STUDY OF THE DEUTERON PLUS ALPH A PAR TIC LE BREAKUP REACTIONS

by

Kunio Nagatani

(B.S. , Tohoku University, 1956)

(M.S. , Tohoku University, 1960)

A Dissertation Presented to the Faculty of the

Graduate School of Yale University in

Candidacy for the Degree of

Doctor of Philosophy

1965

To My Mother

ABSTRACT\

Alpha particle induced deuteron breakup reactions have been

studied in single counter measurements at incident alpha particle energies

of 41. 6 and 29. 3 MeV. Simultaneous differential and total cross section

measurements were carried out on protons, deuterons and alpha particles.

Unambiguous evidence for final state resonance effects in the alpha nu­

cleon interactions was obtained, particularly from the proton energy spectra;5 5

the Pgyg a*P*ia-nucleon resonances corresponding to the He and Li ground

states play important roles.

Phase space factor and zero range Born approximation calculations

failed to reproduce the proton energy spectra. A more exact treatment which

involves the resonance effect in the alpha nucleon interaction explicitly does

provide good agreement with the experimental results. It is demonstrated

that study of these breakup reactions may provide a relatively sensitive probe

for details of the deuteron wave functions used.

ACKNOWLEDGMENTS

The author is deeply grateful to his advisor, Professor D. A. Bromley,

for invaluable guidance not only in this work but also in all phases of his study

at Yale University. He would also like to express sincere thanks to Professor

T. A. Tombrello, Jr. for his active participation and guidance in this work.

Grateful thanks are owed Professor J. C. Overley who has been very helpful

in the experimental aspects of the program, to Dr. H. Nakamura for his help

in the distorted wave theoretical analyses and to Professor K. R. Greider who

has contributed many suggestions and discussions relating to the theoretical

analyses. The author would also like to thank Professor C. K. Bockelman

for his critical reading of the manuscript.

He is indebted to Drs. C. Zupancic, P. F. Donovan, and J. F. Mollenauer

for useful discussions, to M essrs. J. Birnbaum and J. E. Poth who assisted in

the taking of data, to Professor R. Beringer and the staff of the Yale Heavy Ion

Accelerator who have been most helpful in making available the facilities of the

laboratory, and to Mr. J. N. Vitale who has provided a subroutine program for

the numerical calculation of differential equations and who has helped in all

phases of the programming.

Finally, gratitude is expressed to the United States Atomic Energy

Commission for financial support of this project.

T A B L E O F C O N T E N T S

ACKNOWLEDGMENTSPage

I. IN TR O D U C TIO N ................................................................................. 1

II. EXPERIM ENTAL PR O C E D U R E S ................................................... 8A. In troduction ............................................................................... 8B. Scattering C ham ber............................................. 10C. Target S y s te m .......................................................................... 12D. Detector A s s e m b lie s ....................• . ........................................ 15E. D e tec to rs ................................................................................... 17F. E lectronics ...................... 19

IE. EXPERIM ENTAL R E S U L T S ........................ 23A. Alpha Partic le S p e c t r a ........................................................... 23B. Proton S p ec tra .......................................................................... 24C. Cross S e c t io n s ...............................................................................25

IV. THEORETICAL ANALYSIS AND DISCUSSION................................ 28A. Kinematic Consideration...............................................................28B. General Considerations of Theoretical Formulations . . . . 31C. Zero-Range Born Approxim ations..............................................32D. Distorted Wave T re a tm e n t.......................................................... 35E. D iscu ss io n ............................ 41

V. S U M M A R Y .................................................................................................46

APPEND IX I. KINEMATIC R E L A T IO N S .......................................................48A. Three body P r o c e s s ......................................................................49B. Two body sequential p r o c e s s ................................................. 53

APPEND IX II. ZERO RANGE BORN A P P R O X IM A T IO N ............................ 56

A B S T R A C T

REFERENCES 59

I. INTRODUCTION

During the past decade, studies on the structure and behavior of atomic

nuclei have provided an enormous body of information. In large measure, these>

studies have by-passed the fundamental question of detailed knowledge of the

characteristics of the strong nuclear force and have taken advantage of the fact

that d irect study of the solutions of the nuclear equations of motion is possible

through study of nuclear configurations in the excited states. It is clear that

determination of the basic nuclear force is one of the most fundamental problems

in modern physics; however, it should be kept in mind that even i f such a force

w ere completely specified, which it is not at present, it would not be possible

mathematically to solve the equations of motion for situations other than those

involving two bodies, fo r example deuteron or nucleon-nucleon scattering situations,

or fo r the approximation of infinite nuclear matter.

One of the m ajor open problems in nuclear physics beyond the determina­

tion of the nucleon-nucleon interaction itse lf is that of finding the importance or

lack of importance of nucleon interactions of higher order. In particular, empha­

sis thus fa r has been centered on binary interactions and it w ill be particularly

important to examine to what extent the higher order interactions, involving three

or m ore nucleons simultaneously, may be required fo r proper understanding or

consideration of fin ite and complicated nuclear systems.

The same problem has occurred in the study not of nuclear structure but

of nuclear interactions where by far the m ajor part of a ll study has been devoted

to systems involving interaction of two nucleons or nuclear systems. Both the

experimental and the theoretical complexity involved in the case of m ore than

two interacting particles has tended to preclude their study except in isolated cases

until v e ry recently when newer techniques both of data acquisition and of data

handling have made possible extension of the reaction studies to m ore complicated

systems. As described in the follow ing paragraphs, it has been suggested that

three body exit channel nuclear reactions may be treated by utilizing two body

2

interactions sequentially. Beyond interest in the study of the reaction mechanism

itse lf, there are a number of practical advantages in using nuclear reactions

producing three particles in the final state. A single experiment can provide

information about many intermediate systems produced through final state inter­

actions between different constituents in the exit channel. One can also study a

ve ry short lived metastable nuclear system which does not live long enough to

reach a normal detector and which are otherwise inaccessible. In particular,

fo r exam ple, it has been possible in this way to obtain information regarding4 1

the highly excited states of He .

It is necessary to define the term "fina l state interaction" used in this

report herein. We use "fina l state interaction" fo r the interaction, in a three

particle exit channel reaction, between any two partic les among three exit parti­

cles before they reach their non-interacting asymptotic region. This definition2

is not necessarily the same as the original definition of Watson and some other

authors have used the present definition in s im ilar situations as described in

the succeeding paragraphs.

It should be noted that there have been two re la tive ly well separated avenues

of approach utilized in study of multiparticle exit channel reactions. In the firs t,1 3 4

particu larly involving light nuclei as illustrated by the work of Donovan et a l. ) ’

it has been possible to use re la tive ly well understood sim plified interaction

theories to extract spectroscopic information. In the second, emphasis is directed

toward the interaction mechanism itse lf using available spectroscopic inform a­

tion; in this latter case ve ry little work has been reported as yet in contrast to

the enormous activity involving binary reaction channel.5 6

Phillips et al. ’ have suggested that the cluster model of the nucleus must

provide that three or m ore body decay may be treated as a time sequence of two1 3 4

body interactions. On the other hand, Donovan et al. ’ ’ have found that in three

body reactions involving light nuclei two body sequential processes are la rge ly

dominated by d irect knock-out processes without taking two step mode. These

two extrem e points of v iew may of course be interpreted as special aspects of

3

a m ore complete treatment of the entire interacting system (see Chapter IV

for detailed discussion). It is possible to evolve rather simple illustrations

fo r the extrem e cases; in the form er case, a simple physical picture occurs

when a re la tive ly long-lived intermediate state is involved and the final state

interactions force the reaction to proceed via this intermediate state. A rather

simple picture fo r the d irect knock-out process occurs when one of the initial

nuclei interacts predominantly with only one component or constituent of the

other nuclear complex in the entrance channel and final state interactions do

not dominate the reaction.

Perhaps the simplest system beyond the two body one is that involving

interaction of a nucleon with a deuteron, both because the total number of parti­

cles is v e ry restricted and because the structure of the loosely bound deuteron

is re la tively well known. The characteristics of deuteron breakup reactions

induced by protons have in fact already been investigated by several

authors.7 ,3 ’ ^ ’ *0 ’ 11 ’ Evidence fo r significant effects attributable to final> ’ 44state interactions have been found in these studies. Frank and Gammel

have attempted to calculate this breakup reaction including the n-p final state

interaction and have been successful in obtaining quantitative agreement with15

available experimental results. Heckrotte and MacGregor improved this

theoretical treatment further by taking the p-p final state interaction instead of

the n-p one. These latter authors used a 6-potential approximation in the Born16

approximation. Recently, Phillips has attempted to estimate the effect on the

cross section of the deuteron breakup reaction induced by nucleons which reflects

consideration of both the n-n and p-p interactions in the exit channel. The 1 3 4

Brookhaven group ’ ’ has attempted to analyze experimental data from coinci­

dence measurements, applying a simple zero-range Born approximation, and

has found that a strong d irect knock-out amplitude is involved at the particular

kinematic angles selected fo r study. No detailed scan of parameter space has

as yet been attempted in these coincidence studies.

4

Beyond this v e ry simple three body system , it might be anticipated that

deuteron breakup reactions induced by alpha particles might also be amenable

to simple analyses, of equivalent nature, reflecting the fact that the high binding

energy of the alpha particle perm its it to be regarded as a single entity as long

as the reaction study is restricted to re la tively low energies. Indeed the use of

the incident alpha partic le to replace a nucleon makes certain simplifications in

the interpretation of the experimental results since the alpha particle has an

intrinsic spin of ze ro , and, in the final state of the alpha induced deuteron break­

up reaction, all three products are distinguishable while in the nucleon induced

case not only are two of the particles identical but they have non-zero intrinsic

spins. This alpha induced breakup of the deuteron has also been investigated

by several authors?’ I 3 ’ V ’ 19’ In these studies, evidence has beenJ 20, 21presented for re la tively strong final state interactions between both the proton

13and neutron and the outgoing alpha particle. Rybakov et al. have attempted

to fit the experimental data with theoretical calculations based on the theory22 2

developed by Migdal and by Watson for treating these final state interactions.

Although they have obtained excellent agreement with the experimental results,20

it has been pointed out by L e fevre et al. that certain ambiguities remain in

their theoretical treatment.

Despite considerable intrinsic interest in understanding these re la tive ly

simple final state interactions, detailed examination of the reported m easure­

ments indicates that they are of v e ry lim ited scope and provide quite inadequate

information for any m ore complete theoretical analysis. In reference 8,

product neutrons were observed at only 0° using a U235 fission counter and

the neutron production yield curve was measured fo r deuteron energies up to

approximately 6 MeV. In reference 17, emulsion techniques were used to ob­

tain proton and alpha particle distributions at a single incident deuteron energy

of 10. 3 MeV. In reference 18, a magnetic spectrom etric examination of product

protons was carried out at three separate angles at 18°, 19.5° and 24° at a

single incident deuteron energy of 14. 8 MeV; examination of product spectra

5

was restricted to the region of the peak resulting from the ground state He5

(d,p) He reaction. Reference 19 was again an emulsion experiment at a

single deuteron energy of 20. 2 MeV and an attempt was made to obtain crude

energy spectra from emulsion scans. In reference 20, a precise product

neutron spectrum was obtained using tim e-o f-fligh t techniques but only at a

single angle of 0° with an incident 9 MeV deuteron beam. In reference 13,

neutron spectra were again obtained using tim e-o f-fligh t techniques at 0° and o

180 , while in reference 20 coincidence detection of the alpha particle and the

proton in the exit channel was involved.

In consequence, it was considered of importance to attempt a much more

detailed examination of this re la tive ly simple reaction in order to obtain the

experimental input data fo r a hopefully m ore complete examination of the extent

to which quantitative predictions might be made based on the known free alpha-

nucleon interaction. It is fa r from obvious that the two body elastic scattering

data can be utilized d irectly in final state interactions where contributions from

well off-the-energy-shell may be important. Indeed one of the goals of the

present measurements was to examine the extent to which the off-energy-shell

contributions were important in determining the reaction characteristics in addi­

tion to elucidation of the reaction mechanism itse lf in this hopefully rela tively

simple case. The Yale Heavy Ion A ccelera tor has been used to provide the

alpha particle beam since this accelerator does not norm ally accelerate deuterons

and adequate shielding is not available fo r their use. Although it yields a d is­

advantage resulting from the la rge center of mass motion, it should also be

noted that background problems were ve ry much reduced through the use of

the high energy alpha particle beam instead of a deuteron beam. This beam

has an energy of some 40 MeV while low er energies can be obtained with lesser

intensity follow ing the insertion of absorbers in the beam and subsequent m o­

mentum analysis of the reduced energy alpha particles. Full use has been

made of semiconductor detectors in studying re la tive ly high energy charged

p a rtic les , which resulted from the deuteron breakup reactions, with high

4

6

effic iency and good energy resolution. A (dE/dx) x E , all-sem iconductor,

particle identification system , has been used in combination with a 20,000

channel V ictoreen multidimensional pulse height an a lyzer, available in this

laboratory, for simultaneous study of both proton and alpha partic le spectra

from the breakup reaction. Studies have been carried out both at an incident

alpha particle energy at 41.6 MeV and to lesser extent at a low er energy of

29. 3 MeV. A t these energies the only open channels are those producing alpha3 3

partic les, deuterons, protons and neutrons; He , H and all other charged23

particle channels are energetically closed. In the experimental measurements,

spectra of not only alpha partic les and protons but also of deuterons recoiling

from elastic scattering were simultaneously examined at angles in the range of o o

12 to 110 in the laboratory reference system.

These experimental results have been subjected to a variety of theoreti­

cal analyses of increasing complexity. For orientation, the spectral shape was

firs t compared to the simple phase space predictions fo r a three body system ,

although this was not expected to give satisfactory results since evidence for

final state interactions already exists. A zero-range Born approximation calcu­

lation was carried out in a manner ve ry s im ilar to that developed by Zupancic16 12 3

fo r the treatment of C (a, 2a)C reaction. Finally a much m ore complete*treatment fo r the reaction m atrix element was developed by Nakamura which

includes final state alpha-nucleon interactions represented by a sem iem pirical

potential fo r the alpha-nucleon interactions. An extensive numerical calcula­

tion has been carried out using the IBM 7040-7094 computer system at the

Yale Computer Center.

As expected the phase space calculation failed to reproduce even the

gross structure of the spectrum , and it was not confirmed that the zero-range

Born approximation could reproduce any significant parts of the experimental

* H. Nakamura, private communication(to be published).

7

energy spectrum. The m ore detailed calculation includes the alpha-nucleon

final state interactions and it is much m ore successful, qualitatively, in r e ­

producing the whole energy spectrum. Several discrepancies found in the last

treatment can be partly understood by considering the approximations applied

in the formulation as w ill be discussed in greater detail in Chapter IV.

In the present report, the experimental equipment and techniques in­

volved are described in Chapter n while the experimental results are presented

in Chapter ID. Chapter IV presents a discussion of the theoretical analyses d is­

cussed above. The details of these analyses are presented in a number of

Appendices.

8

A. Introduction

The helium beam produced by the heavy ion accelerator at Yale

University hhs an energy of approximately 42 MeV. This beam has been u til­

ized in a study of the breakup of the deuteron in interactions between alpha

particles and deuterons. In this Chapter we shall describe the experimental

equipment used in these studies.

The beam energies have been obtained from the previously calibrated

magnetic analysis system which is used to provide momentum analysis of the

output beam of the accelerator. The maximum energy beam used in the present

work was determined to be 42. 08 M eV in energy. A low energy beam was ob­

tained by inserting absorbers within the momentum analysis system and subse- •...

quently re-analyzing the alpha particles transmitted through the absorber fo il.

This beam was determined to have an energy of 29. 88 MeV. In both cases, it

is estimated that these energies are known to an accuracy of better than 0.6%.

It has been recognized from the outset that nuclear reactions having

three em erging particles in the exit channel are much m ore complex than the

m ore thoroughly studied binary reactions, because of the kinematical com pli­

cations introduced by the additional partic le. A m ore complete mathematical

treatment of these kinematic problems is summarized in Appendix I. The

m ajor problem in considering such experiments arises in the fact that many

exit channels are involved. (In fact, an infinite number of exit channels may

be expected because some if not a ll of the em erging particles have continuous

energy distributions).

A s is indicated in Appendix I, it can be demonstrated that the three

body exit channels can be completely determined if complete measurements

can be made on two of the three exit channel particles. This assumes that the

detectors used can identify and select the d ifferent particles. In order to obtain

m ore complete information on the products of the reaction of in terest, it is

n. E X P E R I M E N T A L P R O C E D U R E S

9

highly desirable to carry out coincidence measurements. However, there are

difficulties inherent in perform ing coincidence experiments. If a gas target

is used, ve ry careful geom etrical alignment is required to insure that the two

detectors are sensitive to reaction products emerging from the same volume

within the gaseous target. Unfortunately, an overrid ing consideration in the

work to be reported herein is the fact that the extrem ely low duty cycle (2%

macroscopic and about 0.08% m icroscopic) of the heavy ion linear accelerator

entirely precluded coincidence measurements in the reaction channels of

interest.

The presently studied reaction , deuteron breakup by alpha p a rtic les ,

is expected to produce alpha p a rtic les , deuterons, protons and neutrons, in

addition to any other secondary radiations which may result from contaminants

or from secondary induced interactions. M oreover, each of these product parti­

cles has a complex kinematical distribution both in energy and in spatial coordin­

ates. It might be assumed that a simple experiment which would eliminate a

number of undesired particles would be one involving measurements on neutral

products alone, where all charged particles can be simply discrim inated against.

Several attempts at such measurements in the (a + d) reaction have in fact been 8 20 13

reported. ’ ’ It is , however, difficult to obtain adequate energy resolution

and angular information unless one uses tim e-o f-flig jit methods which again are

e ffective ly precluded by intensity consideration in the case of the heavy ion linear

accelerator. M oreover, within the target v icin ity, the neutron background from

the accelerator is particularly intense and also argues against neutron studies.

Fortunately, recent development of semiconductor detectors of rela tively

la rge thickness and fast electronic circuits have made possible identification

systems fo r the simultaneous separation and examination of a wide variety of

charged particles. In this laboratory, such an identification system using a24

m ultiplier circu it has been developed. Although the logica l basis fo r the ident­

ification of charged particles involved herein is the same as in the above men­

tioned system , unlike that system , with the availability of a multiparameter

10

pulse height analyzer, it has been possible to eliminate the complexity of analog

pulse multiplication system and to apply the two element detector signals d irectly

to the multiparameter analyzer. This provides much greater efficiency in the

collection and handling of data. Thick lithium drifted silicon semiconductor25

detectors developed in this laboratory have made it possible to detect r e la ­

tive ly high energy light nuclei with high resolution and with high detection

efficiency. Thin surface barrier semiconductor detectors have been used to

detect the differential energy loss ( dE/dx); these are hereafter re fe rred to as

trahsmission detectors. Using these together with the thick detectors and the

multiparameter pulse height ana lyzer, a particle identification system was

assembled which made it possible to simultaneously observe the energy spectra

of protons, deuterons and alpha particles. M ore complete descriptions of this

system follow iin succeeding sections of this Chapter.

B. Scattering Chamber26

The basic scattering chamber used here has been described previously.

As originally designed, the chamber accepted only solid targets while in the

present experiment it was necessary to carry out sufficient modification to

perm it use of gas targets. The target system evolved is described in Section

C and the detector assembly is described in Section D of this Chapter.

The chamber is shown schematically in Fig. I I—1. The chamber body

consisted of a 25" x 25" x 2" blbck of aluminum which has a 20 5/8" diameter

aperture bored through it. Top and bottom lids, of 1" thickness, are indepen­

dently rotated using m otorized drives. The chamber can be evacuated either

through the beam pipe itse lf or through additional pumping ports which are not

shown in Fig. I I—1. Within the top lid, a 4" diameter central aperture is provided

through which the target ce ll has been mounted. Several additional ports at

a greater radii provide access to and support for a number of detector assemblies.

As used herein, the particle identification telescope was attached to the bottom

lid and a single monitor counter which consisted of only a thick detector was

attached to the independehtly rotatable top lid.

Figure I I - 1. Schematic diagram of the scattering chamber.

S C A T T E R I N G C H A M B E RM O N ITO R D E TE C TO R GAS T A R G E T C E L L

SCALE

11

The beam entered the chamber through collim ator s lits , traversed the

target ce ll at the center of the chamber, and passed into a Faraday cup where

the delivered charge was collected by an E lcor Model A309A current indicator

and integrator. The absolute intensity of the incident beam flux was measured

with this Faraday system?

Several slits were used in the collim ator system which perform ed two

separate functions. The firs t was to define the optical character of the beam to

be as nearly parallel as possible and to be of a definite size while traversing the

target. This f ir s t function was obtained by using two parallel rectangular slits

0. 049" x 0. 098" held 12" apart along the beam axis as shown in F ig. n-1. The

second function was that of reducing the background radiation in the target and

detector region resulting from interaction of the scattered beam with m aterial

in the vicin ity of the beam handling system. This background was particularly

troublesome because the thick lithium drifted silicon detectors have re la tively

la rge effective volume and significant sensitivity to gamma radiation, electrons

and neutrons, through the (n,o) reaction induced in the silicon itse lf in the

latter case. The latter function was achieved by inserting a further circular

s lit of 1/8 inch diameter at the exit end of the accelerator system , hence about

2 feet beyond the firs t beam defining slit (This circu lar s lit is not shown in

F ig. n-1.). Beyond the circu lar s lit, a 1 foot long paraffin cylinder having a

1 1/2" hole bored along its axis was fitted into the standard 4 inch beam tube

to m inim ize transmission of neutrons. Striking reduction in ambient back­

ground was achieved by this system.

The alignment of the chamber as well as that of the collim ator s lits ,

with respect to the defined beam axis from the accelerator, was carried out

using standard optical and surveying techniques. The procedure used was ve ry

*The identical system had been used previously and a detailed discussion of calibration is given in the ref. 27 where the absolute e rro r was estimated to be a few percent, fo r a solid target.

12

sim ilar to that described in a previous reference ( Ref. 26). The alignment was

checked before and after each run, each normally of a few days duration. No

significant e rro r in final results can be attributed to problems of alighment.

C. Target System

A gas target was chosen for the present study prim arily because of its

convenience and the possibility of maintaining the required target purity. An

additional advantage was that comparisons of the reactions induced by alpha

particles on hydrogen and on helium gas targets were readily obtained. These

were particularly useful for calibration purposes throughout the experiment

described herein.

The m ore standard, fo il- fre e , differentially-pumped, gas target was

precluded in the present case because of practical considerations— in particu­

lar, the physical diameter of the accelerator beam ( approximately 1/8 inch

diameter). In consequence, it was decided to design a gas cell having adequate

fo il windows. The design necessarily satisfied a rather stringent set of requ ire­

ments. Included among these are the following:

1. The target system had to be readily constructed and utilized in

the existing scattering chamber.

2. The target ce ll had to be of adequate s ize to allow observation of

the entire angular range of interest without permitting the detectors

to observe either the entrance or exit of the beam from the gas

volume where reaction yields from the fo il windows themselves,

or more probably contaminants on these windows, would completely

obscure the results of interest.

3. Adequate reaction volume and pressure w ere required to provide

reasonable reaction yields for feasible detector solid angles.

4. The energy losses, both of the incident beam and of the emergent

particles in the target volume, had to be m inim ized to facilitate

the analysis and reduce e rro rs introduced by absorption corrections.

13

5. The temperature and pressure of the target gas had to be main­

tained and measured with a precision enabling reasonable m easure­

ment of absolute cross sections and accurate normalization of

measurements made at various angles.

With these requirements in mind, a cylindrical ce ll was constructed in

a manner perm itting insertion through the 4" hole in the top lid of the chamber.

The cell had a 3" diameter at the leve l of the beam, and was some 3" in depth

thus fillin g essentially all the available volume in the existing chamber. Two

separate windows 0. 375" wide covering an angular range from 0° to 180° and

from 200° to 340° as measured from the center of the ce ll, were m illed in the

cell. The target cell was mounted in the scattering chamber, and the vacuum

inside the chamber was sealed by atmospheric pressure on the top lid. Thus,

there was no restriction on the angular position which the target ce ll could

take with respect to the top lid and the two window openings; the entire angular

range fo r the detectors was made available through proper rotation of the

target ce ll rela tive to the beam axis.

The cell windows were covered with 0. 00013" Havar fo il (produced by

Hamilton Watch Company; Co 42. 5%, Cr 20. 0%, Fe 17. 9%, N i 13. 0%) which

was held in place and sealed gas tight with epoxy cement. The chamber system

was tested at pressures up to 50" of m ercury equivalent and it was found to

show no leaks. During actual experimental measurements, the pressure was

maintained in the range between 20" to 30" of m ercury equivalent. Under such

conditions, the energy losses of the incident alpha particles passing through

the entrance fo il and through the in terior gas from the center of the target was

calculated to be 0. 48 MeV at 42 MeV incident energy. Since the detectors

viewed a finite length of the beam line (see Section D ), the energy corrections

fo r the incident beam necessarily must account fo r the evaluation of energy

losses along this length. Fortunately, it has been calculated that such co rrec ­

tions are sm aller than would be significant within the accuracies attained in

the present experiment; therefore, they have been neglected. As the result

14

of gross corrections, the energy of the incident alpha particle beam effective

at the target center has been taken to be 41.6 MeV for the higher energy case

and 29. 3 MeV fo r the low er energy beam.

The target cell had three valved gas connections, two of these used for

outlet and inlet of the target gases and the third connected to the volume of the

scattering chamber itself. Using the latter valve it was possible to evacuate

the target cell and the scattering chamber simultaneously without establishing

a significant differential pressure across the fo il windows.

The purity of the deuterium target gas was checked by mass spectros-2

copy* and the only significant impurity was found to be H at less than 0. 3%.

This finding was also confirmed by observation of the elastically scattered r e ­

coil proton peaks in the proton spectra obtained by bombarding the gas with

alpha particles. As a further check of the possibility of contaminant reactions,

purposely contaminated gas fillings containing a ir and CH^ were inserted in the

cell. In both cases, spectra of all charged particles were strikingly d ifferent

from those measured with the deuterium target. These measurements made it

possible to set the upper lim its on the possibility of any significant contamination

from any of the normal expected sources. Such considerations have led to the

finding that all target contaminations if present were completely negligible in

the work to be presented.

The target gas temperature was determined with a thermometer system

in intimate contact with the target cell. Temperatures were determined at

regular intervals throughout individual data rims; and since the variations were

re la tive ly sm a ll, it has been assumed that in each case the average temperature

represents the value adequate for use in data reduction. The gas pressure was

measured using a precision W allace and Tiernan mechanical gauge and using a

Cartesian manostat. The pressure variation during the course of each run was

maintained to less than ± 1%.

*We are indebted to Dr. R. Paul of Yale University fo r making this check.

15

D. Detector Assem blies

A s is always the case, one of the m ajor problems in the use of gas targets

is that of finding the solid angles effective ly subtended by the detectors. This

of course reflects the fact that the gas target does not have a confined effective

volume but one that is determined jointly by the solid angles characteristic of

the detector and that of the collimation system. This problem has been considered

in greater or le sse r detail by a vast number of authors in the past. In the de-28

sign of the system used here, the calculations of Silverstein have been used

and in the follow ing paragraphs, we present a b rie f description of these calcu­

lations.

The reaction yield Y at an observation angle 0 may be written as

y =sin 0 '

Here n and N represent the number of incident particles per unit time and the

number of target particles per unit volume. a (0 ) is the cross section at the

angle of observation. The ratio G/sin 0 takes care of the solid angle subtended

by the detector and must be calculated for any given system. The geometrical

factor G has been calculated by Silverstein fo r a variety of cases.

In the present experiment, a vertica l parallel s lit in front (the side c loser

to the target) and a circu lar s lit in rear (the side c loser to the detector) were

chosen. The factor G is then expressed in term s of the dimensions of the slits

and their separations from the target center, together with correction terms

representing the finite s ize of the incident beam. It also depends on the va r ia ­

tion of the cross section ct(0 ) with angle at the angle of observation; and in

general, it is not possible to solve fo r G unless <7(6) is determined. This

geom etrical factor may be expanded in a power series as

° = Gool1 + Ao + A i ^6) ^ +a2 ^5> + - • -1 <n-2>

16

where can be calculated fo r a given setting of the experimental s lit system.

In the course of the present experim ent, two different sets of detector

slits were used to obtain the actual energy spectra shown herein and one fo r

the less restric tive requirements of the monitor counter. A t forward angles,

i. e. less than 90° in the laboratory coordinate system , a (dE/dx) x E -

counter telesecope was used with a 0.160 cm wide vertica l paralle l slit in

front and 0.164 cm diameter circular slit in rear. A t backward angles, i. e.

m ore than 90° in the laboratory coordinate system , a single energy counter was

used with a 0. 317 cm wide vertica l and a 0. 318 cm circu lar s lit in the same

relative positions. The radial separation of the front and rear slits from the

center of the target in both cases were 5. 24 cm and 15. 24 cm, respectively.

The corresponding values of G appearing in eq. (II-2 ) are approximately -5 -4

2.22 x 10 and 9. 88 x 10 , respectively. F or both sets of the s lits , the

coefficients of the correction term s A and A in eq. (II-2 ) are approximately- 2 - 3 o

10 and 10 , respectively, at 6= 12 and fo r la rger angles both coefficients

decrease rapidly. The angular aperture subtended by the detector was ± 1 .0 °

with the f irs t set of s lits and + 1 .8 ° with the second set of slits.

The monitor counter was also collimated by a system composed of a

front vertica l parallel s lit and a circu lar rea r slit. The width of the form er

was 0. 238 cm and the diameter of the latter was 0. 237 cm; their radial distances

from the target center were 10. 36 cm and 20. 36 cm , respectively , so that the“5

factor G for this counter was 5.12 x 10 . M oreover, since the monitor

counter was used only for establishing the rela tive beam intensities during

runs, the correction term s were not necessary provided the counter itse lf was

not moved during the course of the experiment.

In all cases, slit dimensions were measured with a x20 linear m icro ­

scope and it has been found that the e rro rs in the measured sizes are approxi­

mately ±5% due prim arily to mechanical irregu larities in the edges. The

corresponding e rro r in the geom etrical factor G is about ± 15%.

17

The physical structure of the detector assembly is as follows: Two

slits w ere mounted on either end of a brass block 10 cm long in which a 0. 5"

hole was bored along the long dimension through which the particles traversed.

A ll slits were interchangeable, and anti-scattering slits were placed between

the defining slits described above. The block in turn was connected to an

adjustable mount which was attached to the top or bottom lid of the chamber.

The transmission detector was approximately 1/4" from the rear defining s lit

of the collim ator system and was followed by the thick lithium drifted solid

state detector separated from it by approximately 1/4" by a teflon spacer.

Signal and power connections w ere taken directly through the detector ports

of the chamber lids and to pream plifiers mounted immediately adjacent to these

ports. Standard optical surveying techniques were utilized in aligning the detec­

tors with angular inaccuracies of the entire assembly estimated at less than o

±0.5 .

In order to m inim ize noise, particularly from the thick lithium drifted

silicon detector in the counter telescope, it was found necessary to provide

local cooling by attaching this detector to a copper arm passing outside of the

scattering chamber volume to a heat exchanger cooled by circulating re fr ig e ra ­

tion fluid. It was found that lowering the detector temperature to 10°C provided

adequate reduction in the noise level, but caused condensation on the detector

when the chamber was opened.

E. Detectors

As discussed previously, most of the energy spectra to be presented

herein were taken using the ( dE/dx) x E-counter telescope. A t backward

angles, however, i .e . greater than 90° in the laboratory coordinates, kine­

matic arguments exclude alpha particles and deuterons in the ex it channels, and

in consequence a single energy detector sufficed. At forward angles, an ORTEC2

Au-Si surface barrier detector, with 50 mm circu lar area and thickness of 50

m icrons which was totally depleted at an applied bias of 50 vo lts , was used as a

1 8

transmission detector. The energy resolution of this detector, including all

electronic contributions, was approximately 30 keV fo r 5.5 MeV alpha parti­

cles. A thicker lithium drifted gold surface silicon detector was used fo r

measuring the residual en e rg ie s this latter unit had a resolution of the order

of 250 keV fo r 5. 5 MeV alpha particles and it was used in the counter telescope.

Again at backward angles, kinematic considerations lim it the available energy

so that the single energy detector used was an ORTEC Au-Si surface barrier2

detector with 50 mm circu lar area and a depletion thickness of 170 m icrons at

50 volts applied bias. This detector stopped protons of energies less than 4.5

MeV and provided an energy resolution of 30 keV fo r 5.5 MeV alpha particles.

The monitor counter operated at forward angles also consisted of a thick

lithium drifted silicon detector s im ilar to that used in the counter telescope.

Energy losses associated with system absorbers were both calculated

and measured experimentally through observations of e lastica lly scattered

particles from the reactions (a + o), (a+ d) and (a+ p) at selected laboratory

angles to provide a range of known energies for the various particle species.

As a result of these calibrations, the linearity of the system was established

to ± 3%. The lowest observable energy fo r protons was found to be approxi­

mately 3 MeV if measured in the elastic scattering from (q;+ p); but in the

actual reaction in this study, the background in this energy region pushed this

lim it up to roughly 3.5 MeV. S im ilarly, the energy resolution of this system

for various proton energies was determined by examining the pulse height

spectrum from the reco il proton peaks follow ing elastic alpha scattering on a

hydrogen target. Such results are shown in Fig. n-2. It has been established

that the m ajor lim itations of the energy resolution, listed in order of importance,

are the energy spread of the incident alpha beam itse lf, the energy resolution

of the detector, and the kinematic broadening associated with the finite angular

resolution of - 1° of the detector assembly itself.

?We are indebted to Mr. B. L iles of the Yale Electron Accelerator Laboratory for perm ission to use several detectors fabricated by him in part of this work.

Figure II-2 . Measured energy resolutions for elastically scattered

protons are shown at various energies. Measurements were made

utilizing a -p elastic scattering at selected observation angles.

E N E R G Y R E S O L U T I O N for P R O T O N P E A K S from a - p E L A S T I C S C A T T E R I N G

E N E R G Y

19

In the work to be reported here, the spectral shapes w ere of particular

interest and it was, therefore, essential to examine the low energy section of

the spectrum to search for spurious effects due to multiple scatterings, slit

edge scattering, etc. A number of such measurements w ere carried out using

the elastic scattering reactions. A typical spectrum of protons from the reaction

(ty +p) is shown in Fig. I I -3. The low energy tail itse lf is sm all, but the low

energy regions of energy spectra are necessarily distorted. Unfortunately,

quantitative estimates of these distortions could not be made, because the

characteristic features of low energy tails w ere variable in different runs.

( Although the ratios of the low energy tail heights to the peak heights stayed

below approximately 2% in the worst cases, they changed by a factor of 4 in

different runs.) These changes were considered to be effects due mainly to the

beam geometry and different characteristics o f the detectors used. Following

these arguments, no attempt has been made to correct the measured energy

spectra for these effects, however, it should be kept in mind that the low

energy regions of the energy spectra presented herein do contain this distortion

and corrections w ill be made for this in later work.

F. E lectronics

Fig. I I -4 is a block diagram of the electronic instrumentation system

used in the experimental measurements. The E and ( dE/dx) ( denoted by A E

hereafter) detector outputs w ere firs t am plified in thermionic Beringer p re ­

am plifiers with gains of 30 and 60 for E and A E channels, respectively. White

cathode follow ers ( WCF) were used to couple the outputs of the pream plifiers

to roughly 125 feet of RG114U signal cables joining the target and control room

areas. Outputs of the double delay line am plifiers* (DDA) trigger discrim inators

(DISC) in order to lim it low energy electronic noise. Scalers (S.CAL) reg istered

the total counts in each channel and in addition the total coincidence counts.

*A m p lifiers , discrim inators, single channel analyzers, and coincidence circuits were developed by C. E. Gingell. G .E .L . Gingell, IEEE International Convention Record Part 5 ( 1963).

Figure H-3. A typical energy spectrum of protons from the a -p elastic

scattering. The ratio of the low energy spectrum tail to the peak height is

about 1.3%.

Figure II-4 . Schematic diagram of the electronic instrumentation used.

Abbreviated notations for each unit are given in the text.

COUN

TS

per

CHAN

NEL.

3 0 0

200

0N1CM

100

2 0 MeV

T Y P I C A L P R O T O N

E N E R G Y S P E C T R U M

J

10 20 30 40 50 60 70 80

C H A N N E L N O .

| S Y S T E M S C H E M A T I C D I A G R A M TARG ET A RE A | CONTROL ROOM

20

The am plifier outputs w ere also fed to window am plifiers where a selected

region of interest could be selected through proper adjustment of the gain

and backbias in these am plifiers. These in turn were used to provide inputs

to the multiparameter pulse height analyzer. This analyzer is a 20,000

channel unit constructed to Joint Yale-O RNL specifications by the Tullamore

Instrument Division of V ictoreen Instrument.

Using the internal coincidence circu itry in the analyzer to re ject all

non-coincident events, spectra such as those shown in F ig. n-5 were obtained.

In these p ictu res, the brighter spots represent higher counts in the correspond­

ing channels. Comparison of the pictures on the right with those on the le ft

side illustrates the function of the window am plifiers used here only in the

AE channel. In plots (1) and (3), alpha particle groups extend to the extreme

right, exhibiting greater energy losses in the transmission detector. Deuterons

appearing in the upper portions have small energy spreads since the deuterons

are essentially monoenergetic follow ing elastic reco il scatterings. Proton

groups are located below the deuterons. Pictures (2) and (4) show only deuterons

and protons while alpha particles overflow to the right? It is demonstrated here

that the separation between elements is quite adequate fo r particle identification

without introduction of ambiguities. It is also noted that no counts besides

spurious events were observed in the region between the alpha particle group3 3

and the deuterons. This confirmed the absence of He and H in the products

of the reaction under the study. The analyzer m em ory configuration utilized

was such that 200 channels were used in AE channel and 100 channels in E0channels with total memory capacity of 10 counts in each channel. Data

readout was accomplished on IBM compatible magnetic tape as well as on a

CRT display system. The magnetic tapes were then edited and printed out in

♦The spots at upper right corners in pictures (2) and (4) are the counts fed from the pulser used fo r the stability check described in the succeeding paragraph.

Figure II-5. D irect oscillographic plots of dE/dx against E for the

bombardment of deuterium by 41. 6 MeV alpha particles.

(1) Without a window am plifier, at 18.0° in the laboratory system

(2) With a window am plifier in dE/dx direction, at 18. 0° in the

laboratory system.

(3) Without a window am plifier, at 30 .0° in the laboratory system

(4) With a window am plifier in dE/dx direction, at 30. 0° in the

laboratory system.

dE/dx dE/dx

(3) (4)

E E

dE/dx dE/dx

Direct Oscillographic Plots of dE /dx Against E for the Bombardment of Deuterium by 41.6 MeV Alpha Particle.

(1) Without a Window Amplifier, at 18.0® in Lab.(2) With a Window Amplifier in dE/dx Direction, at 18.0° in Lab.(3) Without a Window Amplifier, at 30.0° in Lab.(4) With a Window Amplifier in dE/dx Direction, at 30.0° in Lab.

FIG.n-5

T E STPU LSE

21

standard two dimensional m atrices using the Yale IBM 1401 and 709 computers?

No addition of the E and AE pulses was carried out e lectrica lly since when

required this was readily done from the individual channel numbers of the two

dimensional display. Energy spectra were obtained by adding counts in each

E channel fo r the selected isotope.

One of the difficult problems in obtaining the absolute cross sections

was that of correcting fo r the dead time in the electronic instrumentation. In

particu lar, the multiparameter analyzer itse lf has a long dead time given by

(25 + j ) fjsec , where n is the highest channel number associated with the coded

event. In order to obtain a reliab le estimate of a ll these dead tim e corrections,

a further data channel was established in parallel with the analyzer as follows:

The signal to the analyzer was also connected to a single channel analyzer

whose window was adjusted so that only the e lastica lly reco ilin g deuteron

peak was accepted and the corresponding signals were passed through to a

coincidence circu it scaler. In this channel, the deuteron peaks were scaled

with no significant dead time introduced except fo r residual deviation of the

coincidence effic iency from 100% due to im proper tim ing in the channels,

feeding the coincidence circuit. H ow ever, in actual measurements this latter

trouble was easily avoided. A t forward angles where a higher counting rate

was reg istered the deuteron peak is located highest in energy, and thus a

discrim ination of lower energy signals provided the selection of the deuteron

peak alone and the coincidence channel was reestablished easily in different

runs.

The overa ll stability of the instrumentation system was checked every

30 minutes during long runs and any slight deviation was corrected. This

check was accomplished by routine coupling of a pulser source.to the p re ­

am plifier inputs in the target area and observing the pulser output simultaneously

*We are indebted to Mr. J. Birnbaum for preparing the program s fo r this process.

22

with the data of interest on the multichannel display. Gain drifts observed

were usually of the order of ± 1 channel in 30 minutes and it can be demon­

strated that these drifts contributed no appreciable e rro r to the results.

The monitor data channel was established entirely separately. Not

only was the entire spectrum recorded on a 400 channel analyzer fo r moni­

toring the performance of this detector but also a single channel analyzer

and a scaler was established on the e lastica lly scattered peak to avoid the

dead time problem inherent in the analyzer.

23

A. Alpha Partic le Spectra

As anticipated from the kinematic calculations, a ll open exit channels

contribute to the y ield of alpha particles at forward angles; this is of course a

disadvantage resulting from use of alpha particles rather than deuterons as the

incident projectile. Because of a large center of mass motion, large kinematic

broadenings of the energy resolution reflecting the finite angular aperture of the o

counter ( ± 1 as discussed in Chapter II) are inevitable. Furthermore, the

overlap of many components, each individually contributing a continuous energy

distribution at any given angle, further complicates the total alpha particle

spectrum.

The alpha particle spectra at the incident energy of 41. 6 M eV were

observed at various angles in the range from 13. 5° to 30. 0° in the laboratory

coordinates. These w ere obtained using the ( dE/dx) x E-counter telescope

described in Chapter II; Fig. I l l -1 shows the typical spectra thus obtained. In

this set of figu res , the laboratory angle at which each spectrum was observed

is shown on the right of the figure. Tae curves have been arb itrarily normalized

for the display and the rela tive cross section is shown on the left hand scale. The

energy scales, shown in terms o f channel number, have been corrected for energy

losses and linearized.

The elastic peaks are readily identified from the characteristic large

cross section and high energy and, in addition, from their kinematic behavior.

The direct breakup process as anticipated yields a continuous distribution. Two

body sequential processes, on the other hand, can produce isolated broad groups5 5

contributing in certain energy regions; in particular He and L i ground states

can contribute alpha particle groups with widths corresponding to the ground states

themselves at the locations in the spectrum indicated. The double arrows in the

figure indicate the extent of the expected kinematic broadening because of the finite

detector aperture. In a ll cases, the statistical e rrors in the actual data counts

III. E X P E R I M E N T A L R ESULTS

Figure III— 1. Alpha particle energy spectra from deuteron breakup

reactions induced by alpha particles of 41.6 MeV. Laboratory observation

angles are shown on the right of the figure. Energy scales are shown

in term s of channel numbers, and the differential cross sections have

been arb itrarily normalized. Kinematic predictions for various p ro ­

cesses are indicated by the arrows together with estimated kinematic

broadenings and are discussed in the text.

AR

BIT

RA

RY

U

NIT

S

FiG.ni-i

24

themselves were kept less than a few percent in the region of interest and are

such that no significance should be attached to the fine structure in these curves.

Although an attempt was made to extend these measurements beyond 30°, no

significant counts were observed consistent with the kinematic predictions.

It should be noted that these spectra are in good agreement with the

expectations on kinematic and open channel grounds; however, the anticipated

complexity makes it difficult to extract specific information in any simple; way

from these rela tively featureless spectra.

Measurements at the reduced alpha particle energy of 29. 3 MeV were

also performed. It was found that apart from the obvious kinematic sh ifts, these

spectra had exactly the same characteristics as those shown in Fig. I l l—1.

B. Proton Spectra

From the kinematic calculations, a continuous energy distribution of

protons from the three body direct breakup process and from the two body

sequential processes might be anticipated. In particular it would be expected5

that the reaction proceeding through the He ground state should appear as a

monoenergetic proton group. *

This is the only known two body sequential process which should yield

a re la tive ly sharp line; the others might be expected as much broader distribu-5

tions resulting from kinematical arguments involving formation o f the L i ground5 5

state or excited states of either He or L i .

Fig. I l l -2 ,3 and 4 show observed energy spectra at a variety of angles,

a ll measured at an incident alpha beam energy of 41. 6 MeV. In the angular

*Here the use of the word "m onoenergetic" is of course re la tive , since the natural resonance width of He® ground state itse lf is not negligible. Warburton and McGruer have measured this width utilizing a magnetic spectrom eter in the He4( d,p) He*7 reaction-*^ at the deuteron energy of 14. 8 M eV, and found it to be 0. 55 ± 0. 030 MeV. Even these measurements are in question regarding the possible contribution of breakup processes to this width.

Figure III—2. Proton energy spectra from deuteron breakup reactions

induced by alpha particles of 41.6 MeV. The d ifferential cross sections

have been arb itrarily normalized. E rro r bars include statistical e rro rs

only.

Figure III—3. Extension of Figure HI-2 for la rger angles of observation.

Figure HI-4. Extension of Figures III—2 and III—3 at la rger angles where

only a single proton counter was utilized.

ARBITRARY

UNITS

LAB

E, in MeVLAB F IG . IO -2

AR

BIT

RA

RY

U

NIT

S

F l G . m - 3

ARBITRARY

UNITS

0.2r

0.1 -

0.1 -

(D3bN*o

UJT3cs-o 0.3-

0.2-

0.1

••

•V-• . x f

• i•V *90.0°

>r

\ , 96 °‘PROTON SPECTRA

Ea=4l.6 MeV

• •f

X 109.3°I

m<CD

E L a b i n M e v - F I G . m - 4

25

range from 12. 6° to 66. 0°, i. e. Fig. I l l -2 and 3, the ( dE/dx) x E-counter

telescope was used. Three spectra taken at angles greater than 90° were

obtained with the single counter as described in Chapter II. On these figures

the indicated e rrors are statistical in nature. Cross sections w ill be discussed

in greater detail in the next section of this Chapter.

The energy of the prominent peak at the high energy end of each spectrum

was found to coincide precisely with that expected from a two body sequential5

process with the He ground state acting as the intermediate. It should also

be noted that the peaks in the intermediate energy range in these spectra at

forward angles, which decrease in energy with increasing angle, may also be

considered to correspond to specific proton groups. Although it is difficult to

determine the exact shapes of these peaks because of overlapping of other

components resulting from direct breakup, etc. , they do fa ll in the range which5should be populated by two body sequential processes with the L i ground state

as an intermediate state. These spectral shapes are discussed in greater

detail in Chapter IV.

In Fig. I l l -5, the energy spectra obtained with the reduced energy

beam at 20. 3 M eV are shown. Because of the great loss in beam intensity

inherent in this method of energy reduction, it has only been possible to carry

out measurements at re la tively forward angles. The spectral shapes shown

in Fig. I l l -5, however, seemed to have almost the same features as in the full

energy case.

C. Cross Sections

An attempt was made to obtain the absolute cross sections from the

data presented herein. The flux of the incident alpha particles was obtained

from the integrated current as reg istered by the beam integration system and

collected by the Faraday cage. Calibration o f the absolute cross section deter­

mination was obtained by examining the elastic scattering cross section for alpha

particles at variety of angles and comparing these experimental data with previously

Figure HI-5. Proton energy spectra as in Figures HI-2 and III—3 but

at a reduced alpha particle energy of 29.3 MeV.

3PU

P

■Oz P

measured absolute cross section determinations. ’ The monitor counter

was used only in corroborating the constancy of the integrator in taking measure­

ments at various angles.

Calibrations were perform ed during each sferies o f runs ( Four different

s e r ie s , each of which comprised some four days o f running time are included

in the data presented herein). In detailed studies of the alpha particle elastic

scattering on alpha partic les , four angular distributions obtained from these

individual series showed internal consistencies in the angular positions of

maxima and minima in the diffraction pattern to within ± 2° in the range from o o

25 to 140 in the center of mass coordinate. The absolute values of the in­

dicated cross sections in these four runs, however, fluctuated between individual

runs. The possible e rro rs in the absolute cross sections, as defined here, are

estimated to be of the following magnitudes.

1. ± 15% reflecting uncertainty in the detector geometry

2. a 3% reflecting statistical uncertainty

3. ±2% reflecting uncertainty in target density

4. ± 1% reflecting uncertainty in the corrections o f the correction

terms in the solid angle calculations.

Compilation of these individual contributions indicates ± 16% erro r in the

absolute cross sections quoted here. This estimate still does not satisfactorily

explain the fluctuations mentioned above. Although the exact cause of these

fluctuations has not been established it has been found that they result from

changes in the beam optics, which in the case of the heavy ion linear accelerator,

re flec t the large number of operational parameters which must be continuously

*In Ref. 29, angular distributions in the incident energy range from 36. 8 to 47. 3 M eVare reported. The comparison o f the present data has been made to the ones atEg = 40. 8 and 41. 9 MeV. The variations o f the cross sections with energy arenot rapid, and it may be justified that an interpolation of these two curves toestimate the values at E = 41. 6 M eV is acceptable.

a

26

29 *

27

adjusted to maintain adequate performance. In consequence, the individual

series were normalized to the previously measured data, and it is estimated

on the basis o f internal consistency and other criter ia that an assigned un­

certainty of ± 35% appears rea listic in the case of the absolute cross sections

reported here; this is indicated by the e rro r bars in the various figures in

this section.

The proton production cross sections were readily calculated knowing

the cross sections for deuteron elastic scattering, since the protons and deuterons

were reg istered simultaneously on the same ( dE/dx) x E display in the multi -

parameter analyzer. The absolute cross section for protons was evaluated

from the ratio of the proton to deuteron counts. A t very backward angles, i. e.

more than 60° in the laboratory, this procedure could not be used because such

low energy deuterons w ere not observable. In consequence, the monitor

counter readings were used to evaluate the intensity of the incident flux.

The result obtained for the elastic scattering of deuterons plus alpha

particles is shown in Fig. I l l -6. Again the same fluctuation as seen in the

case o f alpha particles plus alpha particles w ere seen and indicated by the

e rro r bars. Some other data30 >3 i - 32,18, 330btained at slightly different

energies are shown together with the present result. F ig. n i-7 shows the

integrated cross sections, as functions of angle, from 3. 5 M eV to the maximum

energy studied. The estimated erro rs are largely due to the deuteron cross

section uncertainties.

Figure III— 6. Angular distributions of deuterons from d-a elastic

scattering. Other data obtained at slightly different energies are shown

for comparison.

Figure H I-7 . Angular distributions of product protons, above 3. 5 MeV

from the alpha particle and deuteron interaction at the energies shown.

E rro r bars show values discussed in the text.

A N G U L A R D I S T R I B U T I O N S d - a E L A S T I C S C A T T E R I N G

lOOO

blcs

100

10

m b s r

21.0-- .21-3

24.8

Ed =24.8 MeV by Van Ors et al. 2I.3»----- » Erramuspe et al.-+ + 4- 2 1.0 ••----- » Brock et al. 2 0.3"-----H Artemov et al.minimi 19.0 n----- « Freemantle et al.o o 20.8ii---- " Present Work.

0° 20° 40* 60° 80* 100* 120°

0 c. m.

F iG . n r - 6

A N G U L A R D I S T R I B U T I O N S

F o r P r o t o n s F r o m a - d

R e a c t i o n .

} i f fa = 4 1 . 6 M e V

a = 2 9 . 3 M e V

28

As noted b rie fly in Chapter I, three body exit channel nuclear reactions

have already been subjected to a number of both experimental and theoretical

investigations. D ifficulties inherent in the problem have lim ited both the amount

and the quality of the experimental data as well as the extent of the theoretical

analyses. A number of approximate treatments have been developed, however,

as is usually the case, the approximations have been such that each treatment

has emphasized a different, rather isolated, characteristic of the whole mechanism

and as yet no satisfactory theoretical treatment has been developed.

It is the intent of this Chapter to present more detailed analyses of the

experimental results obtained herein, in order to elucidate further details of the

reaction mechanism and to examine with what precision the final state interaction

may be approximated using known phenomenological alpha-nucleon particle poten­

tials.

Examination of the experimental results from a purely kinematic view

point, i. e. from the view point of phase space analysis, w ill firs t be presented

followed by a zero-range Born approximation treatment and then by a more

form al and complete treatment including explicitly the final state interactions.

Some attempt w ill be made to evaluate the various approximations which have been

utilized here or elsewhere; the mathematical details associated with the analyses

are summarized and discussed, in Appendices in order to prevent their obscuration

of the physical principles invblved in the discussion.

A. Kinematic Consideration

The only quantitative experimental information obtainable in non-coincident

single particle measurements which may be compared directly with the kinematic

calculations, including the known reaction Q-values, is the maximum possible

energy of the product particles at any given observation angle. No detailed

information reflecting reaction mechanisms is d irectly calculable, but the phase

space factors provide gross structures of spectral shapes below the kinematically

allowed maximum energies.

IV. T H E O R E T I C A L ANALYSIS A N D DISCUSSION

29

Both the alpha particle and proton spectra presented in Chapter II show

maximum energies in excellent accord with the predictions of such simple ca l­

culations as indeed they must, if, as assumed throughout, the spectra do result

from three body rather than four or higher body reactions. This is itse lf a useful

datum since it justifies the im plicit assumption that the alpha particle remains

a bound entity throughout the entire dissociation reactions studied.

As noted in the previous chapters, attention must be given to the two5

body sequential decays leading respectively to a He intermediate state in the case

of ot-n resonance interaction and to a L i intermediate state in the a-p resonance

interaction. The p-n resonance interaction, which would correspond to an exicted

state of the deuteron is not involved in the present reaction to the extent that the

isobaric spin selection rule holds; in the entrance channel only T = 0 is involved

and the known excited state of the deuteron of course has T = 1, and the treatment

is being restricted a p riori to nelgect these term s ( This is discussed in the

following Sections of this Chapter). In the alpha particle spectra shown in

Chapter III, slight characteristic increases in cross sections in the spectrum5 5

energy regions corresponding kinematically to the He and L i intermediate

state processes are seen; there is, however, considerable ambiguity in making

detailed assignments for reasons previously discussed in Chapter in .

In the case of the proton spectra, on the other hand, striking agreement

is observed between the energies of the rela tively sharp peak at the upper end

spectrum and that calculated for the proton group resulting from an <*-n resonance5

interaction producing He ground state. It is more difficult to identify the protons5

corresponding to the decay of the L i ground state, since the protons themselves

are being measured. This is evident from the kinematic diagram shown in

Appendix I, F ig. A l-3 , 4, and 5; the protons resulting from this sequential

reaction channel have a finite energy spread and are thus distributed over a

finite energy range in the proton spectra. The intermediate energy ranges in the

observed spectra, in which broad peaks occur at forward observation angles,

do correspond to the calculated energy ranges within which contributions from

30

this two body sequential channel would be expected to fall. These characteristics

have already been noted in Chapter III and are in complete accord with expecta­

tions based on simple kinematic considerations.

It is known that a reaction cross section is proportional to the phase space

factors for the product particles. Consequently, i f a ll other dependence could

be neglected, or, equivalently, i f the reaction matrix elements were constant,

the energy spectra of the product particles would be given purely in terms34

of the phase space factors. Such an approach was firs t developed by Ferm i

and applied by him to m ultiparticle production reactions within a framework of

the statistical theory of nuclear reactions. The phase space factor for three

particles in the exit channel may be readily calculated; detailed consideration

of this calculation is given in Appendix I and we here simply quote the result

in the form of the predicted differential cross section in the laboratory coordinate

system.

inp inpdcr = const. E |(M +M W E . -0 1 -M E -(M +M +M IE

5 ^ 1 i P « r , tt v oc P V PP P

+2 M M E mC‘ cos 0 E 1^2 (IV -1 )*a p a. P P

The notation used in this equation is se lf evident; Q is taken as the binding energy

of the target deuteron. This equation has been programmed for computer evalua­

tion and a typical result is shown as curve 1 of Fig. IV -1 , where the calculation

is compared with the experimental spectrum. Both curves have been normalized

to the same area, i. e. the integrated cross sections Er max

- 1 (‘5 T 3 i - ,dEpdE E p pP o

for both curves are the same, where the lower lim it Eq has been conveniently

*In the Appendices, kinetic energies are denoted by T ’ s to distinguish these from total energies, however, we use E ’ s herein because all treatments are based on non-relativistic approximations where no distinction is necessary.

Figure IV -1. Energy spectra corresponding to the phase space factor

calculation (curve 2), to the zero-range Born approximation calculation

(curve 3), and to the three separate components of the zero-range Born

approximation calculation (curves 4, 5, and 6). The experimental

spectrum (curve 1) is shown fo r comparison. Curves 1, 2, and 3 are

normalized to have the same area above 5 MeV.

ENERGY SPECTRAL SHAPES at flr = !2 .6 °

1 EXPERIMENTAL SPECTRUM2 PHASE SPACE FACTOR3 ZERO-RANGE BORN, I=Ip+IN + IpN4 « '• 11 IN ; Eq. (AH-16)

Ep in MeV

31

taken to be 5 MeV. As anticipated, the lack of agreement apparent here provides

evidence, for m ore complex reaction mechanisms which produce a matrix element

which does not have the constant value assumed in this simple analysis. In the

succeeding Sections in this Chapter, we shall consider some of these contributing

processes and attempt to evaluate their importance.

B. General Considerations of Theoretical Formulations*

In this section, we shall attempt to consider the transition matrix element

in general and investigate some possible approximations which w ill be actually

calculated in succeeding Sections.

The Hamiltonians in the asymptotic initial and final states may be

written as

a and d, and the deuteron internal wave function, while the final wave function

Lippman-Schwinger integral equation, the complete wave function asymptotic

H. = T + V (IV-2)1 np

Hf

T (IV -3 )

where

T = T + T + T a n p

The wave functions are correspondingly defined by

(IV -4 )

(H. - E) 4>. = 0 l i (IV -5 )

(Hf - E ) <f>f = 0 (IV-6)While the complete Hamiltonian of the system is

H = T + V + V + Vnp an ap

(IV -7 )

One must note that the initial wave function 4>. is composed of plane waves of

is composed of three plane waves of a , n, and p. With the help of the

to is expressed by

(IV-8)

* We are indebted to P ro fessor K. R. G reider for useful discussions of the theoretical considerations described in this section.

32

M = < * f ( - ) |V + V | $ . > (IV -9 )^ f an ap 1

The integral equation for the complete wave function asymptotic to

The transition matrix element is then written as

isl*..W > = [~1 + (\

np an ap

Consequently, the matrix element M can be rewritten as

*,<->>= +V + V U--- ] l * r >f L np an ap E - H - ie J f(IV-10)

M = < * I t l * > (IV-11)

whereby Eqs. (IV -9 ) and (IV-10),

t = f l + ( V +V + V ) - — ^-------- 1 (V + V ) ( IV -12)L np an ap E - H - ic J an ap

The calculation is now reduced to the problem of finding an approximation

for the three-body t-m atrix which is not only physically reasonable but also can

be calculated in term s of two body operators. In the present work, two different

approximations have been applied, and they w ill be described in the following

Sections.

C. Zero-Range Born Approximations

One of the simplest calculations is an application of Born approximation

in which only the firs t term in Eq. (IV-12) is taken into account, i .e .

t « t B = V + V ( IV -13)an an

Consequently, the matrix element M becomes

MB = < $ J V + V | $ . > (IV -14)f an ap

The initial state wave function 4>. is written explicitly as

iK • r iK • r ,Qij a a d . ^ . /ttt i r \

4>. = e e - r ) (IV-15)i d n p

where ,(r - r ) represents the deuteron internal wave function. The final d n p

state wave function is just the product of three plane waves.

iK ■ r iK • r iK • r* f = e “ f “ e n n e P P (IV-16)

33

In the calculations, only the rela tive S-state in the deuteron internal wave

function is considered and all effects of tensor forces are neglected. Under

these approximations for 4>,(r - r ), the fam ilia r Hulthen function could easilyd n p

be used. However, without introducing significant change in the calculation

and for reasons of sim plicity, the wave function actually used in the calculation

is the asymptotic form of the Hulthen function given by-a r ,

$ ,(r - r ) a —- p

Lnp

d n(IV —17)

np

where r = I r - r I , and the constant a is the characteristic wave number np n p

a = s/ -MQ /ft2

Using these wave functions, the matrix element of Eq. ( IV -14)

-ar

(IV-18)

A - d:

01 -n

f e iq“ r“P V d3r felq“•J 4r “ p “ J J

fe^ ^nv d3r I fJ a a J

n n _

np enp

lq ‘ r p pn e

np

-dr

d3rnp

(IV - i9 )pn

d3rpn

pn

where q. represents-the momentum transfer defined as

= K - K"a a

qP - j - Kp

q = ^ - K n 2 n

and the rela tive coordinate position vector r „ is given by

(IV-20)

rap

i

111

—. (IV-21)ran

= r - r a n

A further simplification has been introduced in the present calculation;

the interactions V and V are taken to be 6-function potentials. In fact, ap an

these interactions can be taken to be more rea listic potentials, and would be

expected to provide a better approximation; on the other hand, Zupancic et al.1,3,4,21

34

have shown that the same approximation satisfactorily explainsttheSr experimental

data with the present simplification. M oreover, simple kinematic considerations

have indicated that there may be strong effects due to the resonances in the a -n

and a -p systems and the Born approximations could not provide those resonance

effects in any case so that a more detailed calculation is not warranted. JJThe zero-range potentials fo r V and V give the matrix element M

an apin the form

Tt 1 1■ - ■ ■ ■■ + — ------- (IV -22)2 2 2 2 ' 7 q + a q + a _

n p

BM = const.

The cross section is expressed, using this value fo r the matrix element, as

l q + a q + a v n p

a = const. \ / ■■■ ■A n + — " x - - > 6(U - E - E - E )i \ o o 2 I a p n

3 3 2 (IV - 23)6(P - p - p - p ) d S d p d p a p n a p n

A calculation using this expression to predict the differential cross section as

observed in a single particle observation (in this case for proton measurement)

is presented in greater detail in Appendix II. The calculation has been carried

out using a computer code corresponding to Eqs. (A l l - 13) - (A II-16). An

example corresponding to an angle of observation 0 = 12.6° is shown inP

Fig. TV-1 where curve 1 is the observed experimental spectrum, curve 2 is

the phase space calculation discussed in Section A of this chapter and curve 3

is the present calculation. Curve 3 has also been normalized, as curve 2, to

the area of curve 1. Curve 3 is also decomposed into three components

corresponding to three term s in the square of the matrix element in Eq. (IV-22).2 2 2

Curve 4 corresponds to the term including l/ (q + a ) , curve 5 to the2 2 2 2 n

interference term l/ (q + a ) • l/ (q + a ), and curve 6 to the term including, 2 2 2 n P !/(q +a ) .

A s is obvious from examination of these figures, large discrepancies

between the zero-range Born approximation predictions and the experimental

data exist. This, of course, is not unexpected since this prediction has ignored

35

the previously recognized re la tively strong resonance effects in V and V .OP od?

The highest energy peaks in the observed spectra are given greater credence

as a further signature for the existence o f strong resonance effects in the

interaction V . M oreover, the zero-range Born approximation does not anappear capable of explaining the broad maxima occuring inThe proton spectra

in the intermediate energy regions; these broad maxima fcfoenualspiretnain:.as

possible signatures for the resonance effects in the interaction V, involving5 l6*p

L i intermediate state. In the same sp irit, the shoulders which appear between5

these two peaks may be interpreted as corresponding to the process via the He

excited state which is broad; the kinematic considerations do not exclude this

possibility. On the other hand,'the zero-range Born approximation appears to

have a sim ilar broad peak in the same energy region. However, as discussed

in the next section concerning the calculated predictions of a more complete

reaction model, unless resonance term s in the operator t given by Eq. (IV-10)

are involved in the calculation, the spectra cannot be reproduced.

Although the calculated spectrum described in this section failed to

reproduce any structure of the experimentally observed spectrum, the Born

approximation can be improved and may provide better information on the

reaction mechanism. The actual calculation can be easily extended to a

fin ite-range Born approximation in which the interactions V and V inan ap

Eq. ( IV-19) should be replaced by more rea listic potentials. The deuteron

internal wave function has been assumed to be asymptotic in the present ca l­

culation, while the complete Hulthen function could be used without introducing

difficulties in the actual calculations.

In conclusion, any Born approximation treatment is not expected to

provide a satisfactory interpretation of the experimental data, and we shall

develop a m ore complete analysis in the next section.

D. Distorted Wave Treatment

The experimental evidence, along with the fa ilure of the Born approxi­

mation, has indicated the importance of resonance effects arising from the

36

5 5interactions V and V leading to He and L i as intermediate state. It is,

an aptherefore, necessary to reconsider the matrix element so that the calculation

w ill exp lic itly show these effects.

A better approximation to the exact matrix element can be obtained by

retaining certain terms in the Eq. (IV -10). A physically reasonable and ca l­

culable form fo r t is obtained if the following assumptions are made.

Either (1) V^p in the final state can be neglected.

(2) V in the final state can be taken to be zero when the interaction ap

proceeds via V only, and vice versa . . . ( i .e . V is set equal an ap

to zero throughout Eqn ( IV -12).

or (3) V in the final state can be taken to be zero when the interaction ap

proceeds via V , and vice versa . . . ( i .e . V is set equal to ap ap

ze ro only in the propogator in Eqn.{ ( IV -12).

The assumption (1) is based on the argument that after the scattering due to

V (or V ) the neutron and proton are not likely to be found within their an ap

mutual interaction range for scattering in continuum states. Therefore, the

interaction V^p in the final state is likely to be unimportant unless there is a

n-p resonance. However, the only n-p resonance occurs in the T = 1 state

which cannot be rea lized in this reaction.

The basic justification for assumptions (2) and (3) lies in the fact that

the incident alpha particle and the target deuteron have widely different binding

energies; while the deuteron is rela tively loosely bound structure, the alpha

particle may be considered as a rather tightly defined entity. Quantitatively,

the separation between the proton and neutron in the deuteron may be taken

crudely as 1/^ -MQ/ft2 s: 4.3 fm, while the a-nucleon interaction is assumed

to have a significantly sm aller range than this (for example, in the Gammel- 37

Thaler potential , at an interaction radius of 4 fm , the interaction potential

has already fallen to less than 1/20 of its value at the edge of the core). Conse­

quently, it appears obvious that there should be many instances in which the

alpha particle interacts with one of the nucleons in the target deuteron, but

37

not with both, leading to more or less direct knock-out or quasi-free process.

Therefore, term s in the matrix element involving both V and V togetherap an

w ill be neglected; that is, term s due to assumption (2) are assumed to be

la rger than those due to assumption (3).

Following assumptions (1) and (2), Eq. ( IV -12) can be written as

where

D D D t ^ t = t + t

an ap

tD s an

1 + Van E - T - V + ie

anV

an

(IV-24)

(IV-25)

tD _ap l + V

ap E - T - V + i f ap

Vap (IV-26)

Under this approximation, the matrix element is then written as

M ° = < <f>f | tD I * . >

= <4> |tD I 4>.> + <4> I t ° I 4>. > f an i f ap i

It is noted that t^ n is ^ ^ ^ k o d y t-m atrix for free a -n (a -p ) scattering

(IV-27)

and illustrates a quasi-free process due to the presence of <f>. which includes

the initial deuteron wave function. The actual calculation using this matrix

element was made by Nakamura, and here only b rie f description is given to

understand the physical feature of the calculation.

It is noted that in Eq. (IV-27) t13 (t13 ) does not involve any interactionan ap

concerning p(n), thus in the firs t (second) term of Eq. (IV-27) the proton (neutron)

part can easily be factored out resulting in

M ° = <4> | t ° |4>. > 4> ( p ) + < ^ . I tD 1$. > $ (q) (IV-28)f, an an i , a n ^ d f, ap ap i, a p ^ d

*d(p> = y -ip • r ^ 3-» .e 4>,(r) d r, etc.

d(IV-29)

where p and q are the final momenta of p and n respectively, and <f>d( f ’) denotes

the deuteron internal wave function. # and $ are the free wave functionsan ap

38

fo r the a -n and a -p systems. This calculation may be considered to be a type

of distorted wave treatment or alternately, a quasi-free two-body t-m atrix

approach. It should be noted, however, that because the two-body t-m atrices

in Eqs. (IV-25) and (IV-26) are used in the three body problem, the off-energy

shell matrix elements must be known. Furthermore these matrix elements

are required fo r a wide range of initial and final state momenta because of

the phase space integral, Eq. (IV-23). The model used to obtain the off-energy

shell behavior is described by Nakamura.

The actual calculations have been perform ed under the following

approximations.

(1) The interaction potentials V and V are taken to be equal toan ap H 37

each other and given by the phenomenological Gam m el-Thaler potentials

V = + 00

V c(r ) + V LS(r ) L

r < r — c

r > r — c

(IV-29)

(IV-30)

v c(r ) = (IV-31)

n2 hV (r ) = - —---- — V (r )

LS r dr V ' VLS

(IV-32)

for the S-State

r = 0.5 x 10 c

•13cm

-13R = 2 x 10 cm

D = 1 x 10-13

cm

V = -55.4 MeV c

(IV-33)

for the P-State

-13r = 0 .1 8 3 x 10 cm

c

-13R = 2 x 10 cm

-13D = 1 x 10 cm

V = 46. 6 MeV c

(IV-34)

V = 22.8 MeVLib

39

(2) A ll Coulomb forces are neglected.

(3) The partial waves appearing in the expansions of t ° and tD area n ap

terminal at p3/2 -

(4) The deuteron internal wave function ^ ( r ) is again assumed to be

the asymptotic form of Eq. ( IV -17).

(5) The Gam m el-Thaler potentials, although of phenomenological

origin are assumed to be valid for calculating t and t off the energy-shell.an ap

The assumption (2) is made for sim plicity and some discrepancies

should be expected (see next Section of this Chapter). Since the partial waves

higher than D-state are known to have very small phase shifts and particularly

since p 3//2 resonances are most likely important, assumption (4) is expected

to be reasonable. The deuteron internal wave function is again approximated

by its asymptotic form , however, this should be subjected to further test to

examine the detailed dependence upon the choice of deuteron wave function

used. In the actual calculation, an attempt has been made to use also a

6 -function for the deuteron internal wave function in order to check the effects

due to the deuteron form factors.

Under these approximations, the calculated results have been obtained

for the energy spectra at an observation angle of 12.6°. The results are

shown in F ig. IV-2, where the experimental spectrum (curve 1) is again

shown for comparison. Curve 2 shows the calculated spectrum with an assumed3

deuteron internal wave function 4>^(r) = 6 (r ) while curve 3 corresponds to

that with (r ) = e /r. These three curves have been normalized in d

F ig. IV-2 such that the peaks in the intermediate energy region have comparable

heights.

In order to provide intuitive understanding of the calculation procedures,

we shall attempt to sim plify these procedures and illustrate the physical meaning

in the following paragraphs. One should bear in mind, however, that the

illustration is not quantitative as a result of the simplifications introduced for

discussion purposes.

Figure IV-2. Energy spectra calculated in a distorted wave treatment (curves 2 and 3). The points shown are actually the calculated ones. The experimental spectrum (curve 1) is shown for comparison. All three curves are normalized to have the same ordinate at the middle energy peaks.

ENERGY SPECTRAL SHAPES at 9 = 12.6° ■ ■■ ■- p = =

1 E X P E R I M E N T A L C U R V E

2 C A L C U L A T E D C U R V E with (7) = 8 3( 7 )

3 C A L C U L A T E D C U R V E with exp ( - a r / r )

E p in MeV

40

Considering the kinematics of the outgoing particles at a particular set of observation angles, of say the proton and the neutron, the kinematic relation­ship of their momenta p and q is given by an elliptic locus in the p - q plane.An example of such an ellipse at a selected set of angles for a, p, and n is shown in Fig. IV-3(a). It should be noted that to each point on this ellipse there corresponds a unique momentum of the outgoing Ol . A negative value of p does not, of course, correspond to a real situation, but does correspond to that in which the proton is emitted in a backward direction. Assuming a case wherethere are particular resonances in V and V these resonances occur whena n a pthe interacting particles have the relative momenta appropriate to excite theresonances. Thus it is noted that in the laboratory system the resonancesoccur at two different momenta for each resonance point. For example, inthe a-n case the first is that at which the alpha moves faster than the neutronand vice versa for the second. Moreover, characteristic of double values inthe p - q relationship, four resonance yield points are noted for each actualresonance in V or V .In Fig. IV-3(a), the eight resonances in the a-n an apand a-p interactions under consideration, particularly the P , state resonances

5 5leading to He and Li ground state, respectively, are shown superposed on the p - q elliptic locus. In this figure, however, all other details such as non-resonant parts in each partial wave or the interference parts are ignored for reasons of clarity although they play a significant role in the actual cal­culation. If these resonances are projected onto the p-axis as shown schematically in Fig. IV-3(b), a momentum spectrum is obtained reflectingonly the contributions from \P and t^ in Eq. (IV-28) in this simplifieda n a psituation (noted by I's in the figures).

As shown in the matrix element Eq. (IV-28) the deuteron form factors2 2 2contribute to the cross section in the form of I $ ,(p) I = l/(p + a ) or

2 2 2 2I = 1/(<1 + ot ) (this is obtained from Eq. (IV-29), and may beainterpreted as a reflection of the momentum distribution of the proton or neutron in the deuteron). The curves shown in Fig. IV-3(c) are approximate

41

shapes for each component where a semi-log scale is used for the I Iaxis versus q or p as abscissa. The cross section is obtained by combining

2 2 2 2 2 2 these two components, I /(q + a ) and I /(p + a ) . Such a cross sectionP ^curve is shown schematically in Fig. IV-3(d) where the momentum scale phas been converted to the energy scale E .P

The final cross section must of course be obtained by folding in thephase space factor with the curve just described. In the single particlemeasurements, the cross section is further modified by the integrations withrespect to the unobserved variables in the cross section calculation discussedhere. At a different set of angles, there may be constructed a differentdiagram corresponding to Fig. IV-3 and the integration gives the finalmomentum (or energy) spectrum equivalent to curve 3 in Fig. IV-2. If theprocess corresponding to Fig. IV-3(c) is omitted in the above, the prediction

3becomes that for <i> (r) = 6 (r) and the final result is shown as curve 2 in dFig. IV-2.E. Discussion

From a comparison of these results with the experimental data, the following conclusions may be derived:

(1) The broad peak in the middle energy region which has been anti-5cipated to reflect the Li ground state intermediate process can be well ex­

plained, although curve 3 shows a peak at a lower energy. This energy discrepancy arises in a quite obvious fashion from the fact that in ignoring the Coulomb interaction in the a-p scattering calculation, the energy has been shifted (in fact the phase shift in t*16 Present calculation shows theresonance at approximately 1.1 MeV instead of 2 MeV as know experimentally).

(2) The calculated spectra show the peak at the highest energy end 5and this indeed corresponds to the resonance of the a-n interaction (He ground state) in which the Coulomb difference does not come in and the observed energy is in good agreement with the experimental data. However, in curve 3 the peak is considerably depressed compared with the lower energy

2

Figure IV-3. A schematic illustration of aspects of the distorted wave calculation.

(a) Kinematic interaction locus in the space of the neutron momentum (q) and the proton momentum (p). p3/2- resonances are shown on this locus as superimposed peaks.

(b) Momentum spectrum corresponding to (a).(c) Deuteron form factors as functions of proton or neutron

momentum.(d) Final energy spectrum. This figure illustrates, at a

given angle of proton observation, how the various factors appearing in the cross section expressions contribute.

42

region. This is because the deuteron wave function contributes a (l/(p + a ) ) term which is very small at this high proton energy (it is shown schematically in Fig. IV-3(c)). This effect also explains the shoulder appearing at about 20 to 25 MeV in the spectrum. These discrepancies may suggest that the deuteron internal wave function used here is not entirely appropriate for the situation involving high momentum transfer. It would be of interest, for example to re-examine this situation using a more realistic deuteron wave function.

(3) It is noted that the P, resonance in V and V interactions1/ 2 an ap

cannot be expected to yield a resonance type spectrum in the present cal­culation since the Gammel-Thaler potential used did not give a phase shift 6pi/ 2 more than about 65°, instead of the 90° characteristic of the resonance. Consequently, it is necessary to reconsider the a-nucleon potentials in anyattempt to more correctly analyze the spectral shape at the energy regions

5 5corresponding to the He and Li excited state as intermediates. Following these arguments the broad shoulder appearing in the experimental curve at about 20 to 25 MeV still remains to be identified unambiguously.

(4) At very low energy, the calculated curve 3 of Fig. IV-2 shows a sharp rise, and it has been found that this rise primarily reflects the lower resonance peak of Pg/ 2 ^~3(a) the a-p resonance peak (g)). Sincethe low energy part of the experimental spectra has considerable uncertainty in any case a further check of this discrepancy should be made to a more accurately measured experimental spectrum or one obtained with higher incident energies where this resonance is expected to fall in a higher energy part of the product proton spectrum.

(5) The a-p resonance peak appears lower in curve 2 than in curve 3,and this has been understood from the fact that the deuteron wave function

3(r) = 6 (r) does not depress the resonance peaks on the upper branch indFig. IV-3(a) (the a-p resonance peaks (e) and (f)). Thus, the resultant peak appears at a lower energy than the peak (h).(When the integral is performed,

2 2 2

43

those four peaks combine and diminish the valley appearing in Fig. IV-3(d).i)This is, of course, not realistic since the deuteron is well known to have a large spatial distribution instead of the point distribution assumed in this case.

(6) The calculated total cross section from 5 MeV to the maximum energy is found to be approximately 1. 2 barn/str. compared with the estimated experimental value (3.3 + 1.2) barn/str.* This difference may be understood partly from the argument in (3) above.

In summary, it may be concluded that the theoretical calculation referred to herein has at least provided a qualitative understanding of the dissociation reaction mechanism. Certain remaining discrepancies reflect inaccuracy of the model taken in this calculation as well as several technical approximations involved in the actual calculation, and these should be improved. Several suggestions for such improvement of the calculations are mentioned below for possible extension of this study.

(1) The model used in the present calculation has neglected some interactions which obviously exist.

(a) Coulomb interaction - this can be included without essential difficulty.

(b) The assumption (3) instead of (2) in the construction of t -matrix may represent a significant component of the process.

(c) The final n-p interaction - as also mentioned previously, this may be considered to be an insignificant correction but one which should be examined again in a more accurate analysis.

(d) The internal deuteron wave function - as discussed above, this should be subjected to a more extensive test and other variants including those with hard cores should be included.

* For this reason, the calculated curves in Fig. IV-2 are not normalized in terms of the absolute cross section but are plotted such that the peaks in the intermediate energy region have comparable heights to the experimental curve.

44

(e) The higher relative momentum states in the interactions - for example D-waves in the a-nucleon scattering are known to be small but the interference terms might be important. The present formula is adequate for their inclusion.

(2) The numerical calculation in the present work was relatively crude; for example, the phase shifts in the a-nucleon scattering were not in complete agreement with the experimentally known data, in particular, (>Si/ 2 differed by a relatively large amount from experiment reflecting remaining unsatis­factory behavior of the potentials used herein as indeed suggested by Gammel and Thaler. This can be improved by adjusting these potential forms. The numerical integrations, as well as differentiations, may have introduced relatively large errors. These can be improved by extending and modifying the computer program.

(3) From an experimental point of view, the following may be suggested to enable a more complete check of the theoretical and experimental aspectsof the problem:

(a) An accurate measurement of the absolute cross section - this provides strict information regarding the importance of other components included in the reaction process.

(b) Re-examination of low energy tails as discussed in Chapter II, Section E.

(c) Accurate measurement of the low energy part of the energy spectra and the spectra at backward angles - here the a-n final state interaction plays a more important role whilein the present case the a-p final state interaction is of major importance.

(4) Coincidence measurements are essential to determine uniquely the kinematics of the three body reactions. It is, therefore, of interest to consider them in some detail.

45

For example if a coincidence measurement with two directional detectors is carried out to pick up the kinematical configuration such as shown in Fig. IV-3(a), and if those two detectors (in this case one for the neutron and the other for the proton) are energy sensitive, one can measure cross sections along this elliptic locus. This measurement would provide individual information for each resonance region. If there were no resonances in the final state inter­actions, the cross section would be directly given by the curves shown in

—0l rFig. IV-3(c) provided that the deuteron wave function (r) = e /r is adrealistic form (of course, the phase space factor must still be folded into

13 4these curves). This latter case has been found by Brookhaven group,to agree with the experimental results obtained in the study of deuteron breakupprocesses induced by protons and it has been found that the zero-range Bornapproximations discussed in Section B in this Chapter provide the agreement.Consequently, the discrepancy found in Section B in the present work is quiteunderstandable if one considers the existence of strong resonances in theV and V interactions in the a + d reaction, while the p + d reaction, an apno strong resonances are expected in the final state interactions. If there isno resonance in the final state, interactions, I and I are smoothly varyingn palong the elliptic locus shown in Fig. IV-3(a) and the spectral shapes are

2 2 mainly determined by the I 4>d(q) I and I $d(p) I terms.

46

Deuteron breakup reactions induced by alpha particles have been sub­jected to a comprehensive study, both experimental and theoretical, in an attempt to elucidate the reaction mechanism. Incident alpha particle beams of energies at 41. 6 MeV and 29. 3 MeV have been utilized to perform single par­ticle measurements. Energy spectra of alpha particles and of protons have been observed at various angles using a particle identification system.

As anticipated the energy spectra of protons have provided information concerning the reaction mechanism but the energy spectra of alpha particles were too complex to be analyzed. The process was identified as a three body breakup reaction from the maximum energies and the continuous energy dis­tribution observed as expected from simple kinematics. Several characteristic peaks in the energy spectra of protons have been analyzed. A simple kinematic consideration could assign peaks which appear at the high energy end of each

5spectrum to a two body sequential decay via the He ground state as an inter­mediate state. A zero range Born approximation calculation was not successful in confirming any characteristics of the experimental results. On the other hand,this simple calculation was in excellent agreement with the experimental data

12 3of deuteron breakup reaction induced by protons. ’ ’ However, the character­istics of the final state interactions of those two reactions are expected to be very different and this difference may explain the disagreement in the present work.

A distorted wave calculation including the final state interactions between a-n and a-p but not p-n has been performed. In spite of a crude treatment of the actual calculation, the evaluated spectral shape provided good agreement withithe experimental result over almost the entire energy range of the energy spectrum. Quantitatively, however, it has been found that there are discrep­ancies, especially in the high energy region. Several suggestions have been noted to improve the theory such as applying a more realistic deuteron internal wave function or including a three body interaction. It would also be desirable

V. S U M M A R Y

47

to make more accurate experimental measurements including a total cross section measurement, measurements of the very low energy region of the spectra and coincidence measurements.

48

Since the nuclear reaction studied in the present work deals with three emerging particles in the final state, the total number of participating particles in both incident and emerging particles is 2 + 3 = 5, and the kinematics be­comes very complicated. In this Appendix, the kinematic relations used will be investigated.

The energies and cross sections in a reaction with three emergingparticles in general depend on 3 x 3 = 9 variables specifying the three velocityvectors of the merging particles, and, on the initial energy. Momentum andenergy conservation laws expressed by four equations reduce the number ofindependent variables. The fact that the beam and target are not polarizedprovides the beam direction as an axis of azimuthal isotropy and reduces thenumber of independent variables by one. As a result, there are 9 + l - 4 - l = 5independent variables.

It is thus understood that a complete determination of the reactionprocess cannot be achieved by detecting only one of the emerging particles; adetection of two of them in coincidence is needed as the minimum requirement.A single particle measurement such as that in the present report is necessarilya measurement of values integrated over all other undetected kinematic variables.

As a special case, if a reaction is assumed to be a two body sequentialprocess, i.e., M + M - M + M , M -» M + M , the kinematics may be 1 2 3! 5 3f 3 4solved exactly by combining the usual two body treatments. The case for the three body process is considered in Section A and the case for the two body sequential process in Section B of the Appendix.

* The kinematics of multiparticle reactions is discussed in greater detail in C. Zupancic, Hercegnovi Lecture, 1964, Appendix I (to be published).

A P P E N D I X I. KINEMATIC RELATIONS*

49

A . Three body processThe momentum and energy conservation laws are expressed by

P = P, + P = P + P + P (AI-1)1 2 3 4 5 v

U = T + T + Q = T + T + T_ (AI-2)1 2 3 4 5The subscripts 1 and 2 denote the incident and target particles, respectively,

—+and 3, 4 and 5 denote three emerging particles. P's are linear momenta and T's are kinetic energies as usual.* Q is the Q-value of the reaction considered and defined by

2Q = (m1 + m 2 - m g - m 4 - m 5)c (AI-3)

m's and c being the masses and velocity of light, respectively. It is noted that the above equations are still exact in relativistic theory consideration.The non-relativistic approximation** is introduced when the kinetic energy T is replaced by

1 2 PT = -7 mv = — (A 1-4)2 2mThe energy of the observed particle may be evaluated from the above equations. As discussed above, one would not expect to have a unique relationship between energies and observation angles for a single particle measurement, but a continuous distribution from zero to a certain maximum energy. This maximum energy is calculated later in this Appendix (cf. Eq. (AI-23)).

The effective kinematics in cross section will be investigated next.The cross section is in general written in terms of perturbation theory as:

d6 = VEST- <STK)5 I I MI 2 6(U - T3 - T4 - t 5 ) 63 ( p - P3 -P 4 - P5 > d3p3 d 3p4 d3p5

* Hereafter kinetic energies are denoted by T's to distinguish them from total energies.

** The relativistic corrections have been estimated for the energy equations discussed in this Appendix and it was found that the errors introduced were less than 0. 3% which can be neglected in the present experiment.

(AI-5)

50

M is the matrix element for the reaction under consideration. Omitting the matrix element, we define a quantity I as

The quantity I is the phase space factor for three final particles. The 6-functions insure the conservation of momentum and energy and determine the regions of integrations. Furthermore, by having the 6-functions in the integral, any coordinate system may be chosen. The straight-forward cal­culation is described in a number of references.*

— »

Integrating over p in Eq. (AI-6) and using (AI-4),D

where dft is the solid angle for the j-th particle under consideration. A conventional expression for an integration involving a 6-function has been

a schematic diagram of momentum vectors is shown in Fig. AI-1. The argument of the 6-function in Eq. (AI-71) is rewritten as

* The phase space factor can be calculated more easily by using the standard Fermi's statistical treatment of multiparticle productions. However, since the treatment described here is useful for when the matrix element is not equal to zero, the alternate rather detailed description has been included.

(AI-6)

(AI-7;)

used, i.e.,(A I-8)

where xq is a solution of g(x) = 0, and f(x) and g(x) are arbitrary functions, g' (x) denotes the derivative of g(x) with respect to x.

Defining momentum p by

(AI-9)

5(A I-10)

51

P is determined uniquely when p is fixed and is not a function of T , henceo 4the derivative of (AI-10) with respect to is,

6 m 4 P4 ' P5— ^ ( D - T - T ^ T > ] - ! - — — j - (AI-11,

4 5 P4

The integration for in Eq. (AI-7') using Eqs. (AI-8) and (AI-11), becomes

■J £ 3 3 4 rn 25 P4

dT dfi dSL (A I-12)

While a coincidence measurement of particle 3 and 4 provides enough information for the form (d6/dT dfl dfl ), a single particle measurement,u u 4say particle 3, gives information for (d6/dT dfl ). It is, therefore, necessaryO uto perform the integration for dfi . In order to evaluate this, it is easier to start from Eq. (AI-7) in a slightly different form.

9 ' 90 P4 P= o o

1 -J 6(E - 2=■- S-J' d P3 d 54 (AI-13>4 5Here E = U - T and Eq. (A 1-4) has been used. The problem is to integrate othe above equation over p for a certain p . From Eqs. (AI-2), (AI-4), ando(AI-9), 2

,1 1 * 2 pcosG p t, .. T(5 ? r + 3?r,p4 + ( *4 5 5 5

This is the equation that gives the allowed region for p^. Rewriting the coefficients in the above equation as

,1 1 ,

A ~ (2 ? ^ + 2^7,4 52 B = E cos6

m 52

C - E2m 5

52

Let the double solutions of this quadratic equation be and P,„.41 42B- B -AC

41 A

B + B2-AC42 A

(AI-15)

Defining I' byQ—♦

r d p32 2

( V P4 P5 . ,3-«= 3 6(E-i^r-i^r)dp4

(AI-16)

4 5The integration with respect to 0 in Fig. AI-1 which is the azimuthal angle of P4 around p yields 277 and I' becomes

rI' = 277 \ 6(Ap42 - 2Bp4 + C)p4dxx° „ (AI-17)

f 1 P4"J.. IAp, - BIx 4 o

where x = cos 0, 0 being the angle between p4 and p, and xq corresponding tothe cosine value at the maximum 0 allowed at which point x = AC/D.oFig. AI-l(b) shows the vector diagram of p , p_ and p, the plane defined by4 5these three vectors. The two intersections of p4 on the circle correspond tothe double solutions of Eq. (AI-14). The integral extends from 0 to 0.maxThe integral may be written in the following fashion by splitting it into two parts. i 2 2

oAfter trivial calculation, this can be reduced to the form

I« = -2- (D2 _ AC)1/2 (AI-19)A 2

where D = Bx or D = P/2m_. Finally I may be expressed by integration with5

Figure AI-1. Momentum diagrams in the coordinate system chosen inthe general calculation reported herein. Part b is the projection of Part aon a plane defined by p and p_.

4 5

1 *

FIG.AI - 1

53

only the coordinates of particle 3 as variables

1 = 4 v (■m„ m 4__5_ ,2m + m 4 5

(---- +— ) ( U - T ) - -----------2m. 2mr 3; 2m, 2mr4 5 4 5

1/2 ,3-»d Po

: ni, m. „ ~= 4ff (— V 2 m \ ( T,m + m 4 5 j (— + — )(U-T )--- --m, m,_ 3 2m 2m4 5 4 5.

ll/2(A 1-20)

dT3 dO,

Again the calculation may be carried out in any coordinate system. If the center of mass system is chosen, I is

,. 4, (2 . 1/2 W « rm. + m. v m + m ' J 1 3

m, + rm— --- U-Trn + m + m. 3

a / 2

(AI-21)dT dfl 3 3

since p =2m„T„. ^ 3 3If the laboratory coordinate is chosen, I is

(m4 + m 5)U-m1T 1 -(m3 + m 4 + m 5)T3I 4„/ 2 ,1/2 ,m 3 + m 4 + m 5 3/21 = 471 ( „ : • ) ( ' , _ )m +m_ 4 5 m, + rn 4 5 i t

+ 2 /m1 m^ T 1 cosSg /T^ U/2 (A 1-22)dT3 dS23

smce p = px - p3-The maximum energy allowed for particle 3 is obtained by equating the

integrand to zero. mmax

/ m lm 3T l COSe3 +{ m lm 3T l COs203 + (m3 + m 4 + m 5)

(m4 + m 5) U _ m iT l1/2 (AI-23)

B. Two body sequential processThe process may be expressed by

M , + M -* M + M + Q1 2 3' 5M . M + M + Q3 4 c

(step 1)

(step 2)

54

The energies of particles 3, 4 and 5 are necessarily calculated in terms of their scattering angles. The usual two body non-relativistic formulas are applied. *

For step 1 in the laboratory coordinate system,

for given T and 0 .■J- OFor step 2, choosing the coordinate system in which the particle 3 is

at rest, the energies of particles 4 and 5 are

The prime identifies the coordinate system chosen. From these energies, thevelocities of particles 3' and 5 in the laboratory coordinate system, andparticles 3 and 4 in the center of mass coordinate system can be obtained.From vector addition of these velocities, it is possible to obtain the velocitieshence energies of the observed particles in the laboratory coordinate systemas indicated in Fig. AI-2. The final results in the laboratory coordinate aredifferent, i.e. V and V ' o o

1/2(A 1-24)

(AI-25)

0 = cosooos03' (AI-26)

(AI-27)

(AI-28)

(AI-29)

03 = 03' + C°S (A I-30)

* See footnote ** on page 49.

Figure AI-2. Velocity diagram corresponding to the calculation of the reaction kinematics shown in Figures AI-3, AI-4, AI-5, and AI-6.

FIG.AI-2

where 2TV 3' (AI-33)3' m 3'

(A I-34)

2 m 3m„ rn + m 4 3

Qc (A 1—35)‘4

V3 = V 3'2 + V3'2 + 2V3. V COs6 (AI-36)

(A 1-37)

and 0 < 0 < 2it.The actual computations required for several cases of interest here

were carried out using the IBM 709 computer at the Yale Computer Center.Some of the results are shown in Figs. AI-3, 4, and 5.

It is important to note that in these two body sequential processes,emerging particles have definite angular relations. For example particle 5and either particle 3 or 4 are correlated by 0 and 6 or 6 , while 0 and 05 3 4 3 4are spread over a finite angular region. This effect precludes utilization of the one to one correspondence involved in simple binary coincidence studies unless the counter for particle 3 or 4 is of sufficient aperture to cover the entire angular range of interest. In general, such an angular aperture auto­matically results in kinematic broadening of the peaks and makes their identification in the spectra impossible.

Figure AI-3. Kinematic diagram for various exit channels in the a-d reaction at an alpha particle energy of 41.6 MeV.

Figure AI-4. Kinematic diagram for protons from various exit channels in the alpha particle plus deuteron interaction at an alpha particle energy of 41.6 MeV.

Figure AI-5. Kinematic diagram for protons from the alpha particle plus deuteron system for an alpha particle energy of 29.3 MeV. The legend is the same as that of Figure AI-4.

( T) d; E l a s t i c

© a ; E l a s t i c

© p; a + d = H e | s+ p + Q , = a + n + p + Q', Q , = - 3 . l 8 2 M e V

© a ; a + d = H e g S+ p + Q| = a + n + p + Q ' | Q j = + 0 . 9 5 7 M e V

p ; a + d = L i 6.S.+ a + Qg = a + p + n + Q2 Q2 = — 4 . 1 9 2 MeV

a ; a + d = L i g g . + n + = o t n + Q*2 ° 2 = + * - 9 6 7 MeV

KINEMATICSFor a - d Reactions Ea = 41.6 MeV

® p: a+d —►He5 +p—*-a+n + pG.S.

© p: a+d —►Li0 s +n—►a+p+n

p: a+d — ►He5 (2MeV) + p—►a + n + p

© P- a + d5*

—►Li (2 MeV) + n—►a+p + n

KINEMATICS for PROTONSF RO M a - d R E A C T I O N S

Ea = 2 9 . 3 MeV

® P

P

a + d — ► He _ _ + p — ►<* + p + nG.S.5

a + d — ► L i + n — ► a + p + n

: a + d — ► H e (2M eV ) + p —► a + p + n

: a + d — ► L i (2 MeV) + n -^ a + p + n

80° 100° 120°

56

A P P E N D I X n. Z E R O R A N G E B O R N A P P R O X I M A T I O N

In Chapter IV-C, the reaction cross section has been considered within the framework of the plane wave Born approximation and the expression for this cross section has been reduced to the form given by eq. (IV-22), which is not susceptible to analytic solution. In consequence, a numerical calculation has been made using the computer. A brief description of this calculation is given in this Appendix.

Eq. (I;W-22) is

dcr = c o n s t . \ < -5^ — 2~ + ~ 2 ~— 2 ) 2 5 ( U - T - T - T ) f i 3 ( P - p - p -?p )^ L S + qp J a p t? v a P‘ V

d3 p d3 p d3 p (IV-11)a p 17

Using eqs. ( AI-16) and ( AI-17) in Appendix I the cross section can be expressed, omitting the constant in front, by

(An-1)2

f (p4i> -

f ( P41) + P42 f(P42) Id x (AII-2)|Ap42-E| J

12 2 + z 'l ) , i - 1,2 (AII-3)

- V +a qPi J

\ r i « J = Um1

v = ' v 1 “ lp3i' <A n -4)

—* Kd K d ■”* —*smce q^ = — _ ’ qp = ~~2— ” p ’ and d = G abora ory system.And p^ should satisfy eqs. ( AI-14) and ( AI-15) in Appendix I. The subscripts 3,4 and 5 designate proton, alpha particle, and neutron in the final state, re­spectively. The notations are the same as in Appendix I, Section A. I' can be

57

written in an alternate form as 1

I'AC P412 F < V + P42 V 2> dx (AH-5)

This is still an inconvenient form because it dcverges at x = x . This divergence can be avoided by performing a partial integration. A function F (x) is introduced by a definition

F (x) = P41 f ( P41* + P42 f ( P42*P41+ P42

2BThen I’ is written as

p.,2 f<p41> + p422 £<p42)41

I ' = 471 \P41 + P42

Xo 7 b 2-ACF (x) dx

(AH-6)

(All-7)

8nAD 'D2-AC F(l) - / o2x2-AC dF(x)

Xo(AII-7')

It can be seen that if f ( p ) = f ( p ) = 1 is substituted, I' is reduced to thefr 1 14phase space factor discussed in Appendix I. Since f ( p ) and f (P42) CDntain three terms, each of which has different physical meaning, an attempt has been made to separate the function F (x) into three corresponding terms. Also one more term (x) is defined corresponding to the phase space factor. The com­putation of the term F (x) provides an opportunity to check the program by com­paring the results with hand calculations given from eq. ( AI-22). The explicit forms are as follows

F (x) = F (x) + F ,(x) + F (x)p ’ 7]T)

y x )=^ - 41(P2 + a3)2

42(P32 + * “)

(All-8)

(AII-9)

58

2 p 2 2 p 241_________ + P42(Pg + O54) (P51Z+ Of2) (Pg2"1-^) (Pgg^0 )

% (X) 2B 41 + P42

and Fq ( x ) is taken separately as

Fo<x> = I F -

The eq. ( AII-7) is thenP + P P41 42

I’ = I + I + I P V

I =p AD 7d 2-AC Fp(l) - J d 2 x2-AC dF p(x)

8 it

I =

AD

01

/d 2-AC F (1) - \ y D2x2-AC dF (x) P~ P„V J

8nt] AD

and the phase space factor I

— X

Jd 2-AC F (1)- \ v/dx-AC dF (X)

(A 11-10)

(An-ii)

(All-12)

(An-13)

(All-14)

(An-i5)

(AII-16)

I = -§*- o AD /d 2-AC F (1) - \ J D2x2-AC dF o' J - o (x) (An-i7)

These were calculated using the computer and a simple program written for this purpose. A simplified Simpson method was used for numerical integrations.The results are shown in Fig. IV-1.

59

1. P. F. Donovan, Gatlinberg Conference, Oct., 1964.2. K. M. Watson Phys. Rev. 8j8 1163 ( 1952).3. C. Zupancic, Gatlinberg Conference, Oct. , 1964.4. P. F. Donovan, J. V. Kane, C. Zupancic, C.P. Baker and J.F. Mollenauer,

Phys. Rev. 135 , 1361 ( 1960).5. G.C. Phillips, T.A. Griffy and L. C. Biendenharn, Nucl. Phys. 21,

327 ( 1960).6. F.C. Barker and P. B. Treacy, Nucl. Phys. 38, 33 ( 1962).7. W.H. Barks and M.G. White, Phys. Rev. 513, 288 ( 1939).8. R. L. Hankel, J.E. Perry Jr. andR.K. Smith, Phys. Rev. 99, 1050

(1955).9. M.P. Nakada, J.D. Anderson, C.C. Gardner, J. McClure and C. Wang,

Phys. Rev. 110, 594 ( 1958).10. A.T. G. Furguson and Morrison, Nucl. Phys. £L 41 ( 1958).11. L. Cranberg and R. K. Smith, Phys. Rev. 113, 587 ( 1959).12. C. Wang, J.D. Anderson, C.C. Gardner, J. W. McClure and M. P. Nakada,

Phys. Rev. HJB, 164 ( 1959).13. B. V. Rybakov, V.A. Sidrov, andN.A. Vlasov, Nucl. Phys. 23_, 491 (1961).14. R.M. Frank and J. L. Gammel, Phys. Rev. 93_, 463 ( 1954).15. W. Heckrotte and M. MacGregor, Phys. Rev. Ill, (1958).16. R.J.N. Phillips, Nucl. Phys. 53, 650 (1964).17. J.C. Allred, D. K. Froman, A.M. Hudson, and L. Rosen, Phys. Rev.

82, 786 ( 1951).18. E.W. Warburton, and J.N. McGruer, Phys. Rev. 105, 639 ( 1957).19. K.P. Artemov and N. A. Vlasov, JETP 12, 1124 ( 1961).20. H.W. Lefevre, R.R. Borchers, andC.H. Poppe, Phys. Rev. 128,

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R E F E R E N C E S

60

24. For example, a summary of the system is described in M. W. Sachs, Doctoral Dissertation, Yale University ( 1964) ( unpublished).

25. C. Chasman and J. Allen, Nucl. Instr. and Meth. 24 , 253 ( 1963).26. G. T. Garvey , J. Allen, Yale University ( 1963).27. J. Hiebert, J. Allen, Yale University ( 1964).28. E.A. Silverstein, Nucl. Instr. and Meth. , 4, 53 ( 1959).29. H.E. Conzett, G. Igo, H.C. Shaw, and Slobodrian, Phys. Rev. 117,

1075 ( 1960).30. W. T. H. Van Oers and W. K. Brockman Jr. , Nucl. Phys. 44, 546 ( 1963).31. H.J. Erramuspe and R. J. Slobodrian, Nucl. Phys. 49, 65 ( 1963).32. H. W. Broek and J. L. Yntema, Phys. Rev. 135 , B678 ( 1964).33. R. G. Freemantle, T. Grotdal, W.M. Gibson, R. McKeague, D.J. Prowse,

and J. Rotblat, Phil. Mag. 4JL 1090 ( 1954).34. E. Fermi, Prog. Theorc Phys. 5, 570 ( 1950).35. B A. Lippmann and J. Schwinger, Phys. Rev. 79, 469 ( 1950).36. B.A. Lippmann, Phys. Rev. 102, 254 ( 1956).37. J. L. Gammel and R. M Thaler, Phys. Rev. 109, 2041 ( 1958).