(ir'nited <iis;trillion)

International Atomic Energy Agency

Kation^ L'uvcdtionul Scientific rind Culture! Oj'f.a^iaQti

IKTER,S';J.'1O::AL CEKTRS rcu 'niEOH'TiCAL PHYSICS

ELEHEHTS OF NUCLEAR PHYSICS *

Bnj K. Gupta

International Centre for Theoretical Physics, Trieste, Italy,

and

PhjBics Department, Punjab Uuiveroity, Chandignrli, India.

M1RSHARE - TRIESTE

July 1981

* PrelLminary version.Hot to be submitted for publication.

.CO S.TUTS

CHAPTER 1: BASZC PROPERTIES OF NUCLEI •

- l i l Tile nucleus •

1.2 Nuclear mass and binding energy

*1.3 Aston mass spectrometer

l.k Nuclear radius

*1,5 Mirror nuclei

•1.6 Nuclear spin and energy level diagram

EXERCISES

•Questions and problems

CHAPTER 2:

2 . 1

•2 .2

•2.3

2 . 4

NUCLEAH FORCES

Introduction

The deuteron

Nucleon-nucleon scattering

Properties of nuclear forces

2.5 Exchange character of nuclear forces; The meson theory

EXERCISES

•Questions and problems

CHAPTER 3: STRUCTURE OF THE NUCLEUS

3.1 Models of the nucleus

3.2 Liquid drop model1: Semi-empirical mass formula

•3.3 Collective model

3.1* Shell model

EXERCISES

•Questions and problems

CHAPTER U: NATURAL RADIOACTIVE DECAY OF THE NUCLEUS

k.l Natural radioactivity

1*.2 The decay law

*4. 3 Alpha decay

*k.h Beta decay

*1*.5 Gamma decay

EXERCISES

•Questions and problems

* To "be included in the final version.

- 1 -

CHAPTER 5: NUCLEAR REACTIONS

5.1 General considera t ions

5-2 Reaction mechanisms

5-3 Compound nucleus model

*5-'+ Reciprocity theorem

*5-5 The laboratory and the eentre-of-mass co-ordinates

5-6 Nuclear fission

5-7 Nuclear fusion

5.8 Nuclear energy

EXERCISES

•Questions and problems

Summary and things to remember

Other reading material

* Table of constants

- 2 -

CHAPTER 1: BASIC PROPERTIES OF NUCLEI

1.1 The nucleus

A nucleus is a positively charged mass sitting at the centre of an atom.

Knoving that the constituents of a nucleus are protons (positively charged

particles) and neutrons (neutral particles), to an inquisite mind, a large

number of questions would immediately arise: What is the mass of a nucleus;

how do we measure it; how do these constituents keep themselves bound; what

is the nature of their binding force; what is the size and shape of a nucleus;

is it always stable; does it decay under some natural or induced conditions;

how do the protons and neutrons arrange themselves under stable conditions;

can two nuclei react; what happens when two '."iuclei collide, etc., etc.

A large number of experimental and theoretical studies over a period of

several decades have led us to answer most of these questions in a rather

"satisfactory" *) way, which is the major aim of this and the next four

chapters of this write-up.

1.2 Nuclear mass and binding energy

A nucleus with mass number A, consisting of Z protons and H neutrons

is denoted as

where X is the symbol for the element name whose nucleus we are considering.

Z is also called the atomic number because the number of electrons in the atom

are also Z. Since

Z + H (1.2)

only two of these three symbols are enough, and generally H is dropped, there-

by symbolizing a nucleus as

(1.3)

Most of the time, even the Z is also dropped, since X is a universally

agreed symbol which is different from each 1 value.

•) We have used a quotation mark on satisfactory in order to caution the

reader that he should not get an impression of all being so well with the

present state of art in nuclear physics. There are still many open questions

and unexplained problems.

The mass of the nucleus (M.) should therefore be the mass of 2

protons (Z.m ) plus the mass of W neutrons (N.m ). This, however, is not true.

The mass of the nucleus, as determined experimentally, is shown to be less

tho-n the mass obtained by summing the masses of its constituents. This difference,

called the mass defect ( t o ) , can be very small, Of the order of 10 -1Q a.m.u

(see fig.1.1 and Exercise l.l), such that precise measurements are required.

The mass of the nucleus can then be written as

H. ™ 7jn + Nm - AmA p n Z(1.1.)

A large number of instruments, called irass spectrometers are devised which canQ

now measure the mass of a nucleus to an accuracy of 1 in 10 . Aston's workthe

is. pioneer in this field and we shall describe^Aston <rass spectrometer in the

following section.

The physical scale for nuclear masses (and also the atomic masses *')

is customarily given in terms of the atomic-mass-unit (a.m.u) which is so

defined that the mass of some abundant atom is given by a vhole number. In

case of carbon-l£ to have an exact mass of 12.000000 a.m.u.

1 a.m.u 1.6601. x IQ 2? Kg (1.5)

16,,Prior to I960. 0 was used to define an a.m.u. With the present definitionIP -i / •

of a.m.u,chosen vith respect to d the mass of 0 atom is 15.991.915 a.m.u.

In what follows, we shall always give the nuclear masses in a.m.u. The masses

of the proton and neutron, respectively, are then

m • 1.0072766 a.m.uP

m • I.OO8665!* a.m.u (1.6)

In units of energy (the electron volts, eV) an a.m.u is given apparently by

the Einstein energy-mass equivalence equation (E = me ) to be

The mass of an atom is the nuclear mass plus the mass of Z electrons

(Z.m ) which is very small,of the order of 0.025 percent. For the calculations

of the mass defect, however, the mass of electrons cancel out such that one

can use either the nuclear or the atomic masses. From Eq.(l.lt)

Am - m ) + Nm - (atomic mass - Zm ) * ZMH + Nm - atomic mass.

-k-

1 a.m.u = 931.1*8 MeV (1.7)

(see Exercise 1.2; also note that 1 MeV = 10 eV ana 1 KeV = I03 eV).

Fig.1.2 gives a plot of the proton number Z ana the neutron number N,representing aportionof the Unclear Chart, f i r s t prepared by Segre, whereeach square corresponds to a nucleus. This chart contains an enormous amountof information on the properties of nuclei, but in the following we shallpoint out only a few:

1. Three possibi l i t ies are apparent. The nuclei lying along the

horizontal W-axis, having same Z t>ut different K, are called the isotopes.

The nuclei lying vertically along the Z-axis, having same H but different Z,

are called isotones. Also, the nuclei lying along the diagonal of same A,

for different Z and N combinations, are the so-called isobars.

2. We notice that the masses of a l l the isotopes are nearly whole12numbers. This is the characteristic of using C for defining the physical

scale. It is relevant to point out here that the simple choice of the protonmass (hydrogen nucleus), M = 1.0000, does not give nearly whole numbers forthe atomic masses of other isotopes.

3. Another interesting point of observation in this chart is therelative abundance of the different stable isotopes of a nucleus, called theisotopic or percentage abundance. For example, oxygen has three stableisotopes, each having a different percentage abundance in nature. We noticethat relative abundance of 0 is 99-759 percent which means that the majorityof oxygen atom available in nature has i t s mass close to 16.

h. Some of the nuclei have" only a single stable isotope; likeBe, F, "Tla, etc . There are twenty such nuclei known so far and the

reader is advised to look into a complete Buclear Chart.

5. Isotopes vith even Z are always shown to be more than with odd Z.

This tendency is known as pairing effects in nuclei (see also section 1.6).

6. A look at the complete chart shows that stable and long livedisotopes with Z —20, 28, 50 and 82 and N = 20, 28, 50, 82 and 126 are mostabundant. This result is known as shell effects in nuclei.

Both the pairing and shell effects are known to result in extrastabil i ty of the nuclear system and will be discussed in a l i t t l e more detailin Chapter 3.

Considering the complete Huclear Chart and compressing i t into a

small N-Z plot (Fig.1.3) so that now each square reduces to a dot, we notice

a few more interesting results : _e_

1. In stable nuclei N:Z is unity only for very light nuclei (A < 20).

2. The fact that — > 1 for heavier nuclei has a consequence for thestabi l i ty of nuclei. The proton being a charged particle will exert an electro-s ta t ic repulsion on another proton inside the nucleus. This is called Coulombrepulsion. Since the neutrons are otherwise neutral par t ic les , the only otherforce preBent inside the nucleus is the weak (short range) attractive nuclearforce (see, Chapter 2). Hence for heavy nuclei to be stable, the extra at tractiveforces must be provided through extra neutrons in order to compensate the long-range disruptive Coulomb forces. Hence the N-Z plot deviates more and more

the thefrom M = z line as^mas3 of^nucleuB increases. The neutron-excess of the stableheavy nuclei has been shown to satisfy the empirical relation

N - Z - const. A'5/3 (1.8)

The const. » 0.0060 as determined from the Liquid Drop Model of the nucleus(Chapter 3). One can also show that Coulomb energy is also, approximately,proportional to A (see Chapter 3, Eq.(3.3)), thereby justifying the abovediscussion.

3. There is a natural limit to the stabilizing effects of Eq.( l .8) ,

since the nuclear forces are in any case short-range forces and act only ™°ngst the..nucleons

neighbourihg'/ whereaa the Coulomb forces act throughout the nucleus. This209 *

is represented by g-jBi. All nuclei heavier than th is , decay to the l ighterones by an emission of one or more H4 nucleus (called alpha par t ic le) .This process of reduction in charge.and neutron number of the nucleus by twoeach(is called a decay (more in Chapter k).

h. If a nucleus has too small nfy r a t io , the proton-rich nucleusbecomes unstable-and goes to the H:Z l ine (called 6 s tabi l i ty l ine) by anemission of a positively charged electron, called 6+ decay. On the otherhand, i f this rat io is too large, the nucleus is neutron unstable and returnsto the 0 s tabi l i ty line by an emission of a negative electron, called S~ deeey.Hence, the name B s tabi l i ty l ine . The 8 decay process in which a proton (ora neutron) changes into a neutron (or a proton) with the emission of 0+

(or 8 ) par t ic le , forms the subject of Chapter k.

The binding energy of a nucleus is intimately connected to i t s massdefect, which means that i t is an important quantity in the study of nuclearphysics. Using the Einstein mass-energy equivalence relat ion, the mass defectbecomes on energy that is required to bind the nucleons in a nucleus or isreleased when a nucleus i s broken into i t s constituents. I t is therefore calledthe nuclear binding energy (B,E) of the nucleus and is apparently a measure ofthe s tabi l i ty of the nucleus:

- 6 -

or explicitly *)

3.E =

3.E = Nm -n

(1.9)

(1.10)

This is apparently a positive quantity. In literature, the "binding energy is

sometimes written with a negative sign, which then is more commonly called

the "nuclear potential energy". Tables of the experimental nuclear 'binding

energy are published and a large amount of effort has also gone into calculating

this quantity theoretically. We shall discuss in Chapter 3, the first attempt

"by von Weiszacker in 1935 in terms of the now veil-known, liquid drop model

and later on more microscopically in terms of the shell model of the nucleus.

That is where the physios the structure of the nucleus will enter our discussion.

(An idea of the units involved and its magnitude compared to the atomic binding

energies (a few eV) can be had from Exercise 1.1.)

In calculating a nuclear (or atomic) mass in terms of the masses of

its constituents, we must therefore add the binding energy of the nucleus,

such that

1. * Am + Sm *r fa.-A p n ? I . [ I '

The exp-rimentsl nuclear binding energy b.r varies from i'.iJ'i MeV forthe lightest stable system, the deuteron, (the di-neutron and di -protonsystems are unstable) to l6ho MeV for the heaviest stable nucleus o^Bi.Theoretically, one can calculate the binding energy of any system, by using,say, the liquid drop model formula (Eq.(3.T)). Though the term "binding energyis very commonly used, i t is more customary to talk of an average binding energydefined as the binding energy per nucleon (B.E/ft):

^ - j Tzm + Hm - M,l c 2

A A I- p n AJ

= jj [ p + (A-Z) mn - M^

= [~m - \ (m - m ) - M. l<- n A n p AJ (1.12)

In terms of the atomic masses, the nuclear binding energy

B.E. = [ZM + Hm - Atomic mas si c

-7-

This quantity leads to some very interesting results on the nature of nuclear

forces and also on the amount of energy released in the making or breaking of

a nucleus (Chapter 5).

Fig.l.U shows a plot of the experimental B.E/A vs. the nuoleon number

A i tself . The following interesting results are apparent from the structure

of th is curve:

1. The deuteron has a very small binding energy (B.E/A ^ 1.11 MeV/nucleon).This means that i t is a very loosely bound system and i t s study should be usefulin learning about the nature of nuclear forces acting between a proton and aneutron {Chapter 2).

2. In the light mass region, the curve arises steeply with sharp peaks

at Sie, aBe, 12C, l 6 0 , 2°H ana Mg, i . e . at A - Itn nuclei. This can be

interpreted to mean that a l l these nuclei can be built-up from a par t ic les ,

which i t se l f i s shown (Exercise 1.1) to be a very tightly bound system. In

nuclear structure physics, th is interpretation is known by the name of an

aApha-particle model of the nucleus, which has met with a reasonable success

in explaining many properties of light nuclei. The letaiLs uf this model are,

nowever, beyond the scope of this text.

3. For A ^ 16, the b.'v.A ? nearly orintttnt vh.i'ti refersto thesaturation property of the nuclear forces. Thus, as for the chemical bindingenergy between atoms in a liquid, we have in the nuclear case, to a reasonableapproximation,

B.E eC A . (1.13)

This means that the nucleons in the nucleus interact with only a few nucleons

in its immediate neighbourhood which makes the nuclear forces short ranged and

hence saturated. On the contrary, if each of the nucleons interact with all

other nucleons in the nucleus, the attractive nuclear force would then be

proportional to

•a ^

It. The maximum value of - j - =8.8 MeV/nucleon lies at A ss 60, which

represents the nucleids of Fe, Hi or Co to be most stable.

5. After arriving at the maximum value of 8.8 MeV/nucleon, the

binding energy curve falls slowly to 7.It MeV/nucleon. Though the fall is

apparently very small and gradual, this has far reaching consequences in the

sense that when a heavy nucleus like 2 °U with B.E/A = 7.6 MeV/nucleon is split

into two equal A = 120 nuclei, a large amount of energy is released (for actual

calculations, see Exercise 5.^)- This is also true even if the splitting

fragments are of unequal masses (Exercise 5.5)- This process is called

• nuclear'.fission. Similar-ly, i f ;two .vary light nuclei are made.to fuse into "•a ' '

an iriteimediata mass nucleus j/laxge ainc>unt of energy vou-l-d. again, toe released*

Theorder of energy released' in this so-called nuel«ar fusion process' is

i l lustrated in section 5.7-

6> The downward trend in the B-E/A curve suggests that nucleids with

A r 2kO have smaller "binding energy which makes them energetically less stable.

In other words , a natural limits is set on maximum value of A for the stable

nuclei. For heavier nuclei, other effects like Coulomb forces also cone into

Play.

1.3 Aaton rcass spectrometer *

To be included in the final version.

the.nucleus uniform, so tfiat one can ..de-fine i t simply as the nuclear massdivided by its vo-Lujne • . ,

(1.16)

Here it is assumed that m = m * m . Eq.(l.l6) shows that the density isp n

essentially the same for all nuclei, as shown pictorially in Fig.1.5' This,be

however, is not found to be true, though it is shown to)nearly constant at

the centre of the nucleus,which decreases gradually in the surface region.

Depending on the method of measurement, one gets tile mass or the charge

density of the nucleus. One charge density that is commonly used is given in

terms of a Fermi distribution function:

1.4 Nuclear radius

The very association of the concept of radius to a nucleus, pre-

supposes that the nucleus has a finite size, specific shape and a constant

density. The fact that nucleus has a finite size was first realized by

Rutherford through his historical experiments on the scattering of o particles

on thin metallic foils of Au, Ag and Cu. His experiments showed that for the

Coulomb law of repulsion (Eg..(3-3)) to be valid, the nuclear size must be-Ik

greater than of some 10 m. In principle, a radius can, however, be defined

only if the system has a spherical shape. Though all the nuclei are certainly

not spherical, all the measurements, using different methods, consistently

support the interpretation that the nuclear radius is given by

R = .1/3(1.14)

(r-R )/zn(1.17)

1 + e

which is also illustrated in Fig.1,5, p is the density near the centre of the

nucleus, chosen to give the correct total charge Ze of the nucleus, H is

defined as the value of r

related to t as

for p = p./2 and

kz Jin 3

z gives the surface thickness

(1.18)

where t gives the distance between the points at 10 and 90 percent of the

maximum density p . One can calculate the mass or nucleon density by con-

sidering that the nucleonsoccupy the same space as the charged protons,such

that

where the value of r is given to be (slightly) different by the different

methods. A recent analysis of the experimental data on nuclear sizes givesneutron

constant (1.19)

r = 1.15 10

= 1.15 fm

1 fm = 10 m is the unit of length used in nuclear sizes.

(1.15)

The definition (1.14) for the nuclear radius apparently assumes that

the nucleus has an effective spherical shape and its density is uniform through

its volume, since the volume (V = r— B ) is then proportional to the mass

number A. The question once again arises: Is the density of nucleon inside

Experimentally, the charge distribution in a nucleus is measured by the

scattering of ch&rgeS particles and the electrons (energy JJ, 100 MeV) *> from

a nucleus and the scattering of neutrons ( > 20 MeV) ' give the nuclear matter

distribution. The nuclear matter radius (R ) is found to be larger than them

nuclear charge radius (R ) by about 30 percent. The electron scattering

measurements give

*) At these energies the de Broglie wavelength associated with electron (or

neutron) must be comparable to the radius of the nucleus bombarded (see Ex.1.3).

-9- -10-

and

R = (1.07 ± 0.02) A 1 ' 3 fin

t = 2.H ± 0 .3 fta

(1.20)

for nuclei vi th A = 1*0 to 209-

The fact that nuclear density is not constant throughout the nucleus

is further reflected in a detailed analysis of the experimental data on nuclear

sizes. I t i s shown that a l l nuclei are not spherical and the distribution

of charge in a nucleus is not spherically symmetric. The deviation of the

charge from spherical symmetry is expressed in nuclear physics by defining the

so-called electr ic quadrupole moment of tue nucleus. This gives the non-

sphericity of the nucleus, auch that

(1.21)

where R is the radius for a spherical nucleus and SR is related to the

quadrupole moment and is a measure of the deformation of the nucleus which

can be aa much as 25 percent. Empirically, a better determination of the

nuclear radius for both the spherical and deformed nuclei ia given by

1.28 A 1 / 3 -0.76 + 0.8 A~ 1 / 3 (1.22)

The deformation effects are observed in nuclei with mass number A ft 2k,

150 ^ A ^ 190 and A >,22\, Nuclei with positive quadrupole moment are calledprolate deformed (major axis greater than minor axis) and those vith negativequadrupole moment are called oblate deformed, where the major axis i s smallerthan the minor axiB. Both the prolate and oblate shapes are shown schematicallyin Fig.1.6. A prolate deformed nucleus in three dimensions is like anellipsoid (egg-shaped or an smerican foot-ball shaped), whereas an oblatedeformed nucleus can be compared vith a pan-cake (Fig.1.6a). In two-dimensionalrepresentation, the prolate and oblate deformed nuclei are, respectively,shown as spheroid (an ellipse) and cigar-like (Tig.1.6b) shapes. Nuclei withprolate deformation are found to be more abundant in nature.

1.5 Mirror nuclei *

To be included in the final version.

1.6 Huclear spin and energy level diagram

To be included in the final version.

-11-

EXERCISES;

Ex.1.1: The atomic mass of the deuterium is 2.014102 a.m.u and the nuclear1,

mass of 2He is 1*.001502 a.m.u. Calculate the mass defect (in MeV) and comment

on the significance of the number obtained in each case.

Solution:

Deuterium m = 1.0072766 a.m.uP

m = 1.0066654 a.m.un

M • atomic mass of deuterium - electron mass (m )

= 2,011*102 - 0.000549

= 2.013553 a.m.u

= lToO72766 + 1.0086654 - 2.013553

= O.002389 a.m.u

1 a.m.u = 931.l»8 MeV (Ex.1.2)

.*. Am = 2.225 MeV .

This means that a deuteron can be broken into a proton and a neutron with only

2.23 MeV of energy provided by, say, y rays. This gives the binding energy

of deuterium, shoving i t to be loosely bound system.

Helium \e :

p

• 2(1.0072766) + 2(l.OO8665l») - 4.001502- 0.0303ol a.ra.u

= 0.030381 x 931.U8 = 28.239 MeV .

This gives the amount of energy required to break a helium nucleus which is

very large. Thus helium is a very stable system. 28.299 MeV is the binding

energy of helium nucleus.

Ex.1.2: Show that 1 a.m.u • 931.l|8 MeV

Solution:

1 a.m.u io"2T m* 1.6604 x 10~2 7 x (2.99793 x 1O8)2 Joules

, l .66oh x i n " 2 7 x (2.99793 x 1 0 8 ) 2 e V

1.60 x i o ~ 1 9

1.6604 x 1 Q ~ 2 7 x (2.99T93 x l p 8 ) 2 ,MeV

106 * 1.60 x 1O~19

= 931.48 MeV.Q

In terms of E • me , t h i s means t h a t 1 a.m.u i n MeV i s obta ined "by mul t ip ly ingft 1

1 a.m.u of mass with square of t h e v e l o c i t y of l i g h t c = 2-99793 * 10 m s e c "

CHAPTER 2 : HUCL£AB FORCES Properties of nuclear forces

2 . 1 Int reduction

A nucleus, being an aggregate of large number of par t ic les , is a many-body system. The system is complex because the particles are both neutral(neutrons) and positively charged (protons) and that the mass of each of theseparticles is very small. Furthermore, the number of charged particles is notalways equal to the number of neutral par t ic les . The interesting thing, however,is that the nucleus s t i l l has a sharp boundary and the nucleons are held to -gether t ightly inside the nucleus. This reminds us of the fact that nuclearbinding is too strong for i t s very small size. The nuclear forces must, there-fore, be very strong forces. This in fact is true. Representing the strengthof nuclear forces by unity, the strengths of the other known forces are of theorder of 10~2, 10~ and 10 , respectively, for the electromagnetic forces(•between charged par t ic les) , the weak interactions (involved in B decay) andthe gravitational forces (which depend on the masses of the bodies).

We have seen in the previous chapter that in addition to. their strongattractive character, the forces between nucleonsare also short ranged. Inorder to overcome the difficulty of the many-body aspect of the nuclear force,i t s short range property is used to define that nuclear forces are two-bodyforces. It is due to this approximation that , instead of exact many-bodycalculations, the physics of the structure of the nucleus is reduced to thevarious nuclear models (Chapter 3). Apparently, the aature of the two-bodyforceB can he best obtained from the study of the two-nucleon systems; thedeuteron (di-proton and di-neutron systems are not stable ) and the nucleon-nucleon scattering. Since i t is difficult to make neutron targets , the nucleon-nucleon scattering experiments are done for the scattering of neutron onproton and proton on proton at various energies. This is what we shall studyin this chapter.

2.2 The deuteron

This is the first place where an application of the Schrodinger

equation is nade in nuclear physics. The reader may skip this section if the

relevant background of the mathematics is not yet acquired.

2. 3 Mucloon-nucleon scattering *

To be included in the final version.

-13-

In th is section we summarize the information obtained on the nature

of nuclear forces from the study of the properties of various nuclei (Chapter I)

and the two-nucleon systems studied in the previous two sections.

For the heavy nuclei (— > l) to be able to res is t the disruptive effectsLi *

of the repulsive Coulomb force, the extra neutrons must provide an attractiveand short-ranged nuclear force. Such a nuclear force has also to be a saturatedforce since the binding energy and nuclear volume, being proportional to thenumber of nueleotis inside the nucleus, allow the nucleons to interact only ini t s neighbourhood. Mirror nuclei and even isobars show tha t , except for theCoulomb energy, a change of proton to neutron and vice-versa gives r ise to thesame energy level spectrum. This is called the charge-independence of nuclearforces which means that the forces "between the neutron-proton, neutron-neutronand proton-proton are the same. As we shall see in the next chapter, energylevel spectrum of some nuclei also requires the nuclear forces to be spin-dependent (the spin-orbit coupling force in the shell model).

The attractive and short-range character of nuclear forces also follovsfrom the low energy ( <i 10 MeV) n-p scattering experiments and the bound n-psystem (the deuteron). The binding energy of the deuteron is given reasonablywell for an attractive (-30 MeV), short-ranged (<v2 fin) square well potential.The high energy (300 MeV to 1 GeV *)) p-p scattering data, however, requirethe forces to be repulsive at very short distance (0 to O.U fm). In thelanguage of quantum mechanics, the forces between two nucleons can berepresented by a potential , which is schematically of the form givea. itt. fiff-2-1.Since this potential depends only on the separation distance r , i * Is calleda central potential. Iluclear forces are also shown to have a sm«ll velocity-dependent component, called the Tensor force, which is said to make the nuclearpotential as non-central potential. This is required for the proper explanationof the electrical quadrupole moment and the magnetic moment of the deuteron(not studied in this text ) .

The spin-dependence and the charge-independence properties of nuclearforces are also supported by the nucleon-nucleon scattering experiments. Thepotential is required to be spin-dependent for the low-energy n-p scatteringcross-sections. The charge-independence is apparent from the similarity ofthe n-p and p-p scattering cross^sections of both low and high.energies.The charge-independence of nuclear forces has led to the introduction of anew concept of iso-spinof the nucleon, which distinguishes a proton from a neutron.

*) 1 GeV = 10 MeV.

-Ik-

Finally, the nuclear forces are shovn to have an exchange character

which follows from both the properties of complex nuclei and the nucleon-nucleon

scattering at high energies. The exchange character in nuclei results from

the presence of many other stable elementary particles inside the nucleus

(Chapter 6) which give r ise to both the attractive and repulsive force, thereby

producing saturation. The angular distribution measured in the scattering

of high energy n from p can he explained if a charged pion is assumed to

be exchanged between the two interacting nucleons. We study t h i s , in a l i t t l e

more deta i l , in the next section.

For the neutron-neutron or proton-proton interaction, a ir is exchanged:

2.5 Exchange character of nuclear forces : The meson theory

We have already introduced the concept of a potential (Fig.2.1) between

the two neighbouring nucleons inside a nucleus, without having gone into the

question of how these nucleons interact. Meson theory deals with this

question, vhose basis lies on the fact that a nucleus must contain inside it

elementary particles other than neutrons and protons. One such evidence was

already available in 1931*, that electrons are exchanged in the weak interaction

process of 6 decay. In 1935. Yukawa proposed that, just as for weak inter-

actions the electrons are exchanged in 6 decay, the nuclear forces act via

the exchange of me30ns (H-mesons, called the pions) between two neighbouring

nucleons. A charged pion {IT o n ) is exchanged if the interaction is between

a proton and a neutron and a neutral pion (ir ) if the interaction occurs between

two neutrons or two protons. In other words, the nuclear force between a

proton and a neutron or between two protons and two neutrons is due to a

short-range meson field acting like a cloud of mesons around the nucleons

which the nueleons keep exchanging continuously. Thus, a neutron (proton)

changes to a proton (neutron) by a emission of it {" ) , which in turn is

absorbed by a neighbouring proton (neutron),giving back the neutron (proton)

and so on:

p + it

(2.1)

p + ir

p •+ n + ir

n + IT' •+ p

(2.2)

(2.3)p + i

This whole process of exchange of pions between the nucleons takeSplace in a time

short enough to conserve the energy within limits of the uncertainty principle

(see Exercise 2.1). In view of t h i s , the mesonseould be called "vir tual"

mesons which remain confined within the known range of nuclear forces (see

Exercise 2 .2 ) ,

The potential produced due to the processes^2.1), (2.2) or (2,3) is called

the one-pion exchange potential pJPEpJ* . Extensions of these processes are the

exchange of two- or more pions or of other heavy mesons. A large amount of

research work is going on in th i s f ield.

Exchange forces can be both at t ract ive and repulsive. Though there i s

no easy way to show t h i s , a commonly given analogy i s that during any exchange

process the bodies wil l come closer (a t t ract ive) i f the exchange is physical or

theywi l lgo apart (repulsive) i f the exchange IB throwing of the "object at 'each

other.

The processes (2.1) and (2.2) together, where a nucleon changes to

another nucleon and back to the seme, by exchange of mesons,can be considered

as the exchange process of the position and spin (one or both) of the two

nucleons. The exchange of space (posi t ion) , spin and both space-spin are

termed as Majorana,B art lett and Heisenberg exchanges. When DO exchange occurs,

the potent ia l i s given the name Wigner potent ia l . All the four poss ib i l i t i e s

are shown in Fig.2.2. Hotice that the exchange of both space and spin together

is equivalent to charge exchange, which can, therefore, be called an iaotopicor iso-spin

spin,exchange. The two-body central potential i s , therefore, a sum of

the original Wigner potent ia l , vw ( r ) (which may be of the type shown in Fig.2.1

or the square well) and the exchange contributions due to Majorana (Vytr)),

Bart le t t (VB(r)) and Heisenberg (VH(r)):

V(r) Tw(r) VM(r) (2.It)

For the relative signs and strengths of these various exchange interactions

between the nucleons, we refer the reader to other reading material.

-15-

*) Also, called the boson exchange potentials JBEPJ since mesons are spin 0

and spin 1 particles (the bosons).

-16-

CHAPTER-3: ; STRUCTURE OF THE HTOLEUS '•.

.: " X ir-meson-of mass 212 A" is .exchanged,between the two aualeao-s iosi ie '.

a-tuitleus, daicui^fce feiiJe/t.iiie'tAkttn-for -the exchange•.proee^s to /etcur - f orT tin*

energy .,lcss-to te-within" limits of tha utwertsin-ty prifie±t>i'e <

Solution.: AV.pe'r .the:uneertsirity principle ' ' • • • ' . : ' - " • • ' •

If-the ,.em£s;Jion of a .meson [represent-a. an-uncertainty .in' th« ."energy; imbalance

of the process, then . • ' '

AE =m . c2 . .K • .

Atm e

Ex.2.2: Given the time taken for the exchange of meson as in Ex .2 .1 , calculate

the range of the nuclear forces.

Solution: I f i r is the range of nuclear forces, then considering that the

meson t rave ls with veloci ty .o :

i r • c. i t

* 1.5 fin •

3.1 \ Models "of'the nucleus - ••' ' - ' ' ' '

Eisrttir;iroBiILy,Jitrtoblc:some-3Jt. years (.iBSfG-to r9"3a):ibefore-the'.

.awe'ptea model of the nucleus, that we have used in t.h« previous two chapters,

could te realiz-eu. It is nOw.establ.i&h^/b-eyonda^-doulJt.-eii&t.

made up aolely of protons and neutrons. Obe early obs-ej?*a;t"i-t)n;,, frofe

Cllart'j;'wfis that.the "ir&fiit-opie,masses o"f -ligut..,nu£Lfei are-n,eai-!ly'in.tefer:«i muTtipie

of 'tic pffjWn tmxi. This possibility of the tau<Meils to bejmade up o/ only! ' irom .

prottsns was., hovever, soon ruled out for Heavier nuclei aadjjtiie faet that -the •

atom is.on the whole neutral. In 1896,. Thomson -proposed his "melon-seed-" .- /

modei *) (Fi«-3.l(a)) where electrons were, considered-to'be preeent a-ike thTe - .•

seeds in the proton matter of the -neion. At this time, some nuclei v'ere • ' ,

already, known to te emitting electrons (,ff decay). However, the .eJecrtTons

emitted in 0 decay have energies of only 2-3 MeV, whereas the elactrons cominy .

out of the nuclear confinement of 10 m should have enargiea of at lejist

of the order of 20 MeV (sefl Bx.3.1), This model also fails to give the other

properties of nuclei like the ground stats spin, the magnetic moment.and the

strong nuclear binding energy effefcts. The first break through in the knowledge

of the structure of the nucleus came in 1911 when Rutherford gave a "positive

core" model (Fig.3-l)b)) of the nucleus on the basis of their o scattering

experiments. The protons were considered to be sitting at the centre of the

atom with the electron cloud around i t . This model remained in youge t i l l

1932, when1 Chadwick postulated the presence of a neutral particle, neutron,

insiae the central positive core of the Rutherford picture called the nucleus

and Niels Bohr satisfied everybody by making the electrons revolve in their

specific orbits (Fig.3.l)c>). The neutron had evaded i t s detection for a l l

these yeara because a free neutron'is not stable. I t has a half l i fe of only

10.8 min. and decays to a proton.Plua an electron and an antineutrino.

The neutron-proton model of the nucleus is now here to stay and the

next question that arises i s , how these nucleons are arranged inside the

nucleus and give the various observed properties. As already mentioned in

the previous chapter, neglecting the many-body nature of the nucleus, many

models are used for describing the structure of the nucleus. The first one

is the liquid drop model of the nucleus which, instead of considering the

particle nature of the nucleus, takes i t as a hydrodynamical body of continuous

matter, made up of protons and neutrons. This aodel gave very nicely the

*) Also fcnown as the "plum-pudding model".

-17--18-

binding energy of the nucleus, explained the nuclear fission phenomenon, at

least, qualitatively and later became the basis for a more complete model of

the nucleus, called collective nodel. The particle nature of the nucleus is

simulated in another model, called ifaell n.odel of the nucleus, which explains,

at least for lighter nuclei, many properties of nuclei including the binding

energy. It also contributes to a quantitative description of the fission

phenomena, studied in more detail in Chapter 5. Since, these three models

form the core of the nuclear structure physics, we study them in the following

sections.

3.2 Liquid Drop Model: Semi-empirical Mass Formula

von Weizsaeker was the first to calculate the binding energy of a nucleus

by comparing it with a liquid drop. This analogy follows from Eq.(l.l6) showing

that density is the same for all the nuclei and the fact that in liquids also

the density is independent of the size of the droplet. Using this constant

density droplet model, he constructed a formula for the binding energy which,

when substituted in Eq.(l.ll), gives the mass of the nucleus. The empirical

nature of this formula stems from the method of fitting the constants with

the experimental masses. Different authors have later tried to fit the

constants of this formula at different times due to the availability of

improved data, and we shall give below the values aost accepted at this time.

This model has also been used by Bohr and Wheeler for explaining the fission

phenomenon (Sec.5.6). and since the potential (negative of the binding energy)

is an integral part of the Schrodinger equation, it is now being used in its

modified forms (with shell correction included) for the quantitative description

of many nuclear problems, including the fission phenomenon.

A liquid drop has volume and surface which will naturally contribute

to the binding energy of the system. If we consider that this drop contains

protons and neutrons, then the protons will give rise to a repulsive Coulomb

energy and the neutron-proton difference would give an additional contribution,

called the asymmetry energy. By adding those four terms, the volume-, surface-,

Coulomb- and asymmetry-energies, we get the von Weizsacka- form of the

expression for the binding energy. There are, however, additional effects

of pairing and shell (Sec. 1.2), which also contribute to the bindirrg of

nucleons inside the nucleus. In the following, we discuss each of these terms

separately and then give the final form of the semi-empirical mass formula

with the recent values of its constants:

-19-

Volume anergy: For the constant density model of the nucleus (Sec.l.ii), we

have seen that the volume is proportional to the number of nucleons A inside

the nucleus. Therefore, the dependence of the binding energy on the volume

of the system should be proportional to A:

vol.(3.1)

a is the constant to be determined empirically. This forms the mainwhereterm of the semi-empirical mass formula. In the proton-neutron picture, thismeans that every nucleon in the nucleus interacts with a certain number ofneighbouring nucleons (the short-range effect of nuclear forces). However,the nucleons at the surface can interact with nucleons only on one side;thereby pointing out that Eq.(3.1) over-estimates the binding due to the nucleonsat the surface and their contribution (the surface energy) must be subtracted.

Surface energy: The surface area of the nucleus is

,1/3

and hence varies as

trl:> (since R =therefore, be proportional to A

). The surface effect on the binding energy must,2/3 .

E,2/3

surf.(3.2)

The negative sign in (3-2) is suggestive of the fact that the nucleons at the

surface of the nucleus are not as strongly bound as the nucleons inside and

this effect contributes in the opposite direction to the volume energy

contribution. For light nuclei, this term could be as large as the volume

term since in light nuclei most of the nucleons are at the surface.

Coulomb energy: This is due to the mutual repulsion of Z protons in the

nucleus. T,he potential energy due to the uniformly distributed Z protonsQ Z e^

in a nuclear volume of radius H is given as — — — • This gives the Coulomb5 2 1/3

self energy of the nucleus a3 proportional to Z and inversely to A :

_zL *ECoul. ac Al/3

The negative sign here signifies that the Coulomb force is a disruptive force

and acts in the opposite direction to the stabilizing effects in the nucleus.

Under the very crude approximation of A being proportional to Z, the

Coulomb energy becomes proportional to A y which equals the force due to

neutron-excess in heavy nuclei (Eq..(l.8)).

*) In quantum mechanical description, we must substitute Z = Z{2-1).

-20-

Asymmetry energy:*) We have seen in the discussibn of Tig.1.3 that for

N > Z nuclei, the excess neutrons, (H-Z) , have to provide an attractive nuclear

force to compensate the disruptive effects of the Coulomb force. This means

that H = Z nuclei are more stable. Since S = 1 is true only for light nuclei,

we must have a term in the 'binding energy to allow for the (N-Z) difference in

heavy nuclei. We do not want here to go into the detailed cause of this term

but observe that neutrons and protons must obey the Pauli Exclusion principle

in their filling of the single particle states, which give rise to the reduction

in binding energy proportional to the square of the difference (N-Z) and

inverse to A:

asy. a

In view of relation (1.2) • this is also sometimes written as

Ea

Easy. a A

Fairing energy: We have noted in Fig.1.2 that nuclei with even Z are moreabundant than with odd Z. In order to account for this pairing effect, aterm & is added, which is given to be of the following form:

+ 36.5 A for even Z-even H nuclei

0 for even Z-odd N or odd Z-even,H nuclei

-36.5 A for odd Z-odd N nuclei

(3-5)

Apparently, the odd Z-odd B nuclei are least stable.

Shell effects: This effect gives a most important contribution to the liquiddrop formula that has come to be realized only very recently. Since the pairingeffects are related to the f i l l ing of the energy levels, the shell effects aredue to both the shell structure of the nucleus (next section) and the additionalpairing effects. This correction is calculated by using a method, due toStrutinsky, which uses the liquid drop formula (with the above-mentioned fiveterms) and a shell model of the nucleus. In view of this method, this contri-bution is taken as a correction to the liquid drop (or the semi-empirical)massformula, rather than a term of the formula, A description of this method i sbeyond the scope of this text ; but i t is relevant to t e l l that th ismodification has almost revolutionalized the field of nuclear fission andheavy-ion collisions (Chapter 5).

Combining all the terms, the liquid drop formula for the bindingenergy becomes:

B.B.

Explicitly)

(3.6)

(3.7)

Using E4. (1,11), the so-called "von Weizsa'cker mass formula" for themass of a nucleus is

MA - Z* + (A-Z)% - \ /.yA - ac v

. (3.8)

The empirical values of the constants, as obtained by Myers and Swiatecki, that

are used widely at the present time are

* T • 15 MeV

* B • 17-9U39 MeV

afl • 0.7053 MeV (which corresponds to rQ = 1.221*9 fm).

For the symmetry term, these authors define the relevant coefficient as

aa - a1k - a2k A " 1 ^ . ( 3 . g )

k - 1.7626 .

with

A better accepted value of k = 2.53, as obtained by the Lund group (Johansson et

The constants in Eq.(3.8) can also be derived theoretically but thiscannot be done here. The reader is therefore referred to the other readingmaterial.

The binding energy per nucleon, on the "basis of the semi-empiricalmass formula, can be calculated by using Eq.(3.7). Fig.3.2 shows the contr i -bution of different terms of B.E/A and the to ta l B.E/A as a function of A,given by Eq..(3-7). A comparison of the tota l B.E/A with the correspondingcurve in Fig.1.It shows that except for the resonance structure (the peaks),the two match nicely. The peaks are now shown to be due to the "shell effects"that must be added to the liquid drop energy to obtain the actual ( real is t ic)binding energy of the nucleus.

In some texts this is also called symmetry energy.

-21- -22-

3.3 Collective model

To tie included in t he f i n a l vers ion.

3.1» Shel l Model

Both the licjuid drop model and the collective model treat the nucleus

to have only a bulk structure. The liquid drop model is given to calculate

only the binding energy of the nucleus by considering the nucleons inside a

droplet to interact only with their neighbouring nucleons (see the volume energy

term in Sec.3.3). The collective model develops this picture of a hydro-

dynemical object into a complete model of the nucleus by allowing it to both

rotate and vibrate. However, the liquid drop model is shown to give large

deviations in the binding energy of certain nuclei having the number of

nucleons S or Z = 2, 6, 20, 26, 50, 82 and H = 126. Nuclei with tbese numbers of

neutrons or protons or both are found to exhibit extra stability and vith zero

electrical quadrupole moment {means,spherical in shape). The collective model

is also silent about any characteristic nature of these nuclei. Since the

existence of these nuclei could r:ot be explained till the advent of the shell

model, *fife»ft-ffvnBi«rB of nucleon have become known as "magic numbers" and the

nuclei with these numbers of nucleons are called "magic nuclei" or "doubly

magic nuclei" if both M and Z are the magic numbers.

The shell model of the nucleus was given exclusively to explain the

magic numbers in nuclei, though it is now also found to describe many other

nuclear properties. The model makes the fundamental assumption that the

nucleons inside the nucleus move as independent particles and considers the

problem of arranging these nucleons in certain well-defined orbits (called

"shells", and hence the name "shell model"). The idea of the motion of

nucleons in specific orbits, however, require a centre of force, like the

nucleus provides it for the electron orbits in the atomic model but this does

not exist for the nucleus itself. However, the compelling similarity of

stability between the magic nuclei and the inert atomic gasses forced nuclear

physicists to> invent a. centre of force field. It is assumed to be provided

by the nucleons themselves such that each nucleon moves inside the nucleus in

a fixed orbit under the influence of a central field of force (or the central

potential V(r)) produced by the remaining A-l nucleons. The shell model does

not worry a*out the question of how this potential is produced but simply assumes

its existence. We can, however, draw upon our experience with two-body systems

(Chapter 2) and consider this potential as some sort of aggregate of the two-

nucleon potential. Various phenomenological forms that have been tried are

(i) infinite harmonic oscillator, (ii) infinite rectangular well and (iii)finite harmonic oscillator or square well rounded at the corners. One oftenused fora of this potential is the Woods-Saxon potential. Once the form ofthe potential V(r) is fixed, the motion of each proton (or neutron) can beobtained by solving the Sckrodinger equation. In the following, we give thesolution of the Scbrodinger equation for the different potentials to reproducethe magic numters. We shall consider the motion of only one of the particlesthat can be a proton or a neutron.

The harmonic oscillator potential has the form

V(r) - | k r3 , (3.10.)

where k ia the force constant related to the nucleon mass m and the

oscillator frequency u> as

(3.11)

The potential is shown in Fig.3-3(a).

The Schrodinger equation

•i / • V(r)] (3.12)

solved for V(r) to be of the form given by Eq>(3.1l), gives the energy eigen-

values

where

(H + |> 1W

B = 0,1,2,3,!*,...

(3.13)

This shows that all the states in the harmonic oscillator are equi-spaced.

The wave function i|i(r) has both the angular part and the radial part.

The angular part of the wave function requires that the oscillator quantum

number N is related to the orbital angular momentum quantum number I as

B = Sn + t - 2

where the radial quantum number

n - 1,2,3,1*,...

(3.15)

(3.16)

-23- -21* -

• Eqs'.fj'.rii) 't*> {3^!&)'• apparently shows 'that ior $ V 2,- more, than one value of' ,

i satisfies E| . (3-15.). The -various- possible values of SL-ior different,S are: •

0

1

2

3

h

1

l

1,2

1 ,2

1,2,3

0

1

2 , 0

3 , 1

"t,2.0

2

6

12

SO

30; and so on (3.IT)

The states for a given N but for different (n,i) are said to he degenerate

since they all have the same energy, given by Eq.(3.13). These states are

labelled by the quantum numbers n and 4, where for I the following

spectroscopic notation is used:

Notation:

0, 1, 2, 3, It, 5, 6,...

s> p, d, f, g, h, i,... (3.18)

The possible states, given in (3.1T), for the harmonic oscillator are shown

in Pie.3.3(a), in the notation of (3-18). Each states can allow 2(2A+l)

nueleons of one kind. This is also shown in (3.17) and Fig.3.3(a). Thus,

adding the nucleons from below, shell closures occur at the neutron (or

proton) numbers

2, 8, 20, 1*0, TO, 113,, (3.If)

Apparently, the harmonic oscillator gives only the first three magi,e numbers.

Taking the potential to be a rectangular well of the form

V(r) = -VQ for r f E (nuclear radius)

= 0 r > E (3.20)

also gives the same first three magic numbers, though a l l the states are nowobtained to be non-degenerate (see Pig.3.3(b)).

The tfoods-Saxon (real part) potential is given by

V(r) = - (3.21)

where ..R is the value of r at which the depth of "the: potential, is one half'' ot the. total and a is a parameter of founding off of ttre potential at the

surface of the nucleus. This'also gives a situation similar to that for the

rectangular well, as shown.in Fig.3-3(c). .

. . 'Afctlris stage, M.G. Mayer and; also-independently.; 0. :H«xel",. J.HiD. -Jensen

and H.E. Suess in 191*9 made a suggestion, worth a Noble Pri2e *' -that the above

potentials modified by a term (1.1), representing the interaction betveen the

spin and orbital angular momentum of each nucleon (the spin-orbit interaction),

would give a l l the observed magic numbers. These authors also did not vorry

about the origin of this correction term but added i t only empirically. An

addition of this term led to the spl i t t ing of each state into two states with

the to ta l angular momentum

1 + | and I - | (not for Jt= 0)

(where s = —) with j lying lower. This is shown in Fig-3.3(d), which depicts

an exact reproduction of all the magic numbers. The states are now labelled

by the ituantum numbers (nlj). This result is independent of the type of

central potential taken, though for other nuclear properties Woods-Saxon is

found to be invariably better and harmonic oscillator is used more often.

The number of nucleons in each state, is now given by (Sj+l).

Extension of shell model to higher shell closures has been of special

interest recently. This is related to our desire to extend the table of known

elements. The above calculations shov that after Z = 82, N = 126, the doubly

magic nucleus should occur at Z * 126 and N = 184. However, some other studies

related with the stability of nucleus against fission and a decay predict

nuclei in the vicinity of Z = lilt to be more stable. Such elements are called

Super heavy elements and their synthesis in the laboratory or availability

in nature will settle many questions in the structure physics of the nucleus.

finally, let us mention that the shell model is able to give a good

account of many other nuclear properties like (i) stability of even-even nuclei

as compared to odd- and odd-odd nuclei, (ii) the ground state spin of closed

shell or nearly closed shell nuclei, (iii) the magnetic moments and (iv) the

binding energy of the closed shell nuclei.

ESBRCISES:.

Ex.3.1: What should be the kinetic energy of a particle moving with velocity

c, confined inside ai.box vhose dimensions are uncertain by 10 m.

•) Awarded the Nobel Prize in physics for the year 1963

-25--86-

S o l u t i o n : CHAPTER k: NATURAL RADIOACTIVE DECAY OF THE NUCLEUS

If the uncertainty in the measurements of the dimensions of the box is

the uncertainty in the position of the par t ic le , then

-lh -12Ax =10 m. = 10 cm .

The uncertainty in momentum is then by the uncertainty principle

The kinetic energy of the particle in the box = Ap.c = T^-

x 1Q~2T e rg , s ec . * 2.9979-25 * 10 ° cm see"

1.Q5UU9U 2.997925 * 101 Q

1.60206 * 10 » 10~1 2

19.T3 MeV .

MeV

l*.l Hatural Radioactivity

In Fig.1.3 we have seen that a l l nuclei have a natural tendency to goto the 6 stabi l i ty line by emitting a B , £ or an a part icle . Also, if thenucleus is in an excited s ta te , i t will go to the stable ground state by emittinga y ray. Nuclei are said to be "radioactive nuclei" and decay radioactively ifthey go spontaneously *) to other stable nuclei by emitting an a particle(He nucleus), the 8 particle (electrons) or the y ray (photons) or anycombination of these part icles . When a radioactive nucleus decays by two ofthese processes, the decay is called a dual radioactive decay or the radioactivebranching. The branching occurs more often between the a- and fi-decay. I t isimportant to note that the radioactive decay occurs naturally and spontaneously(means no reaction i s involved). In other words, radioactivity is a propertyof some (unstable) nuclei l ike the mass, size and shape, etc. Radioactivenuclei can also be produced ar t i f ic ia l ly (called, a r t i f ic ia l ly produced radio-isotopes) in nuclear reactions (see, e.g. Eqs.(k.l) and (5-28)}. Thear t i f ic ia l ly produced radioactive nuclides decay in exactly the same manner asthe natural radioactive nuclides. A radioactively decaying nucleus is calledthe parent (or mother) nucleus and the decay product is called the daughternucleus. The daughter nucleus can be a radioactive or a stable nucleus; ifradioactive i t will spontaneously decay further and the parent and the daughternuclei are then said to be genetically related. A radioactive nucleus can havemore than one generation before i t finally goes to a stable nucleus.

The property that radioactive nuclei can have generations beforereaching to a stable nucleus leads us to show that most of the radioactive nucleifound in Nature and produced ar t i f ic ia l ly can be classified in four radioactiveseries. These series are given the names, thorium, neptunium, uranium-radium

232 237 238and uranium-actinium series after their parent nuclei Th, Np, U and235U, respectively. Notice that the nuclei 232Th, U and 5U are thenaturally occurring radioactive nuclides with their half lives of the order of

10 237the age of the Earth (~10 years) and 'Up is produced ar t i f ic ia l ly in, say,

the nuclear reaction

K ^ 238T,92

I237.

a decay

92 B decay237W

93(U.I)

*) The other method of achieving a new nucleus is through nuclear reactions,

studied in the next chapter.-28-

- 2 7 -

— • • - S? "

The ar t i f ic ia l ly produced 23TNp has a half-l ife of a.25 x 10 years which is

much shorter than the estimated age of the Earth and hence explains why the

members of this series are not available naturally.

The successive daughter products or generations in a radioactive series

must he connected through an a or/and fi emission, as the y decay is related

only to the excitation of the nucleus. Since both S- and y-decays do not

involve any change in the mass of the nucleus, the mass number of the series

must be related to the mass of the a part icle . In fact, the number of series

being four is a result of the fact that the mass of a particle is four. The

mass numbers of the four series are

A = Un, ltn+1, ltn+2 and hn+3 (k.2)

where n i s an integer . Members of the thorium, neptunium, uraniuza-radium

and uranium-actinium ser ies have t h e i r masses given, respect ively , by Eq.(U.2).

The sequence of decay of a l l the four ser ies i s shown in Figs .^ . l to 1*.!*,

depicting the. masses always given by Eq.(it .2), the brancbing^and the stable

end product. The end product of a l l the three natural ly occurring series i s

lead (of course, different isotopes satisfying Eq.(it.£)) and that of a r t i f i c i a l

ser ies i s Bi. In proof of the existence of the three natural ser ies one

can mention a, fact that helium and lead are always found in a l l the natural

radioactive ores. All the important charac ter i s t ics of the four ser ies are

summarized in the following t ab le ;

Table it.I

Important charac ter i s t ics of the four radioactive ser ies

Name of series

Thorium

Neptunium

Uranium-radium

Uranium-Actinium

Type

Natural

Artificial

Natural

Hatural

Massnumbers

Itn

ltn+1

ltn+1

ltn+3

Parentnucleus

232Th

^ N p

2 3 ^

235U

-Stable endproduct"'

2 0 8Pb

2 0 9Bi

2°6Ph

20TPb

Branchingoccurs for

"- '• members

2 1 2Bi

£13Bi

ss •

Half l i f etime (yrs)

1.39 * 101 0

2.25 x 106

U.51 * 1O9

7.0T i 108

k.2 The decay law

In Table U.l we have given the half-l ife times T,/g of the radioactive

nuclei without having actually defined i t . The definition of T^.g follows from

the decay behaviour of the radioactive nuclei. In a large number of experiments

i t was found that the strength of radiation or the activity of a radioactive

nucleus decreases linearly with the number of radioactive nuclei present at

a certain time t in the given sample. In other words, the rate of (spontaneous)

decay is proportional to the number of nuclei N present at that time t :

dN

at (It. 3)

-XV

The negative sign in (It.It) shows that N decreases with t. Eq.(4.1t) can be

written as

fL . -x dt (It.5)

such that if Ng is the number of radioactive nuclei present at t = 0, then

integrating (It.5) in limits of Jt_ to B and zero to t, we get

V-it (it.6)

This shows that the radioactive nuclei follow an exponential decay law. The

proportionality constant ^ is called the radioactive decay constant which,

from Eq.(lt.5), can be defined as the probability for decay per unit time.

Experimentally, however, we always observe the activity, the rate of

decay ~ , by measuring the counting rate in an instrument called a counter,dt

Denoting

dtthe activity {It.7)

we can show that activity also follows the exponential law. This follows

from Eqs.(lt.!*) and (it.6):

A = AN

-At

If the activity at t = 0 is AQ, Eq.(it.S) gives

U.8)

(It.9)

-29--30-

Substituting (It.9) in {It.8), we get the exponential law

A = AQ e • (^.10)

Fig.lt.5 shows the decay laws of both H and A. The fact that radioactive decay

is exponential justifies the stat is t ical nature of this phenomenon.

The definition of half-life time T1 / 2 now follows from (It.6) or (!t.lO).

It ia the time during which the number of radioactive nuclei or the activity

reduces to one-half, as shown in Fig.lt.5. Thus,

("t.16)

t = T 1 / 2 when

Substituting (U.ll) in (4.6) or (It.10) we get

which, on taking logarithm on both sides, relates T ,„ to X

T = *JLj. _ 0.69?1/2 X ~ i

(1*.11)

( I t .12)

Since X is fixed for each radioactive nucleus, the half-life time ! . , „ isalso a constant of a radioactive nucleus. It is evident from rig.It.5 that wecan start counting the time at any moment and when the activity of the radio-active nucleus reduces to half, the measured time gives i ts half life time T

•1/2"

In addition to the half-life time of the radioactive nucleus, we

define i t s mean life-time T , which comes out to be equal to the inverse of

decay constant X . The mean life-time T is

£- I t(-dN)0 J0

Substituting for dN from (It.6) , we get

which in terms of T , becomes

Ct.13)

I t a lso follows from (It.6) t h a t for

U.15)

get

rget \

which shows the* the number of nucleons^reduced by a factor of — during themean life-time of the radioactive nucleus. This is also shown in Fig.U.5,

i l lus t ra t ing the differenoe between the half life-time T ,_ and the mean l i f e -

time T .

An interesting application of the decay law is to the branching processand to the calculation of activity of a radioactive sample from the knowledgeof i t s mass, atomic mass and decay constant. This is done in Exercises U.I andIt.2, The anits of activity A are Eurie (Ci), where

Also

1 Ci = 3-TO x 10 disintegrationsJsec.

1 millicurie (me) = 10 Ci

1 microcurie (ye) = 10 Ci (U.17)

- 3 1 -

In the following sections, we try to answer the very fundamental

question: why some nuclei decay spontaneously. In other words, we study the

physical processes possible for the alpha-, beta- and gamma-decays.

I*.3 Alpha decay

In this section, we shall try to answer two questions: (i) From our

discussions in Chapter 1, we know that the forces acting inside a nucleus are

the long-range repulsive Coulomb forces due to protons and the short-range

nuclear forces due to the excess of neutrons. A. balance of the two keep the

nucleus stable. Therefore, the nucleons inside a nucleus can be considered to

move in an aggregate potential produced by these forces. The question then ia:

how do the nucleons or a combination of them (the a particle) come out of

such a potential 1 - Is the potential for heavy nuclei (for which the a decay

ia observed) mainly repulsive Coulomb potential that the a particle simplyhas to

runB aown it or it has some other form that the a particle/overcome it some-

*\how? ( i i ) The f i r s t question pre-supposes t h a t in heavy n u c l e i , two neutronsand two protons combine t o form a s t ab l e a p a r t i c l e before e j ec t ion . We know

an &i

from Exercise 1.1 t h a t / a p a r t i c l e ( He nucleus)has la rge binding energy of

28.3 MeV and hence i s a s trongly bound system, perhaps, formed at the surface

Of the nucleus j u s t before emission. However, the question t h a t s t i l l a r i s e s- 3 2 -

is: ' why is i t that/a particle is preferred in the spontaneous decay of

• 3radioactive nuclei than the emission of other light particles n, p, d or He '.

Answering the above-mentioned two questions explains the physical

processes involved in the o decay of radioactive nuclei. We shall first analyse

the second question since i ts answer is based simply on the binding energy

considerations. The first question requires to invoke the quantum mechanics

and hence a solution of the Schrodinger equation.

We consider the general decay of a parent nucleus with mass Mparent

to a daughter nucleus of mass M accompanied by the emission of a

particle of mass m., where the particle could be n, p, d, He or He. The

conservation of mass requires us to write

H = M , + m. ± Q (It.lo)parent daughter i

Q, the energy equivalent of mass defect, is the amount of energy released(+ sign)

or required (- sign) to be supplied from an external source. In the case of

+ Q, the energy released is in the form of kinetic energy which is shared between

the daughter nucleus and the emitted particle, in ratio of their masses. Since

the mass of daughter nucleus is very large compared to the mass of emitted

particle, the energy carried by the daughter nucleus (called the recoil energy)

is very small. The Q value in (It.l8) therefore appears mainly as the kinetic

energy of the emitting particle- Taking M as the mass of a heavy

nucleus and for m. as the masses m , m , M^, M, 'or M, , we find that Q isHe He

positive only for the <x particle emission. This means that only the a decay

of a heavy nucleus is energetically possible. For all other light particle

decays, energy has to be supplied externally. Actually, we shall see in the

next subsection that instead of p and n emission, electron or positron

emission is more spontaneous.

« decay of a nucleus can be considered as a process inverse of the

scattering of a particle from heavy nuclei (like fission can be taken as an

inverse of fusion reaction, discussed in the next chapter). Rutherford in

1911 and later on many others have shown that the scattering of a particles

z Zae2from nuclei follow the Coulomb law •— I a is the chaige of a particle

and Zo of the nucleus) rather beautifully upto a distance of approach r of-12

10 cm, the radius R of the nucleus. For r { R the scattered a particle

meets a hard core attraction which balances the Coulomb repulsion and allows

the a particle to be absorbed by the nucleus. The resulting potential between

the a particle and the nucleus is obtained to be of the form given in Fig.^.6.

For heavy nuclei, the height of the potential, called the Coulomb barrier, is

found to be nearly 25 MeV.-33-

In a decay, the energy E of the emitted a particle (the kinetic energy)

is measured and found to have a fixed value (discrete value) for each nucleus.232 212

This varies from 1* to 9 MeV in going from Th to g^Po . This is represented

by a dotted, line in Fig.l4,6. The problem is then evident: a particle has

energy E, much leas (25-1* to 9 = 21 to 16 MeV) than the barrier height of 25 MeV.

If the analogy of a decay being a reverse process of a scattering is true,

then apparently the a particle is in a deep well of 25 MeV and has energy of

only k-9 MeV (depending on the nucleus). The question then reduces to: What

is the probability for such a particle to come out? Classically, the answer

is zero. However, quantum mechanics always associates a finite probability

(how-so-ever small) with every process. For a decay the probability of finding

a particle outside the Coulomb barrier is found to be finite, though it takes

millions of years ("10 Jirs ) for it to come out. If the time taken for the

a particle emission were not so large, than it would have been difficult to

find any naturally occurring radioactive nuclei. Actually, the a particle

does not climb over the barrier but, as Gamow and independently Gurney and

Condon have shown, it tunnels through the barrier as a wave, called the quantum

mechanical tunnelling effect. For simplicity, the wave that tunnels through

thebarrier is shown as arrows (of diminishing magnitude) in Fig.l4.6. The

solution of the quantum mechanical tunnelling problem requires the solution

of the Schrodinger equation which we do not wish to attempt here. However,

in the following, we give the main steps involved and the results obtained

thereby showing the success of quantum mechanics.

Solution of the Schrodinger equation for a simple potential step of width

x (Fig,It.7) shows that (see Exercise h.h) the probability of transmission of

a particle with energy E < V can "be approximated as

-2 k x

where

'*% (E - VU))

Ct.19)

(4.20)

Since k is imaginary, the wave produced on the right hand of the potential

step is an imaginary wave.

For our actual problem of the a particle being in a potential function

of Fig.it.6 {this is three dimensional,whereas Fig.k..7 is one dimensional),

we can divide the potential into a series of small steps, each of width dr

(as shown dotted in Pig.k.6) and carry out the probability sum *' over all

the steps:

*) When the interval of summation is small, it can be carried out as an integration.

-3k-

where now

with

P = ek dr

% (E - V(p))

V(r)

E =

(4.21)

(U.22)

(4.23)

(b.24)

We can also define the barrier height

„ 2

B (It.25)

R and R are the values of r at which the line representing E meets the

barrier. Solving the integral in Eq.(4.21) is then simply & mathematical

exercise which we shall not do here. For our purposes, it is required to be

solved for |j- = |- < 1.Hl B

In an attempt to come out, if n is the number of times the a particle

hits the walls of the potential barrier in one second, the decay probability

per unit time is

X = n P (U.26)

The a particle of energy E moves with velocity

inside the potential well of radius R, such that

(4.2T)

(it.28)

(see Exercise 4.3 for an estimate of these quantit ies). Rewriting Eq.(U.26)with (4.26) in i t , we get

log A » log ^ + log P (4.29)

Substituting for the solution of P from Eq., (4 .31) , we get

-35-

i

log X = log - 2m B 1/2 |-fB(l/2

TOwhich verif ies the empirical Geiger-Nuttal law

log \ (It.3D

Thus the answer to our first question is the quantum tunnelling of a particle

through a potential barrier.

As an additional success of the 'barrier penetration theory of a decay,

we notice that the excitation energy E of the o particle, which on the potential

barrier diagram (Fig.4.6) can be measured in terms of the width of the barrier,

gives correct trends of the half-time times of the radioactive nuclei. We know212 232

that Po has the largest excitation energy of 8-95 MeV and Th has the

smallest excitation energy of It.05 MeV which corresponds to the barrier widths

AB and CD in Fig.It.8. Since AB is shorter than CD, the decay probability X212 2^2_ 212

for Po should be larger than for T?h and hence the life-time T ,„ of Po232^should be smaller as compared to that for Th. I t is found that T ,„ forp i p _T p^p in

Po is 3.0 x 10 ' sec and that for Th is 1.3 * 10 yrs.

I*.It Be ta decay

it.5 Gamma decay *

EXERCISES:

212Ex.4.1: Use the decay law to explain the radioactive branching of Bi in

203 212thorium series to Tl and Po by 35.4 and 64,6 percents, respectively.

212 20S 212Solution: Branching of Bi to Ti. will occur by a decay and to Poby B decay. Let the decay constants of the two processes be denoted by X and

212 *\. , respectively. Then from the decay law, the rate of decay of Bi is

vwith the mean-life time

-36-

The probability of a decay is 35-^ and that of 0 decay is 6k.6$ In terms

of X and A this means thata p

212 232Sx.U.3: Theat particles emitted in the radioactive decay of Po and Th

have energies 8.95 and -05 MeV, respectively. Estimate the order of probability

ofot particle emission in each case. The half life times for Po and Th-7 10

are, respectively, 3-0 x 10 see and 1.3 x 10 years.

and

35.

6k. 6

These ratios are called the branching ra t ios .

One can also define, very approximately, the mean life-times for the

two processes, separately, as

Solution:

212.Po : E = 8.95 MeV

2E1 He!*.OO26 x 931.1*8 MeV

= 3728-331 MeV

2 x 8.95 x (2 .998^ 8 -1 r w .3Y2fl 331 x 10 m sec LNote: for proper uni ts of

?.O78 x 107 m s e e " 1

v, if E is in MeV, m has tobe in ^

ThenRadius of the nucleus H = 10 m

22ltEx.1>.2: Determine the activity of 1 gm sample of ggEa when it looses two

percent of its mass in 25 years.

= 2.08 , 1021 sec"1

10"

T1/2 = 3.0

Solution: Let S be the i n i t i a l number of nucleons in Ra and H be the number

left af£er 25 years. Then

100

Then

" 1

T l /20.231 U 0 T sec

X_ _ 0.231 * 10T sec"1

n 2.078 x io21 see"1

= 1.11 x 10

Using the decay lav' | - = e~At = O.96 with t = 25 yrs = 25 * 365 * 2k x 60 x 60

seconds. Substituting

= 2.56 x 10 1X sec X .

Then activity

with

A « XN

230

mass number

23number = 6.02217 * 10 . a t o m f , = 2_gQ x 1(J21 a t o f f l S

b 224224

A = 2.56 X 10~11x 2.69 x 10 atoms sec

o10i o 1 0

c t = 1 - S 6 c . i

13.70 x 10

10

232,Th : E = It.05 MeV

/H _ /

Jma ' J

2 " U.O5 » (2.996)^3728.331

v 1.398 " 10T

10 m sec X = 1.398 x 10™ m sec - 1

1.398 x i o 2 1 s e c - 1

O 1 0-1.3 *10

= 1.3 x io 1 0 x 365 x 2lt x 60 x 60 sec = It.100 x 1O1T sec

P .

T l / 2

. . 1-690 x

1 0 - l 6 s e c - l

1.398 x 102

212P212This shows that probability of emission of a part icle in Po is much larger

compared to that in 232Th, as expected from Fig.It.8.- 3 8 -

CHAPTER 5 ; NUCLEAR REACTIONS

5-1 General considerat ions

In this chapter we shall study the collision of one nucleus with another

nucleus, or with gaisma rays (the photons), in order to learn more about the

nucleus itself or about the reaction mechanism. Nuclei do not react chemically

since the nuclear binding energy is much larger (7-8 MeV/particle) compared to

the chemical binding (a few eV). Also, the nuclei can be looked at as billiard

balls so that a nucleus can be accelerated (i.e. given energy) to bombard on

another nucleus. The accelerated nucleus is called a projectile and the

bombarded one, a target nucleus. Projectiles and targets can be anything from

protons,neutrons, deuterons, a particles, Li, Be to U. The charged particles

and nuclei are accelerated in machines, whereas the neutral neutrons are

produced as secondary particles in nuclear reactions or reactors. The machines

used are the Coekcraft-Walton, Van de Graaf, eyclotrons, synehrotons and linear

accelerators, etc. If the collision is such that the projectile enters the

target nucleus, then in the exit channel we generally observe some light particle

or y ray, called the emitted particle and the residual nucleus. This is shown

pictorially in Fig.5-1 and is written as

Target nucleus(projectile, emitted light particle) Residual nucleus .

This means that a projectile nucleus or particle hitting a target nucleus leads

to an expulsion of some light particle leaving behind the residual target nucleus.

For example, a proton hitting a Mg nucleus gives

23Na+ P Ha + a (5.1)

which can be written as

26Mg (p ,a ) 23Ha (5.2)

The residual nucleus, Na, may l ie in the ground state or in the excited

state {j ). The excited residual nucleus is shown with an asterisk(*)and with

the spin-parity Jv of the state such that (5-2) becomes

26Mg<P.a) 23Na» (JT) . (5.3)

The excited residual nucleus always goes to the ground state by an emission of

one or more y rays. The study of such emitted y rays gives information on

excited states of the residual nucleus and is an important subject in nuclear

-39-

• .iw^-v^Mfc^rl"

physics, called gamm-ray spectroscopy. Any further detail on this subject is,

however, beyond the scope of this text.

When two nuclei collide, a nucleus being actually a many nucleon system,

many different kinds of reactions are possible. We first classify these

reactions into two categories:

1. Particle-induced reactions.

2. Heavy-ion (heavy nuclei in ionized states) reactions.

In the first class of reactions, the projectile is a y ray p, n, d,tC or some

light nucleus like Li, Be upto at the most C; whereas in the second case nuclei

as heavy as U are used as projectile. In both types of reactions, the target

can be any nucleus. Since the physics of heavy ion reactions is very new,

not yet fully understood both theoretically and experimentally, we shall

consider in the following the reaction mechanism associated with the particle-

induced reactions only.

A particle-induced reaction can be classified chiefly as one of the

following:

1. Reaction of transmutation or substitution.

2. Scattering (elastic or inelastic).

3* Photo—nuclear reactions -(radiative capture or photo-disintegration).

k. Fission or fusion.

In a normal particle-nucleus reaction, the incident particle gets absorbed in

the nucleus and a new particle (sometimes particles) is emitted, like in

reaction of Eq.(5-2) and we call i t a reaction of nuclear transmutation or

substitution, since a residual nucleus of new species is given in this reaction.

If the emitted particle is the same as the incident particle, like, e.g-. anC piT

prcrtontombarded on Mg givesrise to proton and Mg with, of course,

different energy,

Mg (p.p'l 26Hg (5.U)

we designate it as the scattering process. This is shown schematically in Fig.5.2.

The prime (') on p means to show that it could be a proton different from

the incident proton. A nuclear reaction is always accompanied by some scattering.

In other words, there can be a collision where only scattering takes place

but there cannot be a collision where only a reaction occurs. Like in nuclear

reactions, the residual nucleus in scattering process also can either remain

unexcited (i.e. in the ground state) or go to an excited state (J ). If it

occurs in the ground state, we call the scattering as elastic scatterings

-kO-

Oft.the GtMer hand, if i t goes.i-n an excited s ta te , from where i t vil-l decay •

to the.ground state imaedifit-eXy through art emission gf.,sue or more Y rays, we

refer to. l t as inelaetic .scattering. The inelastic scattering reaction i s ,

similarly, denoted as

2 W 2 W * • (5.5)

Apparently, the inelastic scattering also gives information on the excited states

of the residual nucleus. In photo-nuclear reactions, there is always a y-ray *)

in the entrance or in the exit channel. If the Y ray occurs in the entrance

channel, we call i t a photo-disintegration process; like

27 (5-6)

On the other hand, if a. y ray appears in the exit channel, we get what is

called a radiative capture process; like

(5.T)

I t is evident from Sq.s.(5-6") and (5.7) that radiative capture and photo-tttsintegrations are inverse of each other. Also, the radiative capture andinelastic scattering must involve a similar process of forming an excitedresidual nucleus. Finally, there are special reactions of fission and fusionin which either a nucleus breaks (fission) into two or more nuclei or twolighter nuclei combine (fuse) to fora a new heavy nucleus. Both theseprocesses are of special interest since they are accompanied with the releaseof a large amount of energy. We shall discuss these in Sees.5.6-5.8.

All these reactions must satisfy the basic laws of conservation ofnucleon number, nucleon charge, momentum, energy and mass. This means thata l l these quantities in the entrance channel must balance in the exit channel.For the momentum conservation, ve must remember that this is true only whenno external forces act. Also, the Einstein energy-mass equivalence relationrequires a simultaneous conservation of energy and mass, which leads us toto the definition of the Q value of the reaction. Following the calculationsof the binding energy of a nucleus, a nuclear reaction should be written as

•) I t is relevant to remind here that a l l Y rays whether of atomic origin orfrom X-rays or those obtained during the de-excitation of the nucleus, arephotons of different wavelengths.

-1*1-

Mg'-t-:p -•- Ba + a + Q-value .

Thus.,. one can • in fact say that

( 5 . 8 ) •

Q-value = - B.E (5.9)

This is i l lustrated in Exercise 5.1. I t is apparent from the discussion that

depending on whether the mass on the left-hand side of reaction (5-8) is larger

or smaller compared to the mass on the right-hand side, Q is positive or

negative. Reactions with positive Q-value mean that they are accompanied with

the release of energy and are called exQ-thennie or exo-ergic reactions.

If Q-value is negative, we require at least that men of energy to start the

reaction and such reactions are called endo-thermic or endo-ernic reactions.

5.2 Reaction mechanisms

In the reactions exemplified above, we have seen that though the target

and projectile are always Mg and proton, respectively, in one case the proton

is scattered elastically or inelastically from Mg, whereas in other cases we

get "TJa + B particle or ki + y ray as the reaction products. This simply

illustrates the fact that there is no a priori way of telling the type of

reaction a particular collision is going to result in. Different reaction

models have been proposed which all give different predictions. A comparison

of the experiments with these models allow us to determine the nature of the

reaction and hence the reaction mechanism associated to a particular reaction.

Reactions induced by the particles or very light nuclei can be explained

either as a one step, direct reactions or as a two step, compound nucleus

reaction. Being a one step process, the direct reaction is assumed to take-22

place in a much shorter time of about 10 seconds as compared to a compound

nucleus reaction which requires some 10 seconds. In the direct reaction

process a particle is assumed to be either removed (stripped off) from the

projectile by the target (called, stripping reaction) or inversely, the

projectile is allowed to pick-up a particle from the target nucleus (called,

pick-up reaction). The two processes are thus exactly inverse of each other

and the typical examples are (d,p), (d,n) or (p,d), (n,d), respectively. Hotice

that these models are warranted by the fact that the deuteron is a loosely

bound system. As an alternative description, direct reaction process is

also considered as a knocH-out process where the incident particle goes

straight into the target nucleus and knocks out some nucleon from it. The

three processes of stripping, pick-up and knock out assumed in direct

reactions are shown in Fig.5.3.

In the compound nucleus model, the low-energy projectile is first

assumed to be absorbed by the light mass target nucleus thereby forming a nev

excited compound nucleus which then decays by emitting a nev particle (or-nev

particles). This two step process of (i) formation of an excited compound

nucleus and (ii) decay of the so-formed excited compound nucleus, is called a

compound nucleus reaction. The two steps are Shawn schematically in Fig.5.!t.

The requirements of large reaction times for the compound nucleus reaction,

as compared to that for direct reactions, is self evident. We shall see in

the next section that in the compound nucleus model, the formation cross-

section, i.e. the first step of the formation of the excited compound nucleus

is calculated by assuming that it goes to a well defined, isolated energy

level (the resonance theory). Hovevii1, If the energy of the projectile is

large suetLthat mor»..than one level gets excited in the process of formation

of compound nucleus, then ve use the so-called continuum theory for the

evaluation of the compound nucleus formation cross-section. Furthermore,

together with the high excitation energy of the projectile, if the target

nucleus is also a heavy nucleus, then it is not possible to reach either the

excited levels individually in the resonance model of the compound nucleus

or collectively in the continuum theory. In such a situation, a new approach,

called statistical theory, is used. This method is, however, approximate

since for the laws of statistics to be used, the number of particles in a

nucleus is rather small.

Still another model, called the optical model is introduced where

the target nucleus is considered as an opaque body which absorbs only a part

of incident wave (the projectile is treated as an incident wave) and allows

the remaining part to be transmitted and reflected. In this language, the

compound nucleus model is a complete absorption model. Thus, the optical

model is applicable mainly to the inelastic scattering phenomenon where the

reaction takes place only at the surface of the target nucleus.

It will certainly be of interest to dwell on these models in detail.

However, in this text, we limit ourselves only to a discussion of the compound

nucleus model and refer the reader to the other reading material for details

on other reaction mechanism.

5.3 Compound nucleus model

Guided by the concept of liquid drop model of the nucleus (Chapter 3),

Hiels Bohr in 1936 introduced the two step compound nucleus model for the

collision of two light nuclei at relatively low energy. The model is sketched

in Fig.5.U depicting the tvo steps to be independent of each other. The two

steps are:-1.3-

First step: Formation of an excited compound nucleus due to the complete

absorption of the incident particle (the projectile) by the target nucleus.

The essential condition is that the colliding nuclei are light and the

bombarding energy is sufficiently low. The mass and charge of the compound

nucleus is the sum of the masses and charges of the colliding partners:

A22 A + Z~a21 22

vv (5.10)

The excitation energy, E*, of the compound nucleus is given ~ay the kinetic

energy and the binding energy of the projectile and, in the language of

s t a t i s t i c a l mechanics, is called i t s nuclear temperature • ) . This energy i s

shared equally amongst the /L + Ao nucleona, which accounts for the large•L _jc reaction

reaction time of the order of 10 seconds. Since the^time is very small

on the physical scale, i t is not possible to observe a compound nucleus.

However, i t s half l i fe time T can be inferred from the observed width r

(at half maximum) of the level (see Fig.5.5 and Exercise 5.2) in which the

compound nucleus is formed. This is given by the uncertainty principle

(5.11)

Thus, in other words, the compound nucleus model assumes that the probability

of formation of the compound nucleus i s maximum when the excitation energy

of the compound nucleus i s such that i t goes into a particular energy state

of the resulting compound nucleus. Furthermore, since a compound nucleus C*

with mass A, + JU and charge Z + 1^ o a n b e formed due to the various different

target projecti le combinations, the compound nucleus model does not bring into

consideration the way i t has been formed. Therefore, a l l that the compound

nucleus model in i t s f i rs t step cares for is i t s to ta l mass, charge and the

excitation energy and is independent of i t s mode of formation. This assumption

is supported by many experiments. For example, the reactions O(d,p) 0

and M(a,p) 0 designed to obtain the same compound nucleus F* at the same17

excitation energy, gave, r i se to the same final products '0 + p, proving the

above assumption of the independence of the compound nucleus on i t s mode of

formation.

•) The excitation energy E* and the nuclear temperature/are related through

a statistical relation

E ' sk t 2 ,

where for nuclei k = A/9; A being the mass of excited nucleus.

•") The width r corresponds to uncertainty in the energy of the state.

-UU-

t

Second step: The second step of this model is simply the decay of the excited

compound nucleus

C*-*B + b . (5.12}

The energy of the compound nucleus can occur/in one or more than one way depending only on its excitation energy and completely

independent of how it was formed. The lavs of conservation of the total number

of particles, angular momenta, excitation energy, etc., have, of course, to

be satisfied. The decay is assumed to occur when one or more nucleons gather

enough energy to escape out of the compound nucleus, which in the quantum

mechanical language is always finite. In general, the mode of decay of the

compound nucleus is specified since the probability for the compound nucleus

to be formed in a particular energy state is maximum. This hypothesis is also

supported by many experiments. For example, in the reactions Ki + a and

Cu + p, the bombarding energies of o and p are,chosen that the compound6k • *

nucleus Zn has the same excitation energy. It if observed tha t , independent

of i t s mode of formation, the compound nucleus Zn formed in two reactions

shovijthe same decay processes and the cross-sections for the emission of one

neutron, tvo neutrons o r a neutron-proton pair in two reactions is same; within

about 10 percent.

Putting the two steps together, a compound nucleus reaction can be

represented as

A + a-*C* -»B + b (5-13)

which means different targets A are bombarded by different projectiles a to

give the same compound nucleus C at the same excitation energy E* vhich decays

in, say, a particular channel of heavy E and light b nuclei. Since the two

steps of formation and decay processes are independent of each other, the

to ta l crass-frection o(a,b) of the reaction A(a,b)B can be written as the(f)product of the compound nucleus formation cross-section a (E*,a) and the

decay probability of the compound nucleus P (E*,b):

<r(a,b) = <J(f)(E»,a) p'd )(E*,b) .c c (5. l i t )

In Eq.C5.lit) the compound nucleus formation cross-section cr is

calculated by the use of the Breit-Wigner resonance theory. Without giving

any detai ls , the cross-section for the formation of the compound nucleus in

one isolated level of energy E- for the reaction A(a,b)B is

(E-E0

(5.15}

where X is the vavelength of the incident particle a, E its energy, EQ theresonance energy, Jt the spin of the level {assuming zero intrinsic spin),

r the tota l level width and T& and 1^ are the part ial widths of the

compound level. In order to calculate the decay probability P= for the

reaction channel B + b, we consider that the compound nucleus C» decays into

different exit channels B + b , B1 + b.^ Bg + bg, e t c . , where b , b ^ b 2 , b3 >

e t c . , can be n, p, T , a , e tc . Then the probability of the decay of the

compound nucleus in the specific channel b is given by the rat io of the decay

width T (the particle width) for the exit channel *o to the total decay

width T for a l l the exit channels b, b^> b^, b ^ , . . .

with(5.16)

(5-17)

The summation in (5-17) is over the various possible exit channels b, b , h ,

b^, etc. Eqs.(5.15) and (5.16) completely determine the total reaction cross-

section a in the compound nucleus reaction model.

5-k Reciprocity theorem *

5-5 The laboratory and the eentre-of-mass co-ordinates *

5.6 Nuclear fission

Nuclear fission is a special kind of nuclear reaction where a heavynucleus breaks into two (sometimes three) l ighter nuclei of equal or unequalmasses and is always accompanied by some promptly emitted neutrons in theexit channel. The reaction can occur either spontaneously or when a heavynucleus is bombarded by a light particle like n,p,d or a and the y rays.For example,

+ (slow) n 2 3 6 U» •* X + Y + v ( f a s t ) n + Q-value (5.18)

where the two outgoing fragments X and Y are the pairs of light and heavy

nuclei with masseB lying between about 70 and 170. v = 3 (actually 2.1*7 on

the average) is the number of "promptly emitted" neutrons. The fission

fragments X and Y are also unstable and go to the ground state by emitting

further neutrons (called "delayed neutrons", which come out due to the larger

H/Z ratio in heavy fragments than in the lighter ones), the 6 rays and/or

Y rays, etc.

The distribution of masses of the fission fragments (after the emission

of delayed neutrons, etc.) is as shown in Fig.5.6{a) for U, where the

probability of production (called percentage yield) of different elements is

plotted as a function of the mass of the fragment produced. The light and

heavy m a s s distributions are shown to "be almost mirror symmetric. Since the

yield is maximum at A is 95 and 139. the distribution is said to be asymmetric.

Similar asymmetric mass yield distributions are observed for the fission of230 232 2^2 25U 258

Pu, Th, J Cf and Fm. POT J Fm, however, a symmetric mass distri-

bution is observed which means that the yield is maximum at A ft 129, corresponding

to the symmetric division (Fig.5-6(b)). Some elements like Ra, show even

a triple humped mass distribution, as reproduced in Fig-5.6(c).Different isobars of the fission fragments X and Y are also observed

and the charge distribution is typically of the form shown in Fig-5-T for almost

all the fission fragments of different .fissioning nuclei. Neutron distribution

yields are rarely plotted explicitly.

As the nucleus breaks, the fission fragments leave the site with some

velocity and hence carry some kinetic energy. From the conservation of

momentum, apparently the light fragment will carry larger kinetic energy and

the heavy fragment will move out with smaller kinetic energy. The kinetic

energy distribution of fission fragments from U is shown in Fig.5-8 which

is identical in shape to the mass distribution curve in Fig.5-6(a).

Reaction (5.18) shows that slow neutrons (or thermal neutrons of only

0.0S5 eV energy) are enough to start the fission of U. The same is true

of U and Pu or, for that matter, of all the heavy nuclei with even Z2~ih 236 238 232 231

and odd H. On the other hand, nuclei like * ' U, Th and Pa with

both Z and S even, undergo fission only when fast neutrons (of >f 1 MeV) are

used. This is due to the fact, observed in Fig.1.2, that pairing effects are

large for even Z and even N nuclei which make such nuclei extra stable.

Another interesting aspect of reaction (5-lS) is that the fast neutrons

produced in the fission process can be thermalized (slowed down) through the

help of the so-called moderators like H,0, D O (heavy water) and C (graphite),235

which in turn can be used to initiate fission in another atom of U. See

Fig.5.9. The only condition for such a chain reaction to continue is that

at least one neutron is produced during each fission process. On the other

hand, if the number of neutrons available at each stage of fission is more

than one, the chain reaction will be an uncontrolled reaction and a large

amount of energy (the Q-value) will be released almost instantaneously (in

less than a micro-second). That is what happens in an atom "bomb. However, a

-kl-

chain reaction can be carried out under controlled conditions inside a nuclear

reactor (section 5-8).

The Q-value in reaction (5.18} gives the amount of energy that is

released in the fission process. By using the definition (5-9), this can be

calculated exactly for a particular reaction, keeping in mind that the fission

products themselves also decay further (see Exercise 5-3). However, as already

noted in section 1.2, a very good estimate can be made from the structure of

the binding energy per nucleon curve of Fig. 1.1*. Exercise 5-1* shows that about

220 MeV of energy is released in the symmetric fission of a nucleus of, say,

mass 2^0, which is close to the maBs of U nucleus. This energy is carried

by the fission fragments and other particles like n (both prompt and delayed),

B rays and the y rays, involved in reaction (5.18), The fission products

(fragments) alone carry about 85 percent of it t** 200 MeV) which is apparently

very large compared to a few MeV of energy involved in most of other nuclear

reactions. Similar energies are involved if the fission is asymmetric (see

Exercise 5-5). One can realize that in a chain reaction, the amount of energy

released could actually be enormous and lead to an explosion, if uncontrolled,

Noticing that fission is a process different from other nuclear reactions,

it is now of interest to find out the physical conditions under which a nucleus

should fission. The first, simplest and still acceptable, explanation of the

fission phenomenon was given by Bohr and Wheeler in terms of the liquid drop model

of the nucleus. Just as for the semi-empirical mass formula (section 3..2),

they considered the nucleus to be a charged liquid drop. Knowing that the forces

present inside the nucleus are the short-range attractive nuclear forces and

the long-range repulsive Coulomb forces, Bohr and Wheeler equated the nuclear

forces with the forces due to the surface tension of the liquid drop. There-

fore, in their liquid drop model, the presence of nuclear shape is assumed to

be due to a balance oetween the {attractive) surface energy and the (repulsive)

Coulomb energy. In this picture, when a neutron is added to a nucleus like

U, the compound nucleus, U, formed is in an excited state and with a

shape different from that of U. Thus, surface oscillations are said to be

set up, with the Coulomb energy further trying to distort the compound nuclear

shape. Then, if the excitation energy of the compound nucleus is such that the

disruptive Coulomb energy exceeds the surface energy, the compound nucleus

formed would split into two (or more) fragments. Fig.5.10 gives the various

stages of the fission process in the ahove model.

Mathematically, the semi-empirical mass formula (3- T) gives for a

spherical charged liquid drop, the surface energy

-1*6-

and the-Coulomb energy 5-7 Nuclear fusion

E0 - I Z e&C 5 R

(5-20)

In the above model, the nuclear shape is kept up due to a 'balance between

these two energies. Then for the fission process to begin, the shape of the

nucleus must get deformed (as shovn in Fig.5-10). This means R becomes larger,

resulting in an increase of surface energy and a decrease in Coulomb energy,

Without any proof (see the other reading material) the net change in the

energy due to deformation $ of the nucleus is, to a first order,

Apparently, AE = 0 if

fiE = | S 2 {2

2 E"

Z_A vr.a

(5.21)

{5.22)

(5.23)

Thus, the nucleus is unstable and could fission if AE is negative, which means

(5.210E° > £E° or f- > 1*7.8

Remember that the condition (5.2*0 is only for the induced fission. The

spontaneous fission can, however, occur even for Z /A < 47.8. Actually,

spontaneous fission does occur for all the trans-uranium (Z > 92) elements

and is considered to be due to the barrier penetration, as for the a decay.

The liquid drop model, however, does not give any prescription to

explain the fission properties like the mass distribution, charge distribution

and the kinetic energies of the fission fragments. More refined theories,

using liquid drop model with corrections calculated from shell model

(Strutinsky method) are now worked out which give quantitative account of some

of these properties. Another approach, giving a satisfactory description of

asymmetric fission of U, is the statistical model. Any further detail on

these theories is, however, beyond the scope of this text and the reader is

referred to other reading material.

Fusion is inverse of fission. Fusion is, therefore, the process of

formation of a heavy nucleus from two lighter nuclei. Once again, it is

evident from the binding energy per nucleon curve (Fig.1.1*} that if two very

light nuclei like H, *D and T etc. (A < 60) are allowed to fuse, the reaction

occurs in the direction of increasing E.E/A (as in fission) and a large amount

of energy will be released. On the other hand, if two heavier nuclei- ("A > 60) •-

combine to form a new nucleus, energy will be required to be supplied externally.

Whereas in the first case, if fusion is carried out under controlled conditions,

we get a new source of energy, called thermonuclear energy, in the second case,

new elements are produced both in the known region of the periodic table and

beyond. As we shall see in section 5.3, even in the first case, some external

source of energy is required to start the reaction.

Much effort has gone in, both experimentally and theoretically, towards

the production of new elements as well as energy through nuclear fusion. The

production of new elements, in particular, the very heavy nuclei in the region

of 2 = 110-llli (called, super-heavy elements) is shown to be important not only

for the natural instinct of extending the periodic table of elements, but also

for the further understanding and extension of the present range of validity of

the basic liquid drop model and the nuclear shell model. Large accelerating

machines are developed in different parts of the World in order to study the

possibility of fusing two very heavy nuclei. This field of nuclear physics,

called heavy ion physics, is an important subject of study these days. Since

this most modern subject is beyond the scope of this text, we shall discuss,

in the following, only the fusion of two very light nuclei with a view to obtain

an energy source and also the possible occurrance of this process in natural

sources of energy like the sun and the stars.

The sun gives some 3.8 x 10 joules of energy each second. Since the

chemical reactions cannot produce so much of the energy, its origin must lie

in, perhaps, the fusion process that could be going on inside the Sun. Two

different theories have been advanced assuming that in Mature all elements

are formed or might have been formed by successive fusion of light elements

(or nucleons) to heavier nuclei. These processes are accompanied by a release

of large energy. The theories are known as the Big Bang theory ana" the

Steady State theory.

The basic process of how the matter (various elements) might have been

created in the Universe and how the sun and stars work as great sources of energy,

can be understood simply by analysing the fusion of two hydrogen nuclei (the

protons). This beginning is pos»ible since the hydrogen gas is known to exist

-1.9- -50-

in the interstellar space. The process of the fusion of two hydrogen nuclei

is called the proton-proton cycle *) shown schematically in Fig.5.11. Two

hydrogen nuclei combine to give a deuteron, which combine with another hydrogen

nucleus to give TRe nucleus. This part of the cycle is repeated to get another

gHe nucleus such that the two He nuclei can then fuse to give He nucleus and

two hydrogen nuclei. These two hydrogen nuclei allow the process to repeat

and make it self sustaining. The actual reactions taking place at each stage of

the cycle are:

^H + *H -» 2D + e + + 11 + 0.142 MeV

Synthesis of elements beyond He in Nature can be expected to continue

in the same way. Two He nuclei fuse to give Be (though only for -^10 sec.)1. . 12

which in turn combine with another He to give C; and so on. This is how

the matter in the Universe seems to have been created and the stellar sources

work as continuous sources of energy; though a complete picture must certainly

be far more complex.

The question of feasibility of the fusion process as a controlled

source of energy for the world needs on Earth is discussed in the following

section.

He + f + 5.1;? MeV (5.25)

iH + 12.86 MeV (5-26)

This is the process used in the H bomb. The total energy released in each

cycle ia

2(0.1*2 + 5.1*9) + 12.86 • eU.68 MeV (5-27)

An additional 1.02 MeV of energy is released in each annihilation of the

positron (e ) with an electron. Since some energy, is always lost to the

surroundings and neutrinos carry avay about 10 percent of the energy produced,

the self-sustaining character of these reactions can be maintained only if

i t is mode to occur under very high pressures and temperatures (~10 K ) .

This allows the reacting hydrogen nuclei to have enough energy to overcome

their own repulsive Coulomb field. Therefore, such reactions are difficult to

be produced in the laboratory. Even if temperatures as high as 10 K° are

attainable, there are other associated problems, as discussed further in the

next section.

Another self-sustaining process that is assumed to take place in the

sun and stars is the so-called carbon cycle. This cycle also uses four1,

hydrogen nuclei to form a He nucleus and gives out exactly the same amount

of energy. The carbon nucleus, therefore, works simply as a catalyst, as

shown in Fig.5-12. This cycle is found to be more efficient at s t i l l higher

temperatures.

*} The word cycle means a self-sustaining process.

-51-

5.8 Nuclear energy

As already stated in earlier sections, nuclear energy can be obtained

by letting nuclear fission or fusion to occur under controlled conditions.

To be able to achieve th is , opens up a completely new subject of fission-and

fusion-reactors. It is interesting to note that if the fusion reaction in the

H bomb alone could be controlled, this could support the energy needs on Earth

for at least a few generations to come.

For fission reactors, we have seen that the chain reaction can \>e

controlled by controlling the therm&lization process of the fast neutrons

produced in the fission. The only condition for the chain reaction to go on,

or in other words for the fission reactor to operatt, LS that at least one

neutron is,produced during each fission event. Thus, very many different types

of fission reactors are built in the World, depending on the type of fuel

(fissionable material) and the moderator (for slowing down of neutrons) used.

The most commonly used fuels are '"JJU, "~J'IU> tJUU, '"JJPu and Th and the

2 3 3 u , 2 3 5 u , 2 3 8 u , 2 3 9 Pmostly used moderators are water (H£0), heavy water (Eg0), graphite (C) and

polythylene ((CN.) ). In the case of natural uranium, which contains 99.3$

of U and O.TSt of U, as a fuel the fast neutrons are slowed down outside

the fissionable material in order to avoid their capture in U, since U

gives a non-fission reaction

236,92,U 92

239,93,Hp

:239-9kPu + e (5.28)

This reaction actually shows the possibility of producing fuel ( Pu) inside

the reactor. Such a reactor is, therefore, called a breeder reactor.

-52"

In the design of a fission reactor, however, one needs to have, in

addition to the safety shielding, (i) the control rods to stop the reactor,

(ii) a coolant to keep the temperature within safe limits and (iii) the heat

exchanger to transfer the energy produced in the fission process. The control

rods are, generally, of Cd, B, Li or their alloys, which are absorbers of slow

neutrons. The heat produced in the reactors is used for producing steam, which

in turn is used to turn the turbines for generating electric power.

Generation of thermo-nuclear energy today is more of a ooncept, though

the feasibility of the process itself has already been demonstrated in the

explosion of the H bomb. The initiation of the fusion process in the H bomb

is again done by using the fission process. This renders the use of hydrogen-

hydrogen fusion very cumbersome. Instead, its heavier isotope, deuterium ( D ) ,

which constitutes only 0.03? toy weight of the hydrogen in all waters on Earth,

seems to be the simplest element that can be used in fusion reactors.

Deuterium reacts in one of the following two ways:

^ ?T +

+ n + 3.27 MeV

^ If .03 MeV

(5.29)

(5.30)

He and tr i t ium T react further with D :

+ his He + H + 18.3° MeV

MeV

(5.31)

(5.32)

The neutrons generated in both the reactions can be thermalized by lithium:

Ki + (slow) n — * f? + gKe + lf.60 MeV (5-33)

7,Li + (fast) n _ > ^T + He + (slow) n - 2 . 50 MeV . (5-3>t)

Apparently, lithium here breeds (produces) new tritium for the basic reaction

(5.32). The lithium so used to thermalize the neutrons is technically called

"lithium blanket". Summing up the above reactions, we notice that, without

breedings, in each reaction

i (3.27 + 18.30) or i (it.03 + 17-59) * 7-05 MeV . (5-35)

of energy is generated for each D atom converted into a. stable He nucleus.

This corresponds to 3.55 * 10 Joules or 10 kilo-watt hr. of energy from one

gram of deuterium (see Exereise 5'5).

Actually the reaction (5-32) involving D-T mixture (50? each) with

17-59 MeV energy output is found to be a favourable reaction compared to the

pure D-D reaction (5.31). In reaction (5.32), 3.5 MeV of energy is carried

by He (the other l l t . l MeV is taken away by neutrons) which'heats up the fusion

material ana converts i t into a hot matter, called plasma. This creates a

major problem in the design of a thermo-nuclear reactor.

The plasma is electr ically neutral and would expand in vacuum due to

i t s internal pressure. When i t comes in contact with the walls of i t s

container, i t cools down by heating the walls. Therefore, for any efficient

use of the fusion reaction, th is plasma has to "be confined for some minimum

time so as to keep i t s temperature and density at some optimum value. Very

recently, the containment of plasma for a sufficiently long time is being

t r ied by simpjy placing i t in a magnetic field. This method, however, has not

been vary successful and i t i s now believed that i t may be possible to build

a fully controlled thermo-nuclear reactor based on another technique of the

so-called " iner t ia l confinement" of hot, compressed deuterium (the plasma) by

high intensity laser radiations ( laser , stands for ^ight amplification due to

stimulated ^mission of radiations), The whole question of interaction of laser

radiation with plasma i s , however, s t i l l neither fully understood nor properly

treated in the laboratory; though some conceptual designs, namely, Blagcon,

Wetted-Wall and Dry-Wall reactors based on the concept of "laser induced

nuclear fusion" already exist . I t i s , therefore, apparent that though the

principles involved in thermo-nuclear reactors are internationally accepted

both from scientific and economic feas ib i l i t i e s , there is s t i l l a long way to

go before we can use the fusion energy as a controlled source of energy.

Once realized, th is i s certainly the cleanest and inexpensive source of energy,

since deuterium is abundantly available in the oceans and seas of the World.

EXERCISES:

Ex.5.1: Calculate the Q value of the reaction Mg(p,a) Na.

Solution: The reaction can be written as

Mg + p —» 23Na + a + Q-value

-53- -5k-

we use the atomic masses

= 25.982593 + 1.007825 - 22.969T71 - it.002603

* -0.001956 a.m.u.

= -0.001956 * 931.1*8 HeV

= -1.8? MeV .

The Q-value is negative and hence this reaction is an endothermic reaction.

123Ex.5.2: Calculate the mean l i fe time of the compound nucleus Te formed inJ • 1/2 state when a neutron is bombarded on Te. The resonance width ofthe level is observed to be CIO1* eV,

Solution:- 15.

r

1.051*60 * 10 Joules sec0.1014 eV x 1.60 x 10~19 Jbules/eV

6.31* " 10~15 sec.

Ex.5.3: Calculate the total energy released in the reaction

B + 235U * £36U + lU°Xe +

9USr * 2n

knowing that the fission fragments themselves decay further by successiveB ray emission

and

1.3m 2Om

Solution: The total energy released (Q-value) is given by the difference

of the masses of the entrance channel nuclei ( U and n) and the final stablelUo gl*

products ( Ce, Zr and 2nj :

235U n Ce 9 Zr D

= 235.01*3915 + 1.008665 - 139-905392 - 93.90631**

- 2.017330 a.m.u.

= 0.2235H * 931.U3 MeV

= £08.2 MeV-55-

Ex.5.1*: Using the information from B.E/A curve (Fig.1.1*}, calculate the

energy released in the symmetric fission of mass 2l*o nucleus.

Solution: Symmetric fission means that the nucleus with A = 2h0 breaks into

two fragments of equal masses of 120. From Fig.1.1*

B.E/A for A » 2l*0 » 7-6 MeV

B.E/A for A - 120 = 8 . 5 MeV .

Difference in B.E/A in going from A - 2l*0 to A = 120 = 8.5 - 7-6 = 0.9 MeV/A.This ie the amount of energy that is released per nucleon.

•'• B.E or the total amount of energy released

= A X 0.9

• 2l*0 X 0.9 = 216 HeV.

Ex.5.5: As for Ex.5-1*, calculate the energy released in the asymmetric

fission of A » 21*0 to fragments with masses 100 and lUO.

Solution: From Fig.1.1*'-

£ j ^ for A - 21*0 - 7.60 MeV

^Y" f Q r A = 100 = 8.65 MeV

—•' for A = 1U0 =* S.!*0 MeV .A

.', Difference in ^—- in going from A - 2I4O to A = 100 - 8.65-7-60

= 1.05 MeV/A

anddifference in —=• in going from A = 2l»0 to A = ll*0 = B.U-7.6

=0.8 MeV/A

,', B.^ or energy released in the asymmetric fission of

A - 2U0 to 100 and 11*0 • 100 x 1.05

+ ll*0 x 0.6

= 217 MeV .

Ex.?.6: Show that 7 MeV/A of energy is equivalent to 1.6l « 10 K cal/Kg,

Solution:and

1 MeV - 1.60 * 10 1 3 joules

-h1 Joule = 2.39 * 10 K cal

-56-

1 a.m.u = 1 . 6 6 X 10~ Kg

MeV 1.60 x 10 1 3 Joules

1.66 x i o " 2 T Kg

1.60 * 10-ititg.39 x io K ea l

MeV

1.66 x 10 2 T Kg

7 X 1.60 x 10 J x a.39 x 10 K ea l

1.66 x io" 2 7

= 1.61 x io 1 1 K cal/Kg.

This is apparently a large energy compared to energy released in other

mechanical systems. For example, heat of vaporization of water is only

5l*0 K cal/Kg.

SUMMARY ATO THINGS TO REMEMBER

In Chapter 1 we have l e a r n t t h a t t he mass of t he nucleus (£x) i s not

simply the sum of the masses of i t s constituents but is less by an amount

called mass defect (Am):

M, =Zm + N m -AmA p n

and can be measured very accurately, with the help of the Mass Spectrometers,12in units of C mass, such that

1 a.m.u. = 1.66Olt X 10"27 Kg = 931,1*8 MeV .

-Tn terms of the Einstein energy mass equivalence relation

E = m c2

the mass defect, gives, the •binding energy of the nucleus

B.E = A a.c

which is an important quantity in nuclear physics. The smallest bound (though

very loosely) system is the deuterium and the di-proton and di-neutron systems

are unbound. As a natural consequence of B.E/A curve and also the balance

between the nuclear forces due to neutron excess and the Coulomb force of209

charged particles, the heaviest stable nucleus is g-jBi.

Nuclei are both spherical and deformed with their radii best given by

R - 1.28 A1 /3 - 0.76 + 0.8 A"1/3 fin

which in the limit of the equivalent spherical nucleus gives

E = rQ A1 '3 with rQ = 1,15 fm .

The identity of nuclei inside the nucleus is not uniform, and is best given

by the Fermi distribution

1 +r-E

- 5 7 --58-

In Chapter 2 we have seen that the nucleons (protons and neutrons)

inside the nucleus keep themselves bound through strong nuclear forces. The

nature of the nuclear forces, derived mainly from two-body systems (the

deuteron and nucleon-nueleon scattering), is found to be strongly attractive

(about-30 MeV) , short-ranged (~2 fm, such that i t acts only amongst the

neighbouring nucleons), saturated, charge-independent, spin-dependent and

repulsive at short distances [Q.Qh fm). These two-body forces are also shown

to have seme velocity-dependent (tensor force) component and the exchange

character. The exchange nature of the two-body nuclear forces is explained

by the meson theory by assuming a constant exchange of TT mesons (ir~,ir )

between the nucleons, which is equivalent to the exchange of space (position)

or spin or both the space and the spin between the protons and neutrons

inside the nucleus.

The question of how the nucleons arrange themselves inside the nucleus

is studied in Chapter 3- We find that the nucleus can be taken to behave as

a hydrodynamical body of continuous matter (the liquid drop model and the

collective model) or as individual nucleons mixing in the effective field

produced by the other nucleus (the shell model).

The liquid drop model was introduced to calculate the binding energy

of the nucleus which for a. liquid drop, containing protons and neutrons, has

the volume, surface, Coulomb, asymmetry and pairing energies:

B.E. = a A - a Av s

2/3c Al/3

a's are the constants, determined empirically. An important correction to the

liquid drop formula comes from the shell model of the nucleus.

Shell model considers the nucleons to be moving in well-tdefined

orbits, given by the solution of the Schridinger equation. For the harmonic

oscillator, the rectangular well and Woods-Saxon potentials, the closing of

the shells is obtained at U or Z = 2, 6, 20, kot TO and 112, whereas the

characteristic stability of nuclei is observed at

N or Z = 2, 8, 20, 28, 50, 82 and N = 126 ;

called the "magic numbers". Shell model was essentially introduced to explain

these magic numbers, which became possible only by adding the spin-orbit

coupling term Cs\T) to the potential. Extension of shell model to higher shells

gives rise to new regions of stability at Z = 110 * llU called the superheavy

elements.

In Chapter k heavy nuclei are shown to decay under natural conditions

by emitting a, g or y rays. Such nuclei are called radioactive nuclei.

Most of the known radioactive nuclei are found to be the members of one of the

four series: thorium, naptunium, uranium-radium or uranium-actinium series.

The mass numbers of these series are ha, lin+1, ltn+2 and in+3, where n is

an integer. The end products of the first three naturally occurring series209

are Pb isotopes and that of the art if icial naptunium series is Bi.

The radioactive nuclei having N nuclei at t = 0, follow the exponential

decay law

H =

X is called the decay constant and is related to the half-life time T , as

"1/2 X

and to the mean life time T as

T = X •

The physical process behind the a decay of radioactive nuclei is the

quantum-mechanical tunnelling of an o particle through a potential barrier

provided mainly by the Coulomb forces.

Finally in Chapter 5 i t is shown that nuclei do not react chemically

but with a release or absorption of energy, called Q value of the reaction.

A reaction mechanism is fixed by comparing the experiment with different

models proposed.

Direct reactions are one step processes of stripping or pick-up of-22

a particle in about 10 sec, whereas the compound nuclear reaction involves

the formation of an intermediate compound nucleus and the whole process can

take about 10 sec. The compound nucleuB cross~section for the reaction

A(a,b)B is

O<ab) = O( f ° ™ a t i o n ))(E*,a)

The reaction time is given by uncertainty principle

-59-

-60-

Fission is a special kind of reaction where neutrons are always

produced in the exit channel and Q value is very large. In terms of the

liquid drop model, the condition for the induced fission is

Fusion is inverse of fission. Whereas the fusion of very light nuclei

result in a new source of energy, the fusion of heavy nuclei produce new

element s.

Both fission and fusion processes are shown to be great sources of

nuclear energy. Though fission reactors are already in use, the fusion

reactors are as yet more of a concept.

ACKNOWLEDGMENTS

The author would like to thank Professor Abdus Salam, the International

Atomic Energy Agency and UNESCO for hospitality and support as an Associate

Member of the International Centre for Theoretical Physics, Trieste, where

most of this work was written up. He would also like to thank Professors

U.S. Hans and V.B. Bhanot for encouragement to take up.this work.

OTHER HEADING MATERIAL

1} A. Beiser, Concepts of Modern Physics (McGraw Hill, 1973).

2) W.E. Burcham, HUC1«?T- EhyBJcs;.An I n t r o d u c t i o n (Longman, 1973) .

3) H.D. Evans, The Atomic nucleus (McGraw H i l l , 1955) .

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1, 377 (1970) .

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Phys. 2 8 , 3U2 (1956) .

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Figure Captions

n MeV) as a function of massFig. 1.1 The mass defectit

number A.

Fig. 1.2 A portion of Segre1 Nuclear Chart.

Fig. 1.3 A neutron-proton plot representing the complege

Nuclear chart as well as the various possible

decay processes to the line of p-stability.

The expected regions of superheavy elements are shown.

Fig. 1,4 The experimental "binding energy per nucleon as a

function of mass number A.

Fig. 1.5 A Fermi type of charge density diatribution inaide

the nucleus. The constant density distribution is

also shown.

Big. 1.6 A pictorial representation of prolate and oblate

deformations in two-dimenaiona.

Figure Captions

Fig. 2.1 The schematic two-body nuclear potential derived

from the two-nucleon systems, the deuteron and

nueleon-n-ucleon scattering.

Tig. 2.2. The various possible forms of exchange forces

between a. neutron and a proton.

-63-

Figure- Captions

Fig. 3.1 The various early models along with the.finally

accepted model of an atom.

Fig. 3.2 The binding energy per nucleon calculated on the

•basis of the semi-empirical mass formula ( f i rs t

four terms in Bq. (3.7)) and the contributions from

the individual terms, as a function of mass number A,

Fig. 3.3 The shell model states for (a) Harmonic oscillator,(b) Rectangular \»ell,(c) Wooda-Saxon Potential and(d) Woods-Saxon plus spin-orbit coupling; showingin each case, the configuration, number of nucleonsin each state and the shell closure •

-65-

Pigure captions

Fig. 4.1 Decay of the Thorium ( A = 4n) series.

Pig. 4.2 Decay of the Neptunium ( A = 4n+l) series.

Fig. 4.3 Decay of the Uranium-Sadium (A = 4n+2)series.

Fig. 4.4 Decay of the Uranium-Actinium (A»4n+3) series,

Pig. 4.5 The exponential decay law for the number N of the

radioactive nuclei and the activity A, depicting thehalf-life T, and the mean life time T .

•t'ig. 4.6 The potential offered by a radioactive nucleus for thea-decay process. The dotted lines show the divisionof the potential function into a series of potentialsteps. The tunneling process is exhibited by an arrowwhose length represents the amplitude of the wavetransmitting through the Coulomb barrier.

Fig. 4.7 A potential step of width x and height VQ.

232,i)'ig. 4.8 The potential barrier showing the a-decay of J Thpi p

and Po interms of i t s widths.

-66-

Figure Captions

•"'is'.1 5.1 A pictorial representation of the nuclear reaction.

Fig. 5.2 A schematic representation of the scattering process.

•c'ig. 5.3. A pictorial representation of the various direct

reaction processes.

Fig. 5«4 A sketch of the formation and the decay of the compound

nucleus.

Fig. 5.5 Reaction cross-aection s.s a function of the incidentparticle energy shc.ving a resonance state (level)of the compound nucleus.

Fig.5.6 Measured mass-yield distribution of the fission fragments

for (a) 256U (b) 258Fm and (c) 226Ra.

Fig. 5.7 A typical measured charge-yield distribution.

•"'ig. 5.8 Measured kinetic energy distribution of fissionfragments from 256U.

•t'ig. 5.9 A schematic chain reaction.

Fig. 5.10 Various stages of the fission process in the liquiddrop model picture.

Tig. 5.11 The proton-pro tonjcycle showing the fusion of two

hydrogen nuclei. The wavy arrors show the reactionand the straight arrows give the products of thereaction.

Fig. 5.12 The carbon cycle showing the fusion of four hydrogen12

nuclei (protons) with C as a Catalyst. The wavyarrows show the reaction and the straight arrowsgive the products of the reaction.

80 "

60

40

20

-60

-SO

-100 _L50 100 150

Mass number A

F i g . 1 . 1 .

200 250

-67- -68-

200

120 h

Possiblt regions ofSup«rhwvy nuclei

160 200 240 280 320

•3 8 h

Fie.*

80 120Mass number A

160 200 240

-69- 1.4 .-70-

0.9 f0

2

01 ?o

^Constant dens'ity

with Z, = t / 4 l n 3

r(fm)

Pig. 1.5.

Symmetry axis

Prolate

c

Oblate

F i g . 1 . 6 .

-71-

30

20

10

9 °2-10

J-20-30

r (in f m)

Pig . 2 . 1 .

U Exchange

Majorana Forte

Bartlttt Force

Heisenberq Force

Q Wigner Force

Fig. 2.2.

Energy/Mueleon (in MeV)

CD

CO

5: * / '", ' i . ' N

1g 2d3s

3 I f 2p

2 Id 2s

1p

-70 —3s --68--2

-58—'S-

-40--2p-

--20---2S---I8-—H-

--8---1P-

-~Z I s -

J£L18

J0_

r

10

-20---:,:-'t8-..l(t5,2_

82

50.

28

20

IP 3ft

--2--.it.

N Configuration

Co)

HarmonicOscillator

Shell closure

Cb)

RectangularWell

Shell closure

Woods

Shell

-Saxon

closure Shellclosure

fd)

PlusSpin-orbit

P i e . .1.3.

144

80 82 84 86 88 90 92

126 -

124

144

142

140

138

136

134

132

130

128

126

124

-

2O9

-

p /

Pb\o\

1

Neptunium Series 237,,Np

a Jf

233 /P a \ p

a/

y

i2°*6i

1 J r 1 1

80 82 84 86 88 90 92 94

Ftp. 4 . 1 .

-75-

Flg. 4 . 2 .

-76-

[ 136 -

146

144

142

140

138

136

134

132

130

128

126

124

URANIUM-RADIUM SERIES

-

-

-

222pm /

x l4«

\ fii

<X jr

Pbi i i . i

2 3 6

1 1

144

142

140

138

136

134

132

130

128

126

124

URANIUM-ACTINIUM SERIES

a /

231 T h X\PX>"'pa

227 /

/211 /

i i i i i i

80 82 84 86Z

88 90 92

80 82 84 86 88 90 92 94Z — Fig. 4.4.

Fig. 4.3. -77- -.T8-

0-25

0.125

Time

Fig. 4.S.

Coulomb repulsionZ, Z, eVr

50 r(fm)

FiE. 4 .8 .

-79-

Flg. 4.7.

J

25

2 0

15

10

54.05

-

-

A

c

\

\

• \ D I J1Th

1 1 i * L

10 20 30 40 50 r(fm)

Fig. 4.8.

-8o-

O

A..

01

N

>zo

oI

3DO

CJ) SomtReaction Mechanism

oi

S3' O

A-n

STRIPPING PICK-UP

DIRECT REACTIONS

A+n

KNOCK-OUT

Fig. 5.3.

Step 1 : Formation process

A A + aaO +•

Projectile Target Compound Nucleus(.Low Energy) (Low Mass) (Excited)

Step 2 : Decay pci ' tss

Excited NewCompound Nucleus Nucleus

OLight

Particle

COMPOUND NUCLEUS REACTION

Fig. 5.4.

0 Incident Particle Energy

Pig. 5.5.

-83-

o <J3

«3

v>nt

2

O

oCM

'O 'O00

•A

-flk-

io2h

10°

id

410

,66

i o 8

io1

>-

63 83 103 123Mass Number A

Fig. 5.6 (c)

34 36 Zp 38 40

Charge Number 2

Fie. 5.7.

Counts

(O

5

cn0)

1

100

8A)

160180

rooO

220

3 O1

•

-

O ifl O

o o o—1—-1——1—

^ — —

^ — • _

. —: —

o o1 1

—

— — — -

Xn Oo o

ruum

C*"\

Fig. 5.10.

<D~—"

Fig. 6 . 1 1 .

-87-

®—-

Fig. 5.12.

* i t* * >:{" s #- J; -