbinding energy is small, W& = 2.22 MeV, and this nucleus has no excitedbound states. The study of the deuteron plays no less an important rolein nuclear physics than the study of the hydrogen atom in atomic physics.For the discovery of heavy hydrogen (deuterium), composed of a neutron,a proton, and an electron, the American physicist Urey was awarded theNobel Prize in Chemistry in 1934.

Besides its own angular momentum (spin), an electron has also its ownmagnetic moment /ze = -\e\h/(2mec) = -9.274 • 10~24 J/T. The quantityMB = \e\h/(2mec) is called a Bohr magneton. The proton and neutron alsohave their own magnetic moments — \iv = 2.79/J.N and fin = — 1.91/nw— where fi^ = \e\h/(2mpc) = 5.05 • 10~27 J/T is a nuclear magneton. Itcan be seen that the proton magnetic moment is positive, i.e., is directedalong the spin, while the neutron magnetic moment is negative and directedagainst the spin.

Since a nucleus consists of many nucleons performing complicated mo-tions, it also possesses a magnetic moment. This can be written asM/ = 9^NI, where g is called the gyromagnetic factor. If / = 0, thenfii = 0; that is, magnetic moments of all stable even-even nuclei in groundstates are equal to zero.

One of the most important characteristics of an atomic nucleus is its


electric charge Ze, which defines the chemical properties of an element.However, charge does not give a complete picture of the electric character-istics of a nucleus, which is an extensional structure where knowledge of thespatial distribution of charge is important. That is why a nucleus shouldbe characterized by multipole electric moments.

The dipole moment of any nucleus in the ground state is always equal tozero. Thus, much attention is paid to quadrupole moments of nuclei. Thesimplest model of a quadrupole is a system of two equal and oppositely-oriented electric dipoles situated at some distance from each other. Electricquadrupole moments of nuclei with 7 = 0 and I — 1/2 are always equalto zero. Large quadrupole moments are usually observed for nuclei havingnonspherical form.

3.2 Shell Model of Nuclei

In the model of nucleus as a liquid drop (hydrodynamical model) intro-duced before, it is assumed that nucleons in a nucleus are highly correlated(tightly bound) with each other. The opposite approximation is a modelwhere nucleons in a nucleus are assumed to move independently of oneanother; in such a model, the nucleus is similar to gas contained in somevolume. However, it is not a common gas but a quantum one. Since nu-cleons possess spin 1/2, they are fermions, and the gas of nucleons is aFermi-gas under very low temperature. This gas is called a degenerateFermi-gas, having properties substantially different from those of commonclassical gases. The Fermi-gas model, like the drop model, turned out to betoo simplified to describe several properties of such a complicated systemas an atomic nucleus.

Experimental investigations have shown exceptional stability of nucleihaving numbers of either protons Z or neutrons N equal to one of thenumbers 2, 8, 20, 28, 50, 82, or 126. These numbers of nucleons are called"magic" numbers. First let us consider some experimental data reflectingthe exceptional stability of magic nuclei with either Z or N equal to amagic number. If some particular nucleus has Z and N both equal tomagic numbers, that nucleus is called double magic.

Data on the relative abundance of different nuclei in Nature indicate thegreat stability of magic nuclei. Usually the relative abundance of isotopeswith even numbers of nucleons among other isotopes of the element does notexceed 60% (except the lightest nuclei). However, the following exceptions


exist: 40Ca (N = Z = 20) at 96.9%; 52Cr (N = 28) at 83.8%; 88Sr (N = 50)at 82.6%; 138Ba {N = 82) at 71.7%; 170Ce (N = 82) at 88.5%.

In Nature, there are usually 3-4 nuclei with a definite number of neu-trons N and different Z. The number of such nuclei is greater if the numberN of neutrons is magic: N = 20 (36S, 37C1, 38Ar, 39K, 40Ca); N = 28 (48Ca,50 T i ) 5 1 V ) 5 2 C r ) 54pe ) . N = 5 Q ( 8 6 K r ) 8 7 R b ) 8 8 ^ 89 Y ) 9 0 ^ 9 2 M o ) . ^ = 82

(136Xe, 138Ba, 139La, 140Ce, 141Pr, 142Nd, 144Sm). The Z = 50 element(tin) has 10 stable isotopes — more than any other element.

The probability of neutron capture by a nucleus having a magic numberN of neutrons is substantially smaller than that for other nuclei. This isdue to the anomalously small binding energy of a neutron in nuclei whosenumber of neutrons exceeds the magic number by unity. The energy of thefirst excited state of a magic nucleus is substantially greater than that forneighbor nuclei.

The properties of radioactive decay also attest to the special stabilityof magic nuclei. All three natural radioactive series of nuclei experiencinga-decay finish on lead isotopes (Z = 82). Alpha-particles with the greatestenergy are emitted by nuclei whose a-decay transforms them into a nucleuswith Z = 82 and N = 126. Beta-decay that forms a magic nucleus ischaracterized by the greatest energies.

Therefore, the existence of magic numbers of nucleons should be consid-ered as experimentally confirmed. The problem of theoretical descriptionof magic numbers then arises. This description is given by the shell modelof the nucleus, which in many respects is similar to the shell model of theatom.

The existence of magic numbers of nucleons in nuclei was first noticed byBartlet in 1932 and Elsasser in 1933-1934. However, these works remainedin obscurity because the theory of nuclear structure had just begun to de-velop. Much later, in 1949, the American physicist Maria Goeppert-Mayerand, independently, the German physicist Jensen, explained magic numbersby the concept of the nucleon shells in nuclei. Especially complicated wasthe explanation of the magic numbers 50, 82, and 126. It turned out thattheir existence could be explained only by adding purely quantum forces ofspin-orbit interaction to the common central nucleon-nucleon forces. Theseforces are caused by the interaction of the spin of a nucleon with its angularmomentum. They have no analogy in classical physics.

Forces of spin-orbit interaction were well known in atomic physics, buttheir importance in the theory of nuclear structure appeared absolutelyunexpected. These forces manifest themselves mainly in the surface region


of nuclei. The consideration of spin-orbit forces allowed Goeppert-Mayerand Jensen to develop the nuclear shell model and to explain various experi-mentally established properties of nuclei. Goeppert-Mayer and Jensen wereawarded the Nobel Prize in Physics in 1963 for their discoveries concerningnuclear structure.

At first sight, the nuclear shell model seems impossible to build. Indeed,a nucleus, unlike an atom, does not have any distinguished center of force,while nucleons, unlike electrons, strongly interact with each other. However,a more detailed analysis shows that several bases for the creation of anuclear shell model do exist.

Nucleons in a nucleus are in a state of rapid relative motion with respectto each other, and are separated by a distance on the order of 10~15 m. Itturns out that due to the Pauli principle and repulsion at small distances,the collisions of nucleons with each other in a nucleus are comparativelyrare events. Thus, it can be assumed that the motion of every nucleontakes place almost independently in the specific self-consistent field createdby all nucleons of a nucleus. Provided a nucleus is spherically symmetric,the potential of the self-consistent field will depend only on the absolutevalue of the distance between the given point and the geometrical centerof a spherical nucleus. Due to the short-range character of nuclear forces,the shape of this potential should be similar to the shape of the densitydistribution of nucleons in the nucleus, taking into account that the densityis positive while the potential of attraction is negative:

U(r) = -Cp(r), (3.13)

where C is a positive constant value.According to quantum mechanics, nucleons that move in a self-

consistent field U(r) must be in different energy states, i.e., must fill severalone-particle levels of energy. The nuclear ground state implies a total fill-ing of all levels from the lowest level up to the level with some limitingenergy EF {Fermi energy). When two nucleons collide, one of them shouldoccupy a state with a lower energy. But this is not possible, because allthe low lying energy levels are already occupied, and no nucleon can beadded due to the Pauli principle. This fact, and the repulsive character ofnucleon-nucleon forces at small distances, leads to the comparatively rarecollisions of nucleons in a nucleus. Thus the existence of a self-consistentfield of nucleons in a nucleus is the consequence of the fact that nucleonsare fermions, and that nuclear forces are repulsive at small distances.

As a first approximation, the potential of a self-consistent field U(r)


could be identical for protons and neutrons, since the Coulomb interactionof protons becomes essential only for quite heavy nuclei. This conclusion issupported by the coincidence of neutron and proton magic numbers. Sincein the spherically symmetric field the angular momentum of a particle withrespect to the field center is conserved, 2(21 + 1) nucleons of the same sortcan occupy an energy level with the given I (here 21 + 1 is the number ofpossible values the projection of the nucleon's orbital momentum I onto theaxis of quantization can acquire, 2 is the number of possible values of theprojection of the nucleon's spin on that axis).

During the formulation of the theory of the one-particle energy levelsfor nucleons in a nucleus, it was noticed that the separation between someneighboring levels appeared substantially greater than the typical meandistances between levels. The boundary between nucleon shells lies exactlyin these places. If the given nucleus has all its nucleon shells filled, then it isespecially stable and has zero spin, magnetic dipole and electric quadrupolemoments. The nuclei ^He, g6O, fo^a, lo^a, a nd 828P° a r e doubly magicbecause their proton and neutron shells are totally filled.

If a nucleus consists of a core with totally filled nucleon shells and oneadditional nucleon (the valence nucleon), then evidently the properties ofsuch a nucleus will be defined by this nucleon. For instance, the nucleus17O has one neutron with the orbital moment I = 2 and with the totalmoment j = 5/2 above the closed shell. Therefore, the shell model predictsthe spin of 17O equal to 5/2. This prediction agrees with experiment. Thenucleus 17F has one proton in the state I = 2, j = 5/2 above the closedshell. Therefore, the spin of that nucleus is 5/2 as well.

The shell model allows us to explain the many excited states of nuclei asthe transitions of the valence nucleons to the excited states. Such states arecalled the one-particle excited states of nuclei. However, a nucleus can haveexcited states caused by the changes of states of the core of the nucleus.These states can be explained on the basis of a generalized shell modelconsidering core excitations.

As was mentioned above, nucleons of the same sort in a nucleus preferto form pairs with zero spins. Particularly, any even-even nucleus has zerospin. Even if such a nucleus does not have all shells closed, it can be treatedas a core in the shell model. The generalized shell model also allows one toconsider nuclei having more than one nucleon above a core.

If a nucleus lacks one nucleon to close a shell, then we say that sucha shell has a "hole". Such one hole nuclei are also described by the shellmodel. It appears that the spin of such a nucleus is equal to the total


moment of the missing nucleon. For instance, the nucleus 3He could betreated as the 4He nucleus with one missing neutron in the state I = 0,j = 1/2. Thus the 3He spin is predicted to be 1/2, which is confirmed byexperiment.

Therefore, the shell model allows us to explain magic numbers, spinsof different nuclei, some levels in the energy spectra of nuclei and, in sev-eral cases, the magnetic dipole and electric quadrupole moments of nuclei.Nevertheless, a number of unexplained levels remains. These levels are ev-idently caused by the core excitation when the excited state of the nucleusoriginates from the change of states of many nucleons. These levels havea "collective" character. In many cases, electric quadrupole moments ofnuclei also appear much greater than the predicted ones. These deviationsindicate the existence of collective degrees of freedom connected with themotion of many nucleons in a nucleus. The next paragraph is devoted tothis issue.

The model of nucleon associations, or the cluster model, is a variantof the nuclear shell model. In that model, nuclei (mainly light ones) areconsidered as consisting of several light nuclei that could exchange nucleons.For instance, the 6Li nucleus in such a model could be treated as composedof an a-particle and a deuteron, the 9Be nucleus could contain two a-particles and a neutron, and the 12C nucleus could contain three a-particles.

The statement that some nuclei consist of light nuclei has a deep physicalmeaning. The matter of fact is that the correlations between nucleonscaused by the character of nuclear interaction and the Pauli principle areprobable in nuclei. These correlations are responsible for the formation ofstable groups of nucleons (nucleon associations or clusters) in nuclei. Theformation of clusters increases the binding energy of the nucleus.

Extensive experiments testify to the existence of quite stable clusters innuclei. The most stable is the a-cluster (a-particle). Two protons and twoneutrons forming an a-cluster can occupy the same space and energy state.This results in the great binding energy of the a-particle: Wa = 28.3 MeV.The emission of one nucleon from an a-particle takes about 20 MeV, whilefor the majority of nuclei this value is about 8 MeV.

The light nuclei with Z = N and total number of nucleons A divisibleby four have a binding energy 90% of which is associated with a-clustersand only 10% with interaction between a-clusters. It is clear that theone-a-particle emission for such nuclei takes much less energy than theone-nucleon emission. The a-cluster nuclei are 12C, 16O, 20Ne, 24Mg, etc.(the 8Be nucleus is not stable).


The existence of a-particles inside heavy nuclei is confirmed, in partic-ular, by their a-decay where an a-particle exists inside a nucleus in a readyform. Alpha-clusters are created with great probability in the surface re-gion of a nucleus where the nuclear density is small relative to that in theinterior. Note that a nucleon also is a cluster that consists of three quarks.

The first model that considered the nucleus as consisting of clusterswas the a-particle model. The model assumed that inside a nucleus, thea-particles were stable systems, i.e., a great probability existed to find asystem of four nucleons localized in the nucleus and separated from othersuch systems. The possibility of nucleon exchange between separate a-partides was neglected.

The simplest a-particle model treats the 12C nucleus as a hard equilat-eral triangle with a-particles at the vertices. This model made it possible toexplain a number of properties of 12C nuclei, and to calculate probabilitiesof scattering at small angles of electrons and nucleons by these nuclei. Sim-ilar results were obtained for the 16O nucleus treated as a hard equilateraltetrahedron with a-particles at the vertices.

Further improvement of the a-particle model is connected with the pos-sibility for a-particles to vibrate about their equilibrium locations. In themodel of nucleon associations (cluster model), a-clusters could partiallyoverlap and exchange nucleons with each other. Extensive calculationsmade on the basis of such a model explained various experimentally ob-served properties of nuclei, emphasizing the substantial role of nucleon-nucleon correlations in nuclei.

3.3 Collective Motions of Nucleons in Nuclei

Numerous experiments have shown that many nuclei have electricquadrupole moments much greater than predicted by the shell model. Largequadrupole moments of nuclei are caused by the deviation of these nucleifrom spherical symmetry. Indeed, if a nucleus has the form of an ellipsoid,then its electric quadrupole moment will be proportional to the amount ofdeformation.

Nuclei with all nucleon shells filled are spherically symmetric. If a nu-cleus has one or a few nucleons above the closed shells, these nucleons candeform the nucleus via their interaction with the core. It was theoreticallyproved that one nucleon above the closed shell could have lower energy ifits potential well was not spherical. Therefore, the nucleus consisting of a


core with closed shells and an additional nucleon (or nucleons) can achievea state with lower energy if the core becomes nonspherical.

In other words, the nuclear shape is defined by the competition betweenthose nucleons comprising the closed shells that tend to give the nucleus aspherical shape, and those comprising the unclosed shells that tend to givethe nucleus a nonspherical shape. As a result, the nucleus can acquire anonspherical form that will be statically stable. Particularly, the nuclei oflantanoids and actinoids are nonspherical because their numbers of neutronsand protons lie between magic numbers.

The possibility of the existence of nonspherical nuclei was first pointedout in 1950 by the American physicist Rainwater, who also explained theorigin of the existence of large electric quadrupole moments. In 1950-52,the Danish physicists Aage Bohr (the son of Niels Bohr) and Mottelsondeveloped the closed theory of collective motions of nucleons in nuclei. In1975, for those works, A. Bohr, Mottelson, and Rainwater were awardedthe Nobel Prize in Physics.

In the first approximation, a nonspherical nucleus can be considered asan ellipsoid of rotation. If the deviation from sphericity is not large, thedeformation can be defined by the magnitude (J = AR/R where AR = b — aand R = (o + 6)/2 (a and b are the semi-minor and semi-major axes ofan ellipsoidal nucleus). In that case, the electric quadrupole moment ofthe nucleus is proportional to its deformation: Q ~ f3. Thus the valuesof deformations can be determined by the measured electric quadrupolemoments of nuclei.

Two types of collective motions of nucleons can be observed in nuclei.First, there are the vibrations of the nucleus, namely the vibration of itsshape, which is not accompanied by changes in density because nuclearmatter is in fact incompressible. Nonspherical nuclei can vibrate as well asspherical ones. The energy spectra of many nuclei have levels of a vibra-tional origin. The properties of such levels are predicted by the collectivemodel of a nucleus.

Another type of collective motion of nucleons is the rotation of non-spherical nuclei. If the rotation of a nonspherical nucleus is rather slow,as it is in many cases, then such rotation will not substantially influencethe motions of separate nucleons inside the nucleus and the possible smallvibrations of the nuclear surface. Under these conditions the rotationalenergy can be explicitly extracted from the total energy of a nucleus. Inother words, the energy spectrum of a nonspherical nucleus comprises, be-sides the levels of a stationary nucleus, also the levels of its rotation. The


intervals between the rotational levels turn out to be small compared tothose for the levels of a stationary nucleus (one-particle levels).

If the nucleus takes the shape of a rotational ellipsoid, then such aquantum mechanical system can rotate only around the axis perpendicularto its axis of symmetry. Indeed, rotation of the nucleus around its axisof symmetry does not alter its orientation in space, thus its states are notdistinguishable and the energy of a system does not change. Any rotationabout an axis that is inclined at a definite angle to the axis of symmetry canalways be represented as a result of rotations about the axis of symmetryand the axis perpendicular to the axis of symmetry. Since a change inthe spatial orientation of a nucleus takes place only in the second case,a rotation of a nonspherical nucleus about an arbitrary axis becomes, infact, a rotation about the axis perpendicular to the axis of symmetry. Weemphasize that a spherical nucleus cannot rotate, so it has no rotationaldegrees of freedom. This statement also concerns any spherically symmetricquantum system whose states are indistinguishable upon spatial rotation.

By means of quantum mechanics, it can be shown that the energy ofrotation of a nonspherical nucleus with zero spin around the axis perpen-dicular to the axis of symmetry is

E/ = S J ( J + 1)' (3-14)where / = 2,4,6,... are the moments (spins) of possible excited rotationalstates of the nucleus and J is the moment of inertia of the nucleus.

Explanation of the moments of inertia of nuclei appears to be a rathercomplicated problem, since their values are between the magnitudes pre-dicted by the models of a liquid drop and solid matter. Physical reasonsleading to these values for the moments of inertia of nuclei are discussedbelow.

Even-even nuclei have zero spins. All nucleons in such nuclei are paired.In order to break up one such pair, i.e., to transfer the ground state of theeven-even nucleus into its one-particle excited state, an energy of about2 MeV is needed. The energy of the first excited rotational level of anonspherical nucleus is much smaller than the energy of pairing (almostten times). Therefore, the rotational and one-particle levels are easy todistinguish in the energy spectra of nonspherical even-even nuclei.

If the energy of the first excited rotational level of an even-even nucleusE2 is known, then its moment of inertia can be denned: J = 3h2/E2- Thus


Table 3.1 Ratios E1/E2 for several nonsphericaleven-even nuclei whose theoretically predicted val-ues are / ( / + l)/6.

Nucleus / = 4 / = 6 / = 8 / = 10

160Dy 3.270 6.694 11.14 16.46166Yb 3.239 6.553 10.78 15.76172Hf 3.258 6.635 10.96 16.07176W 3.206 6.434 10.49 15.16186Os 3.17 6.34 10.4 14.7232Th 3.27 6.63 11.0 16.1238U 3.31 6.93 11.6 17.7

/ ( / +1)/6 3.333 7 12 18.33

the energies of rotational levels will be determined by the formula

EI = ll(I + l)E2. (3.15)

In this case the correlation between the energies of rotational levels, whichfollows from (3.15) and is called the rule of intervals, is valid:

E2:E4:E6:E8:E10:... = l: y : 7 : 12 : y .. . . (3.16)

Table 3.1 displays the experimentally measured ratios E1/E2 for severalnonspherical even-even nuclei alongside the theoretically predicted valuesof J ( J+ l ) / 6 .

The contradictions between the predictions of the theory and the ex-perimental data shown in Table 3.1 are explained first by the fact that thesimplest model of nuclear shape — an ellipsoid of rotation — is chosen.The real shape of a nonspherical nucleus appears in fact more complicated.For instance, the shape of a nucleus could be a general ellipsoid, i.e., anellipsoid having all three axes of unequal length. In some cases it couldbe even more complicated. Thus the rule of intervals (3.16) substantiallychanges. It is not always possible to separate rotations and vibrations ofnuclei. Interaction between these two types of collective motions of nucle-ons also changes the rule of intervals (3.16). Therefore, the realistic rules ofintervals considering different properties of nonspherical nuclei allow one tocoordinate the calculated and measured energy spectra of rotational levelsof nuclei.

If the energy of a rotational level and the spin of an excited stateare comparatively small, then the inner structure of the nucleus does not


change, i.e., the ties between coupled nucleons in a nucleus do not break.However, by bombarding the nucleus with heavy ions, an artificial super-rotating nucleus can be created. In such a process, nuclei in high spin statescan be formed. In these nuclei, almost all excitation energy is contained inthe energy of rotation, i.e., the nucleus remains "cold" inside.

Large rotational energy can lead to the breakdown of coupling of nu-cleons which, in its turn, can cause a change in nuclear shape. At compar-atively small excitation energies (I < 15 — 20), a nonspherical even-evennucleus has the form of a prolate ellipsoid of rotation. With an increase ofspin I, the centrifugal and Coriolis forces lead to the breakdown of couplingin nuclei, i.e., at I > 20 - 25 nuclear matter consists of uncoupled nucleons.In this case, the nucleus acquires the form of a three-axial ellipsoid, whileits moment of inertia increases up to the solid matter value. If / > 60,then the nucleus acquires the form of an oblate ellipsoid of rotation. When/ ss 85 - 100, the nuclei become unstable relative to fission. These are themaximum possible values of nuclear spins.

The states of nuclei with high spins and excitation energies, when almostall excitation energy is the energy of rotation, are called yrast-states. Thisterm is connected with the old Norman verb "hvirfla", meaning "to twirl".From this verb the Swedish adverb "yr" originates, the highest degree ofwhich, "yrast", means "dizzy" or "stunning". However, this word shouldbe comprehended as "bearing the most rotation".

Let us briefly discuss some features of volume vibrations of nucleonsin nuclei. If the nucleus absorbs a photon with high enough energy, thenthe electric field of the absorbed photon causes the coherent motion of allprotons of the nucleus along the field direction. Evidently, this leads to theappearance of vibrations of all protons of the nucleus with respect to all itsneutrons. In this case, the nucleus acquires some dipole moment.

If the internal frequency of vibration of such a dipole moment coincideswith the frequency of the absorbed photon, then the effect becomes espe-cially powerful. This phenomenon is called giant dipole resonance. Usuallythe energy of its excitation for different nuclei is contained in the interval15-25 MeV and can be determined from the empirical formula

Eg = 78 • A~1/3 MeV. (3.17)

More complicated types of giant resonances (quadrupole, octupole, etc.)can also be excited in nuclei. But such excitations require special consid-eration on the basis of quantum mechanics. That is why we do not discussthem here.


3.4 Superfluidity of Nuclear Matter

The hydrodynamic (liquid drop), shell, and collective models of the nucleusallow us to explain numerous properties of nuclei. However, importantexperimental data exist that cannot be explained by these models. Thefacts are as follows.

(1) The existence of a gap in the energy spectra of even-even nuclei, andits absence in the spectra of odd and odd-odd nuclei. In other words,in odd and odd-odd nuclei the energy of the first excited level of nonro-tational character is several tens of keV, while in even-even nuclei thisenergy is usually about 1 MeV.

(2) The values of the moments of inertia of nonspherical nuclei are greaterthan those obtained from the hydrodynamic model, but are smallerthan those found from the solid matter approach. Besides, the momentsof inertia of odd nuclei are much greater than those for even-even nuclei.This difference cannot be explained on the basis of the collective modelvia the adding of one nucleon to the even-even core, because in thatcase the nucleus should have too much deformation.

(3) The transition of the shape of the nucleus from spherical to ellipsoidaloccurs when about 25% of all positions in the last unclosed shell of thenucleus are filled, while the independent particle model calculationsshow that all nuclei with nucleons in the unclosed shells should benonspherical.

(4) The differences in masses and binding energies of even-even, odd, andodd-odd nuclei require the introduction of an additional term in theformulae (3.7) and (3.8) considering the coupling of nucleons of thesame kind.

(5) The mean separation between the excited one-particle levels in oddnonspherical nuclei is approximately half that calculated in the modelof a self-consistent field.

(6) The measured probabilities (half-lives) of a- and /3-decay for somenuclei deviate from the values calculated in the independent particlemodel.

Therefore the shell model, which assumes that nucleons move indepen-dently in a self-consistent field (the model of independent particles) anddoes not consider the nucleon-nucleon correlations leading to the coupling ofnucleons of the same kind, cannot explain these experimental facts. These


facts are not explained in the collective model of the nucleus either. Thuswe have to briefly discuss the physical consequences to which the nucleon-nucleon pair correlations lead in nuclear matter.

A nucleus is a system of strongly interacting nucleons having spin 1/2,i.e., fermions. Thus, a nucleus can be considered as a Fermi-liquid. If theforces of attraction act between the particles of a Fermi-liquid, then thephenomenon of creation of bound pairs of fermions occurs that appearsto be extraordinary from the viewpoint of classical physics. In an infinitenuclear matter, the nucleons that are coupled are of the same kind, withequally valued and oppositely directed angular momenta and spins. Sucha coupling is observed, for instance, for electrons in metals at low tempera-tures. It leads to the known phenomenon of superconductivity. As we havementioned, the phenomenon of coupling of fermions is called the Coopereffect.

In a finite-size nucleus, the nucleons that are coupled are of the samekind, occupying the states with the same main quantum numbers and or-bital and total moments, but with opposite projections of total momentson the quantization axis. In other words, the coupled pairs are nucleonswith quantum numbers n,l,j,m and n,l,j,—m (n is the main quantumnumber, I and j are the orbital and total moments of nucleon, and m is theprojection of the total moment on the quantization axis). Cooper's effecttakes place for any (even very weak) interaction between fermions, providedthe interaction is an attraction.

The usual self-consistent field in which nucleons move independentlyconsiders, in fact, only the major part of nucleon-nucleon interaction. Theneglected small contribution is called the residual interaction. However,despite its smallness, the residual interaction can lead to the coupling ofnucleons in nuclei because of its attractive character.

Evidently, the effect of coupling of nucleons in nuclei leads to the super-fluidity of nuclear matter. The superfluidity of a Fermi-liquid, as discussedearlier, is a special phase of matter existing at very low temperatures.

Two coupled fermions form a system with zero spin, representing a sta-ble formation similar to a particle. Therefore, such a group of two coupledfermions (Cooper's pair) is called a quasiparticle. It is a boson, and theFermi-liquid in the superfluid state could be considered as a liquid of suchquasiparticles. The theoretical description of the quasiparticle liquid ismuch easier than that for the nucleon liquid. The point is that nucleonsstrongly interact with each other, while the interaction between quasipar-ticles is very small.


0.4- gp^

0.2 - ° ^ ^ ^

0 0.2 0.4 p

Fig. 3.3 Ratios of the moments of inertia J of atomic nuclei calculated by thesuperfluid (solid line) and hydrodynamical (dots) models to their solid-body val-ues Jo as functions of the nucleus deformation parameter (3. The circles areexperimental data.

Based on the model of superfluid nuclear matter, it is possible to explainmany experimentally measured properties of nuclei. In particular, only thesuperfluid model describes the moments of inertia of nonspherical nucleiand the gaps in the energy spectra of even-even nuclei. Fig. 3.3 shows theexperimentally measured and theoretically predicted moments of inertia ofnonspherical nuclei.

In conclusion, we note that all the nuclear models discussed here origi-nate from rather different and sometimes opposing premises. Nevertheless,all the models describe different properties of nuclei quite well. This meansthat all models contain some unique underlying principle of nuclear struc-ture. This principle is the concept of a self-consistent field in which nucleonsmove independently (the model of independent particles). Different nuclearmodels appear from different approaches to take account of the residual in-teraction. The existing nuclear models allow us to explain a great deal ofexperimental data and understand general regularities of nuclear structure.

Chapter 4

Radioactivity of Atomic Nuclei

4.1 The Law of Radioactive Decay

Human history records but a few great scientific discoveries made acciden-tally. The discovery of the radioactivity of atomic nuclei is one of them.Radioactivity is the process of spontaneous transmutation of an unstablenucleus into another one, which is accompanied by the emission of variousparticles and photons. Several elementary particles can undergo radioactivedecay as well.

In February 1896, the physics professor Becquerel was working at theEcole Poly technique in Paris. He studied the abilities of various crystalsunder sunlight to emit a penetrating radiation similar to the X-rays thathad been recently discovered by Roentgen. Becquerel supposed that crys-tals under the influence of light would emit rays that could register on thephotographic plates covered by black paper. A screen made out of copperwires was placed between the crystal and the plate. Thus, after developing,the plate would be light-struck everywhere except the region covered by thecopper wires.

Among the crystals Becquerel was working with, by chance some ura-nium salts were stored, specifically uranium sodium bisulphate. Also bychance, the weather during the experiments was cloudy so that Becquerelput the plates into the box of the laboratory table together with the crystalsof uranium salt. After a few days when the plates were developed, theyappeared dark and demonstrated a very clear image of the screen, eventhough no sunlight had illuminated the uranium salt.

Becquerel attributed the rays that registered on the plate to the ura-nium. Afterwards it was found that the same rays could be emitted byother elements as well. In 1898, Marie Sklodowska-Curie discovered that

89


the same rays were emitted by thorium; she and her husband Pierre Curiesubsequently discovered radium. The term "radium" originates from theLatin word "radius", which means "ray". The Curies named this phe-nomenon radioactivity. In 1903, Becquerel and the Curies were awardedthe Nobel Prize in Physics: Becquerel for his discovery of spontaneous(unmotivated) radioactivity, and the Curies for their joint research on theradiation phenomena discovered by Becquerel. Marie Sklodowska-Curiewas also awarded the Nobel Price in Chemistry in 1911 for the discoveryof the elements radium and polonium, for the isolation of radium, and forthe study of the nature and compounds of this remarkable element.

In 1899, Rutherford discovered a- and /?-rays. Afterwards he found thatQ-rays are the ions of helium. The rays found in 1900 by French physicistWillard were different from a- and /3-rays. In 1903 Rutherford named these7-rays and assumed they are electromagnetic radiation with quite a shortwavelength similar to X-rays. In 1900 Rutherford introduced the notion ofa half-life period. He was awarded the Nobel Prize in Chemistry for 1908for his investigations into the disintegration of elements and the chemistryof radioactive substances.

Only much later, after the discovery of the atomic nucleus by Ruther-ford in 1911, was it understood that radioactive radiation was emitted justby nuclei. Further investigation showed that many unstable nuclei withfinite life-times exist in Nature. These nuclei change their states by them-selves, and are called radioactive. Many of the known radioactive nucleiare artificial.

Artificial radioactivity was discovered in 1934 by I. Joliot-Curie and F.Joliot-Curie. They were awarded the Nobel Prize in Chemistry in 1935. TheJoliot-Curies bombarded aluminium foil with 5.3 MeV a-particles emittedby 210Po, and among the reaction products they found not only nucleonsbut positrons. Moreover, the positrons were emitted during a certain periodof time after the irradiation had stopped. Thus they observed the nuclearreaction

27Al + a —> 30P + n, (4.1)

in which the radioactive isotope of phosphorus 30P was created to undergothe positron /3-decay (only the isotope 31P is stable). Further, a greatnumber of different radioactive isotopes were obtained, which have wideapplications in nuclear physics and in other fields of science and technology.

Various types of radioactive nuclei exist. As we mentioned, during a-decay the nucleus itself emits an a-particle, i.e., the nucleus of the atom

Radioactivity of Atomic Nuclei 91

^He. If the radioactive nucleus emits an electron and an antineutrino,then such a process is called P~ -decay. If the radioactive nucleus emits apositron and a neutrino, then this is a /3+-decay. If the nucleus (the nuclearproton) captures an electron from the if-shell of the atom, the so-called K-capture, then such a process is /3-decay as well. In this case the electrondisappears while the electron antineutrino is emitted and the nucleus passesinto an excited state. During 7-decay the excited nucleus emits one or morephotons and passes into a lower lying or ground state.

Some heavy nuclei can spontaneously decay into two or more parts. Thisphenomenon is called spontaneous fission and is also treated as a processof radioactive decay. The spontaneous fission of 235U nuclei was discoveredin 1940 by the Soviet physicists Flerov and Petrzhak. There also exist rare(exotic) types of radioactive decay of nuclei. The existence of proton anddiproton radioactivity is possible close to the boundary of proton stability.The process of double /3-decay is possible. The radioactive decays in whichthe nuclei 12C, 14C, 16O, 20Ne, and some others, are emitted have been wellstudied.

The emission of delayed neutrons and protons is also considered as aradioactive decay. These decays are two-step cascade processes, becausethe emission of a delayed neutron or proton from a nucleus takes placeafter the preliminary emission of a positron or electron. Moreover, theemission of a nucleon is delayed by the time of the corresponding /3-decay,because the process of nucleon emission from a nucleus itself occurs almostinstantaneously.

The number of radioactively decaying nuclei decreases with time. Theprocess has a statistical character, meaning that one can speak only aboutthe probability per unit time for the given nucleus to decay. If the initialnumber of a certain type of radioactive nucleus is large enough, the generallaw of its decrease with time can be determined. Actually, the decrease ofthe number of nuclei dN(t) during the interval dt is proportional to boththis interval dt and the current number of nuclei N(t):

dN(t) = -XN(t)dt, (4.2)

where A is the decay constant and the negative sign indicates the decreasein the number of nuclei during time dt.

Formula (4.2) generates the law of radioactive decay

N(t)=Noe-Xt, (4.3)


where No = N(0) is the number of nuclei at the initial moment of timet = 0. Formula (4.3) reflects the random character of radioactive decay,and has been confirmed by numerous experiments.

It is impossible to predict the moment of time when a given radioactivenucleus will decay. One can only evaluate the probability for it to decayat a given moment of time. For a given nucleus, the magnitude exp(—At)represents the probability that it will not decay during the time t. So1— exp(—Xt) is the probability of decay during that same time. For instance,the decay constant for the 226Ra nucleus is A — 1.37 • 10~n s"1. Thus,during each second, on average, one nucleus of this isotope out of a hundredbillion will decay.

The decay constant has a simple physical meaning. It is equal to A =1/i, where t is the mean lifetime of one radioactive nucleus. Evidently,during the time t the initial number of radioactive nuclei decreases by afactor of e K 2.718.... The principal problem of radioactive decay theoryis finding the decay constant for the nuclei of a given isotope, which canalso be measured experimentally.

Formula (4.3) shows that the number of radioactive nuclei decreaseswith time according to the exponential law. The law is valid for all typesof radioactive nuclei. However, it should be noted that this conclusion doesnot consider the dynamics of radioactive decay. Apparently the law (4.3)should be derived not from statistical considerations, but from the equationsof quantum mechanics. Further, it should be kept in mind that the problemof radioactive decay is intrinsically nonstationary, because before the decaythere are no products of decay outside the nucleus.

Therefore, a decaying nucleus occupies one of the nonstationary statesof its energy spectrum. The law of radioactive decay is caused by thecharacter of this spectrum. The nonstationary character of the state leads,in fact, to a deviation from the exponential law. But that deviation takesplace only at very small or very large times, and is not usually observable.In a-decay, for instance, it should be observed at t < 10~21f and t > 50£.

The time ti/2 during which half of the initial number of identical ra-dioactive nuclei decay is called the half-life. This quantity is related to theconstant of decay by

In 2 0.6931 . ,*1/a = - « — (4.4)

Thus we find I = 1.443t1/2- After the time tk = fc*i/2, the number of


1/2 - \

1/4 j. -Nv

1/8 i t -^^r^-^^

0 tm 2tm 3tm t

Fig. 4.1 Number of radioactive nuclei N(t) as a function of time t.

nuclei that have not yet decayed will be (Fig. 4.1)

N(kt1/2) = Noe~Xkt^ = 2~kN0, (4.5)

where fc = 1,2,3,. . . .The values of the half-life periods £j/2 vary widely between radioactive

nuclei. For a-decay, ti/2 varies from 3 • 10~7 s for 212Po nuclei to 1.6 • 1023 s(5 • 1015 years) for 144Nd nuclei. For /3-decay, the values of ti/2 range from10~2 s to 2 • 1015 years. The half-lives for 116In and 115In nuclei are 14 sand 1.9 • 1022 s (6 • 1014 years), respectively.

Sometimes one type of nucleus will decay into another type that is alsoradioactive, and we see a chain of radioactive transmutations. In that case,we must replace (4.2) with a set of two linked equations in order to describethe overall process:

*m=.XlNl{t)i « = A l i V l ( i ) _ W ) . (4.6)Here iVi and N2 are the numbers of the two types of nuclei, and X\ and A2are the respective decay constants. The first equation in (4.6) describes theprocess of radioactive decay of the parent nuclei Ni. The second equationdescribes the decay of the daughter nuclei N2, and contains two terms onits right-hand side. The first represents the number of nuclei IV2 appearingunder the decay of the Ni nuclei, while the second is the number of decaying


N2 nuclei. The solution of equations (4.6) has the form

N^^N^e-^, (4.7)

N2(t) = N2(0)e-^ + M l M (e-Ai* _ e-^t\ ? ( 4 8 )

A2 — Ai

where Wi(0) and N2(0) are the numbers of nuclei Ni(t) and N2(t) at timet = 0.

Equations (4.7) and (4.8) simplify substantially if Ai -C A2, i.e., if thehalf-life of the parent nuclei Ni is much greater than that of the daughternuclei N2. Then we find

JVi(t) «#!(()) , N2(t)wiV2(O)e-Aat + ^7Vi(O)( l -e-A 2 t ) . (4.9)A2

If we have no N2 nuclei at the beginning, ./V2(0) = 0, then

N2(t)*s^N1(p)(l-e-x>t). (4.10)A2

After a long period of time that substantially exceeds the half-life forthe N2 nuclei (A2£ 3> 1), we obtain

XiNiit) = X2N2(t). (4.11)

Formula (4.11) is called the secular equation. This equation means that,if Ai -C A2, then after a long period of time the number of decays of thedaughter substance A27V2(t) is equal to the number of decays of the parentsubstance \\Ni(t). The secular equation is widely used to obtain the half-lives of long-lived radioactive isotopes.

As an example we consider the following chain of radioactive decays ofnuclei: 226Ra nuclei undergo a-decay with half-life i1;/2(Ra) « 1600 year,and turn into radioactive 222Rn nuclei; these undergo a-decay with half-life iiy2(Rn) w 3.8 days. Choosing times satisfying the condition £jy2(Ra)<§; t -C £]y2(Rn), we find the secular equation

A R a % a = *Rn%n- (4-12)

In this case the number of appearing Rn nuclei coincides with the numberof decaying Ra nuclei. The values of iVp a and iVp n can be determinedby weighting the samples, while the value of Aj^n can be found by thecomparatively small half-life period of Rn nuclei. Then the value of Aj^a

for the Ra nuclei with the long half-life can be derived from the secular


equation (4.12). Finally, using the correlation (4.4), it is easy to evaluatethe long half-life of the Ra nuclei.

The activity of a radioactive substance is denned by the number ofdecays per unit time. One decay per second is accepted as a unit of measureknown as the Becquerel (Bk). This unit of measure is too small in practice,however, and 1 MBk = 106 Bk and 1 GBk = 109 Bk frequently appearinstead. For instance, the radon 222Rn contained in 1 m3 of atmosphericair has an activity of about 4 Bk. One kg of uranium ore with 10% ofpure uranium possesses an activity of 0.13 MBk. Cobalt-based radioactivesamples used in medicine for radiotherapy have activities from 75 to 2 • 105

GBk. A nuclear bomb with an explosive power equivalent to that of 20kilotons of trinitrotoluene creates an activity of about 7.4 • 1013 GBk uponexplosion. An earlier unit of activity called the Curie (Ci) is now used onlyrarely; the conversion between these two units is given by 1 Ci = 3.7 • 1010

Bk. Sometimes a unit of activity called the Rutherford (Rd) is also used:1 Rd = 106 Bk = 1 MBk.

When the Earth was formed about 4.6 • 109 years ago, it was com-posed of the isotopes of various elements. Some of these were radioactive.Radioactive isotopes with half-lives substantially shorter than the Earth'sage have long since decayed and are not observed in Nature. Therefore,the radioactivity of natural substances is determined by elements havinghalf-lives greater than or equal to the Earth's age.

238U - 234Th — - 234Pa - — ~ 234U - 230Th 226Raa p p a a a

218At y 2 1 4Po .

_ ^ R n - « p o ^ 2 "B i P " 21°Pb

o N 214Pb " p a ^ 210TI P

, 21°Po

21°Bi P " 206Pb

V 206TI ^

Fig. 4.2 The uranium series.

Three series of radioactive nuclei exist in Nature. They begin from the


, 227Th .

2 3 5 U — 2 3 1 T h - ^ 231Pa — 227Ac 1> «* 223Ra —a p a \ V a

> ^Fr -p

— 219Rn — 215Po — 211Pb -JT 211Bi i> " 207PbV 207TI "p

Fig. 4.3 The uranium-actinium series.

very long-lived radioactive nuclei found on Earth. The uranium series (Fig.4.2) starts from the 238U isotope (ti/2 = 4.5-109 years) undergoing the chainof transmutations (8 a-particles and 6 electrons are emitted) that lead tothe formation of the stable 206Pb isotope. The actinium-uranium series(Fig. 4.3) starts from the 235U isotopes (t1/2 = 7.1 • 108 years) undergoing achain of transmutations (7 a-particles and 4 electrons are emitted) resultingin the stable 207Pb isotope. The thorium series (Fig. 4.4) starts from the232Th isotope (£1/2 = 1.41-1010 years) undergoing a chain of transmutations(6 a-particles and 4 electrons are emitted) leading to the formation of thestable 208Pb isotope.

232_. 2 2 8 D 228 . 2 2 I - , 2 2 4 D 220,-,Th — - Ra -r— Ac —r— Th — - Ra — - Rn — -a p (5 a a a

X-Po

— 2 1 6 P o — 212Pb — 212Bi !> a 208Pb

P V -T, /

Fig. 4.4 The thorium series.

^ N 2 3 3 p g 2 3 3 u _ _ 2 2 ^ _ _ 2 2 5 R a 2 2 5 ^ _ _

a P a a p a

x 2 "Po

— 2 2 1 F r — 217At — 213Bi V a 209Pb ^r 209Bia a ^ s p

> 209TI i

Fig. 4.5 The neptunium series.


The neptunium series also exists (Fig. 4.5), starting from the 237Npisotope (£1/2 = 2.14 • 106 years), artificially created in nuclear reactors,undergoing a decay chain (7 a-particles and 4 electrons emitted) resultingin the stable 209Bi isotope. The series could have existed in the past for sometime under natural conditions after the formation of the Earth. However,the half-lives of elements of the series are small compared to the Earth'sage. Therefore, at present they are not observed in Nature.

Many transuranium elements undergo a-decay and can also be at-tributed to these radioactive series. The 242Pu nucleus (ij/2 = 5 • 105

years) transforms into the 238U nucleus. The 243Cm nucleus (£1/2 = 100years) turns into the 239Pu nucleus (£1/2 = 2.43 • 104 years), which in turntransforms into the 235U nucleus. The 244Cm nucleus (£1/2 = 19 years)turns into the 240Pu nucleus (£1/2 = 6580 years), which then transformsinto the 236U nucleus (t\/2 = 2.4 • 107 years), while the latter turns into the232Th nucleus. Finally, the 241Am nucleus (£1/2 = 470 years) transformsinto the 237Np nucleus. Therefore all the known radioactive series can becontinued to heavier transuranium elements.

Table 4.1 Some radioactive nuclei found in theEarth's crust but that do not enter into the ra-dioactive series

Nucleus Type of decay t i ^ , yeas

40K 0-,P+, K-capture 1.26 • 109

87Rb p~ 4.8 • 1010

113Cd p~ 9-1015

U 5 In P~ 5.1 • 1014

138La p~, K-capture 1.1 • 1011

144Nd a 2.1 • 1015

147Sm a 1.06 • 10 u

148Sm a 8 • 1015

152Gd a 1.1 • 1014

176Lu p~ 3.6 • 1010

174Hf a 2.0 • 1015

187Re p~ 4 • 1010

190Pt Q 6 • 1011

Radioactive elements that do not take part in these series of radioactivenuclei are also observed in Nature. Some of these with large half-lives areshown in Table 4.1. Note that the radioactive isotope 40K is widely usedfor dating minerals via the relation of percent contents of the isotopes 40Kand 40Ar contained therein.


Despite the general character of the law of radioactive decay (4.3), thenature of the different types of radioactive decay is very different. Ana-particle emitted during a-decay is in a nucleus in a ready form. It isformed in a nucleus, and exists there for quite some time before leaving thenucleus. This is not the case for the electron and antineutrino (or positronand neutrino) emitted during /3-decay — these are born just at the momentof decay. In fact, the existence of an electron inside a nucleus contradictsthe data on values of spins and magnetic moments of nuclei. There is aneven deeper physical reason for an electron not to exist in a nucleus: aconflict with the Heisenberg uncertainty principle (Section 3.1). Therefore,an electron and an antineutrino (or positron and neutrino) are born at themoment of /3-decay. A similar situation occurs with 7-decay, when a photonis born at the moment of its emission, because the nucleus does not containa photon in a ready form. The interactions responsible for the differenttypes of radioactive decays are very different. An a-decay is caused by thestrong interaction, while (3-decs.y and 7-decay are caused by the weak forcesand electromagnetic interaction, respectively.

4.2 Alpha-Decay

The mass of a nucleus that undergoes a-decay satisfies the inequality

Tn(A,Z)>m{A-i,Z-2) + mcn (4.13)

where m(A, Z) and m(A — 4, Z — 2) are the masses of the parent anddaughter nuclei. An analysis of (4.13) indicates the possibility of a-decayfor all the nuclei with A > 57, and this is confirmed by experimental data.Most of the kinetic energy released during a-decay is taken away by thea-particle, while the daughter nucleus takes only an insignificant part of it(about 2% for the heavy a-active nuclei). For instance, during the a-decayof the 212Bi nucleus, the a-particle has energy Ea = 6.086 MeV while thedaughter 208Tl nucleus has energy Enuci = 0.117 MeV. The total energy ofthe a-decay of the 212Bi nucleus is therefore E = Ea + EnucX = 6.203 MeV.

The simplest theory of a-decay is based on the suggestion that beforethe decay, an a-particle exists in a nucleus in a ready form and performsits vibration motions. In other words, the probability for the four nucleons(two protons and two neutrons) to configure themselves into an a-particlein an a-radioactive parent nucleus is close to unity. The question ariseshow such an a-particle can leave the parent nucleus.


a bV(r) V(r)

B -K

£ E -Sv^-----.

0 ~ 0 -R r R r

Fig. 4.6 Potential wells of (a) a stable nucleus, and (b) an a-radioactive nucleus.E is the energy of an a-particle in the well.

To answer the question, we consider the potential energy of an a-particle. If the a-particle were trapped inside the nucleus for an infinitelylong time, its potential energy would be a potential well having width equalto the nuclear radius R and height infinity (Fig. 4.6a). The potential en-ergy, in fact, has a finite height H (Fig. 4.6b). Moreover, its external partis the Coulomb energy of interaction between the free a-particle and thedaughter nucleus, because the nuclear forces vanish at distances exceedingthe nuclear radius.

Consider as an example the process of a-decay of the 238U nucleus.Provided the nuclear radius and the amplitude of vibration of the a-particleare of the same order of magnitude, it appears that the a-particle in thenucleus moves with velocity va ~ 107 m/s. This particle strikes the internalwall of the potential well 1021 times per second, the radius of which for thisnucleus is R ~ 10~14 m. Since the half-life of a 238U isotope is ti/2 « 4.5-109

years, before the a-particle leaves the nucleus it strikes the well wall, onaverage, 1038 times. The 238U nuclei emit a-particles with energy Ea = 4.21MeV. The height of the potential barrier, when the angular momentum isequal to zero, is H = 2(Z - 2)e2/R. Since Z = 92 for uranium, H « 30MeV. The barrier height H exceeds the a-particle energy Ea by almost 26MeV. Therefore, from the viewpoint of classical mechanics, an a-particlecannot leave the nucleus, and a-decay cannot be explained in such a way.

The theory of a-decay was developed in 1928 by Gamov and, inde-pendently, by Condon and Gurney. The theory is based on the quantummechanical tunnel effect. In quantum mechanics, the probability that aparticle will penetrate a barrier of finite height and width is nonzero. The


microparticle whose motion is described by the equations of quantum me-chanics can undergo the tunnel transition from the potential well. Thistransition is also called the under barrier [tunnelling) passage of a particle.Calculations based on quantum principles allow us to obtain the probabilityof tunnelling under the potential barrier, and to derive the decay constantA.

The uncertainty in the energy of an a-particle passing through the bar-rier should be of the same order of magnitude as the barrier height AE « H.Thus, for the uranium nucleus, AE « 30 MeV. According to the uncertaintyprinciple for energy and time, we have At > H/2AE, where At is the timethe a-particle takes to pass through the barrier. Therefore At > 10~23

s. The a-particle passes the barrier almost instantaneously, because thecharacteristic nuclear time is about 10~22 s.

The study of radioactive nuclei shows that they possess an interestingfeature: the smaller the half-life of the nucleus, the higher the energy ofthe a-particle emitted. For instance, the 212Po nucleus with half-life t\/2 =3 • 10~7 s emits an a-particle with energy Ea — 8.95 MeV, while the 232Thnucleus with tx/2 = 1010 years emits a-particle with energy Ea = 4.28MeV. The energies of a-particles emitted by nuclei whose half-lives differby a factor of 1030 differ by almost a factor of two. In 1911, data analysisallowed the German physicist Geiger and the English physicist Nuttall toestablish the empirical law that now bears their names. This law relatesthe decay constant A to the mean free path of an a-particle in a substance(gas) under a given pressure and temperature:

( mQR?\ RaIn A—-—) =Aln—+B. (4-14)

V. n ) Ro

Here the constant A is the same for all the radioactive series existing inNature, while the value of the constant B varies by approximately 5%.Also appearing in the formula are the mass ma of the a-particle, the nuclearradius R, the mean free path Ra of an a-particle in air, and a constant Rohaving the dimension of length. The mean free path is the distance betweenthe emitter and the point of complete stoppage (collision) measured alonga straight line.

The mean free path of an a-particle in a gas is related to its kinetico Icy

energy by the correlation Ra = nEa with constant K. This formula allowsone to connect the half-life of a nucleus with the energy of an a-particle


emitted:

K^H- IH ' (4i5)where the constants A' and B' are similar to A and B, while the constantEo has the dimension of energy.

The theory of a-decay based on quantum mechanics permits the deriva-tion of a correct expression connecting the decay constant and the energyof the emitted a-particle:

In i^2^) = C^fE~a + D, (4.16)

where C and D are constants similar to A' and B'. Formula (4.16) givesthe quantum mechanical formulation of the Geiger-Nuttall law.

Formula (4.16) shows that a small change in a-particle energy Ea leadsto a great change in the decay constant A. A 10% change in Ea changesthe decay constant by about a thousand fold. If Ea < 2 MeV, the half-lifebecomes so long that a-decay becomes almost unobservable. That is whythe energies of a-particles (except in some cases) for all the known a-activenuclei are contained in the interval 4 MeV < Ea < 9 MeV. As a rule, in-active nuclei have charge numbers Z > 82; moreover, Ea increases with Z.The exceptions are some nuclei of rare earth elements (gQ4Nd, 626Sm, etc.),7|°Pt nuclei, and some artificially created nuclei.

Table 4.2 The energy spectrum of a-particles created by the a-decayof the 212Bi nuclei

Group of.. , ao ai a2 az 04 as

a-particles

Ea, MeV 6.086 6.047 5.765 5.622 5.603 5.481Percent 2 ? 2 6 g g ^ L 7 ^ Q 1 5 ^ 1A ^ QMQ

contents

Usually the same a-active nuclei emit a-particles with the same energy.But some nuclei emit a-particles with somewhat different values of kineticenergy. In this case we speak of a thin structure of the a-spectrum. Asan example, Table 4.2 shows the energy spectrum of an a-particle emittedby 212Bi nuclei. Fig. 4.7 explains the thin structure of this spectrum. Thea-decay of the parent 212Bi nucleus can take place with the formation ofthe daughter 208Tl nucleus being not only in the ground state but also indifferent excited states. The transition from these excited states into the


2 1 2B i\ V \ V \ \ 6 - 2 0 3 M e V

a\a\aW^aXaX

\ \ \ V \ \ Ei MeV

zoa-pi ° - 4 7 3 ^ ^ \ ^ — 0.327°~~ ^ 0.040

Fig. 4.7 The fine structure of the spectrum of a-particles created by the a-decayof 212Bi nuclei.

ground state is accomplished by emitting photons. Nuclei also exist (e.g.,212Po) that emit, in addition to the main group of a-particles with someenergy, a few a-particles with somewhat greater energy (Table 4.3). Sucha-particles are called long-passing. Their emission is explained by the a-decay of the 212Po nucleus, which is formed in the excited state under the/3-decay of the 212Bi nucleus (Fig. 4.8).

Table 4.3 The energy spectrum of a-particles created by thea-decay of the 212Po nuclei

Group of,. . ao ai oc2 cez

a-particles

Ea, MeV 8.780 9.492 10.422 10.543Percent ^ 1QQ ^ 1Q_3 2 1Q_3 L g 1Q_2

contents

4.3 Beta-Decay

Beta-decay is a process of spontaneous transmutation of an unstable nucleusinto the isobar having its charge different from the initial one by unity.


,,, E, MeVBi 11.195 MeV „ 212-< ^ ^ ^ < ^ . P o ^10.746

^ l ^ S — \ \ \ — 9*75\ \ \ V~~8-949

0\\\VpbFig. 4.8 Spectrum of long-range a-particles created by the a-decay of nuclei212Po.

Beta-decay with emission of an electron is energetically possible if the masscriterion is satisfied:

m(A, Z) > m(A, Z + l) + me, (4.17)

where me is the electron mass.If /3-decay is accompanied by the emission of a positron, this inequality

should be satisfied:

m(A,Z) >m(A,Z -l)+me. (4.18)

For K-capture, the corresponding inequality has the form

m(A, Z) + me > m{A, Z - 1). (4.19)

Let us consider different types of/?-decay. The 10Be nucleus, being extraneutron-rich, undergoes electron /?-decay with a half-life of ti/2 = 2.5 • 106

years and turns into the stable 10B nucleus. The 13N nucleus, having fewerneutrons than protons, undergoes positron /?-decay with t1/2 = 10 min andturns into the stable 13C nucleus. The 37Ar nucleus undergoes if-capture,absorbing an electron from the if-shell of the atom, and turns into the37C1 nucleus (ti/2 = 35 days). As formerly discussed, a free neutron alsoundergoes /3-decay, turning into a proton and emitting an electron and anelectron antineutrino. The half-life of the free neutron is about ten minutes.Note that neutrons in an atomic nucleus are usually stable and do not decay,provided the nucleus is not radioactive.


Some nuclei can undergo two types of /3-decay. For instance, the 48Vnuclei undergo /3+-decay (positron emission) in 58% of cases and ^-capturein 42% of cases, turning into 48Ti nuclei (£1/2 = 16 day) in either case. The74As nuclei undergo /3~-decay (electron emission) in 53% of cases and /3+-decay in 47% of cases, turning into 74Se nuclei in the first case and 74Genuclei in the second case (ti/2 = 17.5 day either way). In addition, thereare nuclei that undergo all three types of /3-decay. An example is the 80Brnucleus undergoing /3~-decay in 92% of cases, /?+-decay in 3% of cases, andif-capture in 5% of cases (£ly/2 = 18 min in any case). The 80Br nucleusturns into the 80Kr nucleus by /3~-decay and into the 80Se nucleus by (3+-decay.

If m(A, Z) > m(A, Z + 2) + 2me then, in principle, the possibility ofdouble /?-decay exists. Here the initial nucleus ( 4, Z) simultaneously emitstwo electrons and two antineutrinos and turns into the nucleus {A, Z +2). Although the probability of such a process is very small, it has beenobserved experimentally. For instance, double /3-decay has been observedfor 48Ca nuclei turning into 48Ti nuclei, 76Ge nuclei turning into 76Se nuclei,etc.

Provided the (3~- or /3+-radioactive nucleus is neutron (proton) rich, thefinal nucleus can be formed in a state having energy that exceeds the energyof neutron (proton) separation. In this case, the daughter nucleus willsubsequently emit a delayed neutron (proton). Since the neutron or protonis emitted by the excited nucleus almost instantly, the delay is denned bythe time of /3-decay.

The nature of /3-decay has presented a number of conceptual difficulties.Actually, before a-decay, the a-particle is already inside the nucleus in aready form. But an electron or a positron cannot exist inside the nucleus,for this contradicts the uncertainty principle. Therefore we cannot say thatan electron or a positron leaves the nucleus.

On the other hand, during a-decay the emitted a-particle has a clearlydefined energy or discrete spectrum of energies if the daughter nucleus isformed in an excited state. Experiments show that electrons or positronsappearing during /3-decay do not have a discrete spectrum but are char-acterized by a continuous spectrum with a definite upper boundary. Atfirst sight, this important feature of /3-decay seemed to contradict the con-servation law of energy and angular momentum. At the beginning of theinvestigation into /3-decay, the idea was even suggested that those lawswere not universal and could be violated, and that the phenomenon of (3-decay was just the process where the laws were violated. Even Niels Bohr


supported this viewpoint at first.The principal solution of the /?-decay problem is associated with the

name of Pauli, who suggested that one more particle, unobservable in ex-periments, is emitted besides the electron or positron in /3-decay. Thismeans that the particle is chargeless, as charge is conserved during /3-decay.Moreover, the particle should have either zero or very small mass in orderto avoid violation of (4.17) and (4.18). The particle acquired the name"neutrino", meaning "small neutral particle", because the name "neutron",meaning "neutral particle", was already reserved for the nuclear particle.The spins of neutrinos and antineutrinos are equal to 1/2 in accordance withthe conservation laws for energy and angular momentum during /3-decay.

Using this idea, Fermi constructed the first theory of /3-decay in whichthe process was considered as a simultaneous generation of two particles— electron and antineutrino. Actually, following the conservation law forthe number of leptons, the generation of an electron with lepton charge +1should be accompanied by the generation of an antineutrino with leptoncharge -1. During (3+-decay, the particles formed are a positron with leptoncharge -1 and a neutrino with lepton charge +1. The theory should beconstructed in such a way that those particles do not exist in a nucleus,but appear at the moment of /?-decay.

Fortunately, at that time the theory of quantum electrodynamics al-ready existed. Here the emission of a photon by some quantum mechanicalsystem was considered as a process of its generation but not its escapefrom the system. Such an approach required the development of a no-tion of quantization for the electromagnetic field. An electron or any othercharged particle or system of charged particles (an atom, a nucleus) can in-teract with this field. Moreover, photons, the quanta of the electromagneticfield, can be generated via such an interaction.

The approach became the starting point for the creation of the theoryof /3-decay. By analogy with quantum electrodynamics, two quantum fieldsare introduced in this theory — an electron-positron field, and a neutrinofield with electrons (positrons) and neutrinos (antineutrinos) as the quanta.Then /3-decay is presented as a process of generation of these particles bythe nucleus, similar to the process of generation of a photon by an atom.


4.4 Gamma-Radiation of Nuclei

If a nucleus is in an excited state, it can spontaneously emit a photon andpass to a state with lower energy. Such a radiation transition can occuronce, if the nucleus emits one photon and passes into the ground state, ormany times, forming a cascade of successively emitted photons. Note thatphotons are always born at the moment of 7-decay of a nucleus, becausethere are no photons inside a nucleus.

Photons emitted by a nucleus can have different angular momenta. Ifthe photon takes the angular momentum I = 1, the radiation is called dipoleradiation. If Z = 2, it is called quadrupole radiation; if I = 3, it is calledoctupole radition, and so on. The electromagnetic transitions of the electricand magnetic types and the corresponding photons are also distinguished.The electric transitions are caused by the redistribution of electric chargesin a nucleus, while the magnetic transitions are caused by the redistributionof magnetic moments and currents.

Another interesting phenomenon is called nuclear isomery. An isomeris an excited state of a nucleus with energy close to that of the groundstate but with a very different spin. Usually the difference is A/ > 4. Thenature of nuclear isomery was explained by von Weizsacker in 1936. Theprobability of the transition of a nucleus from the excited isomer state tothe ground state via the emission of a photon appears to be very small,while the lifetime of the state is large — it can be hours, days, or evenweeks. Such an excited state is metastable. The evolution of the excited(metastable) state of a nucleus can occur in two ways. The first is /3-decayof an isomer. The second consists of the emission of a photon, with thetransition of an isomer into another excited state with lower energy, andthe consequent emission of the electron of internal conversion.

Internal conversion is a process of direct emission of an atomic electronwithout preliminary radiation of a nuclear photon. In such a process, themono-energy electrons are emitted whose energies are determined by theenergy of the excited state of the nucleus and by the state of the electron inthe atom. Usually the .^-electrons are emitted with the greatest probability.

Nuclear isomery is not a rare phenomenon. There exist more than ahundred long-lived nuclear isomers. Some examples are as follows: the107Ag nucleus in the excited state with energy 0.093 MeV ( i ^ = 44.3s); the 113In nucleus in the excited state with energy 0.393 MeV (ti/2 =104 min); the u 9Sn nucleus in the excited state with energy 0.089 MeV(i1/2 « 250 days).


The Mossbauer effect proves useful in the investigation of nuclear struc-ture. Mossbauer, a German physicist, discovered the effect in 1958 andwas awarded a Nobel Prize in 1961. The effect concerns the resonant ab-sorption of a photon by a nucleus "frozen" into a crystal. In this case,the effect of nuclear recoil during photon absorption transforms into theeffect of the recoil of the whole mass of crystal lattice and, consequently,the resonance width coincides with the natural width of the spectral line.In order to observe the Mossbauer effect, the crystal is cooled to very lowtemperatures.

The effect plays an important role in applied physics. It is used for thedirect measurement of the superthin splitting of nuclear levels caused bythe nuclear spin and magnetic moment, and for the determination of theradii of excited nuclei. It allows one to study the superthin fields in metalsand alloys, and is implemented for determining the phases of diffractedwaves in monocrystals, for gathering information on crystal structure, andfor investigating ordered magnetic states. There are many other fields ofphysics, chemistry, and biology where the application of the Mossbauereffect helps us study the structure and properties of different objects.

4.5 Exotic Types of Radioactivity

Many heavy nuclei can undergo spontaneous fission. This usually competeswith a-decay, the probability of which is much greater. Evidently, thehalf-life of a nucleus due to a-decay is much smaller than that due tospontaneous fission (Table 4.4). Thus, spontaneous fission is a rather raretype of radioactive decay.

Table 4.4 Half-lives for some heavy nuclei

Nucleus ti/2> spont. fission (years) *i/2> a-decay (years)

2 3 2 T h 1.3 • 1018 1.41 • 1010

2 3 5 U 1.9 • 1017 7.1 • 108

2 3 8 U 5.9 • 1015 4.5 • 109

2 3 8 P u 4.9 • 1010 89.62 3 9 P u 5.5 • 1015 2.43 • 104

2 4 0 P u 1.3 • 1 0 u 6.58 • 103

2 4 2 P u 7 1 0 1 0 3 .5-105

2 4 1 Am 2.3-101 4 470


If a nucleus is proton rich it can decay by proton emission. In 1970,the isomer states of cobalt nuclei were observed to decay (ti/2 = 0.25 s)with the emission of 1.57 MeV protons (in 1.5% of cases) and positrons(in 98.5% of cases). The competition with /3+-decay makes the process ofproton emission hardly probable. Proton decay was later observed for someother nuclei. For instance, in 1982, the artificially obtained ^ L u nucleiwere shown to be unstable in the ground state. They decay and emit 1.217MeV protons (£^2 = 85 ms).

Some nuclei cannot emit a proton, but decay with the simultaneousemission of two protons. In 1983, the following chain of decays was ob-served:

?§A1 —» flMg + e-+i>e, 22Mg _ ^ 20Ne + 2 p ( 4 2 0 )

Some nuclei first undergo /3-decay and then emit protons and neutrons.Since the emission of a nucleon from the nucleus occurs almost instantly, theemission of the delayed proton or neutron is defined by the time for /?-decayof the nucleus. In such a process the nucleus (A, Z +1) undergoes /3+-decayor if-capture and turns into the nucleus (A, Z) in the excited state, thenemits a proton and turns into the nucleus (A — 1,Z — 1). As examples,note §C nuclei emitting 8.24 MeV and 10.92 MeV protons (t1/2 = 0.126 s),83O emitting four groups of protons with energies from 1.44 MeV to 7.0MeV (£1/2 = 0.0089 s), ioNe emitting five groups of protons with energiesfrom 1.68 MeV to 7.04 MeV (t1/2 = 0.108 s), and some others (^Te,s^Hg). Sometimes the fission fragments of nuclei form in excited stateswith excitation energies greater than the neutron separation energy. Theycan undergo /3~-decay and then emit the delayed neutron. Particularly,these include the 29Na, 30Na, and 31Na nuclei. Double neutron radioactivityis also possible when the nucleus emits two neutrons simultaneously.

Also well known are the exotic radioactive decays with emission of the12C, 14C, 16O, 20Ne, 24Ne, 28Mg, and 32Si nuclei. However, the half-livesof these are very large and this hampered their experimental observationfor a long time. The competition with /3-decay makes these exotic decayscomparatively rare. This circumstance also considerably complicates theirobservation and discovery.

Due to progress in the manufacture of semiconductor detectors, thesetypes of decays were successfully investigated in the 1980s. For instance,in 1984 the decays of 222Ra, 223Ra, 224Ra, and 226Ra nuclei, emitting 14Cnuclei, were observed. In 1985, the decays of 230Th, 231Pa, 232U, 233U,and 234U nuclei emitting 24Ne nuclei, and the decays of 234U, 236Pu, and


238pu n u c j e i ; emitting 28Mg nuclei, were studied. In 1989, the radioactivedecays of 238Pu nuclei were investigated in detail. These nuclides can emita-particles with ti/2 = 88 years, 28Mg nuclei with i j / 2 = 2 • 1018 years, and32Si nuclei with ij/2 = 6 • 1017 years, and also can spontaneously fissionwith ti/2 = 5 • 1010 years. The discovery of other types of radioactive decayis quite possible.

4.6 Application of Radioactive Isotopes

Radioactive isotopes have been widely used in medicine for a long time.Cardiovascular diseases are often accompanied by serious alterations of ves-sels and aberrations in heart function. Knowing the total amount of bloodand the speed of blood flow is very important for correct diagnosis of thesediseases. The measurements of these parameters should be made "in vivo".

To determine the parameters of blood flow, a small amount (0.25 cm3) ofphysiological solution containing about 1 MBk of 24Na radioactive isotopeis injected into a vein in the elbow. This isotope undergoes electron /?-decay with t1/2 « 15 hours. The nuclide transforms into a 24Mg nucleus inan excited state, which relaxes into the ground state with emission of 1.38MeV and 2.76 MeV photons. These photons easily pass through humantissue and are detected by a special detector that allows one to observeblood flow until the radioactive preparation is uniformly distributed overthe total volume of blood.

Numerous studies have shown that in a healthy human, blood flows fromone hand to another in about 15 s, while it flows from hand to foot in about20 s. Passing through the heart, the radioactive preparation is diluted byquite a large volume of blood. The greater the volume of blood flow, thefaster the dilution of the preparation, the degree of which can be estimatedby the change in concentration of radioactive nuclei after the blood leavesthe heart. So by measuring the radioactivity of the injected preparation,one can determine the volume of blood flow. It is about 5.5-6 1/min for ahealthy human, but can be 2-3 times smaller with heart disease.

Diagnosis of the diseases of different organs by radioactive isotopes isbased on the fact that organisms will concentrate definite chemical elementsin these organs. For instance, a thyroid gland is able to accumulate iodine inits tissues, while phosphorus, calcium, and strontium precipitate in bones.

To diagnose the diseases of a thyroid gland, the radioactive isotope131I, undergoing electron /3-decay with ti/2 ~ 8 days, is injected into the


human body. The 131I nucleus decays with the creation of the 131Xe nucleusin an excited state. The 131Xe nuclei transit into the ground state withemission of photons. Detecting this 7-radiation, one can evaluate the speedof concentration of iodine in the thyroid gland and thereby assess the diseaseof that organ.

The radioactive isotope 131I can also be used in diagnosing liver functionif we inject into the human body a special organic dye. This dye concen-trates in the liver. The speed at which it does this, as well as the speedat which it subsequently leaves the liver, allow one to assess liver function.Analogous methods are applied to study the functioning of the kidneys,stomach, duodenum, and intestines.

Radioactive isotopes allow one to reveal malignant tumors. Besidesthe 131I isotope, the radioactive phosphorus 32P isotope, which undergoeselectron /3-decay with i j / 2 « 14.3 days, is used to achieve that goal. Thetumor cells intensely accumulate radioactive phosphorus unlike the cells ofa healthy tissue. This phosphorus isotope emits electrons with a mean freepath in the tissues of the human body of about 0.03-0.08 m. Hence the 32Pisotope helps to diagnose tumors situated close to the body surface, andalso in the cavity of the mouth and larynx, and in the gullet. In this case,the emitted electrons can be detected by a special detector.

The radioactive isotope 198Au undergoes electron /3-decay with ti/2 ~2.7 days, creating the 198Hg isotope in an excited state, the transitionfrom which into the ground state is accompanied by the emission of 0.41MeV photons. That is why the 198Au and 131I isotopes are injected intoa vein with the physiological solution (198Au), or as a part of a specialpreparation (131I) used as a diagnostic for the internal organs (brain, liver,thyroid gland, etc).

Besides diagnostics, radioactive isotopes are widely used for medicalpurposes. Radioactive therapeutics are applied for the irradiation of amalignant tumor itself, with the aim of its destruction, or for the irradiationof the whole body in order to influence the nervous system or immunity.Radioactive therapeutics are usually carried out with the help of a cobalticgun containing a radioactive isotope of cobalt that emits photons.

Application of radioactive isotopes for medical purposes is not restrictedto diagnostics and medical treatment. With the help of radioactive isotopes,one can observe the dispersion of drugs in the body and see how harmfulsubstances reach different organs. One can also sterilize medical instru-ments and materials through the use of 7-rays, use portable 7-ray sourcesinstead of X-rays apparatus, and even obtain marked microflora.


Sterilization by 7-radiation made it possible to replace expensiverepeated-use glass syringes by cheap plastic disposable syringes. Packedalong with needles into the individual packets, the plastic disposable sy-ringes are placed in cardboard boxes and irradiated by gamma-rays (pho-tons) having energy greater than 1 MeV and emitted by the radioactiveisotope 60Co (ti/2 ~ 5.3 years). The 7-rays penetrate the packets andsterilize the syringes and needles, which remain sterilized as long as theintegrity of the packets is not compromised.

Natural radioactive 14C nuclei undergoing electron /3-decay with ti/2 ~5730 years are of great practical interest in archaeology and paleontology.The 14C isotope is created in the atmosphere under the influence of neutronsborn in processes initiated by cosmic rays: 14N(n,p)14C. The quantity of14C nuclei in the atmosphere is 1012 times smaller than that for 12C nuclei.Since the mean intensity of cosmic rays is in fact stable over time, theconcentration of 14C isotope in the atmosphere is actually constant.

Living organisms absorb 14C and 12C isotopes from the atmosphere.Plants absorb 14C as a part of the carbonic acid gas CO2; herbivorousanimals eat plants, and predators feed on herbivorous animals. Thus theratio of the quantity of 14C isotope to the total amount of carbon in aliving plant or animal remains constant. If the plants or organisms die, thequantity of radioactive 14C isotope in them decreases according to the lawof radioactive decay (4.3). Hence the measurement of the relative quantityof 14C isotope in fossil plants or the remains of animals allows one, with thehelp of that law, to estimate the time that has passed since the moment oftheir death. Such a method of dating organic substances is called carbondating. It was proposed in 1946 by the American scientist Libby. For hismethod of using carbon-14 for age determination in archaeology, geology,geophysics, and other branches of science, Libby was awarded the NobelPrize in Chemistry in 1960.

As an example, let us estimate the age of a tree in which the ratio of thequantity of 14C nuclei to the quantity of 12C nuclei is equal to 2/3 of thisratio for modern trees. Evidently, during the time that has passed since themoment of death, the quantity of 14C nuclei in the tree is 2/3 of its initialvalue, i.e., N(t) = 2iVo/3. Equation (4.3) yields t = t1/2 ln(3/2)/ln2 «3350 years.

The first tests of carbon dating were made on organic archaeologicalmonuments of known ages, and gave good results. Further, the methodmade it possible to disentangle many chronological mysteries not investi-gated in other ways. At present, the carbon dating method is deemed one


of the most reliable for dating finds of organic origin.Forensic Science often uses an activation analysis. Its essential idea is

in obtaining the artificial radioactive isotopes after irradiation of a sampleby charged or neutral particles. For example, the irradiation by neutronsleads to their capture by nuclei and the creation of j3~ -radioactive isotopes.The character of decay of such isotopes allows one to identify tiny sam-ples whose masses sometimes are of the order of 10~12 kg! For instance,although the qualitative composition of human hair is stable, the quantityof different substances in the hair of different people differ. Therefore, hairis as individual a feature of a human as fingerprints are.

It is known that the hair of the emperor Napoleon I contained arsenic,the dose of which exceeded the norm by about 10 times. Napoleon wasmost probably poisoned by substances containing arsenic and released intothe air by the wallpaper of his room.

In the USA, a method has been patented for detecting explosives inthe luggage of air passengers, based on the fact that explosives contain thenitrogen isotopes 14N and 15N. Irradiation by neutrons transforms theminto the radioactive 16N isotope with ti/2 = 7 s. This isotope emits 6 MeVphotons. The discovery of such radiation after the irradiation of luggage byneutrons signals that luggage contains substances with nitrogen, probablyexplosive. Since the half-life of 16N isotope is small enough, the applicationof this method does not take much time. Luggage can be simply moved byconveyor past the neutron source and the X-ray detector.

4

Fig. 4.9 The scheme of radioisotope energy generator: 1 — radioisotope, 2 —converter, 3 — isolator, 4 — thermopairs.

There exist energy sources of comparatively small power that use ra-dioactive isotopes as "fuel". The charged particles emitted by radioactivenuclei in such a power supply are absorbed by a substance, and the heat


obtained this way is converted into electricity. Fig. 4.9 shows the scheme ofsuch a device. Charged particles emitted by the radioactive isotope 1 areabsorbed by the converter 2. Thermoelements (thermopairs) 4 are placed inthe insulated cover 3. Their joints share the temperature of the converter,while the outside ends exist at a lower temperature.

In such energy generators the /3-radioactive 90Sr (ix/2 = 27.7 years)and the a-radioactive 238Pu (ti/2 = 87.5 years) are usually used. Thesesources of electric energy are successfully used on artificial satellites, shiningbuoys, isolated meteorological stations situated in inaccessible places, andin pacemaker devices that stimulate heart function.

Chapter 5

Nuclear Reactions

5.1 Conservation Laws in Nuclear Reactions

The various processes of interaction between nuclei and nuclei or nuclei andother particles are called nuclear reactions. They can result from the strong(nuclear) interaction, or from the electromagnetic and weak interactions.The strong interaction can cause a nuclear reaction if the distance betweenparticles is on the order of 10~15 m, because only at such distances canstrong forces act. The electromagnetic interaction is responsible for thenuclear reactions between nuclei and photons or charged leptons. The weakinteraction causes nuclear reactions between nuclei and neutrinos. Nuclearreactions can change the internal states of colliding particles, and can leadto the creation of new ones.

The first artificial nuclear reaction was conducted by Rutherford in 1919.For this purpose he used a-particles emitted by a radioactive bismuth iso-tope 214Bi, then known as RaC. Alpha-particles emitted by that isotope hadan energy of about 5.5 MeV. Passing through a tube filled with gaseous ni-trogen, a-particles caused the appearance of new particles whose free pathsubstantially exceeded that of the a-particles. These long-free-path parti-cles were detected by a scintillation screen coated with sulphurated zinc.Rutherford determined that they were protons.

Rutherford observed the nuclear reaction in which a-particle enteredthe nitrogen nucleus, adhered, and emitted a proton:

14N + a —-> 17O+p. (5.1)

Previously unknown, the oxygen isotope 17O was the first element to becreated artificially. These experiments were difficult to perform becausethe transformation of nitrogen into oxygen occurs very rarely. The twenty

115


registered nuclear reactions (5.1) required a million a-particles to be emit-ted by bismuth. Rutherford, the consummate scientist, solved the age-oldriddle that had puzzled the medieval alchemists who, for many centuries,tried to transform one element into another.

In subsequent years, Rutherford accomplished nuclear reactions with17 light nuclei including boron, fluorine, sodium, aluminium, lithium, andphosphorus. Further study of nuclear reactions required particles withhigher energies and beams of particles with greater intensity than theradioactive sources could give. Thus, in 1932 at Cambridge University,Rutherford's two disciples Cockroft and Walton created a high voltage gen-erator — the first accelerator of elementary particles. Using that device,they observed the reaction

7U+p —> 2a (5.2)

induced by the 0.125 MeV accelerated protons. Thus began the era of theaccelerators that have substantially widened the possibilities for conduct-ing various experiments in nuclear physics. In 1951, Cockroft and Waltonwere awarded the Nobel Prize in Physics for their pioneering work on thetransmutation of atomic nuclei by artificially accelerated atomic particles.

An interaction between a particle a with a nucleus A can occur in severaldifferent ways:

a + A,a + A*,b + B,

a + A —> < ... (5.3)d + g + D,

z + Z+... .

Particles that react are called an initial channel, while particles that resultfrom a reaction are called a final channel. A reaction can have a numberof final channels. The first two final channels in the scheme (5.3) representelastic scattering, when initial particles are identical to the final ones, andinelastic scattering (A* denotes nucleus A being in an excited state). Areaction channel is characterized by energy, spin, etc. A reaction a + A —•>b + B is sometimes denoted as A(a, b)B.

If in the experiment all of the reaction products are completely deter-mined, then this process is called an exclusive reaction. If in the final chan-nel only one particular particle is registered, then this process is called an

Nuclear Reactions 117

inclusive reaction. The study of exclusive reactions gives the most completeinformation about a nuclear process.

Let us consider some of the conservation laws that hold during nuclearreactions.

In nuclear reactions electric charge is always conserved. This means thatthe sum of the charges of all particles in the initial channel is equal to thatsum in the final channel. The conservation of a baryon charge in a nuclearreaction means the following. A baryon is a particle consisting of threequarks. If any baryon is attributed the baryon charge equal to +1 and anyantibaryon is attributed the baryon charge equal to -1, then the sum of thebaryon charges of particles in the initial channel will coincide with that sumin the final channel. Baryon charges of photons, leptons, and mesons areequal to zero. For instance, the reaction p + 7 —* p + n is forbidden becausethe baryon charges in the initial and final channels are equal to 1 and 2,respectively. Also forbidden is the process of annihilation of an electron-positron pair into one neutron, e~~ + e+ —* n, because the baryon charge inthe initial channel is equal to zero while that charge in the final one is equalto unity. Possible, however, is the annihilation of the electron-positron pairinto the nucleon-antinucleon one, e~ + e+ —> n + h, e~ + e+ —> p + p, wherethe line above a letter denotes an antiparticle. It is very important thatthe conservation of baryon charges forbids the annihilation of a hydrogenatom as p + e~ —> 27; to this we owe the stability of our Universe.

The laws of conservation of electric and baryon charges are also valid inthe radioactive decay of nuclei.

The law of conservation of energy in a nuclear reaction a + A —> b + Bmeans that the following equality is valid:

Ea + EA = Eb + EB, (5.4)

where the energy of the ith particle according to (2.10) is equal to Ei =i/p?c2 + TTI?C4. Here pj and rrn are the momentum and mass of the ithparticle.

The energy of the ith particle can also be written as Ei — rriiC2 + Ti,where the first term represents the rest energy of the ith particle while Tiis its kinetic energy. Thus, equation (5.4) can be given in the form

mac2 + Ta + mAc2 + TA = mbc2 + Tb + mBc2 + TB. (5.5)

The magnitude

Q = Tb + TB - Ta - TA = (mo + mA - mb - mB)c2 (5.6)


is called the energy or heat of reaction. It can be either positive or negative.If Q = 0, the nuclear reaction is a process of elastic scattering. If Q > 0,

the reaction is called an exoergic or exothermic reaction. Energy is releasedin such a reaction. For instance, the reaction

2H +3 H —>4 He + n (5.7)

is exoergic with Q = 17.6 MeV.If Q < 0, the reaction is said to be endoergic or endothermic. In this

case energy is absorbed. An endoergic reaction always has a threshold. Thethreshold energy is the minimum energy needed for the proceeding of thatreaction; it is given by

r, = (1 + ^ ) W | . (5.8)

We see that Ti > \Q\ because part of the absorbed energy is turned intokinetic energy of the particles that fly away (their center of mass alwaysmoves in the laboratory system). As an example of an endoergic reaction,we consider the charge-exchange nuclear reaction in which the initial protonis absorbed and the final neutron flies out:

p+ 7 L i—yn+ 7 Be , Q = -1.643MeV. (5.9)

Assuming the nucleon (proton and neutron) masses to be the same and themasses of the 7Li and 7Be nuclei to be equal to seven nucleon masses, wecan use (5.8) to find that Tt « 1.88 MeV.

Here are some further examples. The process of annihilation of a hy-drogen atom is further forbidden by the law of energy conservation becausethe sum of the rest energies of a proton and an electron is smaller than theneutron rest energy (mn-mp — me)c2 = 0.78 MeV, while the kinetic energyof an electron belonging to a hydrogen atom has an order of magnitude ofabout 10~5 MeV. In other words, the process of annihilation of a hydrogenatom would be energetically allowed if the electron in this atom would haveenergy exceeding 0.78 MeV.

Let us now ask why a neutron does not decay in deuterium and in the3He nuclei following the scheme n —> p + e~ + i>, while such a process takesplace in the 3H nuclei (tritium) and causes their /?-radioactivity. Since thereare no bound states for the systems comprising two and three protons, thedecay of a neutron in deuterium or in the 3He nuclei is forbidden by thelaw of energy conservation due to the inequalities (mn — mp — me)c2 < £<$,(mn — mp — me)(? < Eh where (mn — mp — me)c2 = 0.78 MeV and the


deuterium and 3He nuclei binding energies are equal to Sd = 2.22 MeV,Eh = 7.72 MeV. However, for the tritium nuclei, the inequality (mn — mp —me)c2 > et—£h holds where the tritium binding energy is equal to e* = 8.48MeV. Therefore, tritium nuclei appear /3-radioactive: 3H —> 3He + e~ + v(<i/2 ~ 12.5 years).

For the reaction a + A —> b + B, the momentum conservation law canbe written in the form

Pa + PA=Pb + PB, (5-10)

where pj is the momentum of the ith particle. If the target nucleus A doesnot move, then pA — 0 and pa = pfc + p B .

The angular momentum conservation law for the same reaction has theform

Ia + IA + laA = lb + IB+hB, (5.H)

where Ij is the spin of the zth particle, and laA and Its are the orbitalmoments of relative motion of the particles in the initial and final channels.

Conservation laws impose certain restrictions on the physical character-istics of particles emitted due to a nuclear reaction.

5.2 Nuclear Reaction Mechanisms

Let us consider nuclear collisions with slow particles whose energies aresubstantially smaller than the binding energies of atomic nuclei. Most im-portant is the fact that the interaction between the projectile (hadron) andthe nucleus is large. The energy of this interaction is of the same orderof magnitude as the energy of interaction between nucleons entering into anucleus. Owing to that, the nuclear collision problem appears essentiallyas a many-body problem. Note that this statement has nothing to do withnuclear collisions of particles, the energies of which are substantially greaterthan the energy of nuclear interaction.

The strong interaction between a projectile and a nucleus leads to thefact that soon after the collision and fusion with the nucleus, the projectileloses a substantial part of its energy, which is transferred to a nucleonentering into the nucleus. This redistribution of energy has a statisticalcharacter. Thus, none of the particles of the nucleus will have enoughenergy to overcome the nuclear forces of attraction and immediately leavethe system consisting of the initial nucleus and the projectile. Only after


a long time has elapsed, when due to fluctuations some particle acquiresenough energy to overcome the forces of attraction acting on it from therest of particles, will it leave the nucleus.

Therefore we come to a very important conclusion: the projectile andthe target nucleus can be treated as a unified quantum-mechanical systemthat exists and does not decay during a long period of time substantiallyexceeding the typical nuclear time. In other words, the lifetime of such acomposite system is essentially greater than the time the projectile takesto fly through the nucleus. Assuming the velocity of the projectile ~ 108

m/s and the linear size of the nucleus ~ 10~14 m, the typical nuclear timeis estimated as rnuci ~ 10~22 s.

During the time this system exists, its properties do not differ fromthose of ordinary nuclei in highly excited states. Therefore the systemformed by the fusion of the initial nucleus and the projectile is usuallycalled a compound nucleus. We emphasize that the compound nucleus is ina state with positive energy, i.e., its energy exceeds the energy needed toseparate at least some nuclear particles.

After a long period of time the excitation energy of the compound nu-cleus can accidentally concentrate on a single particle, which in this casegets an opportunity to leave the nucleus. It is not necessary that the par-ticle that leaves the compound nucleus be the same as the projectile thatformed the nucleus. On the contrary, it is unlikely that the nature of theinitial and final particles would be the same, because there are variouspossibilities for compound nuclear decay. Still more unlikely is that, evenprovided the nature of both particles is the same, the internal state of thenucleus will not change. More probable is that after the particle's emission,the remaining nucleus will be in an excited state.

If the projectile and the emitted particle are identical but the energyof the final nucleus differs from that of the initial nucleus, then this is aninelastic scattering. If the projectile and the emitted particle are differentby their nature, then this is a nuclear reaction.

During nuclear collisions, in some cases, radiation processes play a sub-stantial role because the emission of a photon requires a smaller concentra-tion of energy than the emission of a particle with nonzero rest mass. Aphoton can take away less energy than a nucleon emitted from a nucleus.Thus, at small excitation energies the lifetime of the compound nucleus ismainly determined by the interaction of nuclear particles with radiation, al-though this interaction is rather small. Hence, the lifetime of the compoundnucleus with comparatively small excitation energy, which only slightly ex-


ceeds the nuclear binding energy in the nucleus, is very long as comparedto the typical nuclear time. For instance, the typical lifetime of the excited52Cr nucleus that emits photons with energy ~ 1 MeV is 0.65 • 10~13 s,which exceeds the typical nuclear time by many orders of magnitude.

The role of radiation processes in nuclear collisions in comparison withatomic collisions is enhanced due to the wandering of the projectile in thenucleus, owing to which the time the projectile spends in the nucleus ap-pears much longer than the typical nuclear time.

Therefore, in nuclear collisions with comparatively small projectile ener-gies, two stages should be distinguished: the formation of the quasistation-ary long-lived compound nucleus, and its decay. These features of nuclearcollisions were first explained by Bohr in 1936. In other words, a nuclearcollision with the formation of a compound nucleus and its decay occursaccording to the scheme

a + A —> C* —> b + B, (5.12)

where C* denotes a compound nucleus being in excited state.The excitation energy of a compound nucleus is equal to

Ec = (ma+ mA -mc)c2+ mATa , (5.13)ma + mA

where ma, rriA, and me are masses of the particles a and A and the com-pound nucleus C in its ground state, while Ta is the kinetic energy of theparticle a.

The lifetime of a compound nucleus is very long relative to the typicalnuclear time. During this lifetime, the compound nucleus forgets aboutthe way it was formed, because the energy of the projectile is statisticallydistributed between the nucleons of the nucleus. Thus the decay of a com-pound nucleus in the final channel / should be treated independently fromthe mechanism of the formation of a compound nucleus in the initial chan-nel i. Hence the probability of the nuclear reaction under consideration Wifis the product of the probability of creation of the compound nucleus Wiand the probability of its decay wf.

Wif = WiWf. (5-14)

The final result of the nuclear reaction is denned by the competitionbetween the different possible processes of decay of the compound nucleus,which are compatible with the general conservation laws.


In order to prove the independence of the processes of creation and decayof a compound nucleus, special investigations of several nuclear reactionswere performed. For the first time this was done using the nuclear reactions

r63Zn + n,60Ni + a - ^ I 62Zn + n + n, (5.15)

i62Cu + n + p,

studied along with the reactions

( 63Zn + n,63Cu + p —» } 62Zn + n + n, (5.16)

i62Cu + n + p.

In reactions (5.15) and (5.16), the same 64Zn compound nucleus was formed.Within experimental error, the angular distributions of the same particlescreated in these reactions coincided independently of the ways the com-pound nuclei were formed.

Among the other experiments conducted, one should mention 12C +63Cu and 16O + 59Co with the compound nucleus 75Br created. The angulardistributions of a-particles (or protons) emitted as a result of the decay ofthe compound nuclei were identical. Many other nuclear reactions wereexperimentally studied, which confirmed the hypothesis of Bohr.

The concept of a compound nucleus as a quasistationary system makessense if the number of nucleons in the compound nucleus, among which theprojectile's energy is distributed, is large enough. On the other hand, if theenergy of the projectile is too high, a nucleus becomes transparent. Thus,the concept of compound nucleus can be used if the projectile (nucleon)mean free path in the nuclear matter is small compared to the linear nuclearsize. Only under this condition will the projectile be absorbed by thenucleus with large probability. Moreover, the nucleon separation energyfrom the compound nucleus e must be large compared to the excitationenergy per nucleon (E + e)/(A+ 1), where E is the projectile energy and Ais the number of nucleons in the target nucleus. Hence we get the conditionE <IC As. Since the nucleon separation energy is e « 8 MeV, both conditionsof compound nucleus formation are usually fulfilled at A > 10 and E < 50MeV.

The probability of a nuclear reaction that occurs via a compound nu-cleus usually has a sharp maximum at a certain excitation energy (theprojectile energy). This energy is the most favorable one for the decay ofa compound nucleus through a certain final channel. With the increase


of excitation energy, the probability of a particular nuclear reaction willdecrease because a new final channel opens.

Like usual stable nuclei, a compound nucleus is characterized by a cer-tain spectrum of energy levels. Since a compound nucleus is a quasistation-ary system, each of these levels has a certain width. The width of a levelF is connected with its lifetime r (more precisely, with the lifetime of thecompound nucleus in this state of its energy spectrum) by the relation

r = - = hw, (5.17)

where w is the probability of transition of a compound nucleus from thegiven state into all other possible states per unit time.

If the decay of a compound nucleus can occur in various ways, then

w = J2wf, (5-18)/

where Wf is the probability of decay of the compound nucleus through the/ th final channel. Then the total width of a level F is equal to

r = ^ r / . (5.19)/

The quantity F/ is called the partial width of a level, corresponding tothe decay of the compound nucleus through the /-channel. Evidently, therelation F/ = hwj holds.

One can talk, for instance, about the neutron width Fn , the radiationwidth F7, and the widths Tp and Ta corresponding to the proton and a-particle escape, understanding these quantities as the probabilities of decayof a compound nucleus with the emission of neutrons, photons, protons, anda-particles, expressed through the energy units.

The lifetime of a compound nucleus is long relative to the typical nucleartime, so the widths of levels of a compound nucleus appear to be smallcompared to the binding energy of a nucleon in a nucleus. However, thisdoes not mean that the level widths are also small relative to the neighborlevels separation D. Two cases are possible: when the level widths F aresmall relative to the level separation D, and when they are of the sameorder of magnitude or even greater than it.

The first case occurs in the region of small energies, while the secondoccurs at high energies. If a compound nucleus is formed through thecapture of a slow particle by the initial nucleus, then the excited state of


this compound nucleus is characterized by a rather small width. But if theprojectile is fast, the compound nucleus is formed with overlapping levels.

Only in the first case F -C D can one say that the separated levelsof the compound nucleus comprise a region of a quasidiscrete spectrum oflevels. In the second case F > D the overlapped levels form a region of aquasicontinuous spectrum of levels.

An important feature of the first case is the strong manifestation of thedependence of the compound nucleus formation probability on the energyof the projectile. This dependence has a resonance character: at certainvalues of projectile energy, the probability of compound nucleus formationand, therefore, the probability for some nuclear reactions to occur, becomesvery large. In quantum mechanics it is shown that the total probability ofa resonance reaction is defined by the simple formula

CY F

where C is a constant, Fe is the elastic scattering level width, Fr is thereaction level width, F = Fe + Fr is the total width of a level, Er is theresonance value of energy, and E is the projectile energy. Formula (5.20) iscalled the dispersion formula or the Brett- Wigner formula because it wasderived in 1936 by American physicists Breit and Wigner.

In the second case (overlapping levels) there is no clearly pronounceddependence between the probability of nuclear reaction and the energy.Instead, the probability of a certain nuclear process is denned by the jointaction of a large number of levels, and originates from the overlappingof the single level probabilities over an energy interval containing a largenumber of levels. The possibility of such overlap substantially simplifiesconsideration of this case.

If the projectile energy is large, E > 50 MeV, a compound nucleusmight not form because the corresponding probability is too small. In thiscase, a projectile hitting the nucleus close to its surface can suffer one orseveral collisions with nucleons entering the nucleus and located near thesurface. Having lost some of its energy, the particle can fly away withoutcreating a compound nucleus. In such a process the projectile can evenchange its nature (e.g., a neutron can turn into a proton, a deuteron canturn into a neutron, etc.). Nuclear processes that occur without creating acompound nucleus are called direct nuclear reactions. Note that these canalso take place in a range of comparatively small energies where a compoundnucleus is formed. Evidently, direct nuclear reactions are surface processes

(5.20)


in which a small number of nucleons contained in the near-surface regionof the target-nucleus participate.

Sometimes processes are observed in which the composite system cre-ated due to the fusion of a projectile with a nucleus decays before the energytransferred to the nucleus by the projectile has been statistically redis-tributed among all nucleons of that system. Those processes are called pre-equilibrium nuclear reactions. They are intermediate between compoundnuclear reactions and direct ones. They are sometimes also called multi-step nuclear reactions.

Let us consider the process of absorption of a particle by a nucleus inmore detail. The energy of a slow particle (e.g., neutron) colliding witha nucleus is first transferred to one or two nucleons comprising a nucleus,which then transfer it to other nucleons. Hence, the process of interactionof a particle with a nucleus can be treated as a sequence of stages, reflectingthe consequent nucleon-nucleon interactions. These stages are characterizedby the number of particle (p) - hole (h) pairs called the exciton number. Ahole is a vacant nucleon state formed due to a one-particle excitation of anucleus, which is the transition of a nucleon from a state with energy lyinglower than the maximum energy Ep (Fermi energy) of nucleon states intoa state with energy exceeding Ep.

At every stage, the emission of pre-equilibrium particles is possible.The probability of emission of such particles decreases from stage to stage,because the excitation energy is distributed more uniformly among thenucleons with every new stage. Eventually the compound nucleus attainsthe state of statistical equilibrium. Then it can emit particles, if possible,until it reaches the ground state. Note that the smaller the projectile'senergy, the smaller the probability of emission of pre-equilibrium particles.

The excited state of a nucleus formed after the first stage is a state inwhich two particles (the captured projectile and the nuclear nucleon in theexcited state) have energy greater than EF and one hole (a vacant nucleonstate) has energy smaller than Ep (Fig. 20). Such a state is called thedoor-way state and is denoted by 2plh (two particles and one hole).

The next stage could be the emission of an initial particle with such en-ergy and angular momentum that the nucleus will return to the groundstate; this constitutes an elastic scattering of the projectile as a pre-equilibrium process. Such an elastic scattering process is characterizedby the final width F^. On the other hand, particles having energies greaterthan EF can interact with one of the remaining nucleons of the nucleus.The composite system so formed will have three particles with energies


a

a

LJ U ^3

Fig. 5.1 Different stages of the nucleon-nucleus interaction: 1 — the nucleusbefore the interaction; 2 — the door-way state 2plh; 3 — the state 3p2h; a —the projectile.

greater than Ep and two holes 3p2h (Fig. 5.1). The width of such a stateis called the damping width and is denoted by F^. The consequent stagesof the formation of a composite system can be considered by analogy withthe first two.

Thus, the long-lived state of a compound nucleus is formed after a largenumber of two-particle nucleon-nucleon interactions, passing through a se-ries of intermediate states that get more and more complex with the numberof stages. In other words, during the formation of a compound nucleus, acertain sequence of states arises, which is called the hierarchy of configura-tions.

As already discussed, the simplest state of a compound system is theinitial state 2plh. By contrast, the states of a compound nucleus, in whichmany nucleons participate, are very complicated. They are usually de-scribed statistically, while the process of proceeding through the compoundnucleus formation is described as "the noise".

Let us consider one more intermediate type of nuclear reaction calledreactions of deep inelastic transfer. These reactions are observed when twocolliding nuclei have energies that exceed the height of their Coulomb bar-rier B = ZiZze21'{R\ + R2), where Ri and R2 are the radii of collidingnuclei. The mechanism of these reactions contains features characteristicof the direct nuclear reactions and reactions proceeding via a compound nu-cleus formation. In deep inelastic transfer reactions, strong coupling occurs


between the initial and final channels, i.e., the products of decay "remem-ber" the process of collision of initial particles, which is also a feature ofdirect nuclear reactions. However, the angular distributions of the productsof these reactions attest to the statistical equilibrium with respect to somedegrees of freedom. In this feature, deep inelastic transfer reactions aresimilar to reactions held via compound nucleus formation.

The mechanism of a deep inelastic transfer reaction assumes that dur-ing the collision of two nuclei a complex called a double nuclear system isformed. It can be assumed that during the collision, nuclei interact as twoliquid drops that roll along one another and stick together for a short timedue to the large viscosity of nuclear matter, making nucleons able to pen-etrate the surface of contact. Despite the partial interpenetration of nucleiduring the collision, their shell structures provide conservation of individu-ality of nuclei under the strong interaction. It appears that even the frontalcollision of heavy nuclei, whose energies exceed the Coulomb barrier, doesnot lead to their total fusion and the formation of a compound nucleus.Note that the double nuclear system changes its state very quickly, whichis different from the case of a compound nucleus with its quasistationarity.Therefore, the deep inelastic transfer reaction occurs much faster than thereaction that takes place through compound nucleus formation. Sometimesthe deep inelastic transfer reaction is called quasifission.

5.3 Nuclear Optics

The phenomena and processes in which atomic nuclei participate are stud-ied by nuclear physics and described by the equations of quantum me-chanics. However, despite the fact that the quantum laws that control thebehavior of nuclear objects differ substantially from those that govern themacroscopic world, there exists a wide analogy between some nuclear andclassical processes.

Detailed consideration shows that various nuclear processes have ana-logues in optical phenomena. These originate from the existence of waveproperties of quantum objects. Such nuclear processes comprise the specialbranch of nuclear physics known as nuclear optics.

Substantial information on the structure of molecules, atoms, atomicnuclei, and elementary particles is gained from the study of their interac-tion with other quantum objects. For this purpose, elastic and inelasticscattering of particles by nuclei, first of all, and then various nuclear reac-


tions are studied in nuclear physics.If the energy of the scattered particle is large enough, the elastic scat-

tering of such a projectile by a nucleus is similar to the scattering of light(an electromagnetic wave) by a spherical liquid drop having a definite re-fractive index and absorption factor. In classical physics (optics), such ascattering process can be described by its complex refractive index.

Let us consider the scattering of a nucleon (nucleon wave) by a nucleus.Outside the nucleus, a nucleon wave is characterized by the wave vectork = y/2mE/h, where m and E are the nucleon mass and energy. Inside thenucleus, the nucleon wave vector is K = \j2m{E — U)/h where U is theeffective potential of the nucleon-nucleus interaction. The refractive indexv of nuclear matter is defined by

" = f = f~l- (5-21)It can be seen that if the refractive index is a complex quantity v — v\ + ivi{i>\ and V2 are real values), the potential of interaction of a projectile withthe target-nucleus should be a complex quantity as well:

f — V — iW r < R

where R is the nuclear radius, and V and W are real and positive. Thenegative sign of the potential indicates the attraction and absorption of theprojectile by the nucleus.

If the energy of the scattered nucleons is large enough to satisfy thecondition E = h2k2/2m > \U\, the expression for the refractive index canbe approximated as

1,-1 U _ mV imW

Hence we have

mV mW

"1 = 1 +*wI U2 = W (5-24)Thus, the specific complex nuclear potential U(r) has a simple physicalmeaning: its real part describes the refraction of a scattered wave by nuclearmatter, while its imaginary part causes the absorption of scattered particlesby nuclei.

The complex nucleon-nucleus potential is called the optical potential,while the model that uses complex potentials to describe the interaction of

(5.22)

(5.23)


hadrons with nuclei is called the optical model. The essence of the opticalmodel of nuclear scattering should be found in the fact that the many-bodyinteraction of the projectile with the individual nucleons of the nucleus (orother particles which comprise the nucleus) is replaced by the effective two-body complex nucleon-nucleon potential. In other words, in the opticalmodel a very complicated many-body problem is reduced to a simple two-body problem. In the optical model a nucleus is considered as a drop ofnuclear liquid characterized by definite refractive and absorption properties.The basics of the optical model of nuclear scattering were first formulatedby American physicists Feshbach, Porter, and Weisskopf in 1953.

The model that considers a nucleus as a liquid drop is a very simpli-fied approach, because such a model does not consider a great number ofproperties of atomic nuclei (e.g., a shell structure). Despite that, however,the optical model of nuclear scattering has turned out to be a powerfultheoretical tool for the description of a great many interaction processesbetween hadrons and atomic nuclei.

As an example, Fig. 5.2 shows the cross sections for the 17 MeV pro-tons elastically scattered by iron, cobalt, nickel, copper, and zinc nuclei asfunctions of the scattering angle: points are experimental data, and curvesare calculated via the optical model. The cross section, da/dfl, is the num-ber of particles ejected into the solid angle element, divided by the densityof the projectile flux. This quantity is measured in mb/sr, where mb is amillibarn (1 mb = 10~31 m2) and sr is a steradian. It can be seen thatthe optical model allows a good description of the experimental data onthe elastic scattering of protons by various nuclei. Data analysis based onthe optical model gives parameters of the complex optical potentials thatcharacterize properties of particular nuclei.

If the energy of a projectile (hadron) is large enough to fulfill the con-dition A <C R (A = 2n/k — 2irh/'\/2mE is a wavelength of a projectile)then the nuclei become strongly absorptive with respect to the projectile.The scattering caused by the strongly absorptive nucleus in the case ofshort wavelengths of particles will be analogous to the diffraction of lighton a black disk in optics. Hence, such a process in nuclear physics is calleddiffraction scattering. Diffraction scattering of neutrons by nuclei, as ananalog of optical Fraunhofer diffraction, was first studied by Placzek andBethe in 1940.

In optics, the Fraunhofer diffraction pattern is manifested when thelight source and observation point are placed at infinite distances from thestrongly absorptive scatterer. In this case the rays from the source to the


Fig. 5.2 Angular distributions of 17 MeV protons elastically scattered by differentnuclei. The curves have been calculated by the optical model; the points areexperimental data.

screen are parallel. The parallel rays also travel from the near and far sidesof the scatterer to the observation point. Therefore, Praunhofer diffractionalters the directions of the light rays that undergo diffraction near the edgeof a screen.

Diffraction scattering is an interference phenomenon. In optical Praun-hofer diffraction from a black disk, the ray 1 scattered by the near side ofthe scatterer interferes with the ray 2 scattered by the far side (Fig. 5.3).An interference pattern results because the phases of near- and far-side raysafter scattering differ: the path length of the far-side ray exceeds that of thenear-side ray by L « 2R9 (the scattering angle 6 of Fraunhofer diffraction


1

Fig. 5.3 Rays of light undergoing interference during Fraunhofer diffraction froma black sphere.

is small).Small scattering angles 9 < X/R « 1 (^ = A/2TT) dominate in the

Fraunhofer diffraction pattern. About 84% of neutrons are scattered bythe strongly absorbing nuclei into that region of angles. The Fraunhoferdiffraction pattern is characterized by the alternation of maxima and min-ima of intensity of scattered particles, while the angular distribution ofelastically scattered particles is similar to the ratio of intensities of scat-tered and incident light in optics.

Real properties of atomic nuclei differ from those of the black disk inoptics. The nucleus has a thin diffused surface layer in which the densityof nuclear matter changes from the value typical for the nucleus center tozero. The nucleus surface diffuseness consideration leads to a more rapid(exponential) decrease in the envelope of maxima of the Fraunhofer patternfor nuclear scattering, unlike optics where the maxima envelope decreasesas 0~3.

The existence of the nuclear surface diffuseness is similar to the so-called apodization (the change of the function of the pupil) in optics. Thescattering of particles by nuclei is also influenced by the refraction of scat-tered nucleon waves in the semitransparent surface layer of the nucleusand the Coulomb interaction of charged particles with nuclei (diffraction of"charged" rays). These two effects cause the partial filling of minima of thecalculated angular distribution of scattered particles, which is confirmed bythe available experimental data. Diffraction scattering of charged particlesby nuclei was studied for the first time by Soviet physicists Akhiezer andPomeranchuk in 1945.

In nuclear scattering, inelastic processes are also possible in which theprojectiles cause the transition of nuclei into excited states. Under the


Fig. 5.4 Angular distributions (mb/sr) of 1 GeV protons elastically (1) and in-elastically (2) scattered by 208Pb nuclei calculated by the diffraction theory; thepoints are experimental data.

conditions of diffraction such processes have a diffractive character. Note,however, that the processes of inelastic scattering have no analogy in optics.Fig. 5.4 depicts the angular distributions (mb/sr) of elastic and inelasticscattering of protons by the lead nuclei. The typical Fraunhofer diffractionpattern (the alternation of maxima and minima) is clearly seen.

Diffraction scattering is also inherent for the composite nuclei such asdeuterons, 3H nuclei, a-particles, and other light nuclei. Besides elasticand inelastic diffraction scattering, the composite nuclei can participate inthe processes of partial or total dissociation of a projectile in the field of anucleus. For instance, a deuteron can break up into a neutron and a proton(diffractive dissociation of the deuteron). The process is possible in whichone of a deuteron's nucleons is absorbed by a nucleus while the other one isreleased. Such a process is called the stripping reaction. The dissociation


processes for complex nuclei have a diffractive character but, like inelasticscattering, have no analogy in optics.

Scattered particles and nuclei can have spins. Collisions of such particleswith nuclei cause the polarization of particles and nuclei, meaning that aftercollision the majority of the spins of particles and nuclei acquire certaindirections. Polarization phenomena in diffraction scattering, as well as spinsthemselves, also lack a classical analogy. Therefore, diffraction processes innuclear physics are more diverse than in optics.

p

Fig. 5.5 Rays of light interfering during the Fresnel diffraction from the edge ofa black sphere.

Besides Praunhofer diffraction, Fresnel diffraction is also observed inoptics. In this case either the light source and the observation point areboth placed at finite distances from the black screen, or only one of thesepoints is. The Fresnel diffraction pattern arises from interference betweenthe direct ray of light 1, which travels from the source to the observationpoint, and the ray 2 scattered by the near side of the scatterer. Possiblevariants of scattering that lead to the Fresnel diffraction pattern in opticsare schematically displayed in Fig. 5.5.

Let us clarify why the Fresnel diffraction pattern is observed in variousexperiments on the scattering of charged particles by strongly absorbing nu-clei. At first sight it seems that the conditions for Fresnel diffraction cannotbe satisfied in nuclear scattering. Indeed, to observe Fresnel diffraction thedistance between the nucleus and the particle source or the particle detec-


tor should be finite. In the case of neutron scattering this is impossible,because one would need to place the neutron source at a distance from thenucleus that should be of the same order of magnitude as the nuclear disk.However, the situation is essentially changed when we turn from neutronsto charged particles.

Because of Coulomb interaction, particles are scattered by the nucleusas if they came from a virtual point source located at a finite distancea from the nucleus. The Coulomb interaction must be strong enough todistort the scattered waves so that the wavefront has a significant curvaturenear the nucleus. If the nucleus strongly absorbs the projectiles and theirwavelength A is small compared to the nuclear radius R, then the scatteringpattern will be analogous to the Fresnel diffraction pattern from a blackdisk in optics. The distance o is defined by the expression

where n = Z\Zi&2 fhv is the Sommerfeld parameter, Z\ and Zi are thecharge numbers of colliding particles, and v is the projectile's velocity.

Formula (5.25) shows that the Fresnel diffraction conditions are fulfilledwhen a strong Coulomb interaction is present: n 3> 1. In this case thestrong electric field near the nuclear surface acts on the incident waves likea diverging lens if the sign of the projectile's charge coincides with that ofthe nucleus, and like a converging lens if these signs are opposite. The focaldistance of this lens is determined by

/ = «-£• (5-26)Thus the Fresnel diffraction pattern in nuclear scattering is formed by theinterference between the strong Coulomb and nuclear interactions.

If we consider the ratio of the intensity of scattered particles to theintensity of pure Coulomb scattering of a point particle having the projec-tile's mass and charge on a point particle having the target-nucleus massand charge, then, under the conditions of Fresnel diffraction, this ratio willbe similar to the ratio of intensity of the light scattering by the edge of ahalfplane to the intensity of the falling light in optics. The diffraction ofthe Fresnel type in nuclear collisions was explained for the first time byFrahn in 1966. As an example, Fig. 5.6 shows this ratio of intensities forthe elastic scattering of oxygen nuclei by lead nuclei.

One can formulate general conditions that allow us to clarify when innuclear scattering the Fraunhofer or Fresnel diffraction pattern is observed.

(5.25)


0(0)«R(8)

0.5 - \ |

i i i ' i~~—^-**< i .

0 20 40 60 9°

Fig. 5.6 Fresnel diffraction pattern for the ratio of the intensity of 170.1 MeV 16Onuclei elastically scattered on 208Pb nuclei to the intensity of the pure Coulombscattering of these particles: the solid line and dashes are the calculation resultsby the diffraction model, taking into account the surface diffuseness and semi-transparency of nuclei and without taking these into account, respectively; thepoints are experimental data.

For this purpose one should introduce the parameter

p=(kR-2n)-. (5.27)a

It appears that when p « l the Fraunhofer scattering diffraction patternis observed, while at p > 1 the Fresnel pattern is observed. Thus the quan-tity p totally defines the type of diffraction scattering pattern. Differentscattering processes having the same values of p give the same diffractionpattern, and the angular distributions of scattering particles coincide. Thisis the essence of the scaling law for nuclear diffraction, and the quantity pis called the scaling parameter.

Another interesting type of nuclear scattering also has an analogy inoptics. This is rainbow scattering. The physical explanation of this phe-nomenon was achieved after the discovery of the laws of refraction and totalinternal reflection of light on the boundary of two media, and also the lawsof propagation of electromagnetic waves. The possibility of rainbow scat-tering in nuclear and atomic collisions was predicted by Ford and Wheelerin 1959.

Recall the essence of the rainbow phenomenon in optics. There we con-sider the refraction and internal reflection of light rays in a liquid drop whenthe linear size of the drop is large relative to the wavelength of light. Fig.


Fig. 5.7 Passage of rays of light through a drop of transparent liquid.

5.7 shows the simplest case of ray propagation through a semitransparentdrop. It is seen that the parallel beam of light rays falls on the sphericaldrop. The beam undergoes refraction when entering the drop, propagatesinside the drop, undergoes total internal reflection from the drop surface,and finally leaves the drop while experiencing refraction once again. Fig. 5.7shows that there exists some maximum angle 9max between the directionsof falling and exit rays (rays leaving the drop at angles greater than 9max donot exist). In the vicinity of the angle 9max the typical rainbow thickeningof rays is observed. The limiting angle for the water drop is 9max « 42°,which corresponds to the scattering angle 9 = 180° — 9max « 138°. Ifthe drop radius is much larger than the light wavelength, the limiting an-gle 9max does not depend on the drop size. The colored rainbow in theatmosphere is caused by different values of refraction coefficients for rayswith different wavelengths, for which the thickening of rays takes place atdifferent values of 8max-

A rainbow scattering phenomenon exists in quantum mechanics. Theanalogy between the scattering of light and particles consists in the exis-tence of the limiting angle (the rainbow angle 9r) around which the classictrajectories (rays) are thickened, which means an increase in the intensity ofscattered particles. The envelope of rays which thicken around the rainbowangle is called a caustic.

In nuclear physics rainbow scattering also exists. In this case the follow-ing behavior of the angular distribution of particles elastically scattered by


nuclei is observed: in a region of comparatively small scattering angles, theangular distribution oscillates — but these oscillations are quickly damped.Then the angular distribution contains a large and wide maximum (rain-bow maximum), after which the intensity of scattered particles decreasesrapidly and smoothly.

The rainbow angle separates two regions. The region with 6 < 9r is theilluminated rainbow side, while the region with 9 > 6r is the dark (shadow)side from the viewpoint of the optics analogy. Note that in classical me-chanics the angular distribution has a singularity if the scattering angle 6approaches 9r in the illuminated region, and turns into zero in the shadowregion.

Fig. 5.8 Angular distribution of 140 MeV a-particles elastically scaterred by50Ti nuclei: the curve is the result of theoretical calculations, the points areexperimental data, and the arrow shows the rainbow angle.

A nuclear rainbow is mainly observed in the scattering of light nuclei3He, 4He, 6Li, etc., with energies E > 25 — 30 MeV/nucleon, by mediumand heavy nuclei. In this case, a little transparency and large refractionof nuclear matter with respect to the scattered particles are essential. Fig.5.8 depicts the ratio of the intensity of the elastically scattered 140 MeVa-particles on 50Ti nuclei to the intensity of the pure Coulomb scattering(as in Fresnel diffraction): points are experimental data, the curve is theresult of theoretical calculations, and the arrow marks the rainbow angle9r. Fig. 5.8 demonstrates the typical pattern of nuclear rainbow scattering.

The scattering of heavy ions by nuclei is characterized by much larger


absorption and much smaller transparency relative to the case of light ions.So the nuclear rainbow is not manifested in those processes. However,sometimes a very small transparency (10-15 times smaller than for lightions) exists. Thus, a hint of the formation of rainbow maxima called therainbow ghost is observed.

Note that, when the rainbow scattering effect is observed in the angulardistributions of elastically scattered nuclei, analogous effects of refractionof scattered waves can be revealed in the angular distributions of inelasticscattering and other quasielastic processes at 6 > 6r. Rainbow scattering isan example of the non-diffractive process of interference of rays propagatingthrough the drop.

The angular distribution of scattering particles calculated with the for-mulae of classical mechanics can also have singularities at 6 = 0° and6 = 180°. In optics and quantum mechanics such singularities do not exist,but there is an increase in the intensity of scattered rays and particles inthe forward and backward directions. This phenomenon is called the glory.An example of the glory in optics is the halo around the head of a manstanding on a small hill when his head shields the Sun.

The different processes of interaction between particles and nuclei athigh enough energies that have analogies in optics (the optical model ofelastic scattering, the Fraunhofer and Fresnel types of diffraction scatter-ing, rainbow scattering, and the glory) and also those that have no di-rect optical analogies (inelastic diffraction scattering, diffractive break-upof composite particles, and some nuclear reactions) considered above areunified in nuclear optics — the peculiar bridge between classical physics(electrodynamics) and quantum mechanics.

5.4 Accelerators

Every epoch in human history is evidenced by some original monuments.The Egyptian pyramids and the towers of Babylon, the sculptures of An-cient Greece and the aqueducts of Ancient Rome, the Mayan temples andthe medieval cathedrals of Western Europe — these are silent witnesses topast times. What monuments of our epoch will find people in a few cen-turies? It is hard to say, but among the most interesting may be the giantaccelerators that have been built in different countries all over the world.What are these grandiose and expensive constructions for? Let us try toanswer that question.


Accelerators allow us to obtain beams of charged particles having enor-mous energies and intensities. The accelerated particles are mainly elec-trons and protons, and, for some special purposes, the nuclei of variouselements (usually light nuclei). Accelerators exist that produce protons of500 GeV. Beam intensities in accelerators can be very large — up to 1016

particles per second. Moreover, these beams can be focused on a target ofseveral mm2.

Accelerators were first created to probe molecules, atoms, and atomicnuclei in order to study their structure and the mechanisms of variousmicroprocesses. In fact, the nucleons in nuclei are bound much more tightlythan the atoms in molecules. The energy cost of separating atoms in themost tightly bound molecule of carbon oxide, CO, is 11.1 eV. If we wantto separate the neutron and proton in the most weakly bound nucleus ofdeuterium (deuteron), we need to spend 2.22 MeV. Therefore, the energiesneeded to split atomic nuclei and to study mechanisms of various nuclearreactions appear much greater than the energies used to study chemicalprocesses.

In nuclear reactions with positively charged baryons (proton-nucleusand nucleus-nucleus collisions), electric forces of repulsion are involved: F =Zj^e2/'(Ri + -R2)2, where i?j and R2 are the radii of colliding nuclei. Thenuclear forces of attraction between these particles start to act at distancesr ~ 10"15 m. In order that these particles come together at such distances,energy exceeding the Coulomb barrier B = ZiZ^e2/{R\ + R2) should bespent. For instance, to collide protons with a 16O nucleus, an energy of 4MeV is needed. To place a proton inside the 238U nucleus, an energy of 15MeV should be used.

The creation of a new particle with mass m in a nuclear collision takesa minimum energy of me2. An energy of more than 140 MeV is neededto create one charged ?r-meson via the collision of nuclear particles. Thecreation of an nucleon-antinucleon pair p — p or n — n requires an energy of1.9 GeV.

High energies are not only needed to carry out nuclear reactions andstudy the creation of new particles. They are also necessary to study thedetailed structure of known microobjects. In order to discern the detailsof a microobject having linear size D, it is necessary to use acceleratedparticles with the de Broglie wavelength X < D. If nonrelativistic particles(e.g., protons) are used, their minimum kinetic energy T should be equal


to

Thus, in order to study the details of an object having a linear size of 10~15

m, the energy of the protons should be about 20 MeV. Formula (5.28)shows that a decrease in linear size of the objects under study requiresa substantial increase in the energy of the particles. Since D ~ 1/y/T, todecrease D by a factor of 10, the energy of the particles should be increaseda factor of 100. The study of smaller and smaller distances requires asubstantial increase in the energy of the accelerated particles.

A charged particle can be accelerated if it passes through some po-tential difference: a particle with a charge q passing through a potentialdifference V acquires the energy E = qV. To keep this energy, the particleshould move in vacuum, otherwise all the energy will be transferred to airmolecules. Besides, a powerful source of charged particles is needed to makethe beam intense enough. Thus, the main parts of any accelerator are theparticle source (injector), the accelerating device, and the vacuum creationsystem (vacuum pumps).

The accelerating system described above cannot accelerate particles tohigh energies. In fact, under several kilovolts the probability of electricbreakdown becomes large. Other technical problems are hard to avoid.Therefore, different types of accelerators were constructed. The mostwidespread of these are considered below.

One of the first accelerators was the electrostatic generator or Van deGraaff generator (Fig. 5.9), devised by the American physicist Van deGraaff in 1931. The generator has a large hollow conductor standing oncolumns of insulator. The strip on which the special device generates pos-itive charges transfers them to the conductor. Positively charged particles(e.g., protons) move from the source (conductor) into the vacuum tubewhere they are accelerated through the potential difference and directedtowards the target by the deviating magnet.

Usually electrostatic generators accelerate protons to an energy of about10 MeV. Tandem electrostatic accelerators where the charge exchange ofhydrogen ions takes place (the ion H~ having two electrons loses them andturns into H+) allows one to obtain particles with energies twice as large.The intensity of a beam of protons accelerated by an electrostatic generatoris very large — it reaches 100 /iA. The maximum energy of the protonsaccelerated by a modern tandem electrostatic generators can approach 30-

(5.28)


Fig. 5.9 The scheme of the Van de Graaff generator.

40 MeV. Van de Graaff generators are mainly used to study the structureof atomic nuclei.

In order to reach high energies, indirect methods of acceleration weredeveloped. In 1929 the American physicist Lawrence proposed the firstcyclotron, which started to work in 1931. In 1939 Lawrence was awardedthe Nobel Prize in Physics for the invention and development of the cy-clotron and for results obtained with it, especially with regard to artificialradioactive elements. The working principles of a cyclotron are as follows.

A particle with charge q and mass m, moving in a magnetic field ofintensity H and with speed v in the plane perpendicular to the directionof the magnetic field, has a circular trajectory (Fig. 5.10). A completerevolution of the nonrelativistic particle around the circle takes a timeT = 2TTm/(no\q\H), where /i0 = 4?r • 10~7 H/m. If v «C c, then the ro-tation time T does not depend on v. Particles are accelerated in a vacuumchamber situated between the poles of an electromagnet. The semicircleelectrodes (duants) are isolated metal disks to which a potential difference


Fig. 5.10 The scheme of a cyclotron.

V is applied. Passing through slot 1, the particle acquires an additionalenergy \q\V. When it approaches 2, the sign of the electric field is inverted,which means that the particle passing through this slot acquires additionalenergy \q\V. The sign of the electric field is inverted every half periodT/2. Therefore, the particle passing through any slot is accelerated eachtime, acquiring an additional energy \q\V. After n revolutions, the particleacquires energy 2n|g|V.

The higher the velocity of the particle, the larger the radius of its tra-jectory R = mv/{no\q\H), meaning that the trajectory is a spiral. Theparticle must move all the time in a homogeneous magnetic field. Thereis a limit for the construction of such a field on a plane. The energy ofthe particle accelerated in a cyclotron is limited by the fact that when thevelocity is comparable to the velocity of light, the period of rotation startsto increase and it is impossible to retain synchronism between the revolu-tions of the particles and the sign inversions of the voltage. This limitingenergy is about 20 MeV for protons and just 0.01 MeV for electrons. Thus,in order to reach higher energies of accelerated particles, it is necessary tochange the working principles of a cyclic accelerator.

In 1945, the American physicist McMillan and the Soviet physicist Vek-sler independently proposed the idea of a synchrotron. In this device parti-cles are injected into the accelerating circle with initial energy EQ. Specialmagnets keep particles on the circular trajectory with the radius R. An-other system of magnets preserves the beam from divergence in space (thebeam is collimated). Particles are accelerated inside special resonance cav-ities operating at frequency w.

When the accelerating devices are turned off, the particle with energyJ5o and momentum poi after being placed in the accelerating circle, moveswith velocity VQ = poc2/Eo (the particle is considered as relativistic) and


makes a complete rotation during the time

T=^f°. (5.29)

The frequency of revolution per unit time in this case is

In order to hold particles on the circle trajectory with radius R, a magneticfield should be created with intensity

Ho\q\R

If the accelerating device is turned on, then its frequency should be largerthan the angular frequency fi by a factor of k times (where k is an integer)in order to add energy to the particle at the necessary times. The frequencycj should increase with the energy of the accelerated particle until the latterbecomes ultrarelativistic, i.e., when the relation pc = E is valid. The inten-sity of the magnetic field should also increase. Thus, in order to acceleratethe particle, we should satisfy both of the following relations:

» - M - * ! § - * ! • "=ism- (5-32)The acceleration process takes place as follows. First, a bunch of parti-

cles with energy Eo is injected into the ring. Then the frequency u> and themagnetic field intensity H are continuously increased in such a way thatboth conditions (5.32) are satisfied at any time. During acceleration, theenergy of the particles increases from the initial value Eo to a final valueE. The time of acceleration is different for different accelerators. For thelargest and most modern accelerators, it is about 1 s.

It follows from (5.32) that in order to shorten the ranges of alteration ofthe frequency w and the intensity of the magnetic field, it is worthwhile toinject particles with maximum possible energy EQ. Therefore, for instance,on the synchrotron in Batavia (USA), where protons are accelerated to300 GeV, the injector consists of an electrostatic accelerator acceleratingprotons to 0.75 MeV and a linear accelerator increasing the proton energyup to 200 MeV. Then the protons enter the booster synchrotron, which theyleave with an energy of 8 GeV, and reach the accelerating ring of a largesynchrotron having a diameter in excess of 2000 m. Proton synchrotronsare also called synchrophasotrons.

(5.30)

(5.31)


Fig. 5.11 The scheme of a collider.

Further increase in the energy of acceleration appears to be a very com-plicated task from both the technical and economical points of view. There-fore the relative energy of colliding relativistic particles should be increasedin another way, e.g., by colliding intersecting beams of high energy. Iftwo proton beams with relativistic energies 21.6 GeV collide with eachother, their relative energy is 1000 GeV. A few accelerators with intersect-ing beams, also called colliders (Fig. 5.11), already exist.

Besides cyclic accelerators of particles, there also exist linear accelera-tors. In these, accelerated particles are driven along a straight line. How-ever, linear acceleration is not free from several difficulties. For instance,the Stanford linear accelerator (20 GeV electrons) has a length of 3000 m.Further increase in energy requires lengthening of the accelerator, which isa difficult technical and economical problem.


5.5 Detectors of Particles

Particles accelerated to high energies interact with targets, after which theresults of reactions should be detected. Charged particles passing through asubstance collide with its atoms. Many collisions lead to atomic ionizations,i.e., to the emission of electrons from atoms and to the transformations ofthe latter into positive ions. A fast particle passing through a substancecreates many ions that can be observed by various techniques.

In 1908 Geiger, in association with Rutherford, devised a particlecounter which consisted of a metal thread stretched along the axis of ametal cylinder and isolated from the latter. The cylinder was filled witha gas under a pressure of about 0.1 atmosphere. The thread was given aslight positive potential. A fast particle passing through the cylinder ion-ized the gas, which led to an electric discharge between the thread and thecylinder, and the emerging impulse of current could be detected. In 1928the counter was improved by Geiger, in association with Muller, and wascalled the Geiger-Miiller counter.

Another type of particle detector is the scintillation counter. The firstsuch device, called the spinthariscope, was invented by the EnglishmanGrookes in 1903. In this device the charged particles hit a screen coatedwith zinc sulfide (ZnS) and caused light flashes on it. Rutherford, alongwith his collaborators and other nuclear physicists at the turn of the 20thcentury, observed these flashes in their experiments and counted them withan unaided eye. This was very tiresome and did not give the needed pre-cision. In 1944 the human eye was replaced by the photomultiplier, whichbreathed a second life into these devices. Instead of the screen coatedwith zinc sulfide, crystals of sodium iodide (Nal) with the addition of somequantity of thallium, or special plastics, came into use.

The Wilson chamber, constructed in 1912, has played an importantrole in the development of the particle detection technique. This cham-ber is filled with a gas consisting of the supersaturated vapor of a liquid.If charged particles pass through the vapor, the ions thus created becomecondensation centers: a track consisting of the resulting liquid drops can beobserved and photographed. For his method of making the paths of elec-trically charged particles visible by condensation of vapor, the EnglishmanWilson was awarded the Nobel Prize in Physics for 1927. The EnglishmanBlackett was awarded the Nobel Prize in Physics for 1948 for his develop-ment of the Wilson cloud chamber method.

In semiconducting detectors, the charged particle creates ions in a solid


semiconductor made of germanium and silicon. Such detectors have highresolution, but they usually require cooling to very low temperatures inorder to reduce the level of noise caused by the thermal excitation of atoms.

In 1952, the American physicist Glaser constructed a bubble chamber.For this invention he received the Nobel Prize in Physics for 1960. Thebubble chamber is filled with a superheated transparent liquid. Passingthrough the liquid, a charged particle causes the liquid to "boil up" at thespots where it collides with atoms; i.e., bubbles of vapour arise around theions. These bubbles form the particle track, which can be photographed.Giant bubble chambers exist to register high energy particles. For instance,the bubble chamber at the Argonne National Laboratory (USA) contains20 m3 of hydrogen. This is a very complex and expensive device, becauseliquid hydrogen boils at -246° C and the necessary pressure in the chamberis several atmospheres. In the early 1970s in CERN (Geneva, Switzerland),a giant (for those times) bubble chamber was built. The chamber was filledwith freon. In this chamber, the results of rather rare interaction eventsbetween neutrinos and protons were observed; this enabled the observationof quarks inside a proton.

The spark chamber was also created to detect particles. This device isa multilayer capacitor with a large number of plates to which high voltageis applied. Ions are formed during the propagation of a charged particlethrough the device, and electrical breakdown takes place in the form ofa spark. The track of sparks can be photographed or registered by othermeans (using special electronics).

In order to register high energy particles, volume nuclear photoemul-sions are sometimes used. A charged particle regenerates silver passingthrough this photoemulsion. After development and fixing, the photoemul-sion shows tracks of particles and "stars" formed as a result of the dissoci-ation of photoemulsion nuclei caused by particle passage. Many discoveriesin nuclear physics (in particular, that of the 7r-meson) have been made inthis way.

Modern technologies for the registration of the products of interactionbetween elementary particles and nuclei possess a powerful storehouse ofdetecting devices, which allow us to investigate events with high precision.This technology is very complex. Detectors of high energy particles aregiant structures, the sizes of which sometimes approach that of a three-storybuilding. They are equipped with complex and expensive electronics, bymeans of which the nuclear and subnuclear processes occurring in detectiondevices are automatically registered and analyzed.

Chapter 6

Fission of Atomic Nuclei

6.1 Nuclear Fission Mechanism

When heavy nuclei capture neutrons, it is possible for a nucleus to splitinto two or more parts. Nuclear fission was discovered by the Germanphysicists Hahn and Strassmann in 1939. Hahn was awarded the NobelPrize in Chemistry in 1944 for that. The discovery of fission was precededby the fundamental works of Fermi on the irradiation of uranium nuclei byneutrons. The Nobel Prize in Physics was awarded to Fermi for his demon-strations of the existence of new radioactive elements produced by neutronirradiation in 1938. The phenomenon of nuclear fission was explained bythe Austrian physicist Meitner and the English physicist Frisch in 1939.They called this new kind of nuclear reaction

"nuclear fission due to the seeming similarity with the process ofcell fission leading to the reproduction of bacterium."

Then, using the analogy between a nucleus and a drop of liquid, in 1939 N.Bohr and the American physicist Wheeler developed the theory of nuclearfission. That year, an analogous theory was independently proposed by theSoviet physicist Frenkel.

Neutron induced fission is observed in goTh, giPa, and 92U, and also inthe transuranium elements with Z > 93. If the nucleus with (Z, A) breaksinto parts, then two nuclei with (Z\,A{) and (Z2, A2) are formed, providedthat Z\ + Zi = Z. Besides the fission fragments, fission is accompanied byneutrons (fission neutrons), the number v of which per fission event variesbetween 2 and 5 for different nuclei (Table 6.1). In uranium fission, themean value of this magnitude is u = 2.3 ±0.3. Fission neutrons do not havethe same energy, but are characterized by a certain energy spectrum. Fig.

147


Fi

103 \

"I I 1 I I I I _

0 4 8 En

Fig. 6.1 Energy spectrum of fission neutrons for 239Pu nuclei (arbitrary units).

6.1 shows the typical spectrum of fission neutrons. This is well reproducedby the formula

F(En) = Csmh^/2E^exp(-En), (6.1)

where En is the energy of a fission neutron (MeV) and C is a constant.If nuclear fission is induced by neutrons, then the conservation law for

the baryonic number gives A + 1 = A\ + A^ + v. The mass number valuesof the fission fragments A\ and A2 are not exactly definite. In uraniumfission, their mean values are in the ratio 2:3. Fig. 6.2 depicts the distribu-tion of fission probability versus the masses of the uranium fission fragmentsformed under the influence of thermal (very slow) and fast neutrons. Fis-sion fragmentation is clearly an asymmetrical process in the distributionof "daughter masses". As the excitation energy of a nucleus increases, theasymmetry of this distribution decreases.

An enormous amount of energy is released in nuclear fission. In actu-ality, at the beginning of fission two fragments are separated by a distancepo = Ri + R2 (where Ri and R2 are the radii of the fragments). Thus,their relative electrostatic potential energy is Z\Z2&2/p®. After fission this

Fission of Atomic Nuclei 149

N

102 - | ^ \

1 0 " 3 - I I

A rt-4 I | I I I I I 1 I ll „

80 110 140 A

Fig. 6.2 Yield of fission fragments (arbitrary units) for 235U nuclei as a functionof nucleus mass number: 1 — fission by thermal neutrons, 2 — fission by 14 MeVneutrons.

Table 6.1 Average number of neutrons v emitted in oneact of fission of some nuclei (at left — fission by thermalneutrons, at right — spontaneous fission)

Nucleus v Nucleus v

2 2 9Th 2.08 ±0.3 232Th 2.13 ±0.142 3 5U 2.407 ±0.007 238U 1.99 ±0.07

2 3 9Pu 2.87 ±0.009 240Pu 2.150 ±0.0152 4 1Pu 2.874 ±0.015 242Pu 2.141 ±0.011

2 4 1 Am 3.07 ±0.04 242Cm 2.51 ±0.011242Am 3.25 ±0.10 244Cm 2.69 ± 0.032243Cm 3.430 ±0.047 252Cf 3.756 ± 0.010249Cf 4.56 ±0.21 254Fm 3.98 ±0.14

energy is converted into kinetic energy of the fission fragments. Simplyputting Z\ = Zi = 46 and R\ = R<i = l • 10~15 m, we find that the kineticenergy of the fragments formed in a uranium fission reaction is slightlylarger than 200 MeV. Gigantic! In order to realize how large this is, notethat the energy of a-particles arising from the a-decay of nuclei is usually


about 4-9 MeV, while the typical energy of a chemical reaction betweentwo molecules is on the order of several eV. The energy released in uraniumfission induced by neutrons is three million times greater than the heat en-ergy released in the burning of coal, and twenty million times greater thanthe trinitrotoluene blast energy, if these substances are taken in comparableamounts. More precise calculations allowing for various effects of nuclearfission give approximately the same value of the released energy: 200 MeV.

Prom calculations of fission energy an important condition can be ob-tained, which shows when fission is energetically possible: Z2/A > 17. Thiscondition is fulfilled for all nuclei starting from 4°8Ag, for which Z2/A w 20.Following these conditions, we find that all nuclei with A > 110 should notbe stable against fission. However, fission is observed only for the heaviestnuclei — Th, Pa, U — and the transuranium ones. This means that simplythe energy feasibility of fission is insufficient to induce fission. As we willsee, fission is prevented by the existence of a potential barrier called thefission barrier.

o oi 2

CO OO3 4

Fig. 6.3 Stages of nuclear fission.

To understand the appearance of the fission barrier, we should con-sider the different stages of nuclear fission shown schematically in Fig. 6.3.Having absorbed the neutron, the nucleus gets excited. Then the excitednucleus becomes deformed, and the nuclear shape begins to vibrate. Sub-figure 1 marks the shape of the initially spherical nucleus of radius R. Thenthe nucleus is deformed into an ellipsoid with semiaxes a and b (subfigure2). If the energy of excitation is high enough, the vibrational amplitudebecomes appreciable. The surface tension forces tend to return the nucleusto its initial spherical shape. The Coulomb force of repulsion between twopositive charges centered at the foci of an ellipsoid tend to enhance defor-mation and separate the nucleus into two fragments. If the Coulomb forces


of repulsion exceed the forces of surface tension, a contraction is formedin the central part of the nucleus (subfigure 3). Each of the fragmentsgenerated tends to acquire a spherical shape, which makes the contractionmore narrow. Further Coulomb repulsion leads to the total separation offragments (subfigure 4). As calculations show, the nucleus energy increaseswith the increase of deformation j3, provided the latter is small, while atrather high deformations the energy of the fragments becomes smaller thanthat of the initial nucleus. In other words, the nucleus energy depends onthe deformation /3 in the manner shown in Fig. 6.4.

PFig. 6.4 The nucleus energy E(/3) as a function of the deformation parameter /3;Ef is the height of the fission barrier.

We see that the states of the initial nucleus and fission fragments areseparated by a barrier of height Ef. This means that the possibility offission is not defined only by the energetic preference (i.e., that the energyof fragments is smaller than the initial energy of the nucleus). A nucleusundergoing the fission process must pass through the potential barrier. Inspontaneous nuclear fission, where no particles are captured by the nucleus,a tunneling through the potential barrier occurs. Due to the substantialheight of the barrier, the probability of such tunneling is small; conse-quently, so is the probability of spontaneous fission. However, for apprecia-ble fission, to be able to break into two fragments, the nucleus must gainadditional energy (for example, by capturing a neutron or other particle)exceeding the barrier height. Therefore, the theory of nuclear fission statesthe fundamental problem of determining the height and form of the fissionbarrier.


0.0 0.3 0.4

0.5 0.6 0.7

0.8 0.9 1.0

Fig. 6.5 The shape of a nucleus in the saddle point for different values of thefissionability parameter x.

In the theory of nuclear fission, a very important role is played bya quantity called the parameter of fission x — (Z2/A)/(Z2/A)o, where(Z2/A)o is estimated to be about 45-50. This parameter characterizes thestability of a spherical nucleus against fission. If x > 1, the nucleus is un-stable. If a; < 1, the nucleus is stable against small deformations; however,sufficiently large deformations can make it unstable. This critical defor-mation which causes fission corresponds to the saddle point state of thenucleus. The critical deformation must be greater as x becomes smaller.Fig. 6.5 shows the shapes of nuclei in the saddle point for various values ofx.

The fragments created during fission are usually /3-radioactive, becausethey are neutron-rich. If the fragment excitation energy is greater thanthe energy of neutron separation, the fragment can emit a neutron. Suchneutrons are called secondary. If the neutron is emitted by a fragmentafter /3-decay, then the time of emission is determined by the half-life of theproceeding /3-decay, because the neutron is emitted almost instantly. Thus,such secondary neutrons are also said to be "delayed".

Fission fragments possess high ionizing ability and, therefore, have shortfree paths in matter. This phenomenon is caused by the high velocity offragments, due to which they lose some of their electrons. Thus, the frag-ments acquire large effective charge, which results in high ionizing abilityof fragments and their short free paths in matter.


Nuclei of some transuranium elements can capture electrons and createlong-lived excited states. The nuclei in these states can undergo fission withthe half-life periods equal to several minutes. These fission processes aresaid to be delayed. The delayed fission can also take place after the /3-decayof some transuranium nuclei.

Nuclear fission into two fragments is the most frequent and probablekind of fission. But fission into three or more fragments is also possible,although rare. For instance, the probability of fission into three or fourapproximately equal fragments taking place due to the capture of slowneutrons by 235U nuclei is no more than one event per 105 events of binaryfission. The emission of a-particles along with two massive fragments ofthe usual type is a firmly established type of ternary fission. It is observedthat an a-particle is emitted directly from the nucleus under fission and,thus, is a fission fragment just like the other two heavy fragments are. Thea-particle emitted possesses high energy 10 MeV < Ea < 40 MeV. Theprobability of such a process is about 400 times smaller than the probabilityof common fission into two fragments. Also possible is ternary fission inwhich the 3H nucleus is emitted along with two heavy fragments. Thisprocess takes place 1-2 times per 104 binary fission events.

The transition of a fission fragment from its excited state to its groundstate is accompanied by the emission of photons (instantaneous photons).The /3-decay of fragments can be accompanied by photon emission. Thetime of emission is determined by the half-life of the fragment, meaningthat such photons are emitted over a long period of time.

Nuclear fission can be initiated by slow and fast neutrons. The like-lihood of fission under the influence of neutrons is different for differentisotopes of nuclei, and is substantially dependent on the neutron energy.Let us consider the isotopes 235U and 238U. The isotope 235U, upon cap-turing neutrons, forms the isotope 236U. The fission barrier for the latteris Ef (236) = 5.8 MeV. The same value for 239U, originating from 238U, isJB/(239) = 6.2 MeV. The energy dowry of the neutron is en+Tn, where en isthe energy of neutron separation and Tn is the neutron kinetic energy. Theneutron separation energy for 236U is en(236) = 6.4 MeV, while for 239U itis en(239) = 4.76 MeV. Thus, the barrier height for 236U is smaller thanthe neutron separation energy, while for 239U it is the reverse. Therefore,235 U nuclei can undergo fission induced by neutrons with arbitrary smallkinetic energies, while 238U nuclei can only undergo fission by fast neutronswhose kinetic energy is greater than Ef (238) - £n(239) « 1.44 MeV.


The calculations of the fission barrier, based on the model of a homo-geneous charged incompressible liquid nuclear drop vibrating due to thecompetition and compromise between the surface and Coulomb forces, al-low one to derive an estimate of the fission barrier and its shape. However,the macroscopic model of a nucleus as a charged liquid drop does not ac-count for a very important property of a nucleus, namely its shell structure.Thus, the fission barrier calculated for the liquid drop should be correctedby adding a term obtained from considering the nuclear shell structure. Inthis case it is assumed that the shell structure is valid for deformed nuclei,but the energies of one-particle levels depend on deformation.

E'

\ ^ /i \ — / \ , \ isomeric

\ PW \y \ \ fission'\ ^ = = / isomeric state \ \\ ~/_ spontaneous \fission

ground state

I t t t ^P. P2 P3 P4 P

Fig. 6.6 Schematic plot of the fission barrier, taking into account the shell cor-rection (solid line) as a function of the deformation parameter j3. The dashed lineshows the fission barrier calculated in the liquid-drop model.

In 1967 the Soviet physicist Strutinsky showed that the shell correctionmodifies the barrier, which can give rise to additional extrema. Fig. 6.6shows the simplest modified double-humped barrier. It has two potentialwells and two maxima, situated according to the extrema of the shell cor-rection. The energy levels corresponding to the states of the nucleus beforefission are situated in the potential wells. The existence of the second lo-cal minimum leads to the possibility of a state in which the nucleus hasdeformed differently from the deformation in the first well. The secondwell situated between two maxima has a depth of about 3 MeV for thenuclei around plutonium. In this well, we have several metastable statescalled the isomers of shape, the lifetimes of which are contained in the in-terval from 10~17 s to 10~3 s. We emphasize that the isomers of shape


have nothing in common with the usual isomers discussed above, whichdiffer from the ground states by high values of spin. The half-life periodsof spontaneous and isomeric fission of several nuclei are presented in Table6.2. Note that nuclear fission can be initiated also by charged particles(protons, deuterons, a-particles, etc.) and photons (photofission).

Table 6.2 Half-life periods of spontaneous and isomeric fission for some nuclei

Nucleus tj/2 (years), spontaneous fision t\/2 (10~9 s), isomeric fision

2 3 5U 1.9 -1017 19.7 ± 4 . 92 3 6 U 2-101 6 70 ± 2 02 3 8U 0.8 • 1016 195 ± 30

2 3 9 Pu 5.5 • 1015 (8 ± 1) • 103

2 4 0 Pu 1.4 • 1011 4.4 ±0 .82 4 2 Pu 7 • 1010 28

2 4 1 Am 2 - 1 0 1 4 ( 1 . 5 ± 0 . 6 ) - 1 0 3

2 4 2 Am 9 .5-10 1 0 (13.5 ± 1.2) • 106

2 4 4 C m 1.35 • 107 1002 4 6 C m 1.7 • 107 10

6.2 Chain Fission Reactions

The neutron-induced fission of heavy nuclei results not only in the formationof fragment nuclei, but also produces secondary neutrons which, in turn, caninitiate fission of the nuclei. These neutrons have continuous distributionof energy and the majority have energies around 1-2 MeV, while theirmaximum energy is about 10 MeV. Therefore, the possibility of a nuclearchain reaction arises.

The notion of chain reaction has been known in chemistry for a longtime. For instance, during coal combustion, carbon atoms react with oxygenatoms and produce carbon dioxide. This reaction is exothermic, because anenergy of about 4.2 eV is liberated for each CO2 molecule created. However,to start burning, coal should acquire some initial energy, i.e., it should be seton fire. The energy liberated during the creation of only one CO2 moleculeturns out to be enough for the burning of neighboring carbon atoms tobegin. Therefore, coal burning is an example of a self-maintained chemicalchain reaction. Under favorable conditions, neutron-induced nuclear fissioncan become a self-maintained nuclear fission chain reaction.

Neutron physics exploits the following (rather conventional) classifica-


tion of neutrons with respect to their energies. Neutrons whose energiesare contained in the interval from 0.025 eV to 0.5 eV are called slow. Ifthe order of magnitude of the neutron energy is the same as that for thethermal motion of particles, then such neutrons are called thermal. Thethermal energy for the temperature of 300 K is about 0.025 eV, while thetemperature of 1000 K corresponds to a thermal energy of 0.086 eV. Neu-trons whose energy is smaller than thermal are called cold. If the neutronenergy is smaller than 10~4 eV, the neutrons are ultracold. Neutrons withenergies lying in the interval from 0.5 eV to 103 eV are called overthermal,while intermediate neutrons are in the energy interval from 103 eV to 0.5MeV. Neutrons are fast if their energy exceeds 0.5 MeV.

Let us consider the conditions for a nuclear chain fission reaction to oc-cur. We proceed from the simplest case of an infinite multiplicative system.Such a system produces neutrons continuously, and the matter whose nucleican undergo fission occupies the whole space. In such a system, we maytake no account of the neutrons that leave a system having limited size.We shall assume that the system contains only nuclei capable of neutroninduced fission. Since nuclear fission is accompanied by fast neutrons, theprocesses of fission and inelastic neutron scattering are highly likely. At thesame time, radiative capture of neutrons whose probability is rather smallcan be, nevertheless, significant.

If the probability of inelastic scattering is small, then in order for thechain reaction to proceed, the condition v > 1 must hold. This condi-tion must hold also in the case where the probability of inelastic neutronscattering is not small, but the inelastically scattered neutrons are able toinduce fission. Fission is possible if the kinetic energy of the majority ofinelastically scattered neutrons exceeds the difference between the barrierheight Ef and the neutron binding energy in the nucleus en, because onlyin this case is the probability of fission induced by the inelastically scatteredneutrons sufficient. If the inelastically scattered neutrons are not able toinduce fission, the possibility of chain reaction is determined, in addition tothe condition on v, by the probability of inelastic scattering and the energyspectrum of inelastically scattered neutrons.

Conditions can be created that favor chain reactions induced by slow(thermal) neutrons. Suppose, due to the high probability of inelastic scat-tering, that neutrons can leave the energy interval in which they can splitnuclei. Then the chain reaction cannot be induced by fast neutrons, yetradiative capture is possible and its role increases with decrease of neutronenergy. For this purpose, the system should be filled with a moderator,


i.e., a light element whose nuclei effectively moderate neutrons by means ofelastic scattering but weakly absorb them. The chain reaction induced byslow neutrons therefore occurs, e.g., in a system comprising uranium andlight moderator (graphite, heavy water).

Fission of the basic uranium isotope 2381J is induced only by fast neu-trons with energies exceeding 1.44 MeV. Thus, the high probability of elas-tic neutron scattering prevents chain reaction by fast neutrons from takingplace for the basic uranium isotope as a material for fission. Nonetheless,the uranium isotope 235U, the concentration of which in the natural mix-ture of uranium isotopes is about 0.72%, is capable of slow neutron fissionand the fission probability appears especially high exactly in the region ofthermal energies.

By the introduction of a moderator whose nuclei are incapable of radia-tive neutron capture at the energies considered, it is possible to decrease therole of the radiative neutron capture by the nuclei of the basic uranium iso-tope, which transforms into plutonium in this way. This means that whenneutron energy is in the thermal range, favorable conditions are formed forthe chain reaction with the 235U isotope as a fission material. In the natu-ral mixture of uranium isotopes, unfavorable conditions for chain reactionoccur down to the lowest energies of neutrons due to the high probabilityof radiative capture of neutrons by 238U nuclei and the small probabilityof fission of 235U nuclei because of their small concentration. Fortunately,the situation changes dramatically in the vicinity of thermal energies whereneutron energies are smaller than the energy of the lowest resonance levelof the 238U nucleus, which is around a few eV.

Now we can formulate the conditions required for chain reaction inducedby slow neutrons. Let us denote by P the probability that an initial fastneutron is captured by an uranium nucleus, which then undergoes fission.Then the mean number of secondary neutrons resulting from fission inducedby the initial neutrons is k = Pv. This quantity is called the multiplicationcoefficient of the system. If the system is infinite, meaning that neutronsdo not leave it, then a self-developing chain reaction can take place whenk > 1. If k < 1, it cannot.

Until now, we have considered an infinite multiplicative system. Inorder for a finite system to support a self-maintained chain reaction, itssize must exceed a certain critical size. The existence of a critical sizehas a simple physical meaning. Indeed, neutron production in a systemis a volume effect, while neutron escape from a system with finite size isa surface effect. If the size of a system is small, the surface escape of


neutrons is more important than their volume production. With increasingsize, the role of surface effects decreases, while volume effects become moreimportant. Surface area increases as the square of the linear size of thesystem, while the volume increases as its cube. The critical size of a systemis denned by the condition that the number of neutrons created inside thesystem coincides with the number of neutrons leaving the system throughits surface.

The condition k > 1 can be not met for a homogeneous mixture ofmoderator and fission material in an infinite system. At the same time,this condition will hold for a heterogeneous system containing separateblocks of fissionable material placed in the moderator. For example, sucha system can be designed as uranium bars placed in a certain order in themoderator.

The advantage of a heterogeneous system follows from the decrease ofresonance absorption in the case when the fissionable material is placed inseparated blocks. The following consideration makes this statement clear.In a homogeneous system, the neutrons are in the vicinity of the 238U nucleiwhich can absorb them all time during the process of slowing down. In theblock-built system, the neutron can move trough the "danger" zone nearthe level of resonance absorption, when it is far from uranium blocks. Inother words, a neutron can be moderated to the thermal energy with agreater probability than in the case of a homogeneous system. Therefore,the radiative capture of neutrons in a system with multiple block domainscan, be substantially lowered relative to a homogeneous system.

Resonance absorption is also decreased in a block heterogeneous systembecause at neutron energies in the region of strong resonance absorption,the interiors of blocks are screened by their outside layers and thereforeare not effectively used. This screening leads to a substantial decreasingof the radiative absorption of neutrons with energies close to resonance byuranium nuclei situated in the block, relative to the radiative absorption ofneutrons by single nuclei.

The chain reaction of fission can be harnessed as a tremendous sourceof energy. This aim is achieved in special devices, nuclear reactors, and alsoduring the explosions of atomic bombs.


6.3 Nuclear Reactors

The reserves of conventional fuel on Earth are limited. According to expertforecasts, oil and gas reserves will be exhausted in approximately 100 years,while coal reserves will vanish in 300-500 years. So it is obvious that theproblem of energy cannot be solved without nuclear energy.

In order to use the chain reaction of nuclear fission to obtain energy,this reaction must be made controllable. That is why nuclear reactors arecomplex devices in which chain reactions are totally controlled. If the chainreaction in a reactor gets out of control, a catastrophe becomes possible —an explosion of the reactor, accompanied by dramatic consequences forhumanity and the environment. Just such a situation led to the explosionof the reactor at the Chernobyl atomic power station (USSR) in April 1986.

The chain reaction in the reactor is regulated by an absorber of neutrons.Boron and cadmium are usually used as absorbers. If a decrease in theenergy released by the reactor is needed, then additional absorbers shouldbe introduced into the reactor and the chain reaction of fission will bedamped. It is clear that the absorbers must be put into the active zoneof the reactor very quickly, otherwise a delay of even part of a second willlead to a catastrophe. That is why regulation of the chain reaction in thereactor is carried out by means of computers.

In a reactor, the neutron generations continuously replace each other.The active zone always contains a large amount of neutrons: any cubiccentimeter of the reactor contains about a half billion neutrons. If at thebeginning we have no thermal neutrons in the reactor, then some (un-doubtedly not all) will cause nuclear fission and, therefore, fast secondaryneutrons will appear. These neutrons then become thermal owing to themoderating elements. New thermal neutrons, in turn, induce new nuclearreactions and the generation of new neutrons. This process lasts until achain reaction of nuclear fission occurs. Let us denote the number of sec-ondary neutrons by n-i and introduce the mean lifetime of neutrons of thesame generation. This lifetime is the time during which one generation ofneutrons is replaced by another. For reactors with thermal neutrons, thistime has order of magnitude 10~3 s - 10~4 s, while for the reactors withfast neutrons it is 10~6 s.

The ratio of the number of neutrons in a generation to their num-ber in the previous generation, taken on the same stage of their tempo-ral evolution, is called the effective multiplication coefficient of neutrons:keff = ni/no. The quantity p — (keff — l)/keff is called the reactivity of


the reactor. Depending on the values of keff or p, three regimes of reactorfunctioning are distinguished.

If keff > 1 and p > 0, the regime is supercritical. This occurs, e.g., atthe start when the reactor is sped up to the required power. If keff = 1 andp = 0, the regime is critical and the reactor works with constant power.If keff < 1 and p < 0, the regime is subcritical. This occurs, for instance,when the reactor is stopped and its power is gradually decreased. Therefore,the functioning of the reactor of a nuclear power station is efficient whenit works in the critical regime. To stay within this regime requires a ratherhard technological task: continuous control of all the parameters of such acomplicated device as a nuclear reactor.

An important problem in constructing reactors is the sink of energy(heat) out of the active zone. Special coolants are employed for this purpose:water, carbon dioxide CO2, heavy water, or liquid (molten) sodium. Thecoolant is heated in the active zone of the reactor and transfers its heatenergy to the external device or secondary circuit. In the active zone, thecoolant is strongly irradiated and its nuclei acquire an induced radioactivity.If the coolant transfers the energy directly to the turbine that generateselectricity, then we can restrict ourselves to a one-circuit scheme to channelenergy out of the active zone. If we have two or even three circuits, thenthermal energy will be transferred to the user by a pure (not radioactive)coolant.

The efficiency of a nuclear reactor is very high. Gigantic energies arereleased in nuclear fission: the total fission of 1 kg of uranium produces thesame energy as that generated during the total combustion of 2000-3000tons of coal. Therefore, countries that lack sufficient reserves of naturalcoal, gas, and oil are forced to build nuclear stations.

The first nuclear reactor was built in Chicago in December 1942 underthe guidance of Enrico Fermi. The first nuclear reactor in the USSR wasdeveloped under the direction of Kurchatov and launched in December 1946in Moscow. Since that time, many reactors have been built to obtain energyand to perform scientific investigations. Nuclear reactors are also widelyused as power sources for the engines on ships and submarines.

A modern nuclear power station is a very complicated structure hav-ing the height of a 10-floor building. However, despite its technologicalcomplexity, a nuclear power plant possesses substantial advantages in com-parison to a conventional power plant. The cost of electricity produced bya nuclear plant is smaller than the cost of electricity generated at a con-ventional power station. A 1 GW nuclear plant spends about 1 kg of 235U


isotope per day, while a thermal power station of the same power burns awhole trainload of coal or oil during the same time period.

In the region of a nuclear power station, ecological impact is kept to aminimum, largely taking the form of thermal pollution. Under normal oper-ating conditions, the probability of lethal irradiation in this region is smallerthan the probability of being struck by lightning. At the same time, eachton of coal contains about 80 grams of uranium. Therefore, the smoke froma thermal power station contains much more radioactive material than thegaseous emissions from a nuclear plant. Moreover, the smoke from thermalpower plants contains many additional harmful chemical substances (e.g.,sulphurous gas).

More than 400 nuclear reactors in 26 countries produce over 300 GWof electrical energy, which is about 16% of all the energy generated in theworld. In France, for instance, nuclear power stations produce more than80% of the electricity supply. Humanity has no current viable alternativesto atomic energy.

In principle, a nuclear reactor could exist in natural conditions. This canbe confirmed by the following considerations. At present, natural uraniumcontains 0.72% of 235U isotope. This amount is not enough for the existenceof a reactor with water as a moderator of neutrons. But the half-lives of235U and 238U are 7.13 • 108 and 4.51 • 109 years, respectively. This meansthat 2 • 109 years ago, the concentration of 235U isotope in the uraniumwas about 3%, i.e., approximately the same as in modern nuclear reactorsworking on enriched 235U fuel. With this concentration, a nuclear chainreaction could occur with groundwater as a moderator.

Analysis of uranium ore mined in Oklo (West Africa) has shown thatit contains 0.64% of 235U isotope. It also contains plenty of rare-earthelements created during uranium fission, and a small amount of 239Pu whichappears as a result of neutron capture by 238U nuclei and the subsequentf3~ -decay. Geological study of the uranium deposit in Oklo has shownthat it is situated in an old river delta. Taking into account the amountof plutonium created, the power of this natural reactor was estimated as25 kW and the duration of its work estimated as 600 million years. Thequantity of the "consumed" 235U isotope gives the age of the reactor asabout 1.8 billion years.

If several pieces of fissionable material, the total amount of which isgreater than the critical mass, are rapidly united, an explosion of tremen-dous power occurs. This idea forms the basis for nuclear weapons. To makethe mutual approach of these pieces rapid, an ordinary explosive is used.


The pieces can be placed close to each other, but separated by a layer ofa substance that strongly absorbs neutrons. To induce explosion in thiscase, it is enough to rapidly remove the neutron absorber or to introducea neutron source to counteract the absorber. The first nuclear bomb wasdetonated on a testing ground in the United States on 16 July 1945.

6.4 Man-Made Synthesized Elements

The list of elements that exist in Nature ends with plutonium (Z = 94).However, plutonium and neptunium (Z = 93) exist in Nature in extremelysmall amounts in uranium ores. For instance, the ratio of the number of239pu n u ciej to t n e number of uranium nuclei in pitchblende deposits inColorado, which contain 50% uranium, is 7.7 • 10~12. This plutonium hasbeen created as a result of the interaction between uranium nuclei and theneutrons of the secondary component of cosmic rays. Therefore, we canstate that uranium (Z = 92) is the last of the elements existing in Na-ture. Elements with Z > 92 are called the transuranium elements. Theirexistence was predicted by Rutherford in 1903, and in 1935 the possibil-ity of their synthesis was proposed by Enrico Fermi. These elements canbe produced artificially using different nuclear reactions. At present, thetransuranium elements with atomic numbers 93 < Z < 109 have been dis-covered (Table 6.3). Let us briefly discuss the production of transuraniumelements.

Neptunium was obtained in 1940 in the reaction of radiative capture ofa neutron by a uranium nucleus with subsequent /3-decay:

238IJ + n _ + 239u + 7 ) (g_2)

239TJ > 239Np + g - + ^ ^ = 33 5 m i n (g 3)

This element was named after the planet Neptune. The 239Np nucleus isradioactive: it undergoes /3~-decay with £1/2 = 2.35 days. This neptuniumisotope was first identified by its half-life, and then its properties were stud-ied. At present, 11 neptunium isotopes are known to have mass numbers231 < A < 241. The longest-lived isotope of neptunium is 237Np, whichexperiences a-decay with £1/2 = 2.14 • 106 years.

Plutonium was obtained in 1940 from the reactions

2 3 8 U + 2 H ^ 238Np + 2 n ) (g 4 )


Table 6.3 Transuranium elements

„ . . _ , , Year First discoveredZ Name Symbol ,. . . ti/2

discovered isotope '

93 Neptunium Np 1940 239Np 2.35 days94 Plutonium Pu 1940 2 3 8Pu 86.4 years95 Americium Am 1944 241Am 458 years96 Curium Cm 1944 242Cm 162.5 days97 Berkelium Bk 1949 243Bk 4.5 hours98 Californium Cf 1950 245Cf 44 min99 Einsteinium Es 1952 253Es 20 days100 Fermium Fm 1952 255Fm 22 hours101 Mendelevium Md 1955 256Md 1.5 hours102 Nobelium No 1957 254No 3 s103 Lawrencium Lr 1961 257Lr 8 s104 Rutherfordium Rf 1968 257Rf 4.8 s105 Dubnium Db 1968 260Db 1.6 s106 Seaborgium Sg 1974 259Sg several ms107 Bohrium Bh 1981 262Bh 4 ms108 Hassium Hs 1984 265Hs 2.4 ms109 Meitnerium Mt 1982 266Mt 5 ms

238Np —> 238Pu + e~ + v, t1/2 = 2.1 days. (6.5)

It was named after the planet Pluto. The 238Pu isotope is a-radioactivewith ti/2 = 86.4 years. Another plutonium isotope, 239Pu, was obtained in1941 as a product of/3^-decay of 239Np. This isotope is also a-radioactive.The large half-life ti/2 = 2.43 • 104 years of the 239Pu nuclei created innuclear reactors in large quantities allows one to use this isotope as a fuel forspecial nuclear reactors and as an explosive in nuclear bombs. At present,15 plutonium isotopes are known to have mass number 232 < A < 246. Thelongest-lived isotope of plutonium is 242Pu, which has ti/2 = 3.8 • 105 yearsand is a-radioactive. For the discovery of plutonium, American physicistsMcMillan and Seaborg were awarded the Nobel Prize in Chemistry in 1951.

Americium was discovered in 1944 from the reactions

239pu + n ^ 24OPu + 7 , (6.6)

240Pu + n _ ^ 241Pu + 7 ) (6.7)

241Pu —> 241Am + e~ + D, t1/2 « 14 years. (6.8)

This element received its name (from the word America) because it washomologous to the element Z = 63 europium (named in honor of Europe).


The 241Am nuclei are a-radioactive with £1/2 = 470 years. At present,we know 11 americium isotopes with mass numbers 237 < A < 247. Thelongest-lived isotope is a-radioactive with t\/2 = 7800 years.

Americium is famous because the 242Am isotope has an isomer of shapewith the half-life of spontaneous fission ti/2 = 1-4 • 10~2 s. This isomer wasdiscovered in Dubna (Russia) in 1962 and was the first observed isomer ofshape.

Curium was produced in 1944 by the reaction

2 3 9Pu+ 4 H e - ^ 242Crn + n. (6.9)

This element, with Z = 96, was named in honor of Pierre and Marie Curie,who discovered actinium — the first element from the actinide series. In thesymbol Cm, the first letter comes from the surname Curie, while the secondis taken from the Christian name of Marie Curie. Now we know 13 curiumisotopes with mass numbers 238 < A < 250. The longest-lived isotope,248Cm, experiences a-decay as its main decay mode with t^ji = 4.7 • 105

years.In 1949 berkelium was obtained in the reaction

241 Am + 4He—> 243Bk + 2n. (6.10)

This element was so named because it was produced at the laboratory ofthe University of California at Berkeley near San Francisco. At present, 9berkelium isotopes with mass numbers 243 < A < 251 have been produced.The longest-lived isotope, 247Bk, is a-radioactive with t^/i = 1380 years.

Californium was produced in 1950 in the same laboratory as berkelium,and was named for the university. The element was created in the reaction

242Crn+ 4 H e - ^ 245Cf+n. (6.11)

Sixteen californium isotopes are known, with mass numbers 240 < A < 255.The longest-lived isotope, 251Cf, is a-radioactive with t^^ — 1600 years.

Einsteinium (Z = 99) and fermium (Z = 100) were first found in theseveral hundred kilograms of corals that remained after the explosion of athermonuclear bomb on the Bikini atoll in 1952. Their names were givenin honor of the two outstanding physicists of the 20th century. Fourteenisotopes of einsteinium with 243 < A < 256, and fifteen isotopes of fer-mium with 243 < A < 258, are known. The longest-lived isotopes of theseelements are a-radioactive: 254Es with ty/% — 480 days, and 257Fm with£]y2 = 79 days.


Mendelevium (Z ~ 101) was obtained in 1955 in the reaction

253Es+ 4 H e - ^ 256Md + n. (6.12)

Named in the honor of Mendeleev, who discovered the periodic table, thiselement illuminated the validity of the periodic law for the elements withatomic numbers greater than 100. Eleven mendelevium isotopes are knownto have 248 < A < 258.

Nobelium (Z = 102) was discovered in nuclear reactions of irradiationof 244Cm nuclei by 13C, studied using the cyclotron of the Nobel Institutein Stockholm from where it acquired the name. Lawrencium (Z = 103)was produced in 1961 at the University of California. It was named inhonor of the American physicist Lawrence, who was one of the creators ofthe first accelerator. The element with Z = 104 was obtained in Dubnain 1964, where it was proposed to be named kurchatovium, and at theUniversity of California in 1968, where it was named rutherfordium. Sincedoubts still exist whether this element was really discovered in 1964, thefinal name for the element is rutherfordium. The element with Z = 105 wasobtained in Dubna in 1968, and its name dubnium is devoted to the townwhere it was discovered. The elements with Z > 106 are also proposed toreceive the names of outstanding scientists: Z = 106 (1974, Dubna andindependently the University of California) — seaborgium; Z — 107 (1981,the accelerator of heavy ions in Darmstadt, Germany) — bohrium; Z = 108(1984, Darmstadt) — hassium; Z = 109 (1982, Darmstadt) — meitnerium.

In 1996-1999 the elements with Z = 112 and A = 283 and also with Z =114, A = 287-289 were found. In 2000-2003 the elements with Z = 116,118and Z = 115, A = 287 and A = 288 were synthesized. However, theseresults must be confirmed in other laboratories of the world. Apparently,in coming years, the creation of superheavy elements will continue.

Let us now discuss the discovery of radioactive elements with Z =43,61,85,87. These nuclei are not observed in Nature, except for the el-ement with Z = 43 which is found in several young stars and in uraniumores in insignificant amounts.

Technetium (Z = 43) was discovered in 1939. Its name comes from theGreek "technetos", meaning "artificial". The reaction was

Mo+ 2H—> Tc + n. (6.13)

Now we know 16 technetium isotopes with 92 > A > 107. The longest-livedisotope, 97Tc, experiences the K-capture with tx/2 = 2.6 • 106 years.


Promethium (Z = 61) was found in 1946 in the fission products of 235Uin nuclear reactors. It was named for the mythical hero Prometheus, whostole fire from Zeus and gave it to man. The name emphasizes the methodof creation of the element with use of the energy of nuclear fission. Thelongest-lived isotope, 145Pm, undergoes K-capture with t1/2 = 18 years.

Astatine (Z = 85) was produced in 1940 from the reaction

209Bi+ 4 H e ^ 211At + 2n. (6.14)

The name originates from the Greek "astatos", meaning "unstable". Asta-tine was predicted by Mendeleev under the preliminary name "eka-iodium".The longest-lived isotope of this element, 210At, has half-life ti/2 — 8.3hours (mainly K-capture).

Prancium (Z = 87) was discovered in 1938 in the decay products of thenatural radioactive element 227Ac. It was named for France, where it wasdiscovered. The element was predicted by Mendeleev under the preliminaryname "eka-caesium". Its longest-lived isotope, 223Fr, undergoes the /3~-decay with £1/2 = 22 min.

Note that all the artificially obtained elements are radioactive with half-lives much smaller than the age of the Earth: 4.6 • 109 years. That is whythese elements are not present on the Earth (except for small amounts inthe uranium ores, where they are produced as a result of radioactive decayof nuclei).

Chapter 7

Nuclear Astrophysics and ControlledNuclear Fusion

7.1 Expanding Universe

There are 287 different isotopes of nuclei known in Nature. They includeboth stable nuclei and nuclei whose half-lives exceed the age of the Earth.All these nuclei have been found on Earth. There is only one element,technetium, which has not been found on Earth but is observed in youngstars. In fact, the half-life of the longest-lived isotope of technetium is onthe order of 105 years, i.e., the nuclei of technetium on Earth have alreadycompletely decayed, but they still exist in sufficiently young stars.

The quantities of different isotopes in Nature differ substantially (Fig.7.1). Light nuclei prevail in Nature; heavy nuclei are the most sparse.Hydrogen and helium by far surpass all other elements in their abundance.Together they form 98% of the substance of our galaxy. The curve in Fig.7.1 shows the relative abundance of the isotopes of nuclei in the solar systemand in the main sequence stars which are close to the Sun by their mass andage. This is called the solar curve. Questions about its shape are closelyrelated to the general problem of the origin and evolution of the Universe.

At the present time, the generally accepted theory is that of the BigBang, as a result of which each particle of matter rushed away from everyother particle. At the moment of the Big Bang, temperature, and densitywere infinite. Immediately after the Big Bang, the temperature of matterwas huge.

The concept of the Big Bang and the picture of the hot Universe at theearly stages of its development were proposed by George Gamov in 1946in the famous paper he published together with his postgraduate studentAlpher. Gamov also persuaded the outstanding physicist Bethe to be acoauthor of this paper — that way, the author list would read "Alpher,

167


H1Q I hydrogen combustion

1 He

8-1 helium combustion£ \ . carbon and oxygen combustionSi fJf \ / silicon combustion

8 \ V4® \ \ l\ — i ron g r o u p

1 4- \CD \ / IQ. \ / \ Z = 5 0

I 2T\ wT N=82£ M ^\ / z=82

W \ / , N=1260 - Li-Be-B \JW\ I

-21 i i i i

0 100 200 A

Fig. 7.1 Dependence of the logarithm of the relative abundance of elements (ar-bitrary units) on nucleus mass number.

Bethe, Gamov" and would correspond to the three first letters of the Greekalphabet: alpha, beta, gamma. Gamov, who possessed a keen sense ofhumor, thought this would befit a paper about the origin of the Universe.

Because of its huge energy, the initial Universe started to expand andgradually became cooler. Gamov called the initial substance ylem which,in translation from the ancient English, can be roughly understood as "aninitial substance from which the creation of elements occurred".

Of course, under the huge temperatures during the first moments af-ter the Big Bang, no bound matter could exist. At that time not onlycould there be no molecules and atoms, but also no nuclei or nuclear con-stituents: protons and neutrons. There only existed photons, electrons,positrons, neutrinos, antineutrinos, and other leptons and, besides these,the constituents of nucleons — quarks, antiquarks, and gluons — whichwere in the state of a quark-gluon plasma.

Immediately after the Big Bang, owing to the huge temperatures (in10~43 s after the Big Bang the temperature was approximately 1032 K) and

Nuclear Astrophysics 169

the correspondingly huge particle energies, there was in fact no differencebetween the various types of interaction: the strong, electromagnetic, weakand, probably, gravitational. Rather, there existed a single fundamental in-teraction. In that case, photons, leptons, quarks, and other particles movedlike free objects and together formed a heat radiation whose temperature,however, was not constant and changed over time.

At the moment of the Big Bang, it is probable that matter possessedall possible symmetry properties: the number of electrons was the sameas the number of positrons, and the number of quarks was the same asthe number of antiquarks. Further, the expansion of the Universe wasaccompanied by cooling of matter. Similar to condensation forming waterdrops during water vapor cooling, during the cooling of the quark-gluonplasma the bound states of these particles, i.e., nucleons and antinucleons,were created as well. The evolution of the Universe after the Big Bang canbe imagined as a succession of four eras (or epochs) leading to its presentstate with an average density of matter p ~ 10~27 kg/m3 and averagetemperature T « 2 . 7 K .

Some 10~10 s after the Big Bang, the density of matter far exceededthat of nuclear matter and equaled p ~ 1018 kg/m3. The temperaturewas also huge: T ~ 1013 K. During this epoch the strongly interactingparticles, hadrons, were created. This hadron epoch lasted about 10~4 s,until the temperature decreased to the rest energy of the lightest hadron,the 7r-meson (T « 1012 K). By the end of the hadron epoch, the density ofmatter was comparable to that of nuclear matter.

The next period in the evolution of the Universe can be called the "lep-ton" era. Its duration was approximately equal to 10 s, until the temper-ature of matter decreased to the threshold of the photoproduction of anelectron-positron pair (T « 0.7 • 1010 K). By the end of the lepton era, thedensity of matter was p ~ 107 kg/m3. At that time, leptons could not al-ready be created spontaneously, and radiation consisted mainly of photons.Thus the "radiation" era began, which ended when photons could exist sep-arately from matter. This era ended approximately 106 years after the BigBang. The temperature at its end was T ~ 107 K. Further expansion andcooling of the Universe led to the "star" era, which lasts until the presentage.

In 1965, the American scientists Pensias and Wilson discovered the ex-istence of an isotropic photon flux in the Universe, having no "seasonal"oscillations, which is called relict radiation. This radiation is uniformly dis-tributed over the celestial sphere, and its energy corresponds to the heat


radiation of an absolutely black body at temperature T « 2.7 K. In 10~2

s after the Big Bang, relict radiation had temperature 1011 K, and in 106

years — 3 • 103 K. Now, relict radiation is isotropically distributed in theUniverse with density 5 • 108 photons/m3. Earlier, it was predicted theo-retically by Gamov on the basis of the expanding Universe model. Relictradiation is a witness to the Big Bang and to the three subsequent epochsin the development of the Universe. Its existence allows us to concludethat the Universe was hot during the early stages of its expansion. For thediscovery of relict radiation, Pensias and Wilson were awarded the NobelPrize in Physics for 1978.

Moments after the Big Bang, the number of particles apparently coin-cided with the number of the corresponding antiparticles, but thereafter theformer became somewhat larger than the latter, i.e., the so-called baryonasymmetry arose. Just from astrophysical observations it is known that theratio of the number of particles (electrons and nucleons) in the Universeto the number of photons is on the order of 10~9. The theory of baryonasymmetry was first developed by the Soviet physicist Sakharov in 1967.

There also exists another viewpoint on the fact that the electron, proton,and neutron — rather than the positron, antiproton, and antineutron — arethe main elementary particles in Nature. In fact, the problem is to indicatethe cause that has led to the separation of matter and antimatter, butnot to their annihilation. In other words, in this approach it is supposedthat matter and antimatter, after their separation, have disseminated todifferent parts of the Universe. Those parts of the Universe observed by usdo not contain antimatter.

It appears that one act of nucleon-antinucleon annihilation leads to theemission of several 7r-mesons (on the average, six 7r-mesons per annihilationact). During contact of a cluster of matter with a cluster of antimatter,the annihilation of particles and antiparticles would occur at the contactsurface, and the 7r-mesons emitted from both sides of the surface wouldcreate a macroscopic pressure called the annihilation pressure. This wouldlead to separation of the media. These processes could proceed in thehadron era when the density of substance by far exceeded that of nuclearmatter. However, this model suffers from various difficulties yet to beresolved.


7.2 Creation of Atomic Nuclei

After the creation of protons and neutrons, nucleons — not quarks —became the fundamental "bricks" of the Universe. Atoms also include elec-trons, and there are also fluxes of photons and neutrinos in the Universe.Gradually, during the expansion and cooling of the Universe, complex nu-clei began to be formed from the nucleons, and an increasingly importantrole was played by different nuclear reactions. These not only led to thecreation of new nuclei, but also became an energy source. The origin ofstellar and solar energy can be explained exactly in this way. An impor-tant role in the creation of light and medium nuclei (up to the elements ofthe iron group, inclusively) was played by nuclear reactions with chargedparticles, and in the creation of heavier nuclei — by neutron capture and/3-decay.

Thus, a question of a paramount importance for studying the creation ofatomic nuclei is the relation between the quantities of protons and neutronsat the early stage of evolution of the Universe. At sufficiently high temper-atures nucleons interacted with leptons, i.e., their mutual transformationstook place:

p + e~ <—> n + ue, (7.1)

p + Pe <—> n + e+. (7.2)

Reactions (7.1) and (7.2) defined the equilibrium between proton and neu-tron concentrations. At T > 1011 K the proton and neutron concentrationswere approximately the same. As a result of the expansion and cooling ofthe Universe, there were more and more protons and fewer and fewer neu-trons since the neutron mass is somewhat greater than the proton mass,and proton creation in the reactions (7.1) and (7.2) is energetically morefavorable than neutron creation.

The velocities of reactions (7.1) and (7.2) rapidly decrease as tempera-ture decreases, so these reactions almost completely ceased seconds after theBig Bang. Then, the relative concentration of neutrons in nucleon matterwas around 15%, which denned the future ratio between the concentrationsof hydrogen and helium in Nature.

Let us now consider the processes for creating complex nuclei. Thesimplest complex nucleus is a deuteron, which consists of a proton and a


neutron. This could be created via the fusion of a proton and a neutron:

p + n—> d + 7, Q = 2.225 MeV. (7.3)

It could also be created from the fusion of two protons, with emission of apositron and a neutrino:

p + p—>d + e+ + v, Q = 1.19 MeV. (7.4)

Because of the Coulomb barrier, the latter reaction can proceed only athigher temperatures than the reaction (7.3). During the further captureof neutrons by deuterons, 3H nuclei were created. Under the /?-decay of3H nuclei, 3He nuclei were created, which captured one more neutron andtransformed into 4He nuclei. However, it is more probable that the 3Henucleus, capturing a neutron, would decay into two deuterons than createan a-particle. Nevertheless, a fraction of the 3He nuclei was transformedinto a-particles via neutron capture.

4He nuclei could also be created in another way, namely, through captureof a proton by a deuteron with creation of a 3 He nucleus and further fusionof two 3He nuclei:

d + p—> 3He + 7, Q = 5.49 MeV, (7.5)

3He+3He—+ 4He + 2p, Q = 12.85 MeV. (7.6)

This possibility plays a fundamental role in the origin of the energy of stars.Together with the fusion reaction of two protons into a deuteron, it formsthe proton-proton cycle. Here the transformation of protons into 4He nucleioccurs as follows: six initial protons are transformed into an a-particle, twofree protons, two positrons, and two neutrinos, i.e., hydrogen "burns out"creating helium. In each cycle, 26.21 MeV of energy is released.

The creation of helium was finished approximately 100 s after the BigBang. At that time, matter in the Universe consisted of 70% protons and30% 4He nuclei. This composition of the Universe remained unchangeduntil the processes of synthesis of heavier nuclei started once stars formed.The ratio between the concentrations of hydrogen and helium nuclei inNature defined the further synthesis of various nuclei. If this ratio weredifferent, the relative concentrations of various elements in the Universewould be different as well. The whole world would be quite different. Thecorrect prediction of the relative abundance of helium nuclei in the Universe


is the second (after relict radiation) great achievement of the model of theBig Bang.

The further creation of nuclei by neutron capture is impossible, as the5He nucleus does not exist. Indeed, the 5He nucleus decays into two partsimmediately after its formation, which creates a insurmountable barrier forthe synthesis of heavier nuclei by the successive capture of neutrons bynuclei. The 5Li nucleus, another candidate for a nucleus with A = 5, is alsounstable. The same difficulty is also encountered further, since there is nostable nucleus with mass number A = 8.

The abundances of Li, Be, and B nuclei in Nature are rather low (Fig.7.1); nevertheless, they are too high to be consistent with the synthesis ofthese nuclei directly inside stars. Indeed, these elements have rather smallbinding energy. For this reason they would be destroyed faster within stars,the higher the temperature of the star. These nuclei, apparently, werecreated partially from stellar explosions and partially in the cold matterfrom fission of different nuclei under the action of cosmic rays. Now wehave to explain how the carbon nuclei 12C could be synthesized.

If the density of 4He nuclei is very large, then fusion of three a-particlesinto a 12C nucleus can occur. The probability of this process is muchhigher than that for triple collisions of a-particles, since it usually proceedsthrough the fusion of a-particles which forms an excited 8Be* nucleus viathe resonance level of the latter.

The reaction a + a —> 8Be is impossible, since 8Be in its ground state isunstable. However, at the energy E ~ 0.1 MeV there exists an excited stateof 8Be*, whose lifetime is T ~ 10~16 s. Although this time seems small,it is large on the nuclear "scale" because the characteristic nuclear time isTnuc ~ 10~22 s. In other words, in an excited 8Be* nucleus two a-particleshave time to perform about 106 oscillations relative to each other beforethey fly apart. This is time enough for a third a-particle to approach themto form a 12C nucleus.

The 4He nucleus plays an important role in the synthesis of heaviernuclei, as well as in the synthesis of the 4He nuclei themselves. Along withthe proton-proton cycle described above, the carbon-nitrogen cycle is alsoessential; it consists of the following reactions:

p+ 1 2 C ^ 13N + 7, Q = 1.95 MeV, (7.7)

1 3 N _ ^ i3C + e+ + l/) Q = 1.50 MeV, (7.8)


p+ 13C —-+ 14N + 7, Q = 7.54 MeV, (7.9)

p+ " N __» i5O + 7 , Q = 7.35 MeV, (7.10)

15O—> isN + e+ + I/) Q = 1.73 MeV, (7.11)

p + 15N _ » 12C + 4He, g = 4.96 MeV. (7.12)

In this cycle, 14N nuclei are created which were absent otherwise and arethe most abundant odd-odd nuclei in Nature. As a result of the carbon-nitrogen cycle, four protons transform into an a-particle, two positrons,and two neutrinos. In this case the number of 12C nuclei is unchanged;i.e., in this cycle, carbon is a catalyst and is not consumed. In the carbon-nitrogen cycle 25.03 MeV of energy is released. This cycle is also a sourceof stellar energy. Since the probabilities of all reactions entering into theproton-proton and carbon-nitrogen cycles are essentially dependent on thestellar temperature, the total energy yield due to these nuclear processesessentially depends on the stellar temperature as well. At temperatures of15-20 million °C, both cycles lead to approximately the same energy release.At lower temperatures, the dominant cycle is the proton-proton cycle, andat higher temperatures the carbon-nitrogen cycle dominates. These cyclesof stellar energy production were established by Bethe in 1938, and in 1967he was awarded the Nobel Prize in Physics for this discovery and for hiscontribution to the theory of nuclear reactions.

After nitrogen, the oxygen nucleus 16O follows, which is formed underthe capture of an a-particle by a 12C nucleus. Within the Sun and mainsequence stars, the most abundant elements among those heavier than he-lium are carbon (0.39%) and oxygen (0.85%). The human body consistsof 18% of carbon and 65% of oxygen by mass (the remainder is largelyhydrogen). For this reason, the determination of the ratio of abundancesof 12C and 16O nuclei created during the consumption of helium is a veryimportant task. This ratio is defined, first of all, by the relative rates ofthe reactions 3a —> 12C and 12C + a —> 16O + 7. If the first reaction willproceed much more rapidly than the second, then very few oxygen nucleiwill be formed by helium consumption. In the opposite case, there will bevery few carbon nuclei.

The ratio of rates of the reactions under consideration and, as a con-sequence, the ratio of the abundances of 12C:16O nuclei in Nature, are


defined by the energy level structure of these nuclei. It is only the 12C nu-cleus, which has a level (excited state) with energy E = 7.656 MeV and spin7 = 0, which is situated somewhat higher than the sum of the rest energiesof 8Be and 4He nuclei (E = 7.370 MeV) and than the total rest energyof three a-particles (E = 7.277 MeV). This means that a 12C nucleus canbe created in the course of a resonance reaction. If the 12C nucleus hadno favorably situated resonance level, then the rate of creation of carbonwould be much less than in the presence of such a level. Thus, the existenceof favorably situated resonance levels of 8Be* and 12C* nuclei has led tothe sufficiently high concentration of carbon nuclei in Nature.

Now let us consider the 16O nucleus. If this, like the 12C nucleus, had afavorably situated resonance level, then the synthesis of 16O nuclei by thecapture of a-particles by 12C nuclei would proceed so rapidly that therewould be very little carbon in Nature. In reality, the 16O nucleus has levelswith energies E = 6.05 MeV (I = 0) and E = 6.92 MeV (I = 2) which areclose to the sum of the rest energies of the 4He and 12C nuclei (E = 7.16MeV). However, these levels are situated lower than the sum of the restenergies of the 4He and 12C nuclei, so that in reality the resonance captureof a-particles by carbon nuclei does not occur.

Thus the evolution of stars, which leads to the synthesis of heavier nucleiand the existing ratio between the quantities of carbon and oxygen nucleiin Nature, is denned by three circumstances: (1) the instability of the 8Benucleus in its ground state and the presence of the resonance level of the8Be* nucleus, (2) the existence of the favorably situated resonance level inthe 12C nucleus, and (3) the absence of such a favorably situated resonancelevel in the 16O nucleus.

In the further evolution process, the absorption of a-particles by 16Onuclei and by heavier nuclei makes it possible to explain the creation of20Ne, 24Mg, 28Si, and other nuclei. Nuclei with the mass numbers lyingbetween the mass numbers of 16O, 20Ne, 24Mg and so on could be formedin the neutron and proton capture reactions. However, along with theseprocesses, the creation of nuclei with A > 20 could also occur as a result ofthe nuclear fusion reactions. At temperatures T « 2 - 1 0 9 K, carbon fusioncould take place:

f 20Ne + a,i 2 C + i 2 C — • ) 23Na + P ) (7.13)

i23Mg + n.

This cycle of reactions makes it possible to explain the abundances of nuclei


with mass numbers 20 < A < 32. The creation and abundances of nucleiwith mass numbers 32 < A < 42 can be explained by oxygen fusion, whichcould occur at temperatures of about 3.6 • 109 K:

C 28Si + a,1 6O+ 1 6 O ^ I 31P + p, (7.14)

[31S + n.

The fusion of neon and silicon allows one to explain the creation of nucleiup to nickel. In this way, one can explain the synthesis processes for thenuclei up to the iron group elements A < 65. However, the binding energyper nucleon reaches its maximum for nuclei of this group, and then begins agradual decrease. For this reason, the iron group elements cannot play therole of a fuel; that is, fusion with energy release ceases as soon as elementsof this group have been created. This explains the sufficiently large relativeabundance in Nature of the nuclei having charge numbers close to that ofthe iron nucleus. In other words, the elements of the iron group are the"nuclear ashes" that have been formed during the fusion of stellar matterconsisting of lighter nuclei.

However, in Nature there exist stable nuclei with mass numbers up toA — 209, and natural radioactive nuclei up to A = 238. It should benoted that the relative abundances of nuclei of the heavy elements arevery small. A typical abundance for the heavy nuclei is 1010 times lessthan the abundance of hydrogen! The extremely low abundances of heavynuclei attest to the accessory nature of their synthesis processes. Theseprocesses could be reactions of neutron or proton capture by nuclei of theiron group as well as by the heavier nuclei being formed. These nucleoncapture reactions could alternate with the /3-decays of nuclei. The nucleoncapture processes go on until neutrons and protons are created in the starsdue to burning or explosion. As soon as the reactions in which energy isreleased cease, the synthesis of heavy nuclei also ceases. The mechanismsof nuclear synthesis described above allow one to calculate the relativeabundances of isotopes and to explain the main features of the curve shownin Fig. 7.1.

7.3 Evolution of Stars

All available data on the observation of galaxies and stars show that theUniverse on a sufficiently large scale is homogeneous. That is, on the average


it looks the same regardless of where and in which direction it is observed.In other words, on the scale of hundreds of millions of light years, theUniverse looks as if it is homogeneously filled with billions of galaxies, thedistances between which are, on the average, equal to several million lightyears (1 light year is « 0.95 • 1016 m — the distance traveled by lightduring one year). The large-scale homogeneity of the Universe leads to thecosmological principle.

In 1929, the American astronomer Hubble discovered that any twogalaxies in the Universe are moving apart with velocities proportional tothe distance between them (the Hubble law). For example, a galaxy 108

light years away is moving away from us with a velocity of about 2 • 106

m/s. A galaxy twice as far away is receding with twice the velocity, etc. Itfollows that the Universe is expanding, i.e., it is not stationary. Hubble'sdiscovery has finally demolished the notions about the static and stableUniverse that were accepted wisdom from the time of Aristotle.

Considering the reverse process, we can conclude that approximately15-20 billion years ago all galaxies (all the substance in the Universe) weregathered at one point: "the singularity". This is exactly the time of creationof the Universe by the Big Bang.

Before the creation of the Universe, i.e., before the moment of the BigBang, the concept of time was meaningless. The Big Bang is the origin oftime; earlier time moments are simply not defined, because the concept oftime is connected with the motion of matter. This circumstance, that theconcept of time is connected with the creation of the Universe, was firstpointed out by St. Augustin (354-430), the bishop of the city of Gippon inNorth Africa, in his work "About God's town".

At present, space is filled with galaxies united into gigantic clusterscomprising hundreds and thousands of galaxies. The formation of galaxiesstarted approximately a billion years after the Big Bang, out of the mixtureof hydrogen and helium that filled all space. Our galaxy consists of about1011 stars, and the Sun is an unremarkable example of a peripheral starwhose age is about 4.6 • 109 years.

The synthesis of atomic nuclei takes place in stars via nuclear reactionsproceeding at very high temperatures. For this reason, these processes andthe explanation of the abundances of nuclei in Nature on the basis of themare closely connected with the problem of the structure and evolution ofstars.

The gravitational forces inside a star tend to decrease its volume, andthe gas pressure inside the star counteracts this. Since these pressure and


temperature values are huge, atoms in stars are completely ionized; thematter inside stars, which consists of free electrons and nuclei, is a plasma(for example, at the center of the Sun the pressure is around 2 • 1010 atm,and the temperature is equal to 1.4 • 107 K). The huge pressure inside astar is sustained by the energy released during nuclear reactions.

A star is in the state of equilibrium where gravitational and internalpressures are mutually balanced while nuclear reactions of hydrogen fusionoccur (the proton-proton and carbon-nitrogen cycles). However, at somepoint, the hydrogen "fuel" will be used up. Then the internal gas pressurewill start to decrease and gravitational compression will begin; this willagain lead to an increase of pressure and temperature inside the star. Atcertain sufficiently high temperatures, new nuclear reactions initiate (e.g.,helium fusion) and equilibrium will be reached again. In this case, thesynthesis of new nuclei will occur. In other words, in the process of stellarevolution, the stages of burning and compression alternate.

The burning process can proceed calmly, as with the Sun, or it can beaccompanied by an explosion (the explosions of supernovas). The energyradiated by the Sun is approximately 3.8 • 1026 J/s. For this, about 630tons of hydrogen must burn every second. Owing to the synthesis of heliumfrom hydrogen, the Sun will continuously radiate this energy during next 5-7 billion years. Then, due to the exhaustion of its hydrogen stocks, heliumburning will start up. The Sun will quickly transfer into the red giantstage, and in only ten thousand years after this it will transform into awhite dwarf.

After the end of burning of light nuclei leading to the creation of nucleiup to the iron group, stellar evolution enters a new stage. Due to gravita-tional compression, the star can transform into a black hole, white dwarf,or neutron star, or it can decay totally. The final result in this case willdepend on the initial mass of the star, which is the most essential parametercharacterizing its dynamics. With a stellar mass twice as large its lumi-nosity is almost thirty-fold greater, and with a stellar mass half as large,its luminosity is thirty-fold lower. There exist stars whose masses are morethan ten-fold greater that of the Sun. Such a star shines like a million Suns,but loses its energy relatively soon because the nuclear fuel in its centralportion burns down rapidly.

If the initial mass of a star does not exceed four solar masses (the massof the Sun is 2 • 1030 kg), then the star is slowly compressed until the densityat its center reaches 1010-10U kg/m3 and its surface temperature becomes104 K. Such a star is called a white dwarf. Its size will be comparable to


that of the Earth if its mass is comparable to that of the Sun. A white dwarfis a "cold" star, since its temperature is insufficient for nuclear synthesisreactions to occur.

If the initial mass of a star exceeds four solar masses, then it explodes viagravitational compression (the supernova outburst). Supernovas are starsthat suddenly explode, become very bright for several months, and thengradually die away. Supernovas are explosions classified by released powerof 1034 J/s. This huge energy is over two orders of magnitude greater thanthe released power of the next category of exploding object — the novas— and is approximately 2.5 • 107 times greater than the power releasedfrom the Sun. Supernova outbursts are the most powerful star explosionsin Nature. Investigation of supernovas is important for the elucidation ofthe evolution of stars, of the origin of cosmic rays, and of the formation ofatomic nuclei of different elements.

According to numerous estimates, approximately one supernova occursevery 30 years in a giant galaxy. The giant galaxy closest to1 us is the An-dromeda nebula situated 2.2 million light years away. This enormous dis-tance complicates the observation of supernovas outbursts in this galaxy.In our galaxy a supernova happens, on average, once every 28 years. Inrecorded history, three very bright supernovas have been observed in ourgalaxy. Their appearance in 1006 and 1054 was described in Chinese chron-icles, and in 1604 the Kepler star was observed. After 1604, more than 600supernovas have been discovered, but all were distant and weak.

The event of the century for astronomers and physicists was the brightoutburst of a supernova in 1987 in the Big Magellanic Cloud, a young galaxywhere the active creation of stars is ongoing. This supernova is relativelyclose to us, only 150000 light years away (the radius of our Galaxy is about50000 light years). The supernova discovered in 1987 resulted from theexplosion of a star whose radius was equal to 30-60 solar radii. Its masswas 10-30 times the solar mass.

After an explosion of a supernova, an inactive stellar nucleus remains;this can become a neutron star, decay completely, or transform into a blackhole. Here the deciding factor is the mass of the stellar nucleus remainingafter the explosion of the supernova. If its mass is less than the critical mass,approximately 1.7 solar masses of the Sun (known as the Chandrasekharmass), then the stellar nucleus will reach its equilibrium state when itsradius will be about 104 m, and the density at its center will be about thesame as the nuclear density, 1018 kg/m3, or even greater. This is a neutronstar or pulsar. If the mass of the stellar nucleus is less than the critical


mass, gravitational compression leads to the formation of a black hole.Only one of five supernovas typically gives rise to a neutron star. A neu-

tron star is formed when, due to gravitational compression, a star of normalmass reaches such density that electrons acquiring very large energies beginto be captured by protons: p + e~ —> n + ue. A neutron star consists of theneutrons so created, which are bound together by gravitational forces.

The existence of neutron stars was predicted in the mid-1930s. How-ever, there was little hope of observing one. The discovery of neutron starsbecame possible after the development of radioastrophysics. In 1967, sci-entists from Cambridge University in England discovered a new class ofcelestial objects, situated beyond the solar system, that radiate periodicradio signals. These objects were called pulsars, even though they do notactually pulse but rather rotate very rapidly. For the discovery of pul-sars, the English radio astronomer Hewish was awarded the Nobel Prize inPhysics for 1974.

Now, more than 100 pulsars are known. Each has its own radiationperiod in the radio frequency range. Periods of all the known pulsars liewithin the range from 0.03 s to 4 s, and over time they gradually increase.There is a pulsar with the very short period of 0.033 s in the Crab nebulawithin the constellation Taurus, where in 1054 Chinese astronomers ob-served an extraordinarily bright supernova. The supposition is that thispulsar arose as a consequence of the observed supernova.

At present, the Crab nebula is an object consisting mainly of gas andoccupying space with a diameter of about 7.5 • 1013 m. It is expanding witha velocity of around 1.1 • 106 m/s.

Pulsars are identified as neutron stars, and their periods are the periodsof rotation of the neutron stars. The increase in the period of a pulsaris caused by an energy loss. The loss of rotational energy of the pulsarobserved in the Crab nebula is a quantity of about the same size as thetotal energy radiated by this nebula. So the pulsar is the main sourceof the energy radiated by the huge Crab nebula. Note that pulsars wereobserved not only in the radio frequency range: there exist objects thatperiodically radiate visible light.

Pulsar radiation is apparently caused by the spiral motion of relativisticcharged particles in the strong magnetic field of the star, which are thrownaway from the rapidly rotating equatorial regions of the pulsar. The asym-metry of the magnetic field relative to the rotation axis leads to radiationemission from certain regions of the pulsar. On Earth, the directed fluxesof radiation rotate together with the star. This phenomenon is similar to


the rotating beam of a searchlight. The huge mass of the pulsar leads toits nearly constant period of radiation registered on Earth.

To calculate time variation of the rotation period of a pulsar, one needsto know its moment of inertia. For this, one requires a sufficient model ofthe structure of a neutron star. It is assumed that such an object has anatmosphere several meters thick. Since the pressure in the external layer isclose to zero, the most stable atoms under these conditions are those of iron,56Fe. Their state having least energy is a crystal lattice. For this reason,the external layer of a pulsar is a "hard crust" consisting of a thin layer ofiron atoms. When moving to the center of a neutron star, the density ofmatter continuously increases. Under the crust of iron a plasma lies — afluid consisting of electrons and nuclei. The density of this layer is about107 kg/m3, and its thickness is about 103 m.

If the density of matter of a pulsar reaches 109-1010 kg/m3, thenneutron-enriched nuclei are formed. Due to the specific conditions insideneutron stars, these nuclei cannot experience /3-decay and are, for this rea-son, stable. When the density exceeds 4 • 1014 kg/m3, a neutron "liquid"begins to form inside the star. With further increase in density, p > 2.5-1017

kg/m3, a continuous liquid consisting of neutrons, protons, and electronsis created, with protons forming only 4% of this liquid. Thus, inside apulsar the major fraction of particles consists of neutrons, hence the name"neutron star". If the density of matter within a pulsar exceeds the nucleardensity, then at the center of the star the creation of mesons and of quitea number of other elementary particles becomes possible.

Finally, let us note that the theory of the Big Bang and the expandingUniverse is based on classical physics augmented by relativity theory. Tak-ing account of quantum effects can essentially alter this elaborate picture ofthe evolution of the Universe. However, the quantum theory of gravitationis not developed yet and it is difficult to say how this picture will change.

7.4 Controlled Nuclear Fusion

The stocks of subterranean coal, oil, and gas are limited. Their combus-tion leads to environmental pollution. For this reason, mankind urgentlyrequires other energy sources. Typical nuclear reactors in which energy is aproduct of fission through a chain reaction of heavy nuclei have two essen-tial shortcomings: first, they yield large amounts of radioactive waste whosedisposal is a difficult issue; second, they lead to thermal pollution. Conse-


quently, the idea of using controlled nuclear fusion for energy productionthrough fusion of light nuclei appears so attractive.

Nuclear fusion proceeds inside stars at very high temperatures (energies)reaching hundreds of millions of degrees. During the collision of two nuclei,fusion can occur if the approach distance is around 10~15 m, where thenuclear forces of attraction already act. For this they must surmount aCoulomb barrier of height B = Z\Z2e2 / (R\ + R2), where Z\, Z2 and R1:

R2 are the charge numbers and radii of the colliding nuclei. In order forthe nuclei to fuse, their energy should exceed the height of the Coulombbarrier: E > B. Since for the collision of light nuclei, the Coulomb barrierheight is about B ~ 0.1 MeV, fusion reactions can proceed at temperaturesT ~ 109 K. Such reactions proceeding at very high temperatures are calledthermonuclear reactions.

At the temperatures necessary for thermonuclear synthesis, matter isin a state of completely ionized plasma which is a mixture of nuclei andelectrons. Plasma is the fourth state of matter; it exists at very high tem-peratures T > 104 K. Plasma becomes completely ionized at T > 107 K.

The particles that compose the plasma have very large kinetic energiesand tend to fly apart from one another. For this reason, the problem ofplasma confinement in a certain restricted volume arises. Since the temper-ature is so high that no material can remain in the solid state, the problemof plasma confinement is complicated by the necessity to isolate it from thewalls of the enclosure (the fusion reactor) that confines the plasma.

To obtain controlled, self-sustained fusion reactions of light nuclei,termed controlled nuclear fusion (CNF), it is necessary that the plasmadensity and temperature are sufficient during a long interval. Then, agreater amount of energy could be produced from CNF than will be neededfor the heating and confinement of the plasma. Furthermore, it is neces-sary to take into account the energy losses due to heat leakage from thewalls of the fusion reactor, brake radiation of electrons, and other effects.One should also remember the huge plasma pressure upon the walls of thereactor. It must not exceed more than a few hundred atmospheres, whichcan be resisted by the construction materials.

The nuclear fusion reaction intensity is defined by the confinement pa-rameter nr (n is the plasma density, r is the time of plasma confinement).A fusion reactor will yield an energy gain, i.e., it will have a positive effi-ciency, upon fulfillment of a criterion established by the English physicistLawson in 1957. For a deuterium-tritium plasma in which the following


reaction proceeds

2 H+ 3H—-> 4He + n, Q = 17.6 MeV, (7.15)

the Lawson criterion reads:

TIT > 1020 s/m3, T ~ 108 K. (7.16)

Huge stocks of deuterium 2H exist on our planet. These can be obtainedfrom the electrolysis of water, since the content of deuterium in water is0.015%. Tritium 3H is unstable. It experiences the /3~-decay with ti/2 =12.2 years, and is therefore absent in Nature. It is usually obtained fromthe reaction

n+ 6Li—> 4He+ 3H, Q = 4.8 MeV. (7.17)

For energies E < 0.2 MeV (T < 2 • 109 K), the dependence of theprobability of the reaction (7.15) for synthesizing 4He nuclei on energy isdetermined by the Gamov formula

wG(E) = -expl-]J-£\. (7.18)

Here C is a constant, EQ = 2TT2Zi^e4\ij'h2, and fi = m-\rn.2/(jn\ + m?)where mi and m^ are the masses of deuterium and tritium.

At high temperatures T, the energies of nuclei in plasma are not identi-cal, but are distributed statistically according to the Maxwell distribution

f(E) = ^expf—j. (7.19)

For this reason, the total probability w(E) of reaction (7.15) for the syn-thesis of 4He nuclei is defined by the product of the quantities WQ [E) andf{E):

v,(E) = ZexP(-yp§-if\. (7.20)

The probability w(E) reaches its maximum at the energy Em =(£0T2/4)1/3. The function w(E) for the reaction (7.15) is shown in Fig. 7.2.Since Eo « 0.02 MeV and T « 108 K, the quantity w(E) has its maximumat the value Em = 0.064 MeV. At E < Em the probability w(E) decreases


w(E) | -

0 E F

Fig. 7.2 Probability of the reaction of synthesis of 4He nuclei as a function ofenergy.

with decreasing energy, and at E > Em it decreases with increasing E. AtT « 108 K the probability of reaction (7.15) is maximum.

To realize CNF, it is necessary to solve two problems: heating theplasma to T ~ 108 K, and confining the plasma at this temperature longenough for an appreciable part of the deuterium and tritium nuclei tomerge. For plasma confinement, strong magnetic fields of special config-uration (magnetic traps) are employed, and for heating the plasma one canuse, for example, radio frequency methods. These are based on the interac-tion of different types of electromagnetic waves with plasma. Wave energy,under certain conditions, is efficiently absorbed by charged particles; thisleads to heating of the plasma.

A magnetic trap usually takes the form of a torus in order to preventplasma from escaping along the magnetic field lines. Then the magneticfield lines are circles, and the charged particles must move along spiralswound around these circles. In reality, it is impossible to create a completelyuniform toroidal magnetic field. For this reason, particles moving along thefield lines will gradually shift (drift) perpendicular to the magnetic field,and this will result in the plasma rejection upon the outer wall of thechamber.

This shortcoming can be removed in special systems called tokamaksand stellators. These devices create magnetic field configurations that sup-press charged particle drift. Hence, plasmas in tokamaks and stellatorscan be confined for a sufficiently long time. The theory of toroidal mag-netic confinement of plasmas has been developed by the Soviet physicistsTamm and Sakharov, who proposed a tokamak in 1951. In the same year,


the American physicist Spitzer proposed a stellator. Besides tokamaks andstellators, there also exist traps of other types for plasma confinement (theadiabatic, ambipolar, and gas dynamic traps, etc.).

In the early 1970s, tokamaks were built in countries carrying out in-vestigations on CNF. These studies have shown that the time of plasmaconfinement in these devices increases with their sizes. Construction ofvery large tokamaks is complicated and expensive. For this reason, it isnecessary to join the efforts of many countries. Note that along with themethodology for CNF in toroidal systems, there also exist other possibilities(the pulse laser heating of an ice grain consisting of a mixture of deuteriumand tritium and having a diameter of several millimeters, plasma heatingby a high-current beam of relativistic electrons, etc.).

If the energy of the fusion of light nuclei is released very rapidly, then itwill not be CNF — but an explosion. The hydrogen bomb is based on thisprinciple. During a fusion explosion, the reactions (7.15) and (7.17) occur.These mutually maintain each other, keeping unchanged the numbers ofneutrons and 3H nuclei. During the explosion of a hydrogen bomb, theenergy of the fusion of light nuclei is released for a very short time intervalof about 10~6 s. In this case, to prevent plasma from a premature escape, itis enough to use a hard shell casing for the bomb. The preliminary heatingof plasma up to the ionization temperature of 107 K is accomplished byexplosion of a nuclear fission bomb. The explosion of a hydrogen bomb, incontrast to CNF, is a thermonuclear reaction of nonstationary character.

Index

accelerators, 138 bosons, 14acceptor impurities, 34 bottom quarks, 63actinium, 164 breakdown voltage, 34actinoids, 25 Breit-Wigner formula, 124alpha-decay, 98 bubble chamber, 146americium, 163amorphous solids, 31 calcium, 109Andromeda nebula, 179 californium, 164annihilation pressure, 170 carbon, 174antiferromagnetic material, 37 carbon dating, 111antimatter, 170 carbon-nitrogen cycle, 173Aristotle, 41 caustic, 136astatine, 166 chain reaction, 155asymptotic freedom, 61 Chandrasekhar mass, 179atomic mass unit, 69 charm quarks, 62atomic structure, 22 charmed particles, 62

Chernobyl, 159Balmer series, 7 classical physics, 1baryon asymmetry, 170 closed shell, 23baryons, 60 cold emission, 31basic groups, 25 cold neutron, 156beam collimation, 142 collider, 144berkelium, 164 color, 60beta-decay, 102 compound nucleus, 120Big Bang theory, 167 conduction band, 33black body radiation, 2 conduction electron, 33black hole, 179 conductivity, 31Bohr magneton, 75 confinement, 61Bohr orbit, 22 confinement parameter, 182Bohr, Niels, 5 conjugate pair, 12Bose condensation, 15 conservation laws, 115Bose-Einstein statistics, 14 controlled fusion, 181

187


controlled nuclear fusion, 182 159Cooper pair, 39 Einstein, 43correspondence principle, 20 Einstein, Albert, 4cosmological principle, 177 einsteinium, 164Coulomb's law, 51 electromagnetic interaction, 51covalent bond, 27 electron configuration, 24critical regime, 160 electron shell, 23critical size, 157 electrostatic unit, 51crystal, 31 endoergic reaction, 118Curie temperature, 37 endothermic reaction, 118Curie, M., 89, 164 energy of reaction, 118Curie, P., 90, 164 energy spectrum, 5curium, 164 energy-momentum tensor, 46

equivalent electrons, 23damping width, 126 europium, 163daughter nuclei, 93 exciton number, 125de Broglie wavelength, 9 exclusive reaction, 116de Broglie, Louis, 8 exoergic reaction, 118decay constant, 91 exothermic reaction, 118deep inelastic transfer, 126 expanding universe, 167degeneracy, 29degenerate Fermi-gas, 76 fast neutron, 156delayed fission, 153 Fermi energy, 33, 78delayed neutron, 152 Fermi, E., 160, 162detector, 17 Fermi-Dirac statistics, 14deuterium, 183 fermions, 14deuteron, 47, 75, 171 fermium, 164diamagnet, 36 ferrimagnetic material, 37dielectrics, 31 ferromagnetic material, 37diffraction scattering, 129 ferrum series, 25diode, 35 field concept, 51dipole radiation, 106 Dirac, Paul, 10 fission,147direct nuclear reaction, 124 dispersion, 12 dispersion formula, 124 domain, 38 forbidden region, 32donor impurity, 34 francium, 166door-way state, 125 Fraunhofer diffraction, 129double nuclear system, 127 free electrons, 31down-quarks, 60 Fresnel diffraction, 133drift, 184 fundamental interactions, 41duants, 141 fusion, 181dubnium, 165

Galileo, 8effective multiplication coefficient, gamma-radiation, 106

final

fission neutron, 147fission barrier,150

fission parameter, 152

final channel, 116

Index 189

gauge bosons, 64Geiger-Muller counter, 145Geiger-Nuttall law, 101general relativity, 43generations, 53giant dipole resonance, 85global exclusions, 13glory, 138glueballs, 62gluons, 61gravitational interaction, 41gravitational mass, 43gravitational waves, 50graviton, 50gyromagnetic factor, 75

hadrons, 59half-life, 92heat of reaction, 118Heisenberg, Werner, 10helicity, 53heteropolar bond, 27hierarchy of configurations, 126hole, 125holes, 34homopolar bond, 27Hubble law, 177

inclusive reaction, 117inert gases, 25inertial mass, 42infinite multiplicative system, 156initial channel, 116interference, 19intermediate bosons, 54intermediate neutron, 156internal conversion, 106iodine, 109ionic bond, 27isobaric nuclei, 69isobars, 69isomer, 106isomers of shape, 154isotones, 69isotopes, 69, 109

K-capture, 91Kirchhoff, Gustav, 2

Lamb shift, 16Lamb, Willis Eugene, 16lambda point, 40lanthanoids, 25Laplace determinism, 16Laplace, Pierre-Simon, 16lawrencium, 165Lenz's law, 36leptons, 53light year, 177linearity, 11Lyman series, 7

Mossbauer effect, 107magnetic substance, 36magnetic trap, 184mass, 42mass defect, 69mass excess, 69mass number, 68mass spectrometer, 70Maxwell distribution, 183Maxwell, James Clerk, 51mean free path, 100Meisner effect, 39Mendeleev, 25mendelevium, 165mesons, 60, 62metals, 31

Minkowski space, 48moderator, 156multiplication coefficient, 157multistep nuclear reaction, 125muon atom, 73

Neel temperature, 37neon, 176neptunium, 162neutron absorber, 159neutron star, 179Newton, Sir Isaac, 2nickel, 176nitrogen, 174


nobelium, 165 principle of equivalence, 43nuclear fission, 147 promethium, 166nuclear interaction, 57 proton-proton cycle, 172nuclear isomery, 106 pulsar, 179nuclear magneton, 75nuclear optics, 127 quadrupole radiation, 106nuclear reaction, 115 quanta, 3nuclear reaction mechanisms, 119 quantization, 9nuclear reactor, 159 quantum chromodynamics, 61nuclei, 67 quantum electrodynamics, 52nucleons, 57 quantum measurementnuclide, 69 irreversiblity of, 18

quark, 59octupole radiation, 106 quasifission, 127Oersted, Hans Christian, 52 quasiparticle, 39, 87optical model, 129optical potential, 128 radio frequency methods, 184overthermal neutron, 156 radioactivity, 89, 90oxygen, 174 radium, 90

rainbow ghost, 138palladium series, 25 rainbow maximum, 137paramagnet, 36 rainbow scattering, 135parent nuclei, 93 rare-earth elements, 25parity, 56 Rayleigh, Lord, 2partial width, 123 Rayleigh-Jeans law, 2particle detectors, 145 reactivity, 159Paschen series, 7 recombination, 34Pauli exclusion principle, 14 rectifier, 35Pauli, Wolfgang, 14 relict radiation, 169permissible orbits, 6 residual interaction, 87phonon, 39 residual magnetism, 38phosphorus, 109 rest energy, 44photoelectric effect, 2, 4 rule of intervals, 84photoemulsion, 146 Rutherford, Ernst, 4photofission, 155 rutherfordium, 165photon, 4pion, 54 saddle point state, 152Planck's constant, 3 saturation, 59Planck, Max, 2 scale invariance, 8planetary model, 4 scaling parameter, 135plasma, 182 scattering, 67platinum series, 25 Schrodinger equation, 6, 11, 15plutonium, 162 Schrodinger, Erwin, 10polarization, 133 scintillation counter, 145polonium, 90 secondary neutron, 152pre-equilibrium nuclear reaction, 125 secular equation, 94

Index 191

self-consistent field, 23 transition groups, 25semiconductor, 34 transuranium elements, 162semiconductors, 31 tritium, 183semimetals, 34 tunnel effect, 31shell model, 76 tunneling passage, 100silicon, 176similarity, 8 ultracold neutron, 156slow neutron, 156 ultraviolet catastrophe, 2solar curve, 167 uncertainty relation, 12Sommerfeld parameter, 134 under barrier passage, 100spark chamber, 146 up-quarks, 60spatial homogeneity, 8 uranium, 89, 147-149, 157, 162spatial isotropy, 8special relativity, 43 valence band, 33spherical harmonic, 28 Van de Graaff generator, 140spin, 14 Van der Waals forces, 64spinthariscope, 145 vector mesons, 59, 62spontaneous fission, 91 virtual photons, 53stars, 176stellator, 184 wave function, 10strange particles, 62 symmetry of, 14strange quarks, 62 wave-particle duality, 9stripping reaction, 132 weak interaction, 53strong interaction, 57 Weizsacker's formula, 71strontium, 109 white dwarf, 178subcritical regime, 160 Wien's law, 2superconductivity, 38 Wien, Wilhelm, 2supercritical regime, 160 Wilson chamber, 145superfluidity, 38, 86supernova outburst, 179 ylem, 168superposition principle, 11 yrast-states, 85synchrophasotrons, 143synchrotron, 142 zero-point vibrational energy, 15synthesized elements, 162

tachyons, 49technetium, 165, 167temporal homogeneity, 8thermal neutron, 156thermonuclear reactions, 182thin structure, 101threshold energy, 118tokamak, 184top quarks, 63trajectories, absence of, 13transistor, 36

18736823 quantum world of nuclear physics

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