DEUTERON STRIPPING REACTIONS

ON CARBON AND OXYGEN:

A COMPARISON OF THEORY

AND EXPERIMENT

By

HAROLD ARTHUR MOORE

A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF

THE UNIVERSITY OF FLORIDA

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

August, 1956

ACKNOWLEDGMENT

The author wishes to acknowledge his preat debt to

Dr. r'. Y. Gordon, chairman of the supervisory committee, without

whose guidance and assistance this work could not have been car-

ried out. He also wishes to thank the other members of the

supervisory committee: Dr. .. C. Williamson, Dr. D. C. Swanson,

Dr. A. Meyrr, and Dr. C. B. Smith.

TABLE OF CONTENTS

Page

ACKNOWLEDGMENT . . .

LIST OF ILLUSTRATIONS . . .. ..

Part

I. INTRODUCTION. .. . .

II. STRIPPING THEORIES. .... .....

III. ENERGY DEPENDENCE OF THE ANGULAR DISTRIBUTIONS.

IV. COMPARISON WITH EXPERIMENTAL RESULTS. .

V. CONCLUSIONS . . ..

APPENDIX. ... .. . ..

REFERENCES. . .. ..

BIOGRAPHICAL SKETCH . . .

* 1

. 6

. 23

. 36

. 6h

. 66

* 7

. 76

LIST OF ILLUSTRATIONS

Figure Page

1. Simple stripping model. . .. 7

2. Stripping model showing momentum K. . 13

3. Variation of A with respect to x for various angles 30

i. Typical stripping peak i 6 versus 9 plot 32

5. Variation of the width of the stripping peak, W, with

y(Ed/0) . ..................... 34

6. Recitation function of the C12 (d,p), ground state

reaction, at G 300 plotted as G' versus EK li

7. Angular distributions of the C12 (d,p), ground state

reaction, at deuteron energies near 3 Y*ev plotted

as 6' versus E . .

8. Angular distributions of the 012 (d,p), ground state

reaction, at deuteron energies near h Nev plotted

as 6' versus . ..

9. Angular distributions of the C12 (d,p), ground state

reaction, over a wide range of deuteron energies

plotted as a/' versus E .

10. Angular distributions of the 016 (d,p), ground state

reaction, at deuteron energies near 3 Mev

plotted as 6' versus E ... 9

16

11. Angular distributions of the 16 (d,p), ground state

reaction, over a wide range of deuteron energies

plotted as 6G versus E ........... 51

12. Angular distributions of the 016 (d,p), first excited

state reaction, at deuteron energies near 3 Mev as

plotted versus E .. 53

13. Angular distributions of the 016 (d,p), first excited

state reaction, over a wide range of deuteron

energies plotted as 6' versus E 5

1h. Comparison of the theoretical angular distribution

curves and the 4.51 Mev experimental angular

distribution curve of the C (d,p), ground state

reaction, plotted as 4'9 versus Ex for a

radius of 6 x 0-13 cm. . ... 58

15. Comparison of the theoretical angular distribution

curves and the 3.43 Mev experimental angular

distribution curve of the O1 (d,p), ground state

reaction, plotted. s C' versus EK for a

radius of 5 x 10"- cm. . 61

16. Comparison of the theoretical angular distribution

curves and the 3.43 Mev experimental angular

distribution curve of the 01 (d,p), first excited

state reaction,pld ted as 6' versus E for a

radius of x 10 cm . .... 63

I

I. INTRODUCTION

The term "stripping" seems to have been coined by Serber (1)

when he attempted to explain the work of Helmholtz, et al (2), who

had found that the bombardment of various targets by 190 mev deuterons

resulted in a beam of neutrons of half-width of about ten degrees in

the forward direction. This beam was of a fairly well-defined energy

in the region of 90 Mev. Serber explained this beam by means of a

simple model in which the deuteron incident on the target nucleus

has the proton stripped off and captured, the neutron carrying away

half the energy of the incident deuteron. Serber's theory applies

equally well to the case in which the neutron is captured and the

proton carried away.

Others found similar results that agreed with Serber's theory.

Hadley, et al (3), and Bruechnor, et al (1), both measured the spectrum

of neutrons from a Be target using 190 liev deuterons and found agree-

ment with Serber's theory to within experimental accuracy. Chupp,

et al (5), made careful experiments on the spectrum of protons from

the (d,p) reaction with the same deuteron beam and a Cu target. They

also obtained good agreement.

Other experiments were done between 1949 and 1951 (e.g., Allen,

et al (6), Ammiraju (7), Gove (8), Schecter (9)) with energies between

the values of 1It and 20 Mev, measuring the angular distributions of

emitted particles of a greater or lesser unresolved energy range, and

the results compared with Serber's theory. The half-widths and even

the shapes of the curves showed violent fluctuations with the part of

the energy spectrum selected. Gove, for example, using lh X:ev deuterons

) on a number of targets (C, Al, etc.) found definite~ peaks for the carbon

reaction which did not fit Serber stripping.

The greatest difficulty that lay in analyzing the data obtained

from deuteron stripping prior to 1?1 was the poor energy resolution of

the emergent nucleon beams; until the energy resolution was improved to

the point that definite nucleon groups could be identified it was not

definitely known that each group represented an energy level in the

final nucleus. Burrows, et al (10), were the first to obtain resolved

3 angular distributions representing the protons from the ground state

16

reaction in 0 using 6 IHev deuterons. These distributions showed con-

centrations of protons at small angles to the deuteron beam expected on

Serber's theory, but the curve for the ground state reaction had an

unexpected decrease very close to the forward direction. The curve for

the first excited state, however, continued to rise with decreasing

angle down to the smallest angles measured. This can be seem in the

06 strioping curves shown in Section IV. Many similar results were

3 obtained after this by others with greater accuracy, especially by the

Liverpool group with which Purrows was associated (Rotblat, Gibson, et

al). It was data from this group that Butler (11) first analyzed using

his stripoin!: theory, satisfactorily explaining the forward maximum in

the angular distributions.

I

-1_

The stripping theory that butlerr developed has been an extremely

useful tool in nuclear spectroscopy. The most important application of

) his stripping, theory lies in the fact that it enables the assigning of

spins and parities to the levels of the final nucleus of the stripping

reaction. It is not always possible, however, to assign a single spin

to a level since the selection rules may be satisfied by more than one

value.

Since 1951, the Putler theory has been quite useful in analyzing

data. Holt and M'arsham (12), in particular, did extensive stripping

experiments with many elements. They were able to assign spins and

rarities to a number of elements ranging from Be to Sr managing

i to resolve proton groups which represented as high as the eighth and

ninth excited states of isotopes. Their results for nuclear spins and

parities fit well with the shell model theory of the nucleus.

Others who obtained Food results were Schectnr and King and

Parkinson (13). King and Parkinson analyzed the C15 (d,p) reaction,

subtracting off the background, and obtained an excellent fit with

theory. Schecter, one of the first to use the Putler analysis obtained

good results with a relatively high deut"ron energy of 20 Mev in the

(d,n) reaction of Pe9 Evans, et al (lb), also obtained good results.

Although the butler theory has had extremely good success it has

had failures. The theory breaks down for the case of the neutron ang-

ular momentum 1 = 0 when the energy of the neutron, En, is approxi-

mately equal to the binding energy of the deuteron. This means a neg-

ative Q value since En = (Q + ed), where ed is the binding energy of

na

the deuteron. An example of this is the analysis of Middleton, et al

(15), of the C12 (d,n) reaction. The results are completely anonl;lous.

The nearest fit is obtained with an 1p = 1 curve (it should be renem-

bered that the stripping theory holds for either a (d,n) or a (d,p)

reaction). The otherwise known spins and parities of the levels in

this case taken together with the selection rules require that 1 = 0.

P

But not all the failures are for the negative Q values. Evans and

Parkinson (16) in taking the angular distributions for the 110 (d,p)

reaction noticed that the first excited state reaction for which Q is

7.13 yev does not fit the required 1n = 1 curve. Also there is an

energy dependence that is odd in this distribution. The curve moves

in closer and closer to smaller angles as the energy is decreased.

Another puzzle is that of the F 1 (d,p) reaction. The strip-

ping reaction fixes the spin of the ground state of F20 as unity. This

value, however, is in disagreement with data from beta decay.

The Butler theory is not the only theory that has been presented

for stripping reactions. The others, notably the ones of Bhatia, et

al (17), and Daitch and French (18) have been successful but in es-

sence are the same as the Butler theory, the only difference being in

the assumption of the region of interaction or extent of interaction of

the captured particle with the target nucleus. The "Bhatia curve" ['ives

essentially the same type of distribution as the "1Butler curve" but the

radius required to fit the Bhatia theory is always approximately the

order of 1 x 10- cm larger than that required for the Butler theory.

The Daitch and :rtench theory is the same na the Putler theory except

that there is a multiplying factor which removes one of the zeroes

of the Butler distribution.

The Butler theory completely neglects such effects as coulomb

interaction and resonances. Tobocman and Kalos (19) have extended

the theory to include interactions with the target nucleus. Others,

such as Glauher (20), who included diffractive effects, and Thomas (21),

who writes the general matrix element for the interaction to include

other effects, have not altered the basic idea of stripping.

Even though the butler theory has had remarkable success in

explaining stripping phenomena and correlating with nuclear shell

models (up to 016 the agreement is complete) the fact remains that the

agreement with experimental data is !,ood only in a very limited region

of the forward peak of the angular distributions. Yost of the angular

distributions have been taken at a single deuteron energy and it has

never been shown that there is any consistency between theory and

experiment at different deuteron enorries.

It is the luroose of this investigation to attempt to determine

the energy dependence of the angular distribution in stripping reactions

and to investigate the internal relationships of the simple (Butler

t:rpe) stripping theories so as to develop a validity criterion for them.

Further, it is our purpose to apply such a validity criterion to perti-

nent data in order to determine whether the energy dependence predicted

by the theory is correct.

II. STRIPPIr(, T'EOl'IEES

A. Simple Model.--Throughout this work we shall consider only

) the (d,p) reaction. Also, we shall let" 1.

A deuteron of momentum kd is incident on a target nucleus, A.

The deuteron, passing close enough to the nucleus, disinterrates. The

neutron is captured and the proton proceeds with momentum k The

assumption is made that the proton does not interact with the nucleus.

This non-interaction characterizes the reaction known as the stripping

process. In a non-stripping or compound nucleus process the proton

would be absorbed into the target along with the neutron and then

3 re-emitted.

The neutron proceeds toward the nucleus with a morrentum given

by

-- p (1)

or

Sp= d 4- _.P ^^ CL 4 (2)

where 9 is the angle between the direction of the incident deuteron

and the direction of the emergent proton (see Fig. 1).

According to the shell model theory of the nucleus, if the

neutron is to be captured by the target nucleus it must be captured

with quantized angular momentum 1n such that the conservation of

angular momentum law is satisfied. That is,

Fii'. 1.--'iimple stripping model.

-7-

0

J f- h + ^ (3)

where J. is the spin of the initial nucleus, Jf is the spin of the

final nucleus and 1/2 is the spin of the neutron.

Classically, for those neutrons having impact parameter R equal

approximately to the radius of A, we should have

Thus, for a fixed kd and k it would seem that equation (2) could be

satisfied only for some definite angle 81 However, even classically

some captured neutrons will have smaller impact parameters than R and

quantum effects will permit further latitude. This means that the

angular distribution will not be concentrated at a single angle but will

actually be a distribution. The cross-section for the reaction will

contain a factor representing the probability amplitude that the

neutron possessing linear momentum k is found at the nuclear surface

with angular momentum 1 .

B. General Expression for the Cross-section.--The formal theory

for the stripping cross-section is derivable from the approximation

methods for time-dependent problems in quantum mechanics.

Consider a deuteron incident on a target nucleus which is

represented by a potential region V(r). The proton proceeds unscathed

at an angle 0 to the forward direction while the neutron is captured into

-0-

the potential region in a bound state. The differential cross-section

is proportional to the transition probability per unit time per unit

flux between the initial and final state of the deuteron as it enters

Sand leaves the potential region. Since the incident flux is propor-

tional to the velocity of the deuteron and hence to the momentum kd,

we can write

(T -T- / (5)

where W is the transition probability per unit time between states.

The transition probability, CI is proportional to the density

I of final states of the emergent proton, (A p) and the square of the

matrix element of the perturbation, H between the final and initial

states of the system. Then,

Sp) I [s A (6)

where

The and are the wave functions of the final and initial states

I of the system. The cross-section can now be written in the form

a 0 I | / (8)

-10-

Since )f(Kp)is proportional to the final momentum of the proton, kp,

GC&) c< K:'r

Assuming the nuclear spin of the initial nucleus is zero, we can

write, following Daitch and French,

d : (Jw I i (10)

?nPP d 7" M )'

lere, md, mp, mn are the spin magnetic numbers for the deuteron, proton

and neutron; m is the orbital magnetic quantum number for the captured

neutron. Since the incident particle and target nucleus are not polar-

ized, 6 is averaged over the initial states and summed over the final

states. The sum over the spin quantum numbers rives simply a factor of

three such that the cross-section becomes

eE-n I / *(11)

where m' and m_ are the reduced masses of the deuteron and proton

respectively.

It is necessary now to arrive at an expression for the matrix

element. We assume the neutron is captured under the influence of a

potential (assumed central) given by V(r) into the one-particle state

with space dependence

m~ Ri)~(~p

(12)

-11-

and with binding energy

tX

EE zrnr (13)

where m~ is the reduced mass of the neutron.

':e assume that as an approximation the initial wave function of

the system is given by

S- N e (14)

where s is the internal coordinate of the deuteron given by

5 S (13)

and

,e-4

1- ^(16)

is the center of mass coordinate of the deuteron. The final wave function

of the system is given by

(17)

The matrix element is then

H, = V(r)

If we make the substitution

P (19a)

I

-12-

and

.r .= r + -S (19b)

we arrive at

Lr: (APY^ 5

S- ) V() IJA-) d (20)

To understand equation (20) more clearly we refer to the simple model of

the stripping process (section A) and consider it as a two stage process.

Firstly, the deuteron disintegrates and, secondly, the neutron is cap-

tured by the target nucleus. When the deuteron disintegrates the neutron

has momentum K with respect to the center of mass of the deuteron. This

momentum is given by (see ?ig. 2),

p (21)

or

e m ntm of te capted ntn (22)

The momentum of the captured neutron is

W = ld i(23)

Fig. 2.--Stripping model showing momentum K.

-13-

-lh-

Thus we see that we can rewrite equation (20) in the form

Hl D( (24)

where

D (25)

and

/ Ci( fd- )'^ -* (26)

D(K) expresses the probability that the momentum K will be obtained from

the disintegration of the deuteron. G(k) expresses the probability that

a neutron of momentum k arrives at the nuclear surface with the proper

angular momentum 1n to be captured.

The cross-section is then given by

6 9 OC I 4) )(L () (27)

Before equation (27) can actually be used for computation a

correction must be made to the neutron momentum. The integral in equation

(26) is taken over the neutron coordinates and not over the coordinates

of the center of mass of the whole system. If the target nucleus were

-15-

infinitely heavy this would be all right, but since we cannot assume

this we must correct for it. Allowing for the finite size of the

nucleus the momentum of the neutron is seen by the center of mass of

the final nucleus is

-J h i= p -(26)

(see appendix for further treatment).

- I ( (29)

7n1.F '0

C. The Deuteron Factor, D(K).--As was mentioned in the previous

section, equation (25) expresses the probability amplitude of finding a

proton or neutron of momentum K in the internal motion of the deuteron.

Using the Hulthen wave function (22) for the deuteron, which is given by

)d =s- [ S

+ as ]

(30)

and where

'-~---

equation (25) can be evaluated to give

Do<) ,'rN [7T& N-

where N is the normalizing constant.

/ / = 70C

(31)

(32)

-16-

D. The vessel Function Factor, G(k).--The integral in equation

(26) expresses the probability amplitude for capturing a neutron into

the state f r). Ty using the expansion

e- (a1PJ (33)

where Pl is the Legendre polynomial and jl is the spherical bessel func-

tion, the integral can be evaluated to five, taking the axis of

quantization along the neutron momentum victor,

This equation can be put into an alternative form. This can be done by

getting rid of the V(r) tenr,. To do this we write the radial equations

for the wave functions. For the radial wave function outside the well

[ +-1- w = 0 (35)

Inside the well

( U R,() = 0 (36)

Ot~ f \ r .1r I<

multiplying equation (35) by RL(r) and equation (36) by jl(kr) and

subtracting, we obtain

-17-

+ t R,,) + M R (. *) (37)

We integrate over r, noticing that the first two terms drop out leaving

us with

ccf

a -

(-r).,r, 0 ) r,(38)

where m' is the reduced mass of the neutron. If we place this result in

n

equation (3h) we get the alternative form for G(k),

(39)

E. Bhatia's Theory.--To obtain Bhatia's result for the cross-

section we evaluate the integral in the G factor by making the assumption

that the interaction of the neutron and the nucleus occurs only at some

particular R which is equal to the nuclear radius plus the range of the

nuclear forces; that is, it is a delta function interaction. Thus, for

Bhatia's theory

V (e) R

-18-

or the cross-section is given by

6(G) DK )( )\ (hi)

F. Butler's Theory.-- To obtain the Dutler form for the stripping

cross-section we again start with the G factor as it is given in equation

(3h). Following the same procedure outlined in part C to obtain the

alternate form for the expression of G we arrive at the equation

( U() (Ar)R ) (42)

However, instead of integrating this equation from zero to infinity we

follow Butler's assumption and neglect the overlap integral from

r 5: r where ro is the nuclear radius. Thus we arrive at

-- -(, -

0 (43)

The left hand side of the equation may be evaluated so that

I

-12-

-i R) ~l

00

ao

-) R

M0

The right hand side is seen to be just the alternative form for G(k) so

that, disregarding constants

In this case

Rt = C^ (h6)

where h1 is the spherical hankel function of the first kind. Then,

6(s) (K)

1 ( )1-4 ( (047)

G. Daitch and French Theory.--Tf the contribution to the over-

lap integral is not omitted for r 5 ro, the Daitch and French form for

-20-

the cross-section can be obtained. We define the quantity, which is

the equivalent of a square potential well,

V(r) 2

(48)

Then it can be shown that the G factor is given by

G () O< [l -

Since the Putler form

gral for r ro, the

term for G. Thus the

equivalent to that of

D !_\ i* t. f .

)( 9)

for G ignores the contribution of the overlap inte-

integral in equation (i9) is precisely the Thtler

matrix element in the Daitch and French form is

Butler except for the multiplying factor

S (0o)

This extra factor does not disturb the essential features of butler'ss

theory (the forward maximum) but it does remove one of the zeroes of the

distribution. This occurs when the neutron momentum k equals an average

wave number which a neutron could have when bound with binding energy

Eh

-_.

- t /

(51)

L, Rr) V6C) P ) lr.

I / 0 '%

;1 -/1

-21-

and with orbital angular momentum 1 in a well of radius r The

n o

Daitch and French cross-section is then given by

3 8) F (52)

where 68 is the cross-section obtained in the Butler case.

H. Serber StrippinE.--Tt can be shown that the expression given

in equation (27) is not inconsistent with Serbpr stripping, i.e., high

energy stripping. Consider, for simplicity, the case of the Dhatia

theory in which case the cross-section is given by

6(G) (53)

where D(K) is given by

1(K) =a L i (54.)

Since & is large ( 8= 7<(), we drop the second term in the brackets

for an approximation. Since high energy stripping is concerned with

energies of the order of hundreds of !Iev, the energy of the emergent

proton will be large. At this energy it is difficult to resolve the

spectrum into a single energy. Thus, since the nuclear levels are

relatively close together, as compared to the proton energies, the G

term in the cross-section will be the sum of a number of adjacent

levels which smooths out roughly to a constant. Serber cross-section

-22-

becomes, approximately,

6 L(55)

which is the form of the observed distribution in high energy

stripninp.

D

III. E;IERGY DEPEn.'DECE OF THE ANGULAR DISTRIITUTIONS

AND VALIDITY CRITERION FOR T'HE

STRIPPING TIIHORITS

A. Validity Criterion.--The stripping cross-section as given

in section II is expressed in the form

) (56)

where D(K) is the deuteron factor and G(k) is the neutron factor. Only

those factors which do not affect the energy or angular dependence of

the cross-section are omitted from equation (4l). We can show that

G(k) is actually a function of K. We start from the equations for

K2 and k2 as given by equations (29) and (22),

J{L =r ^ ~+ CCA Ak c- (57)

K = _+ Id ) e- (58)

where c = mi/mf. If we multiply equation (58) by the factor 2c and

subtract it from equation (57) we obtain

-ck =+ (C-aI (59)

Writing equation (58) in terms of the energies we get

A'- ack = (1- )i'E c c- n E (60)

-23-

-2L-

where m' and m' are the reduced masses of the Droton and deuteron.

p d

From the conservation of energy equation for the reaction we know that

E,

=E, +q

(61)

Substituting this in equation (60) we obtain

-o a= rl- E t (c'- C) (Eq) (62)

or

2-c 1

=Z ., i(- c -cE

(63)

where

7n, = 77 t 77^. = Ire7

is the total mass of the system.

Since c = m./mf we can write

i f

-,C<

(6h)

Thus we see that the coefficient of Ed cancels out and we are left with

ft^ )

-= ^ {[r ; (

-25-

the relationship

? C)9 (65)

The neutron wave number, k, is then directly expressible in terms of K.

The stripping cross-section can be written as

F(K)^ (66)

and for any particular reaction is only a function of k kd and K. If,

however, we define a cross-section such that

w 60D) (67)

we see that

6 o F(K) (68)

W:e now have a validity criterion for checking experiment with theory. If,

instead of plotting ( versus 9 as is done in angular distributions

or 6 versus Ed as is done in excitation functions, we plot GI versus

K, then all curves, either angular distributions or excitation functions

for a given reaction should coincide. Also, since G/ is a function of

K only we can investigate the energy dependence of the angular

distributions by investigating K, which is a function of both Ed and .

An expression for the energy dependence of the width of the forward

maximum can also be obtained in the same manner.

P. Energy Dependence of the Angular IDistributions.--Pecause it

is easier to use energies than wave numbers in actual practice the

remainder of this work will be done in the variable Fk rather than K,

where Fk is defined by

E -

Bk (6 A)

and is an energy associated with K.

If we pick an Fk on the 5 versus Fk curve this will be a

constant no matter how Ed or S varies. Tf the deuteron energy, Ed,

is varied, 9 will have to change to compensate for it. :.e make use of

this fact in investigating the shift of the maximum of the anrulur dis-

tribution (6 versus ) with respect to a change in F..

2

'We return to the equation for K which is ;,iven by equation (58).

'-e can rewrite this in terms of energies as

Y1 Ek

-= I'r, E

- I' E

d4

- E)f ) E

(70)

where the primes denote the reduced masses. Using the conservation cf

energy equation

EP PE + (71)

(69)

-')7-

we get

I

EI

CO- &

(72)

Simplifying and setting

E

6, = we obtain

;L

+ ct9 -V

C .(73)

Since Ek is a function of both Ed and a we can write

dE,- -w + dE

If we choose Ek to be a particular point, we have

- dJe

cLE0

+ E dE

SEK

Ex

eE

(76)

(74)

(75)

= ;( oE +)

-H^

P i

= (-+)F +E

P(1- + Q)

-28-

From equation (73)

E E(Ed ( + ) (77)

and

SE E CE q ) A s F

Then,

(aE,+) '. .

de, VS~ VE (Ed+(?)' ^ (79)

Since the sine is always positive in the first or second quadrant, the

denominator of (79) will always be positive. It is necessary to inves-

tigate only the numerator of equation (79) to determine the sign of the

variation of with Ed.

Since the cosine is negative in the second quadrant, the numer-

ator will always be positive for angles greater than 900. Thus E

will be negative and the portion of the angular distribution beyond 900

will move in toward smaller angles for any increase of Ed'

For angles less than 900 it is simpler to plot the numerator

for various angles and determine the energy at which it changes sign.

To do this we set

-29-

A =-- o--k Ed # o

E (E4 -f-

For convenience we change variables. If we set X r: E.. we obtain

C;

The variation of A with x for various angles is plotted in figure 3.

As an approximation a is taken equal to one-half.

From the ;:ranh we see that for =0 0 A is negative until

x equals one; that is, until Ed = C. Thus would be positive

(the angular distribution would tend to shift to higher G ) until Ed

equalled C, at which point A becomes positive and consequently

negative. The curve would then tend to shift toward smaller ) with

increasing Ed.

The same type of behavior occurs for = 300 and = 600

except that the cross-over point from minus to plus for A is pulled in

toward lower deuteron energies (smaller x). The cross-over point, and

hence the sign reversal for the 9 = 300 curve occurs at approxi-

mately x equals .A; the cross-over point for e = 600 occurs at

approximately x equals .05.

12

Since for the C12 (d,p) reaction the forward peak (for the

ground state) is between 25 and 30 degrees this means that the peak

should move out in the angular distribution until Ed = .hQ or 1.1 Ieov.

Fip. 3.--Variation of A with respect to x for various angles.

-30-

-31-

For ,d greater than this value the peak should move in toward smaller

angles.

Since angular distributions below about two Mev are unreliable

due to high background from coulomb interactions, for all practical

purposes the angular distribution curves would not be reliable in show-

ing this reversing trend. There should be, a slight noticeable shift

toward smaller angles for energies greater than about 2 Mev, but even

this might be difficult to detect since the A versus x curve flattens

quickly after crossing the axis.

C. '.!idth of the Forward maximumm V 'rsus Ed--To obtain an expres-

sion for the variation of the width of the forward maximum in the

angular distribution with respect to 7Ed we consider the curve in Fig. L.

G, and 9. are the angles associated with the half-value points.

t:eturning to equation (73) we can solve for cos e We obtain

cao + E4 + ao (82)

For O9 and 6. we can write

Cd. L t=+ )d -t 0 fE (83)

and

SE a- -+ a.

TU TT EI -f-

Fig. h.--Typical strippinrg peak in 6

versus Q plot.

-32-

-33-

Subtracting,

CI- C. a- E

ra V, E4 (E + 9)

(65)

The above expression is not an expression for the half-width but it does

show how the width of the peak varies with Ed,. Note that E and E

are constants.

'e see from equation (85) that the width of the peak is inversely

proportional to the square root of E4 (Ej + Q) and is always negative.

Thus, the peak sharpens with increasing deuteron energies. To plot

equation (63) :.e change to a new variable, and let

SQ

W =. Q9 can le

Tig. 5 is a plot of the width, '', versus y where '.: has been

normalized to the value it would have when Ed = Q; that is, y = 1 when

d = C. The fonrard peak sharpens quickly with increasing y (Ed).

At y equal to approximately 2.', the width of the peak is already half

the value it would have at the Q value. By the time y reaches 8 the

peak width is almost at a minimum.

D. Summary,.--If the angular distributions and excitation

functions obtained from stripping experiments are plotted in the new

variables, 6 and K, they should coincide, if the simple stripping

theories are correct. There should be only one characteristic curve of

6 versus K for any given reaction no matter what the deuteron energy.

Fir. .--Variation of the width of the stripping peak, i,

h with y(Ed/Q)

-1r-

The angtlnr distributions plotted in the 6 versus & variables

should be observed to shift with an increase in deuteron energy. For

& less than 900 the distribution should shift toward larger 6

until the denteron energy reaches the 0 value. At that point the dis-

tribution should reverse and move in to smaller ( with incrcasingF Ed.

The portion of thl anfulrar distribution beyond 900 should always move in

to smaller 9 with an incre-..se of !.

The width of the stripping peak should decrease with an increase

in the deutrron energy. At deut"ron enfrr'ic's around the Q value the

width decreases rapidly with increasing d. At about d = Q the width

is apnroxirnstely its minimum value.

IV. CO':PARISON 'WITI EXPi.:JIM"'TAL i:I'JILTS

A. Introduction.-- As was pointed out in section III, 6 is

a function of K alone. Thus, for the same reaction, all experimental

angular distributions plotted as 6 versus 9 for various values

of d should coincide when renlotted as 6 w versus K. Also, the exci-

tation functions when replotted in the 6 versus K variables should

coincide with the angular distributions. Ideally then, there should be

only one characteristic curve in the new variables for data taken in a

reaction at any dcutoron energy.

The reactions for which enough data was obtainable to provide a

check with theory over an energy ranFe were the C12 (d,p) ground state

reaction, the 016 (d,p) ground state and the 016 (d,p) first excited

state reaction.

The useable data for both the oxy;ien and carbon reactions was

limited in quantity. Firstly, much of the work done was not expressed

in absolute crosfc-sections which made it impossible to use for compar-

ison purposes with the work of others. Also, data was needed at many

deuteron energies so that a check with the validity criterion over a

ranre of energies could be made, but very little work was done at

energies higher than 8 'Mev, except for the extreme range in the hundreds

of Mev wh're Lerber Stripping is characteristic. "uch of the work done

below 8 'Iev was done near the coulomb barrier. At energies near the

coulomb barrier it is difficult to interpret the data in terms of simple

-36-

-37-

stripping. Compound nucleus effects and coulomb effects destroy the

definite stripping characteristics of the angular distributions.

"or example, the data of Terthelot, et al (23), for the oxygen

(d,p) rcrction is Food but the deuteron enrrry range is from 1.66 to

2.2 :Tev, which is too low; also, he does not [give absolute cross-sections.

Grosskreutz (2h) data is given in absolute cross-sections but again the

energy ranre is too low, between 1.05 and 2."1 Mev. Since the coulomb

barrier for oxyi'en is about 3 V.ev, this data is unreliable for strippinF.

comparisons.

The oxygen data used was taken from the work of Stratton, et al,

(2"), Freemantle, ot al (26), and !urre, et al (27). All curves except

the 3.29 oev curve of Freemantle and the 8 "ev curve of Pur;e are from

SfStratton. His work covers a range of douteron energies between 2.25 and

3.8 :8ev with an excitation curve being taken at 530 (center of mass

coordinates). The data for the first excited state reaction of oxygen

is also taken from his work.

even less usable work was done for the carbon reaction than for

the oxyrnn reaction. Again the difficulty is that the cross-sections

were not given in absolute units or that the energy ranre of the dou-

teron was too low, as in the case of Koudijs, et al(28). His range of

H deuteron energies is from .2 to .(6 V'ev.

The carbon data used here is taken from the work of Ponnerr, et al

(20), iiolmgren, et al (30), and Rotblat, et al (31). The work of Bonner

is excellent in that he works in the deuteron energy ranure front 1.8 to

6 Myv. Also, he takes excitation functions at three angles, 30, 90, and

-38-

15;0 degrees. I'ow'ver, only one of the. excitation curves, th:t at 30

degrees, is used for coniparison nurposes.

P. :nelotting of Data.--All data was transferred into the center

) of mass system. exceptt for the case of the 300 excitation curve of

lonner, all the cross-sections were already in the cent(-r of m.s sys-

tem, as were the scattering angles. The deuteron energies were always

riven in the lah system and had to be transferred into the center of

mass system. The deuteron energies were translated to the new system

by the use of the relation,

E P. (86)

where Ed is now the center of mass energy.

In the case of the 300 excitation curve of nonnor, it was nec-

essary to translate the lab cross-section into the center of mass system.

To do this use was made of the relationship

6 1 +I Y +a 6 (67)

IB Y CX-e

T.ie center of mass anfle was obtained from

)

-39-

where

(mnP

li\

"!ith all the data in the center of mass system, the experimental

data was renlotted. 6 was obtained from

(90)

where

A4

(7.)

Instead of K, Ek was used as the other variable. !,k is the

energy associated with the momentum K and is given by

E ^ ( E +-Q) +

K My-q) I

U&i (9. (92)

(S-1)

For an angular distribution of given rd and C, the values of

were merely substituted in the equation and Ek obtained. !Xcitation

functions were replotted using the same equations.

C. C12 Data.--The Q value of the ground state (d,p) reaction of

C12 is 2.72 Mev. Using this value in equation (92), we obtain

(89)

--

P

I

i t 4-

3m;

)7)F

-\r \I ; (

V^ L J :. Ld +L

-';0-.

E .792 E 1-472 -.736 ECEt cG ((3)

I'rom eonultions (90)and (91) we have

E6-

'ig;. 6 is a renlot of the excitation function for the carbon

reaction taken from the work of "onner and extends in enrrry ran:-r from

Fd = 1.8 to "d = 6 -'ev at an anfle of 300 in the laboratory system.

& ~ven thouFh Ed was taken over a fairly large ranre of values, Fk, when

com:nuted, is over a short range. This is seen from equation (92).

rk is relatively insensitive to 7'd at small a.ngles. N;o stripping peak

would he apparent due to the limited ranro of Fk. The total ranre of

the excitation curve (:i;:. 6) as compared to that of an an;gular distri-

bution plot is shown in Fig. 7. The shaded section is the excitation

function. The dashed line shomn in 'ir. 6 is t.e average over the

resonances and, if the simple stripping theories were correct should

follow the replotted angul,.r distributions exactly.

It is immediately apparent that the stripping theories are inad-

equate since there are a number of resonances showing on the excitation

curve. The simple stripping theories make no mention of resonances.

The resonances shown in Fir. 6 are apparently from the compound nucleus

,'1

Fir. 6.--'xcitation function of the C12 (d,p), Fround state

reaction, at G = 300 plotted as 6 versus E .

-!.1-

00

O

0 o

C' II

0 0

-7

/

(H31S/IIN) _D

'ig. 7 is a replot of the 2.8L ?ev, 3.02 Mov and 3.20 aev curves

of Bonner, et al. These curves bracket the 3 Yev resonance in if as is

shown by the variation of their maxima with Ed. This 3 Mev resonance

) peak can be seen in Fig. 6. It is the larfre peak occurring at approxi-

mately "k = 1.22 Mev. The dotted line in Fig. 6 corresponds to the

bottom of the shaded portion seen in Fir. 7. Theoretically the excita-

tion function should have the same slope as the angular distributions

when plotted as 6 versus F It does seem to exhibit this charac-

tristic to some degree.

The thrre curves in ? i. 7 do not coincide as predicted by the

simple strinoing theories. At the sane Ik, the forward maxima do seem

to he closely aligned. The secondary maxima seem to shift out with

higher Fd. This would tend to indicate a departure from stripping .

These peaks, are probably due to other processes than stripping, most

likely from compound nucleus formation. '.e are most interested in the

forward or stripping peak.

FiR. 6 shows the L.20 "ev, L.09 :ev and 3.75 "ev curves of

Ponner replotted in the new variables. These curves center about the

Lh.0 Mev resonance of ;M. The same generall considerations that apply

to Fip. 7 also apoly here.

9onner attempted to make an analysis of this L.00 !iev resonance

to determine the origin of the resonances. HIe assumed the cross-section

to be the square of an amplitude that had two parts, a non-resonant p ;rt,

6B and a resonant, 6r which obeyed the "3rcit-'.ine(r formula.

lie assumed that thrre was a phase difference, S( ) between th;e two

Fip. 7.--Angilar distributions of the C12 (d,p), ground state

reaction, at deuteron energies near 3 Mev plotted as C versus .

0

CI 0

(U31S / BiN) .

I

12

Fi,. 8.--Angular distributions of the C (d,p), ground state

reaction, at deuteron energies near h Mev plotted as G versus E.

L

O0

0S

(81

components. According to Bonner the cross-section could be written as

6(E, 01 6 (Ea +. r
+ _L- (93)

where E is the energy of the resonance, and 1 the width of the

peak. An analysis of the h.OO 1eov resonance data of the carbon reaction

indicated that the resonance was one in the compound nucleus N1l

Fig. 9 shows a comparison of the l.51 Mev curve of Bonner with

the 8 Mev curve of Rotblat and the 3.29 M'ev curve of Holmgren, et al.

The 4.51 curve of Bonner and the 3.29 Mev curve of Holmeren were used

for comparison purposes because both of them are off resonance. The

intention was to obtain an energy spread to determine the extent of

correlation of the curves over a range of Ed. Again the shaded section

is the averaged-over excitation function, the one mentioned previously

(see Fig. 6).

It is interesting to note that the h.51 Mev curve of Bonner has

a secondary maximum that seems to be out of place. The first sugges-

tion of this maximum occurs in the 3.29 Mev curve in Fiu-. 7 and proceeds

to become more pronounced with higher deuteron energy. :Unfortunately

there is no data available about the h.51 i:ev curve to determine the

extent of the peak. At first glance it might seem that this maximum

is due to some state mixing, but a Butler analysis was made to deter-

mine the existence of this maximum. According to the analysis the

maximum is in approximately the proper place for a secondary stripping

Fig. C.--Anfular distributions of the C12 (d,p), ground state

reaction, over a wide range of deuteron enerries plotted as

versus E

) > > >

O /

3 ro

C\

(U.. 0 / oVl

0/

w

/ -

/w

c\J 0

(W31S/89AI) p

-L7-

maximum. It is probable that at lower deuteron energies other effects,

orodoninately coulonb and compound nucl,,is !-ffects, give a background

that swamps this peak so tht it is not noticeable.

The 8 :?ev curve of -otblat exhibits the characteristics of

3Srber strinning,. A check was made to determine this. The S;rbr:r

cross-section is given by, in the new variables,

6(K) c (94)

where

S= ,3 X 10 (95)

K2 was expressed in terms of Ek by

K = (96)

0 k

The dotted curve shown in "i9. 9 is the theoretical curve for the 8 Mev

data in terms of the ISerber cross-section and normalized arbitrarily at

the point E = 2 ;ev. There is definite indication of Serber type

k

stripping.

It is obvious from the data shown for the carbon (d,p) reaction

that there is an energy dependent factor missing in the cross-section

expression given by the simple stripping theories. Fi;,. 9 also indicates

-!LU-

there should be a transition region hotween the Putler stripping and

the Serber stripping. !!ost probably this transition comes about when

the energy resolution nf the experiment is poor such that the proton

enerries selected are not those due to a single level in the final

nucleus.

016 Data for 'round r;tate ieaction.--The Q value for the

around state reaction is 1.92 !'ev. Using this value, equation (92)

for '- becomes

'k

E .= .7913E,+1.02 -.729 E (E-q C 9 9.s7

The cross-section expression is

6- 6

,1 (96)

"ir. 10 shows the renlotted angular distributions of the 2.65

!ev, 3.01 :'ev and 3.25 :'ev curves of Stratton, et al taken around the

3.00 ::ev resonance of 1 The small curve that cuts through the three

angul'.r distribution curves is the excitation function. Since the range

of deuteron energies was small, 2.25 to 3.8 :r.v, it was impossible to

average over the resonances in the excitation curve for any great vari-

ation of ::k, but it is apparent that the peak of the anguilar distribu-

tions do rise and fall with the first maximum of the excitation curve

Fig. 10.--Anfular distributions of the 016 (d,p), ground state

reaction, at deuteron energies near 3 Hev plotted as t versus E .

-L9-

- rO -

CL C 0 (

xW

W

oLJ

CCQ

0

(131S /81N) ..

I

which is the 3 Nev resonance of ,8 The prim;iry maxima, the stripping

maxima, are' fairly well aligned indicating ith predominance of stripping,

but the secondary maxima are shifted relative to one another.

?'if. 11 shows a comparison of angular distributions over a range

of values of Ed. The 19 -Iev curve is taken from the work of Freemantle

et al, the 7.73 !ev curve from the work of Burge et al, and the 2.65

Korv curve from the work of Stratton et al. The first maximum of the

7.73 M1ev curve is shifted too far to the right which may be an error in

the data of TBrpe rather than the peculiarity of the distribution.

This same curve is reported in two different journals, once as an 8 ?ev

curve and once as a 7.73 :ev curve. The data for the 8 "Xev curve was

renlotted and it does have its maximum at approximately the same Ek as

the 2.65 Mov curve. This would seem more reasonable. It would be

worth pointing out here that a Butler analysis could be made to fit

either the 8 :oev curve or the 7.73 Mev curve, depending on the value of

the nuclear radius taken. This would tend to indicate the arbitrari-

ness of the simple stripping theories and points up the use of the new

6 and E variables. If there is consistency between the theory

k

and experiment then there should be only one Dutler analysis neces-

sary for the angular distributions taken at any deuteron energies.

Thus, poor data could be detected immediately. It would not coincide

with data at other energies.

The 19 Mev curve exhibits some of the characteristics of Serber

stripping. A check was made on this in the same manner as the one made

for the 8 Mev curve of carbon. The results are shown by the dotted

-rO-

Fig. ll.--Angular distributions of the 016 (d,p), ground state

reaction, over a wide range of deuteron energies plotted as 4 versus .

W

w

0

Wtl3 S /8Vry) 9

line in Fig. 11. It is seen that there is much evidence of Serber

stripping.

E. 016 First Excited State Data.--This reaction has a Q value

) of 1.O0 Iev. Equation (92) becomes

Ek = .71 E4 +.553 -7 q E__ C r, & -,L (99)

The cross-section becomes

6 (100)

Fig. 12 shows the replotted angular distributions taken from

the work of Stratton et al, for the first excited state (d,p) reaction

in oxygen. The three curves shown are the 3.01 Mev, the 3.25 Mev and

the 3.A3 Mev curves. These curves are centered around the 3.25 Iiev

18

resonance in FlT There is much evidence of stripping here as is

shown by the fact that all the curves coincide. It is also interesting

to note that the maximum observed for the ground state reaction toward

) higher Ek (around 5 Mev) is missing. This would indicate predominate

stripping for this reaction at this energy.

Fig. 13 shows three angular distributions for the first ex-

cited state reaction at various energies. The curves are the 19 Mev

curve of Freemantle et al, the 7.73 Mev curve of Burge et al, and the

"if. 12.--Anuilar distributions of the 016 (d,p), first excited

state reaction, at deuteron energies near 3 iPev as plotted C versus

EK *

> > >

W W wL

.N 0 .

0 x O

0 x < o

w

C~j

o 0

o I~)

(t~3S/8)J) /

Fig. 13.--Angular distributions of the 016 (d,p), first excited

state reaction, over a wide range of deuteron energies plotted as 6

versus Ek .

> >>

-0 LJ LL LL

a 0 ro in

oO c

0 0X <]

2.65 Ilev curve of Stratton et al. There is a noticeable shift of the

curves toward higher Ek with an increase in '.d. As before we do not

put too much faith in the 7.73 Mev curve of Rurge. Unfortunately the

19 Mev data is not extensive enough to determine whether it would

follow the 7.73 !lev curve or the 2.65 Mev curve. From the secondary

maxima we would infer that there is a definite shift toward higher F,

with increasing deuteron energy. This would indicate there is another

energy dependent factor missing from the cross-section. This is doubt-

ful, however, since this does not show up for the other replotted data.

F. Analysis of the Experimental Curves.--An analysis was made

of the h.51 Mev angular distribution for C12 (d,p) reaction and the

3.h3 'ev angular distribution curve for the ground state reaction in

the 016 (d,p) reaction. Three theoretical analysis were made for each

curve, one using the Butler theory, one using the Bhatia theory and one

using the Daitch and French theory. As was pointed out in section II,

the three theories are derived from the Born approximation for the

cross-section and are similar except in their assumptions for the eval-

uation of the G(k) factor in the cross-section expression. Since the

experimental curves were all replotted in terms of 6 and Ek, the

theoretical curves were also plotted in these terms. Since each of the

three theoretical versions of the theory predict that 6 is a

function of FE only, then one theoretical curve plotted in each reaction

should, of course, fit all the data for that reaction.

Disregarding the constants, the cross-sections are given by, for

the Butler theory,

^ =\ -

~~kt,

o

for the Bhatia theory,

and for the Daitch and French theory,

(101)

(102)

/

BuTL~R (103)

The significance and definition of the various parameters involved were

given in section II.

G. Comparison of Carbon Data with Theory.--The 4.51 Mev angular

distribution curve of Bonner was chosen as the experimental curve of the

carbon reaction to analyze because of the secondary peak that seems out

of place. The theoretical curves were normalized to the peak of the

L-f ~4 d

am,:

-57-

experimental curve so that the -A factor employed in the transfor-

mation from to ; was not employed. A single radius, that

fitting the Butler curve to the data, was used for the three theoretical

curves. The Butler curve was chosen as the primary reference mainly

because of the success of the theory in previous analysis.

The theoretical curves shown are the ones for which 1n 1,

which is the value that 1 should have according to the shell model

theory since C13 is formed by adding the neutron in the Ip shell.

Fir. lh shows the comparison between the three theoretical curves and

the experimental curve. Both the Bhatia curve and the Daitch and French

curve are shifted toward higher FE. The Butler theory seems to pro-

duce a valid curve only for a small region near the maximum of the

forward peak. The Butler curve shows the secondary maximum of the

experimental curve but it is shifted slightly toward higher Ek. The

Daitch and French curve completely lacks the secondary maximum in the

proper range. If the Bhatia curve were shifted to the left (larger

nuclear radius), it would also exhibit the secondary maximum that the

Butler curve shows, but the nuclear radius necessary to do this would

be much larger than normally estimated. As it is, the radius neces-

sary to fit the Butler curve to the experimental data is 6 x 10-13 cm,

Which is much too large compared to the value obtained from neutron

scattering experiments. The Bhatia curve requires a radius of

7 x 10-13 cm to fit the experimental data. The radius necessary to fit

the Daitch and French curve is approximately the same as that needed

for the IBhatia curve. In the calculations of neutron cross-sections of

Fig. lh.--Comparison of the theoretical angular distribution

curves and the L.51 Mev experimental angular distribution curve of the

S12 (d,p), ground state reaction, plotted as 6 versus for

a radius of 6 x 10-13 c

a radius of 6 x 10 cr.

1l Mev neutrons scattered from light elements, Coon et al (32), found

that carbon should have a radius of 3.9 x 10-13 cm. Middleton, in

analyzing the C12 (d,n) reaction found the radius of carbon to be

)4.5 x lo-13 cm. This is the same order of magnitude as that found by

Holt and Marsham from stripping reaction analyses. They indicate that

the radius should be 4.1 x 10-13 cm.

The radius obtained from the t.51 Mev curve analysis is higher

than should be expected, it seems, even from a Butler analysis. This

is not too difficult to understand, however, when it is noticed that

the Butler radius is very sensitive to the position of the forward

maximum. Thus, different data could give different results.

) An interesting point presents itself in the Daitch and French

analysis. The term which multiplies the Butler cross-section and which

signifies the resultant difference between the theories can be used for

an estimate of the nuclear well depth, assuming as Daitch and French

does, a square well. This term is given by

A1%(104)

Zh? V

SThe zero of this is

-0 t (105)

where ko is the value for which the Butler theory has one of its zeroes

-60-

(depending on the shell number). Knowing the value of k we can solve

the equation for V. When, for the carbon (d,p) reaction the value of

ko is substituted, the potential well comes out to be about 10 Mev.

This is certainly too shallow for a nuclear potential well and can only

point up the fact that the theory is inadequate either in its assump-

tions or its evaluation.

H. Comparison of Oxygen Data with Theory.--The same type of

theoretical analysis was done for the oxygen ground state and oxygen

first excited state reactions as was done for the carbon. The experi-

mental curve used for comparison in the ground state reaction was the

3.h3 IMev curve of Stratton. The 1 value necessary to make the Butler

curve fit the data is two, which is as it should be according to the

Shell model theory of the nucleus. The neutron added to 016 goes into

the ld state.

As can be seen from Fig. 15, with the Butler curve adjusted to

experimental data the Bhatia and Daitch and French calculations are

shifted toward Ek. The nuclear radius that was found necessary to fit

the experimental data for the oxygen reaction is more reasonable than

that necessary to fit the carbon data. The radius is 5 x 103 cm.

To make the Bhatia curve fit it is necessary to have a radius of

6 x 10-13 cm. The Daitch and French curve requires a radius of

6 x 10-13, also. The radius of oxygen, taken from the neutron scat-

terihg experiments of Coon is b.3 x 10-13 cm. Holt and Marsham indi-

cate that the radius should be 5.1 x 10-13 cm.

Fig. 1$.--Comparison of the theoretical angular distribution

curves and the 3.h3 Mev experimental angular distribution curve of the

016 (d,p), ground state reaction, plotted as C versus E for

Sa radius 5 x 10-13 cm.

-61-

-62-

The secondary hump does not fall at all within the experi-

mental curve, but this might be expected since at these low energies

the secondary stripping maximum is swamped by the background. Also,

) coulomb effects destroy the predominant stripping characteristics of

the curve. The secondary maximum exhibited here is probably due to

the compound nucleus rather than stripping. It is seen in almost all

stripping curves.

The analysis of the 3.h3 Mev first excited state curve of

Stratton presented somewhat of an anomaly on analysis. (See Fig. 16)

The 1 value necessary to fit the experimental data is correct. It

is 1n 0, which corresponds to the neutron being added in an s state.

p It was assumed that the radius which fitted the ground state reaction

should also fit the first excited state reaction, especially since

the data was taken by the same author under the same conditions. This

turned out to be untrue. The radius of 5 x 1013 cm which is correct

for the ground state reaction is too small for the first excited

state reaction. Fig. 16 shows the comparison of the stripping theories,

using the radius 5 x 10" cm. It was necessary to assume a radius of

6.5 x 10-13 cm to fit the experimental data. The Bhatia curve requires

Sa radius of 9.5 x 10-13 cm to fit. The Daitch and French curve could

not be drawn. The zero that is removed by the Daitch and French term in

the expression for the cross-section could not be computed since it was

not possible to obtain a value for k In this case it is the second

zero and not the first which is removed since the neutron is added in

the 2s state; that is, the shell number is two.

Fig. 16.--Comparison of the theoretical angular distribution

curves and the 3.i3 Mev experimental angular distribution curve of

the 016 (d,p), first excited state reaction plotted, as C" versus

E for a radius of 5x 10-13 cm.

-63-

V. CONCLUSIONS

The simple stripping theories agree with experimental results

only in a limited region around the maximum in the forward direction

of the angular distributions. There are more effects occurring than

the simple stripping as is evidenced by the deviations of the experi-

mental curves from the theoretical curves. These deviations are due to

the exclusion of the compound nucleus and proton nucleus interaction in

the simple theories.

As was shown in section II the best criterion for testing

whether stripping predominates in a reaction is the plotting of both

Sthe angular distributions and excitation functions in the new vari-

ables 6 versus Ek. If simple stripping is predominant the curves

should coincide. In the cases we have just investigated, the curves

could not possibly coincide due to the non-inclusion of the resonances

in the compound nucleus. Nevertheless, in the new variables the

excitation function should fail within the extreme boundaries of the

angular distributions.

We notice there is some alignment d the forward peaks in all

Sthe replotted angular distributions except for the first excited state

of oxygen. There is a definite shift of the curves with energy. This

would tend to indicate there is an energy dependent factor missing

from the stripping theories, at least for this reaction.

The experimental data is not extensive enough to see much of a

general trend in the comparison as shown in section III. This could

be remedied in two ways. Firstly, the angular distributions should be

taken at small deuteron energy intervals over a range of, say, from

the Q value of the reaction investigated to the point where Serber

) stripping becomes noticeable. Also, the excitation function should be

taken over this entire range of energies at several angles from zero

to 1800 so that a correlation could be made with the angular distribu-

tions when plotted in the 6 versus E variables. Secondly, if

data can be taken only over a limited range of deuteron energies, it

should be taken where stripping would be most noticeable. Since, as

equation (85) shows, the width of the forward peak decreases with

increasing deuteron energy the curves should be taken so that the

width is at least twice as great as that at the Q value. Also, data

should be taken so that the angular distribution curves are at

energies well above the coulomb barrier and where the peak cross-section

is large enough so that any secondary maximum due to stripping will not

be swamped in the background.

In conclusion it should be pointed out that even though the

simple stripping theories do allow one to assign spin-parities to

nuclear levels and these agree well with the shell model of the nucle-

us, the theory should be used with some caution.

APPENDIX

A. Initial and Final Systems.--For a better understanding of

the stripping process and the momentum vectors K and k we consider the

initial and final systems of the stripping process.

In the initial system the deuteron is moving with momentum kd

and the target nucleus moving with momentum -kd with respect to the

center of mass of the whole system. The velocity of the deuteron in

this center of mass system is

S :- -- =- -- (106)

and

--4

-- (107)

is the momentum.

Hence,

,--4

(o108)

is the velocity of the neutron in the center of mass system. Now, the

velocity of the neutron relative to the center of mass of the deuteron is

= .o(109)

-66-

-67-

and

V),= =K .(110 o)

Thus, K is the momentum of the neutron as viewed from the center of

mass of the deuteron. pM( l4is the probability that the deu-

teron moving with momentum kd will emit a neutron with momentum

-j

(This is virtual emission since the energy is not conserved though

momentum is.

In the final system the proton is moving with momentum k and

P

the product nucleus, B, is moving with mornentum -kp relative to the

center of mass of the whole system. The velocity of B (final nucleus)

is

-A

S 7 (112)

8 (4+1)mP

The velocity of the neutron is

I- (113)

Then the velocity of the neutron relative to the center of mass of P is

A- 1 -

-68-

and

)- 4 (

44++1 P

--4

is the momentum of the neutron as seen by the center of mass of B.

Hence, Iq (k) is the probability that A moving with momentum -kd

will capture a neutron of momentum kd -k so that B moves off with

momentum -k .

p

B. Deuteron Factor.--If the eigenfunction of a particle in

the eigenstate is x( ) then the probability that the particle in

this state will be found with momentum K is

P (,) -< 2e-^ ZCs) d s (116)

.A

since e is the momentum eigenfunction. Thus the deuteron

factor is proportional to the probability that the neutron in the deu-

Steron will be found with K relative to the center of mass of the deu-

teron.

To make this clear, consider the deuteron as two particle sys-

tems. If the deuteron has momentum kd its wave function is

(1

e.1r ccS) (117)

-69-

where

-J

S

-4 -4

P

(118)

The momentum eigenfunction for the neutron and proton is

* p.,

(119)

Thus the probability that the neutron and proton have k and kn in the

deuteron having momentum kd is

-r

,a.LieA Cj)

4r

Changing variables to

w oa- i-P

we obtain

we obtain

~a n&

(120)

(121)

(122)

p-a--

4( 1')

Ir -J

r t/r

1 P

-J

-1

i~g l/r

PC/,, 'e

-70-

The first integral is proportional to the Dirac delta function which is

zero unless I' = .-4 P thus expressing the fact that

momentum must be conserved. Therefore,

-35

0< P(k) i (123)

which is the same as before. We see that the motion of the deuteron

makes no difference in computing P(K) as long as momentum is conserved.

C. The G(k) Factor.--Consider a free neutron of momentum k

coming under the influence of a potential V(r) so as to be captured into

the state with wave function f ) The wave function before cap-

ture is Qe Hence the Born approximation for the capture

probability is

j-

CM f < (f VWK; J (124)

Here the center of mass of the system (neutron plus A equals to B) is

taken at rest, so k must be the momentum of the neutron relative to the

center of mass of B.

Consider the neutron and A as two particles whose center of mass

is not at rest, but moving with momentum -k Then, the initial wave

function is

function is

-71-

(125)

where the neutron has momentum

and the target nucleus has momentum -kd.

The final wave function is

where

is the center of mass of

vector of A. The matrix

e ,

x

the final nucleus, and rA is the position

element is then

V& S

y(^} e.

e;""1'

(126)

(127)

(128)

(129)

-A*

c|" c 4-

-72-

%' A~A

(130)

and

(131)

as before. The matrix element becomes

x d..

The first

requiring

factor isroportional to the delta function (+-p

-, Thus, we have,

f f (Vr)/110 (133)

where

(132)

=6 A+(

Hence i[fl is proportional to the probability that the target

nucleus moving with momentum -kd will capture a neutron moving with

momentum d$ -- to forn the final nucleus :novinr with

P

(131,)

-73-

momentum -k Momentum is conserved in each step of the reaction, but

P

energy is conserved only between the initial and final states. That

is, energy "borrowed" to form the neutron from the deuteron with momen-

tum I =Rp-i d is paid back when the neutron is captured.

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BIOGRAPHICAL SKETCH

Harold Arthur Moore was born in Brackettville, Texas on

February h, 1925. He was awarded the P. S. degree in physics by

St. Mary's University, San Antonio, in 194h. He served in the

Army of the United States until July, 1916, at which time he en-

tered Washington University in St. Louis, Missouri. He was

awarded the M. S. degree in physics from Washington University

in June, 1908, and joined the staff of the ;issouri Research Labs

in St. Louis. He remained with them in the capacity of Junior

Physicist until 1951 at which time he entered the University of

Florida to do graduate work in physics. In 1952 he joined the

staff of the C-2 Department as an instructor and remained with

the Department until September, 1954, at which time he returned

to his graduate studies in theoretical physics leading to the de-

gree of Doctor of Philosophy. He is a member of Sigma Xi, Pi .u

Epsilon and the American Physical Society.

-76-

This dissertation was prepared under the direction of the

chairman of the candidate's supervisory committee and has been

approved by all members of that committee. It was submitted to the

Dean of the College of Arts and Sciences and to the Graduate Council,

and was approved as partial fulfillment of the requirements for the

degree of Doctor of Philosophy.

August 11, 1956

Dean, College of Arts and Sciences

Dean, Graduate School

SUPERVISORY COMMITTEE:

Chairman

I; L:Leo 4

fy v ^

UNIVERSITY OF FLORIDA

3 1262 08553 6117