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INTRODUCTION TOSUBATOMICPHYSICS

Chary RangacharyuluDepartment of Physics and Engineering Physics

University of Saskatchewan

Saskatoon, SKCanada, S7N5E2

[email protected]

and

Trang HoangDepartment of Nuclear PhysicsNuclear Engineering

University of ScienceHo Chi Minh city, Vietnam

[email protected]

December 2013


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Table of Contents

Chapter 1. Introduction .................................................................................................................................................... 3

Chapter 2. Rudiments of Nuclear Physics ....................................................................................................................... 7

2.1

Energetics .............. .......... .......... ........... .......... ........... .......... ........... .......... ........... .......... .......... ........... .......... ........... .. 7

2.2

Separation energies.................................................................................................................................................... 8

2.3

Time-dilation .............................................................................................................................................................. 9

2.4 Reduced Transition Probabilities ......... .......... ........... .......... ........... .......... ........... .......... ........... .......... .......... ........... 10

2.5 Cross section ............................................................................................................................................................ 11

2.6 Basic properties of nuclei and particles .......... ........... .......... ........... .......... ........... .......... ........... .......... .......... ........... 13

2.7

Parity .......... ........... .......... .......... ........... .......... ........... .......... ........... .......... ........... .......... ........... .......... ........... .......... 14

2.8

Magnetic dipole moment .......... ........... ........... .......... ........... .......... .......... ........... .......... ........... .......... ........... .......... .. 18

2.9

Electric Quadrupole Moments .......... ........... .......... ........... .......... ........... .......... ........... .......... .......... ........... .......... .... 20

Chapter 3. Identical Particles ......................................................................................................................................... 24

Chapter 4.

Isospin ............................................................................................................................................................ 27

Chapter 5.

Isospin Formalism of Two nucleon systems ............................................................................................... 29

Chapter 6. Hadron Structures ........................................................................................................................................ 33

Chapter 7.

Yukawa's theory of Meson Exchange Forces ............................................................................................. 40

Chapter 8. Weak Interactions ........................................................................................................................................ 41

Chapter 9. CP Violation .................................................................................................................................................. 49

Chapter 10.

Electro-Weak Unification ........................................................................................................................ 56


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Chapter 1. Introduction

This course aims to make you appreciate a) the breadth b) the fascination c) the challenges and d)

limitations of the subatomic physics research. We have to develop the background information, realize

the inter-connections of various sub-disciplines and bring the subject to a level where we can relate towhat is happening in the current-day research in this area. While this may sound like a survey course,

we want to maintain some rigour so that you develop some analytical reasoning and apply it toproblems at hand.

The main theme of this course is structures and symmetries in sub-atomic world. People are fascinated

by the symmetries and their implications to physics. People are determined to be able to say what the

ultimate building blocks of matter are. The reductionist principle of Democritus is the driving force forsuch an ambition. While I am convinced that the developments of the 20

thcentury particle physics and

quark-model, in particular, tend to prove that reductionism failed here, we will not elaborate on that.

We will pursue the logic, both mathematical and physics, to convey the present understandings and

challenges of the subatomic physics community.

We may say that subatomic physics began with Thomsons discovery of electron in 1897, even thoug hBecquerels discovery of radioactivity preceded by one year (1896). It was a few years beforeBecquerel and others realized that radioactivity was nuclear transmutations. By 1903, the alpha

particles were recognized to be doubly charged He-atoms. The discovery or identification of protons

was not so direct. In 1886, Goldstein published a paper describing his observation of Canal rays inaddition to the Cathode rays. It led to dedicated experiments by Wilhelm Wien (fame: Wiens

displacement law) and Sir John Joseph Thomson (discovery of electron) among others. It is fair to say

that nearly 20 years passed before people accepted the positive hydrogen ions as one of the basicconstituents of nuclei. Seemingly, Rutherford coined the name proton to this entity in 1919. It is

amusing to note that these observations revive the Prouts hypothesis, as early as in 1815, that all

atoms were built of hypothetical protyle atoms, which he tried to identify with hydrogen atom.

It is well known that Rutherfords scattering experiment, with alphas bouncing off materials, led to the

idea that the entire mass and positive charge of an atom is confined to a very small volume at the

center and it is called atomic nucleus, with charge Ze, where Z is the atomic number, indicating the

location of the chemical element in the Mendeleev table. It turns out that the masses of almost allatomic nuclei are larger than Z times the mass of proton and thus protons cannot be the sole inhabitants

of atomic nuclei. As electrons are very light compared to protons, their contribution to atomic masses

is negligible and it was simplest to assume that an atomic nucleus of mass number A and atomicnumber Z, was made up of A number of protons and (A-Z) electrons, confined to a small volume at the

center of chemical atoms, which has extra Z electrons outside the nuclear volume to render it

electrically neutral.

At around 1920, Rutherford speculated that, at least, some protons and electrons in the nuclear volume

could come very close and be bound together and behave as electrically neutral entities. If they so form

and if they are ejected from nucleus, they can fly around without much interaction to other objects inthe surrounding medium. A search ensued and it was in 1932 that Chadwick discovered neutron. This

neutron, while it is very similar to what Rutherford suggested in terms of mass and charge of the

particle, is an entirely different physical entity. In 1932, one recognized that the neutron cannot simplybe a composite of proton and electron, since it does not satisfy the spin-statistics arguments. A

composite object of a proton and an electron would be a Boson of integer spin, while the neutron is a


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Fermion of half-integer spin. Thus neutron and proton are of same stature in nuclear systems, while the

neutral entity of Rutherford would have been very different from proton.

There were, at least, two arguments against the presence of electrons in nuclei.

1. The spin-statistics: In electron-proton model, a nucleus is made up of A protons and (A-Z)

electrons, i.e. (2A-Z) Fermions. Thus a nucleus of odd-Z, regardless of the mass number A,will consist of odd number of Fermions and it, thus, would be half-integer spin. However, it

was well established that the spin of even-mass nuclei, for all values of Z, is an integer, in

contradiction to the electron-proton model prediction.

2. Nuclear dimensions are of the order of a few femto meters. For electrons to be confined in such

small volumes, the kinetic energy will be at least , with energyexpressed in MeV units and the wavelength in femto meters. Thus the kinetic energies are

about a few tens of MeV for nuclear dimensions. The maximum kinetic energies of electrons

are about 3 MeV or less. This observation is incompatible with electron-proton model.

This period also saw a few other developments of interest to nuclear and particle physicists. First,Diracs theory predicted the existence of anti-particles. Positron, the antiparticle of electron, wasdiscovered lending credence to this theory.

It was well established that -particles (electrons) emitted in nuclear decays are not mono-energetic.One may write the energy released in -decay , where , and are massesof initial (parent) nucleus, final (daughter) nucleus and electron, respectively. It was found that corresponded to the maximum kinetic energy of the emitted electron. The electron energy spectrum

shows a continuous distribution with intensity maximum lying at and corresponds veryclosely to the end-point. The model which attempts to describe the beta decay involving atomic nucleiand electrons (a two-body final state) will not satisfy simultaneous energy-momentum conservation

principles2

. It fails to account for spin conservation too. To rescue the theory, Pauli postulated ahypothetical fermion of spin , devoid of electric charge and nearly zero mass (


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A free neutron undergoes -decay as it changes into a proton, electron and anti-neutrino. This processentails creation of particles. The lifetime is linked to the probability that such energy-matter conversioncan occur while satisfying all conservation principles. To describe this phenomenon, a new force viz.,weak interaction was introduced. In stable nuclei, neutrons do not decay. A free proton cannot decay

to

, since it is energetically forbidden. However, radioactive nuclei, which emit

positrons, are known. These facts are understood as below:

The -decay of a nucleus is governed by the energetics of the initial andfinal states i.e., if the mass of (Z,A) is greater than the mass of (Z+1, A) plus an electron, the decay is

possible, otherwise it is not possible. Also, a is possible if it isenergetically allowed. Neutrons and protons are the sole habitants of nuclei and they are given acollective namenucleons. The notion of electrons as nuclear constituents was abandoned.

With neutrons and protons as nuclear constituents, the immediate question was: What binds them

together so strongly in such small volumes? The ratio of nuclear/atomic volumes are about , andbinding energies of nuclei are about several hundred thousand times the atomic binding energies.

Clearly, electromagnetic interaction cannot be responsible. Coulomb force is repulsive betweenprotons. A strong attractive force, effective only at short distances, was assumed since this force does

not seem to extend to atomic distances. Heisenberg et al.were able to describe the chemical covalence

bonds as due to exchange of electrons between the ions, occurring over short distances. Yukawaextended this idea to nuclear systems and calculated the mass of exchange particle operating over

distances of about a femto meter. It marked the beginning of meson exchange theories.

Initial searches for the meson presented an unexpected guest-muon, which is basically a heavy

unstable electron. It is a transitory particle, weighing 105 MeV/c2(207 me) and its meanlife is 2 sec.

It is weakly interacting with the surrounding medium and it pours down from Heavens as Cosmic rays.

Meanwhile, there was progress in nuclear instrumentation. Cloud chamber, Geiger Muller counters,Ionization chambers, and coincidence circuitry were developed, which were contributing to improved

measurements and more detailed physics information. Particle accelerators such as Cockroft-Walton,

Van de Graaff and cyclotron etc. were invented. These developments enabled nuclear physicists toemploy accelerated particles as projectiles on targets. Artificial transmutation and induced radioactivity

were discovered. The stage for artificial production of isotopes and systematic studies of nuclear

structure, reactions, and decays was set.

It is worth recognizing that nuclei are complex many-body systems where strong and electromagnetic

interactions play important roles. In nuclear systems, the weak interactions are important only for

decay processes. Thus, while these systems offer fertile grounds to test various interactions, they also

pose very difficult, if not insurmountable, problems to theoreticians. People have taken two distinctapproaches. One school of thought considers the nuclear forces to be a sum total of two-body

interactions. The attempt is then to determine nucleon-nucleon interactions for a wide range ofenergies and hope to deduce them for interactions in complex nuclei. The other school of thought

considers this to be a hopeless task. For them, nuclear systems are made up of an average potential in

which a few valence nucleons moving would be responsible for the observable properties such asexcited levels, transition probabilities etc. Here again, two distinct approaches viz., shell model

inspired by the atomic structure models and collective modes where nucleons are considered as fluids

or phonon-excitations are employed. The phenomenological approaches have been more successful in


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their descriptive power of nuclear systematics than those attempting to derive from the basic nucleon-

nucleon interactions, for reasons which we will discuss later.

Sometime in 1940s, a group of physicists separated from the nuclear physics community to pursue

particle physics. It was their hope that, by probing matter at higher and higher energies, one might

arrive at the fundamental constituents of matter and thorough understanding of basic interactions.

Here we have. We might realize our long cherished dream. However, things are more complicated thanthat. Further probing at higher energies resulted in the discovery of more and more resonances which

are identified as mesons and baryons. The economy of building blocks was seemingly lost. Even in

1940s, there were attempts to describe all the particles in terms of minimum number of basic buildingblocks to include normal matter and strange matter, as then discovered. One sorts these building blocks

into a) hadronic matter subject to strong interactions and b) leptons devoid of strong interactions.

It may not be an exaggeration to say that 1955 marks the beginning of a new era in particle physics, as

parity violation, a break down of symmetry conservation, was observed as suggested by the theory,

which itself was prompted by observations in particle physics (decays of strange mesons). 1964 marks

the breakdown of another symmetry known as CP (simultaneous operation of charge conjugation and

parity: particle antiparticle and parity operation) and also the year when the quark hypothesis ofstrongly interacting particles was put forward. As almost everyone knows, quark hypothesis has beenvery successful in offering a description of the subatomic world. The pursuit still continues and thereare many open problems and challenges in the field.

Meanwhile, nuclear physics flourished in 1960, 70s and a bit in 1980s. Phenomenological structureand interaction models as well as implications of quantum mechanical symmetries have been put

forward. While the conventional nuclear physics of 1950-70s has been in decline in the recent years,

nuclear physicists found reincarnation or new lease on life mostly in particle physics.

One might say that particle physics itself went through ups and downs. We might argue that the years

1955 and 1964 marks new beginnings of particle physics. For the first time, the validity of fundamental

symmetries was seriously questioned in the year 1955. The year 1964 marks a new beginning asquarks (fractional charges) were proposed as basic constituents of hadronic matter. Currently, there are

three generations of quarks and also there are three generations of leptons. These structural

organizations accompanied by the 4-types of interactions which are mediated by exchange bosons was,till very recently, considered as the STANDARD model. Several attempts to overthrow this model did

not succeed. However, the particle physics community seems convinced that it is just a matter of time

before the STANDARD model is superseded by a more basic theory. Now, we talk about physicsbeyond Standard Model or new physics. Stay tuned.


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Chapter 2. Rudiments of Nuclear Physics

2.1

Energetics

A nucleus (A,Z) is made up of Z protons and N neutrons and in a stable nucleus, these particles are

confined to small volume. In nuclear physics, the mass number plays as important role as the atomicnumber, since the properties of nuclei are very specific to both numbers. Nuclei with same Z and

different A are called isotopes. Nuclei with different Z and same A are called isobars. We may thus

refer to carbon isotopes (12

C,13

C,14

C etc.) of those with carbon as the chemical element. We may referto an isobar system of specific mass. For example,

14C,

14N,

14O are three isobars of mass number 14.

A good estimate of the nuclear radius of mass number A is . It is of interest to notethat the average space available for an individual nucleon inside a nucleus is nearly independent of the

mass number. As they are bound inside a nucleus, one has to supply energy to the system to free thenucleons. We define binding energy of a nucleus B(A,Z) = M(A,Z)ZMHNMn in energy units

3.

This energy is a negative quantity and it is equal in magnitude to the minimum amount of energy to be

supplied to break the nucleus into Z free protons and N free neutrons. In actual situations, it needs lotmore energy than this estimate to break the nucleus into free protons and neutrons.

At this time, it is instructive to look at the plot of binding energy/nucleon versus mass number.

The deuteron , an isotope of hydrogen is the lightest nucleus and it has lowest binding energy(1.15 MeV per nucleon). Very quickly and almost monotonously B/A increases to a maximum value

of about 8.5 MeV near the mass number A~56, and then slowly decreases to about 7.5 MeV for

heavier nuclei (U, Th etc.). These systematics would provide a qualitative understanding as to why

heavy nuclei undergo fission to lighter masses, while one can gain energy release by the fusion oflighter elements such as tritium etc. to heavier ones.

3Many books use the opposite sign convention for binding energy, where they define

B(A,Z)= ZMH + NMn - M(A,Z). I prefer our sign convention because it is consistent with the

stabilities and Q-value definitions. If 'B' is positive, the free nucleon system has more energy than the

bound system and it is unstable against decay.


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2.2

Separation energies

You might remember work function as the least energy required to liberate an electron from an atom.In the same manner, we define separation energies for neutrons and protons. Neutron separation energy

(Sn) ofA

ZX nucleus is

(2.1)

and the proton separation energy (Sp) [ ] (2.2)are the least energies required to free a neutron and proton from the nucleus X.

We can easily recognize that the stability of nuclei against a decay process is governed by the simple

relation that decay occurs if , where is the mass of the initial system and is the sum ofmasses of particles in the final state. For example

4, the condition that a nucleus of mass M(A,Z) is

energetically allowed to decay by emitting an electron or a positron to the neighboring isobar is given

by with and as the charges of daughter nuclei for the

and

emissions, respectively.

Remember that, here, we are referring to nuclear masses and not the atomic masses. One has to payattention to this fact, as one usually finds atomic mass tables. In case you use atomic tables, the

condition for decay is that and for the positron emission, . Also, when , a nucleus maycapture an electron from the atomic orbits and transform to the daughter nucleus. This process, known

as electron capture decay, plays increasing roles in heavier nuclei since the atomic electrons are closer

to the nucleus.

These energy considerations may be extended to reactions, when projectiles are incident on target

nuclei which would result in either elastic or inelastic collisions, including nuclear transformations. In

the laboratory frame, the Q-value of the reaction for a process , is given by (2.3)

Clearly, the reaction may occur only if is positive. Otherwise, it is energetically forbidden.We can easily calculate the minimum kinetic energy (Tmin) required to render this reaction energetic

possible as

(2.4)

where is the projectile and is the stationary target in the laboratory.4When you are solving numerical problems for stabilities and Q-values, please make sure there are no

round-off errors. As the Q-values are very small (a few keV to a few MeV), while the masses of

participating nuclei are of GeV magnitudes, rounding off the numbers may lead to erroneous

conclusions.


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Of course, one has to satisfy energy-momentum conservation principles simultaneously. For the

overall energetics, it is convenient to work in the center of mass system and employ four momentum

vectors.

A particle of mass and of kinetic energy and 3-momentum , is of total energy ||

(2.5)

and velocity , the four-vector with || .In two-particle collisions, we can calculate the center of mass energy and momentum as (2.6)where , , and are respectively, the mass, energy, momentum and velocity of the particleone. Corresponding symbols for the particle-2 have same meanings. The is the angle between thetwo colliding particles.

Of general interest is the laboratory frame where one particle is at rest (say, particle2). For conversion

from laboratory to center of mass, the velocity of CM frame () is (2.7)The conversion from center of mass frame to the laboratory and vice versa are done easily through

Lorentz transformations. The particle with longitudinal momentum and energy will be found, in aframe moving with a velocity F, to have energy and longitudinal momentum ( )

(2.8)

and the transverse momentum is same in both frames ( )2.3

Time-dilation

We know that radioactive particles obey the exponential decay law and the number of particles oflifetime surviving after a time is given by . If we prepare a beam of theseparticles in the laboratory, then the flux of the particles after a time-interval or at a distance from the source is given by

(2.9)

(2.10)where is the restmass of the particle and other symbols have their usual meanings.It is worth noting that this realization as very important implications for day-to-day nuclear physics

experiments. As one prepares secondary beams of particles such as pions, kaons, muons, and morerecently beams of radioactive nuclei, these expressions have practical value in the designs of

beamlines, experimental setups and detector ensembles.


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2.4

Reduced Transition Probabil iti es

The nuclear and particle physics may be summarized in one sentence as the enterprise of structures andinteractions. The structures involve the properties of energy levels: energies, spin, parity and multipole

moments etc... . What one calls as particles in particle physics are quite often considered as levelsand/or resonances in nuclear physics. For particles too, we are interested in these properties. Forexample, the penta-quark

5if it survives the test of appearing as a resonance, the immediate task is to

determine its spin and parity. Very often, we are measuring the reaction cross sections or decay rates,

which are simply measures of transition probabilities. (2.11)Phase space factors depend on the kinematics of the process and lager phase space factors render the

measurements somewhat easier. They carry no physics other than energy and momentum

conservations. While the phase space is a very important consideration for the feasibility of

measurements, it has little use with regard to statements on the dynamics of processes.

The matrix elements comprise of two distinct components:

a) geometric part: projections of i) the angular momenta (intrinsic spin and orbital angular momenta)of the wavefunctions and ii) the third component of the tensor operator under consideration

b) reduced matrix elements: The reduced matrix elements essentially contain all the dynamics and,

very often, they are the meeting points of theory and experiment.

As an example, say that a nuclear decays by emitting a photon of multipolarity L. The transition rate,

besides other things, is proportional to (), which represents phase space contribution.In nuclear physics and also in atomic and molecular systems, low multipoles are favored transitions,since photon energies in nuclear transitions are a few MeV or less. From the data, one deduces whatare known as reduced transition rates, after factoring out these kinematical contributions. They are

the squares of the reduced matrix elements which come out of Wigner-Eckart theorem that you

encounter in quantum mechanics courses.

When we refer to reduced matrix elements, it is separating out the dependence on projection quantum

numbers, which simply contain the geometrical information from the dynamical information, which

contains physics. These reduced matrix elements contain the dynamics.

One writes

(2.12)where a state of |makes a transition to due to operator . It is important to note that5A penta-quark is a hypothetical subatomic particle consisting of four quarks and one antiquark bound together (compared

to three quarks in normal baryons). As quarks have a baryon number of + 13, and antiquarks of 13, it would have a total

baryon number of 1, thus being classified as an exotic baryon (1).


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the term with double bars , contains the dynamical information. It does not depend on , or . The geometrical information, which pertains to angular momentum conservation principles, iscontained in the Clebsch-Gordon coefficient . While the value of angular momentumconservation cannot be undermined, there is little room to improve or vary the modeling in thatparameter. It is only the reduced matrix element where we can make changes in the physics problem.

We might also take the opportunity to remind that the reduced matrix element, as it simply depends on

, and , it needs to be calculated only once for all the possible combinations of and . Thisfeature has not only the power of calculating various m-states coupled with different q-s very easily,it also enables us to make links with seemingly unrelated processes, say for example, beta decay to the

corresponding gamma transitions in iso-spin formalism.

2.5

Cross section

The experimental results from all reactions are generally presented as cross section. While various

units and dimensions are used to quote them, one basic idea of cross section is that it has dimensions of

an area. Accordingly, we can define a differential cross section as below:

A monoenergetic beam of particles impinges on a target. The beam flux is the number of particlescrossing unit area transverse to the beam direction in a unit time. If there Nbparticles per unit volume

of the beam and all of them traveling with velocity, then the flux Assume that Ntaris the number of target nuclei intercepted by the incident beam. A detector set at an

angle ( ) with respect to the incident beam direction, subtending a solid angle , registers (2.13)or

(2.14)

The proportionality constant, which has dimensions of area, is called cross section.

When the detector subtends a finite solid angle, our measurement yields the differential probability that

the particles go in the direction ( ) ie., the differential cross section. (2.15)

It should be noted that this definition is useful when beam of arbitrary dimensions is incident on a

target as long as the entire beam is intercepted by the target.

If the beam is of cross sectional area , then the number of beam particles incident per unit time is (2.16)and the number of target atoms intercepting the beam (2.17)where is the number of atoms per unit volume of target material and is the target thickness.


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We know

(2.18)here is the density of material and is its atomic number. Then

(2.19)

In the final expression, "" the area cross section of the beam does not enter and it thus free from theuncertainty of beam dimensions. The total number of beam particles are usually measured by the flux

monitors or charge integrator systems.

If and are expressed in units of gm/cm3and cm, respectively, or some multiples of these units,will has dimensions of mass/area. The target thicknesses are usually referred to in units, becauseit is a direct measure of the number of scattering centers, a parameter of basic interest in nuclear and

particle physics experiments.

When one wants to compare relative effectiveness of different materials for the number of scatteringcenters, the units of thickness in cm, mm etc. are not very informative and the are very convenient.As example, an Al (A=27, = 2.7 gm/cm3) target of thickness of same number of scattering centersas Pb (A=208,= 11.35 gm/cm3) of thickness is given by (2.20)Similar units (thickness = mass/area) are used when one is handling problems of energy losses by

charged particles or electromagnetic shower propagation etc.

In the estimations of energy deposits of high energy electrons or electromagnetic radiation in matter,radiation length is a useful unit. Typical of Markovian processes, the energy loss is exponential. The

energy of the particle after it traverses a distance '' in the absorber is given by (2.21)where X0 is the radiation length, specific to the material. Just as above, it is common to refer to thelength is gm/cm

2units or multiple/submultiples of this unit. The X0~ 180 (A/Z

2) gm/cm

2. As Z scales

roughly as A/2, one can see that higher Z materials have shorter radiation lengths and they are more

effective in stopping the electrons or electromagnetic radiation, as we would have expected.

One also estimates a critical energy below which the energy loss of electrons is mainly by ionization

and above which radiation losses become significant. The critical energy is Ec~ 550/Z MeV.


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2.6

Basic properties of nuclei and par ticles

Each particle or nucleus is specified by the classical properties such as mass and charge. We employmultiples of electron volts as units for masses (1GeV = 1.782 x 10

-27kg) and electron charges (e =

1.602 x 10-19

Coulombs). Each particle and each level in nuclei carry a finite angular momentum

(intrinsic spin) and, very often, a well defined parity. Also, in classical terms, these entities carry

electromagnetic multipole moments. In addition, we define certain attributes unique to quantummechanical subatomic world, such as baryon number, lepton number, isospin etc.

First, we should list the commonly encountered quantum numbers of these subatomic entities.

a) Intrinsic spin: Each particle (quantum) has a well-defined spin as intrinsic property6. The spin

can be an integer or a half-integer in units of . The particles of half-integral spin obey Fermi-Dirac statistics, while those of integral spins are subject to Bose-Einstein statistics. As you

know from quantum mechanics, the spins of a composite set of particles are subject to vectorial

additions. As example, for a two particle system | | , where are the spinsof individual particles. The same reasoning is used when we are interested in finding the total

angular momentum of a particle as a vectorial sum of intrinsic spin and orbital angularmomentum.

The nucleons (protons, neutrons), the leptons (e, , and neutrinos) and quarks areFermions. The mesons (, K etc.) and photons are Bosons. Of course, the gauge bosons carryinteger spin.

b) Parity: Parity of a particle/system is the behavior under reflection the correspondence

between the system at hand and its mirror image. For quantum mechanical systems, it is the

spatial inversion of the three axes in Cartesian representation ( ).Under mirror reflection, the left and right are interchanged. When we say a system is invariant

under parity operation, we mean that the system and its mirror image are indistinguishable. In

quantum mechanical systems, we are interested in if the wavefunction remains unchanged ornot under spatial inversion. In macroscopic systems, we know that there are systems which

show a preferred handedness (left versus right). For example, we know optically active

substances which rotate the plane of vibration either clockwise (right hand dextrorotary) orcounterclockwise (left-levorotary). Clearly the system and its mirror image are not the same.

Then, why should one expect that the subatomic systems may be invariant under parity

operations?

There is a bit of history behind this. In 1924, Laporte discovered that there are two classes of atomic

levels and corresponding selection rules for electromagnetic transitions. If you look at two high-lying

energy levels decaying low-lying ones. There are cases where each one selectively populates certain

low-lying ones, which is not populated by the other level. In 1927, Wigner showed that this behaviorfollows from the properties of invariance under space reflections and this is named as parity

invariance. Almost immediately, in 1928, Cox observed that the parity invariance is not obeyed in beta

6 Mathematically and in terms of its manifestation on physical observables, spin is analogous to

angular momentum. Hence, it is called intrinsic spin angular momentum. Some molecules exhibit

even-multiplets of line spectra of doublets and quadruplets. As the multiplicity is , onerequires that the must be half-integer for the even multiplets. The orbital angular momentum cannotaccommodate this requirement. The intrinsic spin was conjectured to accommodate this observation.


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decays. However, he was ignored. In 1955, people encountered a problem, which came to be known as puzzle. A neutral strange meson known, nowadays, as K0meson was found to decay to 2and3final states, while each pion has a negative parity. We will discuss the parity assignment of pion abit later.

The question was how a particle can decay to both a positive parity state and also a negative parity

state and still conserve parity. If the ensemble of particle in question were not one and the same type,there was no distinction among them except for this decay mode. How could a species of particles with

same mass, charge, spin etc., could be different? This was a dilemma. Immediately after that T.D. Lee

and C.N. Yang hypothesized that parity conservation may not be universally valid and it might bebroken in weak decays. They also suggested an experiment, which was successfully carried out Mme.

Wu and her collaborators.

2.7

Parity

What is parity operation? Under parity operation,

and

. We classify physical

observables as scalars, vectors, etc. according to the their behavior under parity operation. Eitherthey change or do not change sign.

Scalars: They are invariant under parity operation (mass, charge)

Vectors: Change sign and Pseudovectors: They are vectors but they do not change sign under parity operation. A common

example is angular momentum, . Under parity operation and .Pseudoscalars: These are scalars, but they change sign under parity operation. Helicity, defined as

|| is an example of pseudoscalar observable.If a physical system has a good parity, ie., if the corresponding wavefunction is an eigenstate of parity

operators, what are the eigenvalues? It is easy to show that the eigenvalues are either +1 or1: (2.22)The eigenvalue of P

2is 1 and thus the eigenvalue of P = 1.

At this stage, we do not know any thing about parity invariance. The wave functions and may be very different.

Say, the system specified by the wavefunction (x) is invariant under parity operation. TheHamiltonian is such that . And as required by Noethers theorem. (2.23)or also satisfies .The wavefunctions and satisfy the same Schrdinger equation with the same eigenvalue .There are two possibilities


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i) The state with energy is doubly degenerate so that two different physical states describedby and have the same energy.

ii) If the state is not degenerate, then and describe the same physical state and thusthey are proportional to each other. (2.24)implies, for

a constant,

(2.25)If the wave function is an eigenfunction of parity operator, we may write , (2.26)where is the eigenvalue.

We have already seen that the eigenvalues of parity operator are 1. (2.27)Thus if a wavefunction is an eigenfunction of parity operator, the corresponding state or particle has

even (

) parity or odd (

) parity.

The orbital parity of particle or system is due to its orbital angular momentum. In three dimensions, not

including the intrinsic spin, we write the wavefunction (2.28)The transformation between Cartesian and spherical polar coordinates is given by (2.29)

The parity operation or is same as , .Thus, the parity operation is replacing by and by in the spherical harmonicscomponent of the wavefunction.We also have

|| || (2.30)

(2.31)

(2.32)

(2.33)

Thus. (2.34)So, the parity is even () if is even and it is odd () if is odd.


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We thus have

l= 0, 1, 2, 3, s, p, d, f,

= +, , +, , For particles, we identify two distinct parities: one is the intrinsic parity of the particle and the other isthe orbital parity, ie., parity due to orbital angular momentum.

Parity is a multiplicative quantum number and thus the parity of a particle is (2.35)Intrinsic parity are characteristics of the individual particles and they are specified once for all.

Protons and neutrons are assigned intrinsic positive parities. Relative to these parity assignments, from

experiments, one deduces that pions have negative (odd) parity. Electrons and strange mesons likekaons do not have well defined parity. We will discuss this later.

Electromagnetic transitions carry parity information. For electric and magnetic multipoles, the parity is

and , where is the multipolarity of the transition viz., for dipole, forquadrupole, for octupole etc.Also, we assign parities to individual nuclear states. Since nucleons (protons and neutrons have

positive parity), and overall parity is the product of individual parities, only orbital parities of nucleonsare relevant.

Thus, for a nuclear state the parity is

Besides the energy, a nuclear state is labeled with angular momentum () and parity () as.Parity conservation gives rise to selection rules in reactions and decays involving particles and nuclei.

Rule:

In a reaction or decay, initial= final.In a reaction A+ B C +D, AB = CD.

Examples

1.

particles have positive intrinsic parity and zero spin(0+). If an particle has orbital angular momentum l, then . Thus, the parity of an particle .If a nuclear level of decays to another level of by -emission, , and ;| | or, we simply have .

(2.36)


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The particle is of even parity. This decay can occur by emitting d-wave alpha particles, satisfyingboth angular momentum and parity selection rules.

Consider, now, a final state JB = 1

+. Angular momentum selection rules allow for or 3 but

the parity considerations will limit the alpha emission to d-waves only.

2.

A nuclearreaction: The ground state of 9Be, 8Be and are of intrinsic spin-parity combinations , 0+ and 1+,respectively. If we can specify the j, a vector composition of intrinsic spin and orbital angularmomentum of deuterons or protons, we can find the allowed spin-parity combinations of the incoming

protons or the emitted deuterons. Also, we can consider the excited levels of8Be, which have different

spin-parity and constrain the angular momenta of the deutrons and protons.

In practical applications, one measures the angular distributions of emitted deutrons, from which one

can deduce the angular mometum transfers in the reactions. This will enable, in favorable cases, toperform the spectroscopic studies.

3.

Electromagnetic transitions:A nuclear level makes a transition to another level by emission of a photon.

Initial parity: in= AFinal parity: where , the parity of the emitted photon. If A and B are of different parities, then is negative (odd). If Aand B are of the the same parity (odd or even), then the ispositive (even).

For multipolarity L, electric transitions have parity (-)L and magnetic transitions have

L+1.

Therefore, if A and B have opposite parities, then parity conservation allows E1, M2, E3 etc.

(Electric odd, Magnetic even) multipoles.

If A and B are of the same parity (even or odd), then parity conservation allows M1, E2, M3etc.

(Magnetic odd and/or Electric even) multipoles.

Angular momentum conservation further limits the multipolarities since | | .Ex:

i) and ii) and iii) and

It is also of interest to note that conservation of parity and angular momentum constrain the multipole

moments of the nuclear levels and particles.


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In case of static moments . Thus, only even parity multipole moments are possible subject tothe angular momentum constraints L 2J, where J = JA= JB

For a systemwith definite parity, odd parity multipole moments vanish, ie. Electric multipole moments

such as E1, E3, or magnetic multipoles such as M2, M4, ... are zero.

The multipole moments of nuclei and particles provide important observables. Any system with non-zero spin (spin 1/2) can have non-zero magnetic dipole moment and systems with spin 1 can also

have electric quadrupole moment. These two properties, especially magnetic dipole moment are wellstudied for particles and nuclei and thus we should discuss them now.

2.8

Magnetic dipole moment

Magnetic dipole moments are manifestations of moving charges and/or intrinsic spin. Classically, anycharged particle moving in a closed path produces a magnetic field which, at large distances, can be

described as due to a magnetic dipole located at the current loop. Thus, all charged particles and

ensembles of them carrying charges evince a magnetic dipole (M1) moment. Electromagnetism tells usthat the magnitude of M1 moment is equal to the product of the current and area of the loop with itsdirection along the normal to the plane of the loop.

(2.37)

where is a unit vector, normal to the current loop, is the area of the loop and is the electriccurrent. If we take the area of the loop as , for a circular path, then we have the current in Gaussian units.Or (2.38)For positive charges is along , and for negative charges it is oppositely oriented.For transition to quantum mechanics, replace by , where with = 0, 1, 2, areintegers. In addition, we have to include the effects of intrinsic spin and replace by , whereis the total angular momentum vector. Also an extra multiplicative factor, known as Landes g-factor,appears. The Landes g-factor is written as

(2.39)

When we are concerned with particles themselves, orbital angular momentum is zero and thus ,is the intrinsic spin of the particle, we expect . For electrons,

(2.40)is known as Bohr magneton.

r


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We would have expected the electron magnetic moment . The electron magneticmoment is found to be 1.0012 B.Thus the effective factor is not exactly equal to 2, but it is . In quantum field theorycalculations, one could account for this deviation from 2 as due to relativistic effects.

Similarly, we can define nuclear magneton as

(2.41)

The magnetic moment of proton is found to be or . The magneticmoment of neutron is or . One assigns the g-factors for the orbitalcomponents as and , for protons and neutrons, respectively.A few things to ponder: First, is much larger than 2 and neutron magnetic moment is notequal to zero. Here is an indication that proton and neutrons are not fundamental particles. Even

though neutrons are of net zero electric charge, this is an indication that there may be equal andopposite charges inside the neutron. Here is a clue for the quark-models of hadrons

7.

In this model, a proton is made up of two up [q=(2/3)e] and one down [q=(-1/3)e] quarks, while a

neutron is made up of one up and two down quarks. Let us consider , wavefunctions of protons and neutrons made from these configurations.As the quarks are Fermions with spin = 1/2, we may write three possible configurations for the proton

wavefunction , with and indicating +1/2 and1/2 projections, respectively.a) u u d , b) u u d and c) u ud

In a) the two up quarks group to give , and it is unique. In the b) and c), the up quarksgroup to , 8Thus the uu ( )d(1/2,1/2) wavefunction is

.

The proton wave function is now written as

(2.42)We can write the quark-magneton (Q) in analogy with Bohr and nuclear magnetons as , where are the masses of quarks and qQ = +2e/3 ande/3 for up and down quarks.The magnetic moment of proton is || . We find the magnetic moment of neutron||

.

7Hadron is a generic name for strongly interacting particles. Hadrons are further classified as baryons

and mesons. Baryons are Fermions or composite system made up of Fermions. Meson are Bosons. In

quark models, they are quark-antiquark pairs. Recently, Belle groups at KEK and BaBar at SLAC haveclaimed that they have discovered exotic mesons made up of complex structures.8 The antisymmetric , , is not used here, since the antisymmetrization requirement is

satisfied by the color components.


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The experimental ratio of proton/neutron magnetic moment is , while the model predicts .We cannot complain. The absolute values of magnetic moments is another story.

2.9

Electric Quadrupole Moments

For parity reasons, only electric multipole moments of even L are non-zero subject to the conditionthat . The multipole moments reveal the shapes of nuclei and charge distributions and thus areof interest in nuclear physics. Working in classical physics framework, we can write the electrical

potential at a point ( ) far from the nucleus as (2.43)

here is the atomic number, and are electric dipole and quadrupole moments, respectively.The same result can be derived as an integration over the nuclear volume. The potential due tocharge

located at a distance

from the center of nucleus in a volume element

at a point

( ) is given by | | (2.44)where | | [ ] (2.45)For , use a Taylor expansion| | (2.46)or the potential

(2.47)Equations(2.43) and(2.47) should be equal for all and . For convenience, set ,We then have (2.48)

(2.49) (2.50)

The charge element in a nucleus is the product of proton charge (e) and the probabilityof finding an ithproton in volume at and sum over all Z protons in the nucleus. is found from the nuclear wave function of all A nucleons and integrating over thepositions of the remaining nucleons.


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| | (2.51)

The monopole moment is simply the total nuclear charge.

The electric dipole moment is

| | (2.52)where is the product of A volume elements.The wave function is proportional to and regardless whether is even or odd, ||2 is afunction of even powers in cos . Thus, the angular integration,

(2.53)for as an integer vanishes.Thus, the electric dipole moment for a nuclear/particle state of well defined parity vanishes.

9 The

quadrupole moment can be written, dropping the primes, as

| | (2.54)

has dimensions of area and it is expressed in units of barns (10

-24cm

2) or fm

2. 1fm

2= 10 millibarns

If charge distribution is spherical, . . It should be noted that spherical chargedistribution gives , but does not always mean that the nuclear charge distribution isspherical. Angular momentum principles require for to be non-zero.In addition to these classical and semi-classical attributes of nuclei and particles, one assigns discreteadditive quantum numbers. Some of these quantum numbers are specific to number conservations and

the interactions they contribute to.

Strong interactions are short-range forces responsible for nuclear binding. These interactions are

effective over distances of a few femto meters (10-15 meters) or less. All particles which participate instrong interactions are called hadrons. Hadrons are further classified into baryons and mesons.

Baryons are Fermions (spin is odd multiples of 1/2). One assigns a baryon number of to each ofthese particles and to their anti-particles. In any reaction or decay process, the total number of9A non-vanishing static electric dipole moment (edm) implies not only Parity break down, but it also

requires that Time reversal invariance is violated. Thus, experiments attempting to measure edm of

electron and neutron have been carried out with ever increasing precisions for over 50 years. So far,

the results are negative. The quest continues.


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baryons remains a constant. Protons, neutrons and the excited states are each of baryon number .The baryon number of a nucleus of mass number A is simply A. The quark models describe theprotons and neutrons as made up of three sea-quarks. Each of these quarks (up and down) carries a

baryon number 1/3, so that the total baryon number of a nucleon is one. The mesons such as pion, kaon

etc. participate in strong interactions. They are hadrons of integer spins. The meson number is not

conserved, just as number of photons is not conserved.

Weak interactions are responsible for decays and they are of even shorter range than the strong

interactions. Strength-wise, they are ~ 10-8

of strong interactions. The Fermions which participate inweak interactions and devoid of strong interactions are called leptons. Charged leptons contribute to

electromagnetic processes also, but they do not contribute to strong interactions. There are three

families of leptons known today. They are electrons (me= 0.511 MeV), muon (m= 105.66 MeV) and

tau lepton (m=1777 MeV). Each of them has an associated neutrino. There was a long search to see ifthese lepton families remain distinct from each other or if they oscillate from one to the other. Also the

question concerns whether the neutrinos are indeed of zero mass. Recent reports from Japan and

Europe indicate positive results, ie ., neutrinos do oscillate, but we await further confirmations10

. In

most processes, each lepton flavor is separately conserved. The electron and its neutrino are assigned a

lepton number and their anti-particles are of lepton number . Similar assignments are made formuons and taus.Strange particles are hadrons (baryons and mesons) and they have the strange property that their

decays proceed through weak interactions. Thus one assigns a strangeness quantum number to them.

Strangeness quantum number is assigned to particles. It is not conserved in weak interactions.

Ordinary matter and leptons are assigned strangeness quantum number zero. The strange mesons andbaryons have a non-zero strangeness number S.

One then defines another quantum number, called hyper charge (Y) . For strange mesons(K

+, K

-, K

0) the hypercharge and strangeness are numerically equal, since

. For strange baryons

( , etc.) and thus .It is of interest to note that there are three generations of quarks and three generations of leptons, asshown below:

They are all Fermions of spin 1/2. In each family the charge difference between the two members

. In first approximation, normal matter is made up of up and down quarks and electrons.

In these models, the proton and neutron, as we have seen before are made up of uud and udd quarks,

respectively.

10As of 2005, particle physics community is convinced of the neutrino flavor oscillations, thanks to the

experimental results from Japan, Canada and Europe. The question of neutrino - antineutrino

oscillations is still open.


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The wave functions for the three pions are , and ()Among the strange baryons, the lowest and the first discovered one is (1115 MeV) of configurationuds.

We will come back to quark models of hadrons later in this course.


24/88


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Only experiment will determine if the particles are Fermions or Bosons. As we have known for a

while, particles with 1/2 integer spins are Fermions and those with integer spins are Bosons. Examples

of Fermions are p, n, e-, etc. Odd A nuclei are Fermions. Particles with integer spins are

Bosons. Examples are mesons, gauge Bosons i.e. Photon, W, Z and gluons etc. Nuclei of even mass

number are Bosons.

Given a system of identical particles, one can construct the wave function following a recipe.

Rules:

1. Know thy particle (Fermion or Boson?)

2. Write a wavefunction for an arbitrary labeling of particles and coordinates, 3. Starting from 2, write a wavefunction, which is a linear combination of all possible

permutations.Note:

a) Symmetric wave functions (Bosons) do not change sign under permutations

b) Antisymmetric wave functions (Fermions) do not change sign under EVEN number of

permutations. They change sign under ODD number of permutations.

4.

Normalize thy wave functions. If you are dealing with n particles, divide 3) byExamples: Two particle wave functionsa) two Bosons b) Two Fermions

c) If we are dealing with more than two particles, it may become difficult to keep track of signs.

No problem. Use determinant approach.

Write | |

| | | |

If we have Nparticles, which give the arbitrary wave function,

| | | | | | | | |


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Now, we obtain a very important result for the Fermions. Say, any two of the coordinates i,j, , n are

identical. For definiteness, say, . Two columns of the determinant are the same. The determinantis zero and thus the wave function vanishes.

Thus, this construction says that no two identical Fermions can occupy the same physical state. This

important result is known as Pauli exclusion principle which you would have come across before.

There is no such restriction for Bosons.

Suppose, we have energy levels available. Let us say that we put N Bosons,one in each of these levels. If we, now, bring in an external cooling agency to reduce the energy of the

system, the Bosons will move down to lower energies. Eventually, they may settle down to the lowest

energy state to so that the total energy of the system becomes N1. This phenomenon is known asBose-Einstein condensation. It has been experimentally observed in atomic systems.

If the particles are Fermions, no such cooling effect can push the particles down to lower occupied

levels due to Pauli exclusion principle. The total energy of the system is . Theenergy level Nbelow which all levels are occupied is called Fermi-level

13.

13At finite temperatures, the filling of levels does not happen such that there is a level below which all

levels are filled and above which every thing is empty. In stead, there are some vacancies in lower

levels with higher levels are partially filled. Here, Fermi level is defined as the level which is 50%

filled.


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Chapter 4. Isospin

Consider a neutron and a proton. Ask what are common features between them? What are the

differences? They both have spin 1/2, of nearly same mass, participate in strong interactions and they

are constituents of atomic nuclei. The only major distinction is the electric charge. In view of this, it istempting to consider these two particles as two different charge states of one and same particle. Just as

a spin 1/2 particle can have two states (m = +1/2, and 1/2), which can be transformed from one to theother by rotations in 3-dimensional space, we may imagine proton and neutron to be two projections ofone single particle, call it nucleon, in a hypothetical charge space, which can be mathematically

transformed into one another by rotations in this hypothetical space, called isospin space. The

mathematical treatment is identical to spin algebra and thus the name was coined14

.

A nucleon has isospin I= 1/2.

A proton is I3= +1/2 projection of nucleon.

A neutron is I3= -1/2 projection of nucleon.

Charge of a nucleon

(4.1)For a nucleus of A(Z,N) (4.2)where I3is the isospin projection of the nucleus A(Z,N).

For Z protons N neutrons

For a nucleus, . We know, from angular momentum algebra | |Thus, for a nucleus A, I = |I3|, |I3|+1, |I3|+2, , A/2.The isospin concept is not limited to nucleons. It is useful for mesons and other baryons. For example,

the pion family can be grouped as an isospin triplet of and I3= +1, 0 and1 for and respectively. We can then write the charge of a pion as . This equation is not the same as whatwe have as relation between isospin and charge for nucleons. It would be nice if we can write one

equation to represent every thing.

We mentioned about hyper charge previously. We can use the definition of hyper charge to write thecharge of a hadron as

(4.3)

14Heisenberg came up with this idea in 1932, soon after the discovery of neutron. Nuclear physicists

use the symbol T, while particle physicists use the symbol I. Also, particle physicists have proton as of

positive projection in isospin space, while nuclear physicists useve sign. We follow the convention

of particle physicists.


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This equation, known as Gellmann-Nishijima formula, applies to mesons and baryons. formesons for baryons and for anti-baryons.With the advent of quark models, we are required to further generalize this formula. It is achieved as

below: All quarks have baryon number , and anti-quarks have . All quarks havepositive intrinsic parity and anti-quarks have negative parity. The up quark is of

, down

quark has . All other quarks have .In addition, the charm quark has , the strange quark has , the top quark has andthe bottom quark has . We may call them as charmness, strangeness, topness and bottomness,respectively. Now, we can write the electric charge of a quark as Generalized Gellmann-Nishijimaformula (4.4)here is the baryon number and is bottomness number.Weak interactions conserve only the baryon number, while electromagnetic and strong interactionsconserve them all. The number conservation refers to the sum, with particles and anti-particles having

opposite sign (and ) for these additive quantum numbers.


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Chapter 5. Isospin Formalism of Two nucleon systems

The total wavefunction of two nucleon system is anti-symmetric under the exchange of the particles:

We write , where , , are radial, spin and isospin components of thewavefunction, respectively.

The treatment of isospin part of the wavefunction is same as the spin part of the wavefunction. For two

nucleons, with , we have resultant . is a triplet with three magnetic substates, is a singlet with one magnetic substate, We use symbols and for the single nucleon spin functions and ,respectively.

We, then, for the spin-part of the two nucleon wavefunctions:

(5.1)

for spin-triplet states and

(5.2)We can construct the isospin wavefunction in exactly same way, call them and .For two identical nucleons viz., proton-proton , for neutron-neutron andneutron-proton system has .Accordingly, we have isospin wavefunction of two protons: (1)(2); isospin wavefunction of twoneutrons: (1)(2)

For a neutron-proton system, we may have an isospin triplet member with and (5.3)or isospin singlet and , (5.4)When we construct wavefunctions, we have to ensure that the total wavefunction isantisymmetric.

For a two-nucleon system with (p-p, n-n or n-p with , pairs) the isospin component of thewavefunction is symmetric under exchange. We then require that the product of spatial and spin

component of the wavefunction is antisymmetric.


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If the nucleon-pair is in spin triplet state ( ), the spin-part of the wavefunction is symmetric and ifit is in spin singlet state ( ), the spin part of the wavefunction is anti-symmetric.For two nucleon system, spatial exchange of particles is mathematically equivalent to parity operation.

The wave function is specified by the relative coordinate

. Clearly, interchanging 1 and 2

results in a ()ve sign in front of the vector, which is same as parity operation.Thus, under spatial exchange , where l is the relative orbital angularmomentum of the two particles.For l even, spatial part of the wavefunction is symmetric, ie., for S, D, G, . is symmetric.For l odd, spatial part of the wavefunction is anti-symmetric, ie., for P, F, H, . is anti-symmetric.

Thus, for , l even, is anti-symmetric. , spin-singlet is the only possibility, For , l odd,is symmetric. , spin-triplet is the possibility, For , neutron-proton (n-p) system, l even,is symmetric. , spin-triplet is the possibility, For , and l odd,is anti-symmetric, , spin-singlet is the only possibility.

System Isospin Space Spin

p-p, n-n

n-p of Symmetric Symmetric (l-even)Anti-symmetric (l-odd) Anti-symmetric ( )Symmetric ( )n-p of

Anti-symmetric Anti-symmetric (l-odd)

Symmetric (l-even)

Anti-symmetric (

)

Symmetric ( )Problem: Write the wave functions of n-n, n-p and p-p systems of arbitrary orbital angular momentum.

Simply write the spin and isospin components explicitly and label the radial component as .Of the three possible two-nucleon systems (n-p, n-n and p-p), only n-p system has one bound state. It

is the ground state of deuteron (nucleus of deuterium, heavy hydrogen).

The properties of the ground state are: .The binding energy = -2.225 MeV

The magnetic dipole moment d= 0.8574 N.The electric quadrupole moment = 0.288 fm

2

The ground state spin-parity suggest3S1, configuration. However, the observation that and quadrupole moment is not zero indicate that the deuteron wave function is not a pure 3S1.

A natural extension suggests that we write the deuteron wave function as| | | (5.5)


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with as normalization condition.One would like to determine the magnitude of from model arguments. This problem was a topic ofstudy for a few decades of research and then abandoned, except to note that its contribution is of the

order of a few percent.

Deuteron has no bound excited levels. In neutron-proton scattering, one observes an s-waveresonances, for an incident neutron kinetic energy of En~ 150 keV, which corresponds to an excited

level of MeV. The resonance is of , and it corresponds to 1S0configuration.It should be noted that deuteron is the only stable A=2 nucleus.

Stable2He and di-neutron do not exist. If they were to exist, the lowest energy state (ground state)

would be of , configuration, corresponding to the unbound excited level of deuteron.One considers this feature to be compatible with the hypothesis that nuclear forces are chargeindependent.

When we think of nuclear forces, we should remember that there are three distinct interactions

contributing to nuclear phenomena. For completeness, we list the interactions and their generalcharacteristics:

Type of interaction Range Strength Exchange quantum

Strong ~ fm 1 etc. mesonsgluons

Electromagnetic = 1/137 photonWeak ~am 10

-- 10

- W , Z

Gravitational 10-

Graviton

In nuclear physics, strong and electromagnetic interactions play important roles in almost all processes.

Weak interactions are essential to describe the -decays and related phenomena. Electromagneticprocesses are well described by Quantum electrodynamics (QED) and the hope is that one has the

theory of strong interactions in quantum chromodynamics (QCD).

If we separately consider the strong interaction part of nuclear forces, we are looking at forces among

identical particles since electric charge is the only difference between protons and neutrons. Thus, the

following hypotheses are put forward:

a) Strong interactions are two-body type.

This amounts to an assumption that in a nucleus or other many-body system, the interaction

between all the particles is sum of the interactions of all possible pairs, acting as if they were

isolated pairs, free from the interactions due to other particles. We may, thus, write the nuclearHamiltonian as

(5.6)

The question whether this hypothesis is correct has been a subject of several investigations.

One asks if we have to incorporate three-body forces of the type and terms of


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higher order. Due to nuclear dynamical complexities and model ambiguities, this question is

not yet settled15

.

b) Strong interactions are charge independent

The strength of interaction between a pair of nucleons does not change whether the partner is a

neutron or proton. Or n-n, p-p, n-p forces are all same. One attempts to verify this statement bycomparing the scattering lengths from the scattering cross section data of pairs of nucleons.

This amounts to say that strong forces are charge blind16

.

According to this hypothesis, in the absence of electromagnetic interactions, all members of an

isobaric multiplet (nuclei of same mass number) will be of identical properties. We know that

they exhibit vastly different properties. These differences may, as first approximation, beattributed to electromagnetic interactions.

Example: c)

Strong interactions are charge- symmetricThis statement asserts that n-n and p-p forces are same and it is implicit in the charge

independence hypothesis. Thus, it is a lesser symmetry.

This symmetry has interesting consequences for properties of mirror nuclei, the pair of nuclei

of same number with proton number (Z) same as the neutron number (N) of the partner

nucleus.

Examples, ; are two pairs. Note that Mass-14 mirror pair is a subset of mass14 isobar multiplet.

Problem: Deduce that the mirror symmetry follows from the charge symmetry hypothesis.

Hint: The p-p pairs of a member are numerically same as the n-n members of its partner, while the n-p

pairs are equal for both members.

In addition, we recognize that they are conservative and they are mainly central forces. They areattractive and they saturate. The last property accounts for, at least qualitatively, that the binding

energy per nucleon (B/A 8.5 MeV). For each mass number, there are a few (Z,N) combinations

which constitute stable combinations. Excess neutrons make the system unstable and it would maketransitions to lower mass nuclei by emitting neutrons.

It is of interest to examine the consequences of charge symmetry and charge independence in some

detail.

15It should of interest to note that the question of two-body versus three-body forces seems to be with

us since Newtonian times. Considering three body system such as Earth-moon-sun or earth-moon-satellite, one asks if the Gravitational potential warrants a three-body potential. To my knowledge, this

is an unsettled question. The problem is more complicated in nuclear physics.16

In quantum chromodynamical description, strong forces are color blind.


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Chapter 6. Hadron Structures

Soon after the beams of pions became available, experiments with hydrogen target were carried out.

The hydrogen bubble chambers were workhorses, as they served the purpose of both target and

detector simultaneously. An important consequence of these measurements was the identification of

several resonances in N system. The structures were observed in elastic scattering and also inelastic

processes such as N N. Angular distribution measurements allow one to determine the spin andparities of resonances. Also, in many cases one can assign the isospin of resonances. For example, a

resonance in or system has . While one cannot unambiguously assign isospin ofresonance in or channels, a comparison with the corresponding channels renders theassignment possible. Indeed, many resonances of are found and for historical resonances, theyare called (delta) resonances, while the are labeled as N* (N-star) resonances. Also,resonance involving strangeness were found, such as (1116) ( ) and multiplet of (1193) ( ) and ( ), etc. Prior to the advent of quark-model, aschematic organization of these resonances

17in terms of octet and decuplets etc was made, which helps

to identify them as members of multiplets with predictable properties.

The figure above shows what is known as baryon octet. The Y-axis represents the strangeness-

quantum number. The two nucleons corresponding to and appear as pair at the twovertices of the top horizontal line. They are respectively of uud and udd quark configurations.

For , with one strange(s) quark there are two families: an isosinglet of mass = 1116 MeVand it is electrically neutral of uds quark configurationand isospin-triplet [ -uus udsand ddswith charge states +, 0 and, respectively.

All these four particles are indicated on the middle horizontal line, with (+)ve charge on the right

vertex, negative charge on the left vertex and two neutrals (and ) at the center of the diagram.

Then, there is the cascade particle-pair [0(1315) - uss and (1321) - dss] corresponding to doublestrangeness ( ).17

Particle physicists refer to narrow resonances and particles as synonyms. Nuclear physicists do not

consider resonances as particles, but rather as excitations of multi-particle configurations.


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It is important to note that change in strangeness of one unit corresponds to mass shift of about 170

MeV, while changing from u quark to d quark amounts an energy difference of 4-6 MeV, except for

the nucleons where the change is only 1.8 MeV.

The Delta isobars with other strange baryons form a multiplet of ten which is known as baryon

decuplet.

The decuplet is made up of four Delta particles (four charge states and ), three sigmaparticles(), two cascade particles s (,) and one omega . An important success ofthis classification was that the mass of was predicted in this simplistic picture and it wassubsequently discovered. is an empirical equation which characterizes this multiplet.


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This event completely characterizes the production and decay of in proton and Kcollisions. Theparticle is produced in strangeness conserving strong interaction process. The later stages involveboth weak process and electromagnetic interactions as specified below:

p + K+ K++ Ko (S = 0 strong interaction) 0 (S = 1 weak interaction)

, (S = 1 weak interaction) pdecays (S= 1 weak interaction) e+e- (electromagnetic interaction)

The mass of is given by the invariant mass of p, -, e+, e-. Frank Close aptly calls this bubblechamber picture as physicists' Monalisa.

Problem: Write down the analytical expression to reconstruct the mass of in this reaction. Assume,the target proton is at rest in the laboratory and Kbeam of energy EK- traverses through the hydrogenbubble chambers.

The quark model offers an excellent description of the configurations of the octet and decupletschemes in terms of isospin lowering and raising operators along a horizontal line for fixed strangeness

family. The changes of one unit in strangeness and one half-unit isospin are achieved by replacing a u

quark by squark. The is a purely strange baryon with .In addition to these baryon multiplets, there is a meson nonet.

The above figure represents the members of , 0 and mesons. K+(493.7) and K0(497.6) are doublet of strangeness . They are of configurations and , respectively. The middleline represents the four non-strange ( ) mesons. The three members of isospin triplet of -mesons[+(139.6), (139.6), 0(135.0)] and iso-spinglet (547.8) meson. As before, the and mesonsare indicated at the right and left vertices with neutral mesons at the center.

Finally, the bottom line represents the , doublet of K 0 and Kmesons at the right andleft vertices. The K 0 and K doublet are also anti-particles of K

0 and K

+mesons, respectively. We

discuss the K-mesons in more detail later as we embark on CP violation. The nonet is complete if we

include the (958) iso-singlet meson at the center along with the other two neutral mesons (seebelow).


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A group theoretical description of these structures was offered, with baryons as three-quark states

involving u, d and s quark states and the mesons as quark-antiquark combinations.

We shall discuss the baryon decuplet in some detail. The occurrence of ++and non-strange baryonsas the lowest energy and states suggests uuu and ddd quark-structuresrespectively to them, with all the three quarks in

1s1/2configuration. This would necessitate that we put

three quarks of same flavor (up or down) in

1

s1/2, in apparent violation of Pauli exclusion principle. Toavoid this problem, one introduces an additional degree of freedom, peculiar to quarks and the strong

interaction carriers, viz. gluons. It is called color degree of freedom18

. According to this, quarks andgluons come in three colors i.e. red, green and blue. Hadrons are color-singlets, such that they do not

carry any overall color. A baryon is made up of three quarks, each carrying a color different from the

other two and a meson is made up of a pair of quark-antiquark of the same type of color-anticolorcombination.

Accordingly, the proton structure of up (u), up(u) , and down(d) quark, will yield a more complicated

configuration. Pauli exclusion principle requires that the overall wavefunctions of baryons areaantisymmetric under exchange of two quarks.

In the above diagrams, without concerning the color degree of freedom, we impose that baryons in a

family have the same symmetry property. The baryon at the extreme edges of decuplet ++(uuu), -

(ddd) and -(sss) are of identical particles and thus they are symmetric under exchange of particles.We prescribe that the same symmetry holds for all baryons, with color degrees being antisymmetric

under exchange of particles.

We may then build the members of the same strangeness by isospin lowering or raising operations.

We build members of different strangenesses by replacing the up or down quarks by the strange

quarks.

Problem: Construct the wavefunctions of the members of the baryon decuplet.

For the baryon octet and decuplet, one sees that the mass change as due to strangeness change issimply due to mass of strange quark and then arrive at a mass formula. We have mass differences

M= 152 MeV, M = 149 MeV and M = 139MeV. One could say Ms= 146 6 MeV. We might

extend this argument for baryon octet to deduce the s-quark mass. We have mass differences MN =

177 MeV and M= 202 MeV.

Clearly, the results for s-quark mass are not consistent. The mass differences are not simply due to

strange-quark mass. It has to do with changes in isospin too. Furthermore the hypercharge changes,

which are due to changes in strangeness may require a bit more complicated formula for masses. A

very good description was found in (6.1)18

The mathematical theory thus formulated is called Quantum Chromo Dynamics (QCD) and isconsidered to be the theory of strong interactions, on the same footing as Quantum Electro Dynamics

(QED) for electromagnetic interactions. For practical calculations, one resorts to several

approximations known as Perturbative QCD, Chiral Perturbation, Lattice QCD etc..


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where, as is obvious, M0, M1and M2are three constants, which can be determined for the two families.

Above, as usual, Y is the hypercharge and I is the isospin of the baryon. This empirical equation is

known as Gellmann-Okubo formula. While it fits the experimental data, no physical insight is gained.

Meson nonet:

Mesons are quark-antiquark pairs. We need to know the result of symmetry under particle antiparticle exchange. We are basically concerned with the result of charge conjugation, which changesa quark to an antiquark is | |or | |?We take guidance from conventions used in the nucleonantinucleon charge conjugation. A protonof positive charge under charge conjugation becomes | |and a neutron | |. It is easierto remember that | (6.2)here we note that a particle of antiparticle, while particle of

antiparticle, under charge conjugation.

The isospin projections are I3= 1/2 for u and and I3= 1/2 for and d. From this convention, we have| |and | |.We should also remember that anti-fermions have intrinsic parity of sign opposite to that the conjugatefermions. With this information, we are ready to write down the configurations of pseudoscalar mesons

of . It is of interest to note that model naturally accommodates the negative parity of pions.Meson I I3 Wavefunction

1 1

1 1 0 One anticipates that the wavefunction

orthogonal to 0 wavefunction represents aphysical state. It is easy to verify that this configuration is an isosinglet of 19.This configuration corresponds to a single pseudoscalar meson of and mass 547.8 MeV and itis known as meson. Thus the u, d base of quark configurations accounts for the 4-pseudoscalarmesons, three among them (+, and 0) as members of isospin triplet and as the isospin singlet

member.

If we include the strange quark (s) into the family, we will have 32=9 mesons, which belong to the

meson nonet scheme. There are 4 strange mesons ( ) of isospin 1/2. They are , ,, mesons we have encountered before.19

Remember the quantum mechanical operations| | | |


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In addition one can make a non-strange meson of , with configuration. One can write itas

. A meson of mass 958 MeV, known as ', is identified with this configuration.The and ' mesons have same quantum numbers. The only difference between them is their masses.Their wavefunctions must be orthogonal. This is achieved by writing the wavefunction of -meson as

. The inclusion of squarks modifies the description of -meson. Thus, the quark

model accommodates the SU(3) scheme of meson-nonet.

Vector Mesons:The meson nonet, described above, was made up of quark-antiquark pairs withspins oppositely oriented. One anticipates meson-nonet with parallel orientations of the spins.Nine candidates are identified and quark-model provides a satisfactory description. The members of

this nonet are K*, , , and mesons. Here K* come as two isospin doublets, mesons are isospintriplet and the and are two singlets.

The meson spectroscopy becomes too complicated or enriched, it depends on one's own views, as we

introduce the charm(c) quarks and bottom quarks. We should mention two characteristic features. One

is the occurrence of J/resonance at 3097 MeV of configuration and its partners and the other isthe (upsilonium) resonance at 9460 MeV as of configurations. Currently, the charm quark isassigned a mass of 1.15-1.35 GeV, while the bottom quark is of 4.6-4.9 GeV. These mesons were

discovered as resonances in electron-positron collisions in 1970s and they constitute the first evidencesfor the existence of charm and bottom quarks, respectively. The evidence for top quark, claimed by

Fermi lab groups in Chicago, came in 1994 from experiments involving proton-antiproton collisions20

.

The current estimate of mass of top quark is 175 GeV.

As an aside, we should mention that this picture is known naive-quark model. One wonders about the

following questions:

a)

Why are baryons made up of multiple of three-quarks? Why not a 5-quark (more correctly 4quark-1 antiquark) and other combinations?

b) Why are mesons pairs of one quark and an antiquark ? Why not 4-quark( 2 quark-2 antiquark)combinations?

c) Is there matter made up of simply gluons and no quarks (glue balls, or gluonium)?

d) Are there hybrid structures, combinations of two or more of the above possibilities?

In year 2003, there were two claims, one about the observation of a penta-quark as an exotic baryon

and also an exotic meson as 4 quark structures. These observations, especially the penta-quark, still

await confirmation. The structure arguments as exotica also need to be ascertained beyond doubt,

namely one has to make sure that there are no other possible interpretations. The experimnetal

questions may be settled in near future. The theoretical understanding, if the experimental signalspersist, will take longer.

20It is of interest to note that colliders involving particle-antiparticle collisions offer some advantages

in discovering new physics. For one thing, all additive quantum numbers such as the total baryon

number, lepton number etc. are zero. Thus, the entire center of mass energy is available to create other

particle-antiparticle pairs.


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Problem: For N scattering, write the wavefunctions in isospin formalism, for the three charge statesof pions and N as neutron and proton, respectively. As the scattering cross section is proportional to

the square of matrix element of the overlap of initial and final wave functions, calculate the ratios of

scattering cross sections of all N channels relative to +p cross section. Make use of the fact thatisospin is conserved in strong interactions.


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Chapter 7. Yukawa's theory of Meson Exchange Forces

In 1932, Yukawa proposed that nucleon-nucleon interactions are mediated by exchange of particles of

finite mass. The exchange particles are Bosons to satisfy spin-statistics. He estimated the mass of

mesons from Klein-Gordon equation and the observation that nuclear forces are of short range (a fewfemto meters). We will present a simple derivation here.

For a particle of mass m, momentum and energy , the relativistic equation is .By noting , we can write Klein-Gordon equation as (7.1)As our problem concerns the static potential resulting in the nuclear binding through exchange

mechanism, we may drop the time-dependent term. Also, as the problem is one of central forces (noangular dependence), we seek the solution for the radial part of the wavefunction. This is written as

(7.2)

The solution for this equation is

(7.3)here, is a strength constant, which is specific to each interaction. The parameter , known as theCompton wavelength

21of a particle of mass m, is the range of interaction involving the exchange of

particles of mass , or , with as the range. It is easy to verify that, according to thistheory, interaction ranges of a femto meters require particles of mass of about 100-200 MeV.

Soon after Yukawa published his theory, Wick presented a simple argument in terms of Heisenberg's

uncertainty principle. According to this principle,

. In the time T, the maximum

separation betwee

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