Direct nuclear reactions with polarized protons : an experimental study of Ge and Se

Direct nuclear reactions with polarized protons : an

experimental study of Ge and Se

Citation for published version (APA):

Moonen, W. H. L. (1986). Direct nuclear reactions with polarized protons : an experimental study of Ge and Se.

Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR242824

DOI:

10.6100/IR242824

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Published: 01/01/1986

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DIRECT NUCLEAR REACTIONS

WITH

POLARIZED PROTONS

An experimental study of Ge and

Se

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. F. N. HOOGE, VOOR EEN COMMISSIE AANGEWJ;ZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP

DINSDAG I I MAART 1986 TE 16.00 UUR. DOOR

WILLEM HUBERT LEONARD MOONEN

GEBOREN TE KERKRADE

Dit proefschr1ft i~ goedgskeurd doo~'

de promoto~ prof. dr. O.J. Poppema.

M<;>tto;

"D~ oot.'"sl'rOnS van al1e. dins:en iB klein." "De F1nibus".

Aan m1jn vader en moeder die dit mogelijk roaakten.

Dit ond~rzoek maakte dee! uit van het oo.derzoekprogramlll8 van de "Stichtinll voor Fundamentee! Onderzoek der Materi"," (F.O.H.), welk" Tinanei""l ondersteulJd ",ordt door de "Nederland"" Orgat11sat~e voor ZuivRr W",tenscbappelijk Onderzoek" (z.W.O.).

This inve~tig"tlon was part of th" research program of the "SCi,chting '.>oor FulJdamenteel Ollrlerzoek der Mater1e·· (F.D .H.), which is

financially supported by the ··Nederlandse OrganL/33 tie voor ZlIl ver Weten.ehappRlljk Oaderzoek" (z.W.O.).

Contents

Chapter 1 Introd~ction ~nd go~l$ of the present atndy

ChApter Z Theoreti~al approa~h

2. Introduction 2.1. The reaction model 2.1.1

2.1.2 2.1.3 2.1.4 2.2.

Cross sectiOn and analysin£ power Scattering theory

The optical model for elastic scattering Distorted Waves Born Approximation (DWBA) The collective model

Harmonic vibrator model Symmetric rotator mouel

Asymroetrtc rotator model

Transition densities in the collective model

7 9 1U 11 12 13 14 17 18 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.3. 2.3.1 2.3.2

Deformation parameters from different kinds of transitions 21 The Interacting Soson Appro~imation (IBA) 24

3. 3.l. 3.2. 3.2.1 3.2.2 3.3. 3.3.1 3.3.2 3.3.3 3.4. 3.5.

The IBA Hamiltonian

Interaction potential in the IBA

Introuuction

Prouuction of the polarizeu-proton beam Scattering chamber and detection Targets

Oet~ctor,;

Monitoring

Out-ai-plane detector.

Monitorins the beam polari~ation

Beam dump and measurement of the beam current

Data acquisition Expe~1mental p.oceuu,e 2S 27 29 30 31 31 33 34 34 35 35 36 33

11 3.6. 3.7. 3.8. 3.8.1 3.8.2 3.8.3 3.8.4 3.8.5

Data handling and daCa analysis

Experimental cross sections and analysing powers

Resolution

Contribution of the energy profile of the beam Contribut10a of kinematical effects

ContriOl.ltion of ttle target Contd but10n of the detectors

Contribution of ttle analog data-a~qui~ition system

Chapter 4 Experimental results and collective-model analysiB

4. 4.1 4.2 4.3. 4.3.1 4.3.2 4.4.3 4.4. 4.4.1 4.4.2 4.4.3 4.5 4.6 Introdu<::tioll The Ge isotopes The Se isot:opes ~xp~rimental dnalyslb

Correction for impuritie~ in ela~Lic ~c~tt~r~ng COHecti("I~ ill inelastic scattering

Spechl rellll11;l;s Optical-model a'lalysis

Para,neter search with O·PTIMO and ECIS79

Volume integrals and rm. radii

Isospln dependence in the optical potential

Generalized-optIcal-model search

Inelastic scattering Discussion and coaclusions

Chapter 6 Final conclUBions and summary

3\1 40 42 42 44 45 46 47 49 49

5>

59 62 62 63 r:.5 65 71 78 84 89 96 125 135 145

iii

Samel1vatt1ug 148

151

Tot beduit 179

C~pter l Introduct1on and goals of the presen~ ~tudy

"When I was direc ting the re?e!lrch wo);k

of students in my days a~ Princeton

University, I always tell them that if

the result of a th"5i8 problem could be foreseen at its beginning it I<as not worth ",orking at."

K.,!,. Compton.

Nuclear physics is still a young, growing field when we compare

it to the history of physics a" a wlloLe. The development of nuclear physics is closely connected With the development of contemporary physics, which started abOl,lt 80 ye3l;"S ago. A br,;,akthrough in physical thinking came in the 1920' s. Since that time we have come to accept that physical n!lture is more complex than ever thoughr before. At the

start of this century Lord Kelvin ?!lid that he understood everything

in phy5ic~. The physica!. sky was very claar to hlm, e""ept for some

very distant cloud". The.,e clou.ds, however, appeared to be [he

s~,,-rctng poiot of a new era· It 1s the quantum mechani!;: .. l ,,-pproach

that underlies a r~volution in physics.

Since the stare of the 20th century our knowledge of m,,~t;e<

developed frOm a cloudy atom via an atom with electro.,s to ao atom with a nucleus and electrons. fox a large part this was ~st"bli;;h"d

due to the work of Rutherford and co-wor~ers around 1910 (llRut). The

properties and ~he constituent", of the ll<.1c).eus were at that tLme almost comJ?!etely unknown. The d~scovery of che <1.ecltron by Chadwick in L932 (32Cha), only half a century "go, WaS a major step forward a.,d is considered as ~he start of nuclear st t<.1ct<.1re physics (S2Cas). Since

rhen the research on ~he nucleu~ has made great progress and reveal~d

a rich variety of simpl" features of nUClear phenomen3. ALso tne

theoretical description became rather de~ailed. Nowad~y~ we know <ha< the nucleus consists of neutrons and protons, ~hich in~eract "'lth each

o~he~ via the strong (nuclear) force, and We can predict a lot of its

2

compl.ete1y d(}es not exist. In the developlDent (}f nuclear t:beory t. ... o !\lodel" play a major role' t.he collect1ve model and tbe ~hell mOd.,l.

Th,,~o models are ~till the !\lost important ones.

N. Bohr (36Boh, 39Boh) conc.eiv.,d the "liquid drop" model, that has beco!\le very fruitful fa!;" the undcrstandin& of nuclear binding energies and (}f the pI"o.;;esa of nuelea!;" fissiOn. His son A. Bohr in co l,l(l.boration wiU. Mot telaon refined this theory quanti tat i ve ly to "'hat is known nowadays as the ··collec.tive model". ln 1953 they

pUblished their f i 1:5t findings (53])oh). With this model i[ is

possible to calculate the energies of nuclear levels and tbe strengths of transi tions be tween those lev" Is. Evan now, 30 jeaLs later, rhis app);"oach is ~t.ill very otten applied in one form or another. ln fact

W~ also 8\\al L empioy [his model in this thesis.

Anothe!: approac.h in the theory of the ()"c~eu", is an analogy of the atom.ic mod"l of electrons, the single-part.icle shell model. As in the el~ctronlc ~a8e we can see n~,-leons as grouped in several shells. The closure. of ~hese shells appear at characteri"tic n"!\lbers' the so-called magic tlumber.s (2, 8, 20, 28, .sO, 82, 126). At first the shell model WaS not able to explain all of these numbel:s. Mayer (49May), and in the aame period Haxel, Jensen and Sue"s (49Hax) found lhat these nl.'mbers could be explained by introducing a strong spin-or:bit coupling

in the shell modeL This model also tclr.n~ out to be very good in

predicting the properties of a nearly magic nucleus, but it fails when

we lry to 00 so of a nucletls witt. a proton and/or neutron nllmber

devIati.ng " lot from ~h", magic numbe~s. Even the lar,ge~t c.(}mputers available at this moment arc not capable of calculating featvJ:"es of

Ute nucleus without introducing seve!:e J:"estric.tions in the model

space. This brings uS to tbe main prob~em in the theory of the

nucleus: the nu<;:leug is a many-body Sjstem with a laJ:"ge number of degrees of freedom. These Cannot be handled explicitly wll",n the number of rrucleons bec.omeg too large. So some simplific.ations are needed.

3

H~ll~day (~OHal, p. 292) states about these two ~odels:

··:rhe basic idea of ahell theory is that the Il.u"leon", b",lHtVe

as though they were c.onfine<i in a CO_OIl. poteIl.tial well 1;tnd the)' iIl.teract which each other slight 1)'. This is directly opposed to the liquid drop idea, which il)lplies a stroIl.g interaction. 80th models are useful. Their bastc

incompatibility simply sho~e the poor st3te of o~r ~nowledge

<;>f nuclear

forces.--Thi8 statemeIl.t is still basically true, despite all ~ind of efforts to unify both models.

A development of the last de~ade to redu"e the degrees of freedom 1" the Int:er.:>cting 8oson Approll:imation (IBA) of Arim~ 3nd I3chello (76Ari, 78Ar1l, 78Ari2, 78Sch). Starting from the ell:perimental fact

that collective ell:citatitions exhibit mainly a quadrupole (L-2)

character giving rise to surface o~cU l3t lons, AriOla all.d Iachello

repl~ced the large numher of Single-particle ~tates by a few collective i.e. bosonic degrees of freedom. They introduced s (L=Q) and d (L-Z) bosons, which can be seen .:>s cotre13ced pairs of nucl~ons.

By meana of group theoretical methods they found three analyti<;",l solutions. These solutions are comparable to the coll@etive limits of

vibration, rotation and y-inet:abiHty. The few degrees of fr.,~dom

result in an easy calcul.:>tion o£ energy levels and transition

strengthS. it is also possible t<;> <;alculate transitions from one limit

tu another, which gives more problems 1n the collective model.

However, also t:he ~8~ 1s not perfect and several extenaions have been

proposed (83Ell).

The present work fits well ill. the framework sket:ched ~bove. For

the experimental physicist it is a chall@Il.ge co discover the

propertie6 of a nucleus which can and those which cannot be de~<;ribed

by ~ 6pecHl.c modeA. In this way we are able to refine the models and to get a better insight in nucl@ar structure. Aa our probe we shall eml'loy po13rt~ed p,otons 3nd measure th@ir sc.attering in dependenc.e of

angle 3nd spin direction. This study is a continuat~on of prev~OllS

work done by Melssen (7aMel), Polane (81Pol) and WasSenaar (82Was). rhe wod~ of Melssen cOIl.centrat.,d On the semi-magic nuclei yttrium, iron, and o.i"kel. rhe description or his experimental findings was dOlle mainly in terms of the optical model, the collective mOdel and

the Distort~d Wave Born ApproKirnation (DWBA). 1he optical mo.del

de~cribe8 the elastic scattering of pro.to.ns io terms of a tew parameteJ:"s. The DWBA then analyses the tnelasti" scattering Wi th the parameter s"t of the optical model. The maIn COnclusion o.f this work

was that in the inelastic ""attering of Po.la~h;ed protoo.s the

defo.rmatio.o. o.f the spin-o.rbit part "an be dependent on the incident energy of the proton when we look at those semi-maSi" nuc~e1. Also. the

wor~ of Polane was centered aruund nickel and irOn {SBNi and ~oFe>. He

especially studied the transfer reactio.ns leading to. the duubl8 m.ag!.c nucl8uS 56Ni (Z=N=28) and the ",emi-magic nucleus 5~Fe (N=28). In his thesis the analysis has been perfo1:"llled wit.h DWBA tor i-step and 2-st.ep processes. The thesis of Wassenaar was also in this directio.n. He

""t~ndcd in hi", work the (esearch of the "uclear physics group at the

EindhO"~11 Ul1:!-versity o.f Technology (J:;VT) to nuclei M:o.un<;l the IIlagic number 50. In Wassenaar' s work there was .~l ready more empha,;;ls Oll a(la1ysis ",ith the coLlective mo.de) within a co.upled-channels approach. This "pproaeh has been used to its full exte"t in the thesis of k'"tit

(85Pet), aince i t haS been devoted to the transitio.n regiOn o.f

vi bratiooa) to p"rruanellt 1y deformed Sm i",utopes (A=lSO, ;,: .. 62).

The p1;esent ir,vestigat ion, togethel; with the work of PetiC, is concern"d with eX(_ited states of !>Ome tran';lit.ional nuclides, ",hieh,

through th~ experiment.al :f.Dlprovemer'lts,. became better .e.cc.essible for

our polarized proton ""'pcrilllents. With the work of Melssen, Polane and

Wassenaar w" have obtained some more knowledge on the behavio.ur of.

"uclei with a pro.ton and/or neutron number in the neighbourhood o.f the 'nagj,c numb"r,;; ZI:l and SO "L\d about th" special h,atures thes~ nuclei e"hibit. Our aim i . now to see how our.lei bohave when they have a proton and/or L1eut-~on number in between 21:l and 50. Another a1m is the eompletion of our pict\lre of even-ev"n nuclei in geoeral where the

re~earch started with the nuclei Fe (Z=26) aod Ni (Z-2B). Therefore

w" have chosen sowe nuclei ",hieh follow this 5eries: - Ge, Z-32, N=38,40,42,44

- Se, Z=34 , N-42,44,46

Our goal is the study of these nuclei in order to fi.nd how the

structure of a o.ucleu5 changes when more and more nucleons are oeing added.

III the past llIo.(!t analyses of 8xpet::!-mental data have been

wQ1cQ c~n be e~c1ted di~eccly from the g~ound atate and tQe excitation strength being not too large (defo~mation parameter amaller then 0.3, see 63Perl), but also not too small. This approach works well for low-lying e~clted states, especially the fi~5t 2+ levels and 3- levels of even-even nuclei. In rec.ent yea~s then" is s trend to investigate higher e~cited levels· These levels impose higher experimental and tneo,etlcal demands, but they also provide a bette~ test ground. for nuclear models. In many CaSeS the exc.itation mechan~sm of these levels is believed to be a two-step process. Possible interference makes these level" very sensitive to details of nuclear models. Then the DWBA d.e6cJ:'l.ptiofi is clearly insufficient and a coupled-chafifiels approach needs to be used. 'these levels also requi re more e~perimental ~kLU. In gefi~ral the cross sections are small and. the levels are situated in a region with inc~easing level density. This requires lon~

measurinS periods together w~th a good energy resolution,

The time need.ed to measure an afigular distribution is mainly determined by the lnteasity of the beam on the target. So it would be wise to uSe a curr""t as high "", poes~ble to collect as Iuuch information as possible in the available time. Since currents of polarized proton beamS are at ~east two orders of magnitude less then those of unpolar1~ed. beamS, the question arises why to uSe a polarized beam. The ~hoice between unpolarized afid polarized protons 1s a choice

be~ween experimental results. With polarized protons we get in return for lower currents a ~econd observable; the analysing pOwer. 1£ only cross sections are cOfisidered it is true that unpolariz"d b"ams give more quickly .lnd oftefi evefi more reliable results. In the past, however, it has been shown that a cross sec.tion give~ only ehe 8_085 fe~t:u~es of a nuclear level (SlVan). With polarized beams st~uct~re effects will be seefi better afid be d~tected sooner. Occasionally evefi

et~on8e, 8tatemefits have beefi put fo~ward. Hanna exp~essed as his opinion that all scattering experiments should be pedo~med with a

pola~i:l:ed beam (8lHan). In fact this 5tatement is wort:n"'hi~e to be taken seriously.

In the next chapter thE th~oretical ingredients are presented. Severa.l models uBed aI:e dlaeus .. "d brhfly. Also a little piece o~

reaction theory '0'11 appea, with formalism. In the third chapter

the DWBA and the coupled-chano.els we will give a survey of the experimental tools to perfo~m polarization expEriments and the special problems involved. Tbe pJ;-ocedUI:e of transposing the experimental data into cr08~ sections and ao.alyslng powers and a discussion on improvement in experimental resolution will be pre;;ent"d there. In chapter 4 the re;;"lts of our eltperiments I.1ill be given. Schematic

struccur~ calculations with I~A-2 is the 8ubject of chapter 5. Fina~~y

7

Cbapter 2 Tbeoretical approac~

2. Introduction

"Two seemingly incompatible conceptions

can each represent an aspect of truth •• _

They may serve in turn to represent the

fac.ts without ever e.ntering in direc.t

conflict ...

1oui~ de Broglie.

In the previous chapte~ ~e have discussed the need for models. 1n

this chapter 'ile shall gi~o;, a shor~ d~s,,-ription of two mod.els for

nuctea~ strvcture. In f8ct nUcle8!; «tnlctlJre rnodel~ (ire In<!nifold b\l~

we shall limit our discussions to the geometrical model of Bohr ana

Motte180n~ its extension by Davydov and Filippov~ and to the Inte~8ct:l(lg Boson Approxim",tion (lEA model) of ",-rima and Iachello. first, howeveJ:", we ",h(ill preeent an abstr"ct of reaction th~ory, whic:h we need to connect the e~pe,1ment(i1 re8\llt~ of proton sc:attering with the nu~lear models.

Z.l. The reaction Model

2.t.t

Cross section and ~naly~lng powe~

Before we present the formalism of scatt~rl[)g th,:,ory \01", will

introduce io. this section two impon8.(lt qllanti ties which are. the

ob~"rvabl"s in our IneaSlirements: the differential cross section and the analysing power. Sinc., \ole ar" dealing with polaritation phellomeaa

it is necessary that we introduce a frame of reference in Ollr

rea<.:cions. 10. this thesis we shall eillploy the M(idiso\1 convennon (71Mad), that defio.es the scattering plane as the ~z plane a~d the y-axis perpendlclll(i~ on the ~c~tter~ng plane (~ee figure 2.1).

da

The first qllantity is the differential CrO~S se~tion

dn'

which

is defined as the number of partlcl",~ emitted per steradian in the

8

b/s~ lO-2E m2/sr). It is a quantitative meas~re of ~he probability that a gi'len n<lclear reac~ion will take place. The 4J.tter",nl;ial cross section is the avera8'" of ~he differential cross secl;ions for each of t he two spin directions of ~he incident proton:

da

1 dO"

+

dO"

-7 (dlT" + <I1T )

wbe~e

"+"

in4i c ates spin-up (in the positive y direction) and

spin-down directiOn.

(2.1)

the

The Second quanti ty is ~he analysing power A, also called aSjmmetry. It is 4et~ned a~ the relative diffe~ence hetween ~he cro~~

sections for the two spin directions in the reacl;ion plane,

A

dO"+ do da

= ( <I1T" -

W

1 /

aTI (2.2)

Th", <lbove expJ:esston$ refer to the differential cross sections for proton beams polarized for 100% in one spin direction. In practice we <llways have al~ admh:t\lre of the opposite spin direc~ion6 in th~ beam and we have to correct the n~mbers found experimentally with the

deg~ee of polarization P of the beam. Having a beam of partic,es with

~pin I and a probability w(M) of finding the spiQ p~OjectiDn H (-I<MG) tn this beam thell the (vector) polari~atioo

r

of Sllch an assembly J.3 4efined as [he average value of H/I:

p 1/1

I:

H ... (M) with

L

w(M) = 1.

In the special case of prorons

(1=.)

expression (2.3a) becomee:

~(+o

- w(-o

p =

w(+i)

+

we

§)

(2.3,,)

(2.3b)

Wi~h thi~ d~finltioo of ~he polarization P toe experimental analysing power becomes (tacitly aa6~m1ng that the deg,ee of polarization is the Sam« in both spin direc;tions):

A exp

1 ... (6) - I (e)

9 wbe~e "+<~), 1_(6) are the normalized number of parti~les dete~ted for, respectively, spin-up and spin-down directions of the beam.

ey

..

kin~

k

_Out e

-ic~ill

y

IkinX

koutl jf - ₊

kin

+ 1<._ ,n (ez) e Ikinl z

F1gu:(e 2.1 The Madison con~ention, with, +

~in

momentum of the incident particles,

momentum of the scattered particles. out

2.1.2 Scattering theory

Tbe process of scattering of a proton by a nucleus c~n b8 described by solVing the Schr~dinger equation:

(H - E) 'I' = 0 tot

The Hamiltonian B

tot conSists of thr"-e plOtrtSl !l

tOt

with; ~ the distance between nucleus and projectile

~ the internal coordinate~ of the nucleus HO<

0

the Hamiltonian of the tarllet nucleus

T(~) the kinetic energy operator of the proj~"til~ +

(2.S)

(2.6)

V(~,~} the interaction potential between proje~tile and target nucleus ...

The Ilamiltoni"n HO hn " ~et- of onhollOrmal eigenfunctions "'n;

50 ~

10

We can expaad f into these eigenfunctions ( f D L Xj(~) ~ .(~» to

j J

obtain a set of coupled equatione in the (l"attering functio[ls X:

(2.8 )

+

Solving tl~e",e equations for the funcUon~ xj(r) w111 sive a complete

picture of the scatterinl.l process. A prerequisite is, however, the

knowledge of all matr.ix elements <jIVI~>. These matrix elem~nts Can be

calculated within a spec~fic model. Still, W~ have to wake a

truncation in the infinite set of equations. 1f elastic ,>cattering is

the dominant process, we can tr~at all other procesees as

p"rturbatlons. The loss of inte.nsity out of the elastic chamlel C<ln

[,ow be accounted for by an imaginary term in the interaction

potentiaL In the next section we shall introduc~ the optieal model

",hidl inte~pret8 the elastic scatterio.g in a phenome(lological way. In c<)nstructio.g equation (2.8) we have tac~tly n(lglecte<i the

antisymmetriz~tion b .. tweeo projectile an<i target. When usio.g the ouclear "hell lllodel i t can be treate<i explicit),y. The phe[lOmenological collective models \lse<i in the an~ly8iB of our results <Iccount for it

effectively by adjusting various st~ength parameters.

2.1.3 The op~ical model for elastic ecattering

The domillilllt p~OCe"'B i.-. the rea.ctio.-.s of low-energy protons with

nuclei is the el,a~tic scatte-.:ing. It genera.lly is described by a

phe.oomenological potential, the (lpttcal-model potential- This model is One of tl:l.e simpiest aod most succes$~\ll of the re",,-tio[\ u\odels. The opti"al potential has the fol1owhlg widely used functional fori\>;

u(r) Vc(r,R,,) +

-V_r f(~,rr,ar) +

-1 Wv f(r,r

i,a1) -1 Ws g(r,ri,a1) +

lm~c12

Vso

~

: r f(r,rso,a_so)

;.i

(2.9s)

11

11 g(r,rj,a j) ~ 4 aj df(r,rj'a

J

)

Or (2.9c) 1 2 2 and Vc(r,R,,) w

~(3

r ) tor r <:

_J\

~o c

~

(2.9d) ,,2 c 1

z

z ~~ r for r > R c

This pot~ntial contains 10 ~nknown pa,ameters which can be fitt~d to

elastic scatterins data. The imaginary part of th~ optical model

potential accounts for the reduction in intensity in the "las tic

channel through other pl;OCeS8eS, Suc.h as inelastic scattering and

nucleon traneter. We have to keep this in mind, i f we take into

account some of thesa processes explicitly. The last term in equation

(2.98) represenU the full Thomas form of the spin-orbit potential

(68She, 72Ray) and is the factor responsible for polarl~ation

phe.nomena by creating a different potential for difee~ent spin

orientations of the incident proton. The potential Vc in equation

(2.9d) is the Coulomb pot~ntial of a unifon;nly ch(l.1:ged sphere with

1/3

radius R~ = rc A . ~he usuaL values of the reduced charge radiu? rc

range from 1.1 to 1.25 fm.

2.1.4 Oisto1:ted Wave Born Approximation (DW5A)

When the elastic scattering is the predomin(l.nt p(l.rt in the

scatter1n8 p,ocess theo all other contributions c::an be treated as

firat-order perturbations. This meaas that the elastically s~attere<;l

waves are oot affected by re8catt~ring into the ela~tic channel, i.e.

the coupU ng !I\a t,iK elements <j

Iv I

0> are gl)fficieatly small. this gives,

o

(2.10a)

(2.10b)

'Ihis approl<iw<l.tion is known in the literature (1.6 t:he Distorted Wave

BOJ:'\l <\pproximation (DWBA). The uncoupling of ebstic and ~nelasttc

channels allows uS an explicit formulation of the relevant transition matrh; T" _ (64Sat).

1J

12

- the coupling between the ground .,tl;lte and the excited state is so strong th~t the elastic scattering is influenced.

- the exc1t~t,on from the ground .,tate to the excited state is forbidden in fir~t orde~.

In both cases we have to intrOduce thB couplings between states explicitly aIld to use a coupled-channels formalism, 1.e. to ~olv"

equations (2.8) in ~ truncated mOdel space.

2.z.

The colleetlve model

The collective '\lode 1 finds its origin in a pilper by N. Bohr (36Boh) whose idea" have been developed in seve,~l ways. Th., approach by A. Bohr and B. Hottelson, Wr~tten down in two p~pers in 1952/1953, is th., one we shall use (52Boh, 53Boh, 75Boh). In this model 011e. looks

at the nu~lcus as ~ collectivB entity, ~nd its features c~n be ell:tcacted from the collective movements of the n,",Cleans. This is i(\ contrast to the shell model whe~e one startS from the movements of the

!.no~vidllal l'"rtlc:les. The collective movement" can b", either of vibratlon"l or of J:otational nature. As the nuclBl studieo in this theRis are predominantly of vibrational ch~racter wB describe f.irst the harmoni" vibrator model in more detail. Next we shall pay some .3-ttention to the rot~tor model and fin~lly we shall discuss sowe elements ot the ext"nsions of both models. The ~ncorporatlon of all these elements in the t~ansltion potentials will be discussed

~hB~eafter. The di.,cuasion about how we can compare the parameters of the collective model (deformation parameters) for di~t~J:ent types of scattering experiments finishes this section 2.2.

2.2.1 Harmonic vibrator model

The concept of a nucle~r auriacB is the basis of the harmoni" vibrator model. 'this Sll~f,,,<;.e is defined by:

where:

(2.ll)

is the rad111s of the spherical nucleus

are the coefficients of expansion tn YA~. They rep.eaent small dynamical deviations from the spherical shape.

for the f~~et1on R(e,$) to be real, one has to require:

We ~an now ~on~tru~t a ~amlltonian of thG form:

H ., T(O,) T Y(a).

with < the kinetic energy operator, and Y the potential energy operator.

In the harmonic approach this ~an be written as:

wnieh represents a set of harmonic oscilh.tor~ w~th e,eqclency: "\ - ICC/BA) 1.) (2.12 ) (Z .13a) (2.13b) (2.14)

<hrough second quantization in the usual Way the Hamiltonian takes the form:

+

with c). and c>. the boson creation and annihilation operators of one

2~-pole

qclantclffi (or phonon) and involving an energy of

~~A.

It is nOW easy to see that the energy sp~~trum will look like:

(2.15)

In this context We also mention the deformation parameter B A, which we shall uS" frE:quently. <nis is just the root m.ean square detormation in the ground state due to zero~pDint 08ci11ations'

<: (2.16 )

The harmonic approacn is valid only in the case of sm. .. ll

de~01;"mat10ns. Whenever the deformations beo;:ome large, teJ:"UlS of order higher than two in the Hamiltonian have to be taKen into ac~Ount. In

14

the "Frankfurter model" of Greiner and co-W'orkers (SOaes, 72Ets) this has been worked out for terms up to the /3ixth order. In the same line and with special at:tention to the inCOrpO'llt~on towards scattering processes is the work of Thijssen (8'Th~j). However, we ~hall only use the pur~ harmonic approach.

~.2,2 Symmetric rotator model

In the casa we have a static deformation, tbe W

Au in equation (2.11) get II different m"anlng. Wh .. n we consider only q"aOq'pole deformatiOns a,ld tral,sform .. quat ion (2.11) from the lllbor;>tory systam into a system of principal ax"", the flv .. collective coordinates "2p are tr"nsformed into Ii Bet consisting of t:he three (time-dependent:)

BulRr angles 8

i(1=l,2,3), which give the orientation of the ngc1eu~ r~lative to the iaboratory system ~nd tne (static) quadrupole deformatioos a_O and R

2' Instead of aO and aZ alternative coo.dinatRs

~ and y ar~ ~sed:

(2 .l7)

(2.l8)

In figure 2.2 ~hcse relat:ionBhip~ are illustrated. Contrary to the ~1

from ~quation (z.~6), which defin~s a dynamic deformation, t:he 6 from equations (2.17) and (2.l8) is a static deformation, meas\lring the deviation hom spherical shape. The angle y gl "<os the oeviat iOn from axial symlllet ry; 1=00

i6 " prolate defot'med shape and y=60· is an oblate deformed "h~pe. ~'or 0" <y<60° there is no aKis of "ymmetry and we ha~" then a so-c~ll"d tri-axial sh~pe with maximal asym[llet:ry at y=30°,

using the above coordinate transto);\D~tion "'ith the oefinitions of

(2.17) and (2.18) the Hamiltonian of (2.l3) becom~s:

H n·19)

wHh: LK the components of the angular momRntum along the principal

+

15

Tk tile effective mOments of inertia witll respect to the

principal axis.

Only the last term of (2.19) gives rise to th@ rotatiOnal motion. The

p~eceeding terms in the Hamiltonian (2.19) are vibrational. giving the so-called Sand y vibrations.

The energy spectrum for a symmetric ri.gid J:otor (T

l" 'fZ- TO i.e. two mOments of inertia ar€ equal) i~ given by:

E h2 1(1-1-1) - K2

+ - -

h2 K2 (2.20)

2 _{TO 2 T3}

with: I th" tot.al I'nglJ.lttJ: momentum

K the projection of 1 along the symmetry axis.

The wave functions

juno

can only exist when the followinll rules I're met:

- for K 0: is even (J=0,Z,4,6,.,.: ground state band) - tor K > 0: I is equal to any possible value of K, K+l, K+Z,

B-), plane

prolate oxis(dgors)

. I .

sphencal pOint

16

2.2.3 Asymmetrle r~~a~Or model

In the previous paragraphs we discussed the harmonic vibrator mooel and ~he a)(ially symmetric rotator. S~nce the models are very simple, several e}(~enSiOi\S have been proposed and worked out. The

deveLopment~ followed two main lines:

- the e)(tens~~n of Davydov and F~lippov and co-workers, who introduced tri-axial deformatioo (58Dav, 59Da~);

the e)(tension of Fa;,;;,;ler and Creiner and co-worlters, who lntrod"ced rotation-vibration interaction (65Fae, 72Eis).

In this paragraph We shall discuss the first extens~on: the asY1lllll.etric rolator.

Davydov and Filippov started their model with the assumption that nue Ie i have no sp,;,cific symmetry i. e. the three mo,"e,lts of inert!" Tit are different:

H

rot (2.21)

Note that hel:e no vibratiOnal degrees of fr~edolll axe used, "nu this must Lead to definice values of a

O

and a

2

(or equ~valen~

a

and y).

this HamiLtonian can be written as that of the Jilymmetric rotator plus a remainder term· So the solutiolls of the symmetric rotator !IMK). are used co solve the Schr~dinge~ equation. The most general solution

U,en has the f.orm:

11M!> ~ K 0,2,4 •••• , I, (2.22)

K

where the band miRing co~fficients AKi depend 00 the asymmetry 1. The

r

index indicates that in general several states wHh spin I may occur. The energy spectr~m is also a function of y and is displayed in

1

figur~ 2.3 for states up to l=5. In ~able 2.1 some relevant

Ai

i values are given for 2+ and 4+ states. Since the excitation energies and the probabilities fo~ electromagnetic transitions between "tates are the same for Y=Y1 and y~60·-Yl' we present the values of various quantities Only in th~ interval 00

and 30°.

A simple relation holds for the 2+ stat~~. sO that from the ratio ~(2!)/E(2t) the y para,"e~er can be deduced. For nuc~e~ with a

Tabl@ 2.1a Some relevant ~oeffi~ient5 ~i I 85 8 function of y.

2 2 2. 2 4 4 4

y _{A01 AZl AOZ A22 AOl A2.1 A41}

25.0 -0.9740 -0.n67 0.n67 -0.9740 0.8516 0.5224 0.042.8 25.5 -0.9687 -0.2482 0.2482 -0.9687 0.8395 0.5412 0.0484 26.0 -0.9625 -0.2712 0.2712 -0.9625 0.8275 0.5589 0.0544 26.5 -0.9552 -0.2959 0.2959 -0.9552 0.8156 0.57$5 0.0610 27.0 -0.9467 -0.3221 0.3221 -0.9467 0.8039 0.5909 0.0682 Z7 .5 -0.9366 -0.3498 0.3498 -0.9368 0.7924 0.6052 0.0759 28.0 -0.92~5 -0.3786 0.3786 -0.9255 0.7813 0.5184 0.0843 28.S -0.9128 -0.4085 0.4085 -0.9128 0.7704 0.6307 0.0934 29.0 -0.8985 -0.4389 0.4389 -0.8985 0.7599 0.&419 0.1031 29.) -0.8829 -0.4695 0.4695 -0.8829 0.7496 0.0521 0.1137 30.0 -0.8660 -0.5000 0.5000 -0.8660 0.7395 0.6614 0.1250

y _AOZ 4 _A22 4 _A42 4 _A03 4 A4 23 4 A43 25.0 -0.5232 0.8425 0.1285 -0.0310 0.1318 -0.9908 2$.$ -0.5420 0.8277 0.1457 -0.0388 0.1486 -0.9881 26.0 -0.5594 0.,3120 0.1666 -0.0489 0.1683 -0.9845 26.5 -0.5753 0.7950 0.1921 -0.0620 0.1918 -0.9795 27.0 -0.58S5 0.7761 0.2239 -0.0794 0.2202 -0.9722 27.5 -0.6013 0.7541 0.2641 -0.1026 0.2549 -0.9615 28.0 -0.6097 0.727Z 0.n~4 -0.1338 0.2978 -0.9452 28.5 -0.6128 0.6922 0.3812 -0.1758 0.3509 -0.9198 29.0 -0.6077 0.6449 0.4635 -0.2310 0.4149 -0.8801 29.5 -0.5904 0.5809 0.5604 -0.2994 0.4871 -0.8204 30.0 -0.5590 0.5000 0.6614 ~0.3750 0.5590 -0.7395

18

Table 2.1b Some reduced electric quadrupole transition probabilities b(EZ), expressed in ~Z Q~/16~ unite eor r~levant values of y On degrees). Y 25.0 25.5 26.0 26.5 ~7.0 27. S 28.0 28.5 29.0 29.5 30.0 ..---,

I~~-" CO ~'<1" VJ +-c :;l '-...' W Figure 2.3

E_Z /E_{Z ];,(E2,2 1}->O) _{b(IlZ,2 z}->O) ];'(E2,22+21 )

2 I 2.4078 0.9575 0.0425 2.3302 0.9627 0.0373 2.2610 0.9683 0.0317 2.2000 0.9740 0.0260 2.1472 0.9796 0.0204 2.1023 0.9850 0.0150 2.06.56 0.9900 0.0100 2.0369 0.9941 0.0059 2.0164 0.9973 0.0027 2.ll041 0.9993 0.0007

Z.OOOO

1.0000 0.0000

40

...

30

20

10

2

-' - - -

-0.8678 0.9406 1.0156 1.091~ 1.1654 1.2.358 1. 2994 1.3531. 1. 3941 1.4198 1.4B()

o

' - -_ _ - " -_ _ _ •. 1 .. _ _ ---1. _ _ _

.1...-o

10

20

Y (deg)

b(E2,2 2..2₁₎ b(E2,2 2->O) 20.4197 25.2368 31. 9990 41.8%2 57.1867 82.5961 129.4348 230.7069 520.1590 2083.3770 '"

2

30

The e!1.ergies of the SC"'tes with I:;5 1n the ,,~y_etrj,c

symmetry axis y will be ~ero ~nd the maxi~u~ asymmetry will be

ob~ain"d fo~ Y = 30·. Tb.is las~ value seems to boO preferen~i"l for vibratlonal nuclei.

2.2.4 Transition densities in the collective model

In th'" previous l?aragraphs On the general r"ac::tion theol:Y we hav~ sh~wn the need for transition potentials in order to indu~e

c.ansitions between levels. The optical pot~rrtial haS been introducad cO describe the elaatic scattering. This 0l?tical potential is connected ~~ the mass distribution. If the maS~ distribution is known then an optical potential Can be deduced following the folding

p~inciplc (see Greenlees e~ a1. (68Gre) for the reformulated optical

~odel). In a simular way we now can derive ch~ transition potentials from the deformed mass distribution. Whefi we use equ«t~on (2.9) and

1/3 replace R

j (= r jA ) by the expression of R o~ eq~ation (2.11) we have « detormed optical potential. In order to have more Suitable expressions this formula is expanded into a Taylor series of multipoles. We sha.ll follow the lifie~ se~ out by TaUlllra for the eomplete expansion up to the seeoond order (65T«m). The final result beo::omes,

wHh 6R ;).s defined In (2.11). Essentially ~his expression h~s the form of,

V a Vdiagonal + VCOUPling (2.24)

'diagonal 16 the normal optic~l potsnriai whiie Vcoupllns Is ehe interaeotion potential up to second order. All cerms of third order a.ld

11~8her wIll be neglected. The first-order term is a tenn whi.:11 represents all. exo::itation of One phonon a.t each interactiofi. So a t~o­

phorron state can be excited by the wor~ing of the first-order ~erm in

~WO seeps or in one step by the S~c::ond-Drder ~erm.

The reduc",d matrix element of the Urst-o"(der term for the

Z

exc~t~tion of one

Z

-pole phonon in an eveIl.-even nu~leus (1=0) beCQ"'es;

20

- ~~ound state ++ one-quadrupole-phonon Btat~:

(2.25)

- one-quacirupoll!-phonoa state ++ two-quacirupDll!-phonon state (with

spin I}:

The secDnci-order reduced ma[rl~ element fo~ e~citins

phonons in An even-even nucll!us from the ground st~te:

two

(2.26)

(2.27)

In the ha 1;-monic ~ibrator model th~re is no diffl!rence between

t; h e Sea -s and we put 13₂ - 1102 = tl2.l = I>or'

Ln practice a pure two-phonon state w~11 0,,11 seldom exist. The

simple fact that the l,evela of the two-phonon multiplet do not

coincide, alr~ady point" to a residual iate,action or an

anharmonlclty. The pu~1! states will be m..1~ed wi th neighbourinS one~

rhonon states. 50 the o;o-(Called two-phonon mul tip leta .... ill have wave f"nct1.ons which are a mixtu~e. In the harmonic vibrator model s"ch mil<,,,d states Cannot "xist. The wave. funcc1ol' of a. mixed state will

.look like:

12-phDnDn'> ~ sin ~ !2-phonon> ~ cos ~ !l-phOOOI'> (2.28a)

Il-Phonon'> = cos u 12~Phonon> ~ sin u Il-Phonon> (2.28b)

A construction of the wave functions ~n this way 1.'111 insure that

21

2.2.5 Deformation parameters from differen~ kiDds of transitions

Aft~r anb~ysis of th~ exp~rimental data ~ith the collective model the deformation para~terB will be a ~e6ultant of this analysis. When one wantS to compare these ~ith those found in the literature, one has to keep in mind the way in which the information has been gained, that is to eay what reaction mechanism is responsible for the transitions. Thi6 lIIeans that we have to be careful with scattering dat:a sine"" the

inte~action strength between like nucleons (p-p and n~n) and between unequal nu.cleons differs approxilllately by a factor of 3. Mads""n, Brown and Anderson (75Madl, 75Mad2) hav" ehown thl;' e><:isteace of essential differenceB in the deformation psn'\lII;'te~

e

for different kinds of

tranBition~ i.e. between those of (p,p') and electromagnetic transitions. When the transition potential V is split ~p in an

~"o"cala1:' Va and ~n isovector part V 1: V .. Vo .,. TV 1 (r is Tl for prOtOIlS, -~ for ne<ltrollS, 0 for deuterons and alpha particles) the[l these differences can indeed be related to the difference in deformation in both parts. So one has to ma~e a compa~igon of parametel:s which result f~om the same reaction or of some other qu.antities which incorporate the above-mentioned differences io

~!\teractioo strength.

On the basis of the same considerations Madsen, Brown ",1d Anderson concluded that for closed-sbell nuclei the quadrupole deformations for electromagnetic transitions (Sem)' for proton (B_pp') and neutron scattering (~nn') should fulfill the following t;e.Lation6h1ps:

a

_em ~ ann' ~pp' for closed-neutron-shell nuclei, Sem Snn' ~ Bpp' for closed-proton-sh~ll nuclei.

(2.29a) (2.29b)

In the case of open-shell nuclei the differences should be amall and the ratio of app.!B

em in tha neighbourhood of on~. Hatoha (79Ma28) made a similar analysis based On mOre recent data and confirmed these results.

In th~ pr~vious paragraph we assumed the availabiLity of Sem values. The results of high quality electron Bcattering experiments, gamma decay and Coulomb excitation a.e usually expt;essed in the form of reduced transition p.obab1Lltles »{~h> 1nstead of Qeformat~on

22

parameters. N@vertheless it is po~~~ble ~o dedo~e SOme ~em values when

we assume th", charge dens:lCy to have a par~icular form. In the

collective model ll(E).) can be derived using th", following expression

(7580h, equation 6-65):

B(EA,

O~A)

g (0.75

~

eZ

R~ ~,)2

ef A

where is ~he deformation parameter, and

Ref 1s an "etf,,-ct:!,ve transititm radius".

(2.28)

According to Owen and Satchler (540"'e) Ref ~an be calculated with a

re~11stlc Woods-Saxon di~tribution f(r,rO,a

O) (2.9b) for the ~harge density: R ef

o

f

'dR

~ d. [ R d£

~+2

Jl/A

o

(2.2.9 )

So if we know the B(EA) value snd the patam«t"rs rO and a

O' lola are able to compute the value of Bem. For lack of accurate measu.ements of the charge del\sity we use the geometry of the re"l part (rr' a

r) of the optical potential.

If one confines oneself to the same typ~ of reactions a direct

(,omp<lrison oj' the deformation paramet",rs or a test of the above-mentioned relations is Ifli!ry well possible and this has been done frequently. We know, however, from optical-model <lnalysls that there

exist some correlations in the opti~al model parameters, resulting in

dlffcre,lt optical pote(lt~(l.ls ~iving ch« same elastic 6C<ltterin~. This will also b" r"flected in the interaction potential deduced from it.

So one also uses a different quantity for cOmpat;~60n: the so-called

deformation lenSth ~lR. I~ is present in the i(lter<lction pot~ntlal as strength and in general it: 1s less dependent on the correlations

between the optical-model parameters. With this quantity also

comparisons bet"een different types of scattering experiments have been mad". The relations of Madsen, Brown and Ande-.:son should also hold for the deformation length.

In an <lttempt to obtain mo.e fundamental quantitle& from

scattering ~xperlment8 M,,~kintoBh (76Maa) proposed to ~ompute the

rnultlpohe moment~ qlO of the transition potential (deformed optlaal

ve.y useful Since it can De measured tor the proton component in a ~ay

which is largely in<iependent of reaction 1I1o<;iels. So it provide" a means for direct compar~60n between results of scattering and electromagnetic processes. A difference between those should indicate either a £a~lu.e of the folding model (e.g. a strong energy dependence not accounted for) or a difference between the neutron and the proton multipole moment (i.e. different deformation for the neutrons and protons in the nucleus). In extracting the multipole moments a theorem of Satchler (72Sat) has been use<;i, which states that the normalized multipole moment

of

the nucleus equals the normalized multipole moment of the folded poteutial. The multipole moments consl.deJ:'ed only have reference to the real field (optical potential), since for this part the foldins model is more likely to be realistic than for the ilIl<lginary component. Mackintosh expressed the el<peCtatto!l that the multipole moments furnish a better mean~ of q~ot1ng deformations. thao do deformatiOn lengt:lIs and deformation parameters. siace they should De less d~pendent on the specific set of optic~l-model parameters. In this $t~dy we shall use the prescdption of Petit (85Pet) fo'( the computatiod of the multipole moments (qAO ; MeA) in 85Pet). Up till now it is not yet common practice to publish these moments, so we will try to calculate them from the given da~a ~henever poss~b~e.

One final remark should be ma<;ie here: the accuracy of the

calc~lated nuclear density mOment~ Bt~ongly depends Od the accuracy of the radius parameter rr. This means that the accuracy ot the other parameters, especially that of the deformation paramete~, is somewhat obscured in the Hnal r;esl.llt. To a lesser extent this remark also holds for the defo,mation lengths.

24

2.3. The Interacting BD~On Approximation (lBA)

In the seventies a mode~ Qf collective state~ in "uclei has been develope(l by Arima aIld Iachello (76Ari, 78Aril, 78Ad2, 78Sch). 1n this llIodel, ol\e 'il-~8umeS that the Qbeerv"d propeJ:"tles of low-lying collective states arise [rom the interplay of two ettects:

- the strong pairing interaction between identical particles

(proton-protOn and Ileutron-neutron),

- the strong quadrupole interactioIl bet:ween nOn-identical particles

(proton-neutron),

Th" strong pairing interaction suggest8 that it may be appropriate to

consider correlated pairs of nucleons as the building blocks of

~oll€~tiye excitat:ions in nuclei an(l to treat these a8 bOBon~. In the lilA only pairs with al).gu1'il-r mOm~ntulll L=O (s-bosoIl) and L=2 (d-boson) are us",d. This crude apptol(imation already provides in most cages a r""sonable description of the collective states. One could, however. illlp1:ove this approl<imation by including other paira (g-bosons .•• ), which has heen u"ed in seve",l reports, see for instance (83Hey).

'fhe Illose salient feature at the IBA-model is th", finite numbe1: of bosons. This contrasts with th'" <;oollectiY" Illodel where the number of phono,\s is not limited and can become iIlfinite. In th" deterroifl.'il-tion of this boson number we r,;,ly on the nuclear shell model. Assuming that

lOW-lying exclteo states re~ult from excitations of v'il-lence nucleons,

it is q\.lJ. toe natural to take for thls boson number half the number of valence l\\.I.c1eons (or holes).

there are two version of the IliA. III the simplest one (IBA-1) one

does not distingUish between neutrons and protons whereas this

differ",\Ce is taken intO aCCOunt in IIIA-2 explicitly. ln the next se<;otion w~ will give a brief discussion of both versions. Though IBA-l is perhaps too s~mple for use in the. analysis ot (p, p') cl(perim",nts (V

pp ~ Vpn) we neverth"less will discuSS it bec'il-uBe of its simplicity and elegance, its application in the G" lsocope~ (84Bau) ano i~a basis for IBA-2. Its ele~aIlce shows up in the possibility to obt'il-in in ~ollle lJ.m~ ts analytical solutions by group theoretical methods. These limits, moreover, correspond to physically relevant situations.

2.3.1 The IRA Hamiltonian

The general Hamiltonian in IBA-, can be written with 6 parameters (we omit the b1na1ng energy part):

(2.30)

nd ~ _{(d .d),} + - _{p =}

₁

(d.d)

_{- ~}

(s.

~),

!:

110 [a+.(I](l),

2

= [dT;<

s

-+- ~+;<

d'(2l_

~17[d+"

d:](2),

!3

'" [d+l{ dj(3),

:&.

=

[d

x

:1] (

4) ,

where creation (s+,d+) and annihilation (5, (I) operators for $ ~nd d

bosons have been introduced.

This fOl:m has been very useful in phenomenological analyses, where it appears th~t only a few t~rms are 6ufficient for an accurate descri?tion of the speCtrum. Through g~oup theoretic~l methods we can now identify 3 limits, which bave an analytical solution. The: group Structure of the Hamiltonian 1s U(6). The three group ehains c~El be identified if one takes into mind that the rotatioEl group 0(3) alway~

ha3 to be part of each of the chainS.

U(5)::) O(S}:::) 0(3):> 0(2) 1 SUeS) limit

U(6) ..

E

U(3):) 0(3}:) 0(2) I I SU(3) limit

0(6):> 0(5):> 00):) O(Z) III 0(6) limit

In each of these cases eOrne patamete,s of the Hamiltonian become zero and the spectrum can be described with only a few pa~amete~s:

SU(5) limit: a O " a2 • 0 1I SU(3) limit: a_O a 3 a4

=

<;: 0 HI 0(6) limit: a 2 a4

" =

0

The~e ~!~!ting C~6e6 c~n be compared with the collective model of

26

SU(3) as the axially symmetric rotator and 0(6) as tbe Y-l,inst",bl" ..-otator. It io. however, not neces,HI);"y tQ stick. to these limits. 1:b1s is on~ of the nicest featu"-"6 of the lEA: we are able to study complex transition regLons in a rath~r simple concept. Examples of Sl,ich tra.nsition studies are the wo):k of scholt.en et al. (785ch) in the case of the transition from SU(5) + SU(3) and the work of Stachel et al. (82Sta), whe~e '" tql.n~ition from SU(5) .. 0(6) haR been found in the Ru nuclei. All calculations ~n the IBA-l can be performed by one programme called PHINT designed and written by Scbolten (80Sch2, SOSc\1l) .

As said before, there i~ a considerable difference betl<le"n the proton-proton and the proton-neutron interaction; th" latter being 5trollger by abol,it a fac.tor of 2·5 (79Von). Thi., means that in our

experiment we hav~ a pretere~ce for neutron excitationS. This le~ds to

the ne"d of pe.forming struct~re C31Cl,ilations witnin the IBA-Z context. The Ha~iltDnian in IBA-2 ls:

H = H + ~ + V ,

'tf V 'IT'\} (2.31)

wherG! rr denotes t1,e prQtons and \) thoe neutrons. Hn and H\) are simple boson Hamiltonians as in J:EA-1, while V"v expresses the .. tronl;: proto[\-neut.on quadrupole force. Mostly Vnv is written as:

(2.32)

wher~ Q" and Q_v are generali~ed quadrupole operators as defined in

(2.30). The factor 1/7 has beea replaced by an additional param"ter

X"'

r""pectively Xv' M represents a Majorana exchange force affecting only ~tates, which are not fully symmetric to the exchange of neutron and proton bas ana.

No,"" the Hamiltonians H". and liv can be tak.en in th" SU(5) Limit. The only parameters that vary significantly within a major shell ar" Xn and Xv' reflecting tne particle or hole charac.ter of the bOSQns (7BOts). Also for IllA-2 calculat;l.Qns there is a programme c.alled

27

2.3.2 Interaction potential in the IRA

In order to I;lpply the U~./I, Illodel in connection with <:oupled-channels calculations, it is nece~~I;lry to specify the transition operators between the various excited states. In a gelleral form we filld for the transition density (84BI;lu):

monopole trans1t1olls: quadrupole trl;lnS1tions: hexadecapole transitions: with: A(2) .. i f ( f (sTd T d+s/ 2₎ Iii> B(O). i f

<

f (d* d) (0) II i (2) Bif = < f ( / d) (2) II i > (4) Bi,f = ( _f (/0)(4)11 _{1 >}

The reduced matrix elements Aif I;lno Bit can be calculated uslllg the !6A programmes With Suitable Hamiltonian •• The coupling factor~ I;l2(r) alld bL(r) callnot b~ derived from IBA principles. In fact, they represellt something lik~ a boson density in the nucleU8, which has not beell defilled ill ISA. Several authors r~port On this matter: Demartel;lu and van Hall (S2Dem), Cereda et al. (82Ce04), alld Morrison and collaborators (80Morl, 80Mor2, 84Amo, 84Bau), with arguments based on analogy to ~he geollle~rical collective Ulodel, which as "tated above, contaills the limits of IBA-I. Their conclusions are essentially the Same· Only in the case of bOer) they ~ive different results and this is mainly caused by Scarceness of data. We shall follow the formulas of Bauhoff and Morrisoll (84Sau), which are:

(1/5) II. R

~

202 dr (2.33)

,,8

(2.35)

Fo~ tl,e boCr) We mention ,he two possibilities uBed by BAuhoff ,,"ad Morrison:

or (the breathing-mode form facto~):

~OZ2

{3 U(r) + r

~f

(2.36b)

This lea.ves 1.16 with four free parameters _{k022' k202 , k222'} a.nd k422'

which are determined by fitting the eXi?e,~mental data, The pal:ameterg

k "roe constants fo,," a ~an!le. of nuclei having \:he same structure. A

change in structllre will also give a change in the value of the

parameters k. For the germanium isotope~ Bauhoff aad Morrison have

worked this out and the resulting values of th" k I S are p,esente.d in

29

Chapte~ 3 Bxperimental setup aud analysis of the e~per1mental data

3. Introduction

"There 1s no hlgh~r or lowe!;" knowledge, but one only, flowing out of experilllentatiorl."

Leonardo dol. Vinci.

In this chapter we shall give a description of "he requiSites to carry Ol,lt pol.a.ized-beam experiments; f~om the production of the

polarized proton beam to the electronic data acquisition. De[ailed

informat~on has been present;,d already in the ~hese8 of MeJ.ssen

(78Me1). Polane (alPol) and Wassenaar (821)1('8). Sinc,;, the time they performed their expeJ:Lwents, sev~ral item~ have been changed or

impro~ed. For the sake of completeness we shall present in this chapter a ~u~vey of all COmponents in the polarized-proton scattering

£a<;~aty at the cyclotron labor"'tOl;"y of the Eindhoven UniversUy of Tecnnology (BUT). In case not ",IL details are given, more i(tfol:'mation can be found in the theses ment!o\\ed above. In th.e next sections we shall die"us~ the following items,

1. ~rodu<;t~on of the polariZed-proton beam 2. scattering chamber and detection 3. lllonttoring

4. data acquisition 5. experimental procedure

6. data handling and data ana1ysis

7. experimental cross sections and analysing pow~rs

l.~. Prodn~tion of the pol~r1zed-proton beam

For th", productiol' of the polarized-protol' beam we use.d an ion sOurc" of the atomic-beam type. The theoreUc.:>l background. and the description of ~u~h a sOur(Oe call be found in g"''',[!l'al p.:>pe!;" a~ for Instan(Oc Baeberli (07Hae), DOnally (71Don), Glavish (71Gla) and Clegg

(70Cle). A comprehensive review of the various techniques ~an be found

in a pil.pe~ by Clausnitzer (74Cla).

The pola~ized-ion sourc" at the cyclotJ:on of the. EU'r has been developed and constructed by Van der Heid., (72Van). Originally it

produ~"d on the ave~age 3 ~A of 80% polarized protons JUBt behind the Wien filter. In th" fall of 1981 a I'ew ionizer (ANAC) together with a cryogenic pump was installed. The imp);oved vac.uum conditions in the ioniz",r (pressure a factor of. 10 lower: nOW on the average 2.10-"1 torr) raised the degree of polarization

ot

the be.am to about 90%. Th", tanhe" itself was responsible for thE! higher curre<}t of 15 J.lA on the average. The "figure-af-merit" 1'21 inc~ea6ed by a fa.ctor of 6. The

6wit~hing b"tween the two tran~versal directions ot the proton spin is performed by reversing the msgnetic and electric f1el.d" in the Wien filter.

Since thc cyclotron of the EUT has no ~acility for axial

lnj"ction, a dlffeJ;ent method had to be Ilsed for the inject ion of the

polarized protons. 6eurtey ~nd Durant (67Beu) developed a radia.l

injection device for the identi~al Sac lay cyclotron. Here the 10ns are

guided through the magnetic field, ~omp",nsating the Lorentz for(Oe by a" electric field which 16 praduc.ed by appropriately shaped

electrodes. The injection system for the polarized beam at the

cyclotJ;on of the EUT is an exact copy of th" Saclay system.

The injection system, howev"r, is not easy to handle. First of all the "l,,(Otrodes a~e c~rrying a high voltage (betw",en opposHe electrodes 20 kV). So the vacuum conditions :l(ls~<;ie the (OJ(llotro" have to be very goo<.l (better then 10- 6 t.orr). This waB realized by Ilsing "a

eold fineer" in the cylotron. A second diffi~ulty is that the space

b"tw~en two major injection electrodes h Dnly 8 1IIIll. The accelerated beam has to pas8 through this gap "very revolution, in total about 300

t~n>e9. So t.he beam wi Ll b", c.ut off i f it is nOt very s table in t.he vertic.al direction. The solution of this problem was to excite t.he lnn"rmost pair of int.ernal correction c.oils asymmetrically. 'together

31

wi th tlle duty factor ... n4 acceptanc(J of the cyclot ron for the ioo.

source thi~ re~ulted in an extracted polari~(Jd bea~ with currents of 100-150 <lA.

The extracted beam can now be guided to ehe scattering chamber. The beam tr?nsport is d(Jsigned to work in either of two modes: doubly achromatic or dispersive (70Hag). We used the last one in view of the imprOVement of the energy resolution (see section 3.8.1).

3.2. Scattering chamber and detection

After the beam has arrived at the experimental area 40 m further downstream the line, it enters the scattering chamber. This chamber has an internal di<lmeter of 560 mm and ... height of 90 mm. It contains the target~ ... nd the detectors (see figure 3.1).

~~A~

__ ._. __

--~

DETECTOR ~OLARI~ATION MONITOR DHEqO_RS;---r-___ \ DETECTORS SCATT~RING CKAMBER

Figure 3.1 S~he~t1~ v1ew of the scatterin~ ch ... mbe. and the

3.;a.l Targets

The targets are placed in the centre of the chamber on a

~otatable disk, which ~an accommodate ei~ht targets. Always one position is re~erved for a diaphragm of 3 mm diameter to be Ilsed for beam posHioning (s",e sectton 3.3.3). A second position Is occupied p<':rmaael1Cly hy a mylar foil [(C_l0H₈0

4

>r..l

for C,jI.lUlration purposes. The target holder is "ontrolled electrically, so facilitating a q,d"k interchange of varioLls tar-gets.

Table 3.~ Isotopic composition in % of the Ge targets a8

specified by the manufacturer.

Target 70 72 73 74 76 =======~~~~_~~=========~c~

__

~~~m~========DD_n~R

__

'IOO_{e 84.62 5.54 1.47 6.36 2.01} 'l~e 0.75 97.85 0.41 0.80 0.19 74Ge 1.71 2.21 0.90 94.48 0.70 nOe 7.69 0.05 1.69 10.0B 73.89

Table 3.2 Isotop!c composition in % of the Be targets as specified by the manufactu~er.

Target 74 77 78 BO

0.23 84.l4 2.99 4.32 7.07

0.06 0.63 0.69 91.74 6.35

82

33

For the measllrements we IIsed several targets with isotopieally

enri~hed material. All targets, for the experiments des~ribed in thi.

tQe6~6, Q6Ve been pro~uce~ by AERE (~6rwel1). Se1f-supportLng t6rgets of germ.anium and selenium were not available. So these targets were ooade by evaporatins enriched ooatet'id On a c<.'\t'bon f011. (thicKness 25

~gf,;;1I\Z). 'l'he !,ve""ge t;QLckneu W65 !'bOUt 200 US/cmZ. The hot;opic

compositions of all target5 used lire li5te~ in t!'blea 3.1-3.2.

Selenium is a diffi~ult target material. Several authors report

On rapid deterioration and eubli_tion of selenium t!'["gets.

Self-supporHrtg selertiUIll target" can ultimately with"tand 10 nA of 50 MeV

protons (79MaZ8). So in most cases one chooses to sandwich the

selenium between two layers of carbon~ In previous experiments we also

tried targets made of selenillm between a s<lndwich of carbon <lnd

aluminium. The contrib~tion of al~mini~m ~nd c~rbon~ however~ was so

<1oJll1n!'~1ng in the spectra thst tQe relevant selenium levels could

hardly be seen· Therefore selenium targets with a carbon bacKing we~e

"sed. Then the problem arises how much curreot these ts~sets CSfl

withstand withollt eV<lporating. To investigate this "e m!'de sever(>l ta._gets of "atu~(>l ",elen~um on ;ii. c!,rbon b;ii.Cking at the KVI

(Groningen). By bomb!'r~ing theae targeta wi~h 22 MeV protons we co~ld

deduce that the maximum current withollt deterioration is !'pproxim!'tely 7'1 riA. For curr@nts higher than 100 nA deterioration and evapor"tion became rtoticeabh. These values are cOllsisteat with tl}e reslILts o£ BOJ;"s(><u et d. (77BolS) of 100 riA for a carbon sandwich typ~ targ~t.

For our experiments we decided to put Ii s!'te upper limit of 50 nA on the current.

3.2.2 De~ector8

The scattet'e~ protons are detected by semi~onduc.tor detec.tors. These ~etectoJ;"s aJ;"e mounted in two detector blOcks with four positions

eacn, so th~t in one run one can measure at eight dngl~s

simultaneously. The tirst block is used ffi;ii.in1y for the forward

dlrectl0rt (20Q

-900) and the second one for backward !'ag<es (706

-165°). In or~e" to keep the counting rate in the forward and the backwar<1

detector block at the same level the solid a~sle of the ba<;kward