Enrico Fermi - Nuclear Physics - Course Notes

Enrico Fermi

Nuclear Physics

Course Notes Compiled by

Jay Orear, A. H. Rosenfeld,

and R. A. Schluter

Revised Edition

Nuclear

Physics

A Course Given by

ENRICO FERMI

at the University of Chicago. Notes Compiled by

Jay Or ear, A. H. Rosenfeld, and R. A. Schluter

Revised Edition

IIA Ulb. T H E U N I V E R S I T Y O F C H I C A G O P R E S S

THE UNIVERSITY OF CHICAGO COMMITTEE

ON PUBLICATIONS IN THE PHYSICAL SCIENCES

*

WALTER BARTKY JOSEPH E. MAYER WARREN C JOHNSON CYRIL S. SMITH

WILLIAM H. ZACHARIASEN

T h e U n i v e r s i t y o f C h i c a g o P r e s s , C h i c a g o 3 7 Cambridge University Press, London, N.W. i, England Copyright iç4ç and iç^o. Copyright under International Copyright Union by The University of Chicago, A ll rights reserved. Published IÇ4Ç, Revised Edition IÇ50, Fourth Impression ipsj* Composed and printed by Th e Un i­

P R E F A C E

This material is a reproduction, with some amplification, of our notes on lectures in Physics 262-3: Nuclear Physics, given by

Enrico Permi, Jan.-June 1949. The course covered a large number

of topics, both experimental and theoretical.

The lectures presupposed a fam iliarity with physics generally acquired by a student who has completed one course in quantum mechanics (this to include a discussion of the Pauli spin ope­ rators and of perturbation theory, both time-independent and time-

dependent). We shall make some use of elementary concepts of

such topics as statistical mechanics and electrodynamics, but we give references, and a reader could probably pick up the neces­ sary ideas as he goes along^ or he could omit a few sections.

Dr. Permi has not read this m aterial; he is not responsible

for errors. We have made some attempt to confine the classroom

presentation to the text proper, putting much of our amplifica­ tions in footnotes, appendices, and in the solutions to the prob­

lems. Most of the problems were assigned in class, but the solu­

tions are not due to Dr. Fermi.

The literature references in the text apply to the l is t on

page 239 . At the end of the book there is also a summary of the

notation and a list of pertinent constants, values, and relation­ ships.

We would very much appreciate your calling errors to our attention; we would like to hear any suggestions and comments that you may have.

May we thank warmly all those who have helped us to prepare these notes.

Jay Orear A.He Rosenfeld R .A . Schluter

, January, 1950

This second printing of thee© notes differs from the first in that corrections and minor revisions have been made on approximately

70 pages in the first nine chapters, and major revisions have been

made in the chapter on cosmic rays. We are grateful to the many

people who have given suggestions and corrections; in particular,

vre are indebted to Prof. Marcel Schein for his suggestions and generous aid in revision of Chapter X.

JO, AHR, RAS

September 1950 An attempt to bring this second printing of the revised edition up to date has been made by adding new footnotes and two pages (257,258) of

recent developments. Corrections and minor revisions have besn made on

approximately 40 pages.

July 1951

Minor revisions have been made, mainly in bringing some of the references up to date.

C O N T E N T S

OHA.PTER I. PROPERTIES OF NUCLEI Page 1

A. Isotopes, Charts and Tables 1

B. Packing Fraction and Binding E n e r ^ 2

C. Liquid Drop Model 5

1. Semi-empirical mass formula 6

2, Isobaric behavior 8

5. /3-emission 9

4. Periodic shell structure 9

D. Spin and Magietic Dipol© Moment· 9

E. Electric Qwadrupole Moment ' . · ‘ 15

F. Radioactivity and its Geological Aspects' · " 17

G. Measurement and Biological Aspects of Radioactivity 18

Appendices

1. Magnetic Moment for Closed-shell-plus-one Nuclei 19

2· Electric Quadrupole Moment 21

5. Mass Correction for Neutron Excess 22

Problems 24

CHAPTER II. INTERACTION OF RADIATION WITH MATTER 27

A. E n e r ^ Loss by Charged Particles 27

1. Introduction 27 2. Bohr formula 27 5. Electrons JO 4. Other particles J1 5· Other absorbers J1 6. Range 51 7. Polarization Effects J2

8. Nature of equation for -dE/dx J2

9. Ionization a gas 35

10. Radiation j4

B. Scattering j4

1. Classical calculation for single scattering 5^

2. Multiple scattering 56

0. Passage of Electromagnetic Radiation through Matter 5^

1. Photoelectric absorption 5^

2. Compton scattering 40

Radiation loss by fast electrons) 4j

Pair formation 4?

5. Cosmic ray showers 49

6. Summary 49

Appendices

1, 2, a n d ’?. Multiple scattering 5I

4, Momentum and pair creation 5^

References >4

CHAPTER III. ALPHA EMISSION

A. Rectangular Barrier 55

B. Barrier of Arbitrary Shape ^6

C. Application of Barriers to «-decay 58

D. Virtual Level Theory of -decay 59

E· c( -ray Spectra 66

Appendix 67

A. Introduction ^

B. Exaaples of /»-prooesees 69

0. Energy diagrame 71

D. Theory of |5-<ieoay 72

E. Rate of Decay 75

P. Shape of Energy and Moxaentum Spectra 78

G. Experimental Verification 79

H. Selection Rules Ô0

J. ? X Tables Ô2

K. Remarks on K-capture Ô4

L. Remarks on the Neutrino Hypothesie Qk

M. Neutrinos a n d Anti-neutrtnos fiÇ

CHAPTER V. OAMKA-DBCAÏ 89

A. Spontaneoue Emission 89

General emission formula 90

Electric dipole emission 9^

Magnetic dipole emission $5

Half lives 96

B. Selection Rules 96

1· Angular Momentum 96

2. Parity 98

5. Improbability of nuclear dipole radiation 99

4. Suoanary 100

5· Dipole absorptionat high energies 100

0. Internal Conversion 101

1. Theory of internal conversion 101

2. Selection rules 104

5· Other processes IO5

4. Experimental determination of conversion ooeff. IO5

D. Isomeric States IO6

Problems 106

CHAPTER VI. NUCLEAR FORCES 111

A. Introduction 111

1. Meson Theory 111

2. Saturation of nuclear forces 111

5. Exchange forces 112

B. The Deuteron II5

1. Non-central and apin-dependent forces II5

2. Ground state of the deuteron II5

0. Neutron-Proton Scattering II7

1. Method of partial waves II7

2· Low-energy solution for d* II9

5. Virtual state of the deuteron 120

4. Evidence for exchange forces 121

D. Proton-Proton Forces I25

1. Pauli principle complications I25

2. Spin fimctione I25

5. Coulomb scattering 125

E. Neutron-Neutron Forces 129

CHAPTER VII MESONS

A. Experimental Properties I5I

B . Theory

References for meson theory I 5 7

CHAPTER IV. BETA-DEOAÏ paga 69

Problems

₁₅₈

A. Notation pag© l4l

B. Cross Sections, General l4l

0* Inverse Processes l4l

D. Compound Nucleus l4l

B. Example of an Unstable Nucleus (l^Be ) l49

F. Resonances; Breit-Wigner Fonmila 152

G. Resonances; Data 157

H. Statistical Nuclear Gas Model 159

J. Fission 164

K. Orbit Model of the Nucleus l67

L· Capture of Slow Neutrons by Hydrogen 171

M. Photonuclear Reactions 175

N. Remarks on Very High Energy Phenomena 177

CHAPTER IX. NEUTRONS 179

A. Neutron Sources 179

1. Radioactive sources 179

2. Photo-Bources 179

5. Artificial sources 180

B. Slowing Down of Neutrons 181

1. Inelastic 181

2, Elastic 181

5. Energy distribution of neutrons from a

mono-energetic source 18J

4. Distance from a point source vs. energy 185

C. Diffusion Theory 187

1. Age Equation 187

2. Distribution of thermal neutrons 191

D. Scattering of Neutrons 194

1. Effect of chemical binding 194

2. Low energy scattering 194

5. Interference phenomena 196

4. Para- and ortho-hydrogen 199

5- Crystalline diffraction 200

6. Index of refraction 201

7· Scattering by microcrystals 205

8. Polarization of neutron besros 204

E. Theory of Chain Reactions 208

CHAPTER X. COSMIC RAYS 215

A. Primary Radiation 215

B. Secondary Radiation 217

1, Protons 220

2. Neutrons 220

5. Mesons 22J.a

4. Electronic Component 221a

0. Analysis into Hard and Soft Component 221b

D. Motion in the Earth’s Magnetic Field 225

1. Trajectories 225

2. Illustration: Equatorial Plane, Shadow Effect 23O

5. Intensity: Liouville theorem 2^2

4, Charge of Primary Radiation 255

5. Latitude Effect 255

REFERENCES 239

NOTATION 240

PHYSICAL CONSTANTS AND VALUES 2A2

index 244

ix

P R O P E R T I E S O F N U C L E I * A . ISOTOPES, CHARTSi?-TABLES

A ll nuclei are composed of Z protons + N neutrons. The

mass number, A , is niven by A = Z + N. Examine an isotope chart,

such as qt the back of tliis 'ooi'lc, and notice that the stable

elements l i e alone o, curve starting out with N /Z = 1 , ending

with N /Z = 1 . 6 . Nuclei with common Z are called isotopes, those

viith common A ?.re called iso b ars, and those vfith common N, iso­ tone s .

Radioactivity

Nuclei richer in N than th e ir stable is’obar(s) are

A. ^ “

P“ -cactlve, i . e . 2^ ) + P

where .8” represents an electron emitted from the nucleus.

Nuclei* relatively poor in N either emit particles or else

capture orbital electrons (Chap. I V ) .

Some nu c le i, for example: ^qCu^ ^ , , can decay

either way or ^

Nuclei near the end of the periodic table also tend to emit particles or to break in two ( f i s s i o n ) . This explains why ele ­ ments with Z > 94 are not found in nature, see section C3, page 9.

Radioactive nuclei emit particles according to the s ta tisti­ cal law:

dn = - /in dt

w hich, inter^.ratod, clvea n (t ) = n (0 ) ^ ^

where n means number of nuclei remainlnc*, after time t , and ^ is the probability of decay per un it time per atom. ^ is called the decay constant.

The mean l i f e , T, Is easily aho\m = l/2 . However, in ta^^les, it is customary to nive the h a l f l i f e , S. = (in 2 ) t = 0 .6 9 3 'T . A fter time the number of nuclei present is l /e times the o r l ‘*lnal number; afte r tlm*^ T, the fraction is

CHAPTER I

^ Much of this material is covered in Chap. I , G-oodman,

y n

I f there are t\Jo competln,^, processes, so that a p article, no, may emit one sort of p a r t ic le , p i , accordln.^, to

dn]_ = - A ^no dt or another ty::e of pr-.rtlcle, P

2

,

dn2 = - ^ono dt then

dno = dni + dn2 = -( ^ i + ^

2

) ^ · Thus mean liv e s onjnbine as the sum of reciprocals.

2

Ch. I Atomic Masses

There are tvro units of atomic mass. In the chemical system,

the "natural” Isotoplc mixture of oxygen Is assicned mass

1 6

.

0

; on

the physical scale, Given the mass

1 6 .0

amu (atomic mass

u n i t s ).

Mass on Physical Scale == 1.000272 Mass on Chemical Scale

Isotope charts generally use the physical scale. The masses

listed are not for nuclei, but for neutral atoms. To get the

nuclear mass, subtract Z x m, the mass of the Z atomic electrons (vihose binding energy can be neglected - see top footnote, p. 3 ) . One amu, that is, the mass in grams of a fictitious atom Mi, of mass 1 .0 0 0 , is given by the reciprocal of Avogadro’ s number

(No = 6.023 X 10^^).

Particle_____________________Mass____________________ Rest Energy

Ml 1 .6 6 0 3 X lO^^-'^s

931

Mev

Electron (1/1822 M^) = O

.9 1 0 7

x

10

"

2 7

g

0

.

5

I Mev Another useful constant: 1 Mev =

1

.601 x 10”^ erg . A table

givlne other quantities may be fo\md on page 840. The following masses are of particular interest:

Mu - MgiW 0 . 7 9 Mev

By Einstein's mass-energy relationship, E = mc^, an isolated system appears to decrease in mass when its energy decreases. Thus the total energy and the mass of two attracting particles decrease as they approach one another, losing their excess energy by radiation.

B. PACKINa FRACTION AMD BINDING ENERGY The packing; fraction, f . is defined by

M (A .2) - A

f = A

V7here M (A ,2) is the mass, in amu, of the nucleus of mass number A , charge 2.

Experiment shows that f is very small throughout the period­ ic table (FIG. I . l ) .

The numerator, M-A, is called the Mass Defect*.

When a nucleus disintegrates, the BINDING ENERGY with vihich xhe daughter particles were bound is defined as the sum of .the

resultant jit-ssea minus ths i n it ia l nuclear mass; i . e .

♦Some authors define tb# maBs defect as -TBE/c^, where TBE te the Total Bind- ing Energy defined on p. J.

Electron ..leO

0 .0 0 0 5 4 8

amu

Proton _iP^

_{1 .0 0 7 5 9}

Hydrogen i h1 I.O O

813

Neutron o hI

1 .0 0 8 9 8

Deuterium iH^ 2.01471

Oh. I

Packing Fraction and Binding Energy

■p o ^ H

S ’ “

M vT Q 80 20 0 N« g Si Te-W 0. ^ _ess 0 “Kr iib T.-R· H,T1 "cThn 20 60 160 FIG* I . l 100 120 140 PACKING FRACTION f 160 200 210 A BE Z U f

-Total' Binding Eaergy is defined as that amount of work we vfould have to supply in order completely to dissociate a nucleus

into its component nucleons. We have been expressing mass in

terms of energy units and vice versa. This w il l be done fre­

quently.

Average Binding Energy per Nucleon

As shown in F i g . I . l above, the mass of any nucleus is very

close to A . Since nuclei contain about the same number of N and

P , the sum of the constituent masses is about 1 .0 0 8 5 A . Thus there is a total BE of about (0 .0 0 8 5 x 931A) Mev, or, dividing

by A , about 8 Mev per nucleon. The a particle has a BE of about

7 Mev per nucleon»*. See F ig . 1 . 2 , page 4 .

The BE of any process ( i . e . of the particles emitted in the process) must be negative for the process to proceed spontaneous­ l y .

* I n calculating B E 's from an atomic mass table, it is only necessary to correct for the orbital electrons in one case,

namely, emission. For the other cases, the correction is

automatic. Thus suppose we wished to calculate BE (p ") for the

reaction, N —* P + ß” . Following our ru le , we would write

BE = M(P) + M(ß~) - M(N) . But, by coincidence, i f you wish, th is is Ju s t*** BE = M(H atom) - M(N) .

A / vA arb/trarz/y

Now, any ß“ process, ' Z + l ( ) + can^be written

as the reaction, /C o n s id e r i n g the inert nucleus (a - 1 .2 ) as

going along unchanged. Therefore BE(ß~) = M(A,Z+1) - M (A ,Z ).

But for P —>-N + ß+

or

we must set

BE(ß·*·) = M(N) + M(ß) - M(P) ^ = M(N) + Miß) - M (iH1) + m BE(ß+) = MCA,Z) - M(A,Z+1) + 2m

**Nuclear binding energies are much larger than those of orbital

electrons. According to the Fermi-Thomas statistical model of

the atom, the to tal BE of all the electrons is 1 .5 5 Z Rydberg.

Binding Energy and Mass Determinations

Ch. I

1

ff

fl

I I I

6

4

1 * 1

1 "

2

0

1

150

Atomic Valght, A

FIG. 1 .2 : AVERAGE BIITOING ENERGY PER NUCLEON

Experimental M£.3S Determinations

The most valuable tool Is the mass spectrograph. T h is w ill

not be discussed. Further data may be obtained from momenta of

particles taking part In nuclear reactions. M(neutron) may be

obtained from the photodlslntegratlon threshold for deuterium.

Problem. Consider the reaction

-bIO Ll"^ + 2He^ + Q

Q Is defined as the kinetic energy (T) of the resultant, minus the T of the I n it i a l , particles. In other words, Q Is the exothermic heat of reaction.

Assume that thermal neutrons ( T ^ O ) , react vilth a f ix e d B

target. Calculate the velocities of the products. In the

reverse reaction, v.'here a particles are shot at a fixed L i target, what Is the reaction threshold energy of the a 's ? S o lu t io n . From mass tables, Q = 0.00304 amu = 2 . 8 3 Mev. Next show that the velocities involved are non- relatlvlstlc. R e l a t i v i t y . For this problem, and for future reference, we set

down some relations from special relativity. For a free par­

t i c l e , moving v/lth velocity v . 2ΤΞ

Throughout t h is text we shall use M as the i^st-mass of a general p a r t ic le , M iC for the relatlvistlc mass, m is re­ served f o r the electron.

T h e n , the momentum force energy

£ = M i ' V "N

F = i

W = Mc^ + T (T = kinetic ener- y)/ 1 . 2

= M y p g ________

Ch. I Binding Ener(

3

y Problem - Rela.tivity 5 W does not include interF.ctlon energy.

Expand T = Mc-(2T- 1) = ,Mc2(|(b2 + (a)

= f . ^ (1 + |p2 + . . . ) (B) At T = , (A) aaya |£ » 0 . 1 , p jai 0 .4 5

(B) says T T d a s s l c a l ^ + 0-15)

2 ^claaaical

Thus for kinetic energies below one-tenth rest energy, the relative error in the classical expressions is on t h e o r d e r

of the fraction, T/Mc*^. Hence the problem at hand m ay~^

considered strictly classical. Reverse Threshold.

In a two-body problem, the only part of the KE that can enter into reactions is the KE of the particles relative to the

center of mass. This is easily shown to be

M

1

M

2

where = reduced mass

r-^2 - relative ■oosltion, r^_ -

£2

Thus Tj.g]_ is Just the KE of the "reduced mass p a r t ic l e ". Now, if one particle ("ta r g e t ") is fixed in the lab co­ ordinates, the total KE of the system is the KE of the bombard- InG particle

Ttotal = 4

where stands for the mass of the bombarding p artic le .

Thus, for fixed target,

T , T

•^rel M]_ total

m _ ^tarr.et T _ wVia-Pe rp. . .

ir the lab system KE of bombardinf, particle

For the problem at hand, T^g^ must equal Q. Tthreshold=ll/7 Q = 4 .4 5 Mev.

C. LIQUID DROP MODEL

Ex-oeriment (scatterinc, quadrupole moment, e t c .) shows that nuclei are rouchly spherical, with volume directly proportional to A , so that a nucleus is analocous to a drop of incompressible f l u i d of very hich density (I0l4g cm "?). We shall use

Atomic Mass Formula Oh. I

Equation

/ ^ f^ucitcn o r

1.5 a'^* X 10-13 cm

•where R is defined by the sketch

V I j

and is derived from the theory of

o< mean lives (Ch. I I I ) . * ^

1. Semi-Empirical Atomic Mass Formula

- fariicic

We shall use this concept, along with other classical ideas (surface tension, electrostatic repulsion) to sel^ up a semi-

empirical formula for the mass of any atom, M (A ,Z ). This

formula can then be used to predict the stability of nuclei against particle emission, also the energy release and stability of nuclei for fission (Ch. V I I I , Sec. j ) .

Naturally the first term vrlll be the masses of the con­ stituent particles 1.00813 Z + 1.00898 N,

Mo = 1.00813 Z + 1.00898 (A-Z),

The first correction term will be the bulk ^heat of cotiden-

sation,” due to short-range nuclear forces. On page ¡5~we men­

tioned that the BE per nucleon was about 8 Mev. Thus we siispect that this correction will be of order of magnitude 10 Mev = .Olamu.

- "“^1 ^

In assigning the same energy of attraction to all nucleons, we have actually over-corrected, since the surface nucleons will

be attracted from only one side. We therefore Introduce a sur­

face tension correction to the large correction. Mi, which is proportional to the surface of the drop.

M

2

= +a2 a2 /3

We next notice that stable nuclei tend to fonn themselves

of N-P pairs. We add a positive mass correction for the number

of unpaired nucleons.

“ 3

=

»3

This form Is derived in appendix 1 .3 , p.aa.. See also Fip,.

1 . 3

and discussion, p. 8,fl.

Next we add a positive term for electrostatic repulsion. The potential energy of a uniformly charged sphere is

0 = I ere = ^ 2 |

so, inserting R = 1.5 x 10“ 13 a^ ,' M

4

= 0.000627

The final correction term will be called <f. It depends upon the stability of nuclei with respect to whether the number of Z and N is even or odd^.

♦Empirical data on the stability of nuclei give the following table, where e = even, o = odd. We therefore construct a correction function <f(A,Z) as shown.

Ch. I Semi-Bnpirical Atomic Mass Formula

Let us now grout) all teims and proceed to fit to eiperi-

mental values S(A,Z) and the three constants aj^·'

M(A,Z) 1.00898 A - 0.00085 Z - A + aglAZ^— —

2

+ 0.000627 _Z, + S(A,Z) 1 .5

This formula must f i t three criteria:

1. It must give the correct A vs. Z curve for the stable elements. See below - discusaion of evaluation of ag.

2. It must give the mass of the odd-A elements on this c\irve.

(For odd-A, Ss 0; S can then be adjusted to give the even-A

masses and (S-energies (see discussion, pp 8 , 9 ) ) .

3. M(A,Z) plotted against Z is a parabola. The third criterion

is that its minimum must fa ll at the stable elements. Thus

we can solve for ag directly by setting A -

z)

= 0 = -0.00085 - 2 a , t— + 0.000627 | | ^*6

3£ Í A AA

Comparison with knovm str.ble elements ^ives aj = 0.0 8 3

Putting a^ back into 1 . 6 , we Qet the A vs. Z curve of stable elements mentioned in criterion 1.

1.9 8 + 0 .0 1 5 A ^

z = ______ 4 ___________ 1 .7

Showing that A/Z starts out at 2 and increases because of the term in k H which comes from the electrostatic repulsion term, M

4

.

Equation I . 7 predicts correctly the stable elements from 1 to

9 2

.

Putting aj and 1 . 7 into 1 . 5 , above, we ne't equation for M(A only) in the tvro unknovms, a]_ and a

2

. From the mass data for the stable elements (step 2 .) \je net as a best fit

a^_ =

0

.

0 1 5 0 7

; a2 = 0 .0 1 4

Equation 1 .5 for the mass of an atom. A , Z, then gives M(A,Z) =

0 .9 9 3 9 1

A - 0 .00085 Z + 0 .0 1 4 A*<^

(A ^ >7)2 «2

+ 0.083 ^2______ + 0.000627 ^ + S (k ,Z ) 1 .8

♦Footnote continued from page

6-A Z H_____Comment_________________________________ (A.Z)

e e e Most stable <f(e,e) = -f(A)

0 e 0? Moderately stable; rourhly ,f(0,e) = 0

0 0 e j equal numbers observed f ( 0 , 0 ) = 0

e 0 0 Least stable <i(e,0) = +f(A)

The theoretical Justificr-.tion for the hehrviour of ¿"is as follows. Use the Permi-r,as model of tho nucleus (Ch. V I I I , Sec. H,).

Because the nucleons have spin l / 2 (hence can have spin up or dov:n), er.ch momentum state of a P or an N i s , rourhly> .tvrofold degenerate. But all our mass terms have assumed that M (A,Z) varies

smoothly every time N or 2 cha.n-,es. corrects for this. See Be

8

Maaa Formula and Isobaric Behavior

and, by step

3

. i(A-even, Z-even) =-0.036 A“

i(A-odd,Z-anythinß) =

0

<i(A-even, Z-odd)· =+0.036 A”

Oh. I 1 .9 Equation 1 .8 describes quite v;ell the atomic masses for A >

1 5

. It can be used to predict the most stable Z^ I Its

numerical value for all atoms from ^ to

9 7

( )^49 has been

published by N. Metropolis, In st, for Nuclear Studies, U. of ChcO·» 1948.

The accuracy of 1 .8 is Indicated by the follovrinc data: Nucleus Experimental Mass

16 .0 0 0 0 0

Maas Formula 15.99615

5 1 .9 5 6

51.959 42^0^®

9 7 .9 4 3

9 7.946

75

AUI

₉₇

197.04 197.04

2 3 8 .1 2

2 3 8 .1 2

For an empii^l correction term for the heavy Isotopes, baaed on decay measurements, see S t e m , R e v . Mod. Phys. 2 1 . 316 (1949).

For a review article, see Feenberg, Rev. Mod. Phys. 19, 239

2 . Iaobari c Behavior "T 1 9 4 7 )

For aome particular odd-A,I

.8

is Given in Fic.

3

, below. Note that the form, a parabola, la Governed by the "unpaired" tenn, M^, but that Z-minlmum need not be an Intecer.

FIG .

1

.

3

, laobars, ODD-A FIQ. 1 .4 , EVEN-A

Arrows are possible decay chains, mentioned below.

In F Ig · 1 .4 , the dashed line represents all the terms of 1 . 8 except i . The solid linea (representine some series of

even-A iaobara) are raised or lowered by f . Thla effect haa

been exaggeruted. The dota are poaalble nuclei. Note that, in

general, there la only one stable odd-A nucleua, but that, in

the case of even-A, there may be two, or even three. Thua In

F i g . 1 .4 , Ni and N

2

v/ould both be atable.

Fig. 1 .4 ahov;a that even-A, odd-Z nuclei are unstable, and

should not be found in nature. A ctual]^, there are a few, but

none heavier than ./v"*·* except for (which is radioactive)

¥e shall shov/ In Ch. IV that p lifetim es Increase as P

energies decrease. Usln^^ this Information, v:e see that the

theory Is corroborated by the follovrlnG chains. Thus, the

odd-A (

1 3 9

) P“ -actlve chain produced by the fission of 92Ü^59

X e i Ü - ^ Ca Ba .9.5- La (stable)

shov:s steadily increaslnr; lifetim e s . The follov/lnG even-A chain

(l40) shows, in a d d itio n , an alternation of lifetim es.

X e I6 S .» Q a Ba La Ce (stable)

6- y

Every second transitio n is from a curve to a curve,

with a short l i f e .

3\ Al^ha Emission

We can use 1 . 8 , p. 7j to compute the BE of neutrons in a nucleus:

BE(N) = M(A-1,Z) + I

0

OO

898

- M (A ,Z )

This always turns out positive for the stable elements, showing

that these do not tend to emit neutrons spontaneously. However,

BE(a) = M(A-4,Z-2) + 4 .0 0 3 9 0 - M (A,Z) 1 . 10

!joes normative in the middle of the periodic table, long before

the "natural a-emitters** are reached. The Intenrenlng elements

are stable a,gainst a-decay only because the a-enerries are so

small that the lifetim es are r)rohlbltlvely lon^. (See problem

on p. rs , Ch. I I I . )

The periodic table ends in the rr^^,lon Z = 90-100, because of the increasingly nei::Atlve valuo of the BE for a-emlsslon and fissio n .

4 . Periodic Shell Structure

In addition to the r'eneral behavior predicted by the liquid drop idea, nuclei are observed to have periodic variations in BE

which are not predicted by 1 . 8 , p.

7

. These variations have

nothlnc; to do vrlth the periodic atomic (chemical) properties. The ll.'^hter nuclei have total BE maxima at A = 4 , 8 , 12, 15, e t c ., shoviin^ the saturated character of sub-units containing 2N + 2P.

A larrser shell structure is also observed. See Ch. V I I I ,

Sec. K, p./>7.

D . SPIN AND MAGNETIC MOMENT^«*

The total a n f^ la r momentum of the nucleus is denoted by I Instead of J , the symbol usually applied to atoms.

^ For more complete d isc u ssio n , see Bethe, A , para. 4 and 5 , and also Ch. V I I I ; Bethe, D , Ch. V . For experimental references, see Cork, "Radioactivity and Nuclear Physics” , Van Nostrand, 1946, p.

1 3 1

.

I = L + S

In the literature, I is often referred to as the "spin"** of the nucleus. in this chapter we shall use "spin** for the

inherent ^non-orbital) part of the angular momentum.

As a general consequence of quantum mechanics, an isolated system (nucleus) has its total angular momentum. I , quantized to

integral and half-integral values, in units of/n. The half­

integral values arise from the inherent spin of the nucleons. For an even number of nucleons, we expect I ~ 0 , 1 , . . . ,

whereas for A-odd, we expect 1 = l / 2 , 1 l / 2 , . . . , . No exceptions to this rule have been found.

In general, nuclei have a magnetic m o m e n t a s s o c i a t e d with I .

We sh al^isc u ss, briefly, four methods of determining I ,

1. Hyperfine structure of optical spectra

This arises either from nuclear magnetic moments or (and we ignore these) nuclear quadupole moments or isotoplc mass d iffe r ­ ences.

Let us calculate the interaction energy,U »between the magnetic

moment,/¿/n, of the nucleus and the magnetic field, , of the

atomic electrons.

This is easily done by setting

U = - / f i n ^ c 1 . 11

where la evaluated at the nucleus and we consider time averages.

We must now express^cin and ^ e in terms of the nuclear and

electronic angular momenta, I and J . The nucleus and electrons

are assumed weakly coupled, so that 1 and J each stay fixed in space (Fig. 1 . 6 ) .

Next we assume that 1.12

where we shall try to solve for and that

1.13 where A' is a constant that must be calculated for each atomic

configuration (Mattauch, p. 3 1 ). A plausibility argument for

the assumption that is parallel to J is

given in Fig. 1 . 5 . Use the vector model, and

assume L-S coupling. ^ rotates rapidly

around J , so that its component perp. to J,

fii, is, on time average, zero. Thus,/^e —Mm

Rather n a t u r a l l y i s parallel t o ^ g , i n all casesj J - L, J - S , J = L -i- S.

Using 1 .1 2 and I.1 3 j iTll becomes

FIG·. 1 .5 +><YvA'i*i T

Electronic Vectors ^ -jy- 1 .1 4

only.

HextjSee Fig* 1 . 6 , we define the total atomic angular

momentum, F, and express in terms of the quantization con­

dition,

111=11 + Jl= I+ J . I+J-1, I+J-2 1.15 classically, the la\i of cosines jives

F^ = + 2 1 .J

or

I . J = ¿(F^ - i2 _ j 2 )

To get quantitative, quantum-mechanical results from the vector model, we must always replace the s q u a r e ^ f angular

momenta, j 2 , , by j ( j + l ) , etc .* Hence I .14 becomes

,^F(F+l) - K l - H ) - J(J+1)

1 .1 6

^

2

v T ( i + i T T T j > i

7

where, as indicated in 1-15, F can take on a m ultiplicity, M, of values

Oh, I

Spin and Moment

11

M smaller of '21 + 1 2 J + 1

Different values of F cive the interval rule for hyperfine multiiDlet structure.

FIG. 1 .6

Whole Atom Notice that merely hy counting the

maximum M in the spectrum of an isotope, I

can be determined. There w ill be, of course, some lines with an

M limited by a low J , but there will always be some lines given

by transitions to or from a J greater than I . Experiment shows that for

Even-A nuclei, I is integral ( 0 , 1 , 2 , . . . )

Odd -A nuclei, I is half Integral ( . . . )

as predicted on page . Note that we would not have this result

i f the nucleus were composed of protons and electrons rather than P + N.

Apart from Hydrogen, unfortunately, we do not know enough about atomic wave functions to calculate A' to within better than 10^, even with the help of empirical fine-structure data?** The values o f^ ^fr o m hyperfine structure are thus uncertain to 10^.

2. Alternation of Intensity of Band Spectra . . .**«·

. . . of diatomic molecules containing Identical nuclei. This

will give I , but not

♦This does not apply to other vectors, llke>u. See appendix,

p. 19ff

* * For more discussion see Mayer and Mayer,»^Statisticfll Mechanics p 172ff.

***Do not confuse this with hyper-fine-structure data, which is what we use to obtain differences of U ( 1 . 1 6 ) .

The alternation depends upon the fact that the state function for the molecule with identical nuclei,

2 (x i, X

2

)

must be either symmetric or antisymmetric with respect to ex­

change of x i and X2* x stands for a ll the coordinates, including

the spin, of a nucleus.

12

^ I n and Moment

Ch. I

/

Experiment shows that a ll even-A nuclei obey Boae-Einstein statistics, and all odd-A, Fermi-Dirac statistics, as is to be expected i f protons and neutrons individually obey Fermi statis-

tlo s. _

For even-A f ( x i , x

2

) = +X(x2»xi)

For odd -A T ( x i , x

2

) = -T(x

2

,Xi)

Now, T may be written ♦

I = Telect ^v^’^ 2 ) v i b r /^ J ^ ® ’^^rot nuclear

defines the position of the electrons. For identical nuclei

it is usually sjrmmetric.

0·^ is S3nnm. because it depends only on a separation distance,

is syTnm. ^^or even J , anti-symm. for odd.

may be either symm. or anti-symm. For 1 = 0 , <3"is symm. Moreover (Bethe, D , p. 1 8 ) , the

^ p. statistical v;t. of symm. to

/ N. symm. <r* states is ( l + l ) / l . Th<^

^ important thin^ is that the syimnotry

s. of determines the symmetry of

I . e . allovrs only certain values

I “or J . The enern;y of a level depends

J . Alternation of

Intensity in Band For example, consider §

2

, v/hlch obeys

Spectra. Fermi statistics, has 1 odd. Each

odd must be combined with an even

, and vice-versa. Because of the statistical vreichts attached

to the J the intensity of t h e ^ e v e n ^^nes will be three

times as great as that of the /^even <3^ odd lines,

3 . Atomic and Molecular Beams

The ^ o f the nucleus is determined by a Stern-G-erlach type experiment, splitting a beam of atoms in an inhomogeneous

Nuclear moments are, hov;ever, hard to observe, since they tend to be masked by electronic moments on the order of 1000 times as great, unless J = 0 .**

4 . Magnetic Resonance (Nuclear Induction)

Rabi has passed neutral atoms through a strong I and J

are decoupled and precess Independently about X at a*”frequency"“ ♦approximately — see Pauling and Wilson, "Introduction to Quantum Mechanics,

page 260

where g is. the gyr o m a g n e t i c ratio Involved. For I this

is the radiofrequency . If a small s e c o n d a r y K ' ,

perpendicular to the m a i n , oscillates at nu?^ ( 1 < n < 2 1+1), it will induce a change in the o r i e n t a t i o n of 1. T h i s flip m a y be detected bec a u s e of its effect u p o n the a t o m ’s trajectory.

This technique of r e s o n a n c e - i n d u c i n g a flip h a s recently b een applied to s t a t i o n a r y nuclei, for example s i m p l y to a drop ot water. The t r a n s i t i o n is observed b y its e m i s s i o n or absorp­

tion of radiation. The relative )i’s of different nuclei can thus

be determined to one part in 10®. Also, n o w that w e know JJproton

very accurately, the e x periment can be run " b a c k w a r d ” to

f\irnish the most precise (and a convenient)method f o r determining

an imknown .*

5. Nuclear Reactions, G a m m a and B e t a De c a y If the spin of an initial nucleus is known, and t h e n there is /9- or 3<^decay for example, we can t h e n t e l l something about the s p i n of the end-

product, or vice-versa. This is discu s s e d in later chapters·

Experimental R e sults

In the same way that electronic m o m e n t s are measured in Bohr maQnetons,

= 2^ = 0.9273 x 10“^^ erg gauss"^, so nuclear moments are measured in nuclear magnetons,

" iÿc = Î537 "

Some observed spins are given below:

P article Spin Moment

E l e c t r o n 1/2 -1.002 B o h r magneton

P r o t o n 1/2 +2.7896 Nuc l e a r

N e u t r o n l/2 -1.9103 ** "

D e u t e r o n 1 + 0 . 85647 " "

^ is considered + v/hen directed alone; s. The electron moment of

one is predicted b y the Dirac theory of the electron, but we have no well accepted theory for the n u c l e a r moment. The ansv/er may lie in meson t h e o r y .

It is rather i n t e r e s t i n g to note that

even thougji the d e v i a t i o n from one is T:ell outside the limits of experimental error.

This Integral difference would be explained b y a naive little fantasy. Suppose that v/hat væ call a "neutron" is really a mixture of states, i.e., that mosttîthe time a "neutron" Is really an "ideal" neutron, w i t h ^ =: 0 , but that, for a fraction of each second, t,

N - » P + 7r“

where P Is an "ideal" proton, v r l t h ^ = 1, and tt" is a negative meson, with m a s s m u c h less t h a n that of a P, movinj^ orbltally aroùnd the P, so that Its orbital m a g netic m o m e n t , , is greater than, and a n t i p arallel to, the u n i t m o ment of the ideal proton.

Ch. I

Spin and Moment

xo

’^Kello£f(and Mlllman, "Molecular Beam Mametlc Resoxiance Method" Rev. Mod.

Phye. 18.525 («46).

Write = -/^

7

t > Atc+ =

The observed would then be given by

(1 -

1 .1 6

Similarly, assume that a **proton** spends most of its time in an ideal proton s t a t e ,/k— 1, but that, for the same fraction of each second, t ,

P- ^N + 7T+

The observed would be given by

= (1 - t) + tyU^ = 1 + - 1)

But from 1 .1 6 , [yU^ - l) =

So X (p = 1 -/Wj, = 1 + 1 .9 1 = 2.91

In the above model, angular momentxim is not conserved. A less fantastic relation in the table on the previous

page is the correlation betv;een/t<p, and/i<jj. I f we assume

that deuteriiim is mainly in a state v^here the IT and P are alined spinijcrallei (momentantiparallei) , we would expect/^ = 2.7896

-1 .9 -1 0 3

=

0

.

8 7 9 3

, fairly close to the observed value. We may assume that the discrepancy is becai^g^^ deuterium is not in a pure S-state, but that there is some/brbital contribution to the

moment. Other evidence bears this out. See Ch. V I, p. 114.

Even-Z. Even-N nuclei, and the effect of adding one nucleon. It was mentioned before that these nuclei are very stable. It is also observed that they all have spin and moment zero. They seem analogous to the closed shell atoms of chemistry.

We can test our ideas on nuclei, and the application of atomic quantum mechanics thereto, by predicting the moment of

closed-shell-plus-one nuclei. The odd nucleon may be either

P or N.

We then assume that the closed-shell part of the nucleus .’'nerely actri as an inert center of mass (central force field) around which the odd nucleon revolves like an atomic electron, complete with spin.

As shov.Ti in appendix I . l ,

p.ltff,

the maximum component of ^ along ^ is Given by

14

Spin and Moment, Mesons

Ch. I

Schmidt Equations* / ^ 2 = I +

2 .2 9

odd-P, I = 1/2

1 .1 7

/« Z = Odd-P, 1 = f-

1 / 2

1.18 /<2 = -1*91 odd-N, 1 = ( + 1 /2 = V i - i ^ = ^ - 1 / 2 *Zelt8. f. Physik 106, 558 ('57)

Ch. I

Schmidt Linea

15

In F ig. 1 .7 (purely schematic, not to acale) the curves rep­ resent 1 .1 7 and 1 .1 8 . The dots are maximum o b s e r v e d /^ ' s for

many nuclei. One might hopefully

say that there is a tendency to­ ward grouping near the two lin e s . Odd-N data is sim ilar.*

F IG . 1 . 7 , ODD PROTON E . ELECTRIC QUADRUPOLE MOMENT

For am plification, see Mattauch, p . 38; also Rosenfeld, "Nuclear F orces,” p. 392.

The hyperfine structure of Europium, for example, shows lines that can be accounted for only by assuming that the nucleus has a permanent electric quadrupole moment, Q .

Rabi has shora that even deuterium possesses a guadrupole moment such that it appears as a spheroid elongated (prolate) along the I a x is . Q(deut) = 0 .0 0 2 7 3 x lO'S'^cm^.

The concept of quadrupole moment stems from classical

electrostatic potential theory. Let us assume that the nuclear

charge is rotating about I . Then, no matter v.-hat its distribu­

tion, it must, on time average, appear cyj,indrically S 3 r m m e t r i c about I . Because, outside the nucleus, ^ 0 = 0 , we can expand

0 in terms of Legendre polynomials.

?n (cos G) 1 .2 0

— time#the coefficient is called

® Q, the quadrupole moment. As

cm'[2 1 .2 1 Nuclear elongation, much

exaggerated

shown in App. 1 .2 p. ai it is Q = y (

32

i

' 2

- r ' 2 ) ^ ( r ' ) d T '

v/here the notation is shora in the F ic . and e is the charge on an electron.

The first term of 1 .2 0 is the ordinary Coulomb 0 . The next term, the permanent dip o le, is not generated by nuclei, in the static case, because of parity considerations (Schiff, p. 1 5 8 ). The first static term giving a measure of the departure from

spherical symmetry is the quadrupole term. A nucleus of the

shape illustrated would have Q > 0 . This is the more common occurence.

Ifilhen 1 = 0 , the nucleus has no preferred axis, the charge

distribution appears spherically symmetric, and Q = 0 . ¥e shall

now use quantum mechanics to prove that Q = 0 also when I = 1 / 2 . In quantum mechanics v;e must replace z* by r* « ej = r* ■

z

’2

= + y .

2

i

2

^ ^ .

23.2

^

2

X-

7

’ (l:,Iy + Iy lx )+ ·.') But because of the coimnutation relations for I , the three teirms of the form 1x^7 + ; also i | ^ j

2

_ i | ^ I d + 1) =

3 / 4

therefore z'^ = ^ (x '^ +

7

'^ + z '^ ) + 0 - rl£ “ 3 so Q = 0 by 1 .2 1

Experimentally, Q ranges from

Q =

7 .0

X 10"24 cm^ for

-I

07

4 /^tfn yaJu€ ·/

to · -0.6 « » ' .0.4* U

Addtndum 1h MticUmr

but* 0 is nior© common. satn«.c S€ri^$ H€p»rt

M« if May /9t9.

Problem. Assume tha.t Lu^"^^ and l^^7 g,re ellipsoids of rotation, obtained by deformins. vfithout chancing its volume, a sphere of radius R - I

. 5

x 1 0 ^ 3 A*'a . Calculate the ratio, a /b , of the axial to the equatorial semi-axes.

Solution. UslnG 1 .2 1 , the intecration over an ellipeoid is straight-

fonm rd. We set o o

Q V I (a*^ - b“^) 1 .2 2

where V = ^ ab^ =

/5V is the charge of the nucleus, ;eZ, so 1.22 gives

1 .2 3

16

Quadrupole Moment

Ch. I

Elimination of Q = ^ (a^ - b^)

b2 = r 3 A

would appear to give a cubic equation. Before vre bother to

solve i t , hovrever, let us hopefully assume that the nucleus is very close to spherical, so that

a = b(l+d) d

« 1

1 .2 4

We shall drop terms in d^ and d^; solve for d. I f d turns out

Indeed to be small, our assumption was Justified, and have

saved much algebra. Insertinc 1 .2 4 , 1.23 goes to

1 = 1 2b2d

Thus for L u , setting b^ «w = 7.02 x

10“ 25

So (a /b )L u = ( a /b ) , = 0.97 with neglisll^le

Ch. I Radioactivity 17

F. RADIOACTIVITY AMD ITS GEOLO-^ICAL ASPECTS*

Presiimably, at the besinninc of the universe ( ^ 3 b illion

years ago) all isotopes were formed. The only radioactive iso­

topes and chains no\i present are those originatinc with isotopes \ihose half l i f e , -t , is appreciable compc.red with 10° y.

Some naturally occurring radioactive isotopes are knoivn that are not parts of chains:

(p~, p·*·, or K capture; -t = 2 .4 x 10® y );

jyRb®'^ (p-, t: = 6.6 X

10

^°y); ¿

2

® ™ ^ ^ ^ y); 7iLu1T6 (p-,

2 . 4

X 10^° y ) ; 75^©^®'*’ (P~,

3

x

10

^^ y)

Also, minute amounts of r.re formed by cosmic rcidlatlon.

However, the three most Important rr‘dlor‘,ctive families c?.re:

Name Parent Half life

(4n - 2) U r anium- Radium

_{4 .4 9}

X 1q9 years

(4n) Thorium 1 .3 9 X lolo II

(4n - 1) U ranium-Ac t i nium

_{7 .0 7}

X 10® If

A nucleus decays do\m the chain by a and P" processes. When the half-lives of these competing processes are comparable, there is "branchinc**, but the daucJiter products all eventually

loop back into one another. These chains end at Pb.

Naturally occurring uranium contains and U^38 ^he

ratio of 1 :1 4 0 , and all their decay products, which are in radio­

active equilibrium, often called "secular equilibrium.** The

ratio of the activities of U^35 and U^38 is 4«g.

By measuring the ratio of Pb^^^ to rocks, we can

estimate the a^e of the earth, which appears to be about 2.5

billion years. Astronomy leads in two or three different ways

to the assumption that the ac.e of the universe is about 3 billion

years. The earth thus appears to have been formed early in the

life of the universe

Terrestrial Distribution of Uranium

One gram of natural U produces 0 .9 5 erg sec“ ^ when in equi­

librium with its decay products. On the average, there are five

parts per million of U in the earth* s crust. One g Th produces

0 .2 7 erg sec~l. The average abundance of Th is about 10"5. Thus a gram of crust produces

7 . 5

x 10“ ° erg sec"^.

The known conductivity and temperature gradient of the earth’ £ crust allow us to calculate the earth* s energy loss due to con­ duction, which is 6 X 1 0 ^ cal sec"^. Assuming a steady state, this requires a generation uf heat, throughout the whole earth, of

only 4 x lO-o erg sec~^.____________ __________________________________

•»^Cork, **Radioactivity and Nuclear Physics**

**See, for example, NRG Bulletin No. 8 0 , "The Age of the Earth. ***Since all the transitions involve a A A = 0 or 4 , one can

This fPvCtor of about 200 is .not very satisfactorily account­ ed for, although there is some geochemical evidence that U should tend to concentrate in the crust of the earth rather than in the core.

G. TVEASIJREMENT AND BIOLOGICAL EFFECTS OF RADIOACTIVITY

CHE CURIE is defined as 3 .7 1 x 10^^ disintegrations per

second. This is rougjily the number of disintegrations per sec.

in one gram of pure radium. The strength of a source is usually

given in terms of the activity of the parent nucleus; thus, i f a source contains a parent nucleus in equilibrium with 7i of its daughters, then there will actually be (n + 1) x 3 .7 1 x 10-·-^

disintegrations per sec. There may also be gamma radiation.

ONE ROENTGEN (r) is that amount of 5c- or T'-radiation which w il l , on passing through pure air under standaid conditiona, pro­

duce one esu of positive and one esu of negative ions cm“’^ . One

r liberates 83 erg g~l air and rougiily the same per gram of water. Typical soft tissue absorbs energy to about the same extent

per gram as does water. A dose of one ROENTGEN EQUIVALENT,

PHYSICAL (rep) i s an irradiation by particles other than v’^iotons such that, again, 83 erg g"^ is absorbed.

Dose^

Biological damage depends not only upon the total dose, but also upon the specific ionization or ion density alonf, the path

of the p article. Particles with hifjier density of ionization

generally have greater biological effectiveness. It is customary

to present the following table.

RELATIVE BIOLOGICAL EFFECTIVENESS

Particle RBE

x-rays, gammas, betas* 1

trotons 5

Alphas 10

Fast Neutrons 10

Thermal Neutons 5

* I n the case of alphas and weak betas, some extra protection is given by the dead, outermost ( ^ 0 . 1 mm) layers of the skin.

Tolerance Dose:

The latest proposal (Jan. 1949) is that the limit be 0 .3 rep,

divided by the RBE, per week, for total body irradiation. Thus,

for example, the limit vj-ould be 0 .3 r per week for gammas, but only

•^Gkpodman, V o l. I I , Chap. 1 ^ ; also S ir i ( see p. 259 of this book)

Nucleonics 4 , ^ 2 (1949) and 60 (1948)

Safety Code f o r the Industrial Use of x-rays'*, Am. Standr.rds Assoc. N. Y . , 1946

Loa, D . E . **Action of l^ d ia t io n on Living Cells" 1947

Tompkins, P .C . **Lab. Handling of Radioactive Material" AEC, MDDC 1414 and 1527

Cantril and P a rk e r, **The Tolerance Dose” AEC MDDC 1100

0 ,0 3 rep per week for fast neutrons.

Ch. I

Radloa.ctivity

19

It mlGht be healthy, hovfever, always to allovr for much larg­ er safety factors than called for by the table, which may imply that more is knovm about the dancers of radiation than is actually

the case. RBE varies with the species of organism or c e ll, the

kind of bioloGical effect studied, dose rate, and probably other factors.

Protons, fast and slow neutrons, really have quite sim ilar

RBE. Accordinc to experience to date, for effects on mammals,

this varies from 3 to 30.

For local exposure, e . ^ . , the hands, a limit of about 1 .5 rep per v;eek is proposed.

A Fev; Sundry Facts

Cosmic radiation is about 1 mr per day, or rouGlily 2 . 5 ^ of the proposed vreekly dose.

One curie of gamma emitter produces on the order of I r lir“ ^ at 1 meter. 8 in . concrete or 2 in . Fb reduce this to one tenth; In terms of neutron flux, assuminc total body irradiation, 40-hr week.

Fast neutrons: allowable flu x , 100 cm“ ^ sec"^

Slow neutrons: '* ** 2000 " "

APPENDIX I . l : MAGIJETIC MOMENT FOR CLOSED-SHELL-PLUS-ONE NUCLEI

In this section, ^ does not mean footnote but, instead, denotes a quantum mechanical v^^ctor.

Interpretation of F ig.1.7. I * is of such a length that, along

"^e x t, ^ maxiimim observ­

able component of I , a h a lf integer.

I f , for example, I were 3 / 2 ,

then may have (21 + l )

different components, given

- 1

1

2 ~2

2

“ 1 = 1

The lenr.th of is p;iven by

= 1(1+1) or I * = i/l (l+ lT

The same applies to the other angular momentum vectors, and s * .

F I G . 1 . 7

... ...mechanical vector —— --- magnetic vector

♦Experimental data for shielding against high energy gammas is found in papers by Westendorp and Charleton, Jour. App l . Phy s . l6, 590 ('^), and Blocker et al., Phys. Rev. 7 9 , 4l9 (’50).

The masnetlc vectors are treated d iffe r e n t ly . In quantum mechanics

A o p e r a t o r ^®operator Therefore, for the vector model, we must write

= k j * = k / s (s + 1 ) S j

where /-observed - / ■ = = k ( s z ) m a x = k s

Thus, fo r a proton, 2 .7 9 = -g

^proton ~ 5 . 5 8 ; N e u t r o n “

Note that y ^ ( ^ + l )

5 . 5 8

and - 3.82 are analogous to the anomalous gyromacnetic r a tio , - 2 .0 , fo r the electron. Because the spin part of the total I contributes this anomalously large magnetic moment,

(notice the fig u re ) the totaly|£* is not parallel to

I f there I s n o ^ Q x t » i * w il l stay fixed in space, andy^*

w ill rotate about with a h ig h frequency, u> . The time

w il l be the component of alo n^ I * .

We call t h is Mr* *

20

Appendix 1.

11

Closed-shell-plus-one

Ch. I

■When ■ is applied, 1 * w i l l start to precess abouti^gxt

with a frequency i<^Larmor (see footnote)

The projection of/(£* a l o n g ^ M f » i ·© · the observable /<· ,

w il l b e : -m

„ # # 'tUl

We sh all nov.·· calculate t h is quantity.

s*2 = I ; f * 2 = Í +1) , S*2 + 1 *2 _ £*2 cos (s ’*· ,1*) = --- 2 s* I * / * V i *2 + T*2 _ a*2 cos ( i * , ! * ) = --2 ¿*^1*

V/orkinr, in nuclear m a g n e t o n s , J = ^ ^ . rotons

C O S ( s * . I * ) = k ° ^ ^ - '*'■ ~ ^—

' 3 — 2 1*

FOOTNOTE: “f c - t a » i n t e r a c t i o n ; i . e . <u 10^7

On tho o t h e r hnnd,

10

. - ^

Laratit

S in c e t h e ^ a t t a in a b l e 10^ Gauss, the two frequencies ■•'.ro v a s t ly d i f f e r e n t .

- „ X + 1 * 2 - s* 2

u * oos I * ) =

---^ 1 P T »

Ch. I

Appendix I.l: Closed-shell-plus-one

21

2 1* For t]ie odd-P case

. * _ ¿ » 2 + 1*2 , a»2 it(s*2 + I»2_

^ I 21*

y « o b = = A i i i = - < ·"> n ,

^ I * 21*2 ^

Nov/ v:rlte the denominator 21 (l + l ) ; >Uobs “ ^x. w ill be given vihen mj takes on the value I .

11

max. = J?*2 + 1*2 _ g*2 + k (a *2 + i*2 _ »2 ) 1 .2 5

y ^o b s *

,

2 (1 + 1 ) nuclear magnetons

We shall now use 1 .25 to .-.©t the first of the four ex­ pressions on p. 14, 1 .1 7 I = t + 1 / 2 ; 1 = 1 - 1 /2 max = (I- i ) ( l + - i ) + K l + l ) - ^A + k L ^ /^+ I ( I + D - ( I - i ) ( l + i )1 / o b s 2 ( I + l ) _ I2-¿ + i2+ I _ | + h;(| j2 ^ J _ ι 2 ^ .¿ ) 2 (1 + 1 ) _ i 2 ( 2 ) + I ( l + k ) - 1 + k

2 (1+1 )

= 1 — + 3«2 9 I + 2 .2 g = 1 + 2 .2 9 Nuclear I + 1 Magneton

The other equations are obtained I n the same manner, remem­ bering, of course, that the fir s t three terms In the numerator of 1 .2 5 . above, are m ultlnlled by zero for the case of the neutron.

APPENDIX 1 . 2 : ELECTRIC QUADRUPOLE MOMENT

Look at P ig . 1 . 8 . The polynomials o f 1 . 2 0 , p. 1 5 , are unity for 0=0, so 1 .2 0 gives, along the z-axls,

^ 1.26

I f we can calculate 0 along the z- a x ls , by any method at a l l , and expand It as a convergent series PIQ·. 1 . 8 : Whole nucleusj In Integral pov/ers of z, we can get

symmetric about I . the c o e ffic ie n ts , a^^, by comparison.

We are Interested only In a

2

· By Coulomb's law

22

Appendix 1.2: Electric Quadrupole Moment

Ch. I

Dewtiii^ rtt< urfil- vtt&r, , I’y S ^

1^ - a'I = (/I®· ♦ A'* - a * · * ' ) = /L I - (■ _>1. ) 3 >t{l - (»)}'/x.

I I- -X

$0

r< 1

For comparison with the ag term in 1.35 , we are interested

only in the two terms in l / p

3

.

and when r l i e s along the z-axis

which, by comparison with 1 .2 6 , Ju s t ifie s 1 . 2 1 , p 15.

APPENDIX 1 . 3 : MASS CORRECTION FOR NEUTRON EXCESS, M

3

, p

6

. Equation V I I I . 46, p./^<^, shows that nuclei can be described as cold Feiml-Dlrac gases, with a Perml-energy (for the neutrons) Now, the total enercy of the neutrons,measured from the bottom of

the w ell. Is ^

-E. · ! » ( « .

So _and

Term p .

6

, corrects e x p lic itly for electrostatic enercy,

so we Ignore It here. Then E]j+

2

;(>“ln) w il l occur when li=Z=^/

2

. The mass correction will be proportional to g.

~ ~ + ? ’ - ^ (

2

) ]

Lel· A 3

rV--

I-Tayloi^expand the first two binomials as far as terms In

as assumed in M

3

·

2 S' a

“ if. T * r

OQ

The solutions, references, e t c ., are not due to Dr. Fermi. 1. Design a mass spectrocraph to measure the mass difference

between Hydrocen and Deuterium. Measure the separation of

the close lines (H2)+ and D+·

References: Mattaucli and Fluegge, ’’Nuclear Physics Tables” 1942; Harn-well and Livingood, »’ Experimental Atooiic Physics”

1933; M.Cj. chapter on Modern Mass Spectroscopy in

’’A^ivances in Electronics” 1948.

2. Use the semi-empirical mass foimila to calculate the energy of

a-particle emitted from 9211^^5. Compare th is with the obser­

ved value.

Calculate the binding energy of a proton and a neutron in U 2 3 5 .

Answers: a-particle— theoretical, 4 . 1 4 Mev; experimental, 4 .5 2 . BE(N) = 6 . 8 ; BE(P) = 4 .8 5 * See Metropolis, "Table of Atomic

masses^ Oak Ridge.

3 . Desic^n a molecular beam apparatus to determine the atomic magnetic moment of Na in the ground state, "-Sx.

Reference: RG-J Fraser, "Molecular Beams" 1937.

Consider: Temperature of furnace, slit dimensions, magnet

dimensions, pressure, beam separation after splitting, width of beam.

4. Problem on r e lativ ity . A cosmic ray meson, mass = 2l6m,

passes throurili two G-eiger counters, 10 m apart. ¥hat error,

dt, in the time, z^t, between the two pulses, is allowable, i f we wish the uncertainty in energy to be less than 10^?. Consider the cases where the "energy" ( i . e . kinetic energy) of the meson is 5 0 , 100, 1000, Mev.

Solution: Be sure to calculate

^ < 1 0 %

where T is the kinetic , not the total^ energy. For lov7-T par-

tiolos, dE/E has little importance.

Ansv:ers: For T rr 50 Mev, dt = l.::!9 x 1 0 “ ^ sec.

1 0 0 " 0 . 7 1 '*

1000 " 0 .0 2 9 6 ”

5 . Design a 10 Mev Betatron. (Three assif^nments; one v/eek‘ s homev;ork)

Points to c o n s i d e r S h a p e of pole pieces (taper, position of stable o r b i t ), ampere turns required and pov/er supply, fre­ quency·, laniinntion, vacuum, d .c . b ia s , injection, extraction. Referencos: W. Bosley, "Betatrons," a review. Jour. S c i . In s t . ,

2 ^ , 277 (1946) ■ , , X

D.W. Kerst, "2 0 Mev Betatron',' R e v . S c i . I n a t . 13 , 387 (1942) alao Phva. R e v. 6 0 . 47 and

63

(1941)

W .F. Weatendorf. se of d . c . in Induction Accelerators," Jo u r. A n p . Phya. 1 6 . 657 (1 9 4 5 ). See also page 581.

For a general reference see M .S . Livingston, chapter on par­ ticle accelerators in "Advances in Electronics," Academic Press, 1948

6 . Design a 200 Mev synchro-cycloti^Dn ( i .e . fm cyclotron). Also one v;eek's homework.

Points to consider: Dimensions, frequency, frequency of modu­

lation, radial decrease of > phase stability, voltage on

dees, electrostatic focussinc, injection, extraction, vacuum. References: Chapter by Pickavance in "Progress in Nuclear

Physics, 1" by 0. Frisch, 1950. The Berlceley machine is

described by Chew and Moyer, Am» Jour. Phvs. 1 8 . 125 (1 9 5 0 ). The Chicago machiEe in the "170- in. synchro-cyclol?rcn,

Progress Report" Institute for Nuclear Studies, Univ. of Chicago, 1950.

24

P R O B L E M S

Oh. I

7 . Design a one Mev Cockroft-Walton accelerator.

References: Proc. Roy. Se e . L o n d . A136 610 , 619 ('3 2 ) To reduce the expense to l /l O by using radiofrequency, see Rev. S c i . In s t. 20, 216 (1 9 4 9 ).

8 . Describe the precautions and apparatus necessary to carry out simple chemical operations upon a one curie sample.

One should not approach within about ten meters of the un­ shielded sample. Thus about 5"Pb would be a reasonable thickness of shield.

9. The activity of a sample is the total number of processes counted per unit time.

A = Z A i = Z / ? i n i

where A is the decay constant, and nj_ the number, of the 1-th sort of disinteciratins nucleus.

Plot, acainst time, the activity of a two-element radio­ active chain.

Solution. The equations are

n, = ”>

1

, e

n,

-W ritt hhU in the farm

h - j i d t = t ? [ ^ n ^ -fPcU-K

= e

iPdUr t

Q

cU· + c]