Alpha-Trans f er Reactions on 2olo"

and 'o*,

By

B. s. (unlver:?Ht:r"Hllt::1il, Davls) reoeM.S. (Flor lda At lant lc Universt ty) L97L

DISSERTATION

Submitted in partial satisfaction of the requirements for the degree of

DOCTOR OF PHILOSOPHY

ln

Phys lcs

in the

GRADUATE DIVISION

of the

UNIVERSITY OF CALIFORNIA

DAVIS

Approved:

/-l

Committee

Deposited in the University Library----...--.--

in Charge

- l -

Date Librarian

Alpha Transfer React ions on 20*.

and'o**

GORDON WAYNE WOLFE

DepartmenE of Physics and Crocker Nuclear LaboratoryUniversi ty of Cal i fornia, Davi-s, CaU-fornia 956L6

September, L976

AB STRACT

Alpha-pi.ckup reactions were investigated on 20N"

and 'Otr targets.

)o 6 16The -"Ne(d, "Li)-"0 react ion was performed at 40 MeV, and di f ferent ia l

cross-sect ions measured for laboratory angles forward of 100" on the

0.00 0*, 6.05 0. and 6.L3 3 doublet , 6,92 2+ and 7.Lz I doublet ,

8.88 2 , g,5g 1 , and 9.85 f and 10.36 4+ doublet states of 16o.

The

'o"r(ar6l i )20*" react ion was also performed at 4a Mev, ancl d i f ferent ia l

cross-sec:t ions for the 0.00 0*, 1.63 2* , 4,25 4* , 4 .g7 2 , and 5.62 3

states of 'o*" , over the same range of angles. The 'ot*(3H.r tu.)20*.

react ion rras done at 55 MeV, and di f ferent ia l cross-sect i -ons for the same

states were measured for two 7U"

spin conf iguraLions over the s€tme

range of angles. An upper l imi t of 3.z lL.L pb/sr c.rn. was establ ished

for the di f ferent ia l cross-sect ion for the'OW1or8Bu;20*. (g.s.)

react ion at a scatter ing angle of 30o. Opt ical-model parameters for

20*" + d at 40 MeV were extracted from the measured dif ferential cross-

sect ion of the 20*" (d, d ' ;

20t1" react ion.

The di f ferent ia l cross-sect ions were compared favorably to both

zero-range and f in i te-range DI{BA calcul-at ions which assumed a direct

interact ion and the descr ipt ion of the target nucleus as a stable core

plus a s ingle alpha- l ike c luster t ransferred in an S=0, T=0 state, and

spectroscopie factors extracted. The spectroscopic factors decrease

rapidly wi th i -ncreasing Earget mass, conf i rming the predict ions of

some ! i lorkers, but are at least re lat ively constant between di f ferent react ions

)oon the -"Ne nucleus. The presence of unnatural-par i ty states, which

cannot be populated by a direct interact ion, was shovrn by means of

coupled-channels calculat ions to be e>rplainable by a two-step process

with entrance-channel- coupl ing, again assuming transfer of a s ingle S=0,

T=0 alpha part icLe. However, a s imi lar calculat ion for ground-state to

ground-state t ransi t ions, which can proceed by a direct t ransfer, gave no

better predict ion of the data than a DWBA treatment.

l-v

CONTENTS

List of Figures

List of Tables

Dedicat i -on

Page.

vi

v i i i

ix

11

11

13

L7

30374L4247

48

I

I

2

4

9

I . Int . roduct ion

1.1 General Problem

L.2 Ev j -dence f or Alpha-Like

1.3 Alpha-Transfer React ionsPrevious Work

L.4 Scope and Purpose of the

Cluster ing

and Summary of

Present Study

II . Exper i -mental Equipment and Procedure

2.L Cyclotron, Beam Faci l i t ies, and Exper imentalArea

2.2 Targets

2.3 Charged-Part ic le Detect ion and Electronics

I I I . Theoret ical Considerat i -ons

3.1 Alpha Cluster ing l4odel

3.2 Direct React ion Theory in Alpha Transfer

3.2.L The DWBA, I ts powers, l imi ts, andappl icabi l i ty

3.2.2 The form factor3 .2.3 Opt ical Potent ia l s3.2.4 Select ion rules and al lowed f inal states3 .2.5 Coupled-channels calculat ions

3.3 Cluster ing in Target Nuclei

29

29

30

IV. Exper imental Resul ts and Discussion

4.L The 2o*" (d, d ' ; 2oN. React ion

4.2 The 2o*.(a,6l i )16o

*."cr ion

4.3 The 'o**(a, 6 l i ) 2or"

Reacr ion

4.4 The 'otr(3H., tu") 2oNu

Reacr ion

4 .5 The 'o**( 0. , Bn.12ott"

Reacr ion

4 .6 Mult i -Step Processes

4.7 Comparisons Between React ions

Append ix

Appendix

Bibl iography

Acknowl edgement s

Vi tae of Candidate

V. Surmnary, Conclusions, and Suggest ions for FurtherStudy

Detect ion Eff ic iencv for 8u"

Data-Summation Program for Two UnresolvedSta tes

51

51

60

66

72

B4

B6

97

101

105

LL6

L20

L24

L26

A

B

vl-

1.

2.

3.

4.

5.

6.

7.

B.

9.

10.

11.

L2.

13.

L4.

15.

L6.

L7.

18.

L9.

20.

2L.

LIST AF FIGURES

Cyclotron and beam l ines for charged-part ic lescatt .er ing.

Gas target.

Gas target conf igurat ions.

Alpha-transfer elec t ronics .

Gas h,andl ing system.

64 x 64 arcay.

ELast ic scatter ing elecLronics.

Harmonic osci l lator potent ia l wel l .

Energy leve1s in nuclei of interest .

Spectrum for 2o*"(d,

d ' )20N..

t0*"(drd ' ;2ONu Elast ic scatter ing wi th DWBA and coupled-

channels f i t .

20*.(drd ' ;20N" inelast ic scatter ing and f i ts.

20u. (d, d ' ;20N" 4 .g7 as z wirh DWBA f i rs .

20Nu (d, d t ;

20N. 4 .g7 DI,TBA comparison for 1 and z .

spectrum of 2o*"(a, 6 l i )160.

Angular distr ibut ion for the 20*.(a,6l i )16o

groundstate and DWBA f i t .

20Nu(a,6l i )16o higher stares and DI^IBA f i t .

spectrr :m for the 'otr(a,6l i )20r. react i -on.

'O**(ar 6l i )20*"

ground state cross-sect j -ons and f i t .

'o*r(a,6l i ) t0*. h igher srares and DI^IBA f i t .

?lL ? 7 )O- -Mg(-Her ' Be)-"Ne spectrum.

Pa ge

L2

15

16

20

IB

23

25

38

44

52

53

56

5B

59

6L

63

64

67

69

7L

75

vl_l_

22.

23.

24.

25.

26.

27.

28.

29.

30.

31 .

32.

33.

34.

35.

'o*r(3H", t r*)20N. ground state distr ibut ion and f i t .

'o**(3H., tu.)20N" 3/2 higher states and DWBA f i t .

'o tr( 3H., t u") 2 0*"

L/ 2 higher s ra tes and DWBA f j-t .

'o*r(3t t . , ' " " )20t t" 4.g7 MeV z srate and DI ' t rBA f i t .

'o *r1 o, Bn.y 2o*" specrrum.

Energy levels and coupl ings to 2 states.

'0*"(a,6l i )16o 8.87 z and coupled-channels f i t .

20Nu(a,6l i )160 8.87 z and DWBA f i r wirh AS=1.

'orr1a, 61i120*u

to 4 .g7 MeV 2 with DWBA and coupled-channels f i t .

Energy levels and couplings to 0+ ground states.

20Nu(a,6l i . )160 0* g.s.1 comparisons of DWBA and

coupled-channel s.

'ourr(a,6l i ) '0*. to 0+ g.s.e comparison of DWBA andcoupled-channels .

Breakup and detect ion of 8r . .

8U" detection geometry for EBGAS and val id detector

schemes.

77

79

80

81

83

89

90

91

95

96

106

108

92

94

vl_l_1

7tth614

l r1,ST Of 'J 'ADLeS

3ag"

I . Calculated Contr ibut ions to Resolut i -on 2L

II . Sourees of Exper imental Error and Uncertal_nty 28

II I . Al lowed Final States for Residual Nuclei for (a,6l i ) 43

IV. Al lowed Final States for Residual Nuclei for (3H., tu") 45

V. Opt ical Model Parameters 54

Vr. The 20*.(d,dt ;20N.

React ion parameters 57

20ru(ar 6l i ) 160 Reacr j -on parameters 65VII . The

yrrr . The 'O**(a, 6 l i )20}0"

Reacr ion parameters 7 z

rx. summary of Resul ts f rom the 'otr(3H"rtu")20*" React ion 84

X. Compari-sons of Alpha-Transfer React i -ons 98

].J(

"Gal i leo, the Tuscan art ist wi th his opt ick glassSaw eterni ty the other ni-ght,And Nature, in the course of th l -ngs,Goes quiet ly along in i ts own true channel . t t

John } l i l tonPav,adi,se Los t

Foz, Cynthia

I . INIRODUCT ION

1.1 GeneraL Pz,obLern

I t should be possible to der ive al l the propert ies of a nucleus of

A nucleons, or of a nuclear react ion in which A nucleons part ic ipate, f rom

the Schroedinger equat ion for the A-body problem.l

o,29'^ L ,2 v( ] . . .?o)+v(f1( - r i= l 2* i " i ' * '1

Such a solut ion assumes that we know

potent ia l and are able to solve the

of these requirements have been met.

- t , \ , , r /* - t . + . A rr t r* - t . , (1)" ' tA/ - r ( r1 ' ' ' rA) =*

5Tt ( t1 ' ' tA/

the speci f ic form of the interact ing

di f ferent ia l equat ion. To date, nei ther

Calculat ions of nuclear propert ies, then, require var ious assumptj-ons

and approximat ions. The usual method of at tack is to postulate a model

which give r ise to a wave funct ion whose propert ies can be tested. The

success of the model- is guaged by the abi l i ty of the model to produce a

predict ion for the value of some observable which is in c lose agreement to

Ehe exper imental ly measur ed. value of that observa bLe .2

Two models which have been qui te successful wi th certain restr icted

groups of nuclei are the shel l rnrdel and the col lect ive model. In the

shel1 model, nucleons of each species f i l l up shel ls in which part j -c les

with mutual ly opposi te spins pair of f , and in the ground state of the

nucleus, she1ls are f i l - l -ed to complet ion forming an inert t 'coret t wi th

t tvalencett nucleons j -n the outer unclosed she11 prescr ib ing the propert ies

of the , r . r"1".r" .3 Despi te di f f icul t ies, the shel l model is qui te successful

in descr ib ing level sLructure of nucl-ei near c losed shel ls, but becomes

very complex away f rom closed shel ls. In contrast , the col lect i -ve model

2

is mosL successful away from cLosed shel ls in so-cal led t tdeformed regionstt .

Here, the col lect ive mot ions of the nucl-eus as a whole are taken into account,

s ince the nucleus may be nonspher ical due to addi t ional nucleons outside

closed shel ls, and rotat ional and vibrat ional states are seen simi lar to

those in exci ted diatomi.c molecul ." .4

A model which j -s much simpler, but st i11 of fers nany of the advantages

of the above mod.els is the c, luster model .5 '6 In th is model, one considers

that the propert ies of the nucleus are due in part to the interact ions

of c lusters of nucleons wiLhin the nucleus, rather than indiv idual nucleons.

The assumption here that the internal wave funct ions of each of the c lusters

is of secondary importance to the wave funct ions of the interact ions

between the clusters is a great tool toward si rnpl i f j -cat ion of calculat ions.

Indeed, ant isSrmrnetr izat ion between the cluster wave funct ions reduces or

el iminates the di f ferences between the cluster model and the other nodels

descr ibed previously for certain "r""" .2

Wigner suggested in L937 that of ten the c lusters in quest ion are alpha-

l ike part ic les, and used this fact to explain the mass systemat ics of

l ight nuclei .T The assumption of a lpha-1ike c lusters has many points

to recommend i t , and i ts use makes certain calculat ions much easj-er. For

example, the 6r,

nucleus is of ten t reated as an alpha part ic le plus a deuteron,

and the level spacing using this model is qui te good.8'9 Theoret ical and

exper imental studies of the structure of 6r i

show cr d c luster ing between

50 and 100 per ""rr t .10

The idea of a lpha cluster ing is, thenr 4rr

at t ract j -ve one and is being used and studied throughout the wor ld.

L,2 Euidence for alpVn-Like elusterine,

Once the idea of srna1*l c lusters of part ic les in nuclei- is accepted,

iE is readi ly seen that alpha- l ike part ic les are good candidat,es for these

sma11 clusEers, especial ly in l ight eyen-even nuclei . Indeedr the

approxi-rnat ion should be ncst val id when the clusters t .hemselves are

t ight ly bound.

From an energet ic standpoint a lone, the alpha part ic le seems to be

the best choice for a c luster. I t has a very large binding energy--28

MeV. The binding energy per part ic le is largest of any stable nucleus

srnal ler than carbon- l2. The f i rst exci ted state of the alpha is 20 MeV

above i ts ground state, and the ground state is very s i -mple, being four

ls nucleons in a total S=0, T=0 state f i l l ing the N=0 lowest harmonic

osci l lator shel l wi th total angular momentum J=0. Thus, the alpha

part ic le is extremely t ight ty bound, and is al-so doubly magj-c.

Addi t ional evidence can be seen by looking at nearby nuclei . I t is

seen that there are no other A=4 nuclei wi th l i fet imes as long as

-)1 - 4--2 X 10

L'- seconds, and A=5 nuclei , formed by adding a nucleon to 'He'

have l i fet imes on the same order.3 fhe 8Be

nucleusr which has been treated

as Lwo alpha clustersrs is unbound by 94 KeV, and decays to two alpha

part ic les in about 2 X 10-16 seconds. Indeed, al l A=8 nuclei quickly

decay to two alpha part ic les.

Heavy nuclei a lso show some indicat i .ons toward alpha cluster ing.

Most isotopes of Z=84 or heavier, &t least those near the minimum of

the curve of energy vs. atomj-c number for constant atomic masst decay

by emission of an alpha part ic le.

I t has been suggested, however, that the l ighter nuclei are those

that are the most prone to c lust . r i rg. l l Shel l -model calculat i -ons tend

to support the idea, €r l though a few studj-es contradict i t .LZ I t has

been shown that resonances and f luctr :aLing cross-sect ions of nuclear

react ions in l ight nuclei can be accounted for by using the cluster

1?mode1.- ' Much of the work done in studying alpha- l ike c lusters has been

done on N=Z even-even l ight nuclei the so-cal led "alpha-part ic le"

4

nuclei because their unusual ly low alpha binding energy would be expected

to favor direct surface react ions. l l Furthermore, Lhese nuclei can usual ly

be treated as a stable even-even core plus an alpha part ic le, ots as in

the alpha-part ic le model, as aggregates of a lpha part ic les only .3 'L4

Eichelberger, et . r1.r15 have shown that the ground state of LZc

could wel- l be approxirnated by an o-part ic l -e-BBe mode1. Wi lk insorr l6

eoncluded that on the nuclear surface there is a higher probabl i l i ty of

f inding nucleons grouped in alpha- l ike c lusters than one r , ,puld expect

from isolated nucleons, because the Paul j - pr inciple plays a smal ler ro le

there. Hartree-Fock calculat ions, which made no assur lpt ions about an

alpha-cluster ing ef fect , showed that in l ight N=Z nuclei , the nucleons

tend to group together in srnal- l c lusters which may be ident i f ied as

alpha cluster" .17 Also, the alpt la c luster model has been shown to account

for deformed ground states and rotat ional bands exci ted by var ious

a-transf er r eact iorr , . 18

L.3 ALrtLm, Tv,ansfev, Reaetions and Suwrwv,u of Preuious Wov,L

One l iay of test ing any nuclear rnodel is to apply i t to the

descr ipt ion of a nuclear react ion. One of the easiest appl icat ions to

vi-sual ize is the addi t ion to a nucleusr or the removal f rom a nucleus

of a s ingle part ic le or part ic l -e- l ike c luster, and the observat ion of

the nuclear levels exci ted.

Sj-nce the alpha part ic le plays such an importanL role in nuclear

physics due to i ts high binding energy, i t was expected that cr- t ransfer

react ions would resul t in nuclear structure informat ion regarding the

correlat ion of nucleons inside ,ru" l . i .19 The transfer of an atpha

cluster f rom the incident part ic le to the target nucleus wi l l g ive

informat ion on states in the resj-dual , r r r"1"r ' r " r20 whereas transfer of

four nucleons as an cr- l ike part ic le f rom the target nucleus to the

incoming part ic le in a s ingle-step process should select ively populate

f inal states which c losely resemble the ground state of the target

nue leus minus an alpha part ic l . .2L

Interpretat ion of the exper imenta1 data and extract ion of nuclear

structure informat ion are much simpl i f ied i f i t can be asstrmed that the

transfer takes place in a pr imari ly di rect process 12'22 as opposed to a

pr imari ly compound process. Distorted-wave Born approximat ion (nWgA)

calculat j -ons can be used to qive theoret ical cross-sect ions f rom a model

Lf d i rect react ions can be assumed. A direct react ion is one in which

the react j -on takes place in a s ingle step, wi th the target and project i le

never losi"ng their d ist inct ion. 0n the other hand, in a compound react ion

the project i le and target form an intermediate nucleus which decays into

the exi t part ic le and residual nucleus. The two types of react ions are

dist inguishable by means of a number of features, but no one feature is

suff ic ient .

Any resonances in the cross-sect i -on of a direct interact i -on should

have a width which is an appreciable f ract ion of the incident energy, at

least at medj-um energies. Indeedr atry osci l lat ions should be qui te wide.

Furthermore' i f the lncident energy is much larger than ei ther the

Coulomb barr ier or the react ion threshold, the total react ion cross-

sect ion should decrease rather s low1y with increasing incident energy.

The di f ferent ia l cross-sect ion wi l l show large f luctuat ions wi th changes

in angle and a general decrease with increasing angle. On the other hand,

compound react ions normal ly behave qui" te the opposi te. They are

character ized by rapid f luctuat ions wi th energy, and the di f ferent ia l

cross-sect ion is more or less symmetr ic about a 90o scatter ing angle.

6

Most of the ear ly work in the study of o- t ransfer react ions was

done using the (6t i ra) and (7t , i , t ) react ions. The pr i rnary purpose in

most of th is work was to study the c luster nature of the Li th ium

project i les, a l though a few tr ied to study the t r iangular and tetrahedral

CI,-part ic le structure of L2c

and t6o,

respect ively. An excel lent summary

of work done before L970 is given with a react ion l is t in the review

art ic le by Bethge.19 Almost al-1 of these ear ly react ions were done on

Li th ium, Beryl l ium, Carbon, and Oxygen targets, wi th a scatter ing of

medium-weight targeLs. The general consensus among these exper imenters

is reported. to be that the react ion proceeds by a direct proc"""23

al thought the resul ts are somewhat inconcl-usive. The exper iments were

hampered by the low energies used, and there i -s some quest ion as to

whether the ineident energies were high enough to al low direct processes

to dominate. The highest energy used unt i l - L970 was 30 MeV, wi th most

less than LZ MeV. The vic in i ty of the Coulomb barr ier compl icated the

analysis of the data. The l -ow energies also al lowed few excj- ted states.

A shi f t in the Li th ium structure wi th j .ncreasing energy was also evident,

indicat ing a part ia l breakup of the project i les.

Since the breakup of L i th ium was a ser ious disadvantage, the

o-transfer was also invest igated with heavy iorr" .19 The invest igat ions

on l ight target nuclei , and only tor , t ' r ,

and 160

have been used, show

the same direct naEure as the Li th ium-induced react ions. Invest igat ions

on heavy nuclei appear to be general ly in the region of the Coulomb barr ier .

In contrast to the above u-str ipping react ions, a number of q,-knockout

and cx-pickup r€ct ions have been done using pro j ect i les of mass equal

to or less than the mass of the alpha i tsel f . The pickup of a lpha

7

part ic les yei lds informat ion about the alpha-cluster wavefunct ions of

the target nucleus i tsel f .

Knockout reac t ions, su ch as (p, op) and ( o, 2 cx) can give, in pr inciple,

the momentum distr ibut ion of the alpha cluster in the target n.r" l . r r" .11

These exper iments are very di f f icul t to perform, however, requir ing a

very high quali.ty bearn and a carefull-y tuned coincj.dence experiment.

Furthermore, the 3-body final state is diff icult to anaLyze, and many of

the exper iments suf f er f rom low incident un"tgy,24 Also, measurements

usual ly are taken in ter .ms of double cross-seet ions wi th uni ts of barns

per steradj-an per steradianr lnaking comparisons with other types of

react j -ons more di f f icul t . In spi te of these di f f icul t ies, however, the

exper i rnents are done because the inforrnat ion is readi ly compared to

theoret ical calculat i -ons. Studies on 20*"

at 78.6 t" t "V25 show ql lasi- f ree

cx-cx scatter ing dominaEes, and consequent ly that cr-c luster ing is important

?n 1Ain the --Ne(qr2o)^"0 (g.s.) react ion and that the plane-wave i -mpulse

approximat ion descr ibes the data wel l for certain condi t ions.

The 1or8n"1 react ion is the least controversial- of a l l cr-pickup

react ions. Both the incident and exi t part ic les are spin-zero, isospin-

zero part ic les. The large alptr ,a-structure ampl i tude of 8U.

should make

this a most useful react ion wi th which to invest igate o,-c lus tut ing.26

However, due to the rapid breakup of the t ru,

detect ion of the breakup

alphas is qui te di f f icul t . A11 the 1o,Bn.; react ions done to date have

been done on tuo, t5*, t4*, L ' r , l t r ,

and 10r.

This react ion also has

the advantage of shar ing an entrance channel wi th the (or2cx) react ion,

rnaking theoret ical- comparisons easier. Brownr €t "* !

f ound the react ion

1Aon

*"0 did not proceed by a direct mechanism, al though at h igher energy,

8

. ,2LWozniak-- has seen spectra which j .ndicate strong direcL processes, and

his theoret ical comparisons tend to agree.

Not much has been done with the (3H", tu") react ion. Vir tual ly al l

of i t has been done on targets wi th A=19 and below, wi th only very few

on A=40 and above. This is also a di f f icul t react ion to work wi th. The

ground-state doublet of 78"

compl icates the spectrun of the react ion

and requires at least 430 KeV resolut ion.22'23 Secondly, the 3".

has

spi-n L/2+, and the 78"

spin 3 /2- or L/2-, meaning that the 3".-o

sysrem

of 7u"

is in a relat ive P-state, as opposed to an S state in 6r i

or 8u. .

For th is reason, i t is d i f f icul- t to perform a DWBA analysis of the data.2

The large size of the 78.

*""rr" that a zer o-range analysis would be suspec t .22

Those exper i rnents which have been dorrul9 '22'28'29 indicate t ,hat the

dominant mechanism is also direct a lpha transfer. Most of these workers

take angular distr ibut ions for the ground state only. Also, only recent ly

have good approxinrat ions for 7I i "

opt ical potent ia l -s become avai lable.

There is qui te an abundance of l i terature on the (ar6l i ) react ion.

10Bethge^' reports the ear l ier work and shows that most of i t is at low

energi-es so that the assumption of a direct react ion is not necessar i ly

val id. McGrathrs survey of the (ar6l i ) react j -on shows populat ion of

states which would proceed by a direct react ion, but he does 1i t t1e else

with the dutr .30 The low energy data of Denes, et a1?1 concurs. His

data is f i t wel l by a DI,EA calcuLat ion, but only l imi ted structure infor-

mat ion is extracted because the nortn"Lizat ion factor for th is react ion

was not known at that t ime. Dorrov"rr2 reported the (ar6l i ) react ion on

heavier elements at 50 MeV and showed a direct a lpha transfer react ion,

but his zero-range DWBA treatment did not f i t the data wel1, re inforcing

Denhardf s conclusion that the zero'range Dtr{BA could not f i t such

9

react ions.32 Denhard also suggested a smal l contr ibut ion f rom compound

processes. Cole suggested that these might be two-step processes rather

than true compound react iorrs.33

The (ar 6l i )

react ion is a nice one to work wi th. 6r ,

is a stable

nucleus' and only the ground state is observed due to 6I , i

breakup at I .47 I" IeV.

Since deuterons and 6f i

are strongly absorbed part ic les, th is react ion is

a surface react ion which is favorable for our purpose i f i t is assumed that

the alpha part ic le is t t ro l l - ing aroundtt the surface of the residual core.

The cx-d separat ion energy of L.47 MeV supports a strong deuteron-alpha

parentage. Furthermore, great ly improved 6r,

opt ical potent ia ls have

recent ly become avai lable. l0

L.4 Scope and Purpose gf the tuesent Sktdt

The present research wi l l at tempt to study the c luster ing of a lpha

part ic les in the 20r"

and 'Otr nuclei by examining the character ist ics

of a-t ransfer react ions on these nuclei . 20*"

is of special interest

because i t may be thought of as a doubly-magic 4""

nucleus bound to a

doubly-magic 160

core. I ts use has necessi tated the development of a

special gas target to minimj-ze losses . 'O*f is a lso of interest i f one

considers i t to be an alpha part ic le plus a f i l led-shel l 20*.

core. Both

these targets have been somewhat neglected by other invesLigators. Both

are even-even N=Z t ta lpha part ic let t nucLei .

To observe whether states consistent wi th the t ransference of an

S=0, T=0 cx part ic le are being preferent ia l ly populated, spectra wi l l be

obrained for rhe 20N"(a,6r i )160

and 'otr(a,6l i ) '0r . , and eirher rhe

20N.(3H.r7nu)160 or 'o*r(3H., tu")20*" reacr ions. These react ions wi l l be

done at suf f ic ient ly high energies to assure that compound react ion

10

effects, i f ory, wi l l be domirrated by direct react ion processes so as

to make a DI^IBA treatment valid.

In addi t ion, an at tempt has been made to obtai-n a spectrr :m for the

'4*r(or8B.)20N" react i -on to supplement the above. This ef for t has required

the development of a 8*"

detect ion system and ef f ic iency cal ibrat ion

heretofore unused in any 1or8S.; react ion.

Angular distr ibut ions of the di f ferent ia l cross-sect ions wi l l be

taken for both (dr6r i ) react ions and a (3t t"r t r") react ion. These angular

distr ibut ions wi l l be taken for the ground state and al l resolveable

exci ted states. The general shape of these cross-sect ions wi l l be

examined to see i f thev are consistent wi th a direct cr- t ransfer. Fini te-

range and zero-range DI^IBA calculat ions wi l l be made assuming a Lransfer

of an S=0, T=0 alpha part ic le, and these calculat ions wi l l be compared to

the exper imental data. Spectroscopic factors wi l l be extracted.

In order to perform the DI^IBA calculat ions, opt ical potent ia ls for

)nthe ' "Ne + d channel wi l l be needed. An angular distr ibut ion for the

elast ic scatter ing cross-sect ions wi l l be obtained and opt ical model

parameters wi l l be calculated.

In the event thaL non-direct processes are observed, coupled-channels

calculat j -ons wi l l test the hypothesis of Denh^td32 and Co1.33 that mult i -

step processes are involved. An at ternpt wi l l be rnade to see i f these

processes are st i l l consistent wi th S=0, T=0 transference.

Last ly, a comparison of var ious cr- t ransfer react ions wi l l be made to

see i f the react ions are consistent wi th c luster ing of a lpha part ic les in

the nucleus.

11

II. EXPffi.IMENIAL EOUIPMEM AND PROCEDURE

2.L C?slotron, Beam FaciLities, anC WperimentaL Arga

The exper iments discussed in th is work used deuteron, Hel ium-3,

and alpha part ic le beams from the 193 cn var iable f requency isochronous

cyclotron at the Universi ty of Cal i fornia at Davis Crocker Nuclear

Laboratory. The deuteron beam was operated at 40 MeV, the Hel j -um-3 beam

at 55 MeV, and the alpha beam at 65 MeV. Beam energies were chosen to

be high enough to assure that di rect processes would dominate any compound

processes. The di f ferent energies for d i f ferent species of beam were

chosen so that the momentum transfer to the target was approxirnately the

same for all beams, with allowances made for ease of tuning and minimum

addit ional beam development. The cyclotron was operated in che f i rst

harmonic mode at a f requency of Lz.B YfrLz f or deuterons, L2.3 ln lz for

hel ium-3, and 11.6 NiHz for alphas. Beam energies were not measured for

the experi-ments, but were calculat,ed from a knowledge of the magnetic

f ie ld strength, f requency, and extract ion radius. Such calculat ions

tr ,ave been done for other such beams at s imi lar energies and a comparison

to t ime-of- f l ight measurements showed the di f ference to be general ly less

than I .3%. Beam resolut ion was typical ly 130 KeV, wi th beam eurrents as

high as 0.50 microampere j -n the faraday cup. Intensi- ty was nornnl ly

0.050 to 0.20 microamperes.

A scale diagram of the beam l ines and beam opt i -cs is given in f igure

1. For good beam resolut ion, the beam was magnet ical ly anaLyzed twice,

f i rst by the LAz cm switching magnet, then by the 50o bending magnet

into the exper imentaL beam l ine. Four quadrupole focusi-ng doublets and

T2

(9ztrtYi l )

tql- J-F<<ISu

c):rJt--'EoorO (,

=oJLrJ

tnII

I*

b0E

'r'ltrc).lJ+J$

(,1

al

r-t

JJt-{$&I

rr{

a)b0trftt

O

${o

l+l

c0oH

'r{Fl

(dc.l

,o

H

H

d

H

o${.lJo

r'loh

CJ

r-l

b0.r{f - .

(9zL_iLJQZl-- 12=<J;=

zoG,

o)U

(J

]JCz>(9 F^-

-'( v

l l r

UJJoo-frts

:)o

a'.rJz- lLJ

J

[J

"*c0?oo-ZO

(9\ 'Z(rZ9so53HJF(J tr.

L)LL( ' )

trCF

tromJ

trL!JU:fZ.

trLr-J)cUox.(J

13

two sets of steer ing doublets were used to proper ly al ign the beam.

Exper iments were carr ied out in the 76 cm ORTEC scatter ing chamber. A

double col l i rnator made of Tantalum r,uas placed at the upstream edge of

the scatter ing chamber to produce a beam spot only 3 mm dia. on the

target. Beam current deposi ted in the Faraday cup was typical ly 100

t imes greater than that stopped on the second hal f of the col l imator,

so that ganuna background was minimized. Nevertheless, lead bricks \^rere

placed between the target col l imator and the detector housing to stop

beam part ic les scattered of f the col- l imator.

An oi l d i f fusion ptmp with Freon cooled baff les and a 10 cm

pumping line was only two meters from the scattering chamber, allowing

the scatter ing chamber and upstream beam l ine to be kept at a pressure

-5of 2 X 10 mm Hg or Less.

The scatter ing angle and target posi t ion were rernotely var iable and

readable by means of c losed circui t te levis ion to a precis ion of 0.1

degrees in each case.

2.2 Taz.qets

A sel f -support ing fo i l of isotopj-cal ly enr iched magnesium-24 was

used as a target in the exper iments. This fo i l - was purchased from

Micromatter Corporat ion, Seatt le, Washington. The foi l was guaranteed

by Micromatter to be 99.94% Magnesium-24, wi th 0.047" magnesium-25 and

0.02 % nagnesium-26 and less than 0.AL% other impuri t i .=.34 Howeverr dD

elemental analysis performed by ion-exci ted X-ray emission at the

Crocker Nuclear Laboratory showed that, in addir ion to the L25 pg/cm2

of magnesium, there ruas 1.1 +/- . l - J- t g/ *r2 of sul fur , L.7 +/- .2 vg/cmz

of chlor ine, 0.8 +/- .2 vg/ cm2 of copper, and 0.19 +/- .05 vg/ cm2 of

L4

zinc. The presence of copper and zine may be explained by the brass

rol lers used to fabr icate the fo i l , but no explanat ion is evident for

the presence of the other large impuri t ies. The sample was shipped in

vacuum and stored in an inert atmosphere. The target th ickness was

chosen to opt imize the t rade-of f between resolut ion and count rate. The

energy loss of a 6rt

in th is fo i l is approximately 100 KeV, wi th a peak

broadening of about 35 KeV, based on calculat ions f rom Vavi lovrs theory.3 '36

The foi l was mounted in a target f rame and placed in a ladder at

the cent.er of the scatter ing chamber. The foi l area exposed to the beam

was approxim at eLy 4 "*2.

The gas used in Lhe 20*"

studies \^/as purchased from Almac Cryogeni-cs,

Inc. , 0akland, Cal i fornia. The gas is of natural isotopic mixture, con-

sist ing of g0.5L7" 'O*. , 0.277" "*" , and g.222 22*",

and was guaranteed.

99 ,9957" pure Neon by the manuf acturer.

A special gas target was designed and constructed for these exper iments.

This gas target is shor^m in f igures 2 and 3. The target is 15 cm in

diameter, and has a 1 mi l p last ic window. Kapton \ , ras chosen fo r the

window because of i ts h igh resistance to radiat ion damage, large tensi le

strength, and Law-Z composi t ion. A snout protrudi-ng 5 cm into the gas

i tsel f , wi th a 1.05 cm aperture inside covered with a U8 mi l mylar

window, reduces the pathlength of heavy ions in the Bssr and with the

thinner window, reduces ion energy loss and improves resolut ion, s ince

the detector is placed behind the snout. Performance of the target was

qui te acceptable at larger scatter ing angles, but below a laboratory

scatLer ing angle of 2A", the snout cuts into the beam. This di f f icul ty

is overcome by a s l ight change in the or ientat ion of the detector,

1..,

a

a

'o

p?Gas Target

L6

NORMAL GAS TA RG ETCON TIGURATION

DETECTORS

D ET ECTO RS

SHEILD ING

SHEILDING

FORWARD ANGLE GASTARGET CON TIGU RATIO N

GAS TARGET

MYLARWIN DOW

S NOUT

GAS TARG ET

Wrrlobw / \ SNOUT

Gas target conf igurat ions

L7

counting events through the Kapton window as shown in figure 3. Forward

of 8.5o, however, e last ic scatter ing of f the entrance window becomes a

problem and a different gas target 31 cm in diameter with a 2 rnil KapEon

window was used.

The acceptance angle wi th the gas target was kept to less than 1.0o

by the use of a s l i t 1.6 mm vr ide, 9 cm from the beam, and a 4 .0 mm

aperture in f ront of the detectors at a distance of 19 cm. The sol id

angle was calculated using the f i rst term in Si lversteinfs " . t i .u37

af ter

i t , was determined that the second term contr ibuted less than L7" to the

sol id angle. The energy loss of a 35 MeV 6tt

through the Kapton window

rias calculated to be 85 KeV but only 9 Kev through the Mylar window.

The energy loss of a l l part icLes of interest through the gas was negl ig ib le.

Figure 4 is a diagram of the gas handl ing system used to control

the gas pressure in the target. Posi t ive pressure wi th respect to the

inter ior of the scatter ing chamber was maintained at a l l t imes to assure

the qual l ty of the seal on the lGpton window. Gas pressures used were

normal ly in the neighborhood of 35 cm Hg, readable to 0.1 en l lg, and

remotely monitored by means of c losed circui t te levis ion.

2.3 Charged-Parttcle Detection arC ELectron'Lcs

The method of ident i fy ing a detected part ic le and determining i ts

energy has been worked out qui te wel l in recent years. Essent ia l ly , the

rate of energy loss in a th in detector is measureC along with the total

energy loss in both the th in and a th ick detector. The lat ter quant i ty

gives the energy of the part ic le direct ly, whi le the former is roughly

proport ional Lo AZ2 lE of the detected part i "1.3B at nonrelat iv j -st ic

energies. For l ight ions, these two quant i t ies are suf f ic ient for

proper ident i f icat i -on.

1B

GAS TA.RG[T

MERCURYMAI{OMTTE R

ToD FUSIOTJ

PUM P

AIR

RTGU LATOR

TOFORTPUM P

SCATTE RIITIG CHAMBT R

NEON GA5

Fig. 4 GAS HANDL_ING SYSTI-:M

L9

For the (a,6l i ) and (3Hur tu.) react ions, fu l ly-depleted sol id srate

surface barr ier detectors were used to measl t re the above quant i t ies. A

telescope t /as prepared with a 25um, 25 r*2 AE detector, a 200um, 100 ,2

E detector, and a 300pm, 100 n*.2 veto detector. In addi t ion, a 3 f l r t r r

)100 mm- l i th ium-dr i f ted s i l icon detector was used as a beam current moni- tor .

)o (ar6l i )160 studies, the veto and E detectors \ . /ere thermoelec-In the - "Ne,

tr ica1, ly cooled to -2A"C, but no signi f icant increase in resolut ion or

decrease in background was observed, so cool ing was not used for other

studies. Table I shows the contr ibut ions to resolut ion in these studies.

A block diagram of the electronics for the (d,6l i ) , (3Hu,tu") , and

o(o, "Be) exper iments is shown in f igure 5. Signals f rom the AE and E

preampl i f iers are ampl i f ied and sent to l inear gates. The l inear gates

are enabled by a s low coincidence pulse, which requires a s low coincidence

from the SCA outputs which were set to cut out low vol tage noise, and a

fast coincidence from the AE and E detectors. The fast coincidence was

set by an SCA on a TAC to L2 ns. Ei ther the veto pulse or the inhibi t

s ignal f rom the pulse pi leup rejector r^rould disable the coincidence.

The outputs of the l inear gates were mixed, and the mixed signal sent to

a Nuclear Data NP02 analogue to digi ta l converter (ADC) as a total E

measurement, and the AE signal was sent to a second ADC. A11 pulse-shaping

electronics were ORTEC npdular 400 ser ies. Count rate was about 4000

counts per second, so pi leup was no problem. The coincidence rate was

typical ly 150 counts per second.

At far forward scatter ing angles, the AE count rate increased str ,arply,

pi leup became a problem. In addi t ion to changing gas targets, i t

necessary to lower the beam current to as l i t t le as 4 nanoamperes

and

was

oY

ADC

INHIBIT

ANTI

COINC

AAITI

TOX

ADC

VETO

AL-PHA-TRANST I RFis.s ILICTRCNICS

22

to keep pi leup problems to a minimum. Both resolut ion and l ive t ine

benef i t ted f rom this move. Live t imes throughout the exper iments were

kept above 99.5,2.

Data acquis i t ion and part ic le ic lent i f icat j -on was handled by both

software and hardware connected to an on-1ine Digi ta l Equipment Corporat ion

PDP-15/40 computer and the two ADCts. The ADC?s converted the pulse

heights to digi ta l informat ion which was stored in the computer core.

The aequi-s i t ion program DACEDE was especial ly developed for charged

part ic le ident i f icat ion and energy r leterminat ion purpose". 39 Informat ion

from the ADCrs is put into a 64 X 64 array. The hor izontal posi t ion in

this matr ix is determi-ned by the total E s ignal , and the vert ical

posi t ion by the AI i s ignal . The array is div ided into f ive data regi-ons

through exterrral paramet, erLzati-on, and are curyed. to f it the AE-MZ2nX/n

relat ionship of the Bethe equat j -or .38 I f an event fa11s into one of the

regions of interest , that event is also appl i .ed to a LA24 channel data

region which wi l l g ive a spectrum. In th is \^/ay, part ic les may be

ident i f ied by the data region they fal1 into. At the c lose of a run, al l

data regions and spectra are recorded on disk. A sample of the 64 X 64

data matr ix, wi th data regions indicated, is shor^rn in f igure 6.

The residence of the data on disk a11ows the informat i -on to be

extracted easi ly by the code DSIGMA, which was developed to assist in

locat ing and integrat ing peaks. I , Ihen a peak is located, the code f i ts

the channel-by-ch,annel informat ion to a Gaussian plus a straight l ine

represent ing the peak and the background. The background is subtracted,

and the Gaussian is i -ntegrated for the area. The centroid of the is

cal ibrated to f ind t l ie corresponding energy, and the cross-sect i .on and

stat ist ical error are calculated in both the laboratory and center-of-mass

23

24MGD,6L I ?ONE 40 MEV 15"

' lf !

t

!

I

a t .

l rr l

t

t

t . PULSERr l t

l r l l

r l l

, t l l

. l l , .

r l l .

t . tD,

o o ) J. . .

\ . . , , s. , , 6l . , , t ) lt t rJrr L l

' t I ' f 'e I la

tr-oFOLIJFL!O

t!

=

l-(no)

\ - \/ \r l \\JIYt-t-

LrJz

rIr+ ' l ! l l ' '. l t l l f .

I r l r l l r '

'r ' tt!!al r l

l l t .I t l l - r ' F

l l tr l l l

r t l tl l l l

TOTAL ENERGY LOST IN ETAE DETECTORSFig. 6 64 X 64 array

24

systems. Gaussj-an f i ts and l inear sums agree qui te we1l , and the f i ts

appear to be good. Chi-sqtrared per degree of f reedom var ies between

.69 and 3.54 f or a l l but a verv f ew cases.

I t was also necessary to take angular distr ibut ions for the

?o )o-" I Ie(d,d )-" l le react ion to obtain inforrnat ion for the calculat ion of

)o-"Ne * d opt ical nrodel parameters.

The exper imental setup for these neasurements was simi lar to that

for the cr t ransf er measurements. The detector te lescope l ras s l ight ly

di f ferent, however. A 200pm, 100 **2 surface barr ier detector was used

for both the f i rst and second AE detectors, AE1 and LEz, wi th a 5mm, 100

))f i r rn Si(Li) as arL E deLector, and a 3mm, 100 mm detector \^/as used as a

ve' t -o, The thicker detectr : rs were needed to stop the less massive

deuterons. At forward angles, where the deuterons were more energet ic,

the ADC for AE measurements were fed a rnixture of AEl + LEz, whereas at

back angles, the ADC was given only AEl , A diagrarn of the electronics

is shown in f igure 7. Note that there is no fast coincidence requi-rement

Because of the Ta.rger numbers of events, i t was fel t that a t ime

resolut ion of 2A0 ns was suff ic ient , so only a s low coincidence was urade.

Beam currents were kept between 5 and 50 nanoamperes for minimum pi leup.

Detect ion of BB.

events f rom the (or BB") react ion was shown to be

di f f icul t in the previous chapter. Both the decay alpha part ic les must

be detected to count a val id event. There are essent ia l lv two \ , , ravs th is

can be done, and i :otL: \ rere at tempted in the course of th is invest igat ion.

aHarney ancl I ,Jozniakts n"p"r40 suggested that

oB" could be detected

with a s ingle AE-E telescope and ident i fy ing the breakup alphas l ike

6_ . 7_.Li or 'L i . This was the pr incipal rnethod at tempted in th is study. A

100um, 300 mm2 AE detector was used" wi th a 400um, 450 mrn2 E detector,

25

,o-3t-

(Jo

oF

rvIlJ

+ mUZ.Cx_FULrJ)

LJ

(}

Z.rrLJJFt--

ULn

ui-*aJLrJ

(J

z6U

26

and a 1 nrn, 300 mm2 veto detector. A s l i t only 2.6 nrn wide over the

whole diameter of the detector was used to l imi t the sol id angle. The

electronics was essent ia l ly the same as that for the other alpha-

transfer exper iments.

Brovmts or ig inal paper on lorBn.; used two detectors s ide-by-side

to detect each of the breakup alphas. In addi t i -on, Woznj .ak has used a

modif icat ion of th is techni"que using a spl i t AE detector and a posi t ion-

sensi t ive E detector wi th good succes " .2 '26'4L

An approach simi lar to

these was also at ternpted. Two AE-E telescopes with 100 Ui l r 300 mm2

surface barr ier AE d.etectors and 300 urr l r 300 mm2 E detectors were placed

one on top of the other wi th the center l ine in the scatter ing plane. A

four-way fast coi-ncidence \^ras required, and the two AEf s were mixed for

a AE' and al l four s ignals mixed for a total E s ignal . This method

gave improved part ic le detect ion and larger sol id angle using the same

2.6 nm s1i t , but the larger sol id angle al-so al lowed a higher chance

coincidence rate, and the separat ion between the detectors drast ical ly

reduced the detect j -on ef f ic iency for UUu,

wi th the resul t that a much

lower s i -gna1-to-noise rat io was seen over the method ment i -oned in the

preceding paragraph. As a resul t , states were not resolvable wi th any

kind of acceptable count ing rate, and the method was abandoned.

In order to calculate the ef f ic iency of detect ion for var ious

conf igurat ions of the BU"

d.etectors, a computer program cal led E8GAS

was d.eveloped. Harney and Worniak40 had developed one also, but i ts use

is l imi ted to the s ingle AE-E telescope, sol id targets, and smal l sol id

angles only. ESGAS wi l l handle twin AE-E detectors, e i ther s ide-by-side

or top-and-bottom, l , lozniaks posi t ion-sensi t ive detector, and single

AE-E detectors. I t is a lso appl icable to ei ther sol id or gas targets,

27

and wi l l accept any size sol id angle up to 2n steradians. The program

uses a monte-car lo technique, and is descr ibed and l is ted in Appendix A.

The author is grateful to Dr. N. King for suggest ing the rnettrod. I t

turns out that the ef f i -c iency for 8Be

with detectors avai lable at t t re

Crocker Laboratory is gr€test for the s ingle AE-E telescope.

Table TT gives the sources of exper imental error and uncertainty,

and est imates as to their s i -ze. Flowever, the f igures in Chapter IV

wi l l have error bars which account only for stat ist ical errors, as is

common pract ice.

28

TABLE II

Souv,e es of ErperLmental Ev,z,oz,and Uneerfnintu

Source

Target th ickness

Farady cup current integrat ion

Pulse pi leup losses

Scatter ing angle

Beam Energy

Peak integrat ion and backgroundsubtract ion (average)

Dead t ime

quadratic sum

Est imated Per CenL Uncertainty2oN"

_ 24Mg._

0 .28

0.2

0.5

0.3

1.1

3.5

1.5

5.0

0.2

0.5

0.3

1.3 (avg. )

5.0

1.5

7 .47"4.07"

29

III . TT1EORET ICAL CONSIDEMT IONS

3 .L ALplm 9lus.lerlnq xaodetr

A cluster may be def ined as a subgroup of nucleons within a nucleus

whose col lect ive mot ion wi th in t t r ,at nucleus may be treated simi lar ly to

the rnot ion of a s ingle part ic le of nass equal to the total mass of the

subgroup. In discussing an t ta lpha clustertr , then, a group of two protons

ant l two neutrons is t rea;ed as a s ingle part ic le of charge *2 and mass 4.

In Lhe course of t -h is study, the extreme alpha-cluster model, a lso

known as the alpha-transfer approximat ion, was used to s impl i fy the

calculat ions. In th is model, i t is assumed that in a t ransfer react ion

of the type A(a,b)B, that at least two of the nuclei involved rnay be

represented as an alpha- l ike part ic i .e and another nucleus. I f the above

react ion were a pickup react ion, then A=B*o and !=4*o,. The alpha-transf er

approximat ion assumes that the s-c luster in the. target and the detected

part ic le are ident ical and the same as a f ree alpha part ic le in i ts

ground state. In al l three cases, the alpha part ic le is t reated as having

al l the internal quantum numbers of a f ree alpha part ic le, spin 0,

isospi-n 0, cTnrge *2, and nass I+. I t is fur ther assumed that the alpha

is the only c luster Present.

For the most part , the assumptions in th is model s l rould be qui te

good. The binding energies of the 5r i

and 7U.

" t" re lat ively smal l , and

thesenucleiwi l l break up into an alpha part ic le and ei ther a deuteron

?or 'He at lower energies than any other form of part ic le break-up. I t

?n

has already been seen that Bu.

decays to two alphas, and that the' t ' }1.

16nucleus may be Lreatecl as a doubl-y magic -"0 core plus a doubly magic

16alpha cluster. Again, breakup to cx * -"0 is energet ical ly preferred

over other forms of part ic le break-up.

30

I t is necessary to point out at th is t ime a very ser ious f law in

the alpha cluster modeI. This major di f f icul ty is Ehe assumption

that only an alpha cluster is present. Whi le the alpha cluster is energet ical iy

preferred, the nucl-eus may spend part of i ts t ime as a di f ferent type of

core wi th a di f ferent cLuster conf igurat ion. Another di f f icul ty is the

assumption that alpha elusters have the szlme propert ies as a f ree alpha

part ic le. This j -s not too drast ic an assumption for the U"rr 'U"r tU",

20--or even -"Ne, but may not be ent i re ly t rue in a heavy nucleus such as

24-. 5- ' I "19, which has a strongly deformed core.

3.2 Ddyect FgaetioL.Theoylt ih, ALpkrn, Tz,ansfey,

3.2.L The DWBA, Powers, L imits, and Appl icabi l i ty

A pickup react ion A(arb)B may be represented as

(B+x)+a -+ B+(x+a)

where B * x = A and x * a = b, and x is the t ransferred part ic le or

cluster. I f a direct mechanism is assumed, the behavior of the cross-sct ion

may be treated theoret ical ly by means of the Born approximat ion. The

appl icabi l i ty of the Born approximat ion is based largely on the assumption

of a direct mechanism. That is, the inLeract ion responsible for t .he

transj- t ion is assumed to occur only once and lasts approximately the same

length of t ime as the t ransi t t ime of the bombarding part ic le across

the target nucleusr or about LO-22 seconds at these energies.

The plane-wave Born approximat ion (PI^IBA) is not appl icable in the

case of Eransfer react i "ons. The PWBA assumes that the wavefunct ions for

entrance and exi t pert ic les may be treated as plane and spher ical waves,

31

respect ively. Since the transfer interact ion ta lces place in or near the

target nucleusr one would expect the r ,uavefunct j -ons to be strongly distorted

by the nuclear potent ia l .

The distorted-wave Born anproxi-mat ion, DI{BA, al lows for the distort j -on

of the vravefunct ions, and j -s therefore a more exact and nslg di f f icul t

to use representat ion of nuclear scatter ing. In the Dtr IBA, scatter ing

cross-sect ions are given in the center of mass system Uy42

dod? - uoug=

L 5 (2-t**1)-_-

l r ^ f(2rl \2)2' r / . . . . i t^ i :2JA+1) (t sr+l) '

^ oB '

where o denotes the entrance a + A channel , and 3 denotes the exi t b + B

channel . u is the reduced mass, K the wavenumber, J and s are the total

angrr lar momenLa and spin of the respect ive nuclei , l ' t r is the number of

permutat j -ons of nucleons in the channel . The transi t j -on matr ix element,

T- oB, is given by

r^=0f jo fot t - ) ' , ] ?.)1u,- , , lv l r . . { - rto3 =)1 o

) B xg (KB,tgr\+sqblv lvA0r)*o ' (Ko,ro) (3)

Here, F ana io are t ,he separat ion vectors between the proj ect i les and

the nuclei for the respect ive channels, ^na

Q is the Jacobian of t rans-./

format ion to these coordinates. X^,+ "rrd

X.,-* are the distorted wavesr t rp

for e last ic scatter ing in the entrance and exi t channels, respect ively.

The term ({ , , 'J ,BlVl . l "Un) is the form factor, and contains al l the nuclear\D

structure informat j -on. The 0ts are the internal wavefunct ions of each

of the part ic les and rv '1 is the potent i -a l causi-ng the interact ion. Integrat ion

is done over al l var iables not included in the outer integrat ion.

Equat ion (3) is the "b"ginning of a l l knowledge" in direct react ion

theory, and has been der ived elser,rhere. Since the der ivat ion is lengthy

and involved, i t wi l l not be repeatecl here .42'43 The notat ion of

Austern, being the more unj-versal ly used, rv i l l be assumed in Lhe

(2)

32

fol lowing pages.

However, to understand the powers and. l imi tat ions of the Dtr^ lBA, i t

is wel l to point out some of the approximaLions made in the der i -vat ion

of equat ion (3). One such approximat ion is that the total wave funct ion

over al l possible exi t channels nay be approximated by entrance channel

elast ic scatter l -ng only, hence the inclusion of X^,+ in the t ransi t iont

matr ix element. I t is certai .n ly t rue that elast ic scatter ing i -s by far

the largest contr ibutor of a l l the exi t channels. l lowever, other channels,

especial ly inelast ic scatter ingr Ray contr ibute large arnounts as we11.

I t is therefore possible that there may be some contr ibut ion to the

transi t ion f rom exci ted states of the target in the entrance channel .

The only other assumption of major importance made in the

der ivat ion i ,s that wavefunct ions of entrance and exi- t part ic les inay be

approxinrated by Coulomb wavefunct ions at large radi i . This is obviously

not a ser ious problem, because the strong interact ion potent ia l approaches

zera just a few fermis f rom the nuclear surface, whi le the Coulomb

potent ia l is ef fect ive for several angstroms.

Equat ion (3) is a s ix-dirnensional integral . The form factor i tsel f

can also be as much as a s j -x-dimensional integral . To simpl i fy the

calculat ion and the appl icat ion of the c luster model to the DI.EA, i t is

helpful to rnake some assumptions and reduce the equat ion to a more

Lltmanageable level . This has been done by both Austern, €t s l . , and

/ ,aBassel* ' , and their t reatment wi l l be sunrnar ized, as hr ief ly as possible

to point out the assumptions and l imi ts of th is t reatment.

The assumption to be made is that a l l potent ia ls have spher ical

symmetry. Then i f r^re wr i te t te angular monentum transf er j , the spin

transfer s, and the orbi ta l angular mom.entrura t ransfer as 9" ancl i ts

z-component as m:

\a' -t' -+j=ro JR

-'t -'t '+s=ss..+aD

3-=?+?m=IL*irL 1 ' ! m

tsbA'--a

Then the angular parts of the internal funct ions in

simply spher ical harmonl-cs, and the form factor can

Clebsch-Gordan coef f ic i -ents.

the

be

33

(4a)

(4u1(4c)

(4d)

forrn factor are

wri t ten wi th

(*rr, I vl'!Au) = f aJaiMR,r{B-MA l -ruuu) (.Q,sm,*"-% | :,q-u)

" (=""b*", *b I s, l l " - r5) $t"r( is,?o,b,

B, &,A)" (- r ) "u- t%-*

factor G can be wri t ten as a sum of a spectroscopic coeff ic ient

a form factor in reduced form t^

= A^ _. f ̂ ( i^ ,7 ) (6). rsJ x,m - tJ ' s '

of standard types of form factors and spectroscopic

A part ia l t ransi t ion ampl i tude f i tsy be def ined.

Br,*(KBn Ko) =r# [ rrJo'iu *r-"(?e,?e)fn*Ciu,7*l

*X o cil*,?*l

and the di f ferent ia l cross-sect ion becomesr wi th no spin-orbi t potent ia l ,

3**Pf 1*=, P luu* i 'Ko 2rA+L/ G(2s"+1 ) #

A number of s impl i f icat ions can be made i rnmediately. The

scatter ing distorted waves are expanded in part ia l waves, their

symmetry is used to aclvantage by i -ntegrat ing over the arguments

spher ical harmonics, and r . re choose the z-axis "1orrg

? and the

(s)

A",f SJ

The

and

/+G" . ( r^rr )! ,s lm' [J- 0, '

Eo al low the use

co ef f ic ients.

(7)

do_=de

u u^0,5

( r tz i l2 (B)

el as t ic

spher j-ca 1

of the

y-axis

34

along ? x fU' so that the part ia l t ransi t ion arnpl i tude becomes

r-Brm (e) =

#L )_, i-l ofls-'?',, 'T+L( Ls.tm, -m I r,oo)

0 b "or,3

, { (L,r-m) ! I &oh) ' . } t lT pl (3 ) rr r 0

(9)t ] tJ L

B "3"o* ,

and

' r , r^ , u= f todto

i tuotBXr, (Ke' tg)Fr,r , - r , ( tB' to)Xr^(Ko,ro)

(10)vt"B,.P-"B"ot0{Jcx

o o

The six-dimensional integral has been reduced to a two-dirnensional

integral over the radi i r . , and rB and a number of f in i te sums. The *Lt"

are the radial parts of the scatter ing wave funct ions xo+ ot xB .

The integrat ions i -n the form factor must st i l1 be performed.:

r-,Fg,

_, (r

U, r o)

= 4 UL *lt, rn-r"i I !,m) x

-" 8"0 r . r u

l l

f ' ! - laf ^ f (r- ? r . ,M ,. : ' m-]4 '" (11)

Joto. , r . i ! ,m- B' to) Y|^(rB)q'-( to)

150

but th is, too, rnay be simpl i f ied by a change of var iablesl the form factor

j -s rewri t ten in terms of the standard var iablu" io and FB. The radius

vector between the cluster and corer or between the cluster and

incident part ic le, can be obtained from simple geometry merely by using

some mass rat ios.

7"s = o(?s f"l

i" =

"(f. n?O)

0 = bA{x (B+b)} -1

Y = a/bg=B/A

rxB = o{r B

*yz't ,. '-rryr o,r i

1/ 2

r--^ = o{ d2r ^ 'n

2-zdr -r ^u}

r / 2xa b 0, otJ

(L2a)

(12b)

(12c)

(12d)

(12e)

(13a)

(13b )

orr nore s i rnply,

35

where u=?cx'?*, the cosine of the angle between the tr^ro vectors. Aust.ern

shows that the reduced form factor i -s given Ayr44

F (? ? r * t . , ,* / - \n/- a o *

-gm\. B, io) = i -U

.q,r( t*B)o(rr*) = i -ggn(t*g)y(to.) i l .Q,,* , ( tax) (14)

ol

where Un r* , ( r ) = LLt rru, ( r )Y.*, r , (?) / tL ' is the normal Lzed wavefunct ion in.x, 'm-

the center-of-rnass of the relat ive c luster-core or c luster- incident system,

V(r^- . ) is the relat ive c luster- incident potent ia l of the exi t part i -c1e,ax

and Lt 1s the relat ive angular momentum of the c luster l "d

core.

I f , in the c luster-core system, the f actor rg ,=r-L uu r ( r ) is wel l -

behaved at r=0, then we may expand w in Legendre polynomials.

@

rs,(r*B)D(ro*) = [" (K+1 /z)sK{ro, r ' )p*(u) (15)ax G.o . -K

orr by inversi-on,

+1BK.[ ' ( ro, ru) G t_, du rg, ( r*B)D(rr*)r*(u) (16)

recal l that r*Uand rax are both funct ions of Ur ro, and rB. Insert ing

equat ions (14), (15), and (16) into equat ion (11), the radial form factor

is f inal ly found Eo be

Fo, T (ro,r^) = oL1 nq2vag f cro)[(-yr^) t - r ( -1)K(zr*r )Eo,(ro,r^),rBro fi rx K, tr

5 cr' -KJc [J- 0,'

x{ (2! ,+1), t / (z^) I / ( 29,-2x) t } *) ,K00 | lo0><s-). , K00 l l^0,

x w(L3l lo, [ - ] . ; K,0) (17)

where 05_151, and W is the usual racah coefficient which comes from the

integrat ion and combinat ion of the spher ical harmonics above.

The problem of calculat ing DWBA cross-sect i -ons has been great ly

simpl i f ied. Instead of s ix to twelve integrat ions, i t has been reduced

to three lntegraEions and eight f in i te or terminatable surmnat i -ons. In

the case that the spectroscopic coeff ic ient does not depend on s or j ,

36

th is reduces to s ix sumrnat ions.

One approximation that is used in rnany DI,I:BA codes is called ttre

zero-range approximat ion. ILs usc great ly s impl i f ies the calculat ion

of the form factor. The assumption is that the j -nteract ion accurs only

when Lhe incident part ic le and the cluster occupy the same point . This

is equivalent to let t ing

v = v(r ) = v 6(7- i i ' { . /M--)ax ' . -ax ' o " x -a 'A"-B'

( 1B)

The interact ion is saidrthen, to have t 'zero rangett . In th is case,

o ( t r*) = Dod (r*-rrMai\)

and

Fn, , ( r - r ) - - -^ = D-t^-26 (r- -6r ' )u^ (r . )

x,rr^rJ -Btra 'zero

range o ct a D y. pt<r l

x{(2L^+1) (2L +L)/+r(2e.+L\L/2 ( l l .OOlrO) (20)' 3 CI, ' 0, lJ '

I - Iowever much the calculat ions are s impl i f ied, th is may not be a

val id approximat ion in alptr ,a- t ransf er react ions. The Urt ,

'U", and

BB.

nuclei are large and di f fuse. I t ur ,ay be that the assumption of zero range

for the interact ion does not hold for these large di f fuse part ic les.

This conclusion has been reached by many workers.2 'LL'45 The approxinnt ion

appears to work qui te wel l for one- and two-nucleon transf " t .43

I t wi l l

probably be necessary to use the exact f in i te-range version of the DWBA.

Another approxirnat ion used in al l t ,he DWBA codes avai lable to th is

worker comes in the proper choice of the interact ion potent ia l V (r^_-) .

This potent ia l should be wri t ten exact ly as

(1e )

! = Va+A ua+A (2 1)

where Va+A is the

on target A, and U

to be the opt ical

37

total interact ion potent ia l for an incident part ic le a

a*A is the elast ic scatter ing potent iaL, usual ly taken

potent ia l . V can be wri t ten approximately as

v = V"+* * Vr+B - ua+A Qz)

and the assumption is rnade that Va+B=Ua+A, so that the opt ical potent ia l

between incident and target is s imi lar to the potent ia l between incident

and core, and V=\t"+*. I^ Ih i le th is approxirnat ion causes only sma11 errors

in s ingle-nucleon transfer, i t has a more substant ia l ef fect on the

transf er of f our , r t r" l .orr" .2

3 .2 .2 The Form Fac tor

The term (%,f , I V I UAU) in the t ransi t ion ampl i tude in equat ion (3 )

is knor,rn as Lhe bound state forrn factor, represent ing the over lap of the

inl t ia l state wi th the f inal state through a t ransi t ion potent ia l V.

This term contains the nuclear structure informat ion for the internal

states of A, a, B, and b. I t is in the calculat ion of th is term that one

rnust apply a part icular mode1.

Since spher ical symmetry has been assumed for the scatter ing

potent ia ls and for the core*cluster and incident*cluster relat ive

potent ia ls, the problem reduces to calculat ing the radial form factor

F0 T r , and in fact , reduces to f inding the form factor expansion- t "Bt "cL

coeff ic ients g ' . in equat ion (16).

I f the mot ion of an alpha cluster is calculated in a lurmonic

osci l lator potent ia l wel l , then some restr ict ions are placed on the

relat ive wave funct ions employed.

Figure I shows the osci l lator energy levels and single-part ic le

osci l lator quanEa for a three-dimensional harmonic osci l lator *u11.48

Flg. 8

HARMCNIC OSC ILI_ATORPCTIN T I AL WtI_ t_

90

72

56

42

30

20

t2

G

2

N O. I DENTICALNUC L EON S

38

I

7

4

3

2

I

o

OSC ILLAT I ONORDER

55,4Do3G r2l , lK

4P,3F ,2H, IJ

\ +s,3D,2G, r l I\ sp,zF,rH I

\ 3s, zD,rG I\ ?P, rF I\ zs , rD I

\ IP I\ rs /

/

39

The alpha-cluster representat ion al lows the ground, state of a nucleus

containing a c luster to be descr ibed as a s ingle part ic le in a harmonic

osci l lator potent i ra i - .2 Therefore, the number of osc1l lator quanta, and

thus the total k inet ic energy, in the c lustered and nonclustered

representat ions must be the same.

As an example, the case of 2nor"

wi l l be examined. 20,r .

has l0

protons and 10 neutrons f i l l ing osci l lator states independent ly.

Recal l ing that E = (N+3/2\k^ for a 3-dimensional harmonic osci l lator,48

the osci l lator energy is given by

n(20N.) = 4(3/2)hco + 12(5/z)h^ + 4(7 /zf i t^ = 50tro. (23a)

In the same w&y,

u(160) = 4(: /z)ho + Lz(s/2f^, = 36f i0rn (4He) = 4 (3 /2 )f*

Thus, the internal

)oand the - " l t re alone

the relat ive q-core

Paul ing and i { i lson

where

Here n is equal to the

funct ion, including the

of the q,-core mot ion.

state energy. Thus, i f

L", i .e. , 1/ to, n=5), (

osci l lator energies of the

to 5Cth,rr, leaving .9h, which

mot ion. This can be done

show that

E = (N+3 /Dh,

cluster*core add

must be accounted

in the fo l lowing

(23b)

(23c )

to 4fr:. u,

for by

way.

(24)

N=Zn*L" Z (25)

total number of nodes in the relat ive wave

or ig in, and L" is the orbi ta l angular nomentum

The factor of 2 includes the 3/zhu of the ground

N=8, there are several possible values of n and

L"=2, TL=4) and so on, The total angular momentum

of the ground state must be zeTar so i f h igher values of 9. t 'are

there must be a contr ibut ion f rom a higher exci ted state of the

having Lhe same intr insic angular momentum. The ground state of

20Nu nucleus can be wri t ten as a sum of terms, then.

'0*" (g.s. , J=0+) = et (n=5,! , ' f=0)*160(Jn=6+) x i lo (g.s.)

* " ;

(n=4 . e. ,"=2, "110 (to=z*l x Uo (s.s. )1A +

* ""

(n=3r9' t t=4)*t""(- ln=4') x Uo (9.s.)J: i

+.. .

40

used,

cor e

Lhe

to j -nclude contr ibut ions f rom as Trnny exci ted states as one wishes.

Idow, i f we examine the expansion coeff ic ienEs gf in equat ion (16) '

i t can be seen that th is inforrnat ion can be included in the nuclear

structure terms w^ (r - )D(r ) . In i ts fu1l forrn, th is becomes

Y" XTJ AX

*g(t*B)D(ru*) = ,uru(t*u) \ -

u( tu*) 0.q, ,* , ( r"*)rxB

(26)

(27 )

Equat ion (26) is inserted into (27) as rn(t*U). I^ I i th spher ical syumetry

and having already perforrned the angular integrat ion, Ug,r , ( r"*) can be

stat-ed as a spher ical Bessel funct ion whose nornal izat ion constant

depend on the form of the interact ion potent ia l v{ t"*)r49

U.q,,* ' ( r"*) = A.Q,,* , jn ' (k;*r"*) ( radial part only)(28 )

Recal l again that 9,1 is the relat ive angular nnmentum of the incident

part ic le and the cluster in the bound state of the exi t channel . For

U"r, .Q, |

-0, and f or ' ,", .Q. | =1 .

In actual pract icen the bound state potent ia ls are chosen to be real

volume Woods-Saxon potent ia ls. This nn.kes l i t t le qual i tat ive di f ference

in the above representat ion, as the Lr loods-Saxon potent ia l may be thought

of as a harrnrnic osci l lator wi th a perEurbat ion.

3.2.3 Opt ical Potent ia ls

In order to compute the radial i -ntegral of

part of the distorted rvaves must be calculated.

model has great ly s impl i f ied the calculat ion of

wavefunc t ions.

4I

eqlrat ion (10), the radial

The use of the opt ical

elast ic sc.at ter ing

The central feature of the opt ical model j -s the representat ion of

the scatter ing nucleus by a potent ia l which is a funct ion of the space

coordinate and incident energy only. The potent ia l is complex, wi th the

real part represent ing the gross aspects of the elast ic scatter ing, and

the iuraginary part account ing for al l inelast ic process." .50

The radial part of the distorted waves sat isf ies the equat ion)

, d- , , ,2 L(T-+l) 2uL , +K-- T - *, (U+U") l"

drz r ryz c XL(Krr) = 0

(2e)

where K is the wavenumber, L the partial wave angular momentum, u the

reduced mass, U the opt ical potentLaL, and U" the Cculomb potent ia l

for a charged sphere of radius r" . The equat ion is solved numerical ly

with the boundary condi t ions xL(Kr0)=0 and xr(k,r) approaching

asymptot ical ly the outgoi-ng Coulomb \ , , /ave funct ion at large t .47

The opt ical potent ia l is a l ,^ Ioods-Saxon form, as fo l lows:3

r r - t Al / t l . f r - r a1l3- lu(r) = v I r*.*p --g-- |

-i + ir+-- | l*exp - :v^ |

-1

L - ao

J vL ' av

J+v^ ̂ (f. il f-r*u*o r-r

-^A1 /=

}-tso'- |

" so_- l - .L/3L % J r r - r"A- ' - - l -1 (301

*i . t+aI . i+ l t* .*o-- lssdrL * as - i

The f i rst term j -s the volume term, account ing for the elast j -c scatter i -ng.

The second term i -s an i rnaginary volume term, which represents inelast ic

ef fects wi th in the volume of the nucleus i tsel f . The third term is a

volume spinrcrbi t potent ia l wi th a Woods-Sa:<on shape. This term in the

potent ia l is only used with the deuteron entrance channel in the (a,6l i )

42

react ion and accounts for any polarLzat i lon ef fects. The last term is

the surface imaginary potent ia l . Tt is assumed that the alpha-transfer

react ion takes place on the surface of the target nucleus, and this term

in the potent ia l accounts for the react ion and deuteron absorpt ion. In

actual fact , the second term in the potent ia l is not used as the fourth term

could account for a l l inelast ic ef fects in these react ions.

Recent ly, good 6r i

opt ical potent i -a1s t rave becorne avai lable.q1

I , ' latson" has used a superposi t ion model calculat ion to calculate opt ical

potent ia ls for ' r r , and apply ing the S-state relat ive nxct ion. He has

used a modif icat ion of l , t ratanabers superposi t ion model for deuteron opt ical

potent ia ls, and calculated

f . )vA (r) = | {v^ (r-R/3) * V, ( r+2Rl3)} l , i , (R) l ' dRol i J c d '

(3 r1

where V is the alpha opt ical potent ia l and V, is the deuteron opt ical0 'd

potent i -a l on the same target. U(R) is the wavefunct ion of the relat ive

mot ion for the ct- and d-clusters.

These potent ia ls have been immensely successful in a number of

appl icat ions, and have predicted correct ly many of the 6t i

propert ies,

including the o * d binding energy of 6r ,

and elast ic scatter j -ng of

6-Li on many l ight targets.

3.2.4 Select ion rules and al lowed f inal states

Alpha transfer react ions must obey al l the conservat j -on laws for

the strong interact ion. Conservat ion of l inear momentum, energy, and

charge trave already been accountecl for in the der ivat ion of the DWBA, as

wel l as the ant isymmetr lzat ion of the wave funct ions of fermions j -n the

clusters, targets, and incident part ic les , a l though this last is not in

the DI,'IBA code DIntrUCK.

43

The conservat ion of angular momentrrn has not yet been total ly accounted

for. The select ion of f inal states is made qui te s imple by the fact that

t .he t ransferred parLic le is assumed in th is model to have the same

j-nternal quanttnn numbers as a f ree alpha part ic le, wi th spin and isospin

both equal to zero. Addi t ional s impl i f icat ion comes from the fact that

both targets used in th is study have Jn=0* in their ground states.

Figure 9 shows the energy levels, wi th spins and par i t ies, of the

var ious nuclei and residual part ic les used in th is study.

Let us examine the (a,6l i ) react ion f i rst . Both the deuteron and

Athe

o] , i have Jn=l* . I f the

6r i can be represented as a deuteron * alpha,

then the orbi ta l angul-ar momentum of the d + o, system is ei ther . [ , r=0 or

9"t=2, and has in fact been measured as.Q, '=0.1 In th is case, there is no

spin t ransfer, and the spin of the residual- nucleus must be equal Lo the

orbi ta l angular momentum transfer. In a direct react ion, and with An=(- l ) [ ,

the al lowed f inal states for the (ar6l i ) react ions on the two targets

used are shown in table I I I .

TABLE I I I

^4 n , 1ALLo*ed. FtnaL States fon Resl,CmL lrluclei fot (d,6ni)

'otr(a, 6r i ) 2o*.

'o*"(a,6l i ) 160

of o.oo o+ o.oo++

2 1.63 0' 6 .05

4+ 4,25 3 6.L3

3 5.62 2+ 6.92

I 5.79

-L0' 6.72

1 7.L2

1 g.5g

+2 9.85

+4' , 10.36

44

ft-r')

I{q Nrery\O t"j tfi sf sf

++ov

r/) N

()

\f,cv

+rv

r,q

++ +r"r)rV cf

rnfv

"t

l'*o'!sf

N\CIr.l.;

oC\J

-€$mF;Ln

r'\vN

v@

tt--

\"oq@

+Jaq)Fro+Jl{

'rl

tl-{o

'r{0)

r{U*t

d

tr.r{

(/)Flq)

or{

h,^n

t{a)tr

rT-l

o\

bo'ril+,

IJ7-

oC\J

o11}

++f \o

+$

++ t+ |cv o\ro -

qH ?qf-- \D \O t9

+o

tn (Drn0 oor r?on @cj,ioid d

++l + l+crJ rtorq $c\J- (\'

4s

_3 1 ?The ( 'H*r '8") react ion is rrpre di f f icul t to descr ibe. ' i fu is L/Z+

in i ts ground. state, and 7u"

is 3/2- in i ts ground state. But 7u"

also

has a 1/2- exci ted state only 430 tseV above the ground state which is

also exci ted at these energies, The 7uu

may be considered to be a 3rr"

plus an alpha with relat ive angular momentum .0t '=1. Depending on the

direct ion of coupl ing of the L" and the spin of the '*" ,

e i ther the 3/2-

or L/2- state may be obtained with th is arrangement. Again, there is no

isospin or spin t ransfer, but there must be a total angular momentum transfer

of at least AJ=1r to give .0"=1 in the 78..

The orbi ta l angular momentum

transfer must be equal to the vector sum of the spin of the residual

nucleus and the orbi ta l angular momentum in the 7U*.

Assuming the

par i ty change to be An=(- l ) [ ,

the al lowed srates in the ' "Oror(3H", tu.)20mu

react ion are shown in table rv for both states of the 7

u".

TAtsLH IV

ALLoued. F'tnal States For Residual NucLet For {3 HnrT gn)

'o*r(3H., to.

3/z--)20*. 'ot*(3H.,78. L/2-) 20*"

0+ C. 00 MeV

2+ L.63

4+ 4.25

2- 4.97

3- 5.62

1- s.78

0+ 0.00

2+ 1.63

4+ 4.25

2- 4.97

3- 5.62

1- 5.78

0+ 6,72 0+ 6.72

The 1o,Bn"; react ion is the s implest of a l l . The spin of the

entrance and exj . t part ic les are each zero, and the two alptr ,as in the BU.

are in an S-state, as in the 6rr .

A11 isospins are zero. The al lorved

f inal states in the'orr(o,BBu)20*" react ion are the same as for (d,6t i ) .

46

For completeness sake, the 20N"(d,dt ;20N"

react ion is also discussed.

Since the same part ic le is in the entrance and exi t channels, no part ic le

transfer and no spin t ransfer psr se takes place. The spin change of the

target nucleus must be equal to the orbj" ta l angular momentum transfer,

wi th the resul ts that the al lowed states of the residual nucleus in th is

react ion are the same as those in the'Otr(ar6l i )20*. react ion, wi th

0&r=(-1)- . In addi t ion, i f the deuteron were to f l ip i ts spin by one or

two units of angular momentum, (which is treated as a spin transfer in

the DWBA code DWTJCK) the 4.g2 MeV 2 state of 20ru

would also be al lowed

with a direct process. The spin f l ip of lK or 2K would have to couple to

an orbi ta l angular momentum transfer of 1or 3 to al low the 2 f i r la l state.

The theoret ical t reatment of par i ty change depends on the DWBA method

used. The zero-range DWBA contains the Clebsch-Gordan coeff ic ient

<L^,L"00 1t0> in the form f actor, but the f in i te-range DWBA contains aol 6 '

Racah coeff ic ient in the form factor which requires that four separate

tr iangle inequal i t ies be met . 43'44

The par i ty of the form factor in

the f in i te-range DWBA is given by An= (-1;Lo+LB, and i t is only in the

zero-rar"ge approximat ion that the form factor contains the Clebsch-Gordan

coeff ic ient above, necessi tat ing An=(-1)! .

I t should be pointed out that a di-screpancy exists in the l i terature

as to the spin of the 4,g7 MeV state in 20*".

Most authors assign a 2

spin and par i ty to th is state. However, one reputable source, the AIP

Itandbook of Physics5z, also assigns a possible 1- spin and par i ty. A 2

state is not al lowed by a direct r ransi t ion in the 'O*r(ar

6r, i ) 2010.

react ion, but a 1 state would be so al lowed.

47

3.2 .5 Coupled Channels Calculat ions

Some rvorkers doing studies of o-pickup and o-str ipping react ions

have not iced the populat ion of the 8.88 I , {eV 2- state in t6n,

and the

4.g7 I leV state j -n 200,u, rvhich is fe l t to be a 2- state by most authors

al so .23 '30 rn some react ions, the 200r"

(a, 6r i ) t '0, for example, the

direct t ransi t ion to th is state is forbidden hy par i ty conservat ion.

There has recent lv been mount ins evi-dence that inelast i -c ef f ects in

entrance and exi t channels can ef fect the cross-sect ions of d i - rect nuclear

transfer react ions when the target or residual nucleus is strongly deformed.

]q i ?oI t has already been shown that the --F(-Herd)-"Ne react ion requires

expl ic i t t reatment of a mult i -step process in which the part ic le t ransfer

is preceded by an inelast ic exci tat ion of the target nucleus.54'55

Shepard, King, and Tr,r"56 have shown this to be the case for s ingle-

nucleon pickup i r , 160

and t ' r ,

respect ively.

A treatment of inelast ic ef fects in the entrance channel and mult i -

step processes requires a coupled-channel-s t reatment in the DI{BA. The

usual method of calculat ing cross-sect ions is to solve nr lner ical ly a

set of coupled di f ferent ia l equat j -ons. The coupled equat ions are:

rt_* 19 +1) /,l .2 t - +G:-uf or rt -CC

\-\ 2,* r"(*" , r") =

\ ,orr"" , xL",(K", , t " , )

"r ](32)

where c and ct are the indices indicat ing channel number, U" is the

opt i -cal*Coulomb potent ia l for channel c, and V"", is the of f -d iagorr ,a l

coupl ing coeff ic ient which depends on the angular nomenta and coupl ing

potent i .a ls betrrreen the var ious channels, and y" is the reduced mass in

the channel . 57

The treatment dif fers from

problenr by a T-rnatr ix approach.

equivalent if only one entrance

48

the usual DWBA approach which solves the

However, the two methods are completely

and one exi t channel are used.

3.3 ClusLerire in Tarqet Nuclei: SpeetToscopic FactLrs

The zero-range DWBA computer code DIfUCK calculates a predict ion for

the angular distr ibut ion of the cross-sect ion, as descr ibed previ-ously

j .n th is chapter. Since i t uses the zero-range approximat ion, the c luster

nature of the exi t part ic le is not used in the calculat ion, and a

normal izat ion constant is undetermined. I f DI, t r t lCK were to calculate a

perfect f i t to the exper imental data, then for every angle, the relat ion

between the exper imenLal and theoret ical cross-sect ions should b"47

do =D2d CI EXPERIMEMAL O

?CS-S^ do

r rK-P

2J* d a DwucK(33 )

where C is a Clebsch-Gordan coeff ic ient coupl ing a change of isospin

(C=1 for ground-state to ground-state t ransi t ions in alpha transfer on

the nuclei of th is work, and for alL t ransfer react ions in which there

j-s no change of isospin.) , J is the t ransferred total anguJ-ar nmmentum,

D- is given by Auster*3 as the volume integral of the normal ized, elast ico

scatter ing wavefunct ion operated on by the t ransi t ion potent ia l in the

zeto-range approximat ion, and So, then, is the spectroscopic factor.

The value of Do is calculated and pr inted by the f in i te-range DWBA code

LOLA' al though LOLA does not speci f ical-Ly use this value in i ts calculat ions.

Knowing this value, the spectroscopic factor can be calculated from the

rat io of the exper imental to theoret ical cross-sect ions at points where

the normal ized predi-ct ion coincides closely wi th the data. SR is the

49

over lap of the exi t part ic le wavefunct ion as measured exper imental ly

with the wavefunct ion of an alpha part i -c le in the potent ia l wel l of the

incident part , ic le. For the exi t part ic l -es used in th is study, th is value

is usual ly presumed close to uni ty, and in keeping with other workers, a

67Rvalue of Sg of .90 for

t l , i , .95 for 'B. , and 1.00 for

oBe was used.

In the case of a lpha transfer, the val-ue of S gives a measure of

the probabi l i ty of f inding the target nucleus as a core plus an alpha

part i -c le, and is therefore a measure of the ocLent of a lpha cluster ing

in the Larget nucleus. So carrnot be greater than uni ty for each alpha

part ic le avai lable for t ransfer in the target nucleus. I t should be

pointed out that So j -s equivalent to r l in equat ion (26) i f ^2r

43, etc.

are al l zer o .

Part of the or ig inal- purpose of the research which this work is a

part was to do a complete survey of a lpha-transfer and alpha-knockout

react ions on the two target nuclej" in th is work, as wel l as other alpha-

part ic le nuclei . I t is unfortunate that f inancial considerat j -ons forced

the abandonment of the project . One aspect of the study was to be an

invest igat ion of the nature of the t rarsfer process to see i f i t was a

true direct t ransfer, wi th the t ransi t ion probabi l - i ty being independent

of the nature of the incident part ic le, or i f i t might be better descr ibed

as a st imulated-emission ef fect , wi th the probabiJ- i ty of t ransferr ing an

alpha part ic le being greatest when an alpha beam was used.

Fermirs second golden rule states that the f i rst order t ransi t ion

probabi l i ty per uni t t ime, Wfi , for a t ransi t ion f rom an in i t ia l state i

to a f inal state f , which is direct ly proport ional to the cross-sect ion

in a scatter ing or col l is ion type $cper iment, is dependent on the densi ty

of phase space p (B) and the transi t ion matr ix element as fo l l -ows:

50

(3 4)

The transi t ion matr ix element contains the structure informat ion, i -ncluding

the extent of a lpha eluster ing. Ideal ly, i t should be independent of

the mass and energy of the entrance and exi t part ic les for a given target

and gi-ven angular momentum transfer. This t ransi t ion matr ix element is

very similar to the T-matrix element in the DI^EA.

The matr ix element of equat ion (34) nny be isolated from other factors

as fol lows:

wri = + p(E) l . r lv l i '12

l . f lv l r , l '= a* "r , .21T p (E)

p(E) = /1 to3n M3/2EL/2 (2s+r)

:n[r '

The cross-sect ion i tsel f may not be independent of mass, energy, etc. ,

but i t would be expected that the t ransi t ion mat,r ix element, would be

near ly so i f the model" under invest igat ion were reasonable. Al though

only a few react ions of those or ig inal-Ly pl-anned have been done, the

betravior of the rat io of the LotaL cross-sect ion for ground-state to ground-

state t ransi t ions to the densi ty of phase space wi l -1 be invest igated.

The densi ty of phase space is qui te easy to calculate. Marmier and

Sheldon calculate the densi ty of phase sFace using box nonnaLLzat ion in

reference 3. They report the densi ty of phase space in th is case to be

(E) = L3MK (so1qr zlJ

where L3 is the volume of the box, M the mass of the project i le, and K

the wavenumber of the projecl i le. I f the volume of the nonnal izat ion

box is approxirnated by the volume of the target nucleus, 413 r ro 'O, and

we recall that K=/ zME/i, , and S being the spin of the f ina 1 state, then

P

(3s1

(st 1

51

IV. EXPERIME}ITAI R"ESULTS AI{D DISCUSSIO}T

2n ,n4 . L The "" It le ( d., d | )"" Ne Reaction-

," " t* t

to generate opt ical model parameters for the entrance

channel of the 20n"(ar6l i )160

react ion, an angular distr ibut ion for the)n 1A' " lTe(drdt ;" 'Ne react ion was taken {or al- l - states up to Ehe 4,97 MeV state.

A spectrum for th is react ion is shown in f igure 10. Resolut ion was

347 KeV in the ground state for the angle shovrn, and al l states up to

9.6 MeV above the ground state were c leanly resolved, except for the

5.63 l {eV 3 ancl 5.78 MeV 1 states. The 0.00 r , lev 0* " tate

of 22*"

is

vis ib le, and the K=2 rotat j -onal 4.g7 l {eV state of 20i{u

is also seen.

The ground state peak at th is angle contains more than 55r000 counts,

and the nurnher of counts in the ground state never fe1l below 10r 000

at any angle. Data were talcen from a scatter ing angle of 9.5o to

104.5' i -n the laboratory coordinate system in steps not larger than 5o,

and as sma1l as 2." .

The elast ic cross-sect ion rat io to the Rutherford cross-sect ion

is shovm i-n Figure 11. To compute opt ical rnodel parameters, the computer

code I ' {AGALI was ,r . "d.58 } l interberger, -e! "1. ,59 have publ ished formulas

for est imat ing deuteron opt ical model parameters on l ight nuclei at 52

MeV' and their f i rst set was used as an in i t ia l value in I4AGAII . Searches

were done on al l parameters, and the t tbest f i t t tvalues are shown in

Table V. For the sake of conciseness, opt ical model parameters for other

react i -on channels used in th is work are also shou'n.

The best f i t for the '0r ," + d channel occured with a chi-square per

degree of f reedom of 9.7A, which is rather good consider ing the data var ies

over f ive orders of magnitude and has a strong angular structure. A

's'9 3 N --cc

+z e9' l

- l

+b-7

8L'g 0 NV-€+o

9Z'FL6' .Vzg's 4-a

z L'9rJ

o\f,

!gr-o

LrJZcC\

.fr.l l

cio\--l

-JLrJZZ

I(J

:_dZoC\J

LNa

sfNI

- l l NNYHf Ulc SI Nf lo l o

GJzN

r-i

rl

c)zOC\

l'rot+{

EH

+J

0)FqUJ

ri

hn

53

o9

a

+J.rJq

(')rla)d-a-AH

V' Ftrj q

t-\ I

HIJEE!.(93trJ L)

OE(d

rg,^ . l ! :UJT

g)F

zg'

+J

(93ZO

oA

V E. ild

r l \vr rJ *l |q

!'- o,

- t3{ : . -(J

Ft(n;;

/^ \ .2X ==?,!Y

Z-"

.FlJ(J.sl'r ,

o

CUOJ USHINU6

YCC

H9t l .J>

9r sri f r0F tJ1 c^\fZ trf rl

; t r 6)CY X(J()<9- ) ) -1J= 3 o rOOJ()

))

4."

( i () i j

/ i 7 '? '/ ,:f| ?.t",

\ \ r ' ,) i'l

/tv''(it . . l\ ' . \

I

:.\

\*

t . \

{e

I

: . \... \

(,

+ooCoot-

lrJ

Z.o(\l*.

Ocl

:--Z

trj

Z.oAJ

o

OI Ot IVU ̂ -NOl r lSS - SSOU l

54

Ln.f,r\

rnN

rt

^la Elcs ql

oloEl

t{ t+r I

OEIt{ (+.{l

oElo iHl

OFIH rr l

o>la ojl -$

I

N(\ lc.) rn co rn co rn N c{

aaoa

rl Gl r-l N Fi N r-l -{

Nr-l -$ erl rl r-{ O \o \oNf\ooo\r{@Lnrn

aaa-a

rl

Ln t-l cn f-' \ONf\rJO\CaCCLnrn

aoaa

r-i rl r-l r{ Fl rl Fl FJ

.ir -ifOO.@.O

'.Oo@...N@O\-{ ' - IOOOtncON\Or{@cn\O

rn Ln -f,

-i' f\ \O t-\ \O Cn \O \Of\F\ \Of\ \O@\O\O

aaaO

_t

+)CDIEJ F >ISr(OJs rf , Elilt\

trl t"-)Fl aD ElFa \ d Hl

3arr

t-)

8' * o El$ tr '{-l I

A++O

Ln '-l

tn @ Ln .if -:fNr l NNNtt \ -$. , t

aaoaaa

rl rl rl Fl rl rl rt ,-l

co o l__ tnLn.GlO..

'F l ' \OFl .OOO-$cn{tLnOOOf\N\ONFltnFlNl t t t t l t l

>lAI

lJXc)

OOOOLNOtn+I- f ,NooNcoc.) \o o

og)

P\b0$'la)l.'l

F]o

t+{q)

.J: '1 'E3g'OF1€\Ocar\dOO

\o++++++++

qJb00)60()b0()zoEzEzEz

O\O*i fO- i fO. i fOc\ ,-{ c\ N (\l N (\l c\]

rlCJl.

Harl

55

DI{BA calculat ion using the zero-range computer code DIrruCK47 was at tempted

with the parameters generated by IVTAGALI ancl wi th the parameters of

HinterberE;er, et a! .59 t roah sets of parameters are seen to f i t the data

wel l at forvrarcl angles, but the calculat ion using pararneters f rom the

data of the present work appear to represent the data better at center-

of-mass angles greater than 55o.

The zero-range DI.'trBA calculation of DI,, l lCK used the extreme alpha-

cluster approximat ion for the target nucleus. The tOn.

nucleus was

treated as an S=0, T=0 alpha- l ike part ic le in a potent ia l wel l of the

16-"0 core. The potent ia l wel l was assumed to have a volume Inloods-Saxon

shape with no imaginary part . The wel l depth wi th 5 nodes in the

wavefunct ion was calculated from the binding energy of the 15n

+ o

system to be -88.91 I{eV, wi th a radius parameter of L.6 fm and a

di f fuseness of 0.6 fm. In addi t ion, a Thomas spi-n-orbi t factor of 25.0

was included in the potent ia l . The alptn part ic le was assumed to have

an angular momentum of zero relat ive t -o the 160

core.

Besides the zero-r ."r lge DWBA calculat ion, Figure 11 also shows a

calculaEion made using the f in i te-range DI{BA code LOLA. The f i t is

qui te good in both cases, but the f in i te-range calculat ion, integrated

to a range of 15 fm, appears to diverge from the exper imental data at

extreme back angles. This fact seems to indicate that the zero-range

approxinnt ion is val id, as expected, for the 20*.

* d elast ic channel

and rnay actual ly tend to negate or compensate for other approxinrat ions,

resul t ing in a better predict ion in some cases.

Figure L2 shows the angular distr ibut ion for the center-of-mass

cross-sect ions for the I .62 MeV 2* and 4.25 MeV 4+ states. These, along

+r.r i th the 0 ground state, are ! i=0 rotat ional staLes.

'oN r(D,D'foN E 40 MEV

O,'O'r

tIItI

I

o, '

.J \ l ,6J 2+r:/

ot . - a

_ \

O+ * '?+<+4+

co

DWU CK--CHUCK

o \

oo \

\\\

o t.o

4+

56

to

roo

?o 40 60 80

C M SCATTERI NG AN GL IDEG REES

Fig. L2 20N.(drdt)20N" inelast ic scatter ing and f i ts

6lUr) l

m=

z.

- to(Jt!(naaoEI(J

o,lroo

57

In the zero-range DWBA calculat ions for these states, the s€Lme opt ical

parameters and the same t6O

* o potent ia l welL were used. The only

essent ia l d i f ferences in the calcul-at ions were the Q-value, the t ransferred

orbital angular momentum, and the total- angular momentum of the final

state of the residual nucleus. (See Table VI.) The f i t f rom DWUCK is

reasonably good and has the same generaL behavior as the experimental

data. The coupled-channeLs cal-culat ions wi lL be discussed later in th is

ehapter. The predict ion f rom DWUCK was normal ized to the data for a best

f i t . Such i .s the case with al l DI^IBA and coupled-channels calculat ions

in th is work.

TABLE VI

m1 20"" /1 ""20__The "- Ne(drd')- 'Ne yeaet ion paratneters

Fina1 state

0. 00

l_.63

4 .25

4.97

J?T A,Q, AJ dold CI nax (mb lgr )

2889

3L.7

2.87

.690

0+

2+

4+

2-

The angular distr ibut ion for the 4.97 MeV negat ive par i ty state is

shown in Figures 13 and L4. Figure 13 also shor^rs the zero-range DWBA

calculat ion based on the assumption that the 4.97 MeV state in 20*"

is

a 2 state. Calculat ions were made with orbital angular momentum transfers

of 1 and 3, and with a change in the deuteron spin Z-component of 1

or 2, I t may be seen Ehat the [=3, s=l case gives the best f i t to the

exPerimental data, although there is some divergence at back angles.

Figure L4 compares the predict ions of DWUCK for the 4.97 MeV state as a

2- state wi th the best f i t above, and as a 1- state wi th .Q,=1, s=0.

58

oo

Bntr-JtrIOt

3if icl$zF

F

(9J,Z. ;

gE:- r !1F' i f

F2<RL).t r )Y

rfi

ojgNdX

INJ

IJ

f-O)

a

$

IJZ.

oN

'63

OhJZ.o

N

rl

.r{

Fq

ooo8

o

o

,/

//

//

r{r l

l , t

, llifoI

/I

IIt

II

/

/,/

/ /

n/

It\\\l

/

o

o

NNft l f f t t l

rn tn t'f)U)

:r-i j t"iI t l l l l l lJJ)J

: (YYY

333 33333ooo oi l i ll l i l

i l i i

),/

/ . /

/ r ", / r ,

I

?\ r\?

\ . r .

,X',//

t i\

a.\. . . \ o

NOII]3S SSOU]as /8rl

59

oo

o@ a

LdLrJE.(, .\

LLJ Eod

I\F i(L '

ot!

q-r

IF-Jo(9.3z.h

arv

( r*zEK.hL!*Forl- -P<Xu.rA Yy. .O

;.z

- .oz- c.l

(J.$F{

bo.r-lr - .

o\r

oCV

l!

f-O)

a

\f,pt-

lrj

Z.ory-.'n

rF

OI-IJZ"

o(\J

aooo 9NOIr] ]S SSOH]

, 'oI

t,oIIr f

/I

Ort\\\

I

II

I

II

//

\I/

fII\\

o

/I

II

/

O -t t '--

v)(

6:: )J

33ocf

l lcv

o

IIt

u s/8r/

60

It is readi ly seen that the lat ter assumption resul ts in a predict ion

which is total ly at odds with the exper imental data, whereas the former

provi-des a reasorrably good f i t . Hereafter in the present work, i t wi l l

be assumed that Ehe 4.97 MeV state in 'O*. is a 2- state, in agreenent,

wi th the large major i ty of other workers.

Table VI shows the energi-es, spins, and par i t ies of the exci ted states

seen in th is react i -on up to 4 .97 MeV, and al- so shows the angular momentum

transfers and the maximum di f ferent ia l - cross-sect ions for each state.

,n R 7A4..2 The "" Ne ( d, " Li ) " " o Reac tion

The f i rst of the alpha-transfer react ions examined was the

20*.(ar6l i )160 react ion. A spectrum from this react ion is shown in

Figure 15. Resolut ion averaged 255 KeV over the range of angles for

the ground state. Three important states l rere c leanly resolved, however,

the rnajor i - ty of states, including the 6.06 0* "rrd

6.13 3 doublet , the

6,92 2* and, 7 .L2 I doublet , and the 9.85 2* and, 10.36 4+ doublet were

not resolved, or only barely resolved. Above this lat ter energy, even

doublets were not resolveable. The 180

ground state f rom the 22N.(ar 6l i )

react j -on is seen as a resul t of the 22*"

impur i ty in the target gas.

The 8.88 MeV unnatural pari ty 2 state j -s also seen, indicat ing

that mult i -step processes nay be important in th is react i -on. The integrated

area of the 2- state reaches a level as high as 2O"l of the area of the

ground state. The exci tat ion of th is state is forbidden by the s ingle-

step cluster t ransfer mechanism usual ly assurned for the (dr 6t

i ) react ion.

The ntunber of integrated counts with background rernoved in the ground

state of the 20N"(a,6l i )160

react ion ranged from 3110, for a stat ist ical

\rr\JO

+v Ez'v

--i!IIItII

Jt ' I's '9 +o --r"

---1t - {

" t- . t

+ z e9' l

- z L6'u_e 29.9 {=, " i- f 8L'9 t -* !q.=

i

[J

C\f,

IJ-2"oC\J

*J(O

(n

O:---z

()

\f.\J

I I

- ) l| | --1r*J I

!I. ' ,I

i. " I.-{ -. 1

l ;

i-r- i_"r.-_ ii ) i

II-*t

itI

O!I lN NVHf U]d SlNNO]

uncertainty of L.87,, to a minimum of

of 5 .57, .

The angular distr ibut ion for the

transi t ion in th is reacLion i -s shovrn

represent only stat ist ical errors ( tN

62

330 for a stat ist ical uncertainty

ground-state to ground-state

in Figure 16. Error bars shown

-1 /7- ' - ) r zrs is the case throughout the

remainder of th is work.

The opt ical roodel parameters ancl the 160

+ CI, wel l parameters

used in the 20*.(drdt ;20t1.

exper innent above were used to calculate the

cross-sect i -ons for the t0*.(ar6r i )160

reacr ion wi th the DI^EA. 160

+ 6rt

parameters were calculated from the work of W"t"orr .51 The zero-range

predict ion f rom DI{LCK is shoqm along rv i th the f in i te-range predict ion

from LOLA. The f i t j -s qui te good in both cases; the predict j -on fo l lowing

the general t rend of the data rather wel l . LOLA appears, however, to f i t

the f iner structure of the measured data better than the zero-range DI^IBA.

This fact is consisEent wi th our understanding of the 6t ,

as an alpha *

deuteron in a relat ive S sta. te. The 6",

is a much more di f fuse part ic le

than ei ther an alpha or a deuteron, and i t is to be expected that a

f in i te-range treatment would better predict t1-re over lap of the alpha

part ic le in the 6r,

and ton"

nuclei .

The wel l parameters in the o, + d system with an g,=0 Z-node wave

funct ion were found to be a wel l depth of -29.LA MeV, radius of 1.3 fm,

di f fuseness of .7 fm, and a Cculomb radius of 1.3 fm. The extreme alpt la2A

cluster inod.el was used for the 6r,

"" wel l as the Ne, as before.

Figure L7 is the angular distr ibut ion for t i re higher exci ted srates

16 )n A 1Aof t , l re ^"0 nucleus in the ' ' 'Ne(dr ' 'T, i ) -"0 react ion. The zero-nange DhtsA

predict ions are also shor^rn. I t \^ras fe l t that the zera-rarrge DI{BA would

provide an adequate predict ion for these higher states s ince i t seemed

63

on,

+J.rlH

Bt-\ A

O€ft

o+Jd

+Jo

11

-!

t{b0

Oorn rOwFl

'rlFl

\o

'tJ

AJz

(\o+J

rO ho

t l-.{

H

o.rl+J

.ri$l+J(.,l

\qt-t

-)b0(..:4

\or{

b0.r{fi

oN

oNC

oI

aLr.JtrJE.(9r!Oe

LdJ(9Z

(9zg.UJFF

L)a

=(J

//

o

a

tI

tII\1

o

\II

II

I

IIt\

, ,

] r "

+/

I

Y

135BJO

,/o

o

/

/

S\

/(a

UJ

o$

+oooo

orJ)

-J

\OG

Ol!Z.

o(\f

o

I ] ]S SSOU]u s/ st/

' totoN.(o, tL; aO MeV

ro

cCa@=t

ZIF(JLrjaaaox.(J

C,M, S CATTE RI NG A NGL E ̂ , DIGREESFig. L7 20Ne(d'6Li)160 higher states ano DWBA f i t

6,05 0+6.13 3-

6.92 2+7,12 | -

9,95 2+fo,36 4+

-DWUCK

65

to give a fa i r predict j -on for the ground state. In the case of the three

doublets discussed above, the DWBA calculat ions shovrn are a sum of the

DWBA calculat i -ons for each state. The f l Ianner in which th is is done and

the computer code used to generate the sum is descr ibed in Appendix B.

Calculat ions f rom this code show that the 6 MeV doubl-et is composed of ,

on the average' 8.2t2.47, of the 6.13 3 state and 91.8t2.47, of the 6.05

+0' state. Simi lar calculat ions also show that the 7 MeV doublet is

0.13t .AI% 6.92 2* and. gg.8gt.02" l 7 .L2 1 , and rhar rhe 10 MeV doubler is

3 .6t .67, 9 . 85 2+ and 96 .4! .6 10 . 36 4+ .

In al l four cases of h igher energy states, the Dtr^IBA predict ion for

the angular behavior of the cross-sect ion appears to fo1low the measured

cross-sect ions reasonably vre1l . The same general behavior of the

predict ion and the exper lment is seen in each case, the only except ions

being srnal l d i f ferences in the detai led structure over short angular spans.

Table vrr g ives the exci ted energy, spin and par i ty, angular

rnomentum transfer used in the calculat ions, and the calculated spectroscopic

factor for each state in the react ion.

TABLE VII

The 20

Nn ( d,6 nt )L6 o Reacti,on parametey,s

Final state

0 .00

6 .05-6 .13

6 .92-7 .L2

g.5g

9.85-10.36

+0'

+0

'3.* ' r -L el

1

J-T2' ,4 '

A0

n

0'3

2rL

1

2r4

0'3

2rL

1

2, /+

J1T AJ

0 .434

dola n nnx (ub/sr)

93

65

37

23

L6

.011 5

66

4 .3 rhn 24 tqq ( dri )20 un nno"tto,

The ldr6r i ) react ion was also

an easy target to work wi th. I t is

i ts energy leve1s and those of the

separated. Both 'o*. and 24tg

^t"

of their behavior can be understood

performed on a 'o** target , 'oW is

a sel f -support ing pure so1id, and

residual nucleus, 2or*

are wel l

strongly-deformed nuclei, and much

in terms of col lect ive rotat ional

motion.

A spectrum for the'o*r(a,6r i )20o. react ion is shor^rn in Figure 18.

A11 the states shovrn are c leanly resolved, wi th the f i rst three states

in the lower K=0 rotat ional band dropping to zero counts between the

peaks. A11 known states up to the 5.78 l , IeV 1 state were seen and

easi ly ident i f ied. The f i rst three states, the 0*, 2*, and 4+ in the

K=0 rotat i -onal band dominate, as expected. The 4.97 Mev z K=2

unnatural par i - ty state is also seen, conf i rming the work of McGrath,

?net al . "" This state cannot be reached by direct t ransfer of an S=0

alpha part ic le to a deuteron, support ing the content ion of many worker s46,-60

that mult i -step processes may also be important in alpha-transfer

react ions. InCeed, the cross-sect ion for th is state reaches a value

which is tuLLy 35% of the cross-sect ion for the ground state.

Several sets of opt ical-rnodel parameters were t r ied for th is

react ion. Unfort t rnately, no elast ic scatter ing data was taken or was

avai lable to compare elast ic predict ions because there was fel t to be

suff ic ient opt ical model data in the l i terature. The opt ical model

parameters f inal ly used r ,uere those which gave a best f i t to the 0+

grouncl s tate when used in a zero-range DWBA calculat ion . The d + 'Ot*

parameters were those used by Djaloeis, et a l - . , at 80 Muv46, and the

70 6 A1-" I t re + "Li parameters were the avera€le of the 20 MeV data of Bethgeor

IJ

Ct'

's '9 0 8l

;3; i3 I-*-:r_1 P8:e -i- l 69'6 1+zs8'e# $

+s Bo't oNv -o sd'o-r -z-;AJr -=

4- ;

-gt"- |-€-

-

-Q-

J',-?

-Tft$f.o(o

ft,

-J(OG

O\--l

L!Z.oC!

oLO

a

$t-O

_Jl,!ZZ.{^

ILJ

cv

t lNNVHl Ujd qINnol

68

and the 2A MeV calculat ions of In l" t "orr5l using Bethgets data. These are

al so shovm in Table V.

As j -n the 20u.

work above, the'orr nucleus was assumed to be capable

of representat ion as a 20r.

core plus an S=0 alpha part ic le in a relat ive

s state. A In loods-saxon wel l was used for the 20*u

* o binding potent ia l ,

wi th a depth of -113.75 MeV, radius parameter r"=1.6 fm, di f fuseness 0.6

fm, and a Coulomb radius of 2.5 fm. The o-d wel l \^ /as the same as in the

?o 6, 1A-"Ne(d, "Li)*"0 react ion.

In the test ing of the var ious opt ical model parameters and combinat ions,

i t was noted that the zero-range DWBA calculat ion is qui te sensi t ive to

the values of the parameters used. Indeed, i f the parameters are not

very c lose to opt imum, large deviat ions of the calculat i -ons mav resul t .

I t is fe l t that th is rnay he the nnjor source of error in the analysis

of th is react ion.

Figure 19 shows the angular distr ibut i -on of the cross-sect ion for

the ground state in th is react ion. The value of the di f ferent ia l cross-

sect ion at the f i rst maximum is very near ly 400 ub/sr. The zero-range

DI^IBA calculation from DMICK and the finite-range DI^IBA from LOLA are

ai--qo shovrn. I t is somewhat surpr is ing to note that the t \^ /o calculat ions

have di f ferent types of behavior, but both f i t the data equal ly wel l .

DWUCK has the advantage at forward angles, whereas LOLA sei iuns to do

a better iob at the back angles. However, in nej- ther case is the f i t

except ional ly good, probably due to the opt ical model parameter problem

discussed in the previous paragraph. I t nray be that the tendency of the

zero-range DWBA to average errors toward zero dominates at forward

angles, whereas the di f fuseness of the 6",

becomes more important wi th

69

]J.rlLH

B

H

oa

C)dr-6 Ov .r{

+Joo(nIoaot{o

oz

N

srJ\o

o0Ft

-+c!

o\t-l

Ob;N; i

ogus/8rl

- ' t ' ' - - -

-

-o-

(

\_

II./+

r!

o\f,

+

+ooo

tol!Z-o

N

' - l

@a

(f(9

.t(\f

+

(,t !UJE.(9t !OI

rljJ(9z

(9Z(.

tr-JF

<L(J(,r)

j(J

ll-'t +

q1=sc)- t

\.\

(

- - :*)

o?

/It\

NOII] :S SSOU]

70

-6_a slower -Li at back angles, so that a f in i te-range DI{BA calculat ion

is more representat ive at back angles. 0n the other hand, nei ther may

be a Lruly representat ive process, and mult i -step ef fects may be

important. This idea wi l l be invest igated later in th is chaprer.

The angular dist ibut ions for three of the higher energy states of

) f ithe ""Ne resj-dual nucleus are shovrn in Figure 20 along vr j - th the

predict ions of DWUCK assuming the same si-mple model . The calculat ion

for the 1.63 MeV 2* , tate fo l lows the exper imental data to a reasonably

good degree, displaying the same general behavior, especial ly at forward

angles. However ' the data and. theory do not agree wel l at a l l for the+

4.25 MeV 4 state. This lat ter resul t is very surpr is ing, consic ler ing

the f i t to the 0* and 2*, which are in the same rotat ional band. Since

the only essent ia l d i f ferences in the t reatment between these states are

their energi-es and rotat ional angular momenta, one would expect that a

simi lar t reatment would give a s j -mi lar ly good resul t . The resul t j -s

especial ly surpr is ing in v iew of the fact that the 4+ state consistent ly

had the highest nurnber of counts of a l l observed states. Stat ist ical

count ing errors would be expected to be smal l . One of the more str ik ing

aspects of the data is that the cross-sect ion appears to drop of f at

fat forward angles. This fact suggests that perhaps a di f ferent angular

momentum transfer would y ie ld better resul ts. A t ransfer of As=2, wi th

L9.=2r4, and 6 coupled to AJ=4 were tr ied, but these calculat ions gave

no signi f icant improvement to the f i t over the zero spin-transfer case.

The 5.63 MeV 3 state is also a rotat ional state wi th K=0, but in a

di f ferent band. Here, again, theory and exper iment agree very wel l .

Higher states were observed or resolved only infrequent ly, and wi l l not

be di-scussed.

Z4ILg(d, 6Li) 20Ne higher

toMc(D,tLi) 'oNE 4c MEV

?+

DWUC K

cct,

d):t

zoFUL!aaaoLJ.(J

oooC,M, SCATTTRING

states and DLIBA f i t

ooooANGLE^, DEGREES

Table VII I , below,

angular momenta used in

for each state.

TLte

Final state

l ists the t ransferred

the calculat ions and

72

orbi ta l and total

the spectroscopic factors

TABLB VII I24. R

'n/ a vT. rUV j_yrg ( 0, L,L ) Iv e Hu.etLon

1TJ A,q, AJ

Payunetez.s

S do / {t max (ub I sr)CX,

0 .00

1.63

4.25

5.63

-t-0'

2+

4+

3

.LA2

.537

.560

.3 26

235

249

425

135

0

2

4

3

n

2

4

3

24,. ,3_, 7^ .204 ,4 The " ' l tq (" Her ' Be )"" Ne R

Another type of a lpha-transfer react ion is the (3H"rto") react ion.

This exchange was perforrned on 'Our, as the react ion has already been done

at 30 MeV on 200t"

by DeLraz, " , ^ t .?2

al though anglular distr ibut ions

have only been taken for the ground. state of t6o.

The react ion has also

been performecl on 'o*r by Pisanor et " t99

but their count ing stat ist ics

were extremely poorrand the main thrust of their r ,vork is involved with

a coupled-channels t reatment of the compound processes that dominate at

their 25 .5 MeV energy.

I t has already been seen that mult i -step processes may play a role

6in the (d '"Li) react ions studied in that unnatural par i ty states are

observed.. Cotfot t63 quest ions the d. i rect character of the (ar6l i )

react ion in general , and concludes that there may be substant j -a l contr ib-

ut ions f rom mult i -step processes in the react ion at a l l energies. This

is supported by most workers, a l though the high energy invest igat ions

of Djaloeis, eI uL.46 show on1-y states which would be populated by

di . rect interaet ions, in opposi t ion to McGrathts ear ly f inding".30

73

However, their DWBA f i t does not represent the data wel l aL al l .

On the other hand, theoret ical predict ior ,"64 and the work of

. 22,65 3 7Detraz show that the (-Her 'Be) react ion is more l ikely to proceed

vj-a a direct a lpha pickup. I t would be r ,sorthwhi le, then, to compare

these two react ions on the same nucleus.

?7The ("He, 'Be) react ion is di f f icul t to do both exper imental ly and

theoret ical ly. The 7O*

nucleus loses energy in matter so rapidly that

very th in targets and AE detectors are required. In addi t ion, both the

0.00 3/2- and 0.43 L/Z- states of 7u"

rr" detected and seen as separare

peaksr so good resolut ion is also required.

7The 'Be rnay be thought of as an alpha parLic le wi th spin zero and

a

a spin L/2 'He in a relat ive P state. Indeed, the 3/2- and Llz- states

can be thought of as only di f ferent coupl ings of orbi ta l and spin

angular nrcmenta. There is considerable evidence that these two states

have the same spat ia l wavefunct ion separated only by a smalJ- spin-orbi t

- 66,67torce. -

However, theoret ical comparj-sons are rnade more di f f icul t

because most DWBA codes, including DWUCK, and most coupled-channels codes,

including CHUCKT make use of the zero'range approximat ion, which assumes

that the t ransfer interact ion has a del ta- funct ion behavior. In the

-6-8case of a -Li or a "Be, these approxirnat ions have less of an ef fect

because these part ic les are S-state combinat ions. But a7r" is in a

relat ive P-state, and the zero-range interact ion occurs at the point where

_77the 'Be wave funct ion goes to zero I fhe 'Be is also a much more di f fuse

part ic le than ei ther of the othersr so that zero-range calculat ions

become real ly suspect.

Fortunately, the DWBA code LOI A assumes an interact ion of f in i te

range' and has been used in th is work for theoret ical comparisons. I ts

74

use was kept to a minimum, however, because it is twenty t j-mes more

expensive to use than DWUCK.

A spectrum for the'ot*(3H"rtu.)20r. react ion is shown in Figure 2L.

The doublet nature of each of the peaks, corresponding to di f ferent 7Bu

states' is easi ly seen. Peaks are v is ib le and easi ly ident i f iable up to

the 6.72 0+ state of 20*. .

The 4.g7 Mev 2 state is seen here again, as

6in the (d ' "L i ) react ion, but in th is react ion i t is not a par i ty- forbidden

direct t ransi t ion. The orbi ta l angular momentum g."=L of the 7Bu

"rrd

a transferred orbi ta l angular momentum of Lg"=Z can combine to form the

required 2 state of 20oru.

Furthermore, i t wi l l be recal led that the

par i ty change is no longer restr icted in the f in i te-range DWBA to

A0Ari = ( -1) "- as in the zero-range DWBA .

Againr several sets of opt ical model parameters were t r ied for the

theoret ical calculat ions. There is a vast amount of data avai labl-e for

) / , a

the --Mg +'He elast i -c channel , and i t is not surpr is ing that the values

which gave the best f i t to the 20*.

ground state were those of Drrhr68

which were done at an energy c losest to that of the present work. The

resul ts did not appear to be very sensj- t ive to the choice of opt ical

model parameters for the elast ic channel .

By compari-sont there are ef fect ively no opt ical potent ia ls avai lable

- ?f i 7 7for the -"Ne + 'Be exi t channel , probably due to the fact that 'B. beams

are not yet avai lable. However,

" ,

beams have been run, and i f i t is

assumed that the strong interact ion force is charge- and isospin-

independent, then 7" i

opt ical potent ia ls may be used to a fa i r ly good

degree of approximat ion. Perhaps a calculat ion s imi lar to that done

f or t r r latsonrs 6r,

potent i " l "51 would y ie ld acceptable 7u"

potent ia ls as

wel l . Again, the 20*.

+ 7

u" potent ia l done at an energy c losest to that

of the present work gave the best resul ts. This potent ia l is reported

J

LOLO

tt

tI

JJIJzzr(J

i?-m, \bIJ

f:ne +, g?'v

-\--z-g- l+O

()

-sC\J

'S'9 +O+z cg' l

z9'gg z'gz L '9

- lSNNVFl f Uf c Sr Nn0l

by Detraz, . t "1.65

was highly dependent

smal1 changes.

76

However, for th is channel , i t was seen that the f i t

on the choice of parameters, and very sensi t ive to

The angular distr ibut ion for the 'o*r(3H", tu.)20*.

0+ ground state

react ion is shornrn in f igure zz. The distr ibut ions for both the 7""

z/z-7

g.s. and 'Be L/2- 0.43 MeV states are shown. In Keeping with stand.ard

pract ice, these wi l l henceforth be cal led the 7u.(o)

and 7u.(1),

respect ively. Resolut ion was about 395 KeV, so these states were just

barely resolved. The angular distr ibut i -ons ran from 12.5" to 90"

in the laboratory system, and the unximum value of the cross-sect ion for

?o 7Ehe -"Ne g.s. and 'Be(O) was 66 ub/sr.

The f in i te-range DWBA predict ions are also shor,rn in Figure 22.

Consider ing the approximat ions made, the f i t is qui te reasonable. At

far back angles, as wel l as far forward, the f i t is very good, and only

in the 4A" to 60" region does the predict ion vary considerably f rom the

exper iment. Also, LOLA predicts that the 7n"(o)

and 7r"( t )

cross-

sect ions should have the same behavior, wi th only a constant factor of

2 between the due to the zJ+L magnet ic di f ferences in spin between the

3/2- and Ll2- states, s ince they are otherwise essent ia l ly ident ical .

I t can be seen, however, that the 7u.( t )

data is s l ight ly lower than

expected at forward angles. Indeedr orr the average, the 7gu

(O ) /7 Se (1)

cross-sect ion rat io is L.26!.60, much lower than would be expected.

The extent to which the data fo1low this s impl-e relat ionship is an

indicat i -on of the val id i ty of our assumptions. At center-of-mass angles

less than 30", where direct processes would be expected to dominate any

compound processes, th is rat io j -s almost exact ly ZzL, in agreemenL with

this expectat ion and with other exper imenters. At backward angles, the

rat io r ises to about 1:1, but the distr ibut ion keeps the same shape.

77

E.LN

@

-

ZaFUlJ-Jf)

(naoK-(J

'oMc$r E,tBr)'oNr c.co e+

c 20 40 tu# 6s i00CM. SCATTTRING ANGLE

Fig. 22 'O*r( '*" , tu.)20N. ground state distr ibur ion ancl f i t

+

L OLA

t

+tB. ,

++

78

Other exper imenters have noted r ises to as high as L:2 at much lower

. 60,62energr_es.

Figures 23 and 24 show the exper imental angular distr ibut j -on and

f in i te-range DWBA calculat ion for the 1.63 2*, 4.25 4*, 5.75 1 and 6.72

0+ stares of '00r. for both rhe 'otr(3H", 'u"(0))20*r. and 'o**(3H", t r . ( r ) )20ne

react ions, respect ively. In al l cases, the behavior of the cross-sect i -on

is approximately the same, dropping more than an order of magnitude over

the range of angles, and showing almost a l inear relat ionship on a

semilog plot wi th l i t t le structure. In mosL cases, especial ly for the

1.63 2+ state, the theoret ical predict ion fo l lows the exper imental data

qui te wel l . The only gross di f ference between exper iment and theory

occurs, again, in the 4.25 4* " tate,

as before in the 'O**qd,6t , i )20u.

exper iment. The 7g"(O)

data has a srnal ler s lope than theory would

predict , and the slope is larger for the 7U.( t )

data than theory.

The top curve of Figure 25 is the exper imental data and predj .ct ion

for the 4.g7 I" IeV 2 state for th is react ion and the 7U"(O)

state, and

the middle curve shows the same state of 'O*. , but for the 7g"(1)

part

of the react ion. For scneunknown reason, resolut ion was much poorer

for th is state, and the 2 doublet was not always resolved.. The upper

two curves show only the exper imental data for those states where the

doublet was resolved, and the lowermost plot shows exper imental data

and theory for the sum of the two 7U"

states over al l measured angles.

I t appea.rs that theory and exper iment agree rather wel l again for th is

state, wi th the except ion of only a few data points.

Table IX is a summary of resul ts obtained from this react ion. I t

shows, for each state, the t ransferred orbi ta l and total angular momentum,

the spectroscopic factor for both 7u"

srates, and the tu.( q/7 Be(1)

'oMc(.HE,'BefiNr

I

aa

a

5,78 l -

+ +-o

6,72 0f

+_ LoLA

1.63

TY

(n

m:t

zoFL)r!aaaoK.U

o?o40C,M. SCATTERI NG

Fig. 23 z4MB€ He,7 ne)20Ne Z/z-ANGL f -higher states

DEGREESand DWBA fit

'oM c(tn r,tc [,,)tnN roN 55 MEV

ro?

t

I

: f

o1,53 2r

\ r r+

t I

t

Ia

4.2 5 4+

^

+ +5.79 l -

+lrA I

to

foo

NC(n

d:t

ZoF(JLtJaaao(.

U ++

o

Fig.

ato

6.72 0+

LOL A

C.M. SCATTERINGANGLE ̂, DEGRTTS24 'oW(

3"., 7u") 2o*.L/2- higher srates and DWBA fir

ACa

dl=t

zol--Ur l

YIU)

LO(,

otU

to M c(tH r,tB E)toNeTo 4,97 2-

tBEo

tBE,

SUM

+LOLA

C,M, SCATT.ERI NC ANGLI ̂, DEG REESFig. 25 '*Mg( 'H", '8.)"Ne 4.97 Mev and DWBA f i t

82

Given the parLial ly indirect nature of the (ar6I , i ) react j -on, and

the fact that 78"

exists in a P state, the (or 8g.) react ion should be a

very good tool for studying alpha cluster ing. The 8Uu

rr . r"1eus is easi ly

treated theoret ical ly as two alpha part ic les in an S state to an

extremely high degree of agreement wi th exper imental data . r '2L,27,69'70

It has been seen in Chapter 2 that the di f f icul ty ar ises in the

detect ion and ident i f icat ion of the 8u".

Since the nucleus d.ecays in a

very short t ime to tv io alphas, and since elast ic alpha scatter ing is

going on at the same t ime, i t is necessary to detect both alphas

simultaneously and ident i fy them as 8u".

The methods to do this were

descr ibed in chapter 2. In addi t ion, the two alpha part ic les f rom 8U.

ident i fy almost ident ical ly to a ' r r , so extremely good resolut ion is

required. Fortunately, the 1or 7t

i ; react ion on 20*"

has a Q-value of

-2L,5 MeV, compared to the 2Or.1or 8n.;

Q-value of -4.8 MeV.

)N R 1AThe -"Ne1o,r"Be)*"0 react ion was at tempted with several d.etector

conf igurat ions. 8U" events were seen, but the gas target and high

elast ic count rate and subsequent pi leup degraded resolut j -on to such a

degree that no states were ident i f iable. In addi t ion, the high accidental

coincidence rate ' coupled with the snnl l cross-sect ion and detect ion

eff ie iency increased the background to a point where the events of

interest were almost masked by noise.

In the hcpes of improving resolut ion, the exper iment was repeated

for the th in 'Otr target. Since there were no windows or long gas

pathso the resolut ion improved signi f icant ly. Figure 26 shows the spect l lm

87for "Be and 'LI events at a laboratory scatter ing angle of 30o. By

not ing the channel locat ions of the ground states of the 'O*f( o, cr) 'Otf

rat io, both on the average,

rat io at forward angles is

al l angles, the average of

and at forward angles. The average

not very di f f erent f rom 2.00 at 1.83,

the rat ios is only L.32.

84

of th is

but for

Swnrnazry

sLate A.a, AJ

TABLE IX

of Results of the

s0(o) sC,t (1)

'n rn {3 Hn,7 an)20 Nn

7n" (o ) /7 s"( t )<30"

2.47

1.61

L .66

1.56

2.57x

2.02r\

1.34

1 .93

Reaction-7 -7'Be (0) / 'Ye1t1

average

L.26

L.26

L.36

L.7 6

1.37*

1.41*

.79

L .32

+0.00 0' 1 1

+1.63 2' I 1

+4.25 4' 3 3

4.97 2 2 2

5.62 3 2 2

5.78 1 0 0

+6.72 0 1 1

average

;knot always resolved

,0423

.0959

.LLz

" 06L2.

.0604

.05 20

.0718

.052L

.108

.130

.07 64

,09L4

.03 42

.0820

2.4. R 'n4.5 The "=Mg ( or 'Be)" Ne Eeaet@

Part of the or ig inal purpose of th is research was to invest igate the

- 8 ?tL )n(o '"Be) react ion on ei ther ' -Vlg or "N", or both. Ind.eed., fu l ly hal f of

the exper imental ef for t was devoted to both at tempts. I t \^ /as a grave

disappointment to th is worker that the react i -on could not be observed

)n )/,on -"Ne, and the --Mg studies had to be abandoned, because heretofore

angular distr ibut ions for the 1or8n.; react ion have only been taken by

one researcher, Gordon WorniakzL, and only on si-x very l ight nuclei for

Z=5 to Z=8. This study would have been a s igni f icant contr ibut ion to

exi-st ing knowledge.

B5

?IL 6 ))and - 'Mg(s ' "L i ) - -Na react ions, i t was possible to calculate the expected

channel locat ion for the 'or ,*1o,8n.;20*u react ion ground. state. The smal l

peak shor,rn by the arrow i-n the figure occurs within two channels of the

expected locat ion. There are only eleven counts j -n th is peak, which

represents more than three hours of count ing. Some simple f igur ing shows

that for a minimum of 100 counts in the ground state, to give a stat ist ical

uncertainty of L}Z, a minj-mum of 30 hours per angle would be required.

Since this exper imenter lacked the bearn t ime funds for a 10-15 day

irradiat ion, i t was necessary to terminate the exper iment.

However ' some informat ion can st i l l - be extracted from the data

avai lable. The computer code EBGAS gave a detect ion ef f ic iency of 0.045

for the detector conf igurat ion used. This al1ows one to calculate a

laboratory cross-sect ion of 5.7+I.7ubf sr for the cross-sect ion at th is

angle. This value should be treated as an upper l imi t to the cross-

sect i -onr 3s i t was di f f icul t to renove the background from the p€k with

so f ew counts. A f in i te-range DI,EA calculat ion was made using the

conputer code LOLA and the 'O** + o opt ical potent ia ls in table V.7I

The same potent ia ls wi th twice the wel l depth were used for the 20tt"

+a"Be channel . The same binding potent ia l appl ied in other studies on

?IL- 'Mg \^7as used. These calculat ions give a spectroscopic factor of 0.076

for th is react ion, i f i t can be assumed that the D' ,^IBA f i ts the data

perfect ly at a l l angles, and there is no transfer of orbi ta l angular

momentum. This calculat ion also al lowed the extract ion of the total-

cross-sectLon, to be used later in th is chapter.

I t seems appropr iate to comment on the method of detect ion of BUu

at th is point . The i r ra jor d i f f icul ty encountered was a 1ow count ing rate.

The detect ion ef f ic iency for sma1l sol id angles using single AE-E

86

telescopes is shown in reference 40 to be proport ional to the sol id angle.

This means that the count rate is then proport ional to the square of the

sol id angle, i .e. , to the square of the detector area exposed and

inversely proport ional to the fourth power of the target-detector distance.

Resolut ion, moreover, is proport ional to the width of the detector and

inversely as i ts distance. For reasonable count ing rates, not only wi l l

resolut j -on degrade, but the accidental coincj-dence rate and pulse pi leup

wi l l a lso increase, resul t ing in more background and poorer resolut ion

st i11. I t should be noted that two low-energy alphas in accidental

coincidence wi l l ident i fy l ike a 8u..

The use of a s ingle AE-E telescope

for the detect ion of Bo.

has only been used once befor.70, and in that

1) Rcase the (*-Cr"Be) react ion was being studiedr so that no o-o,

accidental coincidences were seen. I t may not be reasonable or even

possible to use this method of detect ion for the (o,88.) react ion.

4 .6 Wtt , - step Proces seg

In the case of both (ar 6t i ) react ions above, the unnatural par i ty

2- states of 160

and 20*u

were observed, even though these states are

forbidden by the direct t ransfer of a spin zero alpha part ic le.

Whi le the populat ion of these states is weak, i t is st i1 l large

enough to cause concern over the val id i ty of the direct a lpha- t ransfer

idea. The weak populat ion of these states and the 4 state in 20*"

6from the ("Lird) react ion (which is the t ime reversed react ion to one

done in th is work. ) is f requent ly c i ted 19' 30

as evidence that the

tuo (ut i , d)20*" has a s lmple direct nature. comfort , g

"1 .613 h".r . done

exper iments which would suggest that th is supression may be merely the

resul t of rotat ionaL band select iv i ty in a mult i -step process.

87

Three of the more 1ike1y possible explanat ions for the exci tat ion

of these 2 states by the (ar6l i ) react i -on are: (1) compound nuclear

processes, (2) t ransfer of the four nucleons in an S=1 non-alpha

conf igurat ion, or (3) inelast j -c processes in the entrance or exj- t channels.

Whi le calculat ions of compound nuclear processes have not been performed,

the lat ter two possibi l i t ies have been addressed direct ly.

Udagawa and o1".rr72 have shor,rn that, j-n the l irnit of a vanishing

spJ-n-orbi t interacLion in the opt ical potent ia ls, the mult i -step cross-

sect ion involv ing the t ransfer of an S=0 cluster to an unnatural par i ty

state goes to zero at 180o and 0" scatter ing angle. Even addj- t ions of

qui te reasonable spin-orbi t interact ions st i l l resulEed in very sur,al l

forward angle cross-sect ions in the case of (pr t ) react ions.

Recal l that in the der ivat ion of the DI,EAT €rs discussed in chapter

3, one approximat j -on made was that the elast ic scatter ing of the

entrance part ic le domj-nated al l other processes to the extent that a l l

other processes could be neglected, and the total wave funct ion over al l

channels could be replaced by the elast ic scatter ing wave funct ion. I f

one is going to consider possibi l i ty (3) above, th is approximat ion should

be modif ied. The next most dominant processes are those inelast j ,c

processes which exci te the low-ly ing states of the entrance target

nucleus and resul t in inelast ic scatter ing wi th no part ic l -e exchange.

I t seems reasonable, then, to consider mult i -step processes coupl ing the

lowest- ly ing states in the entrance channel .

The mult i - -step process was calculated using the coupled-channels

57code CHUCK"' and inelast ic exci tat ions in the entrance channel . The

dipole and octupole deformat ions 8. , and g, . of the 20*u

and 'o*r nuclei ,z4

necessary to extract coupl ing strengths to the 2* and,4+ states in each,

B8

were taken from the proton scatter ing data of Swinarski , et aL.73

Figure 27 shows the energy level d iagrams and coupl ing schemes used

to calculate the exci tat ion of the 2 states in 160

and t0*" .

The 2+

+and 4 states were exci ted by two-way coupl ing f rom the ground state of

the entrance channel and from each other. The 2 in the exi t channel

was exci ted f rom the 2* and.4* Uy a t ransfer of 1 and 3 uni ts of angular

momentumn respect ively. I t should be noted that the spectroscopic

strengths for the alpha-pickup from the 2* and, 4+ states were taken to

be equal and of the same sign. With th is s impl i f icat ion, the calculat j -on

can be rel ied on to produce only qual j - tat ive features of the react ion.

The resul ts of rhese calculat ions for rhe 20*.(a,6l i )160

(2-)

react ion are shor^m in Figure 28. The calculat ion shows a drop in the

cross-sect ion which is also present in the data. By comparison, to test

the theory that a spin t ransfer of l i . may cause the same effect , a zero-

range DWBA calculat ion was made assuming a direct t ransfer of four

nucleons w-i th spin l f1 . These resul ts are shown in Figure 29. These

calculat ions show substant ia l increases in cross-sect ions at forward

angles which are not present in the data.

The data and calculat ions for the 'otr(d,6r, iy20N. (z-) reacr ion

are shown in Figure 30. The sol id l ine is the resul t of coupled-channels

calculat ions as above, wi th the two dashed l ines represent ing a s ingle-

step DI{BA calculat ion t ransferr ing a spin- l c luster. I t is readi ly seen

that the coupled-channels calculat ion f i ts the data extremely wel l , whi le

the direct calculat ions bear 1i t t1e resemblance to exper iment. Agai-n,

note that the coupled-channels calculat ion drops of f at forward angles.

These resul ts, whi le not surpr is ing, do raise quest ions as to

whether mult i -step processes are important in the remainder of the states

89

2-RTCOIL

NUC LEUS

TARGE 1NUC LEUS

Fig, 27 Energy levels and coupl ings to 2 states

90

roo

to

i ,^.ld

111l\m:L

gT

\b

T

o,l

o ?o 40 60 80 l@

Fig. 28 20**(a,6l i )160

8.87 2 and coupled-channels f i t

t oNE(D,t r , ) ' to ro 8.87 Me V 2-

r t

COUPLED CHANNELS

t

t! . t {

i

o,ol

O c.M,

toN.(O,tL ' ) ' tO ro &87 MeV 2-

Il -=f- l

DWBA AS=l ,AL=l

DWBA AS= | 'AL = l

t

r--,iF{ I

\\

JI

T+I

o ?o 40 60 80 loo

9L

) i 6 16r ^ .A

Fig . 29 ' "Ne 1dr "Li) -"0 to 8 . 87 2 and DWBA f i t wi th A s=1IU\J

ro

d,

{d)a

G.b

b-b

o,l

O c,M.

o.ol

92

9as / ar1 '^. / N Ol I l l S - SSOUI

]J

'r{tH

ar-loH(a

d

oIc)

t-'lg

oCJ

Fr-.i

H

d

B

#

d$

Ic\

|.\o\.$

o+J

q.)

&N

'r{

drJ

b0

-TN

ct')

b0'r-l

oo

, / ,1

-/ (t

td

Ct

IC\J

t\O)$Ptr,

,n7lY:-.

J\OG

O\---"/

(9='/-

vC\I

Gtr.Jt!,G,(9trj

t-9

r!J(9Z

IzELrJFF

Ut|)

jU

+i l l l \ f ,

i r ln i

jdTl l l l+JJ+<< o

)<yYfJU()f f r=3ioou

t - /

-);--

'lI

/

(\

)

(

)

I

\

\

'1^l)

93

for alptr ,a- t ransfer react ions. I , lh j - le i t was not possible to do coupled-

ctr ,annels calculat ions for a l l states of a l l react ions, i t was possible to

perform them for one addi t ional state in each react ion for which the

coupled-channels code CH.UCK is appl icable.

First of a l l , i t is instruct ive to see i f entrance-channel coupl ing

affects any of the elast ic and inelast j -c scatter ing processes. In

Figure 11r the coupled-channels calculat ions are also shovrn for the

ground state of the 20*"(drdt ;20t t . .

Forward of 30o center-of-mass, the

predict ion of CHUCK and DI/UCK are ident ical . However, at angles greater

than 35", the coupled-channels calculat ic ln gives a much better representat ion

of the exper imental data. A1so, in Figure L2, the coupled-channels cal-

culat ion is displayed alongside the DI,JtsA, and in both cases seems to

f i t the exper imental data s l ight ly better. I t must therefore be concluded

that mult i -step ef fects are important in the entrance channel of 200,.

* d

react j -ons at th is energy.

Addi t ional coupled-channels calculat ions were made to see i f these

mult istep ef fects would af fect states in the exi t channel which were

otherwise wel l -descr ibed by "

d i rect t ransfer process. Figure 31 is an

energy- level d iagram showing the coupl ings used to calculate the t ransfer

+to a 0 ground state. The sane two-way coupl ings in the entrance

channel used for the exci tat ion of the 2 states were used in th is

calculat ion. A11 three states were then al lowed to t ransfer to the 0+

ground state of the residual nucleus. Again, a l l spectroscopic strengths

for the t ransfer were the same value and sign, so only qual i tat ive

resul ts wi l l be obtained. Figures 32 and 33 are the angular distr ibut ions

')nfor rhe "N"1d,6t , i )16n 0+ ground stare and rhe 'o**(a,6l i ) t0*" 0+ ground

state, showing exper imental data, coupled-channels calculat ions, and for

94

O+

AL =/

TARGETNUC LEUS

Fig. 31 Energy levels and

A L=4 RECOI LNUC LELJS

AL=Oor nEcr)

couplings to 0* ground state

95

oN

aLrJLr-JE.(9LrJO

?o trJ( .OJ

L)z

(\cn

'r{h

10ooousTerr NO|I ISS SSOUI

(9ZE.tr-JFF(Jo

oI

o

o\il

(/)F{

oH

Hdt

oI

Ft

a)F{g9

O

t?t

ld

$

E

t+{oa

oa

'r{trftl

g

tioo

o

b0

+O

\orl

'r{Fl\o

€0)zN

o.

(J

OtIIl

ol\II

I

it/

/.

a\

/

+v/}.t

++C\N

$t++oo

Y YYu c.l crf : :3:ExOU(J

//

III\\\\\

tI

I

/I

\

Y

I42'f"

o

o

f?-{o

fi','l l tt ]

-1i

\\

./rt?/ /

f (r)

\ )

/I

\

LJ

o\f,

+oooc

or.lD

-J\^o

rD

cLJJZ.

ocv

96

\

-a- t

\ \

-__

+

\.

+oCocoF

L!Z.o

N

z^.

j\0

a\

O:--Z

(9

\rr\'

s=\_1=/)) LO

IJLrJE.(9tJ.JOt

IJ)(9Z

(9zt-lJJFF(Ja

5(J

+$

0+c\lLT

v+o

L)f

U

+c\l0+o

r.t :)3ro(J

o

\1-s..

--. \_\_.*--_\o

tr,

c\r\ .z?.

-liti--./r'

\..

{-

. ( \

\*

+/

5*?-z-

a

-la)dH

cde

IT,0)

rl

@

H

tr

o*\OE;

t{i

oH

o(rl'r{Hd

a

U

v.C')

b0

+

o+J

a)vC\

/^\ 'r{

ru6

b0A*$C\

c.)cO

b0.r{

r

o9as/ea - N0l I l lS SSOU l

97comparison, the DIAIBA calculat ion f rom DWUCK. Consider ing other coupled-

channels resul ts above, i t is somewhat surpr is ing to see that in nei ther

case do the coupled-ehannels calculat ions give an appreciably better

representat ion of the exper iment. Indeed, in both cases, the f i t is

s l ight ly poorer. The conclusion to be rnade in th is case is that the

direct process domj-nates the mult i -step process in t ransfer to the

ground state.

4 .J__ Compaqi.s ons Wtween Reactlons

I t is worthwhi le to compare the var ious react ions performed in th is

work. In the comparison of cross-sect ions, in order to mj-nimize angular

ef fects, the total cross-sect ions wi l l be compared instead of the

di f ferent ia l cross-sect ion at any angle. The total exper imental cross-

sect ion for the 'O*r1o,8n";20*. react ion was cal-culated from equat ion

r 33) using the known exper imental spectroscopic factor and Lhe total

)n ? 7 16cross-sect ion calculated by the DI, f ,BA. The --Ne(-Her 'Be)^"0 total cross-

sect ion was est i r tsted from reference 65 for comparison purposes.

The comparisons between var ious alpha-transfer react ions are

sui lErarLzed in Table X. Total cross-sect ions and spectroscopic factors

for ground-state to ground-state t ransi t ions are shovm.

I t may be seen that the total cross-sect ion for a lpha-transfer

react ions on 'Ot* decreases rapidly wi th increasing mass of the

project i le. I t is unknown whether th is is due to the var iat ion in mass,

or whecher i t i -s a k inemat ic ef fect due to an increase in energy of the

incident part ic le. For the 'O*", target, the same effect is seen, but

when the two states of the 7

B" are added, the cross-sect ions for the two

react ions become about equal .

9B

TABLE X

Companisons of AlpLw-lTansfer Reactions

Rea ct io nG.S.-G.S.

Total o o/p (E) Soexpt So shel1 modelub

'o*r1d, 6r, i) 2ort. 44L

'o*r(3Hu, tu") 2o*"

O LzL1 89.3Both 200

'o*r( o, Bn.) 2oN.

25 .4

'o*r avera ge

2oN.(a,6l i )16o 334

2o*.(3H.,7nu)160

0 225I 116Bo th 34L

20_,Ne average

L526t244 .0681

L99L

16001650L6L7

.434

, L02

.0423, a52r.0/47 2

.07 6

.50 (approximate)

L57 7

1170L7 2A1350

L637

, oLgT 5

L747 !2L2 .47 .Lg265

99

No such rapld decrease, however, is seen in the rat io of the total

cross-sect ion to the densi ty of phase space. These rat ios are given in

the table in arbi- t rary uni ts. The standard deviat ion in the value for

) lLthe - 'Mg target is only L6" l of the aver i ] .ge. I t is only 122 of the average

in the admit tecl ly srnal l sample for the 20N.

target. Recal l that i t was

shown that th is rat io wi l l be proport ional to the t ransi t ion matr ix

element. I t appears, then, that the probabi l i ty of the alpha transfer

taking place to the ground state is more or less independent of the

project i le in th is range of project i le masses and energies. IL can also

be seen that the rat io for the t0r"

nucleus is s l ight ly larger than that

,/,for the ' -Mg nucleus, which might be interpreted as evid.ence that alpha

cluster ing is present to a s l ight ly larger degree in the 20*"

nucleus,

j -n agreement wi th expectat ions suggested by the doubly-magic character

of the c luster and core in th is nucl€us.

The f indings of the previous paragraph are supported by the

exper imental spectroscopic factors. The 'Ot* spectroscopic factors, based

on an alpha cluster assumption, al l agree within a factor of two,

and are universal ly larger than the she11-rnodel predict ions interpolated75

from the calculat ions of Yoshida, €t a l . However, the average of these

exper imental spectroscopic factors is about three t . i :nes the predicted.

On the other hand, the 20*u

exper imental spectroscopic factors agree

very wel l wi th one another, and are about twice as large as that predic-

ted f rom she1l nrodel calculat ions. Whi le at f i rst g lance, the agreement

of theory and exper iment may be poor, i t is actual ly rather good as

such comparisons go.

Last ly ' the exper imental spectroscopic factors f rom this work show

a rapid decrease with increasing target

theoret ical conclusions of Gutbrod, €t

work of Steele, et ^L. ,64

whi le i t is

Detraz, et ^r .6J

ntass. This

- 65.7 531., - - ' ' - and

in opposi t ion

100

fact supports the

the exper imental

to the f indings of

101

V. SUMI',IARY, CONCLUSIONS, AND SUGGESTIONS FOR FURTHER STUDY

Alpha transf er react ions have been performed at mecl ium energies

suff ic ient ly high to al lor ,o direct processes to dominate on 20N.

and

?L- 'Mg targets, whj-ch are bel ieved to have fair ly high cr-structure parentagles.

Both zero-range and f in i te-range DWBA calculat ions were done which

assumed the target nucleus to be a stable core plus a four-nucleon

cluster in a state resembl ing an alptra part ic le, and the resul ts were

compared to exper iment.

In the (ar 6l i ) react ions, states were populated in a manner consistent

with a predominately direct interact ion, a l though in each case a s ingle

unnatural par i ty state was seen with a faLrLy large cross-sect ion,

indicat ing that mult i -step processes may contr ibute s igni f icant ly. Both

these f indings are consistent wi th ear l ier work done by McGrath30 and

67Cornfort . "" DI^EA analyses show that the cross-sect ion angular distr ibut ions

for states which can be populated by a direct t ransi t ion are consistent

with the direct t ransfer of an S=0, T=0 alpha part ic le. A comparison of

zero-range DWBA and f in i te-range DWBA calculat ions show that the zero-

range DWBA is suf f j -c ient for a reasonable f i t , but that the f in i te-

range calculat ions give better resul ts, as expected.

The transfer of a four-nucleon cluster i -n an S=0, T=0 alpha- l ike

state is conf i rmed by a coupled-channels DI^IBA calculat ion for unnatural

par i ty states. This lat ter resul t suggests that ef fects involv ing two

steps might contr ibute s igni f icant ly j -n the t ransfer of an alpha to states

which can be populated by a direct t ransfer. Comparisons to elast ic

scatter ing data showed that entrance-channel coupl ing was signi f icant,

but no improvement in the predict ions was evident for ground-state to

ground-state alpha pickup react i -ons, suggest ing that di rect processes

dominat e .

r02

A11 states observed in the 'ot*(3H",7p," ;20Nu react ion again conf i rm

that the observed character ist ics of the react ion are consi-stent wi th

the direct p ickup of an alpha- l ike part ic le. This fact is supported by

the comparisons cl f the angular distr ibut ions of the exper imental cross-

sect ions wi th that predicted by the f in i te-range DI^IBA. The behavior

of the cross-sect ions for the two 7U"

states observed was very s imi lar ,

and, as expected, the rat i -o of the cross-sect ions for these two states

was close to that which is predicted for 7

Bu as a 3H"

plus an alpha in

a P state.

The spectroscopic factors determined from the exper iments do not

appear to depend signi f icant ly on the nature of the incident part ic le

or i ts energy in th is mass and energy region, and agree within an order

of magnitude to spectroscopic factors calculated from shel l -model theory

based on an extreme alpha-cluster model for the target nucleus. The

content ion of Steele64 and GutbrodT5 that the spectroscopic factors

decrease sharply wi th increasing target mass is supported in disagreement

with the f indings of D"t t^r .65 r f th is fact is t rue, then alpha-part ic le

nuclei in th is nass region should exper ience decreasing cluster

parentages with j .ncreasing mass. This f inding j -s supported to a sma1l

degree, al though the resul ts are somewhat inconclusive, by a compari-son

of the rat ios of the total cross-sect ion to the densi tv of avai lable

phase space.

opt ical potent ia ls for the 20*.

+ d channel at 40 MeV were obtained.

Last ly ' a method for calculat ing the ef f ic iency of detect ion ofo' "Be nuclei for var ious d.etector conf igurat ions wi th ei ther gas or sol id

targets was developed.

103

It can be seen that the alpha-transfer react ion is a useful react ion

for obtaining spectroscopic informat ion and nuclear structure informat ion.

Due to the large mass of the exi t part ic les, i t is most useful on l ight

nuclei whose energy levels are wel l separated. Because of the t ransfer

of an S=0, T=0 alpha part ic le, calculat ions involv ing the react ion are

easy to do and the angular momentum coupl ings are s i rnple.

Spectroscopic informat ion f rom alpha-transfer react ions is st i l l not

on a f i rm basis. Alnnst al l of the work that has been done has been at

low energies and with ei ther no DWBA analysis or only zero-range DWBA,

which does not al1ow the extract ion of spectroscopic factors. More work

needs to be done at h igh energies wi th a f , in i te-range DWBA analysis,

especial ly wi th the nuclei 20*"

and 44r i ,

both of which ?rave doubly-nagic

cores. More coupled-channels calculat ions should be done on react ions

exhibi t ing mult i -step behavior to determine i f other states besides

unnatural par i ty states include mult i -step ef fects. (orBU.; react ions

have only been fu11y invest igated for s ix targets to date, and the ( t ,7t i )

react ion has not been done aE al l . The former is potent ia l ly the least

controversial , wi th al l part ic les having spin zero and the 88"

in an S

state, and the lat ter j -s the isobar ic analog of the (3H.r tU") react ion

and the t ime reverse react ion Lo the (7Li , t ) , boLh of which have been so

popular wi th invest igators of a lpha-transfer.

I t appears ' fur ther, that the present work is the f i rst to direct ly

address i tsel f to comparisons of d i f ferent alpha-transfer react ions on

the same target. Addi t ional work in th is area is needed. A complete

survey of a lpha transfer react ions on several a lpha-part ic le nuclei

should yei ld much informat ion on the alpha structure of these nuclei .

The major di f f icul ty wi th the theoret ical aspects of th is work was

the inavai labi l i ty of good opt ical potent ia ls. The major i ty of opt ical

104

potent ia ls avai lable have a volume imaginary term as opposed to a surface

imaginary term. Furthermore, most potent i -a ls assume a real wel l depth

of -50 MeV, yet workers in alpha transfer studies seem to prefer a wel l

depth which is approxirnately -{Q MeV t imes the nurss of the project i le.

Opt ical potent ia ls for 78"

rrrd BB.

"r" essent ia l lv nonexistent. A

theoret i -ca1 calculat ion along the l ines of that done by l^ Iatrorr5l for

6"Li could possibly y ie ld acceptable potent ia ls of immense va1ue.

105

APPENDIX A

DETECTToN EFFrcrEIqcY ron Bnu

The detect ion and ident i f icat ion of a medium-energy BB.

part ic le is

somewhat di f ferent f rom the detect ion of other part ic les such as 6rt

or

7 R -1 6'Be. "Be is unbound by 94 KeV and decays with a hal f - l i fe of 2 x 10

seconds to two alpha part ic les. To avoid being confused with a low-

energy alpha part ic le, both the decay products musl be detected with some

reasonable degree of s imultanaei ty and ident i f ied as having come from a

B..be.

I f standard part ic le ident i f icat ion techniques involv ing a s ingle

AE-E telescope are used, then i t is possi-bl-e that the path of the center

of mass of the two alphas, which is the same as what the path of the 8r.

would be i f i t had not decayed, would intersect Ehe acEive port ion of the

detector system, and yet ei ther or both of the decay alphas would miss

the detector completely. Figure 34 wi l l assist j -n explaining this fact .

The top f igure is a veloci ty diagram. Vg is the veloci ty of the 8*"

part ic le and the path of the center-of-mass of the two alphas. The

breakup occurs at point 0. In the laboratory f rame, the alphas have

velocj . t j -es Vol and Y oZ,

but in the center-of-rnass system of the two

alphas, they each have Uot. The two middle diagrams show the center-of-

nass intersect ing the detector at the point marked bV E , but an inval id

event occurs because one or both of the alphas do noL enter the detector.

The bo ttom diagram, for compar ison, shows a valid event .

The detect ion ef f ic iencv is def ined as the rat io of val id events to

events where the center-of-mass of the two-alpha system intersects

act ive port ion of the detector.

al l

the

106

\

INVALID EVENT

INVA L IDEVENT

O ALPHA

a 8BE C.M.

VA LIDEVENT

RFig. 34 Breakup and detect ion of "Be

L07

A computer code named EBGAS has been developed to cal-culate th is

detect ion ef f ic iency for both sol id and gas target using a lv lonte-Carlo

techni-que.

The program works as fo l lows: Referr ing to Figure 35, the

assumed to be produced any,uhere aloug the l ine A B on the beam

is

the

gas target ' which is def ined by the acceptance angle of the col l iuntors

between the target and the detectors. A random number generator chooses

the locat j -on along AB. The random number generator also chooses the

dj-rect ion of the 8u"

b, assumi-ng the center-of-nass rv i l l fa l l somewhere

in the act ive region of the detector col l imator.

Since the 8u*

has a hal f - l i fe of only 2 x 10-16 second.s, which j -s

several orders of rnagni tude larger t t r ,an the t ransj- t t ime across the

nuclear diameter, but very snsl l compared to the t ransi t t ime across the

physical d imensions of the targets and detectors, the approximat ion is

made that the BU"

decays immediately upon being produced. Furthermore,

the direct ion of f l ight of the two breakup alphas in their center-of-rnass

system is assumed random and is generated by randorn numbers.

Ordinary non-relat iv ist ic k inemat ics then determine the spat ia l

posi t ion of the alpha part ic les at the f i rst and second col l imators.

Those events which do not fa l1 wi th in the acceptance area of the col l im-

ators are rejected. After a large number of t r ia ls, the rat i -o of accepted

events to total events is calculated and given as the detect ion ef f ic iency.

Several types of detector conf igurat ions can be used. A single

AE-E telescope nay be used with standard part ic le ident i f icat ion

techniques to detect 8u"

part ic les, as suggested in the paper by

l larney, ut "1.40

Figure 35 also shows other methods which can be descr ibed

R-Be

in

by this code. A 2-detector system, requir ing detect ion of an alpha

108

schemesFig. 35 BB"

detect ion geometry for

FRON TCOLLIMATOR

ESGAS and val id detector

DETECTORCOLLIMATOR

BEAM

DETEC

MOUNTI NGDETECTOR

109

part ic le in each of two detectors, e i ther s ide-by-side or top-and-bottom

in fast coincidence may also be used. In ei ther of these casesr €Lo addi t ional

requirement is made that one of the alphas enter one detector, and the

other alpha enter t ,he other detector. I t should be pointed out that

th is code wi l l be useful in comput ing ef f ic iencies for the posi t ion-

sensi t ive detect ion svstem of I rJozniak .2L'26'4L

In the case of sol id targets, the calculat ions are s impl i f ied. Here

the product ion of BB. j -s presumed to occur at a s i -ngle point where the

beam intersects the target fo i l , and no front col l imator is used. The

calculat ion is much faster in th is case, al though minor errors may

resul t f rom fai lure to consj-der the f in i te s ize of the beam spot on the

target.

The resul ts of th is code were compared to the resul ts obtained

using Harneyf s code EFFI4O for a s ingle AE-E detector system. There was

l i t t le s igni f icant di f ferenc, and i t is fe l t that e i ther code would give

acceptable resul ts in th is case.

A l is t ing of the program f o11ows. The program is wr i t ten j -n

FORTMN and is designed to be accessed from a Eerminal . Minor al tera t ions

in the program would permit input f rorn cards. The subrout ine ESCAT

for scatter ing k inemat ics and the random number generator RAIIDM are also

gi-ven. In the lat ter , the number 131071 should be replaced by the largest

single word integer the part icular machj-ne wi l l take i f machines other

than a DEC PDP-15/40 is used.

Last ly, i t should be noted that th is calculat ion process would be

useful for any react ion in which two part ic les are to be detected, as

long as the k inernat ics of the two part ic les is wel l understood.

110

' l

f ,- \

. i

, l ?

. i

I

111

LL2

a!.)

t

113

/ ' : - n t 'n

t ,

LT4

t : t r

CCC

115

FLNCT ICN ESCA T(Y1, t t?, l t3,M', , G,El ,THETA)FUNCTICN CO\4PUIES ENERGY 3F .MII I :D PARTIC. lElN [ iLCLEA' i REACTI0I ' {

RtAL P1,V?,M3,Y4E=[1+QAi3=F 1* y3 / Uq1 +FZ) / (M3+i l4) * L l lEA24=M2* t44l (141+M?) / (M3+' , |4 ) * (1 .+M1't i l / i4Z/ E)

IF(A13.GT.AZ4)GC TC 1? ESCAT:E*A1 3' t (COS ( IhEIA) +SQRI (A24/ \ i3-S I \ ( I . I t i A) IS I N( IH; iA ) ) ) **Z

RETt-RN1 TMAX:SQRT( A?4/ A13)

T lv AX=ATA N (T |4AX / S( iRT( 1 . - TMAX *T I 'A X) )IF(IHETA.LT.TYI AX)GO IO L

3 TfvAx=TpAx*1 80. / 3.1r+1592dRi TE( 45,5) TI4AX

5 FCRI/AT(2i I REACTIOI{ I14POSSIBLE ABCVE ,F6.?,8H DEGREES)STOP

4 E S C A T : E * A 1 3 * ( C C S ( T H E T A ) - S 0 R T ( A? 4 / A13- S I N ( T hL T A ) * S I N ( T H r T A ) ) ) * * ZRE TURNEID

SLBRCLT IN T f iAND P ( I X, i Y, Z)C SUBR0UTTINE CSMPUTES RANDOM NJM3ER 3EIwEr\ 3 AND 1c AND SlORTS IT IN T}1E LOCATIOI\ Z

ly=iX*ui l+131171+17=ABS ( FLCAT ( IY) ) /131 a7 1 .1X: lYRETLRNEND

116

APPENDIX B

DATA-SUMMATION PROGMM FOR TI^IO TINRNSOLVED STATES

In the course of th is work, i t was found that some pairs of low-

ly ing states were not resolved. In order to compare the DWBA predict ion

for each of the states to the unresolved measured cross-sect ions, i t was

necessary to add the predicted cross-sect ions together.

It was assumed that the summation \,ras l inear i-n nature and

independent of the scatter ing angle. The addi t ion parameters should

be adjustable to get a "best f i t " to the data. I f " i

is the value of

the measured cross-sect ion for a given angle 0 i ,

and Xl . and X2. are

predicted values of the cross-sect ion for that angle for each of the two

states, then the relat ionship between them should be as fo l lows:

Y. = A.X2. + A Xl .l_I l_ol_

(43 )

The values of \

and Ao are constant for a l l i . A s imple div is ion

puts th is relat ionship into a l inear form.

Y.l-

Y)

= A, l - +A

IO (44)xl .

1 xl .l_

The values of A, and Ao can then be calculated by a l inear-

regression techniques using the method of least-squares, to give a | tbest

f i t 'o for a l l values of i ,

A FORTMN computer co de C SC SIM is given which perf orms this summat j-on.

The program reads the input data, calculates At and Ao, calculates the

predicted values of Y- and compares them to the measured va1ue, and

L17

pr ints them on the l ine pr inter. The program also gives the coeff ic ient

of determinat ion for the f i t , the absolute and per cent error i -n the

calculat ion for each point , and the average and maximum per cent error.

Last. ly, the program plots the log of the cross-sect ion sum on the

pr inter to t i re same sJ-ze as 11" x L4" 3-decade semilog graph paper.

The program is wr i t ten to be accessed from a terminal , but is

easi ly adaptable to card input. The program fol lows and is sel f -

descr ipt ive.

118

( ] PRCGRAI. , I CSCSUMC tRCcR/r IC FIT AND PL0I CR0SS-SECIICNS i{hlCH ARt SUl l CF STATES|:

D l lv Et\S I CN Y1 (2 ' l ) ,x1 Qg),x2(2.A),Y\ (20 ) ,XDA IA ( 5) , I I TL E ( 1 0) . ANGL (20))ATA XDATA/5Hr , iH * ,5H r ,5H * .5 ' t r lDATA ELAN(/5I . I

cC FEAD I I , I /EASLRED DATA UB/SR

drt lTE(14.1)1 FCRf,AT(1lh 11 4,1 , 0R Z )

RtA)( I id,2)LUNTT2 tcRrAT(t1)

LUf iTT:r-911|+4JZ6 NtN:1

SUr4X=J.S Lti Y=ri l .SUMX2=r.SLlvXY=9.SUMY2:0.hRITI(LUIIT,3)

3 FORI, IAT(54H iNPUT i I4EASURED AN) CAL:J. iATTD }AIq--S J14,1 ,2,A\SLE qFi)hRITE(LLNTT,4)

1 FORTlAT(1 7H SSSSl 111???2AA\A)11 fcR!AT ( | t4. t '5 REAI (LUNTT, ' , l 1 ) Y1 (NIN),(1 (N iN), X2(\ l \ ) , A\Gt l ( \ i N)

lF(Y1 ( t \ lN).E0.0.)GC T0 5NiN:NIN+1CC TC 6

cC CALCUTATE TINEAR CCEFFICiENTS

5 dRiTE(LUNTT,3')3l , fCRf"AT(11I- TITLE 1OA5)

READ ( L UN TT,31 ) ( I ITL E ( i ) , I =1 ,11)31 iCRIVAT(10A5)

I F ( N I I i . L E.1) STOPNif t=N I l { -1)0 7 I=1 ,NiNx:x2(I>lx1( l )Y:Yl ( i ) /X1 ( l )s LtJ x=sul ' , x +xSIJMY=SUMY+YSLtrX2=SLlvXZ+X*XSUI'1Y2=SU\, lYl+ l+Y

7 St" t XY=SLFXY+X*YIN=Fu0AT(lr | lN)

UI ' IXY- SUtlX*S.JMY / Ti l I ) / ( S UMXZ + SU\ '1 Xt SU14X/ EN)YY /Et\-A1*SLYX/EN* ( SU!1XY-SUMX*S JMY/ EN) / ( SJ \ , IY2- S Jt lY TSUIIY/ : N)

)0 I I :1,NIN8 Y Y ( t ) =.4 : t ,2945*ALCG (A1 ' X2 ( I ) +A0*X1 ( I ) ) +1fr t , .

A1:(SF[=SL;R2=A1

CL

119Ct, FRiNT RESLLTS

12 dr. i ITE(45,13)A1 ,Ad,Rz13 FCRIJAT(23I.1DATA SUTVt ' l !ATiCN PRCCRAIVI /11,X,3NA1=,F10.5,10X,

1 3n A l=, F1 0. i ,1 I X,3f iR2= ,Ft .3 / / )nR iTE(q6, i??) (T ITLE( I ) , I =1,?i)r .JR i IE( q5,1 i )

t22 FCRtvAT(1x,?tA5)1t+ F lRl4AT ( / / 7 X, Z H X1 ,8X ,2HX2,3 X,2' l Y1 , 5X,5 H YCAr- iC, i X, / r lP Ci ERR

1,4X,5HAt\CLE//)AVG:0.XPAX=0.)u 15 I=1,\ l f lty CAL C=A1 *\Z( l )+A0 *X1 ( I )PCTTRR=( YCAL C-Y1 ( l ) ) /Y1 ( i )AVG=AVG+FCTERRr F ( AES(PCTERR) .GT.XMAX) X14AX=AlS (p: IERR)

15 hRITE(q6,16)X1( I ) ,X? ( I ) , Y1 ( I ) ,YCALC,FCTERR,ANGL (I )15 FCRMAT(1 X, 4F1 0.2,F1 A.4,F1 0.2)

AVG=AVG/iNiRITE(45,17) AVG,XMAX

17 FCRYAl(I '1 9h AVERAGT PCT1F7.4)

TRR00R=. F7 .1/15F, MAX PC T ERROR=,

PI-OT SUt4!IED CROSS-SECTI ONIl , ,AX=-1.E4)0 1d I : -1 ,NIN

1 8 lF(YY( I ) . CT.XMAx)XtvAX:YY(i)NMAX=iFIX(Xr lAX)+1thETA=ki .XMiN:FL0AT(NMAX-5)nR lTE(46,22)(TI TLi ( i ) , I=1,19))0 1) I=1, i \ iNiNDEX= i FIx ( (YY( I ) -XlYiN)*33.)K:d

?1 INDiX: I ITDTX-5IF( iNDEX.LE. i l )60 IO ?T|<=K+1G0 T0 21

2fr l t \DEX= ihDEX+5i F ( r NDEX. EQ. 0) INDEX:1

57 iF(ANGL(i) .CT.THETI)bC TC 9G0 T0 1X

t hR iT[(46,2i)T l ' i iA,BLAl iKTdETA=THETA+1. b557CC TC 5ViF(( .EQ.J)G3 TO ?4FCRI AT ( t h1 ,1 l rx,3 ( 33H -+-a +-- *) / 1 X,10 A5 ), i lRi TE( q5 ,23) THETA, ( ELAN(, J =1, () , X) AiA( I \DEX)FCRilAT ( F1g.?, 1 H*,71 A5)THETA:Trt 8TAr1.5557iF(THETA.CT.120.)GC T0 26G0 T0 19

?t, r rR i T E ( ,16,7:)T HE TA,XDATA ( iNDEX )G0 T0 l i

19 CCNT lNirEt j0 T0 25t t \D

1g2?

?325

L20

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L24

AC KN OWLEDGEILENTS

This research would not have been possible wi thout the assistance

of a large group of people. I wish Lo express my deep and sincere

grat i tude to the fo l lowing:

To Dr. Thornas A. Cahi l l , my advi-sor and mentor, for suggest ing the

project in the f i rst p lace, for h is pat ience and understandirg, h is c lear

explanat ions and enthusj-asm dur ing the guidance of th is project , and

his help in obtaining funds.

To Dr. Nicholas S. P. King, who helped me get going, explained things

when Tom wasntE around, taught me fast e lectronics and most of what I

know about distorted-waves theory, and pract , ical ly ran the f i rst

exper iments s ingle-handed. Also for point ing out innumerable valuable

references.

Thanks alng simi lar l ines to Dr. James Shepard, for helping wlth

the exper iments and some of the coupled-channels calculat ions.

To var ious agencies, especial lv Associated Western Universi t ies,

and also to the Cal i fornia Air Resources Board, The Nat ional Science

Foundat ion, The Energy Resources and Developrnent Agency, and the Physics

Department at UCD for providing salary support dur ing these long years.

To the Crocker Laboratory Advisory Counci l , and CNLrs director, Dr.

John Jungerman, for providing the necessary beam t ime to do the later

exper iments.

To the Neutron research group at CNL, especial ly Dr. Paul Brady,

Dr. Wi l l iam True, and Dr. Niek King, for a l lowing me to use so much of

their computer funds.

To the Chancel lor of the Universi ty of Cal i fornia at Davj-s, and the

Commit tee on Fel lowships and GraduaLe Scholarships for the grant f rom the

Chancel lor ts Patent Rrnd to br.ry suppl ies.

L25

To the staf f of the Crocker Nuclear Laboratoty, especial ly Gene

Russel and Joel McCurdy for their operator support and many suggest ions

dur i -ng the exper imental phase of th is work, and to Dr. James Harr ison,

for developi-ng the acquis i t ion codes.

To the Analyt ical Services/Air Qual i ty group on the f i f th f loor, for

providing of f ice space, salary support , f r iendship, and forcing me to

learn something about charged-part ic le detect j -on and electronics.

To the Gibson Banjo Company, for making such a f ine product that

enabled me to stay sat1e.

And, last ly, a most special and heart fe l t thanks to my dear wi fe,

Clmthia, for her warm understanding, unceasing pat ience, and industry to

support me al l these long years. Also for br inging me dinner dur ing those

long nights aE the 1ab, and for helping me with some of the references.