\<2>

Qa.sc-

1

BOSTON UNIVERSITY

GRADUATE SCHOOL

Thesis

NUCLEAR RESONANCE REACTIONS

by

Duane Uson Gaskell(A.B., Atlantic Union College, 1942)

submitted in partial fulfilment of therequirements for the degree of

Master of Arts1948

BOSTON UNIVERSITY

COLLEGE OF LIBERAL ARTS

LIBRARY

w

Approvedby

First Reader....I'essor of Physics

£9.

Professor of PhysicsSecond Reader

X....... ; t

• • ;

4 lo -io

ac : ’v*cl 1o - Ao-i

TOPICAL OUTLINE

I. INTRODUCTION

A, Purpose of Paper

B* Historical Background

C. General theory of nuclear reactions

1* Definition of nuclear resonance

2* Particle interaction

3. Formation of compound nucleus

4. Probability of nuclear reactions

5* Importance of resonance in nuclear studies

II. DISCUSSION OF NUCLEAR RESONANCE

A, Potential Barriers and Energy Levels

1. Barrier heights

2. Reactions with charged and neutral particles

3* Penetration of barrier and resonance

4» Level widths and spacing

5* Orbital momenta and selection rules

B. Absorption and Scattering by Nuclei

1. Neutron as incident particle

2. One level formula

3* The l/v law

4* Selective absorption due to resonance

• Radiation damping5

. .xa:'

•:.

;

- •

*0 ‘O , X

: J ::••• -J ...

:•

. . oo - :

. . : err.

• \

. a

1

. ....

;I^itaen bna bsjri 'do

. ... • i .

'

<j : / e ' .. .;' vv

la.' -

> i' .* I

.

.• Sjsi iv - fi

v

r. f•

‘

> 6 . 'Vi .

2

6. Reaction products

a. Gamma-rays (simple capture)

b. Re-emission of neutral particle (elastic scattering)

c. Re-emission of neutron (inelastic scattering)

d. Emission of alpha-particles

e. Emission of protons

7. Charged particles

a. Lifetime vs. resonance sharpness

b. Absence of resonances

8. A geometrical interpretation of absorption

9. Scattering and the resonance formula

a. Contributions of resonance scattering

b. Contributions of potential scattering

10. Resonance in fission

11. Fast neutrons

a. Theory predictions

b. Experimental Evidences

12. Doppler effect

III. CONCLUSIONS

A. Value of Slow Neutron Studies

B. Comparison of most important Theories of Nuclear Resonance

C. Effects of Recent Experiments on these Theories

Digitized by the Internet Archive

in 2016

https://archive.org/details/nuclearresonanceOOgask

3

Nuclear Resonance Reactions

I.

Introduction

A. It is the purpose of this paper to present a non-mathematical

summary of those principles of nuclear reactions which have been presented

in complete mathematical form by various well-known authors. Only those

principles which have withstood further investigation or research are

discussed. Special attention has been given to the resonance phenomena

connected with nuclear reactions. The survey is not completely exhaustive,

in that only the most important type reactions can be touched upon in a

paper of this length.

B. It is over 50 years since the discovery of the electrical nature

of matter and the atomic nature of electricity. Since this time men have

proposed many theories and interpretations of the known properties of atoms.

Of those who sought to explain the structural nature of atoms, Rutherford,

in 1912, was the first to make the most decisive step. With his now famous

scattering experiments he showed that the positive electricity of the atom

is concentrated in a "nuclear” diameter of approximately 10"^c Thus

the name nucleus was given to the central core of the atom. This funda-

mental assumption has been upheld in nearly all current theories.

Planck's studies of the continuous spectrum, in the early years of

the century, introduced the particle nature of light and the associated

theoretical developments which followed were known as the quantum theory.

Niels Bohr used the quantum theory to explain the structure of the

line spectra of atoms, considering the atom as having a central core

.1

nr, id ; pcnr.^al

.

.

-

-

s r no- bar i 3 :iOis -x 9t-/J 3t\ stio- l j-.sc- i -t ^Irro Jter.i ni

-,iud lo -----

J' 90 .

-rad -. .+5 eJtdd edni . tio±s<Joei.9 'to s-ixriiwi oj

.

. ..-i ,b c i :.v 'i,. J m ojr ariJ” nl- x t J- v l. cu • -

''0

;

? .’ . O'

• TJ '1 ./ ;?f.‘ ' ’• .1 to vi • iJtb ’l Of ttn r. t ./

.

r'

'-

'

-.

•. 9

.

o'too % •?>: •

. uofa o . -.al. >bi -o ,i 1r> xiost,

4

surrounded by electrons. The motion of these electrons occupied the

attention of chemists and physicists for the next fifteen years. Unfruit-

ful applications of Newtonian mechanics, with perturbations thought

necessary, were made during this period in an attempt to find a solution

to the atomic problem. With the introduction of quantum and wave mechanics

by Heisenberg and Schrodinger, a satisfactory solution of the atomic and

nuclear problems became possible.

In the field of nuclear physics, the modem period began soon after

1915 with the application, in that field, of the quantum theory by Gamow,

Gurney, and Condon independently. These authors explained the natural

radioactive disintegration of the heavy elements by making use of the fact

that alpha particles were emitted. Their theories, essentially the same,

showed that the great differences in the average life of different

uranium isotopes and the relatively short average life of radium atoms

could be explained by the application of wave mechanics. With this theory

of the wave nature of particles, many phenomena in nuclear systems can be

explained.

In 1919 Rutherford succeeded in disintegrating nitrogen atoms by

bombardment with alpha particles. In 1933 Curie-Joliot discovered

artificial radioactivity by means of the radioactivity created when alumi-

num was bombarded by alpha particles from Polonium. With the development

of the mass spectrograph, the cyclotron and many other accelerating and

useful instruments, much progress has been made in the field of nuclear

physics. However, present experimental results are far from adequate to

make certain the assumptions underlying the present theories.

.

-, . t .. /'Li ' t . .o'. r.o r; . ft;

':

:

:

- * ,L. rott * r ;u i-TiC, c. L rirufc oi ... -io':

..•(/. •

.•

.

iM i oi • 3J t • ! *J-C rxL . 0. . r >L: ..

. .. : nr.

. j . . j ; . i"•

. :•' r ov. .v'

- -

1

X 1C?? : - S «, m .'. 1 IKihlV '

;It'.

1 r .... m •;./ >. vv -'-n- - tc c..>

>’

i -Tr

.

j

v . .<•'

-

, . ..

I r //. j; J . ior y llx tawig ;

--

. s a .! 1o c1

t: e ^ •'•ms •••v- .2 j .... -:t

•

j x rX r- '

t'. Lrx lo v

.£j +.-Vf v>' ; L

. n.

vf -.v rt v n.v.' -c /£ ; .; / «oc;. '

-'Vi 3 rtf vi-'.f r

• ... arfqXs jttiw taemfciadlncxf

if. r o.’/ vt J > r . :

’ V '

.

lit© ;rP £•. 0 olov.: > •r til :h - * XO

.

mmk • i.) sni^Xiabaa enoltqxj/tte.f? tfci

5.

Breit and Vigner^ in 1936 gave a formula for the probability of a

nuclear reaction in terms of a "virtual" or resonance state of the

excited nucleus* In the following year, Bethe and Placzek^ generalized

the formula to take account of other possible resonance states of the

nucleus* This has become the well-known dispersion formula, because of

its analogy to the formula for the dispersion of light in atomic systems.

Bohr was the first to show that the one-level formula of Breit and

Wigner^, which was correct for atomic collision theory, would not work

for nuclear collision* He showed that a light particle in collision with

a heavy nucleus will form an unstable temporary compound nucleus and that

the lifetime of such a nucleus is usually very long, measured on a nuclear

scale* Thus these compound nuclei have fairly well defined energy or

resonance levels* The position of these levels and the properties of the

meta-stable states belonging to them, determine the cross sections for all

nuclear reactions.

C. In order to arrive at a conception of nuclear resonance, we may

make use of a mechanical analogy* A tuning fork with a given frequency can

be made to produce resonant vibrations in a second fork of the same fre-

quency. This phenomena is termed resonance. A somewhat Simula* phenom-

enon occurs in nuclear physics. It manifests itself by an abrupt increase

in the cross section of a reaction for a given excitation energy. While

there is a considerable difference between classical and quantum mechan-

ical resonance, nevertheless there is a similarity in their mathematical

treatments

•

In the case of atomic collisions, the "Hartree" formula, which has

,

'• *

bre .ti •. ic

• > >1 A /;• - edl -

'f vto'-s >1 JvtXI ©rL* .. rffio<3

;•!• . Xi f t’.-.i

'

- J . :: i- . » • *3 >1 JTo tiod a.f r

.

•t 9 1: • n j ci £>e*u •'Win trroX i-~r. / vi xraxj i 2 i' nun .

rro ygt.ytiB bsr.ileb IXev ; -il uvv, I Iown ruio^noo 9B©ni aitfi? .sltioe

1

-lol jui'JOoa Ecjorcs-

( '.t © ,,t .. ,' I 'd ;

: .fa X' i M

•

, .- r . iv. r.

vo,;. j t 9f>r..'jsc»e©i *t.» iiawi io tqaor: o .9 .? r wf n i ol Tf'

-ro r .

,.v>V

- *1 3 >1 ; O ''•• '

' * f

'

.

-a )ly.' i -win l; • ' - ->oJU . .10 ' " r »cf o-vr V*. " iXd -i- I:'. :•

'

XoOxJ .1 •* - -Xi II ic a! f .• li/tVi. . ,

or

, « \1it-: 3*rJ

. -L.;; • 1 . < j bn'

'io v

6

come to be knovn as the method of distorted wave functions, holds true

whenever the interaction between any two electrons is small compared to

the interaction between an electron and the rest of the atom, or small

compared to the incident particle energy. These do not hold true for

nuclear collisions^, first because the interaction between one particle

and the whole nucleus is only of the same order of magnitude as that

between two nuclear particles, and second because the interaction is gener-

ally large compared to the energy of the incident particle. In atomic

collisions inelastic reactions are rare because of the small interaction

between the incident electron and the atom, while in nuclear collisions

the particle cannot pass through the nucleus without interaction with the

individual particles. The incident particle therefore loses some of its

energy as it enters the nucleus and finally its energy is completely dis-

tributed among the nuclear particles. Thus a compound nucleus is formed.

Since the energy of the incident particle is distributed, no one particle

will escape, for a long time. If the energy happens to become concentrated

in one particle at a certain time, that particle will escape. The energy

of this particle may be different from that of the incident one and hence

the nucleus will be left in an excited state. If it has the same energy,

we speak of the reaction as elastic scattering, this being a rare event.

There are, however, several possible kinds of inelastic collisions, where

either the nucleus is excited, or where a particle of different kind is

emitted. The latter is known as transmutation, the former as ordinary

inelastic collision. In atomic processes the time spent in the atom by

the incident electron is very small (diameter/velocity) , while the time the

,,:;1 \ J , , n itor <lev J>. •, •: >J to ta 63 fl-wrc-

*r xj-ai -I t:( i i J

t

:

3.pifosq ono' c-swdod c ttaii-'fai sxfl eeu ozd fnt1 t *o.nc twliloo xseloi.-a

f r . i- '

•

'

• •'

• • '

: i

ol . J »• d .. 7, j i ir* f * cl. ri u. ey:.' : v

r „ T j. i.'j

1

. i r «J5 f- J gnr •ir.il. r

....

. i j . 1 j J u . . 1 ; >j " CO. i" '*. 3

•'

. ..•

.

I t$i or .'Oti f oj n :;v ' \„i'i • .-c* &t-l -l ’*- ‘ - -

.

eoi & bn . 9oo iir >13of sl& to 1 ril siotI InaneTt.’fc %;• 3 !%• :

toxe n* nJt Itel ©

f

.

,.11 ... to ifcol? f»XdiB900 I 'leva ti3vevori ,***

si ,ix;[ Insieltib to loJtlisc; r £1 dv to ti 9&lvx* exroloira on# - r tl

*x *

j. om ril 0 ollatfieni

(enl .> o 3X 1 ol. . ,

(^jtioc v ;i :i ..sib) 1 ai n jo. I-o 1 -.orl r.

7

nucleus is in a "compound” state may be quite long in comparison. Thus it

is quite probable for radiation to occur during this time. In summing

nuclear collisions characteristics, we have seen that there is a very small

probability of elastic scattering, a much larger probability of inelastic

scattering including transmutations, and a comparatively high probability of

the emission of radiation. Bohr used the term "closed system" for the

nuclear process and "open system" for the atomic process because of the

absorption of the incident particle in the former and because in the latter

an electron can traverse the atomic system without any difficulty.

The probability of any nuclear process is equal to the probability of

formation of the compound nucleus from the initial particles, times the

probability of its disintegration into the particular final particles in

question. Once the compound nucleus is formed it must disintegrate in some

manner. The probability of a particular mode of disintegration is given by

the partial width of that mode divided by the total width, averaged over a

sufficiently large number of modes or levels. The probability of formation

is the geometrical cross-section of the nucleus times a "sticking pro-

bability"1 .

The study of resonance processes in nuclear physics is of great impor-

tance, from the standpoint that the spacings between adjacent levels of the

excited state of the nucleus may be determined from the resonances. Also

these spacings, determined as a function of the mass number and of the

excitation energy, will contribute to the understanding of nuclear structure.

Furthermore the widths of the levels are also of great importance. They

enable us to determine the probability of

ynx.-nu.f8 nl .enid eldi *nl'rusl xirooo od noltaihsz iol si r do*rq sdltrp a£

’**© * si- d Ad a -on sv .J owc lOidBxisdosiariD enoi&Ki.oo looloirn

.r orii -o v.til; -'-yc-: v

.

x.\C dour c ^rixsodsoa oid>s!e lc rdilidrdoig

, £K>ldsdxJBn3£0B<xd ynlfcuXorri gnJbreddaofe

esir idoS .xioxdexboi

• .• lo ssxbos ( sasooiq oA sdn arid 10I "medgye ns- o* ions t- -.-iociq iseXoun

rid nl • susoetf bns laarxol edd nl elol

J

I . :

Ts \H Cx:

cfo-rq exio od Lau s x aasocrrq ixjsIolu Tcns "to 'cdllidsdoiq adT

»*•* -33-- -'dt act's id :.r*i Aider:.. -tuid moil . >J o;.-n X::uo-. -k -jdd lo noid fcr/tol

/ . xsloxti q I, I Iir idinc -.id otrnx fioidPigednied . x-:.: Jo ^dxi Xdxsdoiq

.

'

S wo be c TOT'* <ddbiv I.-d'-d odd ^d befcxvxb soon daid lo ddblv I<xdi£c srid

. 1 lo igdiXld&dovq edT • aleveX to eebom lo led’:.ox •'!-*• rioi i

“0®I 3ir-hfoXdBw aoaiid sueXo-un sdd lo noxdosa-aeoio Lsoiidexaoeg sxid el

.-^diJLidxsd

--£> x • -r to il soxv^i.q i. ,'sxrrr xii ses . >$c.iq eonanoeon lo ode *df

vox j naos’r ba aeawded agnlt. .

.osoasaoesi odd noil benlarrsdob sd. v.<r ax/9loi/n sxld lo eddda be&lox?

- or oc xj s< lo old o.' A • .3 > ;v£flrxed vjt a; tloac o >8S'.d

. xrt ' i < to :d‘. i a.dnd Ad od Jxdixdnoo liivt'7

. noldBdlox9

lo ’ ’ tiJ -oi- -.Ad 7i-'uin iflr od xa sldnne

8

the concentration of energy on any one of the particles in the compound

nucleus. The probability of a nuclear reaction as a function of the

incident particle energy is given by the dispersion formula first

derived by Breit and VJigner as previously mentioned. The original formula

is justified if the widths of the resonance levels which contribute

strongly to the process under consideration, are smaller than the distances

between the levels Only slight modifications of the one-level formula

need to be made to extend the proofs to the general case of actual nuclei.

Thus allowance can be made for the coulomb field in case of charged

particles, or the centrifugal force in case of particles having an angular

momentum and also for the possible presence of magnetic forces within the

nucleus

•

It is possible to arrive at a formula of the type of the dispersion

formula without any assumption concerning the mechanism of the reaction.^

However, such a formula would be of less practical value than the dispersion

formula because of the greater number of parameters it would contain and the

greater complexity of interpretation.

• -

' V'! a>' X i ' 10 a ."*„•£ o

-

-.

4 r l-;ji L• • r & . IvO’.'- K 1X1 * • 0.1

<

. • :

. i :3i]i C -.r o cj •' * : Blu •: t-

‘

. .. & tai CO .

•' '- 05 t r:-

9

II.

Discussion of Nuclear Resonance

Nuclear potential barrier heights are primarily derived from exper-

imental data and corrected for nuclear motion and penetration through the

barrier. Barriers show marked differences for various incident particles.

The barrier heights for charged particles, alpha particles in particular,

increase with atomic number. Also the nuclear radius at the top of the

barrier is proportional to the cube root of the atomic number. Corres-

ponding values of barrier radii of the neutron and radioactive elements,

agree with experiments of Dunning and Gamow. It seems that the attraction

force present in alpha particle collision is only a second order force and

that the irregularities in proton barriers are due to a first order force

effective much further outside their tops. The energies of resonance

levels increase approximately linearly with atomie number. In reactions

with charged particles, the nuclear barrier is a smooth coulomb rise which

decreases for close approach by a potential term of opposite sign.

There is much experimental data available which furnishes information

on the nature of barriers. It is quite well known that high energy particle

incident on light nuclei are not scattered according to the Rutherford

formula, also that many light nuclei disintigrate when bombarded by high

energy particles. We must assume that the particle penetrates the nuclear

barrier in order to explain these phenomena. In the first case, the

incident particle escapes while in the latter a compound nucleus is formed.

From scattering experiments it appears that the components of angular

s

:.o?f -t sicur i > • x !

^ZQqxe amt fcarJfcrsb ^XiiBciiq aia aWsied taimBcf laidaed’oq *i*elairtl

jtorji ex/oi'isv to*! eaonarrslt'iii) - • -

.1

y

,.*. , . v ,-!) ,0 n; iio .x/ r. t lo - « *

"• < to r

-

.

'- 0 t

.

Ic r: "A i t ; -

l

-' yi - T

>

. ,• I

id. vd.’fcerrt dpd neiiv, Q.7ssi$iJcitt : islom digll ^a^.- ^le «Xx.ft

i si;

.

.

.

j £jju;ut \o £ .tfi. noqr a ?iij aid’ :• r.aex- :U £tdflc>rji*i ' :>.ar ! ' fc ir"^

10

momentum higher than zero are of great importance and must not be

neglected, while disintegration experiments indicate that higher order terms

are insignificant* Thus for the absorption of a particle it must make a

1 fthead-on collision with the nucleus . Most experiments are carried on with

alpha particles and are limited generally to disintegration and scattering

phenomena. An approach of approximately 5x10 v cm. is necessary for dis-

integration to occur. Approximately 10~^ of the total number of incident

particles on the target pass within the above distance. A similar situation

is true of the scattering processes. Thus little is observable without

considerable penetration.

Gurney first pointed out that the probability of entry of a particle

is much greater if it has an energy lying within an unoccupied (resonance)

level in the nucleus. Pose was first to present experimental evidence. The

interest in energy levels is in their mean energy and in their energy widths,

the latter affecting the yield of disintegration products. When plots are

made of resonance level and barrier height against the atomic number, the

barrier heights lie on a smooth curve and the resonance levels lie on a

17straight line passing through the origin. This means that resonance

occurs at the same radius for different nuclei. The radii at the top of the

barrier increase with increasing atomic number. Gamow has suggested the

relationship that the cube of the radius at the barrier summit is proportion-

al to the number of particles in it. Experimental information has shown, to

some extent, that the energy of a particular resonance level increases with

atomic number.

13Margenau and Pollard have given an explanation of this linear pro-

^ ,„t„„ -'.rijlf: i. ‘ : ‘ 'u » “'• ' U "

.

„‘

'

'

..• > ' ^

;

.

,*oU «d» U « «-vX,— «*• *>'• - —»<«

11

gression in terms of the energy change due to the addition of a charged

particle in going from one element to the next higher. Smaller energy

increments per unit change of Z, the atomic number, at lower levels are

accounted for by the greater coulombian forces, since these forces decrease

the potential energy due to the added charge. The resonance widths are

slightly more complicated to explain. They would, according to Pollard,17

obey the relation A Ex

AE2Eo

where the suffixes refer to any two levels in one element. Levels at

£. _68.9x10 and 5x10 erg are observed for aluminum, where the level widths

are in a ratio of at most 2:1. This means £l should not be greater than

p21.3. However this ratio is actually found to be 130^. Therefore the

supposition that the levels differ as to angular momentum is not sufficient

and the precise nature of resonance must be sought elsewhere. Suggestions

have been made that the alpha particle exchange force may be an important

factor.

Linear increase in barrier height implies that the radius of the

inside potential wall is, at least approximately, a constant. An explan-

ation was offeredx

in terms of standing waves within this potential wall.

Thus a more adequate explanation in terms of interaction was given.

The resonance levels correspond to energies of unoccupied alpha

particle states inside the nucleus. The probability for the penetration of

,n

t I a 1 rl r. ,1389T?

.

o*i : cXoveX towoI (,'i9dmua oiroi . ecJ *lo ayierio d“lw

.

i!ci.

.

t : . t

,

oiien

:& 31 - 4 1 o i lo it 1 n li. :09*i

'

'

. CTO•

..hi tfenoo a t^a^.p.tnl

..

X ; ?j isa , > . • i too • I‘-'> 'I • * * -•• •

-

12

of the approaching alpha particle of these energies , rises to unity in the

resonance case. It is not necessary to determine the actual positions of

the virtual alpha particle levels for a given nucleus, if the change in

energy of a particular level from nucleus to nucleus is considered. It has

been suggested^ that the energy difference between two corresponding

resonance levels in elements of atomic number Z and Z-t-2, is in general due

to the interaction of two alpha particles. First, the alpha particle added

to the nucleus in passing from the first element to the second, and second,

the alpha particle, the virtual presence of which gives rise to the reson-

ance. Since the radius inside the barrier is sensibly constant for these

elements, this interaction accounts very simply for the approximately

linear increase in the resonance energy as we pass, say from Boron to Nitro-

gen. It seems also that the coulomb energy produces the correct order of

magnitude for AE, but the results are too high. This is in line with the

well known circumstance that the deviations from the coulomb interaction

between alpha particles diminishes the repulsion.^ The correction amounts

to about 50% of the coulomb energy itself.

The shift of the resonance level from one element to the next, e.g.

Beryllium to Boron, is to be understood in a similar manner. The incremental

energy is here due to the interaction between a proton and an alpha particle

and this should be approximately 1_AE. The progression of the levels with2

atomic number seems to be explained. Experimental evidence also seems to

support such a supposition.

1 ftPollardx also offered an explanation in approximate form. He made

use of the fact that light nuclei contain equal numbers of neutrons and pro-

•

:t

• > n I l -no '

.

t i , :oj-

nd( y"

•'

t*

.

hr ,1 rvso o •

. .

'' -t

. -‘

- -<* '• '

, '

:

-

...

: . , ,.iv- :

.. .

.

-lOXti.- T - '.0 -

13 .

tons. By letting the attractive force between neutron and proton be

represented by a potential V of the form

V . Kef.

rP

where (r) is the distance between them, (k) is a constant and (p) is an

unknown index. For the potential barrier around the nucleus he suggested

the form

v =(^X-^br)

r

The potential V is proportional to Z, for a given (r). Considering the

resonance level as one for which the alpha particle forms a standing wave

within the potential barrier of the nucleus, as before, and assuming a

constant radius for all light nuclei, then the potential at which resonance

occurs would vary with the atomic number. The wave-length of the alpha

particle must therefore be a constant. Its wave-length is given by

h

(2M(E-V))

where (V) is the potential energy of alpha particle inside and (E-V) its

kinetic energy of approach. Since the energy increases in increments, of

7*

the order of 10 ergs per alpha particle, as such a particle is added to a

light nucleus, the wavelength of the alpha particle will change slowly with

(V) and can be treated essentially as a constant. Thus the potential of

the level of the standing wave is a linear function of the atomic number.

* Binding energy of alpha particle

'

;*ax » v

;

n.sct-ecoo & al (;0 txw* rf .-iuT ai (?) " ozottr:

: v - v

f . ilchcoqcrtq aJt V

•U'tol -XoMx.

i O .f ;:>i •x > I! •••‘X -<oa •

e -v. t3'to": d . : n ’

- .• v 1

'•!=

'• > *

:•

L .-I'.-m cu‘ 5 •.

’’ -''

-

iJ-I (V-c) [)!• ..M.;.ocX aloltesq •

' lo as* A 'tot re .1 arii X (V) e-: <-

*:or. J; ;J go r-tOi i T ;.-I - r: w-JtZ «'•->' ** oJ

*

1;'

• od .-I-cXoiti ... -or. t 3 «s:r>ir 4 is "CX 'o t ' ' ' ^

- r<

r, "D; 3 ^.i;-;Xr

•: cixriT

r

14

Chadwick and Feather^ have pointed out that disintegration, with alpha

particles, should occur at any alpha energy higher than the barrier, while

at lower energies disintegration can occur in the resonance regions. The

mode of disintegration determines the width of the levels. In most reactions

the particles produced can go over the top of the barrier, even for low alpha

energies. The widths of the levels will then be determined by the type of

emission, that is, whether a proton or neutron is emitted. This is especi-

ally true if the penetrability for the alpha particle is small. Thus there

should not be any variation in the widths with barrier height even above the

top of the barrier.

However, below the barrier, the orbital angular momentum is small,

while above the top much higher moments are possible. Therefore below the

barrier only levels of small angular momentum j will give resonance effects,

while above, compound levels of large (J) will be important. The levels

above the barrier will thus be closer spaced than the resonance levels below.

The resonances above the barrier will not generally be observable since the

levels will overlap at these energies, that is, unobservable as maxima and

minima. Thus the disappearance of pronounced resonance will be an approxi-

mate method of determining barrier heights. Experimental evidence supports

the theory that the nuclear volume is proportional to the number of parti-

cles.

As the nuclei become heavier, the resonance levels become more dense

and therefore more closely spaced. For nuclei ranging form C^ to P31, the

average spacing is slightly more than one Mev for the lightest, and about

1.2 Mev for the heaviest nuclei.

V, ' ,i •••; Oi ,

9it iJ- 'I-.'. X'wOO *'!«:> I,o it*.

3t. -•'-&»

,

.

. lJ. j; f . i \.n : ,*•* ' v ^

"• ''

> '..c : ,

. <f *.

.

t3^9':'i9 -r &vig ,

r ih t refloat“•

.

# -.

r <•}'.: o, WI J -i T* ’• -

"' v

t

!

..*j - 1 i:>• jn r;

f O X,. a ix/'ic•••

'

,

.

-

t t .i'v *

15

Sharp resonance effects are observed in many reactions with light

nuclei.^ This would seem to require strict selection rules to limit the

decay rates of the corresponding compound nucleus. Such selection rules in

several cases follow from the slowness of the interconversion of spin and

orbital angular momentum. The sharpness of the resonances is due to the

narrowing of the widths. Selection rules, combined with the small pene-

trability of the potential barrier for particles of high orbital momentum,

will prevent all compound levels of high momentum from being effective in

disintegrations caused by alpha particles, as long as the alpha particle

cannot go over the barrier. Therefore, above the barrier there will be no

marked resonances observed. Probabilities of disintegrations which show no

such marked resonances are also affected by selection rules.

There are two types of selection rules,^ those which should hold

generally for any coupling scheme in the nucleus, and those which hold

only for Russell-Saunders coupling. The first is concerned with the parity

and total angular momentum of the initial nucleus, incident particle, final

nucleus and the outgoing particle. The second type will give information

on the behavior of orbital momentum and spin separately. However, this

second class is restricted to the lightest nuclei, with atomic weights from

15 to 20. The selection rules for total angular momentum and for orbital

momentum will involve the orbital of the relative motion of the incident

particle and the outgoing particle. The existance of the useful selection

rules depends on the existence of restrictions on this orbital momentum.

Such restrictions will exist for very slow particles for which only the

orbital momentum zero corresponds to a high probability of entering the

JrfSxX uJ* ex: cit» 3arr a 31 ni 0ov~i d. vie • 1 anac &% cr . '.i

.

j

I.- aol. x.,ir >tn : .1 .. :c ;

m b e '; I .

.

,•

r.- . t .0 .. X: I- : 1 t-ir. i 1 ild’ny.i . ftriJ 1

nJt t! oC 1 "*.© iu i u .•-ov o bo: IJU duev 1 j r*

,(• - . 7/S jeBif&D SrtQlf&Z£

.

. „• • i U-:

..• L :

•

.

.

s '> ** i/•

•I

' -il '

.

.

.

r

16

nucleus. They will also exist when the two particles are identical. .They

may be two incident particles or two product particles, taken in pairs. In

this case the symmetry requirements for the wave function will forbid cer-

tain values of the orbital momentum. Because of the Bose statistics and

zero value of the spin only even orbital momenta are possible.

Breit has pointed out,^ that the energy dependence of the disinte-

gration cross section is not appreciably affected by the orbital momentum

for slow charged particles, in contrast to slow neutral particles. The

ratio of the penetrabilities of the potential barrier for various orbital

momenta is practically independent of the energy of the particle if this is

small compared to the coulomb potential and the centrifugal force.

For resonance disintegration, the selection rules are more stringent

than for non-resonance processes.^- Parity and angular momentum must be

considered. This means quite generally that either only even or only odd

values of the orbital momentum are possible when a compound nucleus in a

given state breaks up into two final nuclei. The parity is determined for

the compound state as well as for the final nuclei. Therefore the parity

of the motion of the final nuclei with respect to each other is also given;

and this parity determines whether the orbital momentum is even or odd.

An application of this parity rule is evidenced in t he excited states

of Ne^® formed in the proton capture by F^. Other applications are

connected with gamma-ray emission. A further application may be the capture

of slow neutrons by B^, thus^

10 1 n 7 .

5B + oN 3 5B = ^hi 4 2«e

,hl

. I. L j , s , -r'i .1.1 -••-•±xiv 0.1 1 » ...M '-’ ulV - •

. jt n J ,/v f ‘

.sldxaeo . -i i '&tr.,ao[r l.i'M'r. c • v‘ • i v'ru> ai. ? •"* J -rtar:

’

.

tj ,ifoKv t -I9l M I • ,n 1 Cj - .dtil'o - w 1 ••- <-•

.•;•

nor *•- •• :.n ^ e *$einieli> eftoauowi *tci

-

. Coui i

.

. ...•

::

; •. •

.,- • ^ *

: • t '

' L'Xi '

h\ ? ni 1 'OvztrsQ r' ©lift V - ra(4 ei i

'

8noxiF3lXq.]B lerfiO ."%

xrf noJoiq

V -' \'

• -oiir

'

17.

Another important consequence of the even-odd rule for the orbital

momentum is that the angular distribution of the disintegration products

must be symmetrical with respect to the plane perpendicular to the direction

of the incident beam. This was found by Kempton, Browne, and Massdorp,^

for the disintegration

H2 4 H2 = H3 4 H1

The radii of proton barriers, for light nuclei, do not show

correlations with the rules for alpha particle barriers. For a given pair

of nuclei, the ratio of radius to barrier height for alpha particles, is

only half as great as the corresponding ratio for proton particles. In

4-

accordance with Heisenberg's theory the light nuclei are most stable with

equal numbers of neutrons and protons and if two such nuclei approach each

other the first order force of attraction is zero. Thus the second order

force between neutron and neutron will predominate at the nuclear surface anc.

will vary rapidly. This is what happens in alpha particle collisions,

according to this point of view, where the conditions stated are fulfilled

for incident particle and nuclei being bombarded. These conditions will

never exist for proton particles. There is, besides an impinging proton,

an extra neutron in the bombarded nucleus, if the reaction is to take place

at all. Therefore first order attractions appear between unbalanced proton

and the balanced part of the nucleus and between the proton and the odd

neutron. Studies of proton barriers reveal information concerning the

*Nuclei consist entirely of neutrons and protons with noncoulombian poten-

tial functions between neutron and proton, and between neutron and neutron.

.

> O' •, oo-flr / , t v )DC C '2Xr< 5 Jx 'J '-> . X *X: >XlA

aoiicrvxlb sriJ xt ". DjUtaaq'iexs >n«

f

q add c.t d*»et»*:X • .±7 acf J’BXB'"

^tcpob . • j

r'-r' bastol 8$w alxiT «tfAed dnab-toni arid to

, .utt -ro>

lR 4-eH - % f SH

wodi Jon "'-i JrixJU: id t c/xyjtYie<f ao- >.<q lo xi xvi sriT

t ... l>

—' 7.0

lol >•;

1- 1 . 9<J :ixib••' to old .ji J t i9±,un 'to

.

• .

slow ' '

d •; ».; rr i.li ,.•): .• .'•« • mriJxr^n '-c

.

.: U jt 91/ ho&tJ.t joiJi' noO arid 9-c ct» , .

'

r to Jolo iJ Id cd nibr* >c

S i- ;t ro •

* .. sdiiiocf 31’ J: d

.

00 9 . ' hv j.jcJ-yrc: erU ro i»v.ted bin ixjoio/m arid to Jteq bson.-I’

•; l•

.

18 .

potential energies of a neutron and a proton.

Barriers against deuterons should be higher than those against protons,

since only second order attraction similar to that for alpha particles should

take place between deuterons and other even nuclei. It will be recalled that

light nuclei tend to contain equal numbers of neutrons and protons.

The neutron, being neutral in charge, can readily enter the nucleus

without barrier repulsion. The probability of capture of a slow nuetron

varies greatly from element to element and shows pronounced resonance effects

as a function of the energy of the slow neutron. The level widths are gen-

erally quite small compared to the spacing and are of the order of 1 ev.

If the particle is fast it will be impossible to define its energy

very accurately and hence resonance effects cannot be observed. Fluctu-

ations of the incident particle energy from the best sources are not less

than one percent, which means 10 Kev for 1 Mev. If the bombarding nucleus

is heavy this will be particularly true because of the closeness of the

levels. For nuclei of more than about 50 particles, the level spacing is

less than 10 Kev and is only a few volts for elements of atomic weight

greater than 100. For fast particles and heavy nuclei, only the average

cross section of a nuclear reaction, averaged over a large number of reso-

nance region, can be expected.

The production of artificial radio activity was discovered by Fermi*

*Some authors credit this discovery to Curie-Joliot, see page 4* However

they both worked independently.

' :-; A n o «- ai 1 *Si - r

,

.

*r

. I '

'i

-ltfo: !•" .f/ rxe-de e-i dW-'.ao r-sJ . d^«i: ..<&. a 1

.

.

r:> ;i ic . 9 ' cj ^ 3 “'i

$1 ,vd ::£l& .at .> nr vw 3 a IcW* ;

d‘ ‘ i - -OC I rr,

*

.

19

Qin 1934* Many radioactive isotopes have been produced since their dis-

covery. The production of stable isotopes can only be implied by the

absorption of neutrons. However, gamma rays from capture processes have

been found and measured. It was soon discovered that the slowing down of

the neutrons by a substance containing hydrogen, such as paraffin and water,

would greatly increase the production of radioactive isotopes. Cross sec-

tions more than a hundred times the geometrical cross section of the captur-

9ing nuclei were observed.

The one-level, or one-body, model gave a capture cross section inver-

sely proportional to the neutron velocity, which appeared to explain the

large increase in the cross section. Also the absolute cross section for

slow neutrons was explained reasonably well. It further explained the

fluctuations of the capture cross section from element to element, by means

of the differences in the neutron wave functions in the respective poten-

tials. Now the one-level theory predicted that the scattering and the

capture cross sections should be of the same order of magnitude. The exper-

iments of several investigators^ showed that the scattering cross section

was actually less than ten percent of the capture cross section. Others

found selective absorption to exist, in that neutrons which made one sub-

stance radioactive had very little effect in activating another substance

and vice versa. This is in contradiction to the _1 law mentioned above.v

Thus Bohr, Breit and Wigner were led to the more correct many-level concept

of nuclear processes. The selective absorption of neutrons of particular

energies must be considered as a resonance effect, the temporary addition of

a neutron in the nucleus producing a virtual energy of the now ’’compound"

p•i i v r >o ,<>.: rl&o .1 rt vn : bs

-

> Jo i f-J 1 Lit o . 'T ,vt "O

LVOj • DOa i

t 11 i d L< , 3 J iBvfflOO Ji * V u*

o ; : 2 0 r :J 7• -:m : i r ,

*’ ’ • •

• It

s. B \'*.i . ;

• r

i*£U+ •

• 7 ' I V. f-« tO 10 , io .->1- i ' •?

'

. " r J ' i- •

’'-’i

.

-

•: : • *1 ».CCf r .. '.r r E -nf Iq> • f (r ’ .'i a

, )

: v : .. .. . -

« IlJ y

.to r,r jJ-zio t. ti / orfe nod l>tev "t ech •

. .; • 0 r.r .

• -j U 'v.-.T M rfOH 3 '

>:v.ul Jus a j-oats xJ.J - ,.J cl Joe $ el. ti i .1 r had evi^c oil- 1 roi j

.-^vjde ; n • • .it'•

1:: 0 :fv\ !t :> tj l aii T .©a ov bo1b fr »

T ... -

•7. .

~

xiriv • 30x01 rq au^Lova id rri

20

nucleus. Neutrons corresponding to the energy of these levels have a high

probability of entering the nucleus, and these energies will vary from

element to element*

Radiation damping, due to the emission of gamma-rays, broadens the

resonance and reduces scattering in comparison with absorption, by a large

factor. Interaction with the nucleus is, according th Breit and Wigner,

most probable through the (s) part of the incident wave. The high pro-

bability of having levels in the low energy region makes the explanation

seem reasonable.

Virtual energy levels are supposed to exist in the compound nucleus

in the region of thermal energies as well as somewhat above that region.

The excited compound nucleus will then "jump” into a lower energy level

through the emission of gamma-radiation or perhaps in some other fashion.

It will be recalled that the occurrence of scattering is also included, since

the original neutron may be returned to its free condition during this time.

If the gamma-ray emission from the compound state were negligible there

would be a strong scattering at the resonance, the scattering cross section

being then of the order of the square of the wavelength. It has been shown4

to be reasonable to assign 12 volts to the half-value breadth” of the level

due to radiation damping and the corresponding "half-value breadth" for

passing back into the free state is about 1__ of that above. All this means40

is that the incident neutron, after entering the nucleus and giving up its

energy to the particles of the original nucleus, has little chance of being

re-emitted with the same energy. The ratio of these two "half-value

breadths", is essentially the ratio of the elastic to inelastic absorption.

- SV i. Jo fvn' Vi 9X19 clJJ" dt '£3lbacq£oYX< . ...

. : r r.o o* dr •

.

tI'.::'-

« I

.• 10 a' ;. : . Vi

-ortc .ov -• .v1 '. to: ' lo ci.Y. (a) jdJ-

. z >. ! : iV t; ;

. .

• i-

. -I'isal ". u -Of-. :lc e rrfxv rro aO-Mxil: ; ur irn cVolrv? -u'.i .cf,. . it

:

- ’'too r tttfea 0 il - M pxv t*" y •. aoiJx> .i

r oi^io ) »riJ

.

t’X «i

r. h o-.t . .tlov- a; i oX ->Xd .no^ ei f otf

xo - - t. -.: :i gxiibnoqeoTtoo bne %aiqiPRb noljelbar of eub

•.

r<

.

•'.

r

'

: ‘

anied > .lonrado olXJil arrit 3x/>Ioiixx I irtf 1c sloi;.:

:edf of vj.*xea»

.

21

A high probability of gamma-ray emission is of primary importance for a

large ratio of absorption to scattering. This is indicated by the hardness

of the emitted gamma-ray. It is quite likely that the incident state occurs

through the (s) state In virtue of the head-on collisions at low energies.

The probability of observing effects by means of the (p) state is not ruled

out even though it leads to much smaller cross sections. Breit and Wigner^

have made calculations which show that with resonances of the type con-

sidered here, one may obtain appreciable probability of capture at energies

of the order of 1,000 volts. Large cross sections are thus possible show-

ing it is not necessary to limit large cross sections to energies of ther-

mal velocities.

In connection with gamma-ray emission from the compound state of the

excited nucleus, Rice^O vas the first to make use of the quantum mechanics,

since Dirac's first approach in 1927. Similar processes were used by

others

•

In a more general view of the problem, the nuclear process can be

likened to the absorption of light from a level (1) to a level (3) which is

strongly damped by radiation jumps to a third level (2)

•

The absorption

from (1) to (3) corresponds to the transition of the neutron into virtual

state and the jumps from (3) to (2) correspond to the emission of gamma-rays

in a transition to a more stable level of the nucleus. The absorption pro-

babilities can be obtained by using the principle of detailed balance from

the solution which represents emission from the level (3) to the levels

(1,2) or else by a direct application of the theory of absorption. The

usual theory as developed for either process is not very accurate.

,

.• 2 obcJp lo oil:

* IX b©+2

-

9c (q)•

i jj£ d , ‘y '*

• 1;

fl . T:;- f: 'oxro It f. ».•.

-

e CroiJ: a .0*1 ••

.

,I3»; . -I -'* DCO«I lo *r . .tj

J I: 3*1 tot 3 € :oio 0 if v .21 .

*

,rr ? :i-, 'in a r

;

(Si -t o :.u 3^051 o~ Js-ti’* • it w u'aaif- «eir -I./tfis on

ifllliniB nx rio.QorqqB isiil: s'oerciC eotile

.•Tv id

t ,10©IdOTC .5- lo • B.’v I. Tenoj: STOffi

.( Tsv .: nJ It 3 O' /(, t •sr \jcf‘

<r r ! prrcnrt

:

.

• ' -'

- (•

.

•>; •I,.:-

-

-•t >•.

''

c Is r ; .f J-.S err ^j >

. ?! -e '.

•

: s •• f no ; . . . ;*

i"• 1

• ' 0 12

^ <X)r

ftI oril f-n't jioleelr > ?2a ri 9*x r ‘- ; I.ti T.c j •>

••:'•: ••]' ' j " '

'

’ ‘ r -• T r • : I v To ( v. •'

22

*>

A more formal manner of discussion may be given starting with wave

functions which are solutions of the wave equations for an infinitely high

barrier 3ome distance outside the nucleus. The difference between infinity

and the actual height of the barrier, can be treated then as a perturbation

essentially responsible for the matrix elements of the interaction energy,

which are in turn responsible for transitions between corresponding levels.

The nucleus may also produce heavy particles from the excited state.

This was discovered by the production of heavy particles from the nucleus.

The Boron and Lithium reaction, especially, show large cross sections for

these products including the production of and He^. Such reactions are

often used as sensitive methods of detecting neutrons of thermal energies.

The production of protons is energetically possible in two processes with

slow neutrons, namely, with and B^. With fast neutrons the probability

is much greater. In the production of alpha particles the cross section for

BlO is larger than for Ll^, therefore the resonance for Lithium should be

farther away from zero neutron energy than the corresponding Boron level!

Since the protons produced are of low energy compared to the potential

barrier, the penetrability of the potential barrier for such protons is

about 1 .

30We have seen that with slow neutron bombardment, only one kind of

particle can be produced with any probability. In the Breit-Wigner^ one-

i level formula for nuclear resonance, the total level width contains a term

corresponding to neutron emission and a term corresponding to emission of

other particles. The neutron width varies as the square of the neutron

energy. Usually the neutron width is small compared to the other contri-

.

. o'- jj-’iso- - .. ( rj .V.*i r ..i

< a.V '.j ' Lsjrtoo bi i .t i./is’ *

'1

,

... .

'

, .- *

.•

•

. ; Lsr :/:J -cr - nttt- ,

•*, '£

..r-

.- cr- :

- 7 -

: ""

. :-:'i aelaichCBcj sxi *io nof^oufep’ ; I .

B

-.

•

. m ' ?: .

'

r/. ..7

1

-i

'• •oi

- 0X .ned’Bwg dour: ex

.

to vj £ Ltdamfoa&q n'i97

. '.

•; '.-r

• -n t•' ms . ; w . £ - t : e

-

So* >;1

t o c *3 loirn ic't alun

oi :ral>” ‘00 ?• 1 «n -7 t floi < t • u • nr >

rr~. j-• ’• J

. x ' m-riuoa s .T .a^lolchu - r'i ->

:'

. %

23

buttons to the total width because of the low energy of the incident neutron

With these considerations, together with the fact that experiment measures

the maximum cross section at exact resonance, we have a means for the direct

measure of the ratio of the neutron width to the total width. Here the

width of the resonance is small compared to the resonance energy, and the

shape of the resonance level is like that of the optical lines.

For sufficiently small neutron energy, there will always be a (l/v)

region in which the capture cross section is inversely proportional to the

neutron velocity.-*- The position and the width of the first level deter-

mines how large the l/v region is. The cross section will have a maximum

at zero energy for all nuclei, besides the corresponding ma.xi.ma. for various

nuclei at the resonances. Experimentally, this effect is usually eliminated

by use of suitable absorbers such as cadmium to absorb the very low energy

neutrons. These neutrons are known as "C" group neutrons. If the first

resonance level is quite close to zero energy and is "strong” the one-level

formula is equivalent to the many-level formula. Thus the l/v law holds if

the neutron energy is small compared to the energy of the first resonance

or compared to the level width. With the spacing and width of the energy

levels of light nuclei, the l/v law holds up to quite high energies, while

it holds for only a small region in heavy nuclei where the levels are very

dense. In fact, the first resonance for heavy nuclei is often so low that

the l/v region does not extend beyond thermal energies.^ On the other

hand, the spacing for light nuclei may be of the order of several hundred

thousands of volts and hence the first level will, ingeneral, be at this

energy, and in any event the widths will be large. Therefore

h si: 3 V - . •; ••*! it') i

•dJti

i

•net tL -'J . o it 1c i- ri-r L i v I eor. “i :

r;*

;_«d.

<

•’

.

".. , .V ..

/' ... -v::

,• ;'V, oxss ,

v "

Dells eixit rr--

tatrlffibjao 833 doua arc&dzosdB

.

11 eblorf wsl v\l edct awdT . , - yjasr uii ol XoelBvlifpe ei jbIi/

-X ttr.-ri' . In Y Ic -m 'it J ...r£. 3, - lima ..

r

c .. '.i-/ a. o •. • i.j v.' ! . *.-t ’ x-.: W >t ace -a ,

qj/ , bio. v VJ rt xloLot:n <r ilL ' alsval

t? ? r-\ i r 3 '• ielru /v.r-1, ;t; AOs?-- lien? 3 y;i - io'; .:

•'

.>d t.<

. r.ar 3

1

TTO3 ‘tO ISbTO

f •'• tXJ •- i t- Ct ':

1 ’

•.t

for these regions the l/v law will hold up to quite high energies as is

borne out by experiment.

24

By bombardment with protons, some nuclei yield alpha particles

following Gamow's prediction on the penetration of the barrier. The yield

increases steadily with increase in energy. Certain other reactions show

sharp resonances with the emission of high energy gamma-rays. Particularly^

Li7 , bombarded with protons, shows a sharp resonance at 440 volts, and

Fluorine also, where there are three pronounced resonances. The process is

quite complicated, some reactions yielding gamma radiation, and othere not.

Some show gamma radiation at only one level, indicating there may be other

levels. Experimental information is not available for enough reactions to

make any reliable assumptions. In any case, the sharpness of the resonance

is a measure of the lifetime of the compound nucleus formed by the capture

of the incident particles.

The absence of any resonances in some reactions with proton bombard-

ment, shows that the total width may be large for those nuclei. The gamma

width as well as the proton width, is always small for low bombarding

energies because of the barrier. The sharp resonances in the other

reactions indicate that either the nuclei cannot disintegrate by alpha

emission at all, or they do so, at a much slower rate than other nuclei. An

example of the two reactions is given by:

Li7+H . Be8* - 2He4 Li7-fH = Be8* • Be84.r

For the Be® case, two unexcited alpha particles are necessarily described

by a wave function of even parity, with odd parity there would be no decay.

^

.

rj gc, r>9 hgt?,at . d%M ediop o1 qu bXorf il.br w«I v\l erfd aiiols©* sasd" rol

•di • •

. o. ;.w vd tx .

•' t o'i

solo id*isq Bdql-s bleb: iaXoi/n a*oa t anodo'xq rfdb dnerirradmod ^8

:-Xe erfT .lebrtad erfd lo n sidfrideneq srid no noWolbrin ft«*.*onaP gn-Ncriol

. . ddo f:x£?bi90 *Y3~sn9 ni

C,rrT^ffffi rttaS BOSKS - SBilEB ©rid" rfdiv WWMD8OT qiBdS

bne t e,+Iov dB eoaenoaei qxBiie s awxte t anoio'tq rfdbr bsfrxedfijod t U

,hx aa9o<yxq erfT .aeo' omrofKrrq eevrfd eT© eiedd eiexiw toaLs eniioi/r?

•don e-rerfdo boa ,noi' "

,.;do scf yixn evsrfd sfl.rd.~oibni t level eno vino da noxd ubm bbw*‘& vro-

J- 3 5:i;£''-

od BfloldDBei ri^none *iol elcfBliavB don at aot&acmolal I eo.rrwcitxeqx3 # aX9veI

eor^nosei ex.d lo eeenqisrfe add 1 shqo ^n* nl . anoidqaxiaas exi' ila*i xtv- ‘-nan

eiu&o&o end ^cf bear 1 Losia bnnoqffloo 9dd lo 9i.iidelil MU’ - -

.eoXoid:.-- daebi onjt 9dd lo

-bx-ccteod nodoiq rfdiv anoido& i eros ai eeDnsaoeer \'n^ lo sonsade eriT

annas eriT .leloufl o cofid lol 93IBI ed ' -r 8

sr.s fpwlB ei «rfdbivr flodoiq erfd bb Hew bb ddbiw

^ftddc t rxi B^ojaeuor-erc quarts erf? .lebrtBd ad d lo esj/aoed aelgrexis

adqla -^d sd ^sedoiebb donneo isloxm arid isddie drrfd edaoibni afloidOB-n

nA .islone *rerido aadd edat T9woIe rionm b da t03 ob TCerfd 10 t IXs da noxeebns

; , d novi§ a* ticido.ser ov + a d lo 9X.;a

1Aa - A»8 s H^xJ s sP «

- •

. i'

' *

a sd blxjov «rt9dd ^dbinq bbo t^dbi.se neve lo noidomil evsw b

k

25.

Thus we have the qualitative explanation for the two reactions. However,

in line with what has been said, such considerations of parity are not

always applicable. For example, the reaction of (T- and Ne^ , since the

disintegration products are not identical in these cases.

The analogy to the dispersion of light is not commensurable with the

strong interaction between the impinging particles and the nucleus, as pre-

viously pointed out. Another approach to the problem from a geometrical

JOpoint of view can be expressed by a qualitative discussion of waves.

From the very existence of quantum mechanics, particles have a wave nature.

Let "k" denote the wave number of a wave outside the nucleus.

When the incident neutron approaches the nuclear surface, this wave is

joined smoothly with equal value to the wave function inside the nucleus,

which is the solution of the many-level problem. The wave function on the

inside of the nucleus is dependent on the coordinate of the incoming neu-

tron. Since the neutron will, in general, on the average have a high

kinetic energy, the inside wave function will have a larger wave number "K".

Therefore these two waves of different wave numbers must be joined together

at the surface of the nucleus. This can only be accomplished when the

amplitude "A" of the inside wave, is smaller than the amplitude "a" of the

outside wave. "A" is of the order k/K to accomplish this. In the res-

onance case, the derivative of the inside wave functions is zero at the

surface of the nucleus and the two waves are joined with equal amplitude.

This corresponds to excitation energies of the compound nucleus for which

the neutron can enter the nucleus and exist in certain narrow intervals of

the energy. The widths of these intervals are the resonance widths.

t

Q&St

. 21”*r g vjjtiaq lo ;> >id. e iBblsi fetffi tb±38 /it.-ad e irf ^Idw ndxw Mil

• ..r eonie t

° As5l boa 1c noidosai arid ,

"• •

.30330 aoerid ni Isold rabi don e-xp *do. bote noidsT/jsdnxaib

o j rltirt s-Tdait/eaewnioo don ex .drisil lo aoia’ieqeib add od Troians ariT

oiq as t anaIona add bns aoIoid'tBq gflisaiqini add naavded noxdorrednx anorrde

uni ureldorrq add od rio.3o*tqqjB TsridonA .dno badflioq i^Xswcxv

^.aovBw lo noieauoeib avidadilanp 3 yd beaseixpca ad oao walv lo doioq

*2 rfnaffl r ednrxp lo armadalx© VidV 9ii * raorr'

7

9dd ebiadno ©raw b lo rredienn avsv odd sdonab ndM dej

sx .-v , si. d ,v? hira iHeXotm add ea/loso*! • 3 nordnen dnabloat end nsr.'.

t snalour: arid abirrx noidonnl av w add cd anl

v

Ifinp© d+lv -^Ifidooine ben.ror,

arid no noxdonnl svbw sdT .r ' add lo noidixfoa and Si doi

animoonx arid lo adBnibtooo add no dixabneqab ei snalonn arid lo ©bleat

. .

' •

.

t'dds^od -nxofc 9d damn aisdimn avsv dnaiellib lo esvuw ovd 93©rid oTols'isriT

fl--,o.w nsi'isilqxsooo ' acf \Ia° nso sixlT .enalonn add lo aoBlma and ds

arid lo "c" abndilcxnB arid asrid teXX-me i ,©Ymw abxaox arid lo "A” abndiXqiae

- add nl *sirid .ailqrnooos od }l\ri lebnc end lo ai "A" .avjrw abiadno

arid dB crxas si anoxdooul svbw abxan and lo avldBviiofc add ,©eao aonano

•sbudilqias Isnpa ridiw beniot 9*xs aavnw ovd arid boa anaXotm 9/id lo aoxlwa

- - *rol -•jjaloirn bioioqjBoo ©rid lo aeigfiaa© noidadiosc© od shooqaamoo aiiiT

lo eXBVTcednx w***n aUdiao nx deix© bns enalonn arid iadn© nso nondnon arid

. 9oarnoaa*x 9rid a*is aiB^nadni aaarid lo eridbiw ariT •'iCS’isn© arid

26 .

Assuming that the phase of the wave function inside changes smoothly at the

surface with energy, then the intervals which give rise to a large "A" are

proportional to the separation of the resonance levels. In other words,

the widths of the resonance are proportional to the distance between them.

This is, of course, only an approximation neglecting any absorption inside

the nucleus and thus represents only the neutron width. This is seen to

be a good approximation even in a more quantitative derivation.

In the consideration of the scattering, the wave function outside

the nucleus has small values of energies except at resonance. The situation

at the surface is therefore similar to the scattering of an impenetrable

sphere the size of the nucleus, where the wave function vanishes at the

surface. Thus we have a potential scattering between resonances whose

cross section is similar to the one of an impenetrable sphere.

QFermi and Marshall7 have measured the change in phase of the scatter-

ing wave with respect to the incident wave for several heavy elements near

thermal energies. The phase change was always practically 180 degrees.

This is in agreement with the assumption that the potential scattering is

the scattering by an impenetrable sphere. This phase will be constant

where resonance does not exist, in the resonance regions the phase will be

constant only for neutron energies higher than the resonance energy.

The largest elastic scattering cross section has been observed for

hydrogen (12xK)”^cm ) and the reaction is used for the production of slow

neutrons. Except where "strong" resonances occur, eleastic scattering is

the most probable reaction in light and medium-heavy nuclei. The cross

section ranges from 2x10”^ to 10x10“^.^ Even for the resonance cases the

.as

s ^Iridooxna eeax* rio eblf.ii xxoldomfi evcv e t lo er • q exit dcx-d ^nlxurBA

{yx.. siAn 9 -oXSi B od ©all ©vis xiolxiv zl&nsSrJ. ©rid cent tX2Tt*n* ridlw eoel un

t a£riow csxito nl .©level scnsnoeei exid lo noitn ?.qea e^t oj iBxxoitroqcTq

,merfd nsewded tens**!* ©fid ot laxioIdTOqaajq *M eonecoasT erid lo eridblv ©rid

1 joitqroedB ^os rin.tv*)o:> oDikMfat<«qq« c* •

lo *ai elriT

od neea si siriT .ritblv nondir Cue adneaertqei ai avelotm dt

•floldavlreb evid-Bdldoacp ©Tom b nl aove noldmlxoTqqB boo§ e ed

sMadcc rroitoxu.fl evw erid ^^axiixeddnoe exit lo xroitBreblanoo ©fit ai

: .eonsnoaei da dceoxe asx^iene lo aenlav Xlwce sari anelowi end

eldfnteneqxnl oa lo Sr:lTedtBoa srid ot Tfslimle erolsTerft ax aoaltwe erit ta

sxit ds eexlalnar coltoowl svbv arid eceiiw ,ac9iOirn exit lo ecle oxit c-ierige

. o---' itc-r-tc'. B svsri mr -'ifrfT .eo*lwe

, -

.eldcitecsqail na lo axso ©rit ot T.oXln/a ax nolt

-fiettsoa exit lo easrit • rr ^no arid bevmaaw evaxf ^Hfirier&M tec to

men adnexnole ^ «d i^evea to! evrw daeblocl erid ot dosqeer liib ere* gel

.eeeTseb 081 xf^coXtoorq b^-bwIb bbw ©snario esariq eriT ...

si ^ciiedtcoa Xeldxxedoq exit tent coltqxmxcBB erid rfdiv tneateer^c nx al air.T

• ed HJtw eaariq elriT .ereriqa ©XdBTdeneqffll cb ^d sniTeddaq© arid

-J : ;!?•; e Adq exit axtolj.sT eonex'occ'x exit n. t d-xi e ton ©oo aouom.-van e xoriv

.v :: t eat narid Teriglri eelsTsne coidcsc to! '^Too dtisdacoo

to! boviesdb need b -ri noldoee eeuro ^ciiett !.os oxta.c tc ^;* ' lit

'

03 olda-ale tnrooo aeoasnoaeT "snuide" erariw dqeoxZI .axiOTtxren

©Boro eriT .ieloxm ^ed-^lteiB tea txigll cl xtoldoaw ©Idadorq daoffl add

>1 «*ra€>r'“o;xOJ. Ot ^-OIxS morl ae^cBT colt:

27

elastic cross section will be of the same order of magnitude but small,

compared to other contributions to the total cross section.

According to Bethe, the elastic scattering cross section for neutrons,

including potential scattering, is of the forrn^

TT

2 ( 2i + I

)

(2J+ I) 2R + ir.Pn

E-Er + ^i P4- (4i + I—2 J )4R

4TTR2+ TT ( |

± '

2

4R(E-Er)-t-fl r Pn2i + ,

) ^rPn(E _ Er)

2 + ± p

2

4

where only one resonance level is of importance. is the wave-length at

exact resonance, Ep the resonance energy, R is the nuclear radius. Pjj*

denotes the partial or neutron width, the contribution of the incident

neutron to the excited nucleus. The square bracketed term in the cross

section given above, represents the effect of resonance and potential

scattering for a particular angular momentum j, the other terms represent

the potential scattering for the other angular momenta which are possible

by selection rules, for which there are no resonances. The 4TT term

represents the potential scattering and the last term is the contribution

of resonance scattering and an interference between the two. The behavior

1of the cross section near the resonance level Ep will be controlled by the

ratio ftt-Tn

R P

This deteraiines the ratio of the cross section at resonance to that of the

potential scattering alone. The ratio of the neutron width to total width

is quite small (10~^) for most resonance levels observed for energies near

*The width of a nuclear energy level is defined as Ps= — where'T' is theV

lifetime of the level.

tJIsKi3 dx/cf ©hadingboi lo letno emsa and 1c ad I£i*r noidoea B8o?o Oldsnle

. .

-

iSnoiduen 10I ooido9a eeoio gndteddaoe olde©!© odd «

. .- jj- ,

•• .'i-noasn arid «j3 t9oaBno8©i doaxe

. blood . J lo roldodlidabo ©rid- tridblw adn^mst ip isldrcsq ©rid aedoneb

qqoi© grid r l cried bodoatoaid 9*1 ,-^

.7 t J loxe

1 m - ac<ei lo d d d 1 ;qei tsvocffi nsvig cold?

fined 7.

,;.i-r iJBXifoldiaa a 10I godieddaoe

slcfit eoq e-xB . o'rlw Bjn-tfjoxc 1 -lirj,!.? ler'.ro arid 10I gciieddBoa laidaodoq ©rid

00 id: • ; - rdnoo sii'J si nod das! ait: bn-: parted .: bob Xaidnsdoc arid edn uoiqei

; . be gcJUdddeoa ©ortBoose-x lo

’odd vd bslloidaoo s<i LEJtw S Isvrl so c irtos-i ai d tsen . Idoea 3COip aiid lo

sad lo i -.1 .d ©0/ . ss'i da r. .’doea daoio arid To odd/.i adc- tax K-rxedeh

arid lo o.tdsi 8dT .eaolB gniisddBo::

+ !H +

o.rd•'•

sc oexiil? si Xav»I -^gisne i - Iox/ii .-• lo ridbiw ex^T*

.1 v* i d To id -111

28

the thermal region up to a few volts. The wave-length corresponding to

r such energies is approximately 3xl0“^cra. The resonance scattering will not

be very different from the potential scattering for slow neutron energies,

PnThere are a few cases where the ratio -jj— is quite large and the resonance

scattering will be larger in these cases. For example the A level of Ag

(22").1

In connection with the products of slow neutron excitation Bohr, in

1939, discussed the phenomenon of resonance in Uranium isotopes. The

probability of fission, according to his letter, being of primary interest

at the time. He suggested resonance capture of the slow neutrons by the

nucleus exciting the nucleus into a compound state quite like that suggested

by Grahme and Seaborg. The energy of the nucleus, being distributed in

such a manner so as to split the nuclear surface, would thereby produce

lighter nuclei and fast neutrons, accompanied with the release of a large

amount of energy. The particular number of particles and their binding

energy would then contribute to the probability of the reaction.t

Experiments with fast neutrons*^ 3how that there are reactions other

than potential scattering as predicted by theories. For fast neutrons, the

probability of simple capture is very small compared to elastic scattering.

Other types of possible reactions are inelastic scattering and the emission

of alpha particles. The inelastic process is unimportant where the levels

are small and the neutron energies are not too large (<4Mev). PotentialW

scattering contributes to the total scattering at these energies, 1.8 to

4.0 Mev, due to diffraction of the neutrons by the nucleus. The nucleus is

thus left unexcited in the process. Amaldi has shown that this type of

-SV4SW sriT .adXotr vrel x Xscnerid ©rid

doa XIJw ankwddaoa eorwaoeer ert? .txr>OI~0£x£ \lB&atnJxmqqR ei eeisiene rf<

.ssiyteo* • oiduen vola -ioI .ribieddsoa Xaidaedoq .d cart dno-xellib 'C-9V 9rf

-n

801X11089*1 ©rid bits eg'XBl edlt/D ai ~~ oid*-t ©id eisriw a.oe*o wel .« s-i.r e*i-riT

rel A sad- ©Iqmax© 10I *88880 ©aorid ai •tes'X*! dcf Iliw sufcsoddBOe

'

. ("£•;)

ni ,irioS acid1 diox© noidiien vole Jo adwbetc sxfd rfdiv aoidosri/r .*> nl

dflnsnU ai soasnose*! 1o nanec

j-Qf.TL iai 71 ; - .1'; 1c .’:* >{/ c-dcsJ siri od gnib'iooos i.sR.Li ~c ooi.r

arid- vd aaordiren voXa ©rid lo ©nJdqBO eoasnoe©*! b&S&e^uz eH .

bsdee^saa daild e*ll edi*rp at 00 » oditti eixvloxm add anidloxe essloxra

djjdjfcidaib pitied %8ifel!)vn ©rid Jo xgieae enT *giocf£er baa sodBoO

•ol ...•;•/ xfi:

’*t9i'id t’X/iow . 90^ine *TF:jXoifn 9iii dilqs od a- oh rtcr'xOT p rioj.2

000a t Boo*rdif©n dael bos ieloan isdi

- Co dTtfi ]1o -icdim/n 'rsXxroi^i«q eriT .XS^®«® danofiia

.noidfcBei rid Jo x&±IidB<Scnq ©rid od ©diKlfcrdflo© n

*i9iido aroido^wr en t©i d darfd voile <tVar 'idxrea de.sl ridJtv edr-a* fcteor^

diren da©4

! i</? . asi'iosrid X'd bsdoibaKj & -•:.

'*ifd ; ? ds oids :1© od b^rr.^'-too ' Isca 71 v si eii/d^xo elqf.-ir: *0 ;diXidfiootq

rsoieaicie ©rid hoe IrreddBOE oidfifileal e*B enoidc • Xdiaaoq 1c e©qxd *£©4^0

t

.• - pwi ©rid bos IXflfiia &v.‘s

.

... dj'

'

jo ©q^d eirid darid nworie end iblsmA .es&oo*iq odd ai :. :: i-- -X- d ->

-l'•

29

scattering is independent of the nuclear radius* For Beryllium, the poten-

tial scattering and elastic scattering are found to be of comparable magni-

tude for fast neutrons. Variation in the total scattering has been reported

in several cases showing resonance peaks* There wre no pronounced minima

and furthermore the minima are different on opposite sides of the resonance

peaks. This has been attributed^ to constructive and distinctive inter-

ference between the two types of scattering. The elastic scattering cross

section in the region of very high energies, tends to a limit of where

R is the nuclear radius. The ratio of this resonant scattering to the

potential scattering, while less than l/2 at exact resonance, cannot be dis-

regarded. However, there seems to be no exact agreement between the maximum

of the scattering cross section and the maximum of the probability of the

bombarding particle being in the nucleus. The two maxima are displaced

relative to each other by a distance equivalent to the level widths. There

is no explanation given for this occurrence.

Fast neutrons pass through Lead with emerging energies intermediate

between that shown by theory for inelastic scattering and elastic scattering.

Cross sections for inelastic scattering with large energy loss (2.5Mev) do

not vary with atomic number or atomic weight as predicted. This may be due

to resonance effects for the lighter elements, but does not explain the

phenomena for the close spaced levels in the heavier elements. For the

nuclei which have been studied, there is a close correlation of the inelas-

tic scattering cross section and variation with incident neutron energy.

Also the level widths are shown to be much too high to agree with present

theories. The cross sections for 2.5 Mev neutrons, show a marked variation

.es

-ced’oq Bdt ticulII^l6fl 10^ .etfJfcbiSX saeldtux axftf lo snciaidaoe

ei’S&trvL'OO 'to 3(1 o.t iV.'r't • T" snit9.+^ •-•OB oidaals bap z r-

bed'xoqei need aarf sntetftaMt Xjufa* edit aJt ao&ebutf .easn&UBa feel tol ebut

awfrrfm toOflUOOOTQ Ott. 9W fI9uT . .. B9<J MUBCD80X JfttVOlla

eoi -n . 2 e*i - rid- lo aefcie e.tisoqqo no nollxt etts w-init* ani

eoo'io gnhve.ik+Roe . uiT «g. .

9-i-©i •

t% ;:

\ lo . xmii. 5 0*1 Efcn •! «.=..- :gi9ne tr r1 v 1° n i‘3s '

1 ^ :j -£ flc -td>oe8

*

-eib sd *ofln*o ^eoaAaom tozxe te S\I uadi seel elJtifw t3ffi*i9idB0a IrxJnaioq

ed oj afiseea eratt ,«TO«oH •

9 ,, j- jc 'T+uteBiirrr: ' edf lo rai«iii>...s0 ont '.a.-- noid-ose eacrco ijnii9iwAoa and- 'to

. -f.J- ni snied slpitai

1

*’

, col • •

e&. lbsi Ini esxs'lan© gni^Tar. s dii' b--'©J ii^.i rotri«t s~c sts-.- ifisen &<..£%

. '±i~4&80B 0Jkteal* bn _ni'/ >dd .c, abtealsr -, •:.! ^tositd- ^d flworie tstJ nocwted

ob ) osol 9§Ti?I dd-Jb.' sniiedino. oitaalsni iol rnoxioee s-soi'?

jtriT .‘

- fcl*'• Piffled* io ladncra oifflodfl 1

odd ni >• Ccxe dm e b ,tud -

T r".»rr 5>r.«* ‘XdddgiX eiid* *ioi sine < eo/iKflo.-ia'X o&

, reivasd «ld ni sieve 1 bPP*'

-5 -

r

n.r eiid lo noiisle’noo ©solo b ci eien-i t beibif.le need ev.ni doxi-v ielowr

infejoii n;iv .ri£'i od rl^id cod .ioi® 9d od nvoria -exa edd.biv I-' cv ar'i cel*

I ,

••

30

from element to element and indicate other typical resonance behavior.^

There may be some probability of the incident neutrons reaction with

(p) waves of the bombarding nucleus. We are unable to determine at present,

the amount of this interaction. Also the combining of spins can take place

in other ways than those we have assumed. The problem for fast neutrons is

not unlike that for slow neutrons, if the (p) wave interaction is not taken

into consideration.

Grahme and Seaborg have suggested that a neutron scattering may exist

where there is an incomplete interaction between the incident neutrons and

the nucleus. An "intermediate compound state" which may exist for a time

before the more stable compound nucleus is formed. This would give rise

to scattering with a small energy loss as indicated by present experiments.

So far, the bombarding nucleus has been considered at rest with respect

to the incident particles. This is not the general, and certainly the prac-

tical case. The velocity of the capturing nucleus might be quite large

and the effects on the cross sections of nuclear reactions would be quite

noticeable. Changes in temperature can then be interpreted to give rise

to changes in velocity of the nucleus. The effect is noticeable in the

broadening of the resonance widths. In some cases this Doppler effect may

be larger than the actual level width in the nucleus. If this Doppler width

is comparable with the natural width, which is quite probable for several

cases, the temperature dependence of the cross section may be used to deter-

mine the natural width or the Doppler width if either one is already known.

^

.01

.

<Looas*iq t* Qalm&tsb ot oLd&ass sib sW .aweloxrxx &i±tn& 1o ssybv (q)

a> it xuso aniqa lo gnixiidffoo exit oaXA .aoltOBistixi ai-nt lo tenon-

.befcuesB erax :'bw nerito al

;_,9: - ton si ncitosiexnl vr ,r(q) silt lx ^Gn^itron voXs *xoi t xit ejLr-nr; ten

,xn Ltsxobionoo etui

tzhi'- \fM geinsttsOE non-tirex: s tsdt betEeggxre evsd giocfeae bn a exsrisiO

bn 'it : i :U^ a tnebiocJ: ant xivy/teJ noitos'ietni etsXqmooflx ns si • lerit evarfv

bxu/oqiffioo ^t.elb^a^Ietai ,, tiA .axxsloxm *. .

ssxi ©vis bluo-J aixfT .benrrol ax -sueloun bnwoqi»QO eXdste sioxa ant srrolad

•atcafiLTisqx® tneea^iq betsoibni sjb ssoX y^xsae XlBflta a dtiw . :

’

td9*i xitXV taei t‘

t

—o .*xq vdi yXci.ct'xoo ins tl ‘xens^ ant ton ax aim I •ssj.oitTBq kifi&Lix** a- *; 't

exit lo * x ;

•

.-.tinp ed bXxrov aroitOBSi isoXonn 'to anoitoea asoio exit no xtoolls ent bus

Soli- 9V.;i ot bet©’'.' tatei ed n xit nno srtntB'xex^ist ni 39gn. r0 .eXdoeoiton

a± ©XdBOOiton ei to© tie •axrdldxm exit to \txoolsv ni aagnado ot

X< a •. - l

i'r'ri .ixi- oa airlt II .axxeXoxm exit ni xitbi * level Xnxrtos exit x */It *xe§rroI ed

l£^ov©3 xol eld -d.-t. at .Loo si rioiriw t xitbi*: lyrirtsa exit xidiw ©Xdeisqxroc si

••.» ot S> m ©d •;• x. itoe . aao*ro exit to oo.iebnsqeb ©xntir: st ell « xe' ; o

.

31

III.

Conclusion

The phenomena produced by the bombardment of nuclei by slow neutrons

have been of primary importance in the development of the present theory of

nuclear processes. Much information has been supplied in quite detailed

form. The neutron and radiation widths deduced from slow neutron data pro-

vide the only available information for comparing the probabilities of

emission of various kinds of particles from the compound nucleus. The ratio

of neutron to radiation width may be obtained directly from the cross section

at resonances, provided the resonance energy is known. It is also quite

possible to find the neutron width by methods of Feimi and Amaldi.

The more recent resonance theory derivations of Fashbak, Peaslee and

g 10Weisskopf, is much like that of Kapur and Peierls in many respects. The

mathematical approaches are equivalent, though each started with different

assumptions. The former method offers a more easy interpretation of magni-

tudes involved. Qualitative conclusions can be drawn more readily as to the

behavior of the logarithmic derivative of the wave functions, than con-

clusions concerning a whole series of eigensolutions of a complicated eigen

value problem. However, the essential point of the approximation to reso-

nance by the two methods is the same.

Breit and Wigner^ gave a formula for the probability of a nuclear

reaction in terms of the virtual state of the compound nucleus. They

assumed only one important level, hence the applied name of one-level for-

mula. Bethe and Placzek'*’ ,^> generalized the one level formula to take accodn

of all possible resonance levels. This was analogous to that for the dispers

t

ion.

BOSTON UNIVERSITY'COLLEGE OF LIBERAL ARTS

LIBRARY

noiar'.Dr* I-

ola ’ S leZoca lo in-'-" wxsdtaoi ' e rfd > <?oubox< ’aainoxtadq ax.T

7o i\ -cij set • odd °> v: a n. r rid nr ort-..t*roq;ii rxBxiliq lo read sv d

• aeeaeoottq ‘lealairfl

-

lo asidiild dfv/iq erfj §xix*xn.,*noo xol noidrviolnl ©Id/ XX*v* vino arid eliv

,

....•' .* uxolw xo nojssI ttiB

X oi . i- <x OEO'io er i rnnl ^doei-ib ianlndlo ad ;r r rinbiv rxcic n llxrj; c-d cotissec lo

oalfi el dl •rworti al ^sisna e

noxdxrec odd bull cd aXdlaaoq

£wm eeXase? t2l£dda.flft' vlxsb vxosbd eoaaaon&z dnsoai ercoxn edT

i^ie ci baa *iiJCja2 lo diedd ejLil xioixxe si «lq<

. jda rioBf dj ,dx • tope ei£ aedoso'iqqs XBOldflmeddiMc

.- •

-noo ,rr-ridam;! own odd lo evidi.viieb olmridbi^oX add lo loJtvBded.

bfe.j- oiXq/txoo b lo QAoldfiXoBfid$X8 lo aei*rec sXoxb a gnlmeonoo enoicuXo

lo dnloq Xaldneeee .*' •

#en . si e: •. :.r« - d end vd 9.or 'n

sXoxxa s lo ^dilldadonq edd iol aXxnrrol a evag rtsngivr bna dicntfl

.

dr

i X.- -

-loi li 76/- J lo r- 9 .’ o.O0€x ;

tr©7 dm-sdnc. ml a,no {/xo bv/mea

Sr r

. 9 ' ?.>>,d od ( Jjiimcl XavoX ano exld besXJ>xexie$ 'xafesoaXl bi

; od aixo^olBCB aew eldT .a

.

32

of light and is known as the "Dispersion Formula". In all these derivations,

a perturbation theory was used involving assumptions that certain terms in

the Hamiltonian of the nuclear system may be small enough to be neglected.

Without these assumptions, there would be no excited state of the compound

nucleus

•

The chief difficulty with the dispersion formula is due to the strong

interaction between the reaction particles which prevents the calculation

in analogy to electrons surrounding the atomic nucleus. These theories pre-

dicted wrong effects in the far off resonance regions, which contribute to

the background or potential scattering, and are important in the consider-

ation of scattering considerations.

Recent experiments indicate that the present theories, while quite

satisfactory for light nuclei, do not explain questions of importance for

the heavier nuclei. The complexity of these heavier nuclei make formulations

very difficult. Only by an increase in the experimental data, can better

and more accurate assumptions be suggested and tested.

Itf-BV-treb eao/it IIb d • ‘•bImbseo'S £K>l8i«q8jtaM erid sb mold si bas id^lL lo

ni axinsd ni*dr&o darid enoi^crocraRB soivlovni baaxr sbw '?ro?r!d noiinJiif^zeq b

•bedoelsan 3d od rfsxronp Urraa ed xpJC Eeds^a issloxm erid to nBinodlimaH arid*

i;u;o r .o ©dd to od.da rsdxoxs on ed felwow ©n©dd tenoidqmtresB ©8©;1d dxioddIV

ga^.-ida ©rid od sub si Blumol nolens sib erid ridlv ^dlnoiltlb teirio < xiT

. 3d,

£i6v©*X£ rioirfw ssXoidiBci aoidoflsi scli ®9wt8d noldostsdni

• i• f

oi edntfl'idnoc ioxriv t air>igeT[ 9e»::an<r;i?i tlo nsl ex id ni sdosTte qnonv bedslb

-tobisnoe ©rid ni drad-rroqiri gib bin? ,31:' teddRoix Caldnedoq no iti.'oijplDBd arid

i

9,+ix p six o/nosrd dues* i: sr ' v«rid ^djsoibni adns ineq>* ' dnooeH

Ofiedioqifii. 1© anoidsei/p nlBlqxo don ob tiel9iin d^/xxl not ^xodoBlaxdBa

telox ’ to xdjbcelqjBoo sriT .ieloxm nsiVBari &d&

.;jjf r^; . : -.ineexs erid ni esi;9io«Jt os \d ^IijO .dli/oittib ypev

•bed8ed bus bedasasua ed ecoldqM/88B ed-ertxiooB aioin bos

33

Abstract

Nuclear Resonance Reactions

The modern period of nuclear physics began in the latter part of the

1920's, with the introduction of quantum mechanics to surplant the partially

successful Newtonian mechanics for nuclear reactions. The introduction of

the wave nature of particles, together with the knowledge of the concen-

tration of positive charge in the core of the atom, has been of considerable

importance in the development of present day nuclear theory.

It was soon evident that the one-body formula for nuclear reactions,

which was so successful in atomic collision problems, was inadequate for

nuclear reactions. The nucleus becomes and remains in an excited or compound

state for a relatively long time, thus giving rise to energy levels, the

presence of which affects nuclear reactions. The nucleus may take part in

reactions involving fission, light particle emission as well as gamma-ray

emission.

The determination of the spacing of the energy levels as well as of

their width, is of primary importance in the development of modern nuclear

theory. The study of resonance phenomena is of equal importance, since the

level spacing, as well as other useful information, is obtained from the

resonances

•

For the case of the incident neutron, the effect of the potential

barrier exterior to the nuclear core, will not be evidenced. However, for

charged particles, we have an additional means of probing the nucleus to

learn more of its structure. Many experiments have been performed which

supply information concerning nuclear barriers. Pronounced resonances occur

II —<im

dOBidt cfA

> ojtdC ,6 • SOI. ' V-V

edt ‘to J-i.l-j redds! 9 -i* al negsd aola^iq iPSIoim lo fcoiisq «ebo^ &dT

I oioHriosm jnudnaup lo noidoifbeadni arid riddy t e’0^91

lo ooJtdocfxrxdnl edT *eaojtdo*oa oaoloim nol aolm^osin cataodveH ItfSesoT

~0€Onoo add lo 9gt ®80d «eeIo±duaq lo &Z0dJMX 0VBV

^IctetsblaiioO 5o xisecf sad ,inod.B slid lo enoo odd al 9§'rarfo svxdxeoq 1c aol&srzi

, t • -

.ancldoserr inslona icl .filttfl'xol rboU- '-'lx J irfd ifisblvs soos 8bw dl

•jol edJBfipeiu aj OXW t Idonq floleiXXoo oJtaodb at Illl8B838U8 ot 6W noxdw

ancidoBen TtBsXoirn

C’.X <> ,. -s! v; i- “i od •.• li jni. ! eirr d < ri.' • oi ^lev-dalsa a *iol 1

'

I . •EodldoBso: TBeloun

;. ,s Hey -£ no1- nh a ©Jo2t*ifi *noIsa i:l '-cxvxovfii

.aoiae.a r.

lo t j. lew ei- alovyl v^-isre odd lo gnlob^s erfd .to noidsr-arrced >b eilT

•sa-jJaun rrce >e.m lo Xr-i .rlev- b t ni socndioorJ: ^xsirj*i., lo el <ddbxvr ilifrid

ont real benisddo al tnollapnolni Itflan nerido eu IXev e? t3nloaqa Xav* r

. Oi'•

•: r-

IPX ro:

t.'' lo 4-091 erf

'

t n« dne io -t lo eaao or' * no :

. " : v; od don IX i . o *xae.,. X pd •loi'isdxe *elm«d

us 9Vad ©vr t ESlo.r-j*X

» • nr.:. .sn/doirxde Bdi lo ©aom insel

oxrooo psorunoas'r beorujonoaT . -2i9x*nocf aaelotm , . ,'xolni xlqqixe

34

for incident particle energy below the barrier summit. This is evidence of

the existance of nuclear energy levels, as well as giving information as to

their widths and spacing. With charged particles, the barriers are in the

form of a smooth rising curve as the nucleus is approached, until the summit

is reached. There is then a rapid decline in the barrier as the nucleus is

further approached.

Incident particles which penetrate the nuclear barrier give rise to

effects not evidenced in an analogous fashion in atomic collisions. Experi-

ments show that the height of the barrier increases with the atomic number.

Resonance occurs with a constant barrier radius from element to element,

except at the top of the barrier where the radius appears to increase with

the atomic number. The addition of charged particles into the nucleus, by

bombardment, has been used to explain the increase in barrier height with

atomic weight.

|

The probability of an incident particle of an energy corresponding to

the resonance energy of a nuclear level is unity at exact resonance. It has

been shown by several authors, that the probability of a nuclear reaction

depends upon the incident particle energy. The one-level formula developed

and generally referred to as the dispersion formula, will hold for cases

where the width of the resonance levels is smaller than the distance between

them. Subsequent modifications of this formula were made to take in the

most general case.

We must assume a particle penetrates the nuclear barrier in order to

form the compound nucleus. The incident particles are not scattered as might

ordinarily be expected, this is evidence of nuclear reactions with incident

.

•Jo eoasbivs at eJtriT .dlxjuja TelTXMd ©rid volod XSP9£® elcx.^i sq JneMoat -xol

* rcolfiJt yLtvls a« XI«w es \arrec© tsaloira lo aoofidsl£9 ecd

ri d t:

di, ue s.ld Ii+rJJ t h- . 0"ci-;JB si anyone arid as ev-i ro ^nlei-x ridooma s lo ^ 1

•

*fo< iOBC' • lax'-d*'.

0j- . .'rsd iflelox/ri 9X3d sdctdeneq rioirfc- eeXoidisq dnsbionl

, f. .81 . tsootA nt noideal axrogoXjaoB ns nx beotieblrs don ad-coils

.jv eeaBaiofll i&twsd srid If dda*© •

•’

,f <.r-si3 od d-.i !:• fto'il ei/ifc»n isItisri dnsdanoo e /d xv -.11x000 soxix.ro:^!,

xi. J. a;-£ lonx v ©i/aqc * aribsi ©rid ei9j;:;..» iziiisd sxid lo . od arid d o dqeoxe

. riT .lediaxn olmods eiid

rldiw td^iari *telvied nx eessionl arid zttsXqxs od bosu xr-scf aci» %dnBiixbisdji

,.ti >.3V oJtfc^ds

->.t ihnoc ?exioo •v^iene ns lo ale 'disq dneblonx as lo ydiliclsdoaq •. a C

,• t

' '

noido^i iselonr lo ^dmdvrfoiq end darid t :iorfdir? Ibi©v©8 yp nuotit >.?&d

JbeqoXsreL alifinol Isrsl-sno eri? .•

' isqefc

eseso 10I bXorf XX±v , Ix/raiol nvleisqaxb arid as od be-rxelsi ^ rsien©3

90i, dsifc -.id ri: i»I1 bj.s al alovsl aormnoaoi end lo ridblv ©rid ^©riv

XlUboa dneixpoectoS •xnarid

*e-. so .: i9 X’f- * -•

-,w-

«. ‘yx.o i isin Tcsslcun ©rid ? -> 'eidono; 9loid%*q a 9fi;/eE» d'uoi sW

.- dnafeXoal ©riT .tmluni bxuxoqaioo erid «xo-:l

doe. ioxri xidiv uiv idoeai w Icxrrr lo 90flefciv© . c aliid t bedoeox© ©ri yJ.liBmdno

35

particles, an effect not known in atomic collisions. There are reactions in

which light nuclei are disintegrated when bombarded with high energy parti-

cles. When the incident particle escapes, we have scattering phenomena and

if the particle escaping has less energy than the incident particle, we have

what is known as inelastic scattering, the latter giving evidence that the

incident particle has given up part of its energy to the nucleus, leaving

it in an excited state. In the case of transmutations, the incident particle

would be absorbed, its energy being distributed among the nuclear particles.

Thus the atomic weight would be increased. Once the incident particle is

inside the nucleus, it will have little chance of escaping. Only when the

energy becomes concentrated in any one particle will that particle escape.

It is more probable that a particle will escape with less energy than the

original particle and the nucleus will be left excited. In the phenomena

of fission, tie distribution of energy among the particles is such that the

nuclear surface itself becomes disrupted. The probability of a particle

penetrating the nucleus and entering into these reactions is greatly

increased when the incident particle energy is within a certain energy

region. This region of "acceptance" by the nuclear energy levels is the

resonance region for that particular incident energy.

For heavier nuclei the resonance levels become more dense and

more closely spaced. It is more difficult therefore to determine the

structure of heavier nuclei.

Certain selection rules seem to govern many nuclear reactions,

and are much more stringent for resonance reactions than for non-resonance

reactions. This is a similarity to atomic collisions. Of the two general

tsllloo otmode al nwom ion

-idrraq d^iri rid.tv betriednod nsrtw J T.ngadrtxelb are iel IgJtX rio

: xiq sai*X9ddi!oa erart ev « 8eqao89 aloid'tBq doabxoni arid- xxaiiW «aaXo

evari ew taXol»txsq aoXqBOBe sloixisq arid 12

9! -ab.cvo §n2y23 *xeddBl 9d2 t^axi9»tdooe .

& -r<ieL t oan add od v;-x«r«p. odi lo dx"- qi/ navi •- -j eioi.-r :, dico . tori

iXoi 1 ,anoidBdiraaaflsrid lo7

.

,•

i • laalcxin add i-itwoxx Jexb v.niacf ygxans sdi t &9cfioe&a sd bli/ow

si ef.oi&ieq Srtebionl arid eottQ .beaBexoni sd bluow drf§2aw oi/noda arid sorfT

Bit# : 'XnO .^rtiqaoae lo eoitedo alddil eyad £22w di t eiraXaifa add ebieai

,« B9 b£o. hflq bond XXiw aloidxaq sno at bedBidaeonoc. >ad ^gtSM

{teas seal ddiv sqsose XXiw 9Xo2dxaq b d*rid eXdadoxq e*XO« i-;i di

BflSMHnflftrfg add nl . b*t.tox9 dlel 9<J XXiw ausloxin add boa 9Xoid*iBq I-8fl.b3.b10

odd dadd riora si Baloidxaq add grroaxs v- /v n :

’ lo

o1

. -/rri i vio-~q ••

. ixrxeib aamooed IXsadx 90B*bix/8 'taeloua

Xlie*T& "it aooidoBax aaadd odni gaixsdna baa auaXcim arid snlda^d9n9q

iiedxao 3 airidiw ai ^x&na aloidrsq daebioai add n*rfu boasal al

'

•

. .

' .

. ;/ r -

: x.’.ehioixX xsXiroidiBq drdd xol noigei ao.rr *x

o. r ~-.b art-..- • ©root-;' alavsl sorLoroesx --dd ieXoun rt3iv/ari •xol

-.id *.•*.>;'£» ;eb od arro'-.aiedd dJj/oilllb 9io/:i si dl .beoBgo ^Xssols 3-:oct

.i-eiox : T3X aar; -Tirdoircdf

, :n. 'dof.-n rtBaloWfl ;fl» --.a .rxs^o^ od mt>s3 -i8 cioldo&Ies ndsdieO

so. nose-i-ror? ool !. Jd sc »ldo arc aorrnoaei *xo‘t dr;e-^ai xd-.i arcoxi dom eie baa

.

36 .

types of selection rules, the general type is of most interest since it

includes more elements. However, the existence of useful selection rules

will depend upon restrictions on the orbital momentum of the incident

particles. Because of Bose statistics and zero value of single particle

spin, only even orbital momenta are possible. Only low momenta are permiss-

able below the barrier, while abdve the barrier large momenta are possible,

thus giving rise to many levels which are so closely spaced that they will in

general be undetectable. For resonance disintegration, parity as well as

angular momentum must be considered.

Work with fast neutrons seems to indicate the necessity of further

modifications of theory, since they show phenomena which apparently contra-

dict present theories. The necessary experimental evidence is not available

to act as a guide for more adequate assumptions.

Some authors have tried to illustrate nuclear reactions with an analogy.

to the dispersion of light by geometrical approximations. These hold true

only for a limited number of cases, since the nuclear reactions are not in

general analogous to the dispersion of light. The primary reason for the

failure of this analogy, being due to the presence of a strong interaction

force between tie incident particle and the nucleus in nuclear reactions.

•

£ '

td'i

oil? Ikot r 1 o lo si scTfrfr I oisnog sill » * *Xjrr noiloeXse *o asqvi

eonslsixe odd t*r©xr»woH .sdnscssls yron esbi/Xoni

inoi ioni Mv >o ... • L n-Hc stii no ejroitoiilson noqu bn? jb I »••

• -

eiitrceq sus slneiBOCi vol . . tcfcro n9vs nlrro t niqe

t r ‘.dioaoq sie .sln^ior •• nsl't'ijsd eiil svot' - sXiriv ^isii'too orf.fr wolsd ^X*. r

il Iixv xsril dr. fr baos .3 vXaooXo oa era rfoix / aXsvsI viw ol oerx orrlnri^ aunt

: fcijsq t jacii3Ta-9crul3ii) soxiBnosirc 'to'? .sXdaioeiehatf ed X^ia

.

tcerfii 1 ">o . r.jotjn s.rfi oi arose snoorlx/sn lasl rfilv *coV

, d , Jo anoliBoiliboo:

»xve Xsinsmileqxs •

. . .. r , :;j ; ,.r •• ..'

.1 . % - o jr : ..

'

• 1 - + r

sirxt bfori ssedT .anoii .-. 1 •.

'

'

ni vton sib aaoldossi 'xseloiwi ©rid so > bsllax. • lol vl/io

.

- '

rr Mo ? :sinl nnexta b 1q eonsesiq j oi 9ub gri^dt

" r ,oI -as airfl id surli**!

, obs*i issXojjn nx ausXoxrn add bna sloidnsq inooioni erf nsswtecf sotol

37 .

BIBLIOGRAPHY

1. Bethe, H. A., Rev. Mod. Phys. Vol. 9, pp 71, 108, 155, 208, 211 (1937)

2. Bethe, H. A., Phys. Rev. 47, 747 (1935)

3. Breit, G., Phys. Rev. 53, 506 (1940; Phys. Rev. 49, 519 (1936)

4. Bohr, N., Phys. Rev. 55, 513 (1936)

5. Chadwick, J., Proc. Roy. Soc. A128, 114 (1938)

6. Cockcroft, J. D., Proc. Roy. Soc. 192, 114 (1947)

7. Dunning, Pegram, Fink, and Mitchell, Phys. Rev. 48, 265 (1935)

8. Fechback, H., Peaslee, P., Weisskopf, Phys. Rev. 71, 145 (1945)

9. Fermi, E., Proc. Roy. Soc. A149, 522 (1935) J 146,483 (1934;

70, 103 (1946)

10. Kapur, P., Peierls, R. , Proc. Roy. Soc. A166, 277 (1938)

11. Kempton, Browne, Maasdorp, Proc. Roy. Soc. 157, 372 (1936)

12. Konopinshi and Bethe, Phys. Rev. 52, 122 (1937)

13. Margenau, H., Pollard, E., Phys. Rev.46 , 228 (1934)

14. Massey, R. , Proc. Roy. Soc. A137, 455 (1932)

15. Nucleonics, 1, (1947)

16. Placzek and Bethe, Phys. Rev. 51, 450 (1937)

17. Pollard, E., Phys. Rev. 47, 611 (1937)

18. Pollard, E., Phys. Rev. 45, 2118; 46 , 228 (1934)

19. Rasetti and Segre and Fink, Phys. Rev. 49, 104 (1936)

20. Rice, 0. K., Phys. Rev. 33, 748; 35, 1551; 38, 1943 (1929-1931)

21. Taylor, B., Proc. Roy. Soc. A134, 103 (1931

22. Weisskopf, V., and Ewing, D., Phys. Rev. 57, 472 (1937)

TOAHOOItiaiff

. . . .vaH , *A .

'?!)<-*

-

,r

. . i . y «. t®‘{ •

- .

’

. D •

, . . .ii t

.

. ... . t x .p

(dA^I) COI ,0V

.... . . , . . . . . i

'

v

• f

; ) SVC ,VCI *3o8 . .*t no .11

... }

...* .

'

'

. - t . 1 .

:V ‘

.•

• >.,

. .<- •’ .'I

(^eec) ass t a;. . . t .i .ar

( :

: ?i) y;X ,PA' r

u, '•. .

‘

.

-eser) uei ,ae «cc *a • • • . • i

.

.. . .

...