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

Midway Reprint

Nuclear Physics A Course Given by

ENRICO FERMI

at the University of Chicago. Notes Compiled by Jay Orear, A. H. Rosenfeld, and R. A. Schluter

Revised Edition

IIA Ulb. THE

U N IV E R S IT Y

OF

C H IC A G O

PRESS

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 ic 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 T h e U n i ­ versity OP C hicago P ress , Chicago^ Illinois^ U.S,A.

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 timedependent). 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. JO, AHR, RAS

July 1951 Minor revisions have been made, mainly in bringing some of the references up to date. ¿TO, AHR, RAS

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 ofRadioactivity 18 Appendices 1. Magnetic Moment for Closed-shell-plus-oneNuclei 19 2· Electric Quadrupole Moment 21 5 . Mass Correction for Neutron Excess 22 Problems 24 CHAPTER II.

INTERACTION OF RADIATION WITH MATTER

A. E n e r ^ Loss by Charged Particles 1. Introduction 2. Bohr formula 5 . Electrons 4. Other particles 5· Other absorbers 6. Range 7 . Polarization Effects 8. Nature of equationfor -dE/dx 9 . Ionization a gas 10. Radiation B. Scattering 1. Classical calculation for single scattering 2. Multiple scattering 0. Passage of Electromagnetic Radiation through Matter 1. Photoelectric absorption 2. Compton scattering Radiation loss by fast electrons) Pair formation 5 . Cosmic ray showers 6 . Summary Appendices 1 , 2, an d ’?. Multiple scattering 4, Momentum and pair creation References CHAPTER III.

ALPHA

27 27 27 JO J1 J1 51 J2 J2 35 j4 j4 5^

56 5^ 5^ 40 4j 4? 49 49

5I 5^ >4

EMISSION

A. Rectangular Barrier B. Barrier of Arbitrary Shape C. Application of Barriers to «-decay D. Virtual Level Theory of -decay E· c( -ray Spectra Appendix Til

27

55 ^6

58 59

66 67

CHAPTER IV. A. B. 0. D. E. P. G. H. J. K. L. M.

BETA-DEOAÏ

Introduction Exaaples of /»-prooesees Energy diagrame Theory of |5 - 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

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

=

G-oodman,

yn

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.

Ch. I

2 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 Ml Electron

(1/1822 M^)

1 .6 6 0 3 X lO^^-'^s = O .9107 x 10 " 27 g

931 0 .5I

Energy Mev 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: Electron ..leO Proton iP^ Hydrogen i h 1 Neutron ohI Deuterium iH^ Helium 2He^

0 .0 0 0 5 4 8 amu 1 .0 0 7 5 9 I.O O 813 1 .0 0 8 9 8

Mu - MgiW 0 . 7 9 Mev

2.01471 4.00390

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 f =

M (A .2) - A A

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

mass

number

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 Binding Energy defined on p. J.

Oh. I

Packing Fraction and Binding Energy 80

■p o ^

H

S’ “ M vT

20

N« g

0

"cThn FIG* I . l BE

“Kr iib 100

60

20

W 0. ^ ess T.-R· H,T1

Te-

Si

0

Q

120

140

160

PACKING FRACTION f

160

A

200 210

ZUf -

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 ly. * 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 youwish, 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 ( i H1) + 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. ***Neglecting the 1 3 .5 ev BE of the orbital electron.

Binding Energy and Mass Determinations

1

ff

fl III

6

Ch. I

4 1*1 1 "

2 0

150

1

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 -b IO

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

£

=

M i'V

"N

force

F = i

energy

W = Mc^ + T

(T = kinetic ener- y)/ 1 . 2

= M y p g ________ + p2 c^'

Ch. I

Binding Ener(3y Problem - Rela.tivity

5

W does not include interF.ctlon energy. Expand

T = Mc-(2T- 1) = ,M c 2(|( b2 + = f .^

At

T =

,

(a )

(1 + |p2 + . . . )

(B)

(A) aaya

|£ » 0 . 1 ,

(B)

T

Tdasslcal

2

^claaaical

says

p jai 0 .4 5 ^

+ 0-15)

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

where

M1M2

= 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 bombardInG particle Ttotal = 4 where

stands for the mass of the bombarding p artic le .

Thus, for fixed target, T

,

•^rel

m ir the

M]_

T

total

_ ^tarr.et T

_

wVia-Pe rp.

. .

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

1.5

a '^*

X 10-13

cm

/

V I j

•where R is defined by the sketch and is derived from the theory of o< mean lives (Ch. I I I ) . 1.

^ f^ucitcn o r -

fariicic

* ^

Semi-Empirical Atomic Mass Formula

We shall use this concept, along with other classical ideas (surface tension, electrostatic repulsion) to sel^ up a semiempirical 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 cotidensation,” 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. M2 = +a2 a 2 /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.. p. 8,fl.

See also Fip,.

1 .3 and discussion,

Next we add a positive term for electrostatic repulsion. The potential energy of a uniformly charged sphere is 0 = I

ere

=

so, inserting R = 1.5 x 10“ 13 a ^

^ 2

|

,'

M4 = 0.000627 The final correction term will be called i7 where, as indicated in 1-15, F can take on a m ultiplicity, M, of values M smaller of '21 + 1 2J +

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 bytransitions to or from a J greater than I . Experiment shows that for Even-A nuclei, I is integral Odd -A nuclei, I is half Integral

( 0 ,1 ,2 , (

...) ...)

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. will give I , but not

This

♦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 ) .

12

^In and Moment

Ch. I

The alternation depends upon the fact that the state function for the molecule with identical nuclei, 2 (x i, X2 ) 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. 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 statistlo s. _ For even-A f ( x i , x 2 ) = +X(x2»xi) For odd -A T ( x i , x 2 ) = -T(x2 ,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 , n , ^

21*2

>Uobs “ ^x. w ill be given

J?*2 + 1*2 _ g*2 + k (a *2 + i*2 _

,

2 (1 + 1 )

»2 )

1 .2 5

nuclear magnetons

We shall now use 1 .25 to .-.©t the first of the four ex­ pressions on p. 14, 1 .1 7 I = max =

t + 1 /2 ;

1 = 1 -

(I- i ) ( l + - i )

/o b s

1 /2

+ K l + l ) - ^A + k L ^ /^+ I ( I + D - ( I - i ) ( l + i )1 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 I + 1

= 1 + 2 .2 9

Nuclear 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, ^

PIQ·. 1 . 8 : Whole nucleusj symmetric about I .

1.26

I f we can calculate 0 along the za x ls , by any method at a l l , and expand It as a convergent series In Integral pov/ers of z, we can get the c o e ffic ie n ts , a^^, by comparison. We are Interested only In a2 ·

By Coulomb's law ^(^') dtl

22

Ch. I

Appendix 1.2: Electric Quadrupole Moment

Dewtiii^ rtt< urfil- vtt&r,

, I’y S ^

1^ -a 'I = (/I®· ♦ A'* - a * · * ' )

=

/L

I - (■

)

>1.

I

3 >t{l - (»)}

'/x.

r< 1

I- -X

$0

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, M3 , p 6 . Equation V I I I . 46, p./^“ln) w il l occur when li=Z=^/ 2 . The mass correction will be proportional to g. ~

~

+ ?’

- ^ (2)

]

rV-Lel·

A 3

I-

Tayloi^expand the first two binomials as far as terms In 2

“ if.

S' a

T*r OQ

as assumed in M3 · *Mayer and Mayer ,"Stat 1st leal Mechanics," p 3 7 6 .

Ch. I

P R O B L E M S

23

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 235.

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 ^

<

10%

where T is the kinetic , not the total^ energy. tiolos, dE/E has little importance. Ansv:ers:

For T rr 50 Mev, dt = l.::!9 100

1000

"

"

5 . Design a 10 Mev Betatron. homev;ork)

0 .7 1

0 .0 2 9 6

For lov7-T par-

x 1 0 “ ^ sec. '*

”

(Three assif^nments; one v/eek‘ s

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.

24

PRO BLE MS

Oh. I

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 . one v;eek's homework.

fm cyclotron).

Also

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.

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 /? in 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 -ji dt

=

t

?[^n ^ -fPcUK

= e

iPdUr

t

QcU· + c]

P R O B L E M S

Ch. I

f

-Xb

Je

dL·

25 ^ ]

A , 4.

It can be ahovm by straichforward substitution that the curves of parent and dauchter a c t iv it y cross at the exact time that the daufhter ac tiv ity is a maximum. This is il l u s t ­ rated in the curves below.

’ Parertl·

AdUv't'Cy, A , D a u ^h tir AcJiVily,

A ^^

•Z. Fcrsi·^ Pau^htir^

o.s

/ ^ » /¿>

» 1 -z,

I.S

i.s

PARENT and DAUGHTER A C TIVITIES Two statements can be made about the curves above: 1 . The f in a l decay rate depends upon the lonr.er

X·, X ^

A 2 . The simultaneous Ap max, and cross-over point occurs at a time rouchly of the order of magnitude of the smaller X. This is shown as follov^s: I f the cross-over time is called 0 , then straichtforward alff.ebra gives e Since the lo g . is a slowly v a r y in c function, O varies roughly as the loncer "K ; that i s , as the shorter -C .

Ch.

P R O B L E M S

26

I

t2 3 5 At time t = 0 , U'·''"' Is stripped of a l l its decay products. Plot the build^xp and decay of Actlnlvun X .

10.

Solution. The chain, v.'ith decay constants,/^ , In s e c "^ . Is as follovis (/^ = 0 .6 9 3 /T , vfhere T i s the h a lf l i f e ) . i/£ 3 5

3.l7i^’''

7.6%I0~^

l.sxio'^

4.2

X

to

-7

7.4% to

Tlie f i r s t daurii-tor, Th, vjlll Into secular equlllbrluni with the U vrlthln a matter of d ay s . This Is illustrated by the *‘fast-dau^>liter" (type I ) curve of i:^robleiii 9. A fter a fevr df^.yswe may consider the Th activity equal to the U activity , and .'^.o on to consider hov; the Pa crows in . The Pa viill build up in a period ^ 10^3 sec (105 y e a r s ). 'We may thus neglect the comparatively short Th f,rovth period, and write c4'i^p lt ra I f vre now restrict ourselves to a time considerable less than ti/, we may c o n s i d e r a constant, equal to . The solution to the d ifferential equation is then

0 -

^Pa ^Pa

Pa has a T much longer than any of its dau 'hters. Thus, as the Pa Grows in G^i3.ually, so do a l l its daur'hters, in secular e q uilibriu m . This is another v;ay of sayinc that the AcX activity w ill always equal the Pa ac tiv ity . We have nov; completed the problem, except for a description of the disappearance of the vrhole chain (v;hich is by then in per­ fect eq uilibriu m ) as the U decays. During, this time the AcX activity»· is r.iven by

M

J. O

—

^

The coniplete curve Is riven belovi. Notice that the asyn^totlc lncreo.se of the Pa ac tiv ity after a time on the order of Its h a lf l i f e is important in an anala,^ous problem: irra d ia tio n . One accomplishes l i t t l e by irradlatin,·^ c?. sample for a time lo'icer than, say, twice the h a lf life under c o n sid e ra tio n .

CHAPTER I I ■ INTERACTION OF RADIATION VnCTH MATTER A.

ENERGY LOSS BY CHAR&ED PARTICLES

1 . A charged p a r tic le moving through m atter lo ses energy by electromagnetic in te rac tio n s which r ais e electrons of the matter to excited energy states. I f an e xc ite d le v e l Is I n the continuum of states the electron I s io n ize d ; I f not, the electron is In an excited bound s ta te . I n e it h e r case the Increment of energy Is taken from the k in e tic energy of the in c id e n t p a r t ic l e . I n the following section "i o n i z a t io n " w i l l r e f e r to both degrees o f exc itatio n . Range = to tal distance traveled by the p a r t ic le u n t i l i t s kinetic energy is 0 . Before a formula fo r the range o f a par­ tic le can be d e r iv e d , the rate of energy lo ss per u n it path must be calculated. The f ir s t such c a lc u latio n i s due to Bohr**·, and is essentially classical., i . e . , non-quantum m ec h a n ic a l, 2. Bohr Formula. Consider one electron o f mass m at a distance b from the path o f an incident p a r t ic le having charge ^e, mass M and ve lo c ity V . m (e le c tr o n ) *— b= impact parameter mass M charge velocity V F IG . I I . 1 Assume the electron is free and I n i t i a l l y at r e s t, and moves so sllgjitly d urin g the c o l l is io n that the e le c t r ic f i e l d acting on the electron due to the p a r t ic le can be c alculated at the In it i a l lo c atio n of the electron. The l a s t assumption I s not v a lid fo r an Incident p artic le of v e l o c it y comparable to that acquired by the electron. We shall calculate fir s t the momentum acquired by an e le c ­ tron during a c o l l i s i o n , and from th is fin d the energy ac q u ire d. As the p article p asse s, the e le ctr o s ta tic force F changes d irection. By symmetry the Impulse cLt p a r a lle l to the path Is zero, since for each p o sitio n o f the p a r t ic le to the le ft of A , y ie ld in g a forward contrib ution to the Im pulse, there Is a p osition at equal distance to the rig h t of A g iv in g an equal but opposite c ontrib ution . The Impulse-L to the path la the order o f magnitude of : L. “ ,

« (e l e c t r o s t a t i c

.

We f i r s t estimate

fo rce)X (tlm e of c o l l i s i o n )* « ·^ :

More exact computation: Consider a c irc u la r' c y lin d e r centered on the path and passing through the p o s itio n o f the electron. F i g . I I . 2 . Let & be the e le c t r o s ta tic f i e l d i n ­ tensity due to the particle" The e le c t r ic f lu x i s

/ ê - .,_

~/mY

·

Only values of

another criterion for

,

J.

have meaning, and therefore

is I I .8

The larger of(bjjjin)c< (binin)oM 't'® used in the intecration*. For values of V where bm ax> l^mln. (^min) > (^m in L Lng I I . o in I I .B s * ” and therefore I I . 8 whould be used. UslTng d£

4-ir3-^e^^ a

"oOx

'wiV*

(erg cm“ ^ )

I I .9

where is a suitable average of the oscillation frequencies of the electrons. More precise calculation** leads to the following formula for heavy particles, i . e . , not electrons:

11.10

where I Is the average ionization potential of the electrons of t h ^ aTj^sorber. in ergs. The In term as 9 for 1 mev protons in 3. Electrons. There are two main reasons why 1 1 .1 0 cannot apply to electrons, (l) The derivation assumes that the incident particle Is practically undeflected. But the Incident particle acquires a transverse component of momentum per collision approx­ imately equal to that given to an electron In the absorber, and i f the incident particle is an electron, the transverse velocity corresponding to this momentum w il l not be ne g ligib le . (2) For collisions between identical particles exchange phenomena must be taken Into account**·*. Bethe * * gives the following formula for energy loss by electrons:

^

2 Tre 'oL-X 'tn. ■ V

-1 +173“J

11.11

_________________________________________ (electrons) where T la the average ionization potential of the atoms of the absorber and ~f~ = relatlvistlc kinetic energy of the electron. * I n cutting the Integral of I I . 4 o f f at "bjjjjLn'^O, we have neglected a term , This is Ju stifie d in "Lecture Series in Nuclear P h y s ic s ", LA 2 4 , Lecture X I , printed edition p. 2 7 . * * Bethe, Handbuch der Ph v sik. p . 519 * **M o t t , Proc. Rov. See. 1 2 5 . 2 2 2 , 1 2 6 , 259 (1929)

Ch. II

Bohr Formula, Electrons

An approximation for I

51

is'i-

T ~ (l3 .5 2 ) x l.feoi X 10“'^

(ergs)

1 1 .1 2

The formula 1 1 .1 1 for electrons, applied to a i r , values ^iven in the ta b le . 4.

.Other p a r t i c l e s .

F o r _______ E l e c t r o n s

y ie ld s the

I n Al_r___________

in e . V . per _ dE Enercy, ln incident p articles of identical e . V . ______ __ charge moving in l ik e absorbers -dE/dx is a function of V only. 1 9 .5 X : 10^ Therefore i f -dE/dx as a func­ lo5 3 .6 7 tion of energy is knora fo r , say 10° 1 .6 9 protons of mass Mp it can be 10 J 1 .9 5 found for some otner singly 105 2 .4 7 charged particle B by changing 2 .7 9 the energy scale so that the new lo^o 3 .4 8 energy values are Mb /M ^ times the old. The follo w in g table for protons absorbed in a i r enables f i l l i n g out the table for (l) deuterons, mass 2 , ( 2 ) ^ ^ mesons of mass --^215m. For a par­ ticle B of d iffe r e n t charge as well as d i f f e r e n t mass, the above energy correction is made, but furthermore the io n iza t io n lo ss value is m ultiplied by since enters the formula. In this way column (3 ) for alpha p a r t ic le s is derived from that for protons. The un it for rate of energy loss in th is table is ergs per «^.ram cm” ^ and is denoted by

dE _ d f

I ctE 1 1 .1 3

^

where Is the t h i c k n e s s I n r.-cni“ 2. e r g - c m “ ^ Is o ’- t a l n e d f r o m - d E / d f by o f air, 25 .0012 f - c m “ 5 (STP).

In Air PROTONS (l)Deutero ns -ae Enercy, _dE. Kev . ;?■ e .V . d? r-cirr·^ 106 300 2x106 300 2x107 47 10 ^ 47 10 ° 7 .6 2 x 10 ° 7 .6 2x105 2 .3 109 2 .3 lolO 2 .3 2 x l0 l 0 2 .3

t M s tabl e - d E / d x In m u l t l p l 3;-lnr;, b y t h e d e n s i t y

( 2 ) M meaons d£ - ¿r

( 3 )AlDhas *3-«T 4x10^ 1200 . 1 1 7 x 10 ^ 300 .117x107 47 4x10^ 188 . 1 1 7 x 10 ° 7 . 6 4x10° 30 . 1 1 7 x1 0 ^ . 2 . 3 4x10^ 9 .2 . 1 1 7 x1 0 ^ ° 2 . 3 4 x l 0 l 0 9 .2

I n Lead Protons e.V. -4 ·

106 150 loZ 2 7 . 1 0 ° 5. 109 1 .6 1 0 ^ 1 .6

5 . Other absor b e r s . I f io n izatio n loss were exactly pro­ portional to the density of the absorber, then -dE/df ergs per g-cm“ ^ for a f,iven narticl^^^ would not va ry . But -dE/d/ depends on two further fa c to r s : (l ) The numbor of electrons per atom; ? of an atom does not increase r,L] ft'ist as the v;eirht, thus ^9^/^ in the formula for d E/d? io lens for h e av ie r elements. (2)T , appearing in the Io·" t^rm, depends on tho absorber (Eqn. 11.12)*^*, P-^np;e■ O ften in oy“iorimontn the original data is the range and from t h is the energy is estim ated. We have derived an * Bloch, Z e i t s . f . Phvsik 8 1 , 363 (l 9 3 3 ) ^^See also Living ston and Bethe, Rev .K o d . Phy s .

p. 265*

Ch. I I

Bohr Fonnula, Range

32

equation o f the form -dE/dx = f ( E ) .

Intecratlng, we get

I I .14 ^ ~ 'U X E» Jm ^0 where the limits of Integration have been chosen so as to d efine X as the range of a particle with i n i t i a l energy E q . The Inte­ gration may be performed m unerlcally. For a rough approximation we assume f (E ) oc i/v^ec i / E . Then YicC E q V 2 cC nonrelatlvlstlc a lly . More precise consideration shows that a better approxima­ tio n i s * •^0C E j> *0C V ^

1 1 .1 5

Em p irical range-energy formulas: A rough formula giving the range of alpha p articles I n a i r at 15°C . and atmospheric pressure i s ^ (Mev)"cm . (alphas in a ir )

1 1 .1 6

This is correct to about 1 0 ^ . I t breaks down for r e latlv istlc v e lo c itie s. The general nature of the range-energy relation is shown i n P i g . I I . 3 . I n 1 9 3 8 Feather proposed for electrons the empirical rela­ tio n s h ip : 0 .5 4 3 E - 0 .1 6 0 for E > 0 .7 Mev , where R Is the range in gm/cm^, E Is energy In Mev. Glendenin and Coryell (1946) have improved this and also worked out a low enerfzy relatio n ; for E > 0 .0 Mev R = 0 .5 4 2 E - 0 .1 3 3 R = 0 .4 0 7 E ^ ·^ for 0 . 1 5 M e v < E < 0 . 8 Mev (for Al, but ie close for all other substajices

7.

Polg.rlzation e ffe c t s . I n the d e r iv a tio n of 1 1 .1 0 no account has been taken of the Influence on one electron due to the simultaneous motion o f the other electrons near i t . The electrons In a r e g io n move so as to dim inish the e le ctric f i e l d beyond that region. This p a r t ia l s h ie ld in g effect Increases w ith increase I n density o f e lectrons. The change I n -dE/dx due to th is e ffe c t is u s u a lly s m a l l .* * F IG . I I . 3 I f th e Index of r e fr a c tio n , n , i s not one, th e velocity of l ig h t I s le s s than c . In water, for example, n — 1 . 5 and v e lo c ity of l l ( ^ t Is ~-2/3 c. I f the in­ cident p a r t i c l e has v e lo c ity V > c /n it s electric fie l d Is strongly perturbed and can be lik e n e d to a wake in water. Such a p a r t i c l e produces r a d ia tio n known as Cerenkov rad iation , afte r I t a experim ental discoverer**^' 8. the form

I

N a tu r e of the equation fo r -dE/dx. Equation 1 1 .1 0 has ' oC

* Exponents fo r various p a r t i c l e s and energies in L ivingston and Bethe, p . 2 6 5 . * * T r e a t e d i n d e t a il by F e rm i, F h t s . Rev. 5 7 . 485 (1 9 4 0 ) *·»* C e r en k o v , Phvs. Rev. 5 2 . 3 78 ( 1 9 3 7 ) ; theory In S c h i f f , I . e . , p. 2 6 1 .

Ch. II

Ionization of a G-as

Me*

10Me» FIG. I I . 4

35

\OOMc*

ENERGY

The curve BCD ßives the 1/V dependence. At relativistic energies V chanj^es l it t l e and CD is asymptotic to V = c. relativistic energies, the log term in (V v l- ß ^ ) changes, and increases as V — c , giving the rise in the curve from C to E. At very lov7 energies (region AB) equation 1 1 .1 0 breaks down because the particle has velocity comparable to that of the orbital electrons in the absorber, and the efficiency of energy exchange is much lower. The particle i t s e l f captures electrons and spends part of its time with reduced charge.

At

9. Ionization of a p:as. I f ionizatio n is produced in agas the ions may be collected by charged electrodes, and the amount of charge collected w ill be proportional to the number of ions produced. The change of potential of one of the electrodes will depend on the charge collected (and the external circuit) and therefore on the number of ions produced. This voltage pulse may be amplified linearly and measured q u antitatively , as with an oscillograph. A gas chamber for this purpose is called an ion­ ization chamber^*·.

FIG .

I I .5

In the arranr.ement in FiG- I I . 5 , electrons are collected at the top plate. A negative pulse, of duration determined by R and the capacity of the ionization chamber and associated circuit, is produced at the ^rid of the linear a m p lifie r. It turns out that there is a close proportionality between number of ions produced and total energy lost by the incident particle. For most ^ases one ion pair (electron plus ionized atom) is produced for each 32-34 e .v . lost by the particle, (see table on following p ag e ). Although empirically the result is a References on ionizatio n chambers are: K o r f f , Elec'tron and Nuclear Counters (Yan Nostrejid), Rossi and Staub, Ionization chambers and Counters (McGraw-Hill).

34

Scattering

Ch. I I

simple proportionality betv/een Enerp;;^;^ for one ion p a ir ^ number of ions and energy spent, G-as Energy spent for ____________ one ion pair>e_.v. the explanation is very compli3 3 .0 H cated. Theoretical prediction He 2 7 .8 of the average energy per ion 3 5 .0 N pair involves: (l) calculating 0 the % of a ll primary collision s 3 2 .3 2 7 .4 Ne that lead to removal of an elec­ 2 5 .4 A tron in order to know how much Kr 2 2 .8 energy is ” wasted" on non-ionizing Xe 2 0 .8 excitation of the atom; (2) cal­ culating what fraction of energy carried away by primary ionized electrons i s used in producing secondary io nizatio n. This problem has not been completely investigated. Problem: Design an io n izatio n chamber and specify the char­ a c te ristic s of an associated lin e a r am plifier so that the system is suitable for measurlnç the a energy difference between the (ui) and T u il) a decays. ^ 2 U 2 3 8 _ ^ a + 9oTh23^; The follow ing are among the necessary considerations: {l) losa of energy of particles w hile s t i l l in the em itting substance, ( 2 ) gas to be used, (3) dimensions and electroatatic capacity of chamber, (4) gain and frequency response necessary in the am p lifie r, ( 5 ) rate of emission of p articles by the emitter. 10. Hish energy g p articles lose energy mainly by radiation. This effect I s taken up la t e r , in I I , section C , 3 · B.

SCATTERING DUE TO A COULOMB FIELD

Scattering due to interation o f charged p a r t ic le s with the Coulomb fie ld of nuclei is disting uish ed from scattering in which the incident particle enters a nucleua. Only the former is treated in this section. Scattering due to c ollision s with nuclei la observed for .all charged p ar tic le s in varying degree. An alpha track In a cloud chamber may have a single k in k , ind icatin g one large angle scattering event. Electrons are acattered much more fre­ quently, and th e ir tracks are as ahown: / 3 ------

o ( ------ s

1. C lassical calculation for single scattering. I f the screening of the nuclear charge by nearby electrons is neglected, the force on an incident p article due to one nucleus is the Coulomb force » where H is the charge of the nucleus and ^ the charge of the p a r t ic le . Assume that the nucleus I s heavy compared to the particle so that the center of mass of the system I s almost at the nucleus. Let b , the impact parameter, and 0 , the angle of d e fle c tio n be defined in F ig . I I . 6 , p. 35* For inverse

square forces between p a r t ic le s ,

^R uth e rfo rd , Chadwick and E l l i s , S u b stan c e s, p. 8 1 .

c las s ic a l

Radiations frcm Radioactive

Ch. II

Scattering

35 nucleus

of

1 1 .1 7 This formula ia v a lid at non- relativistic v e l o c i t i e s , V · ^ · ^ c. A relativistiosLlly correct version of 1 1 .1 9 for small angles Q is given in the paragraph containing 1 1 . 2 0 . Exact quantum mechanical c alculatio n g iv es the same formula provided the nuclear fie ld is exactly a Coulomb f i e l d . Both classically and quantum m echanically the formula is v a l i d only i f the distance of nearest approach of the p artic le to the nucleus is larger than the nuclear radius. The cross-section for scattering of the incident p a r t ic le at an angle 0 in the range d© is defined to be the total area _ L to the i n it ia l path o f the p a rtic le such that i f the p a r t ic le passes through this area i t is deflected by anaangle 0 in d0 . Since for given particle and nucleus, b is a function of 0 o n ly , the area corresponding to a given 0 l i e s at a certain radius b ( 0 ) , and has magnitude dC^« 2.Tr bc®> db. S u bstitu tin g for b i n terms of the corresponding angle 0 : dC^ = 27fb(0)b' {Q)dQ. D iv id e by the element of solid angle 2Trsin0d0 to find the. cross section per unit steradian, and substitute for b ( 0 ) it s value from solving 1 1 .1 7 for b . Then the cross-section per u n it solid angle at 0 is cLoj

4-\M V® /

1 1 .1 8

Note that most p articles are scattered at small eCngles. A r e la t iv is t ic a lly correct equation for 0 as a function of b when 0 is small can be derived e a s i l y , u s in g the same argu­ ments used to derive I I . 3 . Since now we deal with nu clear charge Z a n d incident particle of charge we must multipy I I . 2 by the nuclear charge in order to get the transverse Impulse impart­ ed to the incident particle in the c o l l i s io n . T h is gives , =

A/jp =

2 2 ie " V to

1 1 .1 9

If y p is the r e la tiv is tic momentum o f the inc id ent p a r t ic l e , angle of d eflection is very nearly , if

the

11.20 I f we put ^ = M V (n o n - r e la tiv is tic ) t h is becomes id e n t ic a l to 1 1 .1 7 when 0 is small and tan 0 / 2 =«? 0 / 2 . In these formulae »See,

for example. Lindsay. Physical M e c h an ic s, p . 7 6 .

Multiple Scattering

36

Gh. I I

b la limited to dlatancea from the nucleus within which the nu­ clear charre can be felt, I . e . , has not been screened by nearby electrons. 2. Multiple Scatterinc· Particles, particularly electrons, are deflected many times in passing through a foil of metal. The net angle of deflection, denoted 0 , is the result of a sta­ tis tic a l acctumilation of single small scattering events. The de­ tailed theory is complicated . A simplified treatment will be given here. Ve assume that no paths are complete loops, as in b F I B .I I .7. It is plausible and can be shown·*··* that the values of 0 for many traversals are distributed about © = 0 according to the gausslan law. I . e . , probability f o r 0 in the range d © is y^(0)cl©=:Const. 0 For small

11.21

scattering angles.

1 1 .2 2

’..•here © p I s the net angle of defxection for p collisions end the bar meajis the average for many such traversals of the absorber. Since st&tistlcally the individual events do not differ, 11.23 Using II.20_f.o r 0 , averacins over· for values of b from bjgin bmaxj and summins for all collisions in the lencth of path D , we obtlain

Cl F i a .I I .7 0 ^ = 2t t N D

11.25

For thin fo il s , D differs little from the thiclmess of the fo il. JN “ nu . oi‘ atoms per cc. Due to screening of the nuclear charge by electrons, the 2 felt by a particle depends on b, therefore H in the integral is a function of b. We tiike Z outside and adjust for the error by choice of Assuming the absorber is so thin that ^ and V do not change,

O -

11.26

Choice of limitò: bmax: The equation 11.26 would be strictly correct i f at distances beyond bmax-the screening of the nucleus were perfect·;i . e * , no scattering, and for distances within bmax there were no screening at a l l , i . e . , full value of 2 v/ere f e l t . No such boundary exists, since screening inornasos gradually with distance, but fortunately the log term E. J . W illiam s, Proc .Roy. Soc. A169 531 (1939); Rossi and Greisen, Rev. Mod. Plivs. 13 249 (1941). Rossi and Greisen, I . e . This is shown in Appendix I I . 1 Sliown in greater detail in Appendix I I . 2

Ch. II

Multiple Scattering

37

Is not sensitive and we maj;· put* Bohr radios Qc, r-4 The factor i n t a k e s w i t h '2 . radius^/j ).

1 1 .2 7

Into account the v a r ia t io n of the function ^ — (electron charge w it h in sphere of I\

The limit bmin e ffe c t iv e ly ad justs the maximum anyle in a single scr.tterinG process. Since we are not countlnc any values of ® > 1 ,we may Impose the rough restriction that ^ < 1 . T his gives

t

U6HT ATOMS HEAVy ATOMS

F IG . I I .8 (from 1 1 .2 0 ) 1 1 .2 8 M)p Other considerations may n o v e m bmin such as ( l ) , the fin ite size of the nucleus: bniin> 1 . 5 X 10“ ^ ^ , from equation 1 . 4 (page 6 ); ( 2 ) , size of wave packets of p a r t ic l e s in the c o lli­ sion*^. The result for 0 ^ u sin g our choices for bmin ^ax is a ? ^ 8 - i r U D » :i : e ^ ^ aoV^. 1 1 .2 9 The result Is not sensitive to the choice of bmin bmax; the log tei*m is of the order of ma,-x

(b) The region in which the electromaipietlc fie ld is not zero has a length. In the electron frame, of about b/jC the factor coming from the Lorentz contraction. SHAPE OF PULSE:

t*o

SPECTRAL DlSTRlBUTlOr^J

when the electron is in wave-front plane representing nucleus FIG . 1 1 .1 6

The spectral analysis of this pulse may be done by assuming for sim plicity that it Is a Gaussian of time width b/)fx>: =X^

cC 6

1 1 .4 4

Then the amplitude distribution of component frequencies is another gausslan, of width ^ V b # * . The energy per unit area in the disc-shaped electromagnetic disturbance is a function of the distance b . Energy per unit volume is given by ' , ,The volume per unit area of the d isc equals the thickness = b/y(cirn). Therefore the energy per u n it area is “

(c ) Approximate to the gausslan spectral distribution by a rectangle of w i d t h % c / b * * * . This means that the energy carried by photons In the frequency range A j / ^ Is simply proportional to •^Abrnham-Becker, V o l . I I , p.4ti. ■sHf·Stratton, Electromar.netlc Theory, p. 290. *«^*It v/ill turn out late r, (just before 1 1 .5 4 ) t ^ t only frequenci§sc > and^ are all

mast be finite

62

Virtual level theory

Ch. Ill

The W .E .B . method givea for th© general solution in I I

The W .K .B · method gives for the connecting asymptotic functions across r = h (Schiff, page 184)

Thus •jt

p 'R i

-I ^

^

where a is defined by I I I . 3

This gives for Uj^

At large r

__ i - [ C e r " * 4 e e '° ] '■

I I I .7

In the appendix to this chapter, page 6 7 , C_ and C+ w i l l be c a l ­ culated i n the region near a virtual level with the result where

Where £ is the energy separation from the nearby virtual level and ^ ( r ,t ) w h ich at t = 0 gives zero probability or zero amplitude f o r r > b*

Ch. Ill

Virtual level theory

63

Such a state fu n c t io n can be expanded in terms of the knomi eigen functions u ^ C r ), where u^^Tr) is u * (r ) i n region I , Uat (r) in I I , and (r ) in I I I , a ll corresponding to the same energy En. This w ill turn out to give a narrow energy spectrum about the virtual energy. The d is t rib u tio n w i l l turn out to be such that j ^ ] ^ (r ,t ) w ill decrease exponentially w ith tim e. The rate of de­ crease of total p r o b ab ility inside w ill be the reciprocal of the mesin l i f e of the p artic le i n s id e . In s\immary the procedure w i l l be: (1 ) Determination of eigen functions in energy region about the v irtu a l l e v e l . (2 ) Expansion o f > ^ ( r ,0 ) in terms of these eigen fu n ctio n s . (3 ) Determination of rate of decrease of . (1 ) Actually there is a continuous spectrum of eigen values and fu n c ­ tio ns. The problem w i l l f i r s t be treated as one of discrete levels _____________________________ by lim iting region I I I to r< L . 1L Later L w i l l be made to approach In f i n i t y , which approaches the continuous spectrum sit u a t io n . The region is lim ited to r ■

Hi =

'

T]_ and To are the wave functions of the i n i t i a l and final states, resp. Tnis is derived, for example, in S c h i f f , Q . M . . p. 193. It is discussed in more d etail in Ch. V I I I , sec. B.

76

Beta Decay Theory

For r = 0

\

Ch. IT

1 1 7 .9

The mmber of plane wave states having magnitude of momentum between p and p + dp, with the particle aii3n'i'here in S i , is*

Therefore ~ ^ d /a ,J E ’,:h.ere J is the Jacobian. Usinr the rei.ation E = cp„ + ^ , J is found to be l/c.'^ (Mass of v a s s m e d zero for ^ ^ ^ this derivation.) Thus

4?-= ?5^FET

^

Usln^ xhlB to express I V .6 , the probability^ of emission per unit time, P(Pj/,Pp)dpp, Is ? < P , . P g ) d P 6 = ^ ( i l ' m | 8 / ^ ^ ^ ^ / -

Using the relation Pj,c = writing p fo r po from nov; on,

1V.14

- Eo to eliminate p^y , and IV .1 5

Using the equation E^®·^ = '^m^c'^ + c^P^ax define P^ax» 1^*16 we get I ______________ ^______________ _____________________ C« 0. Then the ntimber of states, — t2 IV 18 The number of states having momentum between p and p+dp is

*

,

J^

if'

o

^ t

~

C.

o il.

IV

)

Beta Decay Theory

77

_______________2 +Ka; = p, mcni^^= P^ax· 'W’l’ l t e the intocral in terms of , ^ and e x p r e s s i o n for the lifetime becomes

, and call it F (^ o )·

I V ,2 0 The IV . 21 IV

.22

S-br*aicJitfonmrd Integration leads to «= (T?.)

Vo

V

V h ^ J ^ (9 .+ ‘i i 7 ^ )

=

s.nk"' 7,

IV. 23

L im l t in r , forms for F (7· )· larce compared to 5 ·

Fi'^o ) '

rl t, small compared to .5 : F(T7 ) -t

30

IV. 24

•2.

10 5

'·

(G::*;''/v'’. in {3 the lo ^ term leads to 7‘OViers of'??#\7hich cancel the pov7ers of-:^^ lov/or than in the oxprossion for F(^^) ) Be-bween these extremes, F()Jo) has the values in the table: Vo 0 .5 1 2 3 4

F ( V ,) 0 .0 0 0 12 0 .0 1 1 5 0 .5 5 7 6 .2 3 2 9 .5

’B *i7]o) calculated here is for a plane wave ¥ for the olectron. X T the distortion of tho v/avo function l^y t]io Coulomh field of •blae nucleus is taken into account a factor f(Z,>?o) must bo in- · soopted into the un-inter,r^^-tod probability/ function P (p ). The in b o r r a t o d function F(?7o ) then doponds on Z and shoxild be v;ritten F C z /o ;^ ). For small Z, P(Z,>7o)=5=i F('^o) as dofincd before. The f i m l expression f o r 7;:" is

1

IV . 25

■wlaere I*Vy\I is uncertain, but of order unity for ”allo\;ed” transi"b i o n s . (Allowed and forbidden transitions v^ill be discussed in a e c t i o n s H and j)* From equation I V .25 it is seen that tlie theory pr*ed icts tnat F '^ = consta?'it i f does not cUanr.e. Problem: Look un the^B spectra of the follovflnc; P emitters: lleo^ and f i l l in tho table: Kr.clc’as F(*W^) Max. Enercy %

Olx. IV

Beta Spectrum 78 1? aViaT)e of Bnerrar and Momentum Spectrum

Equation IV.17 for P, the probability of emiaslon of an eleoLn the momentiun range dp depends on p throuf^ the factor which haa tlie form:

curve approaches P = 0 parabollcaliy at 0, since there the expression Cv » ^

is almost

constant. It approaches P = 0 parabollcaliy at p = p^ax ________ there _____ p^ ia __________ PIG. rv.8 because almost constant and a Taylor expn. of the 2nd term about Pmax. gives (P ‘ P ^ ) c^Cp it ^ * Tn*C*· *: P P^»f This plot must be corrected for the perturbation of the elec­ tron or positron wave function by the nuclear charge. Tq_ I s larger than for Z = 0; Tp+ is smaller. The correction is greater for low energies. For negative electrons the correction near p = 0 is rouglily propor­ tional to l /p ; tlius the corrected curve ;for negative electrons is linear near p = 0 (FIG. I V . 9 ) . For positrons the correction is in the other direction.

The corresponding plot against energy Is given in FIG. IV .IO ."' IV

(E stands for kinetic energy here)

For F « since p^dp = 1 /2 p d (p ^) «c ^

.2 6

dE.

The curves corrected for non-zero nuclear corrocted negative electron curve has the (jjmax _ j;j2 ¿g ^ constant x dE, therefore ordinate at E = 0 . G. Experimental V erification. There has alviays been uncertainty in fo r the low energji- part of the spectrum.

charge are shown. The form near E = 0 of the curve has fin it e

the experimental results Improvement in exper-

•Some forbidden^ decays have a different spectrum shape. spectra have been experimentally verified.

These forbidden

Cli. IV

79

Experimental Beta Spectra

F ia. I V . 10 imental teclmique 3aas so far imprbved the agreement between exp­ eriment and theory. The theoretical shape of the curve near depe'nds on the mass of the neutrino. For neutrino mass 0 , there is second order contact; for neutrino mass 0 the curve has a vertical tangent at the point of contact. See Bethe A , p. 191*

Within experimental error, the curve for neutrino mass 0 Is correct. The mass Is certainly small, leas than 10 Kev, In energy units. The point at wJalch the curve reaches the horizontal axla l*s d iffic u lt to determine experimentally because the curve Is tangent to the axis there. It is therefore d iffic u lt to determine EW®·" directly. More accurate determination of I s made possible by the Kurie p lo t. Prom equation I V .1 5 , the intensity of emission at p Is

nr.2T W h e re C (Z,p ) Includes the constants and also the dependence on nuclear charge. This can be written T -\= .\l

‘

I V .28

Now the plot of the radical against enercy should be a atralcJit line whose Intercept with the horizontal axis is eaay to deter­ mine, FIG·. IV . 1 2 ( a ) .

(a)

Eurle Plots FIG. I V . 12

Cb)

80

Beta Selection Rales

Ch. IV

Eurle olots deviate more or less from a stralGli't lin e at low enerrar. These deviations are belnc: reduced by more refin ed exp­ erimental techniques and by refinements In the theory of the rac*^Occ^ionally a Kurie plot has the form in F I G . 1 2 ( b ) . T h is is Interpreted as the superposition of two p decay phenomena with d ifferent values of ES^^, as exemplified by the diagram: Often the camma for the debay from the excited state is observed. Sometimes p spectra have the form Gi‘''*en in FIG. IV .1 3 . The lin e spectrum Is not true ,3 decay, but is due to internal conversion, discussed in Ch. V . Problerr.: Deslcn an experiment to study p decay i n Cu . In d i­ cate vrhat type of electron spectrograph you would use . Assume data Is desired dovm to enercies of about 50 Kev. Specify the thickness of the soiirce. Specify the thiclmess and type of backing for the soi’.rce in order not to have too G^eat d is t o r ­ tion at lovr enerj-iles. Distortion ViocKin^ of the enercy spectrum Is due to back scattering, and absorption. H. Selection Rules In optical (atomic; transitions, the complete and exact matrix element is complicated and rarely evaluated. Rather, the integral of the optical retardation factor exp(-l n»r is expanded in pov/ers of R /^ , vrhere R is the extension of the atom. The -irst term elves the rate of dipole radiation, the second, qtiadinpoie, etc. The succpsslve terras of the FIG. I V .1 3 unsquared matrix element d iffe r by factors o f W expr,nsion Is valid only i f R « / , vfhlch means that the fir s t nonzero terra is dominant. I f the fir s t (dipole) tern; is zero, tho transition is said to be "f o r b i d d e n " . Tlie transition may still •::'roceed, but at the much smaller rate , niven by the hicher order terras. In optical emission, R /X = s lO “ < The probability of transition depends on the square o f the matrix element, therefore the forbidden optical transitions are about 10“ ° as probable as "allowed" transitions, i . e . , t r a n s it io n s for nonzero first order terms. The "selection rules" ci^e the necessary conditions on the chance of state of the atom for the first term to be nonzero. A sim ilar situation obtains in the case of p decay. The matrix element is ^ , / Y *fln a lW T in ltia l^^ ♦ i^ is tne wa^e fu n c ­ tion that is intecrated over the fin it e size of the nucleus. In the calculation of section D it was assumed that the nucleus h ad zero extension. This amounts to Ignoring (in the unsquared m a t r ix element) all taut the first term in an expansion of the wave function in powers of B /.^ , where R is now the extension of the n u c l e u s , (This is analocous to ignoring higher order terms i n the expan­ sion of the retardation factor in atomic emission.) I f the first term is zero, the higher order terms, so fa r

Ch. IV

Beta Selection Rules

ignored, must be considered. Since for a nucleus (R /^)^ ^ ( 10“ = l/lO O (instead of lO""^ as in the atomic case), forbidden transitions are relatively more prominent in p decay than in atomic emission. An additional approximation has been made so far, and is an additional source of correction terms, and therefore of forbid­ den transitions. S t r ic tly , there is a r e la t iv is t ic correction term of order V /c v/hich should be added to the matrix element. V = velocity of nucleons. Since the matrix element is squared, and since V /c = l / l O , th is correction results in terms again about l/lOO as large as for the firs t term in allowed transitions. The selection rules for a given order of transition, i . e · , for a given term in the expansion, give the necessary conditions on the change of the quantum numbers specifying the state of the nucleus in order that the term of given order in the complete matrix element be nonze r o . Clearly there must be different sel­ ection rules for different orders in the expansion of the matlTix' element, otherwise a l l would be zero simultaneously, and there would be no forbidden transitions. The reason that each order of the expansion has characteristic selection rules is that in the expansion in pov/ers of R / ^ , the d iffe re n t orders contain d iffer­ ent pov7ers of R , a polar v e c t o r .* * The q u a n t u m "niimbers" w h o s e chanf!;e in a t r a n s i t i o n are g o v ­ erned by s e l e c t i o n r u l e s are to t a l a n g u l a r m o m e n t u m and p a r i t y . D ue o s a e n t i a l l y to t h e i s o t r o p y o T space, t h e s e t w o are characteristics of th e str'^te of a n y i s o l a t e d s y s t e m . S p i n for a nucleus is den oted by 1 = (a m ^ u l a r m o m e n t u m ) /]?{. P a r i t y r e f e r s to the pr o p e r t y tha t the w a v e f u n c t i o n e i t h e r chanrres s i g n or d o e s not change sign (map^.nitude u n c h a n g e d in botli oases) if t h e s p a c e c o o rd inates are If the wave t rans P ^ r me d b y i n v e r s i o n : ( x , y , z ) ---- ^ ( - x , - y , - z ) . fane M o n chanr.es sip;n, its p a r i t y is odd; if not , even. If the m a t r i x e l e m e n t inter.ral is to d i f f e r f r o m z ero, tlie t o tal integrand must be even. T h i s i m p o s e s a c o r r e l a t i o n b e t w e e n the p a r i t y of ana the p a r i t y o f ; in o t h e r w o r d s , a c o r r e l a t i o n b e t w e e n the D a r i t y o f Til nnd t he chanft;e in p a r i t y o f t h e w a v e function* In the two c a s e s in the n e x t p ar a g ra p h , even, and he n ce t h e r e is no c h a n g e in Dar i t y. Of the s e v e r a l Por>aible forms for in b e t a decay*, two are men ti o n ed here. O n e is t h e "scjilar” form, w h i c h g ives a m a t r i x e l e me n t es^enti-Tlly of tne form:

jn.

f

i

i

i

i

m

l

V

I

.

29

This clYes the follovrlnf, selection r>.ile for allowed tranaitions: I -

I ',

"no"

I V .30

'.There I is tlio spin of tho lnltlr>.l state. I ' the spin of the final stato, and. "no" means no parity change. A second form is tho "tensor" matrix clonient, essentially IV.31 * The various possible forms of interaction , each with different matrix element and selection rules, are fiven in Konopi n s kl s comprehensive article on p decay. Rev .Mod. Phy s. 3^ 209 (1943) **Eeaders unfam iliar with the general ideas behind selection roles may first consult Ch. V , section B , p.

82

PTr Tables

Ch. IV

Here 2" is a generalization of the Pauli spin matrit.es. Tlie sel­ ection rule for this matrix element for allov^ed transitions is l'

=

^ ^

(1=0—^ I * = 0 forbidden) This spin change specification is the same as in optical (dipole) radiation and alvrays results when the matrix element is a vector. **No“ parity change depends on the fact that the vector is an axial vector, that i s , invérsion of space does not change the sign of the vector.* (In optical transitions, parity changes because the vector, (the dipole moment) is a '*polar” vector, that is , its sign changes upon inversion). Tlie second selection rule, I V . 32 is the (ramow-Teller (G-T) selection rule, and is favored by the scanty experimental evidence accumulated so far. The distinction between allowed and forbidden transitions is blurred by uncertainty in the valüe of the matrix element for allowed transitions. The matrix element may give a factor l /l O , and therefore a factor l/lOO in the transition probability, and thusthe transition will appear first order forbidden, but, in principle, be allowed. Situations like this make it difficult to know the selection rales from experimental data. Any influence that increases the curvature of the wave func­ tion at the nucleus will increase the size of the second order terms· relative to first order in the expansion of the exact mat­ rix element. One such influence, particularly in heavy elements, is the Coulomb field of the nucleus. J.

Tables ** Prom equation I V .25 , = (üDÑersal Cov\stayit) jTn|

Therefore

v

I V . 33

Evidently the product F T is a measure of the fbrbiddeimess of a transition. The lowest F T values correspond to trflnsitions per­ mitted by the selection rules discussed here. The very lowest FT values, 1000-5000 represent superallowed transitions, and are allowed transitions between nuclei having similar.nuclear wave functions (see next p^raarRPh). F f values of 10^ to 10 repre­ sent allowed transitions between nuclei not having very similar nuclear w*ive functions. If the successive orders of forbidden­ ness iifferec’ in tr^msition probability by factors of l/lOO, as is predicted by the simple considerations of section H, we would find clusters of points on the F r P l o t at Ft: = 10° to 108, representing?: first forbidden transitions, at F t = 10® to 10 , representinsç second forbidden transitions, and so forth. Actually, the points do not cluster nearly so nicely. In fact, the clusterina is barely discernible with imagination. Rigid classifi­ cation of the empirical F"C values seems hopeless. *

A n e x a m p le

of

an a x i a l v e c to r

is

o r d in a r y

angularly m om entum

r X £_· ♦^ordheim discusses ÎT values and their relation to other selection rules, and to the shell model in P.R. JS, 294 (1950).

83

Tables

Ch. IV

The size of the matrix element for a transition depends essentially on the extent to v/hlch the In it i a l \mve function of the nucleus overlaps the final v/ave ‘ function. In General, the nature of the v;ave functions Is not Irnora, ?.nd not much can be said about the overlap, except by observing P decay rates. But In the case of the Vlf:;ner or "mirror'* pairs of elements, there Is rep.son for assumlnc; that the 'i^ave functions for both members of a pair are very much a l i k e .* (The results of this assumption accord v;lth experiment and serve to confirm the assumption.) Winner pairs are elements such that one results from the other by Interchanr-e of neutrons and protons. The simplest pair Is of course, the neutron and the proton. Tlie next simplest pair Is He^, In general, the nucleus ( Z ) ^ ^ ^ Is the mirror of the nucleus (Z + l)““^*^^. p transitions between mirror elements are superallowed, because In the matrix element Integral, ^ I n it ia l l it t l e from Yfj_j^a.l· ^ 1000 to5000. * The follov;lnc Is a crude arciunent to shovr vrhy the wave func­ tion of one of a mirror pair does not chance much when the nuc­ leus chanGes to Its "Im aGe", Assume that except for the presence or absence of one unit of + charge, all nucleons are the same. We ml ¿tit ascribe to each nucleon two dichotomic variables, one for spin orientation and one for charge. The latter* s two values correspond to charge +1 or 0 . Assume that the nucleus is in its loviest energy state, and that this corresponds to havlnc the low­ est space states of the system f i l l e d . To each space state there may be four nucleons, without violating the exclusion principle· The four correspond to the four possible combinations of spin and charge variables. That I s , each space state may be occupied by two neutrons v/lth opposlnr; spins and by tv7o protons v.dth spins opposed. Now consider two examples of transformation. In the firs t example, the number of neutrons is much cheater tho.n the nu:nber of protons. This means tliat probably some space states are not completely filled with tho full quota of four nucleons: y>cutvx>n X

S^ace ^ Stages 3

X

X

If tVte scheme becomes

^oro-kon ne^utmr? X

X X

X

2

X

K

X

X

X

X

X

X

1

K

X

X

X

X

X

X

X

Vroion t>euirbn New/jo drojB X to lowest available X X X sto-ie X X X X X

X

X

S>Din — y

t 1 1 t t t Now i f onenucleon chanr.os from p to n, or vice versa, it can also chanf.e its sn^.co stato In order to occupy the lowest available to it. Therefore tlie v:avo functions before and after w ill d iffe r in their snace dopcndence. In the second example, the number of neut­ rons almost equals the number of protons, as in a mirror pair. Nov; i f a nucleon changes its chr-^r'^.e, it ^probably w ill not also ncotvt>n |3>roton h«utirttn change its space state: 4· X X 3

X

)C

X

X

X

X X

X

2

yc

X

K

X

X

X

X

X

\ X

X

K

X.

X

X

X

X

>

t

V

1

84

E-Capture

Ch. IV

g.

Remarks on K-Capture As described In section B , K-capture follovis the scheme '’Atom (Z) v/lth usual quota of 2 K-shell e l e c t r o n s )----- ►(Neutral atom (Z-l) with one K-shell orbit empty, plus a neutrino) No observable particle comes out of the atom, but the excited f i n a l atom emits an.X-ray when a bound electron drops Into the empty K-shell state. Thus K-capture I s observed by means of the X-rays emitted. The X-rays are so ft, and d i f f i c u l t to observe; t h i s makes K-capture hard to observe. undergo either K-capture, positron emission, or emission to Referring to P i c . r v .3 , the energy Ipvel of Cu” ·^ occupies a p osition of type C v:ith respect t o ^ N i°'^ ,b u t of type A vrith respect to · K-ce.pture becomes more important at the heavy end of the p e r io d ic ta b le . The reason IB that there the K orbits are small and the probability for the electron to be at the nucleus is lar^e. On the other hand, the positron wave function is small at the p o s itiv e ly charged nucleus, and the probability fo r positron emissitDn liB correspondingly small. Problem.

Discuss methods for observing K-capture.

L.

Remarks on the Neutrino Hypothesis A promising lin e of research on the neutrino hypothesis is the study of the recoil of the nucleus i n p decay, particularly i n K-capture. I f the nucleus and the P particle account fo r a ll the momentim, then of course the nuclear recoil momentum is equal and opposite· to the momentum of the p. Experiments to determine the angular correlation between the nuclear recoil and the emitted p p a r t ic le are at the limit of a v ailab le technique. They sliow th at recoil i s not opposite to the d ir e c tio n of the p p a r t ic l e * . I n K-cafiture only the neutrino I s emitted. I f K-capture o c c u r s , the nucleus v/111 recoil with momentum equal and opposite to that of the emitted neutrino. This experiment is best perform­ ed w it h a lig h t nucleus, such as B e 7, I n order that the recoil energy be as large as possible, and therefore easy to detect. Observed recoil of the nucleus in K-capture does not rule out the hyp othesis that energy is not conserved i n p decay (no n e u t r in o ), provided one also postulates non-conser*vatlon of momentTom. Suoh experiments can discriminate between a one-neutrino theory and a two- or more neutrino theory. In a one-neutrino th e o ry , i n the case of K-capture, the nucleus always recoils with the same energy. In a theory in which more than one neutrino are em itted, the nuclear recoil energy is not alv/ays the same. A d e t a il e d study of the features of recoil would show which is the true situ atio n . Assuming one neutrino, one should find that connects the energy and the momentum attrilauted to the neutrino. Experimental detexmination of E and p , by recoil experiments, vrould then determine the mass of the neutrino in a v:ay indepen­ dent o f the p decay theory. Assujning that experiments above viould jrield tne result that E = op, i . e . , mass = 0 , then a d e ta ile d knowledge of the angular * Such experiments have buen performed by A l le n , Paneth and M o rrlsh , Phy s . R e v. 75 570 (1 9 4 9 ); Sherwin, Pliys.Rev. 73 2l6 ( 1 9 4 8 ) ; and by others. A comprehensive a r tic le on the search fo r the neutrino is by H .R .C r a n e , Rev .Mod . Phys. 20 278 (1 9 4 8 ).

Ch. IV

Neutrino Hypothesis

correlation betvieen emitted p and. 2^ would discrim inate amonc the forms of the p theory, that I s , shovr vfhat In te r a c tio n operator should form the Integrand of the matrix element int© cral. The shape of the spectrum does not d isc rim in ate. The ancular d i s ­ tributions to be expected from the various forms of in te rac tio n are Given by Ham ilton, Phy s . R e v . 71 456 ( 1 9 4 7 ) · A more d ir e c t method to v e r ify the existence of tho neutrino is to observe it by c o llis io n processes. This has been attempted, u sine intense neutrino flux that presumably is ^,ivon o ff by a chain reactinc p i l e . A vory w ell shielded counter v/as placed near a p ile. Tliere were no c o llision s observed. Tlie experiment sets an upper lim it on the cross section for detectable c o l l i s io n s , v.'ithin the counter, of 10” ^^ or 10 “ 32 cm^ . Another p o s s ib ilit y for direct d e te c tio n of the neutrino i s by an inverse P process. Tlie existence of the invèrse process i s assured by a p rin c ip le of d etailed b a la n c in c , stemminc, from the basic concepts of quantum m echanics.* Assumine that quantum mechanics (i n some form retaining th is p r in c ip le ) i s ap p licab le to p decay, inverse P processes are p o s s i b l e , such as \

-1· y

^ ____ -

^

^

U»UAL U C A C TiO H

Such a reaction mifjit take place i f a neutrino im-^inf;es on a nuc­ le u s, and the nucleus simultaneouBly takes an electron from the bound, o r b its, or from the positron sea. The cross section for this process is \ 2 ^

\y r \\

{tL

\c ^

fo r rieotnno

^fcW 'M ev)

For fa.vorablG c a s o s , 1 . p arid V are the momontiim and v e lo c ity of the electron^ Such a small cross section permits a neutrino to cross the sun vrith l i t t l e p r ob ab ility of bciiiG absorbed in an inverse process. Perhaps the most conclusive proof for t]\o e x is ­ tence of the neutrino, and the most remote of attainm ent, would be to observe p decay with recoil of nucleus and momontuin of e le c ­ tron known so as to r:Xvo the d ir e c tio n of tho ncutriiio, and then on the path of tlie neutrino to detect almost simultancouoly an inverse p reaction \rhose energy relation s af.rco with tlio enerfy of the neutrino emitted in the fir s t r e ac tio n . The Greatest source of neutrino flu x at tho earth io the sun. About 10% of tho sun* s onerrv i s spent in p emission, and rour.iily 50 % of th is {5% of the to tal) enerr:y ^scapes in the form of neut­ rinos. Tills corresponds to rou^'Jily 10^^ neutrinos per second per cm^ at the earth*s surface. M. Neutrinos and Anti-neutrinos The theory of the neutrino may bo formulated in a way like that for t]ie olectron, that i s , poatulatinr, noG^^'tive onorry states which are almost always f i l l o d , FIGr. I V . 1 4 . I f tho rest mass of the neutrino is 0 , the snectnim of f i l l e d noGative onnr.Gy statoo Joins that for the u n f ill e d positive ener^y states at enerry = 0 . Holes or vacancies in the noGativo enerr'y neutrino sea are called an tineutrinos, 7/ ^ . The nostulation of of aiitinoutrinos permitó formulation of a l l p processes in terms of one, namely, N

+ p^ + 2^

I V . 34

The reverse reaction is P" + + P —^ N . Nov; since the p" taken on the le ft side leaves a hole in the electron nofAtive enercy sea, we may w rite it as tho production of a positron on the riGht side (the 2mc^ threshold enerGy, section 0 , appears clearly as the * Detailed balance is discussed in G^^^-ter d e ta il In Ch. V I I I ,

C.

86

Ch. IV

Antineutrinos

energy needed to lift the electron from the positron sea. I f It takes an orbital electron as In K-capture, no hole is prodiiced and henc’e no positron appears). Exactly in the same manner, we may write an antineutrino, , on the rl£jit side, to represent the hole left by the neutrino absorbed, and the equation takes the form:

+mc*·

I V .35 Equations I V .34 and 35 can be taken as defining the antineutrino. V and Z/^are equally difficult to detect. They are, in principle, different particles. The difference cannot be detected by electric c h a r ^ , as it eiectnsns neolnnos can for electrons and positrons. FIG. rV.14 Accordinc to theory, 1/ has spin l / 2 . I f it also has magnetic moment^,^ (very small, to accord v/ith very sllcht interaction between it and matter), then the ratio / (angular momentum) would be the same fo r 7^ andV*^, but with opposite si{pi. I f Z/' and Z^^have no electromagpietlc properties, and cannot be dlstinc^ulshed by them, we can in principle only determine the source of a p a r t ic u la r ^ , and i f it comes from a reaction like I V . 34, it is a 7/ , i f from a reaction I V .35, a Double B decay provides a means of findlnc v;hlch form of the P decay theory is correct, the form postulating 1 / andZ^^or the (Majorana) form in^v/’hlch all ^ are, the same. Consider the Iso­ baric triplet Snl^^, Sbl24 , Tei24, whose energy levels may be as shown: ^ a 50 sn s^

Direct transformation from Sn to Te with the emission of two is conceivable. Both forms of the P decay theory say this is pos­ sible, but in second order approximation. Since the constant 5 is very small, transition probabilities arisinc from second order transitions are extremely wealt. The probability of transition depends acutely on which form of the theory is used. For the form in v:hich I V .34 is the fundamental reaction, and 1/ is different from , the double process is simply the sum of two processes, and four particles, two electrons and tv;o neutrinos, come out: — »-p + 3·^ + z/ (either form) Snl24«^^Tel24 + 2p- + The mean life for this reaction i s Y = 10^^ years, by this theory. For the Majorana form, in which all7^*s are the same, the process may go as above, that i s , two separate p reactions with four emitted particles. But in this theory, the emission of either p or p·^ is .accompanied by either the emission or absorptlon of a neutrino. Therefore another mode is possible, namely one in which the neutrino produced by the first reaction is

Oh. IV

Doublé Beta Decay

absorbed in the second: + p- + 7^ V ■ (. — »-p + B~______ Snl24— ►Te^24 + gp-

87

(Majorana form only)

This neutrino is "virtual” ; no neutrino comes out. This mode has much £sreater probability bocause the virtual neutrino has a much larger region pf phase space accessible to it than an emitted neutrino. The mean life under this theory is 10^^ - 10^7 years. Experimentally, a decay rate with = 1 0 ^ years is undet­ ectable. For 10^^ or 10^7 y^ars, it may be just vrithln detection. The double P decay from Sn^^'^ has been reported as observed.* The experiment needs further checking, i n particular, it can be seen from the above that in the last mode, (virtual neutrino), in which no neutrinos are emitted, the sum of the energies of the p particles is a constant. This crucial point should be checked · (The theory of double p decay and summary of the differences between the tv;o forms is in Furry, Phy s . Rev. 56 1184 (19 3 9 ). Also, M. Gosppert-Mayer, Phy s . Rev. 48 512 (1 9 3 5 )). * Note I n PnTS.Rev. 75 323 (1949) h Fireman. and Reynolds, Phys. Rey. 76 1265 (1949).

See also Inghram

Problem. Plan an experiment to find the ann.ilar correlation between nuclear recoil and emitted p, in p decay by He^. The articles by Allen, e t .a l , by Shen^'in, and by Crane contain information pertinent to this problem. Note:

The n e u t ri n o and a n t i n e u t r i n o are sorn.etimes defined

'

tv] —> P p ^

M

+. 7 /

Note: Inghram and Reynolds get half-life of 1.4 x lO^i years for do'jblo beta decay of Te^JO. P.R., JS, 822 (1950).

CHAPTER V.

PIG . V . l Post-a and posb-P gammas. Gamma radiation w ill follow the transitions indicated hy solid l in e s .

GAMMA RADIATION Gamma radiation and the ejectio n from the atom of internal conversion electrons are two of the ways that an excited nucleus may lose energy. The excitation may have ar is e n from "bombardaient,«»»* the decay of some other nucleus to an excited state of the nucleus under consideration (illustrated in PIG . V . l ) , or by the absorption of a photon,

I f the ^-emitting nucleus is formed by a-deoay, it is improbable that w i l l be ?-^Mev. This is because o f the extreme energy de­ pendence of the Gsimow b arrier penetration probability (Ch. I l l , p. 57 ). On the other hand, i f the nucleus is formed by p-decay, then subsequent y s are commonly observed with energies around 2 Mev. We can explain this fact b y remembering that, for pdecay ( I V . 24, p. 77)

f

M

x

^

n ^

This dependence of the tra n sitio n rate, l / C , upon the f i f t h power of E is less sensitive than the exponential dependence for a-emission. In this chapter we shall discuss the different ways in which an excited nucleus can decay to a state of lower energy. We shall determine the mosb probable sort of transition between a given pair of sta tes, and its tra n sitio n probability per unit time. At the end we shall present only the briefest application of the theory bo isomerism. Some of the ideas developed w ill, however, be used in t.!ie chapter on nuclear reactions. For a more complete treatment, not only of radiation, but also of isomerism, both theoretical and experimental (with a discussion of selected cases), see the review article by Segre and Helmliolz, ’’Nuclear Isom erism ,’’ R e v . Mod, Phys. 2 1 , 2 7 1 ,(^ 4 9 ). A. SPONTANEOUS EMISSION The object of this section is to present eqn. V .5 (p. 94·), which gives the tra nsitio n probability per unit time for multi­ pole gamma emission, and eqn. V .9 (p. 35 )for the transition probability per unit time for dipole em ission. We shall need V .5 to discuss the re lativ e importance of dipole, quadrupole, e t c ., radiation by n u c le i. We sliall use V .9 for discussion of selection rules and of the interna l conversion coefficient. The reader who is fam iliar with these equations or who would like to save time by merely accepting them, should ski

89

90.

Spontaneous

Emission

Cli. V.

the intervening review. In any case we shall not prove these equations, but uShall discuss them and indicate how one might guess them from the classical (non-quantum mechanical, nonrelativisti