University Microfilms, A XEROX Company, Ann Arbor, Michigan

70-22,778

SKOPIK, Dennis Michael, 1942-

A STUDY OF THE REACTION 4He(e,e'd)2H.

The American University, Ph.D., 1970 Physics, nuclear

University Microfilms, A XEROX Company, Ann Arbor, Michigan

A STUDY OF THE REACTION Sle(e,e'd)2H

by

Dennis Michael Skopik

Submitted to the

Faculty of the College of Arts and Sciences of the American University

in Partial fulfillment of the Requirements for the Degree

of

Doctor of Philosophy

in

Physics

Signatures of "--- — Chairman

Dean of the College

UNIVERSITY THE AMERICA* UNIVERSITY

MAY 2 9 1970

The American University Washington, D. C.

Hut

Annon edhellen, edro hi ammen.' Fennas nogot hr im, lasto beth 1 ammen.'

- Gandalf the Grey

Abstract

k / \2

The differential cross section of He(e,e'd) H has been measured at six angles and over an energy interval of 35"50 MeV. Since the

center of mass and charge coincide, the reaction is expected to be primarily E2. The measured angular distribution of the deuterons

2 2

is consistent with sin 0cos 0. The measured cross section is a factor of 10 lower than the predictions of earlier theory. Calculations in a direct reaction model have been performed in an attempt to fit the magnitude of the experimental cross section and its energy dependence.

iv ACKNOWLEDGMENTS

This experiment was carried out while an AEC graduate fellow from the American University, at the National Bureau of Standards. I wish to express ray appreciation to Dr. J. E. Leiss and the accelerator

staff for their kind assistance.

I wish to also thank:

Dr. B. T. Chertok for his continued encouragement and advice during this work,

Dr. B. F. Gibson for the cross section calculations and patient counselling about the theoretical interpretation,

Dr. W. R. Dodge who supervised the work at the National Bureau of Standards, Dr. J. S. O'Connell for many illuminating conversations about the data, Mr. W. E. Ewing of the Stanford High Energy Physics Laboratory for

fabricating the gas targets,

Mr. J. J. Murphy for assisting with the data taking, and Elizabeth for her stoicism.

V

TABLE OF CONTENTS

CHAPTER PAGE

1. INTRODUCTION ... 1

2. A P P A R A T U S ... h- Calibration of experimental components ... 6

Solid angle measurements ... 9

Relative Efficiencies ... 10

Calculations of target thickness ... 10

3. EXPERIMENTAL METHODS... 26 h. RESULTS AND DISCUSSION... 3^

5. CONCLUSION... ^7

A PPEN D I C E S... ^9

APPENDIX A. The derivation of the formulae used in determining the kinematics and data analysis... 30

APPENDIX B. Solid angle calculations ... 55

APPENDIX C. Development of the differential cross section and angular distribution in the direct reaction model ... 57

BIBLIOGRAPHY... 62

LIST OF TABLES

TABLE PAGE

1. Values of B and Bn for two different baffle sizes . . . . 13

o 1

2. Summary of the relative efficiency measurements ... 1^

3. Calculated target thicknesses ... 13 k . Cross section values and estimates of contamination from

the (e,e'd)np reaction ... 31 5. Summary of wave functions used in the direct reaction

model calculations... 1*1

vii LIST OF FIGURES

FIGURE PAGE

1. Block diagram of electronics and computer interface . . . . 16 2. Spectrometer energy calibration ... 17 3- Ferrite current monitor calibration... 18 k . Elastic electron scattering from ^He and - ^ F e ... 19 5- Least squares fit to the elastic electron scattering data . 20 6. Proton and triton yields near E ^ ... 21 7. Alpha particle intensity spectra as a function of magnet

entrance s i z e ... 22 8. Alpha intensity spectra for baffle sizes (3 /8 x 3A)inches

and (3/16 x 3/8) inches...23 9. Lateral efficiency function for counter k ... 2k 10. Coordinate system for integration of effective path length . 25 11. Deuteron yields at 6 ^ ^ = 3^° and h8° ... 32 12. Pulse height spectra for signal and background runs . . . . 33 13. Angular distribution of deuterons in the center of mass

system...k2 lk. Total cross section of the reaction He(Y,d) H as a

function of photon e n e r g y ... k j 15. Total cross section measurements with exponential and

Gaussian f i t s ... kk 16. Total cross section measurements with Gaussian, zero

range, and Gunn-Irving fits ... ^5 1 7. Cross section measurements with a resonating group

c a l c u l a t i o n ^ ... k6

Chapter 1

INTRODUCTION

Recently the number of experiments concerned with the photodis- integration of He and the inverse reactions has increased greatly. Thek advent of improved experimental equipment has made possible the deter- mination of the He energy spectrum and its decay modes. The mosth

k 1

recent energy level diagram of He is:

as.5 _al«3_

317.8 a7.iT

a4.3

(a+7vcj}

_ _<r Tkd

O'J\0

~ (

rs ar T=l aa.v a"T=o

0"T=0 0+T=0

< ^‘‘He

r

The dashed lines indicate the states which are uncertain and have not been detected.

Ij.

Because the deuterons and the He ground state are both T = 0 nuclei, a measurement of the d-d capture cross section has been suggest- ed as a possible method to look for the predicted 2+, T * 0 level in Hek

2 2 2 k

near 30 MeV. One such experiment using the reaction H + H = He + y 3

was performed recently by Meyerhof et al. These authors analyzed their data assuming a Breit-Wigner resonance. Their analysis indicated that the reaction proceeds not by a resonance but primarily by a direct reaction.

In addition to searching for 2+, T = 0 states, the measurement I4.

of the two deuteron disintegration of He allows one to estimate the im­

portance of quadrupole absorption in the ground state due to the d-d channel. The allowed transition having the lowest possible angular momentum change is the E2 transition for the predominantly ground

1 in­

state to the D continuum state. Furthermore, by extending the cross section to higher photon energies, one can possibly draw conclusions about the strength of the cross section by comparing the results to the AT = 0 sum rule of Gell-Mann and Telegdi.^

Since the d-d disintegration is not complicated by higher order interferring terms that the (y,p) and (^,0) reactions are, one would hope that information could be obtained on the properties of the ground

lj.

state wave function of He. The energy dependence and the magnitude of the cross section are expected to be sensitive to the asymptotic part of the wave function since the R dependence of the matrix element (which arises from the form of the operator) picks out the asymptotic region.

2j.

Also it is of theoretical interest to study the d-d breakup of He to determine the degree of clustering of the He ground state as two real deuterons. By "clustering" one means the grouping of nucleons within the nuclear ground state to form subunits. Delves showed that the deuteron clustering is expected to be small as compared to the principal

k 5 5

clusters in He, namely the p- H and n- He clusters. Delves estimated that the d-d cluster probability is approximately 0.1 that of the n- He3

5 h

and p- H clusters. In other words the He ground state appears as a d-d cluster only 10$ of the time. A more recent study of the d-d system

7

has been carried out by Thompson using the method of resonating groups.

This method is equivalent to cluster model calculations and also predicts

a 2+, T = 0 level near 30 MeV in He.1+

In this experiment we have used electrons to disintegrate the He k

system. Although the square of the ^-momentum transfer is not zero as in the case of real photons, the approximation of pure photon kinematics to describe the outgoing deuterons is adequate, especially when the final electron kinetic energy is small relative to the initial electron energy. In this instance the momentum transfer to the target nucleus is nearly the momentum transfer obtained with real photons.

If p^ is the final electron momentum and p^ is the initial electron

momentum the maximum angle the momentum transfer can make with the initial beam direction is sin ^(p^/p^) « 3°* This approximation is considered further in Appendix A. Assuming photon kinematics we have the following characteristics of the ground state to the continuum state transition:

(l) a cross section of the order of 100 times smaller than the dominant, primarily El (Y,p) and (Y,n) reactions, and (2) the angular distribution

2 2 of the outgoing deuterons is sin Ocos Q.

Chapter 2

APPARATUS

The experimental apparatus consists of the National Bureau of g

Standards' 100 MeV, kW linear accelerator facility and the photo- nuclear physics spectrometer.

The spectrometer is made up of a k-3-T cm double focusing magnet and a scattering chamber designed to permit angular distributions to be measured from 20° to l60° in steps of lh°. The magnet which is capable of bending protons of 21 MeV, has nine particle detectors

in the focal plane. The detectors are of the lithium drifted silicon g

type with a gold surface barrier to form the "p" window. The detectors were cooled to 77° Kelvin to decrease the noise inherent in junction devices and to decrease the charge collection time. All of the devices were masked to 10mm by 60mm areas and placed at an angle of 60° in the

focal plane so that particles bent by the magnet were incident normally on their surfaces.

The current monitor system is a non-intercepting ferrite core with toroidal windings. The current from the linear accelerator acts as the primary of a transformer and the secondary windings are those of the ferrite. The current from the ferrite is fed to a low input impedance current amplifier with pulse compensation to overcome the droop in the ferrite signal.^ This droop is due to the finite induc­

tance of the ferrite and exhibits the characteristic L/R exponential decay. The amplified current pulse is then integrated using a linear pulse integrator^ gated on during the beam burst. The ferrite

system is calibrated with a movable beam dump located at the exit window

5 of the spectrometer. The net charge from the beam dump is integrated with a Rodgers integrator.12

All of the data compilation was done with an SDS 920 computer.

The on line computer system and its application to experiments at

the National Bureau of Standards' accelerator facility have been describ- 13 1^

ed elsewhere and hence a detailed description will not be given here. ’ Figure 1 shows the interface between the experiment and the computer. The charge liberated during a beam burst in the counters is amplified and sent to the computer for processing. The associated electronics for one of the detectors only is shown in figure 1. The analog-to-digital converters are patterned after the Argonne design.^

The priority system of the computer is arranged so that beam monitor scaling and pulse height analysis are given the highest priority during the run.

To reduce the background to a tolerable level the counters were shielded with minimum thicknesses of 10 cm of lead, 3 cm of iron, and surrounded on four sides by 61 cm of borated paraffin.

The hydrogen and helium gases were contained in cells mounted on a ladder in the scattering chamber which was capable of vertical and rotational motion.

The gas cells are right circular cylinders of height 3»8l cm and radius of 2. 5k Cm. The walls are made of .00&4- mm havar brazed to .6^ cm stainless steel end caps with two outlets on one of the caps for filling and exhausting the gases.

The gas targets are pressurized through a remote filling station located in the counting room. The pressure and temperature of the gas were measured at the end of each run by a pressure transducer and

a chromel-constantan thermocouple, epoxyed to the stainless steel end cap of the gas cell.

CALIBRATION OF EXPERIMENTAL COMPONENTS

Detector Calibration

The detector ladder was calibrated essentially by the method of reference 16; the energy of the observed particle being related to the magnetic field by the relationship:

where the subscript a refers to the alpha particle used in the

calibration and subscript p denotes the particle being measured. The mass, atomic number and magnetic field are given by m, Z, and B re­

spectively.

all counters. Using these results, can be determined and hence the energy calibration for all mass groups is defined. Note that the

2AP/P = AE/E.

Current monitor calibration

The calibration of the current monitor is accomplished by com­

paring the integrated charge of the ferrite to that caught by the movable beam dump at frequent intervals. Due to the noise levels associated with ferrite pickups, the signal had to be integrated on a pedestal. The various "anding" modes for the ferrite signal shown

3

Figure 2 shows the intensity spectra of the 5*305 MeV alpha particle in the decay of 210Po, as a function of magnetic field for

energy resolution AE/E is related to the momentum spread by 2AB/B =

in figure 1 perform the task of routing the signals for the pedestal subtraction of the ferrite signal and ratio forming between the ferrite and beam dump. This ratio multiplied by the appropiate scale factor of the Rodgers integrator directly calibrates the ferrite system.

Due to larger than normal fluctuations of the ferrite to beam dump ratios during this experiment a subsequent check was made on the ferrite-beam dump system. It was found that the ferrite amplifier system had two linear regions of different slope over the current range that was used during this experiment.

In figure 3 the calibration of the ferrite system as determined by a pulser of width 3*5 jisec and a repetition rate of 300 pulses per second is shown. Since the average current was kept below 25 for other experimental reasons (namely a precaution against rupture of the exit window of the scattering chamber) the maximum deviation from the normal calibration line is seen to be approximately 10$. This is in agreement with fluctuations seen in the beam dump to ferrite ratios, and is the limit we place on the uncertainty in the current measurement.

It should be pointed out that while the deviation is less than 10$, the error introduced in the subtraction of a given set of signal and background runs could be as high as:

aQ (0d - °d> * 0

where and are the charge normalized deuteron counts for a signal and background run respectively. Equation 2-2 was found by using an expansion of the current in a power series in AQ/Q. Typical values of and are 50 deuterons and 40 deuterons respectively. It is seen then that the subtraction error could be as high as kQrfo for the above

(2-2)

example. This is not felt to be the case since the ratio of signal to background charge as collected by a second beam dump approximately 30 feet from the exit window, was in agreement with the ratio of the signal to background charge as determined by the ferrite monitor. This indicates that the current was fairly stable over the running time for the signal and background runs.

Incident electron energy determination

The incident electron energy was ascertained by elastic electron scattering from He and the havar walls of the gas cell. The relation­k

ship between the scattered electron to the incident electron energy being:

E , = EinCld^ t (2-5)

scattered f L + 2Eincldent n £

M 2

Figure !)• shows the data from electron scattering at a laboratory angle of 132°. The least squares fit to the data at four linear accelerator energies is shown in figure 5*

Fortuitously, in the forward hemisphere where electron scatter­

ing was not feasible due to the background, the initial electron energy for the angular distribution measurements was such that the threshold for the \te(e,e't)*H reaction was covered simultaneously with the deuteron measurements. Due to the much higher cross section for this reaction, the end point energy was determined with good accuracy and is in agreement with the electron scattering data at the backward angle.

The proton and triton yields are shown in figure 6 at. a laboratory angle of 3^°-

9 SOLID ANGLE MEASUREMENTS

The solid angle subtended by the spectrometer was measured by masking the entrance to the magnet with baffles of decreasing size, and observing the intensity spectra of the ^ ^ P o alpha particles.

In figure 7> it is apparent that there are two distinct regions of different slope. Where the slope is zero the magnet clearly is determining the solid angle and in the region where the intensity of the alpha particles decreases linearly with decreasing baffle size, the solid angle can be calculated from purely geometrical factors.

Figure 8 shows the alpha distribution for counter If as a function of magnetic field for two baffle sizes in the region where the baffle is defining the solid angle of the spectrometer. Obviously the para­

meters B and B, are identical in both cases. We have shown in

o 1

Appendix B that the solid angle of any counter is determined by:

C

AO = A n ' ( - £t) (2-10 if B = B' and B, = B,'. The primes refer to the baffle restricted

o o 1 1 r

measurements and Afl' is given by:

Area of Baffle

A O ' ---p--- (2-5)

(9-5

f

Table 1 lists the results of B and B. for all counters except counter

o I

5. At the time of these measurements detector 5 had developed a suffi­

ciently thick window to prevent the detection of the 5*505 MeV alpha particles. As one can see, the agreement between the two different baffle parameters is sufficient to justify the use of equation (B-2), in Appendix B. Values of AC! for the counters are listed in table 2.

10 RELATIVE EFFICIENCIES

The efficiencies of the counters relative to counter ^ were

determined by forming the product of AQ(AE/e). These results are compared

to counter ^ were formed for the yields in the same energy interval.

With the exception of counter 9 the ratios are in fair agreement. It is felt that A0(AE/^e) for counter 9 with the alpha particle measurements is the better estimate since anomalies in yields from this detector were apparent in other work. In any case the combined data were not

appreciably affected by the difference in the ratios, which are given in table 2.

CALCULATIONS OF TARGET THICKNESS

As in all experiments using extended targets, one of the largest uncertainties arises from not knowing the effective path length of the incident beam in the target. We have measured the lateral efficiency of the spectrometer (figure 9) with the 210Po alpha particles. This was done by moving the source across the horizontal acceptance dimen­

sion of the magnet at beam height and position in the scattering chamber.

The effective path length was then determined by numerically integrating this function over the beam profile which was assumed to have the form:

where the coordinate system is shown in figure 10. The prime system is thus related to the unprimed system through the transformation matrix

k / \ 1

with efficiencies found from earlier He(e,e't) H work where the ratios

pQ -w/2 sS y £ +W/2

0 | y | > W/2 (2-6)

11 by:

(2-7)

Hence:

-W/2 5 x'sin0 + y'cos0 ^ +W/2

| x'sin0 +cos0 | > W/2 and the effective path length Is defined as:

IT p(x ’>y') I(y') dx'dy'

AT = --- (2-9)

J* p ( y ) dy

(2-9) 2 2 4 where the limits of integration are taken to be -R to +R and -(R - X' )®.

max 2 2 4

to +(R - X' ) . The results of these integrations for the lateral max

efficiency function given in figure 9 ar® listed in table 3> for an assumed beam width of ,6k cm. The calculated effective path length was relatively insensitive to the beam width, changing less than 3$

for an increase of 50$ in the width of the beam. The errors associated with excursions off axis are expected to be small since a geometrical estimate at 90° was less than 1$ for an excursion of .65 cm. The error for the effective path length calculation was found by least squares fitting the top and sides of the efficiency function to straight lines.

In an attempt to compare the calculated target thickness with an experimental determination, we proposed to collate the proton yields

from a point target and the gas cylinder. Since BeO was easily ob­

tainable, oxygen was the most likely candidate for the experiment.

However, we were not able to obtain thicknesses of BeO that permitted the yields to be compared in the same energy interval.

A future method for making such a comparison could perhaps be found by fabricating an alpha source to approximate the beam and

comparing the intensity of the alpha spectra at each angle

13

Table 1. Values of Board Bx for two different baffle sizes. Errors are ± .4 Gauss.

Detector (.95X1.91) cm2 baffle (.48x.95)cm2 baffle

B0 Bx B0 Bi

1 12.8 21.6 12.6 21.6

2 13.6 24.4 13.6 24.4

3 12.8 20.8 13.0 20.8

4 13.0 22.0 13.2 22.0

P

6 17.0 24.0 16.8 24.0

7 14.6 23.0 15.0 22.8

8 17.6 28.0 18.0 27.6

9 19.2 29.6 ^19.2 30.4

Det

#

1 2 3 4 5 6 7 8 9

14

Table 2. Summary of relative efficiency measurements.

AE /AE\ Ratio of (4=)Afi to detector 4

A Q ± 6(AO) f* ( f ^ A n _ _ _ _ _ _ _ VE J

Alpha source 4He(y,t) data measurements

5.29 ±.06 1.024 5.42 ±.06 .86 ±.01 .86 ±.01 5.76 ±.06 .832 5.08 ±.06 .80 ±.01 1.05 ±.02 5.88 ±.07 1.058 6.23 ±.06 .96 ±.02 1.13 ±.02 6.18 ±.07 1.022 6.32 ±.07 1.00 ±.02 1.00 ±.02

.932 - - 1.05 ±.02

6.13 ±.07 1.088 6.67 ±.07 1.09 ±.02 1.10 ±.02 6 .6 6 ± .0 7 .850 5 .6 6 ± .0 6 .91 ± .0 2 1.15 ± .0 2 7.52 ±.08 1.166 8.77 ±.10 1.34 ±.02 1.29 ±.02 9.46 ±.11 1.098 10.38 ±012 1.76 ±.03 1.06 ±.02

15

Table 3. Calculated target thicknesses.

0T An TARGET THICKNESS, CM

LAB 7

20 3.69 ± .052

34 3.35 ± .047

48 2.91 ± .047

62 2.52 ± .035

76 2.29 ± .032

90 2.22 ± .031

16

Figure 1. Block diagram of electronics and computer interface

s a N n id n a a 3 j.N i x n a o id

w io <fio «o

Z J X

<L ^{OT _}

3 I D *

o a <

CE 111 UJ

o a. o w o>

o o

1NHHS

© @ ® 13 98 VI

S5

xwvvvv’y v^

V 0

</) o

CL hr UJ <

(9 CL

a o o W GC t

Ui

<r o (9 oy k

u. vSo

uj ae i- uj - <9 K C9

So_Oo CCOLU-<

CL (9

OC fO a: £

U . O

o

33

Figure 2. Spectrometer energy calibration

io in

GO

.01 * ass 0£ /SINOOO

Figure J. Ferrite current monitor calibration

12,000

max ave

10,000

a; 8,000

Z 6,000

ave * Ip x (pulses/sec )x T, 1 [Ep ( VOLTS) x 2l] /x.A 4,000

2,000

1.6

VOLTS

19

4 56

Figure 4. Elastic electron scattering from He and ' Fe

Fe(e,e ) F e

30SVH0 UNn/sNoaxoans

250025202540256025802600

M A G N E T IC F IE L D , G A U S S

20

Figure 5* Least squares fit to the elastic electron scattering data

50

45

4 0

30

25

f L E A S T S Q U A R E S F I T T O

E L A S T I C E L E C T R O N S C A T T E R I N G D A T A

20

E 0= 3 . 4 8 x 1 0 ( V ) - .

X 2 * 1 . 7 5

.02 .04 .06 .08 .10 .12 .14 .16

SHUNT VOLTAGE (VOLTS)

21

Figure 6. Proton and triton yields near

C O U N T S /M e V • E L E C T R O N

0 ^{l a b}

34

• T R I T O N Y I E L D o P R O T O N Y I E L D

-16

-17

3 4 3 6 3 8 4 0 4 2 4 4

E y , Me V

22

Figure 7* Alpha particle intensity spectra as a function of magnet entrance size

C O U N T S /1 20 SE C x 1 0

D E T - 4

B a 5 7 9 2 G A U S S

3 . 2 -

2.8

2 . 4

2.0

3 . 0

1.0 2.0

AREA (INCHES)2

Figure 8. Alpha intensity spectra for baffle sizes (3 /8 inches and (3 /1 6 x 3/8) inches

A L P H A C O U N T S /3 0 S E C

COUNTER 4

®-3/8"x 3/4"

-3 /l6 " x 3 /8 "

4 0 0 0

3600

3200

2800

2 4 0 0

2000

1600

1200

8 0 0

4 0 0

29 0 6 2916

2886 2896

MAGNETIC FIELD (GAUSS)

Figure 9. Lateral efficiency function i(y') for counter !)■

C O U N T S /U N IT T IM E

D E T - 4

1 3 2

spec.

6 4 0 0

4 8 0 0

3 2 0 0

1600

- 7 5 - 50 -.2 5 0 .25 .50 .75 1.0

DISPLACEMENT (INCHES)

Figure 10. Coordinate system for integration of effective path length

R - TARGET RADIUS

W- INCIDENT BEAM WIDTH

I(y ')- MAGNET LATERAL EFFICIENCY FUNCTION

BEAM DIRECTION

- W T

i(y')

Chapter 3

EXPERIMENTAL METHODS

Data taking

Due to the competing reaction Ste(e,e'd)np the deuteron yield from Ij.

He could not be uniquely measured at all deuteron energies at one linac energy. However, 3"body deuterons were discriminated against by using the different thresholds for the two different deuteron producing reactions which differ by 2.23 MeV. The spectrometer was operated such that the only deuterons which were accepted originated from a photon energy in the energy region of E^ to (E^ - 2.23) MeV, where E^ is the linac operat­

ing energy. In this case the contamination of three body deuterons was expected to be negligible, since this region of energy is completely free of three body deuterons for pure photon induced reactions.

As diagrammed in Appendix A the angle between the momentum trans­

fer q = p^ - p^ and the primary electron beam, can have a maximum angle of sin”^(p^/p^). This implies that the energy transfer qQ is uncertain by approximately 170 keV for a final electron energy of 2.23 MeV. There­

fore the energy region of E^ to (E^ - 2.23) MeV could be contaminated by 3-body deuterons from this 170 keV region. The 170 keV estimate was obtained by varying the electron polar and azimuthal angles and calculat­

ing the resulting energy transfer. In addition to this 170 keV region an additional contamination of the "clean" region arises from the uncer­

tainty in the initial electron energy which was known to ± Vjo.

We know from the work of Gorbunov 17 that the (7,d)np reaction -28 2

has a total cross section of magnitude of 2 X 10 cm . We now show that the three body total cross section could be phase space limited

26

27 to a differential cross section of magnitude comparable to that of the 2-body differential cross section. An upper limit for 3"body deuterons

is:

da atotal^,np^ -2 9 2,

~ ____________ ~ 2 x 10 cm /sr

dfl lj-TT

A further reduction of the cross section arises from the phase space limitations for the final deuteron energy. Estimating that in the ex­

treme case only 1$ of the cross section comes from the upper 10$ of the energy distribution of deuterons in the 3-body disintegration for a given photon energy, we could expect a cross section on the order of

-31 2 /

10 cm /sr. This estimate neglects final state interactions which would tend to peak the cross section at higher deuteron energies and enhance the contamination. The requirement that the deuteron energy be in the upper 10$ of its total range (or near E^ - 2.23 MeV) requires that the neutron and proton have to emerge with a very small angular spearation. Thus to a good approximation we can apply 2-body kine­

matics to the 3-body disintegration when the deuteron energy is near the maximum value it can obtain. We have then a differential cross section comparable to the d-d differential cross section. It could be detected at energies less than (E^ - 2.23) MeV, if the 3-body differential cross section is at least equal to the d-d differential cross section.

We have taken data over the energy interval E^ to (E^ - 7) MeV and have observed no break in the deuteron yield that one would expect if the three body deuterons were being detected in a significant

number.

In table ^ the amount of contamination from 3-body deuterons is given assuming equal cross sections for the 3~body and 2-body deuterons.

28

On the basis of yields shown in figure 11 this would be a maximum estimate for 3~body contamination since there is no evidence that the 3-body cross section is equal to the 2-body cross section in this energy interval. The contamination was estimated by solving equation A-5 for

Figure 12 shows a typical pulse height spectra for signal(helium) and background(hydrogen) runs normalized to an arbitrary charge. It is apparent that a considerable fraction of the deuteron counts originate from the walls of the gas target through the process Fe(y,d). The

target was inflated to approximately the same pressure for both signal and background runs in order to insure the same target geometry and thus a realistic background subtraction.

To measure the angular distribution of the deuterons, the linear accelerator was operated at k l . 5 MeV and the spectrometer set to look at the proper energy interval at six different laboratory angles. For the cross section measurements as a function of energy, the limit to which we could extend the cross section measurement was limited by the maximum deuteron energy detectable, which was approximately 11 MeV laboratory energy. The low energy limit was determined by the background and stabil­

ity of the accelerator; 35 MeV being the lowest energy easily obtainable with good stability.

Data analysis

The net counts of various mass groups were obtained by normal­

izing the spectra to the charge accumulated during each run and sub­

tracting the region of interest. The errors associated with the counts were assumed to be purely statistical. After correcting for the

magnet dispersion by dividing by AE the yields were weighted and combined in appropriate energy intervals via the expression:

29 Yield = [ Si Ciai ]/ ^ (3-1) where:

deuterons Ci = AE x electron and:

The energy loss in the target walls and gas was calculated using the Bethe-Bloch equation:

? it

dE 8rr Z N e m , k m ~

P- o log

dx m E m l

e p p

/ *f IU Cl \

(— ) (5-4)

with:

m - electron mass e

Zp - particle charge m^ - particle mass

Ep - initial particle energy

I - ionization potential of the gas or walls N^q - Avagadros' number

To compare the experimental results with other work using real photons, a cross section of ~ qQ was calculated. The experimental yields were divided by the number of virtual photons per unit energy

interval ^ to obtain the equivalent photon cross section. The expressions 19 relating photon to electron cross sections are given in Appendix A by equations A-6 and A-7. The dominant term in the electron cross section is seen to be the log term. All other terms are down by the ratio

P .7

(E^/E^) ~ 10 from the log term. In particular the 0 -* 0 transitions which are possible with electrons are seen to be negligible. To a good

approximation the two cross sections are related by a constant which we identify as N(E^,E^) in equation A-5-

The equivalent photon energy was calculated by equation A—h-, and the cross sections were converted to the center of mass by the equations A-10 and A-11.

31

Table 4* Cross section values and upper limit estimates of contamination from the

4He(e,e'd)np reaction.

Estimate of deuteron c*c contamination from the

reaction 4He(e,e'd)np assuming equal cross

sections.

49.4 ^{3 .1 6} ± 1 .0 2 1356

47.4 2.49 ± .57 ¹²J6

45.4 2.32 ± .96 11$

43.5 6.31 ± 4 .2 0 10$

41.5 5.30 ± 2.06 8$

39.6 4.56 ± .85 7$

37.6 4* 50 ± • 64 5%

35.7 3.54 ± 1.15 3%

EY c.m.

52

Figure 11. Deuteron yields at 0-^ab = 5^° anc* ^8

D E U T E R O N S /M e V - E L E C T R O N

3 4 ' LAB

1t

U - 48

4 0 4 2

Ey, MeV

33

Figure 12. Pulse height spectra for signal and background runs x - background run, histogram - signal run

rnzz>io nmn k h z c o o

1— *

ro 0) 01 cn 03 03 0

o 0 o o o o o O O 0

o 0 0 0 0 o o O 0 0

r

Chapter 4

RESULTS AND DISCUSSION

The salient features of the photodisintegration of He into two h

deuterons are:

(a) The transition is primarily one from a *S ground state to the ^D continuum. Hence the cross section should be nearly pure E2.

(b) The matrix elements should not be complicated by higher

order terms especially at lower energies. One can estimate the importance of higher order effects by first noting that the higher order transitions are limited to even multiples since the over-all wave function describing the outgoing deuterons must be symmetric. A reasonable estimate of the magnitude of the non-vanishing higher order terms in the matrix element can be obtained by considering the energy dependence of the operator given by equation C-l in Appendix C. Since the strength of the transi­

tion operator varies as where k is the wave number of the incident photon, we see that for energies of less than 200 MeV the next non-vanishing

6 2

term in the cross section will be attenuated by the factor (k /k ). This is one reason that attempts to fit the existing data are restricted to energies less than 200 MeV. In addition, at energies above 200 MeV, meson production effects which become important have been neglected.

Angular distribution

We have measured the differential cross section of deuterons at six laboratory angles near the peak energy for the reaction. These cross sections are shown in figure 13- The solid curve represents a

2 2

pure sin 0cos 0 distribution which is consistent with our data.

Total cross section measurements

The total cross section as a function of energy Is shown In figure 14. The data have been combined In 2 MeV energy Intervals. The break In the cross section near k-6 MeV Is not thought to be significant.

3 20-25

The results of other experiments * combined with these data are shown in figure 15. The lower energy data (less than 10 MeV) resulted from deuteron capture measurements. The various radiative capture cross sections are all in essential agreement, and with the exception of the last few points from the work of Meyerhof et al. the trend of the low energy data is consistent with our measurements.

Theoretical considerations

k ^

Flowers and Mandl in their early paper on He used Gaussian wave functions to describe the He ground state and deuterons in the final k

state and used a plane wave to describe the relative motion of the two deuterons. The results of their calculation are given by the dashed curves in figure 15. Shown is the maximum range one can obtain in the magnitude of the cross section by varying the parameter describing the deuteron wave function over the limit prescribed by Flowers and Mandl.

The most obvious observation that one can make is that the magnitude of the cross section can be adjusted but the shape of the cross section is not consistent with the available data.

Asbury and Loeffler, 23 in an attempt to fit their high energy data at = 220.5 and 265-3 MeV, calculated an E2 cross section in the long wavelength approximation using exponential wave functions. The ground state wave function was taken to be of the Irving form

[exp( _Ha{S }^)] and the deuteron was represented by [exp( -n^r)/r].

Their results, using a value of consistent with a deuteron radius of 2.16 fermis, are given by the dashed curve in figure 1 5. As these authors have noted, the exponential wave functions are expected to have better asymptotic behavior. However, this property should not be important at such high energies. A more serious defect in their calculation arises in neglecting higher order terms and meson effects.

Because of the apparent shortcomings of the present calculations

k- f \2

describing the He(Y,d) H reaction, we have performed calculations in

h 2

the direct reaction model using phenomenological He and H wave func­

tions. A plane wave was used to describe the relative motion of the two deuterons.

It is shown in Appendix C that the expression for the differential cross section in the long wave length limit is given by:

he (2rr h)^ pol da o / e \ mPE^

— = I4.TT2 (---- ) 2-

dQ X <f|

x(R-e)(R.ic)

ID

where the natural coordinate system for the d-d disintegration is:

<P>

The vectors r^ and r^ represent the internal nucleon-nucleon coordinate in each deuteron and R represents the center of mass separation of the deuterons. By definition:

— * — » — *

rij - ri - r, whence:

57 From the considerations of the quadrupole operator in Appendix C, it is observed that the operator contains an additional factor of R with respect to the dipole operator. In addition the Bessel function in the L = 2 partial wave of the final state peaks at a larger value of kR than the corresponding L = 1 Bessel function. As a consequence of this we expect to find the quadrupole matrix elements more sensitive

Ij.

to the asymptotic form of the He wave function than the dipole matrix elements.

Table 5 summarizes the wave functions that were used in evaluating equation C~7 in Appendix C, using the National Bureau of Standards'

Univac 1108 computer.

We have sought to obtain agreement between the calculated cross sections and our measured cross sections by varying the wave function parameters. The only constraint imposed upon the parameters was that

If.

they correspond to a total binding in the He nucleus of approximately 28 MeV. The free deuteron parameter was fixed to correspond to the binding energy of 2.2J MeV. It is clear from figure 16 that only the zero range wave function(curve 2) fits the magnitude and the shape of the measured cross section. The asymptotic internal distribution of energies in the zero range wave function for the He ground state is suchk

that each deuteron is internally bound by 6 MeV, while the two clusters are bound together by 11 MeV. The mean square radius of He predicted k

by the parameters for curve 2 is roughly .92 fermi. This compares reasonably well with a value of l A fermis obtained from the electron scattering radius of 1 .6 (by correcting for proton size effects), since the zero range wave function is expected to weight the origin too heavily due to the singularity at the origin.

While the Irving-Gunn and Gaussian wave function parameters are not easily characterized by the binding energy, one can make heuristic estimates of the asymptotic binding between the two deuterons in the ground state. For the Irving-Gunn (curve l) we have asymptotically, a binding of approximately 16 MeV between the deuterons. Forcing the Irving-Gunn wave functions to give a magnitude for the cross section comparable to our data resulted in a parameter that corresponded

asymptotically to a total binding in the He ground state of 35 MeV, ah

result completely unphysical.

For the Gaussian (curve 3) we made the same identification between the binding and the wave function parameter as Flowers and Mandl. k This

results in roughly 15 MeV between the two deuterons in the ground state.

Calculations using a cluster model for the radiative capture of 6 7 deuterons by deuterons have been carried out by at least two authors ’

6 3 3

Of these Delves has calculated the probability for the n- He, p- H, and d-d clusters. His conclusion is that the d-d cluster is relatively unimportant and the d-d cross section is approximately a factor of

10 lower than the cross section calculated by Flowers and Mandl , a con­h

clusion which would agree with our data. Delves' calculations utilize coulomb wave functions and a three by three scattering matrix to describe the different channels. Unfortunately, due to probable printing errors, the analytic cross section that is given does not have the proper energy convergence and no direct comparison could be made to the experimental data.

In another cluster model calculation, Thompson has examined the 7

d-d system by using a resonating group structure. In this calculation the ground state wave function was assumed to be of the form;

39

with a chosen to fit the He rms radius. The wave function describing the outgoing d-d system is calculated from the resonating group formal­

ism. The results of this calculation are given in figure 17* The ex­

perimental points are those of references (3,2 0,2 1) and this experiment.

The parameter y determines the amount of exchange mixture in the two nucleon potential used. As can be seen the value of y = 1 .5 and y = 1.0 both fit our results but fall below the deuteron capture measurements.

It should be noted that the effects of final state interactions are in­

cluded in this calculation, but specific deuteron distortion effects are not. 7 Since the magnitude of this calculation appears to be con­

sistent with our experiment but not with the lower energy points, it would be of interest to use more sophisticated wave functions for the ground state of He, with the hope of obtaining a better fit to the k

low energy data.

Sum rule calculation

It is of interest to apply the results of this experiment to the sum rule derived by Gell-Mann and Telegdi^ for AT = 0 transitions for T = 0 nuclei. Since the d-d channel does not exhaust the sum rule

z

and we have integrated only to 50 MeV, we would expect the result to be less than the theoretical value. From their calculation we have:

A < r2 >

12 me2

= .515 lib/ MeV he

with ( r2 ) = 2 .0 1 fermis.

Experimentally we obtain for this sum rule the value:

(•50 tf(E2) dE _g

--- = 3.52 x 10 iib/ MeV

Q E

y

Thus for the d-d channel we have that the reaction exhausts only h

approximately 10% of the AT = 0 transition sum rule for He.

41

Table 5» Summary of the wave functions used in the direct reaction calculations.

TYPE STATE FORM

GAUSSIAN

If) e 3 II e 3 1=1 ,2

I D

rs . e s s

i<j

ZERO RANGE

I D

~k P-R _ -fir.

e 3 n e i i=l,2’ r.i

I D

-E.a.r.

e 1 1 1 nril

Hulthen

Irving-Gunn

I D

' i p . r „ t

e 3 n ( e e

i=l,2 r. r. '

’ l l

I D

-5 (S. r3 .)^

e 2 i<3 S (r® .)*

i<j

Figure 13. Angular distribution of deuterons in the center of mass system

0C +

2.0

1.0

- . 5

1 6 0

120

4 0 8 0

Figure lh. Total cross section of the reaction function of photon energy

50 42

38 46

3 4

( E r )c .m ., M e V

Figure 1 5. Total cross section measurements with exponential 23 and Gaussian^ fits. ^ - this work, J - reference 3> ^ " reference 20,

tjl - reference 21, reference 22, ^ - reference 23, <j> - reference 2k, - reference 2 5.

Gaussion Wave FunctionsI Exponential Wave Functions.

^5

Figure 16. Total cross section measurements with Irving-Gunn (curve l), zero range (curve 2), and Gaussian (curve 3) fits.

^ - this work, j - reference 3, ^ - reference 20, [j] - reference 21, reference 22, ^1 - reference 2 3, (J) - reference 2k, - reference.2 5.

10.0

0.01

0.001

100 1000

k6

Figure 17. Cross section calculation of reference 7*

^ - this work, | - reference 3, ^ - reference 20, tjl - reference 21

nUb)

y* 1.00

y = 1.50

1.0

0.1

28

2 0

24

( E - Q ) c.m , MeV

Chapter 5

CONCLUSION

We have interpreted our electron disintegration of He as a photo­1).

disintegration reaction using the virtual photon theory as developed by Gibson and Williams. 19 The cardinal points are:

(a) The cross section appears to reach a maximum near a photon energy of kk MeV.

(b) The cross section is an order of magnitude lower than pre- viously predicted. ^ 23* ^ Also we confirm previous determinations of a pure E2 angular distribution for the final reaction products.3 20

Our small measured cross sections show that the two deuteron disintegration channel makes a relatively small contribution to the

1,.

total photodisintegration cross section for He. In particular, the limits placed on the yield of two body deuterons by Gorbunov et al. 27 relative to the (y,p) cross section should be lowered by an order of magnitude. Our small cross sections lend credence to Delves1^ calcula-

tions which claim very little deuteron clustering in the He ground state, and which predict a cross section, of a few |ib. However a more recent

7 k

calculation based on d-d clustering in the ground state of He does pre- 2|.

diet the small magnitude of the cross section for d-d breakup of He that we have measured.

We have calculated the cross section for the direct reaction in the long wavelength limit using three different sets of wave functions for

1).

the He system. Using the binding energy to determine the wave function parameters asymptotically, we have obtained a fit to the data below 50 MeV with zero range wave functions. However, the parameters for this fit

correspond to non-physical values of the binding in He. The inter­

pretation of this fit is consistent with Delves' conclusion that there is little d-d clustering in He.k

The shortcomings of these calculations are that meson effects and higher order corrections to the matrix element have been neglected.

Obviously any calculation attempting to fit all the data would have to include these effects. In addition the calculations have neglected any competition for quadrupole absorption by the (y,p) and (y, n) disinte­

grations. It is quite likely that E2 strength (that is assumed in our calculation to go entirely into the d-d channel) is being depleted by these reactions. Taking this into account would in principal allow a renormalization of the experimental fits reflecting the fraction of E2 cross section going into the (y,p) and (y,n) channels.

k9

APPENDICES to

A STUDY OF THE REACTION Ste(e,e'd)2H by

Dennis Michael Skopik Department of Physics The American University

Washington, D. C.

May, 1970

APPENDIX A

THE DERIVATION OF THE FORMULAE USED IN DETERMINING THE KINEMATICS AND DATA ANALYSIS

The diagram for electron disintegration is as follows:

a.

In four vector notation we have:

pi + pa = \ + pd2 + Pf lj.

where a and d refer to the He nucleus and outgoing deuteron respectively, and p^ are the electron initial and final 4-vector momenta. Squaring and evaluating all terms, we obtain:

2 2 2 2 ®ee1

- bn -m +m m,+m T.+2E.(sin — r- )-p,E.(cos0,-cos0 ,,) 3 a e a d a d i' 2 ' rd i' d e'd' q =(E.-E.)= ---5---

° 1 f - o . l 2 ee'

(A-2) m -m,-T,+(Tj+2m,T ,)^cos0 ,,+2E,sin2--

a d d ' d d d' e'd i 2

For experiments where photons are used to disintegrate the target nucleus, we have:

Py + pa ’ % + \ (A'5)

where 7 refers to the incident photon and a and d have the same meaning as the electron case. Squaring and adding terms again we have:

51 Making the identification that qQ = E^ and letting me = p^ = 0, we

see that the expression for electron disintegration reduces to the equivalent photon expression.

The experimental differential cross section for the deuterons was found by:

i

dn Nfc Q AEy N(Et,Ef) AO AT where:

= number of deuterons from a signal run

= number of deuterons from a background run Nfc = number of target nuclei/cm^

Q = number of electrons incident on the target during the signal run

AE^ = energy interval accepted by the spectrometer AQ = spectrometer solid angle

AT = effective path length of the gas target, and

N(E^,Ej) = number of photons/MeV in the energy interval corres­

ponding to AE^.

The differences C, - C! for each detector are normalized by the ratios a a

of AQ(AE/E) given in table 2. The number of photons/MeV was computed using the results of reference 19. If the photon differential cross section is given by:

da 2 ^{„ p}

— = |e (2)| sin 0cos 9 (A_6)

dn °

Then the electron cross section can be written as:

da r 2 - , E, E . v E.Ej.+p,p.-m 2 0

{ I e0^2^ I [ ( — + ) 108 ■ - 6 sin 0cos 0

dn tt(137) Ei Ef me(Ei_Ef)

+(-6E2 + 8EtEf - E2)sin20 + (12E2 - 8EiEf - E2)

( — - ) (12E2 + 20EiEf - 25E2)sin20cos20 ] 12(Er Ef ) 2 ' E.

+ 2U> )ei(2)|2 -ffc- [ (l - » ii ) . ( * . ) ilA ]

' i f' i i

2 r IfE E( .

+ l^ (2)l ™

In the above equation:

E^ = initial electron energy E^ = final electron energy eQ(2) = <i| Pg(cos0) |f>, and ex(2) = <i| PQ(cos0) - Pg(cos0) |f)

To convert the photon energy to the center of mass system, one has simply to evaluate the k-vector product:

(PY + Pa) (P* + Pa) | ^ = (PY + Pa) (Py + Pa) \ m (A-8) Solving this expression one finds the energy in the center of mass and the value of E^ in the center of mass, i.e.:

55 Substituting E' in E^ we have then:

Ey = "7 “ ^ —Y (1 + 2E /m )»7 7 i (A_9)

V ty,r a

To convert the laboratory cross section into the center of mass system one has to simply transform the quantity (d^^Ag to ^ i s is done by using:

1 sin0'

tan0 =

y ' (cos0' + 0 ') where,

1 V

y ' = ---5 -5 and 0 ' =---

(1 - V'2)* VJ

and the primes refer to center of mass coordinates. We have then:

. da v sin0 d0 , da .

Using the relationship:

d(tan0) _ d0 = sec 0 —

d0 1 d0 1

and the Lorentz transformation properties of the momentum we obtain:

, do. (l + 01cos01)(1 + V'2) , da.

' dn yCM (1 + 20 'cos0' + 0 ' 2 - v ,2sin20')^ ' dn where,

. Ey

V' =

(E + m )

\ y & !

5^

To convert the differential cross sections into the total cross 2 2

sections, we assume an angular distribution of sin dcos 6. Hence we have:

= £ J

or,

a = -- A 15

55 APPENDIX B

SOLID ANGLE CALCULATIONS

We assume a trapezoidal function for the intensity spectra shown in figure 8 and adopt the following notation:

We know that the source strength is proportional to the area under the measured distribution of alphas as a function of magnetic field. Since the source strength is an invarient we can calculate it from the

relationship:

AREA OF ALPHA DISTRIBUTION S =

(^Baffle

(Bo + b;) c;

(AO),

(B-l) 'Baffle

The primes refer to the distribution parameters for the baffle re­

stricting case.

If we can take B ' = B , and B ' = B,, and knowing that the source

o o' 1 1 '

strength is given by B-l, we can calculate the solid angle of the magnet by:

m

AREA OF ALPHA DISTRIBUTION

Magnet (B-2)

56

( 40 W > . t - C?(BJ + bJ) (sn)Bi££le

c

“ ^7 ^ B a f f l e

o

(B-3)

APPENDIX C

57

DEVELOPEMENT OF THE DIFFERENTIAL CROSS SECTION AND ANGULAR DISTRIBUTION IN THE DIRECT REACTION MODEL

It is shown by Sachs 28 that one form of the electric multipdle

operator is given by:

(e.RKk*R) L_1

— (C-l)

L

where § is the polarization vector of the incident photon having a wave number k. We will follow the procedure outlined by Sachs 28 for determining the E2 cross section. The procedure neglects meson effects and is not in an irreducible representation which has been treated in the literature. 29 Furthermore, it is a true multipole operator for El and E2 in the long wavelength limit only.

The basic Hamiltonian describing the interaction with the electro­

magnetic field is given by:

r 6

(P - — A)

H = --- + V(r) (c-2)

2m Squaring and combining terms we have:

2 e e2

H = — + V ( r ) ---- (p.A + A-p) + — ^(A-A)

2m 2mc c

Then:

Hint “ "^fl -- (p*A + A-p) |ilri>

2mc e

2mc

Since p.A -* V-A = 0, we have:

Hin(; = — <f| A-p |i> (C-3) me

Taking the vector potential A to be:

, 2nhn / e v -iic-r

A = c (--- ) ( ) S e (C-h)

u) me

we have upon expanding A in a power series and taking only the second term:

/ \ / ® \ r -

■ * « - • ( — ) ( — ) [ c* - 5><* ~>]

me Rewriting (e-p)(Ie-r) we obtain:

4 {[(e*p)(^-r)+(e-r)(^.p)] + [(k.r)(e-p)-(e-r)(lc*p)] ]■

Taking the first term and using the usual expression for the momentum operator we have:

— t —4

m , dr dr ..

H = --- -f(k-r)(e •-— ) + (e-r)(k ) J

inc 2 dt dt J

m , d d ^

= {(k-r) — (e-r) + (e-r) — (k-r)j

2 dt dt

--- (k-r)(e -r) (C-5)

2 dt Since:

d (AB) (AB)H - H(AB)

dt ih

we can write equation C-5 as:

59

r o d m r (ic«r)(e.r)H - H(ic-r)(e-r) ..

- — (k’r) (S . ? ) - - { --- }

2 dt 2 ih

Then:

2 e r / 2TThnv^-. ra f (k-r)(g.r)H-H(ic-r)(e-r) , £

|<f|Hlnt|i>| I < £ I [c(--- ) ] - { --- -}|t>l

me uj 2 ih

We now use the relationships that H|i) = E^ji) and H|f) = E^|f) where (Et - Ef)/h = u). Then:

2 _ (§-r)(k.r) 2

Hint = 6 2TTh(1)n U fl ---- ~- U>l The differential cross section is defined as:

duj 1 , 2tt 2 «

da = 7 --- i r p(E) '<f' Hint l4>l } flux nc h

which becomes:

1 f 2tT 2 i , («•*)(**•*) , , 2 -l da = — H — p(e) e 2rrha) | <f | --- |i) | r

nc h 2

The density of final states is given by:

1

p(E) dfl (c-6)

(2^h Y

Therefore dc becomes:

da 0 , e2 . mpE (e*r)(k*r) 2

7-^,3 K*l — --- H>1 * t CC-T)

dn he 7 (2TTh)? 2

where the factor of •§ is introduced for spin summation (in reference 4 the spin sum is assumed to be l) and r = j^R for the coordinate system used.

60 We now show that the expected angular distribution from the

2 2

above cross section is sin dcos 0. Taking the initial state to be spatially symmetric with L = 0, we have the following diagram for the disintegration:

r* A n a rs

Since (k-r) °= Y^(k-r) and (e*k) = 0 we can write the cross section, using for the final state angular wave function the usual expansion for a plane wave of:

co L

e-i(P*r)= w £ £ jL(pr) ¥^(k*p) Y~m (k»r) (C-8) L=0 m=-L

as:

da 2

— c= l<f I (e-r)(k-r)| i) I

(c-9)

=

I I I j L(pr)l?(k-p)Yr (k,r)

m A A

Since YT(k*p) does not depend on the varible over which we are inte- Jj

grating we can write the expression as:

61

— “ I I I ^ P * k) I

{ [

dU T

L m

1 a 2

+ — £ Y^(k-r) + Y~^(k*r) Jsin^ j- dQr | (C-10) 1

i

Coupling the spherical harmonics and applying orthogonality we obtain:

d a f ^ A A a ^ A a a a A A ^ A A a A A A ^ A A A Q

— oc | Y ^ p . k ^ P ' k ) - 2Y2(p-k)Y"i(p.k) + Y ^ C p . k ^ ^ P - k ) }(cos<t>e) dn

6 « A A 4 A A 4 A A 4 A A 4 A A 4 A A N Q

+ | Y'^p-kjYg^k.p) + 2Y^(p-k)Y"i(p.k) + Y^(p-k)Yj(p.k) jCsin^)^ (C-ll)

Averaging over polarizations we see that the cross terms cancel since

(sin<De ) 2 = (cos<t>e ) 2 = \

We are left then with the angular dependence of the outgoing nucleon with respect to the photon momentum, or:

— oc | Y^(p*k) | oc sin20cos20 (C-12) dn

62 LIST OF REFERENCES

* W. E. Meyerhof and T. A. Torabrello, Nucl. Phys. A109(1968)1 2 L. Crone and C. Werntz, Nucl. Phys. A154(1969)161

^ W. E. Meyerhof, W. Feldman, S. Gilbert, W. O'Connell, Nucl. Phys.

Al^l(1969)^89

^ B. H. Flowers and F. Mandl, Proc. Roy. Soc. A206(l95l)l51

^ M. Gell-Mann, and V. L. Telegdi, Phys. Rev. 9l(l955)l69 c

L. M. Delves, Australian J. Phys. 15(1962)59

^ D. R. Thompson (private communication) Q

J. E. Leiss, National Bureau of Standards internal report 267 Q W. R. Dodge, J. A. Coleman, S. R. Domen, Nucl. Instr. and Meth.

^

(

J. L. Menke, National Bureau of Standards internal report 306

^ J. L. Menke, Nucl. Instr. and Meth. 64(1968)1

^2 E. J. Rodgers, Rev. Sci. Inst. 54(195)660

^ J. B. Broberg, IEEE Trans, on Nucl. Sci. NS-15.192

^ W. R. Dodge and J. J. Murphy (to be published)

^ M. Strauss, Rev. Sci. Inst. 54(1965)535

16 W. R. Dodge, W. C. Barber, Phys. Rev. 127(1961)1746

^ A. N. Gorbunov and V. M. Spiridonov, Soviet Phys. (JETP) 54(1958)600 1 ft

B. Bosco and P. Quarati, Nuovo Cimento 35(1964)527

^ B. F. Gibson (private communication)

20 A. Degre, Ph. D. thesis, University of Strasbourg

2^ R. Zurmuhle, et al., Phys. Rev. 152(1965)751

22 W. DelBianco and J. M. Poutissou, Phys. Letts. 28b(1969)299 2^ J. Asbury and F. Loeffler, Phys. Rev. 157(l965)B12l4

2k Yu. K. Akimov, et al., Soviet Phys. (JETP) 14(1962)512 2^ J. A. Poirier and M. Pripstein, Phys. Rev. 130(1965)1171 2^ J. O'Connell (private communication)

2^ A. N. Gorbunov and V. M. Spiridonov, Soviet Phys. (JETP) 6(l958)l6 2® R. G. Sachs, Nuclear Theory(Mass., Addison-Wesley, 195$)

29 r. w. Hayward, Handbook of Physics, ed. by Condon and Odishow (New York, McGraw-Hill, 1958)