\

The giant resonance in 12C occurs at 22.5 MeV. Above i t at

approximately 26 MeV is a strong second resonance thought to be bu ilt

upon the 2+ f i r s t excited state o f 12C. Both o f these have been

observed in photoexcitation and radiative capture studies, but the

present theoretical explanations o f C have fa iled to duplicate the

detailed shape o f either resonance.1 5We have investigated the charged p a rtic le decays of C,

3*B(p,pQ) 1*B, 1JB(p,p1) 1JB(2 .14), 11B(p,a ) 8Be to determine whether

structure observed in the (p,Y) and (Y,p) studies implies a ctiv ity in

the p a rtic le channels. The angle and energy dependence o f the

proton and alpha decays for incident proton energies between 7.5 and

21.5 MeV is presented.

P article-hole interpretations o f the 12C structure inadequately

describe the e la stic scattering; whereas, an optica l model for low Z

nuclei shows good agreement with the p0 data. The in elastic processes

are also best analyzed with a d irect reaction theory such as the DWBA.

Two alpha cluster models for 12C are cited and a PW3A which includes

the pickup o f the target core is shown to reproduce the (p,a)

angular distributions.

Abstract

A STUDY OF THE CHARGED PARTICLE EMISSIONS

OF 12C FOLLOWING PROTON CAPTURE BY 71B

by William Eond Thompson

A.B., Dartmouth College, 1965 M.S., Yale University, 1968 M.Ph., Yale University, 1969

A Dissertation Presented to the Faculty of

the Graduate School of Yale University

in Candidacy for the Degree Doctor of Philosophy

-1972-

Table of Contents

• Abstract

Acknowledgements..........................................................................................................

Chapter I: Introduction ........................................................................................... 1

A. Theoretical Considerations..................................................................... 2

B. Experimental Methods................................................................................. 5

Chapter I I : Experimental Procedure .................................................................. 6

A. Detection of the Charge P artic les ....................................................... 7

B. Hardware and E lectronics............... '........................................................ 9

C. Data Reduction............................................................................................. 21

D. Absolute Cross Sections and U n certa in ties .................................... 22

Chapter I I I : Results ............................................................................. 22

A. The Energy Dependence of 1 *B+p Cross Sections. . 23

B. Angular D istributions.......................................................................... 25

Chapter IV: Theoretical Analysis............................................................................ 49

A. E lastic Scattering ................................................................................... 50

B. Analysis of the In e lestic Scattering Data....................................... 73

C. Alpha Decay of C and the (p,a) Reaction.................... 7 5

A. Selection o f a Model for the 1 *B+p Reactions................................ I l l

B. Implications o f the (p,p) and the (p,a) Studies 114 '

11 *Appendix 1: B Target Production.............a ....................................................... 117

Appendix 2: Uncertainties in the D ifferen tia l Cross Section.................. 120

Appendix 3: D ifferentia l Reaction Cross Section in a j - j

Coupling Notation.....................................■........................................... 125

References.........................................( ..............................................................................133

Chapter V: Conclusion....... :............ •................... 110Q

/

With a great deal o f love I dedicate this thesis to my w ife,

Sylvia, to my children William and Melanie, to Gerald Bostock, and

to the smiling people on this curious planet.

Acknowledgements

For introducing me to the rea lity o f nuclear physics, for

suggesting the topic o f th is thesis, and for enthusiasm in both the

laboratory and the classroom, I would like to thank Jack Overley who

was my orig in al thesis advisor. The major portion., o f the experimental

and theoretica l work was supervised by Peter Parker without whose

guidance th is thesis would not have been possib le . His attention to

mathematical deta il in checking derivations proved invaluable and his

knack for solving the mysteries o f e lectron ics fa ilu res saved many an

experiment.

Several fellow graduate students contributed s ign ificant e ffo r t in

my behalf, and I am particu larly indebted to Henry Shay, Marty Cobem,

and Dan Pisano, Three other associates, Marty F ritts , Tony Aponick,

and Bruce Shwiersky\ provided me with much socia l and in tellectu al

entertainment which I appreciate. The particler-hole calculations o f

Chapter IV were derived by Claude Brassard and he deserves particular

recognition for permitting me to use these and h is computer codes.

My Yale career was made especia lly pleasant by the entire s ta ff

o f the A.W. Wright Nuclear Structure Laboratory, The diligence and

devotion o f the ch ie f accelerator engineer, Kenzo Sato, assured the

success o f my experiments and the unbounded optimism o f the laboratory's

d irector, D.A. Bromley, kept up morale.

Introduction

Once the features o f a particular nuclear reaction have been

delineated, an interpretation o f those features requires the selection

o f a model to describe the resu lts. This model must use as few

parameters as possible. The true test o f a model's e ffica cy is not

merely that i t reproduce the data, but that i t y ie ld a reasonable

physical description o f the system being studied.

Nuclear reaction models fa l l into two basic categories,compound and

d irect. A compound model assumes that when two identifiable pieces

o f nuclear matter interact for_a_time longer than a typical nuclear

transit time a new compound system is formed in which the identity

o f the original constituents is lo s t . A further assumption o f

compound theories is that, within certain lim itations, the decay of

the new system is independent o f its mode o f formation. This

compound theory has had considerable success in the study o f ligh t

closed-shell or near-closed shell nuclei at low excitation energies

where there are few degrees o f freedom. The category o f d irect

reaction theories includes plane-wave, distorted-wave, and optical

model approximations. These theories involve a description o f nuclear

systems in terms of macroscopic (and often empirical) parameters such

as well depth, diffuseness, and r e fle c t iv ity . Clearly in nature

A. Theoretical Considerations

d irect theories. v

The present experiment was designed to study the p artic le decay

modes o f excited states in 12C at and above the giant dipole resonance

and to examine their correlation with states observed in 11B(p,Y)12C

experiments (AL64, KE69, BR70). Since compound theories such as those

o f G ille t and Arima (GI64, KA67, BR70) were available which predicted

the gross structure for resonances found in these radiative capture

experiments, it was o f interest to try to explain C particle decay

modes within the same compound framework. I t was also o f interest to •*

determine i f the fine structure o f the (p ,Y ), (y ,p ) , and (y,n) excitation

functions (WU68, BA57) could be explained in terms o f the behavior o f

the partic le channels, since currently available pa rticle -h ole or

coupled-channel calculations (MA69) have not been able to describe this$

fine structure.

A macroscopic analysis o f the 11B(p,pQ) , 11B(p,pJ) , ^BCpjCt)

reactions was also undertaken. This was accomplished with a recent

optical model calculation (WA69) which successfu lly predicts e la stic

scattering results in ligh t nuclei using a minimum o f free parameters.

From these op tica l model parameters one could then predict reaction

and in elastic scattering yields with a distorted wave (DWBA) calculation.

Prior to the development o f the DWBA, pickup and stripping mechanisms

were described in terms o f the* interaction between nuclear clusters

on the target surface and the incident plane wave o f the p ro je c t ile .

Only the reduced widths and spectroscopic factors for these clusters

were needed in this method. Increased computer size and speed caused

there will be an overlap between the domains of the compound and the

this PWBA to be replaced by a DWBA which included the distorting

e ffe c t o f the nuclear and Coulomb potentials on incident and

exiting waves. Recently the PWBA has been revived for use in the

analysis o f multi-nucleon processes involving pickup or knockout

not only o f surface clusters but also o f the core. P ro je ctile -

heavy partic le interactions have only just begun to be introduced

into DWBA because o f the d iff icu lty o f performing fin ite range

calculations for clusters having relative motion with non-zero

angular momentum. In our analysis we have used a PWBA including

contributions such as 7Li pickup to interpret the 11B(p,a ) 8Be

reaction.

We present in what follows a particle -h ole S-matrix derivation

of pro'con e la stic scattering cro^s sec cions , che predictions or an

optica l model and a DWBA for e la stic and in elastic proton scattering

on 11B, and the results o f a plane-wave calculation which includes

heavy pa rtic le interactions for the 11B(p,a ) 8Be reaction. Experimental

data are compared with the various theories in an e ffo r t to determine

how best to interpret reactions induced in a J1B target by a beam of

protons.

B. Experimental Methods

The giant dipole resonance in 12C occurs at 22.5 MeV, and

a second giant resonance (thought to be a coupling o f p a rtic le -

hole configurations to the 2+ 4.44 MeV State in 12C) exists at

©26 MeV. To supplement data accumulated below the giant resonance

(RI68, BL62, BL64, and SE65) and to extend the'available information

beyond this second resonance, we designed our experiment to study

the reactions 1JB (p,pQ ) 1 *B, 11B(p,pj ) 11B(2.14), and ^B C p^^ 'B e

for proton energies between 7.5 and 21.5 MeV corresponding to 12C

excitation energies between 22.5 and 35.5 MeV. .Any large anomaly in

the proton e la stic scattering would e ffe c t the results o f the

available particle -h ole theory—(BR70) for radiative capture and any

correlation o f proton yields with gamma yields- could be analyzed

with this theory. We, therefore, were particu larly interested in

the (p,p ) reaction. The CL decay from 12C was also o f interest 0since the previous gamma ray studies were sensitive mainly to the -

1 2T = 1 component o f C leve ls , whereas alpha particles to the ground8 12 state o f Be originate in T = 0 natural parity levels o f ■ C.

Thus i t was hoped that by studying the (p,pQ) , (P /Pj) and (p ,a0)

reactions we could learn more about the importance o f isospin mixing

and p article -h ole configurations while at the same time investigating

the role o f d irect reaction mechanisms at high 12C excitations.

Experimental Procedure

Figure 1 shows a level diagram for 12C indicating a number of«

pa rtic le emission thresholds. For protons incident on 1*B the

only open, charged-particle thresholds below 25 .0 MeV are those for

protons and alphas. This ‘lim itation on the emitted species allows ■

one to separate decay products simply by a proper selection of

detector thicknesses. The Q-values for the 1 (p ,3He)9Be and

11B (p ,d )10B reactions are su ffic ie n tly negative, 10.32 and 9.23 MeV ,

respectively, that even a fter these channels open their reaction

products never "interfere with the analysis o f the proton or alpha

p artic le spectra. To extract e la stic and in e lastic proton yields

we used single Si(SB) detectors o f between 1000U and 2000U

to detect protons with energies up to 17 MeV. For protons having

more energy than 17 MeV two transmission mounted detectors were

placed in a telescope array and their summed output was then analyzed.

To displace the alpha peaks and the broad alpha background from

regions o f interest in the proton spectra lOOy Al fo i ls were placed

at the front o f each detector. A typical particle spectrum

(figure 2) contains e la stic and in e lastic peaks from 1XB and the

contaminants 1 -B, 12C, 1SN, 1 eO e tc . as well as the alpha peaks8 • 8 associated with decays to the Be ground state and ro the Be

f ir s t excited state which appears as a broad peak merged with the

A. . Detection of the charged particles

breakup alpha particles underlying the e la stic protons. In figure 3

we show a typical proton spectrum with the Al f o i l in place.

Alpha pa rtic le spectra were obtained with thin detectors which

were thick enough (100-500y) to stop the most energetic alpha p a rtic les ,

but thin enough so that the protons deposited only a fraction o f

their maximum energy in them. The alpha spectrum o f figure 4 shows

the alpha background generated by ®Be and d irect 12C breakup in

addition to alpha particles to the ground and f i r s t excited

states o f 8Be.

B. Hardware and Electronics

baseplate o f an ORTEC target chamber at 17°, 32°, 62°, 118°, 158°o oand 168 . By moving the baseplate 20 data were also collected at

37 , 52°, 82°, 98°, 138° and 148° re la tive to the incident protons.

Several o f these angles were selected because in the middle energy

range their corresponding center-of-mass angle is a zero of a

Lengendre polynomial; for example at 12.00 MeV 82° is approximately o .90 in the center-of-mass and hence a zero o f a l l odd, polynomials.

In addition 62°, 118° and 148° are zeroes for polynomials with L = 4,

2 and 5.

The target with which most-of the data were taken v?ae measured

with an alpha gauge (C066) to be 310±20 ygm/cm2 thick. Subsequent

analysis indicated that o f th is 310± ygm/cm2 78% was 71B, 15% 160,

3% 12C, and 2% JCiB. The target fabrication technique and contaminant

e ffe cts are discussed in Appendices 1 and 2.c

The beam energy was varied between 7.5 and 21.5 MeV in 50 keVC

steps. Thus with two detector arrangements, one for the protons and

one for the alpha particles,w e obtained excitation functions at up

to twelve angles for the reactions ^BCpjp^) J1B, 11B(2.14)

and 11B(p,a()) 8Be for 12C excitations between 22.5 and 35 MeV.

Data were accumulated in six o f eight octants o f a 4096 channel

analyzer. Routing pulses were generated by threshold discriminators

and low voltage signals were biased out to minimize the analyzer

In a typical experiment six detectors were placed on the

This dead time was determined by routing the d ig ita l output o f

our beam current integrator into the eighth octant o f the analyzer.

The ratio o f routed to unrouted BCI pulses gave a measure o f the

analyzer dead time. For a typical 300 nA proton beam this dead

time ranged between 4% and 16%. When a fixed charge had been

co llected (100-500 y c ) , data were transferred from the analyzer

onto magnetic tape for future processing.

dead time.

\

To extract cross sections from these spectra the magnetic tape

was read back into the analyzer memory and then sent through a

data link to our IBM 360/44 where i t was rewritten on 1600 bpi tape

for editing and reduction. The reduction consisted o f displaying each

spectrum on a CRT, -identifying the peaks o f in terest, defining lim its

o f integration with a ligh t pen, subtracting a linear background, and

then determining the resultant cross section . To identify contaminant

or unknown peaks, a special routine was incorporated into the data

reduction program. This routine performed an energy calibration o f

the spectrum and was useful in kinematic studies. Two peaks from

known reactions were identified-.with the ligh t pen and the energy

d ifference between these peaks established an energy/channel

ca libration . One could then select with the ligh t pen or enter •

through the teletype other possible reactions; the expected peak -

position for these reactions were then indicated on the display by

two arrows as is shown in figure 4.

C. Data Reduction

D. Absolute Cross Sections and Uncertainties

In addition to the s ta t is t ica l uncertainties associated with

the extraction o f peak y ie ld s , absolute errors include uncertainties

inherent in the measurement o f target thickness and inclination ,

detector position and so lid angle, charge co lle ction , and analyzer

dead time. An error o f typ ica lly ±10% is associated with a ll data

points. For data points without error bars the uncertainty is

less than the size o f the data point. Error bars for the angular

uncertainty have not been shown. These should be taken as ±0.2°

for detector geometries o f between 0.4° <3 1 . 4° o f angular acceptance.

Additional corrections and uncertainties had to be included

in the proton e la stic scattering^ cross sections for anqles foward ofO x 282 where we did not separate the e la stic yields from the C or

160 contaminants. Since the e lectron ics were not setup to resolve

the e la stic peaks o f carbon and oxygen for detector positions o f

less than 90°, the y ields for 10B, 12C, and 160 were included in12 16our integration o f the e la stic peak. The C and 0 yields at these

12 12 16 16angles were determined from th e ' C(p, p )' C and 0 (p p ) 0 angular0 Q

distributions o f LeVine (LE69) and Skwiersky (SK72). Legendre

polynomial f i t s to their data-were normalized to the present data at

backward angles (where the three e la st ic peaks were resolved). The1 0normalized Legendre f i t s were then used to predict forward angle "C

and 160 e la s t ic yields and these were then subtracted from the

unresolved e la st ic data.

<9

Ficrure 1.—

1 2Lowest lying charge.particle thresholds fo r C and selected energy

leve l schemes fo r 1 and 12C. The 22.6 MeV state in 12C indicates

the centroid o f the f i r s t giant resonance.

-14-

l2C CHARGED -PARTICLE

25.19l 0 B + d5.02 3 / 2 "

4.44 5 /2 "

2.12 1/2"

. . . . . ................................ - 3 / 2 ’^ 8 + p 1 5 . 9 5 7

22.6 I’ I

16.11 2 +: !i5. i i . . . . i : , i

9.638 3"7.653 0+

4.439 2+

-0+

T H R E S H O L D S '

26 .28 9 B e + 3 He

2.9 2+

8 Be + C L 7 . 3 6 9

P article spectrum fo r protons on a target. The detector

thickness o f 1500 y was capable o f stopping a l l emitted charged

p a rtic les . Arrows show the peak positions o f reaction productso

generated by the proton beam on the target and its contaminants.

CO

UN

TS

F U L L P A R T I C L E S P E C T R U M T A K E N W I T H A T H I C K D E T E C T O R

. E p = 1 2 . 0 0 Me V , 6|a b= I 18° .

1 0 0 0 0 -

8 0 0 0

6 0 0 0 -

4 0 0 0 -

2 0 0 0 -

0

-— - - ^0) 07CP CP'-'CD CD'o— .

o '’oCL CL•» 0.Q. CLDQ CDO

o>CPOOJT>Cl

inCPO(D'oCL

tnCPa>CDooO

aclCD

—I----------1----------1----------1 I I l4 0 8 0 120 160

C H A N N E L N U M B E R200 240

-17-

Figure 3

Proton spectrum accumulated with a 2000 micron detector positioned

behind a 100 micron Al f o i l . The f o i l was used to reduce the energy

o f the alpha particles su ffic ien tly to remove them from the regions

o f interest.

CO

UN

TS

0

I

An alpha particle spectrum taken with a thin detector(500y) so that

protons which have greater than a 500 micron range in silicon would

deposit only a small fraction of their energy in the detector.

This method produced discrete alpha particle peaks well separated

from the low energy proton background.

CO

UN

TS

Q

T Y P I C A L T H I N D E T E C T O R A L P H A S P E C T R U M

1 00 1 4 0 1 8 0 2 2 0 2 6 0 3 0 0 3 4 0 3 8 0C H A N N E L N U M B E R

Photograph of a typical CRT display which includes a spectrum scaled

according to a square root plot routine. The characters in the two

right hand columns are used to establish the emitted species and the

analyzer dead time, while the V's indicate the positions of identifiedc-

peaks. The characters in the upper left of the display gave the center-*

of-mass cross section after a peak integration had been completed.

Figure 5--

CM1 ..... a . x . m . H 1 1 1 CL.— —

i » c* * • P*0• ** ft.

0 1

i r o t

coOCT L O s Q

SCT

VCT 3 > f

ECT -Sc/ 2CT J ctf

. f a . D s ? o o t t f

ic r OtP

» ft* •*«"TT1 '<■• «0 • ♦ • •• *» > ♦*4 «"»* .• ft

• ♦ft

«/ \ A

0 8 I I 4 A 5 1 4 0 0 : .5 j Hff / Q l / g o

0 ®|frO£Z •9

3

-C h ap te r I I I -

Results

A. The Energy Dependence of J1B + p Cross Sections

*

Figures 6, -7, 8, 9 and 10 show the energy variation of the

11B(p,p0), (p/Pj), and. (p,aQ ) reaction cross sections at several lab­

oratory angles. The 12 C.excitation energy is given at the top border

of each figure and the incident proton energy at the bottom. For com­

parison figure 11 is presented to summarize the available information on

these and other reactions (TH69, TH71, BL62, BL64,'BR67, BR70, KE69,

AL64, SE65, RI68 and OV65). At low 12C excitation energies (18.0 - 25.0 MeV)

curreialioiis Liti'Lwtien tut: par tic j-6 and g&mma- cliannals air c +r»c

implication of these correlations to 12C isospin mixing and to p-h theories

have been discussed by other authors,(e.g. WU68, BE63, and GI62). Proton

emissions for higher 12C excitations (above 2 5 TO MeV) show no significant *

resonant behavior which has not previously been observed; however, the

a Q decays do show minor anomalies at 26.5, 28.3, 30 and 32 MeV. Structure

in the a0 channel is also evident at 25.2 MeV in the 118° excitation

function. It is interesting to note that in the center-of-mass system

the laboratory angle 118° is a zero of the Legendre polynomial , 12and that a 0 , -C quartet excitation has been predicted at approximately

this energy by Arima and co-workers (AR70). Also predicted at such 12C

excitation energies is a 6 member of an a-cluster band built on the + *7.65 MeV, 0 state. According the Reynolds et al. (RE71) this state

should appear at 32.2 MeV. Our data indicate the possibility of a weak

state at approximately this energy. Although none of the a-structure

mentioned here appears in the current energy level literature (AJ68) ,

no energy or width assigments for these levels were made because of the%

lack of significant detail in the excitation functions.

Above 29 MeV the excitation functions for all charged particle

decays begin to show a damped structureless energy dependence. This

absence of resonance behavior is indicative of direct reaction processes

and should best be described by DWBA or optical model theories rather

than microscopic particle-hole theories. For high excitation energies,

with many modes of single particle motion available, the number and

width of the microscooic p-h state becomes large.- and their lifetime

short, so that statistical, collective, or macroscopic descriptions of

the nucleus are needed.

B. Angular Distributions

In figures 12, 13, and 14 the angular dependence for the three

reactions is given for several representative energies. A smooth

curve has been drawn through the data to guide the eye. Note .that the

p Q and p x angular distributions are similar to one another and peaked a

at forward angles; vdiereas, the aQ angular variation is more

oscillatory and peaked at backward angles.

This difference is reflected in the fitted values of the Legendre

coefficients for the p a and a Q data. Reasonable fits to the p 3 data

could be obtained using polynomial expansions with L < 4; the a0

angular distributions, however, required the inclusion of polynomials

l i n + O T — (T rT1'h n o n e . v . - t tr n ra rir3 ait*. H r - , • P f V ,o q o T a P P -/•«■? r.—i ■*" ~ *— I T — - * — W . . _ - W - - — - — — ' - 3 ' _ w . i . w ~ * f f

is given for the p 1 and a Q data in figures 15 and 16. These graphs

represent a smoothed curve through the coefficients and discontinuities

associated with angular distributions having tod few data points have

been neglected. Both sets of data were analyzed using a linear least

square fitting routine which included all polynomials with L between

0 and 6. In figure 15 the coefficients A^ and A^ have been omitted

because, to within their uncertainties, their values were zero. Also

omitted are A_ and A . both of which were constant at 0.3±0.6 and 3 4

-0.15±.3, respectively. The total cross sections and plotted coefficients

of figures 15 and 16 have uncertainties ranging from 10% to 50%. Such

large uncertainties arise when the number of data points is close to the

It was possible to obtain coefficients which gave adequate

simulation of the data throughout the experimental range but which

resulted in negative cross sections at zero degrees. To avoid this

problem we introduced a weakly weighted spurious data point at

zero degrees and thus forced a more realistic fit to the data. This

technique was also used to fit the 12C and 160 proton elastic scattering

results of LeVine (LE69) and Skwiersky (SK72) which we used to extract

interpolated contaminant yields as described earlier.

The energy dependence of the total cross sections for the Pj

and a„ decays (figures 15 and 16) reflects the quiescence of the

excitation functions and shows no distinct resonance phenomena above a19T4 V f t ft + ‘ ^ F O 1 A. r« Al A. l ‘»*a %A Aa F -? X. A - A-% A A’ A*- ' • - '^ j \_/ * <«/ . i V « i”l + L U b >A O X. O X-XtA L- 1^0 M+lwA. W- -A

indicated no d i s c e m a b l e resonant behavior for 0 , (p„) as well.total cThis complete absence of compound structure in all three reactions

for the energy range 27.0 to 35.Q MeV was not anticipated. The data of

figure 11 below 27 MeV, which are exemplary of all particle ando

gamma ray activity in this region, often do correlate well with many of

the compound levels predicted by Gillet (GI63), Marangoni (MA69) and

Arima (KA67) . However., although both T = 0 and T = 1 p-h states are

anticipated theoretically between 27 and 40 I-leV ( v . figure 5-3a of

reference GI63) we observed none experimentally. The main reason that

these states are predicted and not observed is that the initial lp-lh

calculations neglected multiple particle-hole excitations and core

deformations which cause the pure configurations to mix and disperse

their strength.

number of fitting parameters, A^.

Figure 6_.

Center-of-mass differential cross sections as functions of energy

for the 11B( p /p 0) 11B(g.s.) reaction. 'The increased error associated

with the.82°. 62®. and 52° results from a subtraction of t+e estimated

contributions to these cross sections from target contaminants.

8

4

0<10

0■c

6

3

0<

2 0

0^40

2 0

0^

60

30

0-s

40

2 0

02 0

10

0_60

30

0

I 2 C EXCITATION ENERGY 24 26 28 30 32 34

XCITATION FUNCTIONS FOR THE REACTION 118(p,p0)11B(g.s.)

T I I I I I I I I I I I I

^ 0 | a b = l 6 8 ‘

11 ' II I ' ll l . ' l 11!

0 , ob = l 5 8 ‘

1jiik ;»4 ® ,ob ''4 8 "‘rii'i’ i iVii'iiiM.i',! 'Hu..

I "38 °

tV’/V+vi'*.- fllob'HS"

^lab=98'

In, f,|l V l1! S lo b '82'* * * ' (Ujaauii . ....m|i

» , .b '6 2 '

**i*W

10 12 14 16 18 20E p,lab ( MeV)

-29-

P A r t i i v o V*■ — - '___

11B(p,p1) alB(2.14) center-of-mass -differential cross sections foro o o o o32 , 62 , 148 , 158 and 168 . The laboratory proton enerqy appears as

1 2 •the absissca and the C excitation energy is indicated at the top.

of the figure. 12.0 MeV '< Ep < 21.45 MeV

4

2

0<

4

2

0^

2

I

0<

2

I

0<

4

2

0

E X C IT A T I O N FUNCT IONS FOR THE REACT ION I , B ( p f p | ) l , B ( 2 . 1 4 )

,2C EXCITATION ENERGY 2 8 3 0 3 2 3 4 3fi

fl---------- 1-----------1 T

" ^ I a b = l 8 8 <

^ l a b ~ ^ 8

e i o b = ' 4 8 (

Jll

r 0 | ° b s 6 2

O

(w ,»♦

% ®lob = 3 2 °

V - 1

4

14 16 18 E p,lab (MeV)

2 0

The energy dependence of (dcr/d^) c.m. for the Pj's at 52°, 82°, 98°,

118° and 138°. 7.50 MeV < Ep < 21.45 MeV.

Figure 8

(js/qu) UP/-OP

-32-

E X C I T A T I O N F U N C T I O N S F O R T H E

R E A C T I O N l l B ( p , p , ) l l B ( 2 . 1 4 )

l 2 C EXCITATION ENERGY

33-

Fjaure 9

Excitation function showing (da/dft)c.m. for the a Q 's at 17°, 32°,

62°, 148°, 158° and 168°. 11.50 MeV < Ep < 18.85 MeV.

dcr/d

ft (m

b/sr

)

- 3 4 -

E X C I T A T I O N F U N C T I O N S F O R T H E R E A C T I O N " B ( p , a 0 ) 8 Be (g.s. )

I 2 C EXCITATION ENERGY.28 30 32 34

E p j a b <MeV>

-35-

Figure 10

(da/d^)c.m. for 's at 52°, 82°, 98°, 118° and 138°.

7.50 MeV < Ep < 18.85 MeV.

dcr/d

Xl

(mb/

sr)

‘ E X C I T A T I O N F U N C T I O N S F O R T H E R E A C T I O N M B ( p , a 0 ) 8 Be (g.s. )

I 2 C E X C I T A T I O N E N E R G Y2 4 2 6 2 8 3 0 3 2

:- igure li_

Excitation functions for several proton induced reactions using

a 1XB target. Data depicted may be found in the follov/ing references

(TH69, TH70, RI68, O V 6 5 , B L 6 2 , BL64, BR67 and A L 6 4 ) . Note the

damping and disappearance of structure beyond a 12C excitation energy

of 28.0 MeV.

o-(/

ib)

dcr/

dJl

(mb/

sr)

-38-

40

2 0

0

60

40

2 0

0

ENERGY DEPENDENCE OF SOME PROTON INDUCED REACTIONSON A " B TARGET

E x of l2C (MeV)28 30

j , L J i_

32i ■ r

DIFFERENTIAL CROSS FOR THE REACTION

34 36

SECTION

nB(p.a)8Be (g.s.)0 iob= 1 2 0 °------0|ob=H8°---

11B (p.p,)11B (2.14) 0 iab = l2O°----0 |ab = " 8 °-----

,lB(p.p0),lB (g.s.)0 |ab = l2 O°-----0 lab=M8 — - -

IIB(p.n)llC

■ i ■ !i 1 i i i 1 r TOTAL CRO SS■SECTION FOR THE REACTION

B(p,yi)l2C(4.44)

80

40

O

iB(p,/0)l2C (g.s.)

10 12 14Ep.lab (MeV)

16 18 20 22

39-

Figure 12— -

Selected typical p fl angular distribution for the energy range Ep

12.00 to 21.50 MeV. The solid curves are included as an aid to

the eye and show the similarity of these angular distributions to

one another over the entire energy range.

dcr/d

X), (m

b/sr

)

-40-

S E L E C T E D A N G U L A R D I S T R I B U T I O N S F O R T H E R E A C T I O N n B ( p , p 0 ) MB

^ c . m . ( d e g )

Figure 13

Typical angular distributions for the inelastically scattered protons.

The distinct lack of energy variation in the angular dependence is

common to both the p 0 's and p j 's.

dcr/d

£l (m

b/sr

)

-42-

S E L E C T E D A N G U L A R D I S t RI B U T I O N S FOR

#c.m.(deg) -

9

Fiqure 1 4 -

Several a. angular distributions which show considerably more

energy and angular dependence than do the proton angular distri­

butions. *

da-/d

,ft (m

b/sr

)

-44-

SELECTED ANGULAR DISTRIBUTIONS FOR THE "B( p ,a0 )8 Be REACTION

0 c.m. (deg)

45-

Figure 1 5--

Energy dependence of the Legendre coefficients obtained by a least

•squares fit to the 11B(p,p1) 1.1B(2.14) angular distr.ihut.inns with

L max = 6. Not shown are A , A , A and A which were nearly constant3 4 o 6

at 0.30 ± 0.60, -0.15 ± 0.30, ^0.0 ± 1.0 and ^0.0 ± 1.0, respectively,

for the entire energy range.

TOTAL CROSS SECTION AND C O E F F I C IE N T S OF P, AND P2 FOR THE REACTION 1' B ( p , p , ) 1 !B ( 2 . 1 4 )

l 2 C EXCITATION ENERGY (MeV)2 7 2 8 2 9 3 0 31 3 2 3 3 3 4 3 5 3 6

E p ? law ( M e V )

-47-

—. „• nj: _Ly u_l t ; i u

1 1 8Legendre coefficients for the reaction B(pfOic) Be. A more

systematic variation with energy is apparent for these A. than for

the A^ of either the p & or p x data. Note that these coefficients

remain significant even up to L = 6 verifying the greater complexity

of the angular distributions.

TOTAL CROSS SECTION AND COEFFICIENTS OF P| . P 2 . P 3 .P4 . P 5 AND P6 FOR THE REACTION

n B ( p , a 0 ) 8 Be

, 2 C EXCITATION ENERGY (MeV)2 7 2 8 2 9 3 0 31 3 2 3 3

p , I ab ( M e V )

1. Compound S-matrix Analysis

In an effort to understand the results of his 11B(p,Y)12C studies,

Brassard (BR70) developed a modified version of Gillet's (GI62)

particle-hole theory. Brassard contended that although Gillet's12calculations were adequate for predicting energy levels in C, the

theory was not designed to describe angular distributions for radiative

capture. Gillet's calculations were performed with an infinitely deep

harmonic oscillator potential, and therefore the nucleon radial wave

functions have zero amplitude outside of the nuclear volume. This

type of bound state calculation ignores the possibility of nucleons in

the continuum and the concept of a particle reduced width has no meaning.1 2As a thoicc rccil icir ic rr.ccid c*f C, B^sssxsrcl c.r.l c.\? 1 r* tlic. 1r2.cli1.j_

solutions to the Schrodinger equation for a nucleon in a potential1 2which closely resembled the shape of the C ground state as determined

from electron scattering. The resultant radial wave functions were

then combined with angular momentum functions of good & and j to obtain1 2intrinsic single-particle states for C. On the assumption that the

intrinsic states differed from those of Gillet only at the channel sur­

face, Gillet's amplitudes for the contribution of particle-hole states 1 2to a given C level were retained.

This technique was found to be useful for predicting the angular1 2distribution of gamma ray transition to the ground state of C. Gamma

transitions to the 4.44 and the 9.64 MeV levels were not as well pre­

dicted, perhaps because these levels consist partially of a deformed

Theoretical AnalysisA. Elastic Scattering

C core built of multi-particle multi-hole excitations neglected by

the theory.

Contained in the mathematics of this gamma transition theory are

the R and S matrices required to calculate proton elastic scattering from

11B. It was necessary merely to develop a j-j coupled expression for

the 11B(P,P0) 11B cross section to extract results from Brassard's

S-matrix.

The next section describes a procedure for obtaining from an

arbitrary model of a nuclear interior the matrix elements for R and SL

The R-matrix links resonant phenomena- within the nuclear volume to

allowed decay channels at the surface of the nucleus. Once these sur­

face boundary conditions are defined the form of all asymptotic wave

functions outside the nucleus ^re determined. In Section 2 we also

derive the relationship between the R and the S-matrix. The S-matrix

specifies how the incident flux is distributed into all exit channels

and provides a direct measure of the reaction's strength. With the

nuclear information contained in the S-matrix and the appropriate

angular momentum coupling procedure one obtains the angular dependence

of the decay fragments as is seen in Appendix 3.

In Section 3 we derive the radial wave functions for a nucleon in

a realistic spherical potential resembling that of 11£C. The reduced

widths, associated with these wave functions are derived and the assump­

tions of the particle-hole theory outlined.

12

-52-

An S-matrix theory of compound nuclear systems as given by Ma’naux

and Wiedenmueller (MA69) or by Preston' (PR53) is particularly useful

near closed or quasi-closed shells. In this case 12C may be considered

as a quasi-closed shell nucleus, since it contains a full complement of

p 2 nucleons.

For the nuclear systems described by a discrete set of single particle

eigenfunctions within the nuclear interior using the S-matrix theory,

one may calculate the cross sections for both scattering and reaction

processes. The degree to which an internal eigenstate resembles a

specific exit or entrance channel, c determines the probability of

decay or formation through that channel. The R-matrix relates the

eigenstate A to the wave function 'i'^ in the external region where the

remnant subsystems exert no polarizing effect on on® ano+-hev-..

If the internal region can be expressed by a Hamiltonian H having

the eigenfunctions , then the outgoing scattered wave function in

the internal region is

2. The Nuclear R and S-Matrices

s £ (+ ) = L ( E ) X . . C T V -1 )E A A f

QThe requirement that be continuous and have a continuous derivativecat the channel surface S determines the A, . For a specific channelc A

c in the external region in a j-j coupled notation one has

i , J M ) (r ,“2)

-53-

with

y i. 2|j.p y Im

.m J - m (IV-3)

describes the intrinsic residual nucleus final state of A-1

nucleons and O 1 m is the spin function for the nucleon in channel c.

We designate as the surface wave function, (Pc' that portion of equation

IV-2 within the square brackets and the reduced width amplitude y.^ as

2 \ 1 / /2

= (_j?-----) fdstp-X, =a2 (tp 1X.> (IV-4)7 Xc \2M a J J vc X c c A ' c c '

The curved bra indicates an integration over all but the radial co-

■'yrdiri^t»3 whiT_a ++51 —'“‘tat^on .'dS implies an integration over all

channel surfaces (v. MA69 p. 142). Two related expressions which will

be useful are

dx c = 7x = + G s 2- Y 7 s ^ ^ )(po - ^

and

Bc ’ V YXe • (W- 6)

M is the reduced mass of channel c and a is the channel radius. The c cset {b . } is generally real and determines .

■ Suppose flux is incident on the nuclear volume through channel c

-54

then the radial wavefunction U ' , for tha exit channel c 1 isc .

U<? < W > = 0 0 . ( r c . . k o l , + , ' « > i , < r „ v , , , (TV-7)

with

(C) ( c ) ,y C' = . y c 6 co> • p v - 8)

The 1^, and 0^, are the incoming and outgoing waves typically com-(c)posed of Coulomb or Bessel functions. From this definition of U .c

it follows that

P’ tf1> ° c ' " U S ’= - 2 i kc y c'* ■ (^ -9 )

if one assumes plane wave asymptotic behavior for 0 , and I .. Wec c

should also define two more quantities

y (c0., = J A<X yXC P7 - 10'

and

i(c) (c)d . = a ■/ / c’ c’ c'

'O_c<0 /a

- 2 ik , 0 c’.-1

a h

2M

1/2 (c) (i-V-11)

It is now convenient to introduce t'le expression obtained by

applying the nuclear Green's Theorem with Y f and which reduces to

(E -E ) A (c) = Tj y ,d (CT) - y ^ d , . (IV-1 2 )Ii jli c1 n e ’ c* c1 j ic 1 v '

The shift function and the penetrability P are given by

O ( a , k ) = ( k a r 1 / 2 f i p 1 / 2 , ' (T V -1 3 )c c T c c c c c

and

a O , c cI ■ ■ . J m j n a «-»\

C r. \ O - j a — /-' C • C

« c - ) V 2 • ( I V - 1 4 b )c

P c- = 0 , (IV-15a)

where c implies a closed channel,

where c+ im plies an open channel.

When all of these expression are substituted into IV-12 one obtains

e a < c) {(e - e ) 6 + r a - r r ,}= - i h 1 / 2 v 1 / 2 n r 1 / 2 . y (c) ( tv - ig )X X X Xji c' f y c 1 2 c t X j i c 1 c c >ic j c

where *

-5

and

A = r A . (IV-22)A p C ApC

When A_ is expressed as

with

and

x , ■ ■ p v - 2 3 )

’ < I V ' 2 4 )

« V S E V ■ <nf- 25)

then from equations IV-1 and IV-23 it follows that

= i f i 1 / 2 v 1 / 2 y ( C ) a S A r l / 2 X . . (TV -2G )c c c Xp Xp p c A

The collision or S-matrix is defined in the external region as the

ratio of the outgoing flux to that of the incoming flux which explicitly is

s*c- r ( z ) ■J c ’

If one evalutates the overlap between the outgoing wave 0 , rf* ,c cand the exp ression for S’ in both th e in ter n a l and ex tern a l region

one then ob ta in s

This expression for S may be substituted directly into any

formula which relates the differential scattering or reaction cross

section to the collision matrix (v. BL52). Thus if one has a set of

model parameters to describe the nuclear interior he can predict the

results of any scattering experiments with this S-matrix.

The usefulness of the S-matrix in explaining how an incoming

wave is dispersed by the nuclear region into the exic channels is

most easily exploited by a decomposition of the S-matrix into a

product of the R-matrix with several diagonal matrices. If in

equation IV-12 the simplification is made that

. (TV-28)

(IV-29)

then

, % y d .(°) _ y GC1 C1

J.I b E -E (TV-30)

substituting this into equation IV-1 gives

-59-

X A 7 \ c f (o )

By operating on this with where S in the channel surface one has

y (C) = I R , d (,0) . ( T V -3 2 )c c c c ’ c*

with

be be-Rco'= I - ^ T i T • <I' ,- 33>

The reality cf this R-iuStnx is assu-.eci if the proper phases are

chosen for U and to .c ■ cTo obtain a relationship between the R and S-matrix one must

evaluate the amplitudes and in terms of the reduced widths Y-^c *

From equation IV-7 it follows that

<L>= T ( x (C) 0 , + y (? I , ) ( 0 , ( I V - 3 4 )c t c 1 c' J c' c ’ ’ ' c v ‘

(c) 1/SV2 M a ' c c'

lx i c,0. ,+ y(C)

Itf3 (IV-35)

-60-

and

. 2 \ 1 / 2 h a '

^ = \ ~ Y r ) { x(C, ) 0 / + y ^ I 7 | } . ( I V - 3 6 )c* \ 2 M / c* c 1 c 1 c*

c

Combining these two relations with equation IV-32 yields

i " 1 \ p + y J } = S f L a 1 / W + y i ' ) ] - B [ a _ 1 / 2 C x . P + y » ] } . ( i v - 3 7 )oo r v J

All matrices except R in this equation are diagonal; for example,

(a-35) l = ( — ) 35 6 1 . A further sinrolif ication of IV-36 results if= cc a cccone makes the substitutions

and

so that

a 0 ° = a O ' - B O = O L ° • ( I V - 3 8 )

-o' v/ _ T _ _ _ o*4} I = a r i . ( T V -3 9 )

£ = - y 0 _ 1 a - R J j V 1 (1 - R L 0 * ) I ’ ( T V -4 0 )

which gives as the matrix equivalent of equation (IV-27)

S S-1 4 - 8 i;0)-1 g- s if)I f 1 ( I V - 4 1 )

This equation [c.f. eqn. V-93 of (CB70)] constitutes a definition of

the nuclear S-matrix in terms of the known asymptotic parameters and

the reduced widths contained in R. . It allows the direct determination

of elastic scattering angular distributions as seen in Appendix 3.

3. Determination of the Reduced Widths*

In order to utilize the R-matrix in IV-41 it is necessary to have

the eigenvalues for the compound model being, used. What follows is

an outline of the procedure used by Gillet and by Brassard for obtaining

these quantities.

The full Hamiltonian H for A interacting nucleons may be written as

AH = £ H. + E v . . = H + V . (IV-42)

i i i<j iJ °

Rather than performing a Hartree-Folk calculation to establish the

eigenvalues of H , Gillet used a phenomenological technique in which othe energies from pure single-particle or single-hole states were sub­

stituted. For the case of 12C, the 3/2“ state in laC at 18.72 MeV was

taken to represent the lp /2 neutron-hole state while the 1/2+ state at

35 MeV was assumed to be a pure neutron-hole state in the ls^ shell.13 1 1 , ,The proton energies were obtained from N and B. For a specific

2single-particle or hole state one has E„ , = E + £ + m c or 3 e ■ A+l A A n2E . = E + -£ - m e where E is the ground state energy of the

A - l A a n A

A nucleon system and m the nucleon rest m a s s . The eigenenergy of annunperturbed particle-hole level in 12C is then

-62-

CAa = eA + ea (IV-43)

The full Hamiltonian containing the residual interaction is now

diagonalized in the space of the pure particle-hole wavefunctions to

derive lp-lh 12C energy levels and eigenfunctions. The procedure is

referred to by Gillet (GI64) as Approximation I and only lp-lh

excitation of one or two -ftw are allowed. Note that Approximation II

which is known variously as the quasi-boson approximation or the RPA,

permits ground state•correlations and 2p-2h excitations but it provides

only slightly better agreement with experiment. Due to the increase in

complexity of Approximation II, Brassard chose the lp-lh approach for

his work.

Gillet used as his effective two body interaction the potential

V ( r i j ) = V o f ( r i j 7 , i ) ^ + B P a " H P r + M P • - (T V _44)

Here f(r„/})) has a Gaussian form of rangey . The diagonalization of

H in lp-lh space determined the E ^ . . The parameters y , H, W, B and M' 1 2were evaluated by least squares fitting a few known states in C.

They., are obtained by evaluating at the channel surface the Ac

overlap of the wave function for the internal region with that for the

external region. The explicit form for the state vector X is given by

-63-

| x > = r x | p > , P p

(IV-45)

where the |B> are the particle-hole eigenvectors of H defined byo

Gillet to be

l“a W a ' nA < V A»A' « T05

= £ (-1) ' (j j mj -mj |JM)(-1)mtArot a A a AA amjAmja

<tatA mta -mtA lT° ) l<-«as a> Ja “V * ‘ >‘V a W

U m i <la>U n . (r.)a a A A R

t mt > : It.a a 1 .A

( T V -4 6 )

The kets | ( £s) jm> are defined as

|(iE)jm>= S ' (fsm m | jm) |im > |sm >in f m £ 5 1 sX s

(IV-47)

int. > A

-6

with

|i m t > = Y ? l (TV-48)

An improved notation suggested by Brassared creates a lp-lh state for a

proton (p) or a neutron (n)- particle (a) or hole (b) from the vacuum

state (12C g.s.) as follows

|B > = l* j£ 'j'J M ± > = ± a ”imb£"i'm’ 3l0>

( - l y ' V j X J - MTO} = r

(TV-49)

These states are related to Gillet's by

( - l j ' ^ ’- ^ ^ n ' P j 1 n ijjlij i'j 'J M T> . nTi ',1 of it

(TV-5 0)

Reference (GI62) tabulates the X's need to determine y ... .Ac

The new notation was developed so that the state vectors would be

time reversible and would have the proper normalization. Implied in it

is that

|n i j m > = V ( - 1 s j l t |n £ m > | sroiEQj >

l - i - V •mm

and

(IV-51)

The only quantities as yet undetermined are the u which weren

obtained by solving the radial Schrodinger equation for a single nucleon

in a realistic 12C potential V ( r ) . V(r) was selected so that it described

the charge distribution measured by Fregeau (FR55, FR56) through

electron scattering. The analytic form of V is

2 2 , 2

. Vn (r ) = V o ( 1 + 3 ^ ~ ) 6_r 0 * ( I V “ 5 3 )

with a^ = 1.635f and Vq = 65 MeV. The proton potential V p is the sum of

V and v . where V_ n has the form n .Coul Coul

V c o u l ( r ) = 4Tr^ " ^ 2 ^ f - 2 P ’ ( T V -5 4 )V O

and

P ( r ) = k V . ( I V - 5 5 )P n

To establish k one has assumed that the proton and neutron density

distribution are given by and requires that

e 4 tt J Q p ( r " ) r " 2 dr" = Z{ = 7 . 1 9 0 M e V F , , ( I V - 5 6 )P

or equivalently that

-66

<s

e 2ZV , ( r )

c o u l 1 r = 7 . Of

kB |' r=7. OF. (IV-57)

Because the spin-orbit strength has been neglected y0(r)* = ru (r)

may be simply extracted from the Schrodinger equation

y + -? ir (v (r ) - E ) 5 y , / r ) » ( iv -5 8 )

with V(r) = V (r) or V (r). y can be solved for‘numerically with the n p X.boundary con d ition th a t the u are f i n i t e a t r = 0 and y 0 fo r r 0

. X> Xj

for bound states: In the case of unbound states one has that

R u* (R )/u* (R ) = b = -1 at the channel radius R of 4.5f. Thisc c £ . c c c

value o f b was chosen to minimize th e le v e l s h i f t A described e a r l ie rcand hirings the resonance energy close to that of the eigenvalue E ^ .

It is now possible to evaluate the reduced widths Y- c anc hence R.

Recall that Y ^ c represents the overlap of the surface function or |c)

with the state vector |A> which implies that

( t? \ 1 / 2 '

\ l i n r j p * H 0 > a v -a o )

l _ J ? \ 1 / 2 ■

\ c \ 2 M c R o j | y f3c ’ _ <IV" 6 0)

for

¥ = E X |/3 > . (IV-61)X 0 P

|c ) i s composed o f th e in t r in s ic res id u a l nucleus w avefunction and the

angular and sp in w avefunctions for th e e x it in g nucleon

|c) = | I > j i j r a )

(n' - 62>

I f one assumes th a t the pure lp - lh con figu ration |3> may be decomposed

in to two components and resem bling th e d irect-p ro d u ct o f

a nucleon s ta te 1$^ coupled w ith an e leven nucleon system to t o t a l

angular momentum J and th a t

|6 > = U (r) | i jm > ,1 0 1

( I V - G 3 )

then

> < l | 0 n >

'U/3 (r)lRc < I^ ll> * (n7_64>:

Antisymmetrizing £5 and including the isospin coupling yields

/ f .2 \ 1/2b c = ± ( i s V ^ / < I l% >Re \< ne> ' <W-65)

c

where the minus sign is used only for an antisymmetric neutron channel.*The quantity R U„ (R ) can be extracted directly from TV-58.

C P ] C

The evaluation of <l|811> , however,is not straightforward.

<l|B, > vanishes if J differs from or if the hole configurations1 l

of |611> and |I> are not the same. For the channels in which ]I> is

either the 1XC or 1XB ground state the value of <l|B > is a measure of

the degree to which 11> resembles a hole in the ]p shell. Brassard

chos e and 7 B(g.s.) ! — be unity for a B which

represents a p “ configuration. In the case where 13x 1> represents

an Sy* configuration it was contended that if sufficient energy, 35 MeV

is available to create the s,, hole from the p ?.1 initial state of 1 *3Vz Vz

then any of the eight p ^ particles could be promoted to the p orbit

or ejected into the continuum. The possibility of any of ths p ^ ' s

particles contributing to | B c a u s e s an enhancement of Y ^ c

amount 10 for the case of |B11> being an s ^ hole. One then has

an

with an explicit relation for the reduced widths it is now possible

to evaluate the R matrix for the channels of interest. It is of

course necessary to limit the number of states and channels included

in the determination of the R matrix. For the details of this truncation

and for the explicit form of 0 j. and V the reader is referred to

reference (BR70). It should be mentioned here that the allowed set of

reaction channels includes only those involving nucleons in the continium

since other channels will have zero overlap with the ip-in internal

wave functions. As a consequence this theory cannot predict the cross

sections for multiple .nucleon emissions. Also not included in {c} are

those target or residual excitations which are known not ro be pure hole

configurations, rhis is a severe limitation or tne tneory ana prevents

the calculation of proton inelastic scattering cross sections for the

first h~ state in 1]B at 2.14 KeV which probably is a p^, hole configor-

4- ’ 1 2atron coupled to the 2 collective state in ' C.

The S matrix of equation IV-41 which describes scattering from one

channel in {c } to another, can be substituted directly into the last

expression of Appendix 3. In figures 17 and 18 the’predicted elastic

scattering cross sections are given for the energies indicated. The

data and optical model predictions are also given. Note that the

magnitude of the'particle-hole cross sections tend toward agreement with

the data at forward angles and diverge rapidly thereafter. At lower

energies there is slighly better agreement in the shapes of the two

4. Predictions of the particle-hole theory

-70-

predictions (p-h and O.M.). At all energxes, however, there is a great

discrepancy between the magnitudes of.the two theories.

As mentioned before,there is no provision in the p-h theory for

decays into any channel which doesn't resemble a core plus a nucleon in

the continuum. This constraint is analgous to an optical model without

an imaginary channel, in that there is no way to account for a loss of

flux due to decays into channels not included in'the calculation. Tijiusfat backward angles where the influence of nuclear effects is large the

discrepancy is worse. At forward angles the elastic scattering is governed

mostly by Coulomb effects and agreement is better. If the theory contain­

ed an energy dependent background term within the R matrix, the results

would have been improved.

5. Elastic scattering according to the optical model

Recently Watson and collaborators (WA69) analyzed a wide range

of elastic scattering data in lp-shell nuclei. Using a potential

form

V = - V p f ( r ) + i W f ( r ) + 4 i a W V (r) o p t iv ' V I S

+ a • 1 Vs o

f,2 11 1 1- m c • 7T L a 1/3 i

f ’ ( r ) + V , ( r )Coul

( I V - 6 7 )

where

* 1/3 — 1f ( r ) = { I + e x p [ ( r - r QA , ( I V -6 8)

-71

and

Vz Z e

Coui 2R (

zZe

c • 2

(3 - r / R z c

r ^ R = r A1/3

j r >B (IV-69)

they determined a reliable set of parameters which satisfactorily

reproduced proton elastic scattering data from 6Li to 160 for a variety

of proton energies. The solid curves of figures 17 and 18 are the

angular distributions determined by the optical model code JIB3 (SI70)

with Watson's parameters as input. Although the data at 8.5, 9.5 MeV

and 10.5 MeV are sparse, the quality of the optical model fits is

best at higher energies where less resonance structure is exhibited in .

GxciJt. t‘.,ion functions Tiis of tt:C optics.2. rr.ocLol.

results with our data indicates that even for such low Z nuclei as 1*B

direct reaction mechanisms can successfully describe proton elastic

scattering.

Although we present only a few representative optical model

angular distributions, we have made comparisons of the Watson JIB3

predictions with our data at 1.0 MeV increments from Ep„ , = 8.0 tolab

21.0 MeV. Because Watson's optical potentials contain an energy depen­

dence which helps to parameterize non-local phenomena in a local calcula­

tion, the theory’and our data follow one another very well with energy

The slow energy variation of the angular distributions can be seen in puri 0figures 17 and 18, although it is more graphically depicted for B

«j

in figure 3 of (WA69) . Note in the figure of Watson et. al the .similarity

of the 10B elastic scattering data to that of the 1JB data. This\

similarity is an additional confirmation that for proton energies above

10 MeV nucleon elastic scattering has no real resonance behavior. Graphs

of the energy dependent optical parameters (figures 4, 5 and 7 of

WA69) obtained by fitting much-p shell data show nearly a linear energy

variation. From this we conclude that the smooth variation of our data

is typical of that in lp shell nuclei and that above approximately 10 MeV

protons interact with targets of low Z as if these targest were composed of

homogeneous nuclear matter.

B. Analysis of the Inelastic Scattering Data

At the bottom of figure 19 a typical angular distribution for the 11 1 1BCpjPj) B(2.14) reaction is shown. As is implied by the structureless

excitation function for the Pj Legendre coefficients, this angular

distribution is typical of those found throughout the entire energy

region. It is similar to those for the elastic scattering and lacks the

oscillations predicted for the p reaction by the particle-hole theory.o

The success of the optical model in the p 0 case encouraged us to attempt

a DWBA analysis of the p : data using the distorted wave code LvJUCK (KU71)

and the optical model parameters of Watson et al. (WA69). The solid

curve through the 11B(p,p1) 11 B data represents the unadjusted predictions

of this calculations for a pure £ = 2 momentum transfer. Since P. — 0

is the only other momentum transfer allowed and it failed to fit either

the shape or magnitude of the data, we did not attempt to calculate

any angular distributions for the case of mixed £ transfer.

The success of this DWBA for an £ transfer of two may be due to

the similarity of /the h~ (2.14 MeV) i n ^ B state to the 2+ (4.44MeV) state in«

12C. If the h~ state resembles a p y proton hole coupled to the 12C 2'v

state, then proton inelastic scattering to the 2+ level should show a

similarity to our p : data.

To corroborate this, detailed angular distributions were taken at

12.0 and 14.0 MeV for the reaction 12C ( p,p1 ) 12 C(4.44). The 12C data

and DWUCK predictions for £ = 2 momentum transfer are given at the top

of figure 19 and tend to confirm our hypothesis for the 1JB h~ level.

In both the 12C and J1B DWBA estimates shown in figure 19 there has

been no adjustment of the spectroscopic factors to normalize the theory

to the data. We again selected Watson's optical model parameters as

input to DWUCK. The agreement of the DWUCK and JIB3 elastic scattering

predictions with the data of LeVine and Parker (LE69) and the success

of the DWBA code in reproducing both sets of inelastic data even though

they differed from one another by a factor of ©10 gave us confidence

in this parameter set.

C. Alpha Decay of 12C and the (p,a) Reaction

1. Possible Quartet and Alpha Cluster Structure, in 12C

The excitation functions (figures 9 and 10) for the (p,ct) reaction

have more detail than do the (p,PQ) or (p,px) excitation functions.

When distinct structure rises above a diffuse background, it may indicate

that a state of simplified structure having a selectively long life

time is formed at this particular energy. Since the (p,Cl) reaction has1 2more structure than the two proton channels, it is possible that C

has a significant alpha particle consituency at these high excitation

energies.

A recent theory proposed by Arima et al. (AR70) discusses the

eneryies associated with quartet-hole excitations in several N = Z

nuclei. At ^25.2 MeV these authors predict the existance of a 0+ state

corresponding to the excitation of two quartets into the lf-2p shell.

Although the energy associated with this state could be significantly

different from 2.5.2 MeV, it is interesting to observe that at 118°

(where p the Legendre polynomial for Z = 2 has a zero)there, appears

a sharp anomaly at exactly 25.2 MeV. Whether or not this can be

associated with such a quartet excitation is uncertain; it is clear,

however, that there is significant resonance behavior in the a

decays‘near this_ energy.

An alternative to this theory which also might explain the enhanced

structure in the a Q excitation functions was developed by Sheline and

Wildermuth (SH60). They visualized 12C as an alpha cluster orbiting

in the effective central anharmonic nuclear potential of Be. In

this model the 12C ground state consists of 8Be in its ground state and

an alpha cluster in a 3s orbital. A second 0+ state would result from

an excitation of the alpha cluster to the 4s orbital. Both the ground

and first excited 0+ states would have rotational levels built upon them.

Reynolds et al. (RE71) believe to have found the 2+ and 4+ members of

the second 0+ rotational band at 11.16 and 19.39 MeV and predict the 6+

level to be at approximately 32 MeV. This or a similar a-cluster

level built upon the 8Be first excited state (SH60) may be responsible

for the increased detail visible in figures 9, 10 and 16 at excitation

energies beyond 25 MeV.

Although these theories can forecast the postions of a few isolated

resonances, they were not developed to show detailed energy or a.ngular

dependence for the differential cross sections. In the next section a

procedure.for obtaining anqular distributions from a direct reaction

mechanism is proposed.

2. A Plane Wave Theory for the llB(p,a)eBe Reaction

In an earlier section the results of linear least squares fit to the

(p,0() angular distributions using Legendre polynomials showed that

both even and odd coefficients for £ > 4 were non-negligible and that

the reaction cannot be well described by simple compound mechanisms.

Because of the success of both the optical model and the distorted

wave theory at describing our (p,p0) and (p,Pj) data we felt that

perhaps a direct reaction theory should also be applied to the (p,C<)

data. Such a theory has been proposed by Cavaignac et al. (CA71) who

sugcpst • that a PWBA calculation which includes heavy particle pickup is

perferrable to a DWBA approach which omits it.

Our PWBA calculation includes contributions from three of four

possible projectile-target initial state interactions. These inter­

actions and their associated matrix elements are diagrammed in figures 19

and 20. The exact expression for the differential cross section con­

tributed by these matrix elements would have the form

However, Cavaignac and collaborators had determined from preliminary

studies that the contribution to this cross section from the heavy

particle knockout was almost isotropic. Although calculations by

Gambarini et a l . (GA69) tend to disagree slightly with this conclusion,

-73-€

we have followed the suggestion of Honda and co-workers and have set

the heavy particle knockout term to zero. 'A further simplification

was made in which only the alpha knockout contribution, A , , or theJ * Ctkotriton pickup contribution, A , is added coherently to the h e a wtpuparticle pickup. Since A , and A are very similar analytically,ako tpu 2

and the number of free parameters is thus significantly reduced, such

a simplification seems justifiable.

What follows is a detailed calculation of a typical matrix element,

the heavy particle pickup term, A, , and a presentation of thehppuexplicit relationships for A^, q and • The amplitude for the heavy

particle pickup contribution can be written as

= < ^ | V | 0 .>hppu f 1 pc,ri = m r dr dr0dr dr dr V (r J ^ p 2 c pc ac f pc pc i( T V - 7 1 )

with

l c m c

< (=v > a'/ O 'a a

and<s

= <P ( k , r ) X ^ ( r ) r ( S i / i m l j u ) . f a a R o R a oc £ m p p p p up p p

P P I c ^ c

( I m j ( i |L , r n ) X f ( r ) X f ( r ) Y P ( r ) Q . ( j r | ) . c c p p 1 R R p p c c i p c £ r p c '

' P P

m (r/-73)

The <p's are plane waves for the entrance and exit channels, the x's are

intrinsic spin functions, and Y™ ( L gives the angular and radial dependence

of a cluster relative to its core. ip describes the probability of

finding a cluster of four_nucleons resembling an alpha particle in the

target nucleus and from it one can define the quantity

0 = f j/)1 / ? ) X f ( r )* d r a J Yot a a or <a

( T V -7 4 )

The radial dependence of cluster relative motion can be further expanded as/

0 . o =

\ I

R .0 - S o \0 1 a .

1

> f o r r > R . .l

( T V -7 5 )

Q . , has an asymptotic dependence governed by h 0 . (iK.r.) , a HankelX/1 iC 1 X 1

funtion of the first kind, and is normalized to unity at r = R, provided

one assumes a radial reduced width 6 and a soectroscopic factor Sr ..o£i * -1-

With this notation ppu reduces to

© -fth p p u 2/i

p c

/ \

i R 3 R 3 j ' p a '

1/2(477) ■ £ s. S. 0 O

i i I I i o a ap Jp Ac a p a

. ( ^ p - ^ a )

• ~ o — r ~ • J o (K ^ » K R ) J n ( K R , k R ) K 2 + k? p p p P a »a a a a

a a

EM p m p m c m a

(3 y f m lj u ) (j a I m |L,na ) (I m 1 m Ii m ) P P P P wp V u p Fp c c ' T l R 7 K c c a a ' T T .

M n

Yi <Ka>Y/ ( V -a p

( T V -7 6 )

The J's are Wronskians defined by

J i = K R j i - i ( K R ) + c i <K R ) h ( K R ) * ( T V -7 7 )

C i + 1 = ( K R ) / ( 2 i + 1 + K R ) ’ (T V -7 8 )

K - 12 p E j / ft , E = c l u s t e r b i n d i n g e n e r g y ( T V -7 8 )• D* v » / IJ • m2 *

r* m p t*K = k + — o r k

p p m R a ( I V - SO)

■* -* r a a -»K = k + - ~ k

a a ro T p (TV-81)

■»

-81-

Similarly A and A_, aretpu uko

a . = (8 7T)1 /2 (K R )1 /2t p u 2/1 p t t '

_ l t V ' ^ V Y W• s s. o . e . i -

h i t l t ' o t ( K t R t ) 2 + ( K t R t ) 2

for

Y f ( K t ) . t

■+ _ + m R -»Kt " ka " "SF kp CIV-

and

A = - (4 jt)3 / 2 v r 3 r S , S , 0 o f L P“ k , j i p a L i oof Of

V r V W■V a T . W \ u »

S <'Snl'n', r,m i J i T,,In ) ^ . ‘ V n h J V ' V/ ip m p n ig m Q , P P P P P P c c p p rt

P m i m | L m m ) ( i - m I m IL M ) ( - l ) m p ( K 1 c c a oc1 T T ; K p p a a 1 ; \ 1 L R

for

+ M e +^ R ~ M R t . ( I V - 8 5 )

( I V - 8 2 )

8 3 )

(TV-84)

In equation IV-84 the relations V(r ) = -V . 4fr „ 3 - , i > 1 i-> 1 ,p a p a — R* 6 (!rp |-|ra |)

and h^^ (iK, R) h^ 2 (iK2 R).=: hL (lKR)( £ ^ “r ) f°r L = £ l+ 1 z and K = lK i + '

have been assumed.ft

The PWBA estimates, of the angular distributions for the alphas

were calculated for two cases. In the first approximation the coherent

sum of A , and A, ' as in.figure 20 was squared and summed over allowed ako hppu

channel spins as indicated in equation IV-70. For the second case A ;axo

was replaced by Atpu an^ the processes repeated. -In figure 22 the

relative importance of the triton and heavy particle pickup terms is

illustrated. The triton pickup and ako matrix elements give a

predominantly forward contributions v;hile that for the heavy parricie

pickup gives a backward peaking.

At the seven energies where we have taken detailed angular dis­

tributions a least squares PWBA fits to the data were performed. The

cut-off radii for the triton, proton, and alpha clusters, as well as the

triton and alpha reduced widths were the only parameters allowed to

vary in the fitting routine. The spectroscopic factors used are tabu­

lated in Cavaignac1 s article andwTere derived from a p-shell intermediate

coupling model such as Kurath's (KU56). The non-linear least square

fitting routine developed by Bevington (BE69) was permitted to search

until the change in X2 was less that 1%. Results for the two cases are

shown in figures 23 and 24.

In an attempt to further improve the PWBA results we modified the

plane wave code to include first order Coulomb distorting effects as

suggested by Austem (AU64 ) . The wave vectors k -, were replaced by

their localized WKB counterparts in which a Coulomb potential had

been subtracted from the kinetic energy. In the entrance channel, for

example

- - V 2 / X , ( E - V ! P 7 / h . ( I V - 86)P't pt pt cou],

This modification in general produced a slight deterioration of the

agreement and caused the fitting routine to increase the value of the

optimum cut-off radius to compensate for the decrease due to the

Coulomb correction. The Coulomb corrected angular distributions are

shown in figures 25 and 26.

The shift introduced by the Coulomb correction is particularly

apparent when one plots the best fit cut-off radii as a function of

energy (figure 27). The solid and double dashed lines represent those'

radii predicted by fits with A^ppU plus or &hppU plus Aako respectively.

The dotted lines represent their Coulomb corrected counterparts which in

every case are merely displaced from the non-corrected radii by a few

tenths of a fermi.

Figure 28 gives the energy dependence of the reduced widths, 9^ ,

0^ , and of V ^ . These quantities serve mainly to normalize each

matrix element. _ At the top of this figure is plotted the energy variation

of y ) for each case and for its Coulomb corrected version. It is

seen that the best results were obtained when the triton and heavy particle'

pickup terms were combined and there was no adjustment of the wave\

vectors.

The success of this rather simplified model of the (p,a) reaction

is encouraging. Although finite ranged DWBA codes including heavy

particle pickup are now available, this PWBA calculation with projecti e

core interactions may still be the simplest means of examining the

(p,a) reaction for large incident proton energies. The importance of

the heavy particle pick up contribution to the differential cross

section is evident from figure 22. In order to reproduce the angular

distributions any model of the (p,a) reaction should contain some

projectile-heavy particle interaction. The success of the PWBA is,

in part, due to the inclusion of such a term, i.e. the heavy-particle

Optical model (solid curve) and particle-hole (dashed curve) predic­

tions for the proton elastic scattering angular distribution. The

calculations were performed at the energies indicated by arrows on

the excitation function.

(Bap

)'-0

’3©

{dcr/dil)c rn (mb/sr)— oo o

(d cr/d Q.) a frjmb /sr)

-98-

(A3W

) (g

p';

Optical model (solid curve) and particle-hole (dashed curve) predic­

tions for the proton elastic scattering angular distributions at

energies indicated by arrows in the excitation function.

2340i—i— 24 23 26 —i—

27 28 —i—

Ee x (Ci:?)(MeV) 23 30

“ i—31 32

—i—34

— r—33

— r -

30

5 2 0 v / V \«*> H}Si,

B 11 (p,p0 ) B11

0 c . m . * H 2 '

* cccito-H* +i ii ii, i - q

10 12 13 14 15Ep.iob(MeV)

16 17 18 19 2 0 21

i00CD1

©

P T . T i r o 1_Q-----

DWBA estimates for the inelastic scattering of protons by a 1*B

and a 12C target. The DV7BA results which use the optical parameters

of Watson (V7A69) and £ transfer of 2 have not been normalized to

the data.

dcr/d

& (m

b/sr

)

AN G U L A R DEPENDENCE OF P R O T O N S I N E L A S T I C A L Y S C A T T E R E D FROM l2C AND MB

# c .n .(d eg )

-91-

Figure 20

Feynman diagrams, Hamiltonian and matrix elements

particle pickup and alpha knockout terms included

calculation.

for the heavy-

in the PWBA

'32-

MATRIX ELEMENTS

T E A V Y P A R T I C L E P I C K U P A L P H A K N O C K O U T

H = T pT + T a C + V p a + V p C + V aC

H o = T p T - + T a c + V a C

H - H 0 = V c + V

<hppu) = <f|vpC| > ^ako^> = >Kf V o a

Feynman diagrams, Hamiltonian and matrix elements for the triton

pickup and the heavy-particle knockout terms needed for the PWBA

calculation.

P W B A M A T R I X E L E M E N T S

T R I T O N P ICKUP HEAVY P A R T I C L E KNOCKOUT

t

H = TpT + T t C + V p t + V pC + V tC

H 0 = T p T + T t c + V tc

V + V Pt'

(tpu>=(%|vpt|%> <hpko)= <^f |vpC| i)

The forward angle (|tpu|2) and backward angle (|hppu|2)

to the (p,a0) angular distributions and their coherent

for Ep = 12.0 and 18.0 MeV.

J? i q*jur g 22

contributions

sum (dcx/dft)

C O N T R I B U T I O N S T O T H E P W B A D I F F E R E N T I A L

C R O S S S E C T I O N S F R O M T H E T R I T O N P I C K U P A N DH E A V Y P A R T I C L E P I C K U P

M A T R I X E L E M E N T S F O R R E A C T I O N n B ( p , a ) 8 B e

dcr/d^il < t p u ) i 2|<hppu>|2

O . O I

& . m . ( d e q )

30 90 • 1500 R . m . ( d e g )

Angular distributions predicted by a PWBA which included triton

pickup and heavy-particle pickup terms plotted with the data. No

Coulomb correction has been introduced.

(do

Vd

£l)

c m

(mb

/s

r)

M B ( p , a ) 8 B e C R O S S S E C T I O N S F O R T R I T O NP L U S H E A V Y P A R T I C L E P I C K U P

W I T H O U T C O U L O M B C O R R E C T I O N S

0 c . m . W e g )

Angular distributions predicted by a FWBA including alpha knockout

and heavy-particle pickup contributions without Coulomb correction.

Figure 24—

(d

cr

/d

^)c

rn<

(mb

/s

r)

llB ( p , a ) 8 B e c r o s s s e c t i o n s f o r a l p h a k n o c k o u t

P L U S H E A V Y P A R T I C L E P I C K U PW I T H O U T COULO.VIB C O R R E C T I O N S

0 c.m.(deg)

-102-

Figure 2 5 — -

The Coulomb corrected results of a PWBA containing both triton and

heavy-particle pickup matrix elements.

(da

-/d

<n

,)c-

rn.(

mb

/s

r)

1 1 B ( p , a ) 8 B e C R O S S S E C T I O N S F O R T R I T O NP L U S H E A V Y P A R T I C L E P I C K U PW I T H C O U L O M B C O R R E C T I O N S

0 c.m.(deg)

-104-

1? •* m i v n O C~.L 4 .V ) v W

The Coulomb corrected results of a PVJBA containing both alpha knockout

and heavy particle pickup matrix elements.

(dor/d

&L m (mb

/sr)

llB ( p , a ) 8 B e C R O S S S E C T I O N S . F O R A L P H A K N O C K O U TP L U S H E A V Y P A R T I C L E P I C K U PW I T H C O U L O M B C O R R E C T I O N S '

^ c . m . ( d e g )

106-

Figure 2 7 __

The energy dependence of the optimum cutoff radii as determined by

a non-linear least squares fit of the PWBA results to the data. Note

that the cutoff radii obtained by a Coulomb adjustment (CC) of the

wave vectors are merely shifted a few tenths of a Fermi from their

uncorrected (NC) values.

/ - . I t

ENERGY D E P E N D E N C E OF C U T O F F RAD I i P R E D I C T E D BY L E A S T SQUARE F I T T I N G

n B ( p , a ) 8 Be A N G U L A R D I S T R I B U T I O N SW I T H A P W B A

V

------------| < t p u > + < h p p u > | ^ c -----------| < a k o > + < h p p u > | ^ c

— | < t p u > + < h p p u > | ^ c .................. | < a k o > + < h p p u > | ^ c

^ • p j a b . )

08-

9

Figure 28

The energy dependence of y 2 and the cluster reduced widths which

result from a least squares fit to the (p,a0) data using several

forms of the PV7BA.

DEPENDENCE OF X 2 AND THE C L U S T E R "109‘ REDUCED W I D T H S ON ENERGY

-----|<C2ko>-F<hppu>|c |<tpu>+<hppu>lNC |<tpu>+<hppu>|?CC ■|<ako>-r<hppu>|?CC

2 0 . 0 -

e a

\2 13 14 15 16^•p J a b . )

17 18

-C haD ter V

multi-nucleon emission threshold; where the compound nucleus has a

large number of decay modes available to it, and the lifetime of a

compound state (^h/ [ET ]) approximates the projectile transit time

('T'10- sec) . The Bohr assumptions of the compound nucleus are nop

longer valid.

In figure 11 one can see that for 12C the transition from a

nucleus v/ith resonant'features such as at 20 MeV to one in which 17B

interacts as homogeneous matter occurs at 'u27 MeV. The twelve or

so levels below the proton threshold and those upto nearly 27 MeV

excitation have various descriptions, as analogue resonances (AJ57),

particle-hole configurations (GI64), shell model states of intermediate

coupling (KU56) , -or alpha cluster states (AR70) . Also in this region are

states approximately described by these models but which have additional

rotational or vibrational characteristics. There are a finite number

of states below 27 MeV and they all‘can be parameterized by microscopic

quantities. Above 27 MeV levels begin to mix and overlap, and there

is a diffusion of the microscopic information. Given sufficiently

elaborate compound models and much computer time perhaps one could

filter out this information. Its import would be obscured in the

complexity of the model and one must be satisfied with the alternative

approach of direct reaction theories.

Before abandoning all compound analysis in favor of these theories,

the investigator should not overlook the possibility of the collection

of nucleons clustering into a state of high energy but low entropy. For*

such a state (as is observed in nuclear molecules) the excitation

energy is great but the degrees of freedom are few and resonance

phenomena will appear. Such an occurence in 12C may be responsible

for the resonance features observed at approximately 29, 30 and 32 MeV

excitation energy by Shay in his JBe(3He,y)12C studies (SH72).

1. The Elastic Scattering

Although considerable improvement may be possible in the

particle-hole S-matrix /theory, the discrepancies between it and the

data indicate that for the energy range of our experiment effort is

best spent in an optical model analysis. The reason is that an

improved multi-particle multi-hole theory, such as proposed by

Dreschel (DR67).or Goswami and Pal (G063), having ground state and

collective correlations has an associated increase in ad hoc parameters

and its usefulness at high excitation energies is still questionable.

The optical model, however, car. reproduce the elastic scattering cross

sections v?hile indicating some of the properties of **B. The parameters

of the optical model, the real, imaginary, volume surface and spin-

orbit potentials; the nuclear radii’; and the diffuseness characterise

the target in generic rather than intrinsic quantities. Although these

quantities do predict certain of the 1 *B's nuclear properties one v;ould

like to know if they are unique or a function of the experiment performed.

Generally optical parameters are not unique and the problem of finding

an absolute minimum in their parameter space is difficult. They may

also be sensitive to the type of experiment being performed, for

example, the best fit nuclear radius is a function of what projectile,

nucleon, electron, alpha particle-etc. is probing the target. So in

this transition from a compound to a direct analysis of the elastic

B. Implications of the (p,p) and (p,a) studies

scattering there is a loss of information about the internal nuclear

properties, but a gain of information about the specific bulk properties.

This new information depends on the type and energy of the probe as

well as the assumptions made concerning the model. The optical model

results should therefore be interpreted carefully.

2. The Inelastic Scattering

Because the h~ first excited state of 1*B is not a simple particle-

hole configuration, the theory of Gillet and Brassard could not directly

be applied. It is unlikely that an improved calculation performed

with Gillet's Approximation II, essentially on RPA, would have enhanced

the results, since both Approximations I and II gave poor fits to the

first: excited 2~ L~C state which is felt to be similar to the h~ stare

in 1 1 B. The results of the particle-hole work for the elastic scattering

indicated that a direct rather than a compound analysis should be used.

Presumably this would hold true for the inelastic case as well.

The similarity of the angular distributions for the 12C (p,p,) 12C (4. 44)

and the 1*B(p,px)1*B(2.14) reactions is evident in figure 19 and implies

that quadrupole excitation of both 1 and lzC has an identical effect

on the proton flux. Such a similarity is not necessarily a feature of

nucleon inelastic scattering on neighboring nuclei; for example angular* * 7 9distributions for 14 MeV neutrons inelastically scattered by Li, Be,

and 12C have markedly different patterns, c.f. figure 19-3 of Preston

(PR63). ^Although the n B and l z C cross sections differ by almost a

-116-

DWBA predictions, which are unnormalized and not adjusted to fit the

data, to give both the correct shape and magnitude in each case. The

success of the DWBA for light nuclei warrants a further investigation

of nucleon inelastic scattering in this mass region.■s

3. The (p,0t) Reactions

For the proton induced alpha emission there also was no compound

theory which could adequately describe the angular distributions.

Consequently, we adopted an alternative plane wave model of the inter­

action. This was not an unreasonable decision in view of the direct

character of the (p,p0) and (p,p ) reactions. The success of thisc 1

sim^O.o tti.9 t o t z i o n d.s '*5.o.iTons1ii72.Lccl d.n. our itspiroci'ucLi.cLL.£r. n

complicated angular distributions throughout the entire energy range.

The monotonic trend of all the parameters beyond 15 MeV suggest that

the compound manifestations have now become sufficiently damped out.

The particular feature of this PWBA which makes it appealing is the

inclusion of the projectile heavy particle interaction. Since this

technique is not limited to the type of cluster species, it has universal

applicability and could be utilized, at least for surveillance studies,

in other reactions and mass regions. It will also be of interest to

analyze such reactions using a true finite range DWBA code which included

these projectile-heavy particle exchanges.

factor of 10, the optical parameters of Watson et al. (WA69) allow the

0

- A p p e n d i x 1 -

11B Target Production

Thin boron foils with thicknesses between 50 and 500 ygm/cm2

are difficult to fabricate because stresses created during the evapor­

ation induce flaking and curling in the films when they are released

from the substrate. The effect of these stresses can be minimized by

evaporating slowly onto a heated substrate. The processes outlined

below describes a technique to produce stable self-supporting boron

targets.

Boron sublimes at 2600°C (at atmospheric pressure); consequently,

electron bombardment and not ohmic heating methods must be used to

vaporize it. By molding amphorous boron powder enriched in 1 *B

and slurried in methanol into a small conical pellet we could obtain

intense local heating of the boron. The pellet was placed in the center

of a cooled copper crucible at the focus of a 200 milliamp - 1 keV

electron beam. To reduce the contaminant level evaporations were per­

formed with a vaccum of 10- 6 torr or less. The optimum pellet* otemperature for even deposition was ^1900 C, as estimated with an

optical pyrometer. Above 1900°C the boron tended to fuse and the

increased currents associated with this fusing produced showers of

boron fragments. Targets exposed to such a shower showed pitting and

non-uniformities.

The boron vapor was condensed onto a glass slide positioned about

five inches above the pellet. To inhibit stressing film we warmed the

substrate to ^200°C before the evaporation began and maintained this

In addition depostion rates were kept below 1 yigm/cm per min. A

thin layer of NaCl evaporated onto the glass before the boron served

as a release agent. After cooling, a slide would be gradually immersed

in water dissolving the salt and allowing the foils to float freely

on the surface. A typical 2" x 3" slide could be sectioned into six

target squares each large enough to span a 3/8" hole in the target holders.

Target thicknesses were roughly measured with an alpha thickness

gauge (C066). Areal densities derived from this technique were

accurate to 5% for pure targets.

Gauged and mounted targets were stored in evacuated dessicators.

Undessicated targets seemed more fragile and frequently broke during

storage or when subsequently placed in vaccum.

A more detailed description of the vapor deposition equipment and

the effects of substrate heating can be found in references (MR65)

and (AR6 6 ).

temperature for at least six hours after the evaporation was completed.

- I 2 C -

&

r-AppendjLx 2 -

0

Uncertainties in the Differential Cross Section

The differential cross section (da/dft) can be expressed incm

mb/sr as,

dCTd ft cm

Y ■ A • a * cos ij> 102 7 1 ; ____________________(1-D ) • L • Q * T • P dfi, ,t c lab

labcm (Ap. 2-1)

where

Y = measured yield in the spectrum,

A = target atomic weight (gm/mole),

q^ = projectile charge (Coul/projectile),

<{> = angle target incline to the beam (degrees) ,

L = Avaqadro's number -(atoms/mole),

Q = total charge collected (Ccul),

T = target thickness (gm/cm ),

dfi = detector solid angle (steradiam),

d_Qlab

cm

= Jacobain transforming from lab to center-of-mass,

D = analyzer dead time

P = per cent of the foil composed of target material c

dft = 7r(d/2) 2/r2

d = aperture diameter (cm),

r = distance from aperture to target (cm)

The uncertainity Af associated with a function f dependent on the

variables q; is given by

Af/f =

f ni -3

6f<5q-i

2

(A_2 j) 2

3 f 2 (Ap. 2-2)

Tabulated below are typical values of the q_. entering into Equation (Ap2-1)

(|> = 45 ± 2°

Q = 100 - 300 ± . 5yC

T = 310 ± 20.ygm/cm2

P = 78 ± 1.0% c

d = . 2 - .5 ± .0/cm

o o r

A<p/<p = .035

AQ/Q = .01

AT/T = .06

AP/P = .02C

Ad/d = .05

'Ar/r - . 0 1

If a linear background is subtracted from the total counts within a

peak, then the yield Y for X^ counts in the i ^ channel is

NY = I X. - [ ( X ^ J / 2 3-N

i=l(Ap. 2-3)

where Av is generally taken as 1/X. . For a yield of Y at least"i 1

1000 counts the statistical uncertainity is 3% and the total uncertainty

8.5%. From equation Ap. 2-2 one sees that the lower bound on the un­

certainty is 6.5% determined by the quality of the physical measurements

and that the upper bound is a function of the counting statistics. We

adjusted the amount of charge collected to maintain yields at 1 0 0 counts

0or more; thus the maximum total uncertainty could not exceed 13%.

12 1 6It is possible given the cross section for C and O to

determine exactly how much increase in thickness the presence of the

contaminants introduces into the estimated areal density T of the

target. Equation Ap.2-1 can be simplified and rewritten for a pure

target k with N scattering centers /cm (N = T-L/A ) as K K K K

Y q cosUJ da k P ,dQ ~ Q aft N 2_4^k

Since in Equation Ap.2-4 only N is unknown, one may simply solve forJC

T , the areal density of the contaminant k. By subtracting all T 's Jc k

from the estimated thickness T. the oercent by mass, P . of the foil1 - c

composed of the isotope of interest can be obtained. This process

requires a few iterations, however, because the specific energy loss

S (E)' entering into the equation for an alpha gauged target,

IL = P • d • (S(E )/S(E )) , (Ad 2-5)target air air a a t a 1 p ;

varies with the element and isotope involved. For our 71B target

the ratio P of the actual areal density of 1:B to the areal density c

predicted by assuming the range shift, d due t-o a pure 1 targe

was .78 ± .01.

When our cross section for the p„, p , and a reactions werer o r i o

compared with those of other authors .(SE65, RI6 8 , 3E63) and the

above corrections had been introduced, agreement was usually within 5%.

No correction was made for contributions -to the areal density due

to- 10B or 1 5N . Their low yields in the proton spectra and the good

agreement of our.data with other sources indicate that their effect

was minimal.

A p p e n d i x 3

-126

Differential Reaction Cross Section in a j-j Coupling Notation -

In order to utilize the S-matrix elements of Brassard for

estimates of proton elastic scattering cross sections, it is

necessary to derive an expression for da/dSI in a j-j coupling

notation. A procedure similar to that of Blatt and Biedenharn (BL52)

will be followed, except that Wigner 3-j and 6 -j symbols will be

used in place of Clebsch-Gordon and Racah coefficients.

An unperturbed flux of plane-waves is assumed such that

7 ° ^ —i eik“ Z | lv > | s o > . (Ap.3-1)a ± ^ 0 ,

The factor (.IS) * = [ (21+1) (2S+1) ] "* was introduced in order that

the total beam be normalized to unity and not merely each spin state". iTc zThe quantity e & can also be expressed as

e l k a z = (477)1 / 2 £ ? j£ (k r ) i* . . ( A p . 3 - 2 )

In a j-j coupling scheme the system consisting of a particle of spin

s and one of spin I in an eigenstate CL (where a may define the energy-,

the isospin, or several other quantum numbers) is represented explicitlyc

by ’

i n f i v o m

( i s ] \ ( ] I S \ k o: i , V V 9* J W J ■*■ <-> \ o: JL „UJ JL OC \ ^ 1 . .mo-ti) ( w -u ) T T 1 V V i r r - | I y > | s 0 > ( A p - 3- 3)

-127

rj /x J M

Employing the orthogonality relations for 3-j symbols (ME62) one has

k a J- . . m , ,

7 T 1 Y i t n a ) l T 7 r |1 1 ' , s aa v a '

(--l)<1 - E* » V l ) (H+M )l 3- ( ^ 4 ) ( ^ M) ' l«d s)iUM >.n (Ap. 3-4)

The j^(kr) in equation Ap. 3-4 have-the asymptotic form

j , ( k r ) ~ £ [ t e i ( ! ~ - < i + l > < r / 2 ) 3 + { e - i C , r - ( J + 1 ) , / 2 ) j ]

Let f^(kr) contain the radial dependence of the wave function in the*. 4.- n ~ _i(kr-(£+l)TT/2 )external region,then, asymptotically f^ - e and

e ik Z ~ (4tt)1 / 2 £ t ( f * + f £ ) i Y ° ( n ) . ( A p . 3 - 6 )

Combining equations 3-1, 3-2, and 3-4 one gets

r r . / , • ( - l ) ' ^

“ “ “

. ^ jj-I+ M ) fi s j j ( ,0 ( is )j iJM>out+|a(is)jIJM >i n } . (Ap. 3-7)

V OThe asymptotic value of the unperturbed wave function f o r two

particles of spin S and I may also be expressed as

0 y ° ~ £ A ™ ’ VG [ lo£(i s ) j I J M > + \ a ( l s ) j I J M > . n ] . ( A p . 3 - 8 )a '£ j J M c « J ’

By matching the coefficients of the spherical wave |a(£s)jIJM > at

infinity we see that

AJM;pa= I f f < i +j+S+I - p f s j \ / j l J J cdj i s k \ocr-oy \cv-MJ

After an interaction occurs the asymptotic expression describing the

system in channel a with spins S and I and projections C and V

becomes

A t o t a l ) = S [A J f ;yCT | a ( i s ) j I J M > . + B ™ . ;yCT| a ( £ s ) j I J M > } . ( A p . 3 - 1 0 )« i j J M i n c d ] o u t

The incident flux we had defined to be

0 yCT ( i n c i d e n t ) = £ { A J ^ ;I/Cr | a ( i s ) i I J M > . + A ™ ;l/Cr | a ( i s ) j I J M > J . ( A p . 3 - 1 1 )a JM-JM ^ J — 111 OUi'

The flux in channel a 1 due to an interaction is

( r e a c t i o n ) = ( t o t a l ) - 17 ( i n c i d e n t ) . ( A p . 3 - 1 2 )

In what follows the primed notation will refer to final state

wave functions and unprimed notation to initial state wave functions.

The relationship between the incident flux of channel a and that of

channel a" is defined by the transformation

' g J M A J M ; y a _ < A p . 3 - 1 3 )a J s w tt'l'j'v'o'fitlivo a£j

Because there is no flux incident in channels other than a, the sum

over a is superfluous. For the reactions under consideration here

S is independent of V, a , V', and o ' . S also does not depend on M

(c.f. BL52). These conclusions result in the following simplification

of (Ap. 3-13)

J M ; V » a ' J M J J M ; v a .a ’ i ' j * ;a £ j VQ a £ j ( P* ”

Combining equations Ap. 3-4, 10, 11, and 12 gives

M u ' , , . , ~ f , , J M ; V ' O ' | ' ■>lb ( r e a c t i o n ) = S [ A a ' . . . > . + B Ip/. . . > j

v 1 i n a ' l ’ j' r o u t

V ' A 1 o ‘ (- ! ^ , ! „ , l - - ■>. . 'A — *> IST*.t r . , ; U ' * • • ‘ *4. • • • " • V 11, » °

T’ j ' J M a 1 ) m o u t

Where |a'. . . > implies

| a ' - • ■ % ) “ . ■ ( A p . 3 - 1 6 )

( o u t ) . ( o u t )

\This gives

K • ->out.- (Ap.3-17)

or

° ( r e a c t i o n ) = £ ( - 1 ) [ A J M ; b 'Crf 6 6 6 - S ' ™ >“ f } l o d j v W b a ' i « j » ; o r f j i J i * * * o u t '

JMiycr

Replacing |a’. . . > ^ with its actual representation yields

( r e a c t i o n ) = ( - 1 ) £ { 6 . . . 6 6 - S ' ™ } • A ™ ;Z' crj ’ J a ’ i ’ j ' j a i j a i j

J Mua

• [ £ ( - i ) ( i , “ s , + A ’ ) ( _ 1 ) ( J ' - r + M ) ? A . s ’ j* \

( j* I 1' J \ £* m ’ i i* ( n « y t _ M * ~ 7 Z ~ 1 Y n ^ ) i r - Z ----------------- | l V > , s » c r » > | .

Asymptotically this reduces to

IT-,!

(r e a c t i o n > = i T V T / v ■ | r ^ > l s , c;, >01% a 1 Q-t

J" jT

X i / j j t } ^

JM u c rp i ' m '

( l + j + s + I - u ) ( l j j \ / j l J \ , ( i ' - s ’ + j i 1)( _ i ' \ o a - G J \ o v - M J k ’

; (_1) 0. - r + M ) ? f ( , , g ( j ; r g / < . ( v ) ]

(Ap. 3-18)

( A p . 3 - 1 9 )

(Ap. 3-20)

-131

The quantity inside the large set of parentheses represents the reaction

amplitude *3 • v ■ O' * -otVCT 3n< accor< in9 to Blatt et a l . (BL52) is related

to the scattering cross section by

d a = X 2 | q |2 d<n . • (Ap. 3-21)a ' v ' a ' ;a v a a |na ’ f ' , c r ' ; a u a l a * *

Where ^ = k 1. If the reaction is insensitive to spin polarizations

then

dV ; o T fio' (Ap.3 22)V O

Substituting,

q a ' v ' a ' ; a v ( j n , ] y ^ a ' a V £ 5 y f S a xP y - , a l ] ]J M y c r

f i ' m *

r 1 1 1 ,-t,V+j+s+1- v) r j j j 1 Jl i \ o a - a J \ a v - M j

( i ' - s ' + j j 1 ) - G ’ - I ' + M U j ] £ ' s’ i T >

)' r J [ ,) ] . (Ap- 3-23)(j ’ y ’ - M ' l Jt* a '

into (Ap. 3-22) and collapsing Y™'s and 3—j 's wherever possible gives,

-132-V?

d ^ ( a > a ' ) i r X a S ^ , ^ ; c d ^ 3

£ 2 £ 2 ^ 2 J 2L>

T2 , 1 1 2 2 1 1 2 2

^ a ' a P p £ ^ i ’ i ’ * S a , £ ’ i ’ - a l i ^ ^ 2 ~ 2 , -“ “ 2 2 2 2 2 2 2 2 (I) ( s ) /4 7 T

A ^ i A / y i L ' \ / J 2J 1L ’ \ / ¥ l L

JaJie J W U V j K V ,* Y ° ( f t ) . ( A p . 3 - 2 4 )

It is this expression which was used to determine from the S-matrix of

Chapter IV the particle-hole elastic scattering cross sections.

fe re n c

AJ6 8

AL64

AR6 6

AR70

AU64

BA57

BE63

BL52

BL62

BL64

BE69

BR67

BR70

C066

AJ57

CA71

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