direct nuclear reactions with polarized protons : an ...direct nuclear reactions with polarized...
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Direct nuclear reactions with polarized protons : anexperimental study of Ge and SeCitation for published version (APA):Moonen, W. H. L. (1986). Direct nuclear reactions with polarized protons : an experimental study of Ge and Se.Eindhoven: Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR242824
DOI:10.6100/IR242824
Document status and date: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.
De eo-promotor: dr. P.J, vau Hal)..
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
LCVCllsloop 181
iv
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
properties, but still, a single model that: de5cribes the nucle\lS
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 fi 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 [email protected] 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 it 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
performed with the DWBA formalism. This procedure is valid for ~evels
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
in chapter 6 We shall suIMl.3rize our final conclu8i"n~.
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
directiOn e normali&ed to unit incident particle flllx (un1t is
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 "+" in4icates 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" - W1 / 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)
1FT 1+(8) + I_(e) (2.4)
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 Btot
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 ~ n n
(2.7)
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 it 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,ri,a
1) -1 Ws g(r,r
i,a
1) +
lm~c12 Vso ~ :r f(r,rso,aso ) ;.i (2.9s) 11
(2.9b)
11
g(r,rj,aj
) ~ 4 aj
df(r,rj'aJ)
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, if 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
Such a D~SA analySiS ia clearly insufficient in two cases:
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 BA
,
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 WAu
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 8i(1=l,2,3), which give the orientation of the ngc1eu~
r~lative to the iaboratory system ~nd tne (static) quadrupole
deformatioos aO and R2 ' Instead of aO and a Z 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=0 0 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
axis:I
+
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 h 2 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
Figure 2.2 The ~-y plane.
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 r the band miRing co~fficients AKi depend 00 the asymmetry 1. The
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 Aii
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 0 0 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 I
Some relevant ~oeffi~ient5 ~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$5 0.0610
27.0 -0.9467 -0.3221 0.3221 -0.9467 0.8039 0.5909 0.0682
Z7 .5 -0.9366 -0.3498 0.3498 -0.9368 0.7924 0.6052 0.0759
28.0 -0.92~5 -0.3786 0.3786 -0.9255 0.7813 0.5184 0.0843
28.S -0.9128 -0.4085 0.4085 -0.9128 0.7704 0.6307 0.0934
29.0 -0.8985 -0.4389 0.4389 -0.8985 0.7599 0.&419 0.1031
29.) -0.8829 -0.4695 0.4695 -0.8829 0.7496 0.0521 0.1137
30.0 -0.8660 -0.5000 0.5000 -0.8660 0.7395 0.6614 0.1250
y 4
AOZ 4
A22 4
A42 4
A03 4
A23
4 A43
25.0 -0.5232 0.8425 0.1285 -0.0310 0.1318 -0.9908
2$.$ -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
EZ /E Z ];,(E2,2 1->O) b(IlZ,2 z->O) ];'(E2,2 2 +2 1 ) 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..2 1)
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
rotator model as function of the s~ymmetry parameter y.
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 Rj
(= 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
o~thosonality is maintained.
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 (Bpp ')
and neutron scattering (~nn') should fulfill the following
t;e.Lation6h1ps:
aem ~ 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.!Bem
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 aO
' 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' ar
) 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
potential). Such a quantity appears to be rather constant a"d it is
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
(Vpp
~ 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) II 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: aO " a 2
• 0
1I SU(3) limit: a O a
3 a4
= <;: 0
HI 0(6) limit: a2
a4 " = 0
The~e ~!~!ting C~6e6 c~n be compared with the collective model of
Eoh~ and Mottelson. We can identify SU(5) as the anharmonic vibrator,
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 Qv 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
N~80S, w~~~h has been designed and written b1 Otsuka (790ts)'
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) .. if
( f (sTd T d+s/ 2 ) Iii>
B(O). if < 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)
(2.34)
,,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
t~blc Z of thel~ paper (B4Bau).
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
8. resolution.
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 if 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
pola~i~at1on monitor (horizont<ll cross section).
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 [(Cl0
H8
04
>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 __
'IOOe 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
OS)
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 (20 Q -900) and the second one for backward !'ag<es (70 6 -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
detectors is 4 times the solid angle of the fo~w!,~~ <1etector~.
34
Table 3.3 Properties of the detectors.
typ'" det~ctor S:I.(LJ.) 6urf"c:e-b"rr V,r
(8nertec) (Ortec)
<lcdv" area 100 mm2 100 mm 2 100 _2
active thickness 3000 )Jm 3000 U[iI :1.000 urn Cllo:e(lt at 20" <; 1-5 IlA 1-2 \11\ 1-2 vA
current at -s" C 0.2-0.5 VA
ex resolution 20-35 keV lS-21 keV 1.5-20 keV
In O\.lr fir~t experiments \ole used surface-barl"iel" silicon
detectors (Ortee) of a thickn"'ss of 2 _ or :3 mill. Since the proton
energy ""s 22 MeV the 2 mm d€ce:ctors were not tIde\<; enoush to stop
these protons. Thel"efore they were pl"""d at 45" with respect to the
direction of scattel"ing. The8e surface-barrier detectors gave a ~O(lg
tail on the low en.,rgy side of the peaks in the Bpectra, due to
improper charge collection. So in 1982 iii'" tried a new type 3 mm
silicon Li-drifteo detectors (Enertec). These proved to be ml,lch better
alld gave less tail cOlltI'ibution. Tl>e me(>,>urements sillce that time have
beell don",. as illuch as possible, with these detectors.
3.3. Monitoring
3.3.1 Out-ot-plane deteetoI"~
III the scattering chamber there is also a pair of monitoring
detector ... These detectors have been poaitioned at about 35" at both
sides of the reaooUoll plalle (out-of-plane detectors). They are used
for the relative normalization of the c~oss section~. The absolu te
normalization is Obtained from the fit of tl>e optical model
parameters. Another purpose of the ol,lt-of-plane detectors is co
.;:ontrol the ""'itching time for ~eve~sing the polarization direction.
After l",aving the l;I<:-at.ce,Lng chamber the beam enter", a second,
smaller, scat.t.ering cnamber: the polari~ation monit.or. At the entrance
of thi", pola~L~atlon monitor (diameter 20 em) there are some aluminium
toils to degrade t.he energy. We~t the beam hits a pOlyethylene foil in
the cent.re of the chamber. Scattered protons ",re det;ected in two
detectors at 52.5' in the horizontd plane on both sides of the beam
and in one out-of-plane detect.or. The p.;rposc of this setup is to
measure continuousry t.he pola,L;J;ation of the beam.
The analYSing power of 12C is well known at several energies. In
t.he euel:gy range of 10~30 MeV measurement:s have been "erforllled by
Meyer er. ~l. (76MelS) and Gaillard et al. (7QGal, 790a13). It appears
that the analysing power of 12C around 55· is nearly <:-onsta~t 10 the
energy range of 12-18 MeV (677. ± 1%). Therefore t.ne beam energy was
deg,aded to 15.5 MeV by the aluminium foils mentioned earlier. By
measur~ng the asymmetry 1n the cOunt rate tor the spin-up and spin
down direetions We ~an deduce the degree of polarizatiOn of the beam.
The out-of-plane dete~tor is used to correct for possible differenceg
of beam inten6~ty during the measurements. So lon~-t.~rm vari~tLons 10
healll polarl~atlon can be detected.
Consistency checks are made in two other ways. First we alWayS
meas~re t.he ca~bon ~nalysln8 power together with our other
m~aSurement.. These dato", I":e a~so analysed and compared wi th the
analySing power dat.", of GailL~,d (76Ga1). A second check is performed
in the optical model fits. In this ~ea,ch p_oceduro the nOrmalization
of the analysing power ~an be introduced as an additional parameter ~n
the fit, When the difference between this param;,tet" and unity i",
within the limits of th;, error calculated from che nt, we can be
reaeouably sure that our data are reliable.
3.3.3 Beam dump and measurement of the beam eurrent
At t.ne eoct of the beam line w;, find the beam dump. It is located
3 m aw~y from the scattering chamb.;,r. This beam BtD~ cone!sts ot a
circular sr~phLte ctisl<; with a diameter of 5 em and a graphite 1;ing
with ~n internal diameter of 5 em and an outer diamete. ot 7 cm.
36
It se~ve5 three pUrpGs~~:
1. to dump the unscatte.red part of tbe be~m;
2. to measure a fraction of th~ total beam Current
3. to adjust the beam pos~tion.
The measurem~nt of fracttone of th~ beam CU~rent is performed o~
three plac.es:
~ the current on the tsrget
- the current on the beam pipe and walls of the scattering chambers
~ th~ current in the beam dump.
The curro!!nts OIeasv,ed on the h.s [ two place~ ;ore a18G used as
input for a current inte~rator. This integr;otor ~iv~,;; a pulse for
ev~ry 10- 8 C collected charge. By counting these p\llses in a scaler we
can derive the total integrated current after an experimental run and
this ~$ ~leed a8 a ch"ck on the normalization by means oe the out-of
plane detectore.
For the pGsitioning of tbe beam in tbe scattering chamber the
follOWing procedure was \lsed. Fi-r8t a beam stop at the ~ntrallce oe
the scattering c.hambe.r waS put into the beam. the current on this stop
waS maximized. Ne~t the diaphragm of 3 ~ in the target holder was put
into th~ beam. We then minimized the beam curre.nt On the ta~get frame
of tr,e mm hole 8.1).d maximi~~d Simultaneously the current on the
central beam dvmp. The ratiO of the cu~rent Dn the 3 mm diaphragm and
the Cu1:J:eot on the beam dump hac;! to be at least 1 ;400, in ord" r to
sufficiently reduce the background scattering from th" target frames.
Using chis proc.edure th~ divergenc.e of the beam at the position of the
target is at mo~t 8.6 mrad.
3.4. Uata a~quiA1tion
Scacter:ed partIcles are "topp"d in the silic.on detectors. The
detectors give a charge signal from which we can deduce the energy of
the pa"("ticle detect~d. Each detector is tollowed bj an electronic
chain conBisting of standard NIM electron~cs (see figure 3.2). Such a
chain i5 dl'J1ded into two paru;: the energy (1':) and the timing (t)
part. The timing signal i8 u""d for triggering the data-(l."quisition
ayotew and for a three~bit detecto~ identification.
A~ter amplification and shaping ;(n p-re-aI1lpl1fier and maio.
amplifier the e.nergy signal l(l. gat"d thro~gh a linear gate stretcher.
37
All Outputs of the gate stretchers are mi>:ed together and the II>l)Ced
signal is sent to the ADC. The convClrs1o .... and the "torage o~ the
(l"e .... ts in the external 38k MOS lI>e1l\0:r:y J.s controlled by the data
acquisition system (see figure 3.3). the features of this 6y~tem have
been described by De ll.aa£ (7~Der, nDer). These controll(!rs work
independently of the computer. The computer (a PDP 11/23) is used only
for controlling the experime .... t. For this purpose a program EXPO has
been deve),oped by N1jgh (SiNij). This program provide", u'" wHh all the
.... ecessary c.hecks tor starting and termi .... ating an experimental fun. It
stores the data on'disk (RL01) and 0 .... floppy disk (RX01) for back up.
It can perforlll an amplification test, a beam p0131&ion test or a
.;letector leakage current measurement. It also can give the channel~
with maximum contents and. it can display th,:, "peCtoa while being
collected.
Test ~. L---,-p_u'_$e_'_...J1 V· '.-
Routing unit Mixer
F1gur~ 3.2 Electroni~ c.h~in for one .;letector.
1)esides the sigllals of the detectors Dl-D8 also event:s in the
mOllitor detecto~6 a~e processed. Tb.", ",lectJ;onic chaJ. .... s of the
monitoring detectors consist only of a pre-amplifier ~nd a main
amplifier/si .... gle channel a .... alyser (MA/SCA.). The output of the single
channel analyzer is counted in a sca~er, .;leveloped at the EUT. It is a
38
module of the EUROBUS system (79Van), which ~e ~onnected th~ough CAMAC
with. th.e PDP 11 /23. When a"(l experiment is completed the events in the
scalers and SOfie additional experimental data are stored together with
the ;5iIpectra ..
p 0 I a r i m
" t e r
c M c 0 0
A ,., e ,., D D t m t r A C r 0 r " 0 r <) 9 S
I Y I CAMAC EUROBU5
Figure 3.3 Schematic view of the data-acq~i8ition system.
3.5. Exp~rimental procedure
Since the c.ross sectiolls we wished to measure are rath.,r small
and tho! current of poJ.e.r~zed protons on the te.J;"get is [lot too h1gh, we
had to schedu~e Q~r experiments e."(Id preparationS. I~ ge"(leral we
l'l;\lll',ed an experiment every month Qo>:ing a wnole week. Before Buch "
week the whole equipm"ilt was checked and set appropl;iately using an
uopolarized proton beam.
As soon as the pol(>.rized proton beam became available fo. the
eXpel;~\l\ent a final check was performed on electrolli~B and settings.
Next w~ determined the number of counts 1"(1 the control unit in Buch a
way the.t. ji.n experimental ruu would last about one hour and that the
spin ctlrect~on w(>.s swltehed about. every minute. The mes~~rint periods
39
we-re- lopg e-nough to collect a reasonable number of event~ h' the
spectra for one run whithout risking amplificatiOn dri£t~, be~m
qualit.y <:h~nge", o~ loos;\1'8 a whole experiment by SOIlle fatal error
(operational errors Dr problems with th", b",am). The total number of
experi_ntal runs n",eo.ed was d"'termined by demanding the numbo:.r of
events in the levels of intero:.st to be at least a thousand for most of
the dete~tors. In this way we usually measured four angular aettin£s.
thereby obtaining an angular distribution of about thirty points,
running from 20· to 16S·.
Since all targets contain SOme hydrogen, ,,-arbon and oxy£en, aa
["eBulting f~om t:a~get preparation and oxidation, we could use the
correBpond~ng pea~s !n the spect~a for calibrating and checking.
Moreover we always !ll.east.l1:ed the angular distributions of a mylar
target in between every 3 or 4 spectra of the target und~r
inl1~stisation.
We &1"'0 det:",rmineo. t:he <:loHd angle of tl1.e deteccors relative to
ea"h other. To this ",nd we pe~to~med a scattering experiment on a
polyethylene foil, thi"k enough to give " high counting rate. By
meas~ring at: the same angle with differe~t detecto~s we could deduce
the relative normalizations.
3.6. Data handlins and data analysis
After a meag~~ing pe~iod of abo~t one week we had collected quite
a lot of spectra for ea"-h angular setting. For the analysis all these
spect ra co~ld have been analysed seperately. The mos t interes ting
pea\<s, howeve1;-, did not conta!n eI:>ough counts to att:aiI:> 6uUicient
~tati6t:ic~l aCCU1;-acy witch respe-ct too t:he background. So these spect1;-a
had to b~ added to one single spectrum. Since the primary spectra had
been collected over a long period, there might be slight shifts in
amplH1catio(l. In the addition pJ:'ogJ:'am SUMS theJ:'e are pl:"ovisiofls to
correct fo~ ampliiicat:ion shifts. The addition ot a spectrum to
another one is performed on the basis of a chi-squar~d criterion (X Z).
If the value af chi-squared is larger thao a certain limit, the
prosramme tries a shift procedure on the basis of a previous
COmparison of the calibration for both spectra. the addition is
rejecCed if the value of the chi-squared remains above this limiC,
otherwise the shifted spectrum will be added. In correctiog for
40
alllpliUc"t~on Bhift~ we essenti"lly tnaincain the resolution. The
omission of correc~ing procedures would have gi~en rise In so~e
spflctra to a slgnH:l<:,,,nt loss of resolution (especially at forward
angles) •
After having added "U relevant spectra we obt"in four =pectra
corresponding with the four al1gu1". ~e~ting5. These al:e "r".lysed by
means of the pl:ogr"mme BIOTEX. The callbl:"tion of each detector is
obt"ined by analysing the myl"r spectra. The aMlyeis ,.i~h the
programme BIOTEX is an inter-active way of peak processing, by setting
markers, calc"lating a suitable background lind integrating the peaks
by simply adding the content'> of the 9pectrulll chanDels. An indication
of th~ real peak pOSition, peak width and peak asymmetry ie calculated
from the statistical moments. Thi~ is done with both ttle "pin-up and
Ehs spin-down spectr~m of th~ sam~ detector, so it is easy to
calculste "n experimental crosg section and an<llysi!1g power
simultaneouslY. FOr dUficult cases a fit procedure with a ga"ssian
profile was "sed (see. 8SPet). The ~esuHs of the analysis fOI;" each
spe.ctrum are writteo on a result file which com be used for further
processing.
The "ext step is to coml;>ine thes;, r",sult files and to trallsform
the laboratory data ~nto centre-of-mass data. this is performed by the
prograoome CMCONV, ~hieh also takes care of the different
nOl:008.lt~aUon8 of each detector and of possible cor:rectionB in the
analy"ing pow~r. This programme the" produce'> the output data on a
file which can be used for transmission to the EUT central comp"te~
(£7900). Ae~e ttl" data are incorporated into the data file N~WEXPDATA.
This file contains all the experimental scattel.""ing data of the Ol.!clelir
physics group and is accessU>le for th~oretical programme~.
3.7. E~perimen~al cross sections and analysing powere
The analy~iB of the peaks gives th~ number of events in a
spectr~m. TQ convert these numbers into experimental dlfferent~al
crOB8 sections and analysing powers several calculations have to be
performed. For th~ dlfferent1"l cross s~ctlons the following proced~re
is used:
,. normalize th~ number of events by the counts in the out-of~plane
detectors;
2. mak~ appropriate co~~ections for di~~erences in detection
etticiency for each detecto~;
3. multiply with the £acto~ for the transformat~on of laboratory
system to centre-of-mass system.
Fo~ the calculation of the analysing power" the pX'ocedure: is
slightly different. Al~ common factors for spin-up aod spin-down
events, as for iostance detectot efficiency, cancel. This also implies
that the clllcuh,ted analysing pOwer in the laboratory ",y$teOl is eh«
Same a8 that calculated in ehe centre-of-mass system for the
co~responding llngle.
All experimental data extracted 1n this wlly are affected by
experimetl.tal errors. woico stem from vllriolls sOurc",s. The errOrS in
the differential cross sections result from,
- th« stati",tLca! error in the numbeJ:" of ellents ineludh1g the
bll~kg~ound s~btraction;
- the statisti~al e~ror in the counts of toe o~t-of-plane detectors;
- the errors Conne[!ted with the selleral norm"Uzations;
- the "YBtern"Ucal errors due: to mis",Ugnments.
The analysis program aCcount~ fo~ statisti~al errors only. The error~
due to all the normaliZations as well as the systematicsl eJ:"to.s are
nOE known accllrately. We halle chOsen to put a !ower lImit On the total
errOl;S to account for all kind o~ errors We do nat know. For the
differential ~rOS8 Section~ this lower limit has been estimated to be
2% of the experimental value.
The error~ ~n the analysing powers are due to:
- the Statistical error ~n the numb~r of events including the
background subtraction;
- the error in the calculation of the degree of polarization;
- the systematical errors due to m~eaUgntnents.
Also 1tl. this cas!;! the analysis programme. accounts E01: the statistical
errors only. For the systemattcal errors we also use her", a lower
limit. Sin[!e the analysing power is alre",dy a r",lati'le t'lumb<i!r the
lower limit is a tixed number herE, wh~ch we estimate at 2%.
42
3.8. Resol1.ltion
We have studied variolls methods to improve the r,;,solutio"(l ",inc"
we are interested in the so-called two-phOnon multiplets, which are in
most ca6e~ weakly e~cited levels. These a~e located in an energy r~nge
where the l,;,vel spacing already become", small. So it is necessary to
have a good resolution to separate th", variollS levels of lntet:est:. The
final r"solution <)ne obtains ia an e}{perim"nt is a combination of
s,;,v~ral fact<)rs. SOme of these a~e'
- the energy profil,;, of the pr:oton beam,
- th" stability of the _an energy of the proton beam,
- the divergence of the prOton beam,
- the target thickness and the target angle,
- the 'lngular acc"ptance of th", detectors,
- the detector noise, and
- the electronic noise in the data-acquiS~tLon syst"'nl.
3.8.1 Contribution of tbe energy p~of~l~ of the beam
The ex~ra~ted beam of the cyclotron h~~ an energy spre~d of 0.4%
(BO k<.;:V) which he<;omes a dominating c.ontributioG to the worsening of
the resolcLtion. For that l:eason we decided to analyse the be~m through
tt.e di,;persi ve mode of the beam transport system. Here iii'" enCQ\.lnter
th., di.lemma of the competition between r"Ji'olucion and beam intensity.
The decrease in beam 1nt",nsity in the dispersive [nod., ~'" in first
l...,stan<;e proportional to the improvement of the resolution till the
energy profile of the beam has a width of about 0.1%. A further
improvement of the resolution would result in a quadratic decrease in
beam J,(1tensity (70flag). We selected an eneq~y "idth of 0.1% which
resulted in "n """rag'" beam loss of a fa~tor: 4-5. So the increase in
int(c[).sity of th'" ion ,;ource "'as sacrificed for iml'roving t\l'" energy
profil~ of the be"m. Another adva.nta~e of the dispersive mode i~ that
the stability of mean energy of the be"m 1,; much lJ",tter. Any energy
shift of the beam will result 111, jI. noticeable <J.ecreaSe of the
intensity wh1c.h is a s1SI1 thlLt the ,;cttll1Ss should be checked.
43
TarSfJt 0 .. d8sr.elR' t~"9 .. t J S.. dll9PIIIUI
100 100 90 SO 80 SO 70 70
"- 60 "'" eo .. .. .., ..:0: SO 50
J: 40
I: 40 II: '" ,. ,.
"- 30 30
20 20
10 10 0
0 30 60 SO 120 150 180 30 SO 90 120 150 180
a~.b !d~91 1I~.b [dll9 l
r~,..set 30. dtlg"'~Bft farg8t -30 r drJtJ~.t/.tI~
100 100 90 90
60 ao 70 70
»-60
,. 60 .. ..
.>: -'< 50 50
'" 40 " 40 :J: '" '" '" "- 3D "- 30 20 20
10 10
0 0 0 30 60 90 120 150 180 0 30 50 90 l20 150 leo
BLob (dog) BLob I doS'
Figure 3.4 The effect of the kinematics of the reaction on the
resolution. For the ~~l~ul~tion ~l60 the et~ect ot t1n1te
si~e of beam spot, the energy profile of the beam and the
finite solia angle ot the detector a~e taken Into
account. (ll: = 22.3 MeV, Il.E/E = lO-O).
... 3.8.2 Contribution of ki~matlcal effects
A s~cond conttibution to the decrease of resol~t10n can be found
in kinematical broadening due to finite sizes. We can assign
~ontributlons to:
- the beam spOt
- the >:loUd angle of the detector
- the target thickness and the target angle.
The stability of the position of the beam spot on the target i5
of ilnportan.:e not only fox: til", resolution but aLso for to the accuracy
in tile det",rminatiOn of the scattering angles. When the position of
the beam spot is changed a Un 1e bi t, the scattering angle of the
protons changes accordingly. D,.e to the kinematics of the r,;,~ction
this r,;,sults in a "mall change in energy of tli", scatte);ed particlcs.
The final r~sult is b);o~dening of the peaks in th~ spcctra. The remedy
is to check regularly th" position of the be~ffi in the aca[teri~g
chamber. This ef~ect turned out to be not very time dependent, ~o that
this check had to be p",rforme<;l only a few times every day.
Another contribution can be assigned to the divergence of tile
beam. The reaSDn for this divergence is our way of pos i tioning th~
beam. There i~ an altern~tive w~y to pOSitiOn tohe beam. For, whe.n we
p~o<;l"c", a beam f()~us at a distanc!! behind the target equ~l to til"
<;Ietec toor distance, the spread in the seato tering angle wJ,ll be less.
Such a p~ocedure works only well when all <;Iete<;:tors ar~ "t the same
distance from th~ t<lrget. Moreover, in o"r scatte.ring chamber the
distance of the forward detector block 10 twice that of the backwa);d
dete~tor block. In this altern"tive procedure it is not easy to obtai~
roillima1 b"c.kground of the targel holder as in the procedur", actually
used. So we stuck to the old method. The eft~c~ of the divergence in
thi" situation can be seen a6 an additional angular spread of 0.5
degree~.
The fin1 ~e solid angle of the de~E!ctors and the finite Size. of
the heam spot play a ~jO); role in the klnem"tical contribution to the
resolution. TheBe ~ffects can be est~m"ted from kinematical
calculations (see Ugure 3.4). Also here \ole enCounter the cO~nt rate
v"rsus resolution dil~mma. Sinc" the resolutiOn is dominated by the.
e!\e:r:gy width of the b",am and since the targets were very thin, we
chose ~or a oolid angle in ~uch a way that most of the effeCtiVe area
45
Qf the det~ctor ~as used.
3.8.3 Contribution of the ~~tget
I:\,ny worsening of r@solution d\1e to the target is o:!aused by its
th~c~ness. First the finite thickness causes diff~r@nces in path
length of the protons through the target material. The path length ~s
depenQent on the location in the target whe~e the reaction takes place
and the target angle with respect to the oeam (effective target
thickness), Also the Q-value of the reaction has a small influence on
th~s effect. Two situatiOns can occu~;
- Bcatcet"ing in transmission mode i.e. tl1e particle ent",rs the tar,!\et
at One side and leaves it at the OppOsite side;
- scattering in reflection mode i.e. the particle enters and leaves
the target at the Same side.
ln reflection mode the various path length~ can va~y from zero to
twice the target thl~kness. 1n transmtss10n mode the path lengths are
scattered arounC a mean value. 1n tais mode the eerect can be
(Jllnlmi~"d if we create" symmetric situation: tae tar,;et angle is hal~
of the Scattering angle. It is obvious that this effect w~ll g~ve ~ise
to broadening. In figure 3.5 three situations have been worked Out and
this gives a quantitative estimate for the energy d~tte~ences. In our
experiments it was not possible to uS~ the bisector method, sioce we
used ~91."; forward detectors at the Same time. The.efo!:e we set the
targ~t at an ave~age ang~e with ~espect to all four detectors.
Another effect of the finite site of the target is straggling in
the energy of particles travelling through the ta~get material. This
effect is in general very small for thin taq;ets. If we use the
straggling theory of Bohr {48Boh) I<'e find that the atraggUng width W
fo. the nuclei of Zn, Ge and Se is given by:
W D 8.4 It (keV) (3.1 )
with: th~c~ness 1n mg/cm2
SO W~ s(!;, indeed that this effe<;t i", negU ble fox: the Ge and Se
targets,
46
10
" 1 "-.
I
~ 1 /
-'£ I
ill I I -<l
f 1 I
1
1
I I I
1
" I
1/ 1/ J'
II II
0.1 Ii
0 30 60 90 120 150 180
Glob (deg)
Figure 3.:; The influ,,;\ce of flnit.e t.arget thick.ness:
the waxim~l energy diff"rences as a functiOn of sCattering
angle for three target positions: -30°,
(dashed, solid and dot.-dashed respectively).
3.8.4 Contribution of the detectQrS
0° ,
De[~ctor qnal1ty and detector noise are the most probab~e
contributions when we look at the det"ctors. The quality of the
d~t.ectors 1s an i~portant factor for the resolution. Radiation damage
47
is one of the main causes to reject a detector, but also imperfections
in the production of detectors 1s a $ood reason. In the case of Si(Li)
detectors One out of three had to be sent back to the manufacturer.
The use of Si(Li) detectors improved the resolution compared to
8ur£ace-parrier detectors. The main improvement was found in a shorter
and lower tall on the low energy eide of a peak.
Noise is an intrinsic ~roperty snd 1s connected with the leakage
curJ:ent through the barrier. A ~ul;ther improvement "'as realized oy
cooling the detectors down to about -S· C. The leakage current drops
in mast c",ees tl:'Om 3 ~A to 0.3 ),IA. From his measurement" Wakker
concludes a decrease of the nOise width trom 34 keY at 22' C to 13 keY
at 1° C (81Wak). In our e~periment~ the ~ufluence of cooling gave an
improvement in overall resolution, which is ~stimated to a factor of
1. S. To p,event condensation on. the detecto);s and thereby degrading
the ~e50lution, a "cold finger" l<Ias mounted in the scattering chamber.
With the dispersl yO! beam and the new type of cooled datec::tors l<Ie
realized a totd FWHM ~[l the spectra of 35-40 k"V. This figure is a
factor of two better than p.eviously where a FWHM of 70-80 keY WaS
no~roal practice.
3.8.5 Contribution of the analog data-acquisition system
Any CauSe" for deterioration of the' resolution coming from the
"'Dalog data acquisition are not easy to deduce. Noise from the pl:'e
amplifiers is the most probable .ouree. Other contributions can cOlne
from the amplification modules or from bad connections. The only thing
one Can do is to check and recheck. all electronic modules and leads
regularly and replace a part as soon as its performance 15 de8~adlng.
Another caus~ will be drift in amplification. The specifications
of the electronic.. ,.how that this d,Ht can only occur over long
p~riods of time. We corrected £0< ~hi6 process ~u~1ng the addi"ion of
the sp~e~ra af~er the experiments (see 6ect~on 3.6).
The ADC c",n aho g~ve r~se to some bJ:'oaden1ng. Accord1o.g to the
specifica~ions the resolution of the ADC sho~ld be + or - one ch~nnel.
In pract~ce it >!Lppe;;><s that ttle J;'ep~oduct1on of the conversion of the
ADC used is twice the spect£~ed amount.
48
49
Cbapter 4 Experimental r~sult8 and co~~ectlve-model analysis
4. Introduction
"If your experiment M!<!ds statist.ics. you
ought to have. done a bet.t.er expeI;"~ment."
E. Rutherford.
The region of the even-even Ge and Se isot.opes i? a very
interest.in~ field for nuclea~ structure analyse.s. The nuclei in this
mass re$ion escape a complete description in terms of s single model
(80Ver). In these nuclei, protons and neutrons are distributed among
several au~6hells. fo. the valence neutrons these are the lfS/2-2PL/Z-
199/2 orbits and for the valence protons the 2P3/2-1fS/Z state~.
Especially the large number of these .ofr"e·· nucleon5 already mak.es
collective effl!c:ts more p1au.,ible than typical single-particle
characteri~tic~. ,h~6 ~6 also confirmed by the lev~l scheml!" of the
even Ge and Se isotopes. In figures 4.1 and 4.2 these schemes are
p.esented for the positive-parity states up to 3 MeV. All the6e nuclei
show th~ typical ~p~ctral bl!havio~r of an an~aI;"mon~c vibrato., be it
that SOme nu~ll!i h~v~ ~ low-ly~ng 0+ 5t~te, espe~~ally 7~e. Ibis is
displayed in anothet way in figures 4.3a and 4.3b, where the ratio
betloTeen th", excit.:>-tion energy o~ the so-called two-phonon states and
that of th", lowest ",xc~ted 2+ state is clearly in the neig~bour~ood of
2, which fact also favours t~is vibrational pict.url1-. 'l:~~s view is
moreovl!r corroborated by the lowest 3- state (~ot displayed in figures
4.1 and 4.2, excitation energy about 2.5 MeV). In t~e rotator model
this stace should be a member of an odd-spin nl1-gative-pa.ity ba~d. It
is then conslde-.;ed as tbe head of an octupole rotational band. This
inte~pretation of the 3- is not confirmed by expe,lments. Theoretical
analys~s for this state in term~ of the vibrator model are, however,
in good agreement with the e~pertmental data. Except for the behaviour
of the lowest excited 0+ state th~ gener<il pattern of thl! lowest
energy levels with respect to A is ver'j regul.!l.r. There "re, howeve"
indications that tbe peculiarities exhibited by these 0+ states point
to a soft subshell closure at N=40, which wakes the exclusivity of the
50
0.0 68 70 72 74 76 78
A
Figure 4.1 the l~v~l scheme of the positiv~-parity States in Ehe Ge
L~otOp~_~ from the adopted levels in NuCh,ar Oata Sheecs
(78KC<l, 80Kca, 7&Koc, 84Sin, 81Sin).
51
(3+) 3+
2.5 2+ (~ (6+) (5+)==
2+= (2.3.4) ~
(2,3,4)
(1.:L -- (.!dL 6+ 6+ (U)
~ (g,:L ( 2'i------- ~4 &L-
2.0 2+ 4+ (2+) 2+
(2,4) 8,3,,/ (0.2/ 2+
(0,1,2) 4+ =--=
4+ 2+ (3+) 4+
-. 1.5 W)
> ~ (0+) G,) 4+
(0+) 0+
~ 2+ 4+ 2+ 2+ ~
)( 2+
W ~
1.0 0+
~ 0+
2+ 2+ 2+ 2"
2+
0.5
0.0 0+ - 0+ - 0+ -0+ 0+ -72 74 78 82 76 80
A
Figure 4.2 The level scheme of the positive-parity state~ in the Se
isotopes from the adopted levels in Nuclear Data Sheets
(80Kea, 76Koc, 84S1n, 8151n, 8281n, 75Lem).
52
Ge(Z=32)
3
+ ..... ('.J 41 tIl
2. --. ~ W 2+ 2
1. 02
N
FiguJ:"e 4.3a. The ratio of the e)(citacion e(le.gy of the "two-phonon"
states and the ~xcitation energy of the first 2+ state as
a function of A for the even-A Ge isotopea. (In Lh"
harmonic-vibrator limit thi6 J:"atio is equal to 2. and in
eh., sy_etric-rotator limit 3.3).
the collective pictore dO\Jbtf\Jl. Many experiments have been. performed
0.\ these isotopes, but the data give no clear answer on the question
which model "hould be suitable for these nuclei. Our e>tperiments are
me~nt a~ a further trial for the collective moaels for theS8 nuclei and
ill this chapte. we Shall l'resent the results. The ex;perimental elastiC
scattering data will be compared with stand,:lI"a optical-mod"l
pJ:"edic.tiona and also 8eneral.1~ea optic.al-model analyses "ill be
performed. Th" Incl<lstic scatte.lng analysis with collective model~ is
th.:o fiCtal subject of this chapter. First, however, we 6h(111 briefly
review the data available iII the literature.
53
2. 41 rr w 22 ....... € w 02 1.
N
F1gur~ 4.3b The "atio ot the excitation energy of the "t:wo-phonon"
~tates and the excitation enerSy of the first 2+ state as
a function of A to~ the even-A Se isotopes. (In t:he
harmonic-vibrator limit this tatio is equal to 2. and in
the symmetric-rotator limit 3.3).
4.1 Tbe Ge ieotopes
The experiments performed on the Ge nuclei are manifold; for our
purpOSe We have collected the most important gamma-deeay and
scattering: e"pet~ments ana listea them in tables 4.1-4.3. Earlier
experiments ~n the sixties and thc first half of t:he seventies aimed
more on the detection of new levels with their spin and parity
assignments than On the explanation in tetlllS of nuclear scructur(!
(tttb).e 4. J.). The level schemes of the Ge iSDtopes have been well
est:ab1ished by these Experim~nt~ (eee figure 4.1).
In the pa?t mo~t reaction experiments were of the transfEr (p,t)
or (t,p) type. Usually the ttnalyses have been p~rformed with the DWBA
app~oach. ,t became apparent, however, that the Ge nuclei exhibit both
si<lgle-particle as '<1",11 as collective aspects in thei" structure.
Becker et a1. (82Be13, 8211e45) recognized thIs and \1eed th~ more
comple]{ CC8A approach to analyse t:he expe~imentttl data. Collective
aspects can be stuaied better by inelastic-scattering expEriments.
54
Table 4.1 Compilation of gamma-decay data of Ge nuclei.
Reaetion
7~a decay from 71Ca(n,y)7Zca
72AB decay from 72Ge(p,n)72As
70Ge(p,p'y) (E=7.0 MeV)
70A5 o;Ieca.y
72Ga de<;",y from n,(lt"r",l G",(n, y)
Remarks
meaBured Ey~ y-y~
deduced 7Zce levelB, J, TI
measured O(E pl ,6), Ey
deduced lUGe levels, B(E2)
me~~'lred E ~ y-y y
deduced 720e levels, 8(E2)
Ref.
68t;,,20
f>9HiOl
69Mo23
n.(ltu~al Ge(n,n'y) (E=0.5-2.55 Mev) measured Ey' cr(En
, ,6);
Hauser-Feshbach analysis
70Chl5
5 "F,;, ( l~N, pny) oOGe (J::"3e; MeV)
58Ni( lZt;,2py) 68Ge (E=36 MeV)
74Ca decay from 74Ge(n,p)74Ca
68Z n(q,2ny)70Ce (E=24-40 MeV)
7QGe(a,2ny) 68G~ (E~30 MeV)
70Zn(q,2ny)72Ce (E=22-35 MeV)
deduced 70,7Z,74,16Ce levels
me~$u,ed Ey ' y-y, I~
deduced 74Ce levels
measured Ey' y-y, T t ded~ced 76Ge levels
measu~eo;l Ey, y-y, I,
o;Ieduced G8Ge level~
measured Ey' y-y
deduced 7ZGe levels
measureo;! Ey ' y-Y
deduced 72ce levels
measured E Y
deduced 56, GSCe levels
measured E Y
deduced 74Ge levels
meaBured Ey~ y-y
deduced 70Ge level~
Tl of levels in beGe
measured a(E,Ey,e), y(t)
dedoceo;l 72ae level~, J, ~.
T j' B(E2)
measured Ey deduced' DGe T!
B(JlZ), B(Ml)
7lCa06
71Ga39
7lPa32
71Re04
71Re05
74NoOS
75'ta03
76Mo1)
77Mo2.0
79MoOl
82CI02
measured I y ' Ey ' I(conv.el.) 8SPas
deduced 70Ge levels, J, ~
B(EO)/B(E2)
Table 4.2 Compilation of transf~r experiment~ on Ge nuclei.
\(eaction
72'7~'74'76Ge(p.t)
71Gn( 3He.d) 7%e
69'71Ca(3He,d)/O'7lee
09oa( (I, t) 70Ge
69Ga( 3}j",. d) 700e
70G~Cp, t) 08Ge
70·72" 74Ge(p, t)
75As(d, 3He) 74Ge
70' 72(;",(d. 3!le) 59' lICs.
7~' 7SGe(d. 3H",)1S' 7SGa
70- 72, 74. '7 QGe(t,p)
72,74.7Oce(t.p)
70Ce(t.p) 7lee
64,66Zn(6Li,d)68.70Ge
6e'70Zn(~i.d)72'74Ge
74'76ce(p.t)72'74Ge
Energy Remarks
(MeV)
20
25
25
39.35
DWIIA analysis
deduced 70,71,72,74Ge level~
deduced 7lee levels
DWBA analysis
deduced '70' J20@ le~els
22.5 OWSA enalY6~6
deduced 70Ge levele
26 deduced 680e levels
26 deduced 68'70'72Ge levels
26 meas~red o(e)
17
15.0
15.0
15.0
2&.0
2&.0
17
17 15.0
34
34
35
deduced l~Ge levels, J, ]f. S
DWBA analysis
d~duced 7SGe level"
DWBA analysis
deduced 74Ge levels
m-iIlA analy$i8
deduced /80", l",vels
DWBA analysis
~educed 'l60e levels
deduced Ge sround state
proton occupation numbers
measured cr(S)
deduce~ ~ac~o o(Oe~)fo(Ogs)
DWBA analysis; deduced J,L,~
DWBA analy",1s
d~duced 7Zoe levels
measured aCe) 0+ states
DWBA analysis; deduced S
DWBA analysis;
deduced 12'74Ge levels
55
Ref.
74Ba67
75Ar08
75Ar29
75La05
770u02
77Gu12
77Ro22
78Arl2
78La12
78MoZ4
78Ro14
78v,,03
79Le07
79Mo08
SOAr14
SORe04
to be cont:inued
56
Taple 4.2 Compil~tion of transfer experiments on Ce nuclei.
(continued) .
Re~ction Eneq;y Remark.a
(MeV)
7Z'74' 76Ge(p,t) 13 measured d( G)
CCBA analysis
74Ge( t, p) 15 m@asured o( e) ; deduced
n:>tu{"~l Ge(t,p) 15 distriDution of 0+ strength
table 4.3 Compilation of scattering e~pe~iments on Ge nuclei.
Ref.
8ZBe13
82Be4S
84Mo07
(OM = optical model, VM = vibrator model, CC ~ coupled
channels, ARM = :>symmet~ic rotator model, IBM ~ interacting
boson model)
Re~c.tion
(MeV)
70' 72' 74' '1Sce(p,p') ll.S
70' 72Ge{(\,d') 12.0
70' 72Ce( 160, IbO'e) 30-)7.:;
natural Ce(a,a'y) 0.3-1.::;
70,n'"/4,nG<l:(p,p') 14.5
70,72,74, 76Ge(p,p') 11
70- "TZGe(e,e') 80-120
16
deduced levels
measured dee); DWSA aaalysis
dedl,leed 70' 72Ce levels, J, 1r,
defor~tion parameters
me~6ure<;l <;r(Eee)'
deduced 8(E2)
me~6ured <;r(En,.e), Ey
oM/a~use~-~e6hb~ch analysis
measured ~(e)l OM and CC
analysis; VM; deduced ~I
Ref.
67Br10
67KrOl
69Lil2
70Cu03
measured o(e); gen~ralized and 70Pe09
systematic OM/DWBA analysis
measured o(~, ~~,,8)
10'"'%e deduced level~. J, 'IT
a(~2), B(E3), ground stat~
charge den~ity. form factors
measured aCe), tlll
deduced ~I' J and ff
75KllO
78S~08
to be continue.d
Table 4.3 Compilation of sca~tering experimenc~ on Ge nuclei
(c;;ontim, .. "d).
Reaction
7%e( lSO,leO')
?2ce( 180, 180)
70Ge( 0Li, 6Li')
72'74,76Ge(a,a')
70,72.7 ... 76Ge( 160,160')
70,72. 7 ... 7~e( 160.1GO'y)
7D'72'73'74'7~e(a.a)
74Ce(p,p')
70' 72' 74' 76Ce(e,e')
70'72'74'76Ge(d,d')
70'72.74'76Ge(p,p')
Energy
(MeV)
68.6
30-48
10.5
7.0
29.9
36-42
25.0
22
22
R",marks
measured o ( e) 0+ <lnd 2+
measu1:'ed 0"(E,l75", z+ OM/CC analysis
deduc~d deforma~ion parameters
measured Coulomb e~citation
ded~ced static quadrupole
mOment, B(E2, 0++2+)
deduced BeEl), B(Mi)
measured o(e);
systematic;; OM sear~h
measured 0(6); e~10·-100·
DWBA and CC analysis VM, ARM
deduced III
measured Oee); 9=10°-100·
DWSA and CC analysis: VM. ARM
deduced 6r 200-500 deduced t~ansition densities
8,16
11.5
65
2!. 2!. IBM analysi6;
deduced boson densities
measured O(e), iT 1L (6), A(G)
0+ and 2+; oM/ce analysis
VM and SRM, deduced 62
measured o(S), A(a)
CC analysis: systematic ~4' deduc;;ed Q4
57
Ref.
79Es04
80Le16
80Le24
82En04
82Ta16
83Ra32
85Ba~
8SSe05
85Matl
58
Table 4.4 Gamma-decay studies of Se nuclei since 1970.
Reaction: Remarks:
measured E Y
deduced 72,74S e levels
measured E-y' '£t deduced 78Se levels
mea6u1;ed E , -y-y y
deduced 70Se levels
58Ni(160.2py) (E~26 HeV) T, iG 725e
6~Ni(160,a,2ny) (£-42~81 MeV) measured cr(E.Ey'~)
deduced 7~Se levels, J, '£5 measured T~ "'2Se
saNt( 160,2py) 72Se (E=40-58 MeV) measured E y
bONi( 160,2pY) 7~Se (E=45 MeV)
64Ni( 12C,2I>Y) 7~Se (E=39 MeV)
~5Cu(IIB,2ny)7~Se (E=29 MeV)
70Aa decay
74'76'77'78'80S e (n,y)
74,76Ge (u,2ny) (E=25-38 MeV)
75Se(n,y)76Se (E~thermal)
deduced 725e levels, I n , 1!
measured E (6), y-y, T, y •
deduced 7~Se levels
measured Ey ' y--y(9)
deduced 70Se levels
measured E-y' deduced 75,77.78.79.81Se levels
measured E y
deduced 76, 7B 56 levels
m,;,asured Ey
deduced 7aSe levels
measured cr, Ey
deduced 76Se levels, In'
IBM a[1alys1s
Ref. :
70No03
73lHOl
74r1008
7611aOl
76HalO
75Lo08
77L~06
79Pi05
80Ka36
81En07
82Ma4S
82Si07
83To2()
59
ftolll the eompilation in table 4.3 it ean be seen that there ed,,~ a
£ew proton seattering data and only two experiments W'ith pohrLzed
partic1e~: Szalo~y et al. (78Sz08) and recently Sen et al. (85Se05).
During the last four year~ new p~oton scattering data were gathered in
Orsay by a Fren<;.h coUabol;ation (8ZTa16). Their first proton
scattering expe1;1ment (on 7~Ge) was publisned in the eLme we were
planning to do a similar experim"fit. Since their data are obtained
with a magnetic spectrograph and our experimental eetIJ.p works with
semiconductor (ietectors, we decided to perform our polarized-pr;oto[l
experiments ",itn the same labol;atory energy of 22 MeV as they have
us@d. In this way our po~al;~~ed data will gi~e a complementary set to
the"e high-resolution cross section data. /I, year later also data On
nGe were published (83&a32), but untiL now these ti<lO proton
experiments are the only high-resolution proton-scattering d"ts On the
Ge isotopes.
Only S 6ho~t time ago the first results of a syotemsCic ~tudy ot a
Japanese group were reported (8SMati). Theee expe~J.ments were
performed wi~h 65-MeV polsrlzed p~otons on the Ge isotopeS to obtain
",ynelMtic6 on the hexadeeapoh: (6 4) deformation in the mass ~egion
A=60-80. Only the resulting hexadecapole momen~" have been presented
in a figure for the nuclides ranging from Ni to Se. Cro~s ~~~tion Bnd
analYl'lng powe. data have not yet been given. I,e are looking forward
to the publication of these although we were able to get a glimpse on
their data (85Mat2).
Compared ~o Ge the e~pe.iments On Se isotop~s are less ~xtensive.
This i~ prObably due to the difUcultJ.es which :>X'ise in the
preparation and the use of Se targets (see section 3.2.1).
Nevertheless the Se isotopes have been investigated thoroughly in [he
past. Also for these nuclei tne gamma-decay (table 4.4) and oucleoo
transfer experimel'lts (table 4.5) are in the majority. So the leve 1
schemes, the branching ratios and BE/BM values are rather well known.
Several inelastic scattering eltpo!:rimell[S 'have been perfo~me.d (table
4.6) but only a few with protons. Connected to our investigation is
the work of Matoba et al. (79Ma28), of Matsuki et al. (83Ma59) and o~
Delaroche Bt al. (840eOl). The. last two studies also concerned
polarized protons with an laboratory energy of 65 and 16 MeV,
60
TaDle 4.5 Exper.imeutal ~ata from transfer reactions on Se.
R,eact1ol\;
76, 77, 76' eo' ~:aSe(d, p)
76,77.7a.aO, 82S e (d,t)
76'78Se (p,t)74'"!bSe
76' 78' 80' 8zSe (p, t)
Ellergy:
(MeV)
15
1.5
2.5
52
33
1.5
51.9
Remarks; Ref. :
measured E, deduced levels 65LiOB
n,18,n,ijl,ij'Se below:> MeV
OWBA analysis
measured ~(e), DWBA analysis 74Kn02
deduce~ 84Se level~, I~
measured ~(a); DWSA analysis 75ArZ9
deduced 'l6Se levels, I " measured Q(6) 760r02
deduced /4SC levels, I • S 'If
measured 0(8); DWBA analysis 778018
deduced 74.75Se levels, I
me a I>',ll; eo o( e)
OMjOWM an(iLys~s
measured 0(6); deduc~d
systematics go strength
1T 84wal3
85Mi06
Table 4.6 Elastic and inelastic scattering from the Se isotopes
(since 1965).
Reac.tion:
(OM ~ optical model, HVM w harmonic vibrational model.
AVM D anharmonic vibrator model, SRM ~ symmetric rotator
model, ARM - asymmetric rotator model, EARM ~ extQuded
asymmetrl~ rotator model, RVM - rotation-vibration model),
En"rgy: Remarks: R«f.:
(MeV)
76'17'76'80'8ZSe (d,d') 15 measu·red a( e); deduced 65L1l0
:Levels 76,77,78,80,82S e
76'78'80'82Se (p,p') 6.4 mea6\,1J:ed o(e), polar~zat~on 70Hel0
deduced l6'78'8U, eLSe level~.
DWSA analysis, d(!duc.;.d J,", 101 2
to be continued
Table 4.6 Elastic and inelastic scat~ering fro~ the Se isotopes
(continued).
lI.e!lction:
74, 76, 78Se ( 160, 160'y)
80, 82Se( 160, 1.0'y)
7G' B2Se (n,n')
78' 805e(n,n')
eZSe{ a, a' )
7~SeCa,(l' )
74Se( 160, 100')
76,7S,SO.8ZSe (p,p')
76.78.80.8 2Se (!,d)
76.ie,eOSe{a,Cl)
76'S0Se (n,n')
8ll.ergy:
(MeV)
39.2
39.2
6.6,7.3
6,8,10
8
7.3
7.3
33,34
51.9
12
12,16
25.0
65
16
8,10
Remarks: Ref. :
measured Co~lo~b excitation, 748a80
~y' y(9), o(Ea ,); deduced
74'76'78'80' 82Se levels,
B(B2), B(£3), S(Ml), T!
JUe'asured 0'(£,£ "e) n
OM/CC analysis
76La12
deduced deformation parameters
me!l6u~ed Coulomb excitation 77Lell
deduced Qz+, B(£2)
mea6~red Coulo~b excitation 78LeZZ
deduced 74Se QZ+' B(E2.0+~2+)
me!la~~ed 0'(8) 79Ma28
OM, DWBA and CC analysis
HVM: deduced 62 , 6 3
measured 0(6), iT11(6) 0+
CC analysis
meas~red 0(8), Ace)
OM/CC calculations
m<;>",,\lJ;ed (J( a)
systematic OM search
measured 0'(6), A(e)
CC analysis SRM, HVM
deduced B~ and 6~
79Ma41
81NuQl
81Varl
81 Var2
82£n04
83Ma59
measured a(9),A(9); OM and CC 84DeOl
calc~lations: ARM!EARM, RVM,
VM;
deduced deformation param~t@rs
mGOasured a( e) 84Ku09
CC calculations liVM, AVM, RVM,
ARM,
deduced deformation parameters
1'>2
respecti vo;,ly. The present work f1 ts
Especially Matsuk1 ct a1. aim at
1n bet\oi'een t.hese ex:perimeoots.
8 systematic search of the
hex:adecapo1e deformation in th~S mass region, as ~as mentioned before.
The work of Delaroche et al. is more directly comparable to ours. The
analysis of the experimental results in termS of t.he collective model
hJ>s been performed in the same way as ~e do. Thi!j will lead to
frequent comparison to the work of Oelaroche et al.
4.3. Experlmeota1 Bnaly~ls
In the next sections we present the experimental results aod the
analysis in terms of collective models of polarized~proton scattering
from the even-even Oe isotopes with A=70-76 and from th~ even-even Se
isotopes with A=76-80. As this \oi'ork aims at a systematIc study of the
A=70-S0 region the experiments were performed with a labor(l.t<;>ry energy
of 22 MeV as argued before. We shall present the experimental data
together with the model predictions. Before doing ~o, however, we will
first give SOme deta~16 of the exper1ment~l aoalysis in this section,
in order to give some insight in the special problema cOncerning otlr
experiments and the way we handlad them, aB well as how we have to
judge the. quality of the experimental data. In total we have analysed
38 states of 7 nuclides of G~ and Se. It would go too far in detail to
dlsctl65 each state separately. So only the difficult cas~s wIll be
mentioned and the common problems will be handled accordingly.
4.3.~ Correction for impurities in elastic Gcattering
S.ince the target.s used c.onte.f.n.ed small,. in some caSes e.ven 1arse,
co()tributions of other isotopes of the Same element (see t"ble 3.1-
3 .2). it became inevitl>b~e to correct the elastic dat". l'he
kinematical dIfferences in the posltions of the clastic peaks were, in
most cases, too ",mal~; the resolution was not good enough to separate
them. For the co,.ection we deve.loped a method based on 6ubt~action of
the cOrltributions of the other isotopes witll optical-model predictions
based on fits with Otlr own e~perimental data.
It can be expected that the corrected cross s~ction data show s
slightly different normalizatiOn ~hen used in sn optical model search.
So we followed the ne~t procedure,
&3
1. All ellie~i"-:ecat,,ering daLa lire \leed in eepltl;lite optical-model
searches to find a first estimate of the values of the
normalization and of the optical-model parameters as if the data
contained no contributions from impurities.
2. The calculated cross sections and anal)'sinS pOwers are used to
correct the experimental data. (For the final error in the
corrected data ~e assumed an error of 10% in the ealc\llated eross
eeet!on and io the anal)'s1ng-powe~ calculation.)
3. The corrected experimental elastic-scattering data are used in a
new optical-model search to find new (and hopefully better)
estimates of the normalization and of the optical-model parameters.
Steps 2 and 3 are repeated several times until the Variations in the
normalization are small (less then 4%). It appeared that this
procedure converged to a relatively stable ~alue within 4-5
iterations. the normali~ation of the inela"tic scattering data was
deri~~d from the r~sult~ of this proc~dure.
4.3.2 Correct lone in inelastic ~cattering
When impurities give problems d\ll;ing the anal)'si$ of the
inelastic scal:terinS >Ie hav,;: USed another method, This is necessary
since the influence of an impu:r;1t)' is usuaHy _inl), not~cea'ole ~n a
small ang\llar range. The correction ~ith model predictions on basis of
experimental data wo~ld introduce too much model dependency, since ~e
do not know beforehand which model is adequate. Secondly a model
prediction for an inelastic process is usually nOt as accurate as the
optical-model predictions for elastic scattering.
In the anal)'sis of inelaStic scattering we have to deal >lith
three kinds of impurities:
1. There will alwa),s he contributions from the elastic scattering from
12C and 1(0. Since the targets were manufact\ll:ed with a ca(bon
backing the contribution of 12C was Ilery large. Also due to tllis
carbon support even measurable intensities of 13C we:r;e p(esent, in
50me cases 50 large that they were comparable with or even eKceeded
the intenSities due to excited states of Ge and Be. The intensities
from elastic scattering from 12C and 160 were at least one orde~ o{
magnitude larger. For those angles wlle~e an overlap exiete between
the elastic peake of 12c, 19C and 160 corrections would result in
64
errors larger th<!:n th,;: actual values of the en]sa sections. So the
~l1elast1c angular distributions will show SOme gaps in the pattern
of data points.
Z. The second source of disturbances in the analysis is the occurrenCe
of other i~otop~s De ~oe same e~ement. In this case we have to deal
",ith l~v~h' nearly ~q\.la~ ~u e)l;citation ene);gy. Ln most cases this
will not lead to large contrib\.ltions ~ince the cross section
concerned has to be multiplied with the fraction of occurrence. In
SOm~ cas,;:s, however, this can be a rather large contribution making
tIle correc tion n"""ssary. th" diff ic:ul t cases are especially found
for those targets with the :lighest degree of isotopic impurity:
70Ge and 76Ge • For example in 760e w~ hav" to correct the
experimental inten~itie~ of ~he 21 level for a 50% contrlbution of
the 2t Level of 70Ge. This is, by th~ way, also an unfOrtunate
~ase, since a ~t stat,;: will be excited BtrDn&ly, While a 2} state
is the 't"esul t of d. i:iec.ond-order process t whic.h. leads t.o ~ much
weaker excitation. in order to apply /l. prope~ cOI:"~ectioo of the
data at thc~e angleS whe,e significant overlap exists, we dedu~"d
the contribllt"lon of the cootalllinating le'l'el at angles with nO
overlap. This factor was then used to correct the data.
3. The third kind of Impud ties are thos <;! of unknown n1.lcUd<;!s. Ih<;!se
are in gene~al of low A so that th"y move (kinematically) rather
fast through the spectra and overlap with levela of interest exists
ooly at a few angles. When such a case oc,,-urred we tried to
establish the ""p"rimental cross section and analysing power Of
this ,-,nkllowll nuclide in th~ neighbourhood of the angles where
overlap existed. til" COrrectiOn of the data was then performed by
usiog the interl'olateu values. An example of thi<; kind of impurity
W"-l> foun.;! ~n the 72Ge target. Probably due to the targ"t
preparation proced\lre an ~\lIP1.lr;l, ty of A,,28 (Sf.?) was found to
overlap the peaks of 01 at 70"-75", 21 at 80"-85", 2} at 120'-130'
and the 41 at 15S"-160·.
J:he errors introduced by this procedure were estimated on the
basis of th" ,,"pel;i\llental vel\.les and taken into account properly. Th"
overall normalization (and its error) derived from optical-model
analysis i5 of covrse not reflected in the data (correlation), We
will., however, bring the normalh:atioll error into account when we
discuss tile accuracy of certain deduced model parameters 1.c.
deformation parameters.
4.4.3 Special remarks
In spite of all effoJ:ts we made to obta1n the (ststhUC'lll)
sccuracy of 3% (1000 counts) aimed at, this could not alwaya be
achieved experimentally making the corresponding data less usefuL
Especially the anslysing power becomes rapidly of minor significance
when the ~t~t.iBtic.al ~~c.uracy diminishes. There were Iilliiinly thr~e
causes for this low accuracy:
1. Firat of all the targets are rather thin (ZOO ug/<;m2). Thi!;! make!;!
long measuring periods inevitable if the required accuracy has to
be met. It is, however, not feasible and not desirable to have
measuring periods at one angular setting exceeding the time of two
days (see section 3.6).
2. Secondly we were not always able to produce high-current polarized
beamS (~ 20 rtA).
3. The isotopic purity of sOme target. wa5 rather low, ~o that rather
large correc:tioas had to be made. Also the contributioas of unkaown
elements in the targets, even when they were ot high pu~ity, 8~ve
sOlIlet!lI\es r~se to l.:>rl;e corre<;tionlj. The fin,,1 Sccl,lracy in such
data 15 severely <ieterior"te<i by the. errors introduced through
these corrections~
These three facts togethe~ made that in seve.a~ e~pertlI\ents the
accuracy 1s rather poor. Under the. conditions we had to perform our
experiments it was not pOssible to obtain better r<!sults. We must
admit, however, with t.he words of Ruthe.rfo.:d above, that we should
have done a better ell:periment. In OOJJ:: case this \Jol,lld me(l.TI more
specifically that we needed a furtheX' improvement of the polarized
proton beam current aL"ld that acw self-supportiL"lg targets shoOJld be
I,lsed, havinS a thickness of at least , mg/cm2 and au isotoptc purity
of at least 90%.
4.4. Optlca14mOdel analysis
In order to fit the elasttc 5catterinl; and to fiL"ld a set of
reliable optical-model parameters for O~BA and coupled-channels
calculations a search procedure was applied. Fo. the ses~ch on some
66
opt1cal-mo4e1 parameters and the calculation ot e~a~tic scattering the
programmes OPTIMO (72Vos) and ECIS79 (82Ray) were u~ed. The sum of the
sqoared differences between the e~perimental and calculated
observables was minimized (this quantity is known as ~hi-Bquared: X~)·
During the Bearch with th~ programme OPTIMO the normalization of
the eKpertmental data was contill.uously adjusted (see paragraph 4.3.1),
so aete, the ~earch a r .. asonable absolute normalizatiOn had been
obtained. We know that this way of normalizing experimental data can
depend an the choRen optical-model potential (82Was), however, for not
too large variations ot t;he opti"al potential this o.orma.l1zation will
vary with~n 4% of its meall. value. The ambiguity in finding " good
normalization as "ncountered by Alons (SOAlo) was nOt found in our
search. In the code OPTIMO the normalization can b~ searched on
together wltl~ the C>tt><i!r parameter" or adjusted afte1: each run. We uSed
eh", latter option so that tll<i! normalization is o.ot a rea~ly free
parameter. Also the availability of analysing-power data r~stricts the
parameter space which mak<i!s an ambiguity for the n01:mal1zation leBs
pl:obable. The slmll~tan<i!oue fit of 10 paraui",ters gives rise to sOme n
well known ambiguities tn the param<i!terB (£or ill.stance Vr.r and
Ws.ai), wh~ch r<i!sult in ~he sam", observed elastic scattering. We tried
to avoid these by starting from the g~obal potential of Becchetti and
Greenlees (69Bec).
The laboratory Qn<lrgy of 22 MeV i6 just at the point where the
Volum<i! ab6orption becomes non-neglib'e. There eKis~o, however, a
strong correlation between surface and volume ahsorption in this
eo.crgy region which may result in unrealistic values of the surf"c"
and volume absorption. TO avoid this correlation we d<i!cided for the
final 3ea~che~ to fix the value of t;he volum<i! abaorptioo. at the global
valU<i! of Becchetci and Greenlees. Our optical-model sea~ch procedure
was then of the follOWing sequeo.ce;
start from the global potential of Becchetti ao.d Greenlee~,
- ~earch on the well depths,
- s"arch on th~ geometry parameterS,
- search on all paramete~s.
Sio.ce there happen to be differences ~n the procedure of
calcula~ing the elastic Bcatterill.g by the va~iolla programmes i.e. tn<i!
optical-model search code OptIMO alld tbe coupled-cha.nnels progJ:amme
ECIS79 (82Ray), tbe final par~meters of OPTIMO were used as starting
67
values for a new search by the ECIS19 progra1Illli~ (th~ ~xp"rim"ntal
normalization was now fixed). In this way we were able to determine
whether some irregularities exist in the data or the search procedure
itself. It was important to know this before we started our optical
model search within a generalized-optical-model approach (see section
4.4.3).
This generalized-optical-model approach becomes necessary if one
wants to analyse nuclei with strongly excited states, as is the case
for the Ge and Se nucLei. When an explicit coupling to such states is
introduced ~nto " coupled-channels calculat~on H appeare that: the
elastic ecattering is not any longer described accurately by the
odginal optical-model parameters (see section 2.1.3). In coupled
channels calculations including such excited states we have to corr~ct
the optical-model parameters in order to describe the elastic
scattering again properly. Usually one tries to correct for these
explicit strong couplings by varying the imaginary potential only. It
was shown, however, by Petit (8SPet) that such an approach gives only
a partial improvement of the description of elastic scatte:ring. It
appeared that a variation of all parameters again ~as needed to obtain
th", elaJ>tic scattering at the same lev",l of Agreament as with an
optical-model eearch without couplings. This conclusion was confirmed
by our findings.
Another reAson for the application of the general1zed-optical
model approach can be found when the rotator model has to be used for
the calculation of the inelastic channels. In this case the optica~
model parameters found in a normal search are incorrect since the
normal optical potential has a spherical shape. In the rotator model
we have a static deformation., which implies that also the potential
should be permanently de~o:rmed.
Our way of findin~ the generalized-optical-model is based on the
method devieed by Petit (85Pet). Usually one tries to vary the
optical-model parameters together with the deformatiOn parameters in a
coupled-channels calculation and £;lte theee l'arameter~ on all states
(see for instaace SOAlo). we do not use this method, since the elastic
scattering is by far the dominating part; ot the 6catt",ring process
and, even more impo~tant, it is the low~st-order process 1n
scattering. $0 by also fitting agaiast the excited states "e III<lY
expect that tbe paramet:ers are mainly deduced by the form of the
68
angular distribution of the ine).asti.c 6catteI"~ng. This seems
reasonab~e. but there aI"e
(and eorr~spondingly the
some PI"eauml'tions about the reac tion model
iate,action potentiala) pu~ into the
calculations, implicit).y declaring ~he reaction model to be correct.
Another qU<lstioa is about the usefulness of comparing processes of
different ordel:' 1n the scattering proce5';;. ThiS approach 1s
methodologically incorrect. Our FI"oceduI"e is, however, strongly
dependent on the specific elastic data. Since the introduction of
strong coupling", will aff"ct mainly the backward angular range, the
differences will be the laJ::gest in thh resion. A search procedure
will try to minimize these differences and that makes our proc"dure
especially sensitive to the details in th" ~xperimental poiats ia the
backward angular range.
Our app~o3ch toward~ the incorpOratiOn of stroagly coupled states
is the following:
1. start a grid search on the deformation parameters at the two most
i~po,tant states in the model, i.c. e~ and Sa for the vibrational
model aod, aZ and B4 for the ro~ational mod"l;
2. fit all optical model-parameters on the elastic scattering with the
de~Qrmation param~ter~ set to a fi~ed value.
3. compare the calculated angular distributions with the experimental
ones and select the calculation which best represents ~he
exp"rimental data (for 2+ and 3- ~t3te~ poasibly res~ricted to ~he
forward angular range).
Th~ selection of the deformation parameters by the compari~on
with the experimental data Cola be done 1n several rnann~ra. One way is
to use an "objective" criterion i.e. a. chi-squar,;,d criterion OIl the
complete data set. This results in defor~t!on paI"ameteI"s and
calculated angular distributions giving an overall ag~~em~nt with the
eKperilllelltal data. We think that it is better to compare. theoretical
calculations and experimental data mainly in the forward ansulax
region, in view of the applicability of OWBA and couple4-channels. In
tile bac~ward angular r"sion the specific nuclear stucture (which is
also connected to hisher-ord~r processes) influences th~ ~xperimen~al
data rnOI"e stI"ongiy since in this region the scattered particl~ comes
much closer to the nucleus and will feel more of tile nuclear
interaction.
This procedure works well for tb~ v!bx-atoI" calcu1a~ions, where 2+
and 3- states inUuence each other indirectly via the ground~state
69
Table 4.7 B~st-flt optical-model paramete~s, ded\lced from
calcul~t1ons with the programme O~TIMO (72VOB).
Coulomb radius r =1.25 fl!>. c
Each first linE give~ the
pl'l:ameters, the second line tho! errors 1p the~e parameters
deduced trom the fit.
A " r a w W ri !Ii V r a X2
/ N r r r v B so so 80
(MeV) (fm) (£m) (MeV) (MeV) (tm) (fm) (MeV) (fm) ( tm)
Ge
70 55.56 1.14 0.72 0.00 7.82 1.22 0.78 6.23 0.91 0.59 9.z6
0.67 0.01 0.01 0.30 0.23 0.01 0.02 0.25 0.02 0.04
n. 5&.79 1.13 0.68 0.21 8.0& 1.19 0.83 5.00 0.91 0.65 24.63
0.96 0.02 0.02 0.01 0.64 0.05 0.03 0.44 0.08 0.01
74 52.67 1.16 0.68 1.63 6.95 1.25 0.80 6.41 0.92 0.57 lO.OS
0.75 0.01 0.01 0.18 0.60 0.02 0.02 0.23 0.02 0.06
76 54.52 1.13 0.73 1.04 0.61 1 • .33 0.82 5.45 0.86 0.45 12.95
1.57 0.02 0.03 0.67 0.48 0.04 0.03 0.35 0.03 0.07
5e
76 52.54 1.~7 0.68 0.01 8.24 1.20 0.84 5.78 0.98 0.59 15.83
1.12 0.02 0.03 0.01 1.65 0.03 0.05 0.32 0.05 0.08
78 56.43 1.10 0.77 4.14 4.37 1.42 0.73 5.92 0.82 0.50 25.78
1.80 0.02 0.03 0.63 0.50 0.05 0.03 0.48 0.04 0.07
80 54.83 1.14 0.76 2.07 7.41 1.24 0.79 $.98 0.97 0.66 13.87
0.67 0.01 0.01 0.50 0.60 0.02 0·02 0.02 0.03 0.05
~=~~==~~=~3 ___ ~~ __ =========a __ M~._3~=======~~~~_a •• ~m========~=_.~ __ w=
70
0.'
0-1
0.5
o·
0.1
-0.5
30 60 so 120 150 IBO 0 30 60 90 120 ISO 180
Figure 4.4a Optical~mod~l calculatiDn", of toe elastic polarized
prOton scatte~ing from the Ge iBotDp~~.
global optical potential according to 8ecch~tti
and Greenlee.s,;
full best-fit optical-model pDtential (Wv
~ BG);
dotted = fit with an ave rase geomet~, for thc G~
isotopes.
chann"l. ,11 the rotator model thil> ;I.s dHferent for the 2+ and 4+
lev"l.,_ The 4+ state cal) be ~xcited from I;:h~ Lnter£erence between a
tWO-BI;:e.p quadrupole contribution and a one-step hexadecal?ole
COntribution. Moreover the 2+ ~tate is also slightly influenced
toro.;gh th~ dependen<;e oe the potential on e". This make~ a decision
on the basis of a chi-Bqua~ed criterium questionable. the more so as
we kuow that 4+ states ar~ usually characterized by a rather
71
+ Ge(p,p)Ge; 22.3 MeV
" .S
o. •
0.\
alO"R
0.1
-o_s
30 60 90 I~O 150 ISO 0 90
J.'J.sure 4.4& Optical.-mooe1 C(;l.lC\Ih.t:l.one of the elaat:ic l!0laI"ized
proton scatteril1g from the Ge isotopes (continued).
stI"uct~I"elea8 anSular distribution. The decisiol1 which combination of
6z, ~~ values should be the ~ight ooe is here ~ather 5\lbjective, ~hich
~ill lead to considerable \Incertainties, mainly in B~. This might be
d\Ie to the rather low :l.nc:l.dent energy a6 to ttJ.e !,tee o( pJ;OtOn6 of
p~ojectiles. We think that this is a specific property of this type
of experiments.
As discussed above we began 0\It' optical-lIIQdet Ute wiCh a sea>:ch
On all parameters starting from the Secchetti-Gre,;,;).l,;,es values. The
results of these calculations are presented in table 4.7. One fa~t is
72
0.1
0.1
0-1
Se(p,p)Se; E p
22.3 M",V
Figure 4.4h Optieal-mode~ calculations of the elastic polarhed
proton Scattering from the Se isotopes.
dashec1 global optical potential according to 8ecchett1
and Greenlees;
full ~ be~t-fit optical-model potential;
dotted - fit with an average geometry for the Be
isotopes.
73
apparent: tbe X2 v~l~es for 72G~ a~d 7SS@ are co~siderably larger than
those for the otber nuclides. Also th~ resulti~g parameters for these
two hotQpes show a deeper central potential. Th.e imaginary strength.
for 7aSe tefids to be equally divided ov~r volume and surrace
absorptiofi while for the other nuclides the volume absorptiQo
dOminates. The question ~rises if we are to d~ali~g with a~ effect of
specific nuclear ~tructure or with a stro~g correlation between volume
aud surface absorption for this incident efiergy or with some
pecoliaritle~ in the elastic data. We ~afi pinpoint part of the X2. The
elastic data wer" corrected as described in section 4.3.1. For 72(;e
t.hese correctio~s were very slIlall ctue to the high pur!"ty of the targe r
50 that the tinal errors after correction remain@d small. For 7BSe
more data points have been collected. Since the observables can be
determined by a ~erEain number of data points, intermediate data
points ~ill not give any improvement in the fiE but will only enla~8a
the value of chi-squa~ed.
Nevertheless the competition between surface and volum~
absorption will be p,e5en~ (see above). So the most extensive
ca1c.ulations were performed wi ~h a fixed volume absorptioa., Sa t to a
value of 2. n !-leV. These saarches were pertonued not only by the
l'rosramme OP'l'IMO but also by the couplad-channels code ECLS7~ a" a
comparison before we started the genera1i~ed-op~ical-model searches
(see the introduction to this chapter). The results of these
calculaEions are tabul~ted in tabl€ 4.8a (Ge) and 4.$b (Se), where we
have given the results of both programmes. In these tables also the
calculated errors are quoted.
First we can conclude tnat tne differences betwee~ both
programmes (OPTIMO and EC1S79) ~re negligible wh~n fitting the elastic
scattering: all paramet~rs are nearly the same and lie well inside the
quoted e~~Q~e. ThiS gl~es uS confidence that further ~alculatlon5 ~ltn
the programme EClS79 will not gi~e essentially oifferent resul~s just
becau~e the numerical procedures are different. Secondly the paramete~
J>"t'" obl;:a1ned for the individual isotopes are now more comparable,
sin~e the spread nas become small. FrOm these fits it appears that the
geometry parameters ar~ not ve~y rouch different for the four Ge
isotopes and the three Se isotopes. So we trted two other calculations
with the prosramme O~'l'IMO. In the",e calculations all geQmetry
parameters (rj,aj, j=r, i or so) were set to an averase value for one
element as well as to an average v~lue for both elements. The results
74
Table 4.8a Best-fit optical-model parameters with the volume
absorption set to 2.21 MeV (from (69Bec». Coulomb radius
r =1.25 fm. c
Ea.ch first line gives the f·Ham.eters, the
se.CDnd line. t.he errors in these parameters deduce.d f.rom
the fit.
Ce V r a W ri
a1
V r a X2/ N Code ~ ~ r s so so so
(MeV) (to!) (fm) (M,;,V) (fm) (fm) (MeV) (fm) ( fm)
70 55.95 1.13 0.72 6.29 1.26 0.75 6.27 0.90 0.$6 13.02 OPTIMO
0.88 0.01 0.01 0.27 0.01 0.01 0.29 0.02 0.05
55.74 1.13 0.72 6.29 L.26 0.74 6.24 0.90 0.57 13.02 ECIS79
0.94 0.05 0.01 0.29 0.07 0.02 0.29 0.10 0.06
72 56 .58 1.14 0.67 6.56 1.22 0.83 6.16 0.91 0.65 32.95 OPTIMO
1.18 0.02 0.02 0.54 0.03 0.05 0.47 0.04 0.09
56.5! l.U 0.68 6.39 1.22 0.83 6.05 0.91 0.63 31.87 £CrS79
1.22 0.06 0.02 0.34 0.12 0.03 0.42 0.15 0.08
74 52.91 1.17 0.69 1.12 1.24 0.76 b.50 0.94 0.b3 11. 70 OPTIMa
0.70 0.01 0.01 0.26 0.02 0.02 0.24 0.05 0.02
52.42 1.16 0.69 6.35 1.27 0.79 6.42 0.91 0.55 10.02 ECIS79
0.83 0.05 0.02 0.33 0.09 0.02 0.25 0.08 0.06
76 56.30 1.10 0.76 5.85 1.40 0.77 5.72 0.83 0.44 12.64 OPTIMO
2.03 0.03 0.04 0.25 0.04 0.04 0.45 0.04 0.06
55.43 1.),0 0.75 5.80 1.40 0.76 5.61, 0.84 0.44 12.75 ECIS79
1. 7 5 0.10 0.03 0.24 0.13 0.02 0.37 0.12 0.06 __ ~~~~~.~~~.~.~~_~_W~~~~.~~~~R~~~~~~~_~~~M~~_M_~W _____ m~2_m~33m~_=~==~
75
Table 4.8b He5t-~1t opti~~l-model ?arameters with the volume
absorption set to 2.21 MeV (hom (69Bec}). Coulomb radiu,;
r =1.:25 fm. Ea~h first line gives the parameters. the ~
second line the errors ill theee parameters deduced tIO~
tl;le Ut.
Se V r a w ri ~i V r a X2
/ N Code r r r 8 SO SO so
(MeV) (fm) (fIJI) (MeV) (fm) (fm) (MeV) (fm) (fm)
76 52.62 1.17 0.67 6.51 1.25 0.83 5.86 0.95 0.57 18.67 OPTIMa
1.15 0.02 0.03 0.62 0.04 0.04 0.38 0.05 0.08
52.75 1.)'6 0.67 6.23 1.25 0.84 5.86 0.94 0.55 18.79 ECIS79
1.08 0.06 0.02 0.47 0.01 0.03 0.35 0.09 0.14
78 54.43 1.13 0.74 6.19 1.34 0.79 5.55 0.84 0.52 Z7.19 O?HMO
1.16 0.02 0.02 0.31 0.03 0.02 0.48 0.04 0.09
53.90 1.13 0.73 6.15 1.33 0.78 5.43 0.85 0.51 2&.26 ECIS79
1.17 0.07 0.02 0.25 0.12 0.02 0.38 0.13 0.07
80 54.91 1.14 0.76 7.n 1.24 0.79 5.98 0.97 0.66 U .93 OPTIMO
0.64 0.01 0.02 0.48 0.02 0.03 0.22 0.04 0.05
54.87 1.14 0.76 7.,24 1.23 0.79 5.98 0.96 0.65 13.9$ Eels79
0.60 0.01 0.03 0 . .32 0.02 0.06 0.21 0.06 0.11
==~~~~~~~~~~~~~=====================_D_m_.~~~=D~=~===============~=~=_
76
Tab~e 4.9a OptkcaL-model parameters with an average geometry for the
element (Ge or Se). Volume absorption is set to the
Becchetti-Greenlees value and the Coulomb radiu~ r =1.25 " fm. The first line gives the paramete~6, the second line
the errors 1n the varied parameters deduced from the fit.
Parameters with no calculated error are kept fixed.
A V r a VI VI r1 "1 V r a X")N
J; ( r V s so so so (MeV) (fm) (fm) (MeV) (MeV) (im) (fm) (MeV) (fm) Urn)
Ge
70 55.43 1.14 0.71 2.21 6.19 1.25 0.77 6.38 0.91 0.60 12.75
0.16 0.05 0.16
72 55~58 1.14 0.71 2.21 6.89 1.25 0.77 S.71 0.91 0.60 41.18
0.29 0.14 0.26
74 54.86 1.14 0.71 2.21 7.02 1.25 0.77 6.57 0.91 0.60 12.75
0.16 0.07 0.16
76 55.99 1.14 0.71 2.21 7.18 1.25 0.77 5.66 0.91 0.60 18.08
0.27 0.16 0.29
Se
76 54.31 1.14 0.75 2.21 6.83 1.24 0.81 6.01 0.95 0.56 20.98
().26 0.10 0.26
78 55.11 1.14 0.75 2.21 7.07 1.24 0.81 5.05 0.95 0.51) 35.79
0.25 (J.n 0.23
80 55.08 1.14 0.75 2.21 7.17 1.24 0.81 5.69 0.95 0.56 14.58
0.14 0.07 0.14 =~=~~~=~=~~==~=~~~=~~~~~~~~~~~~~~~~~mm~~3aDmD3_~_D~~~~n~~_._~_~ _______
77
Table 4.9b Optical-model pa"amete~e w~th an avera~e geometry for th" Ge an~ Se isotopes. Volume absorpCioo 1s sec to the
B"c~hett1-Greenlees value and the Coulomb radius r =1.25 c
fm. The first linE: gives the parameters, tlle second line
the E:rrors 10 the varied parameters deduced from th" fit.
Parameters witll no calculated error are kept fi~ed.
A V 'r a W W r1
ai
V 1; "'50
X2./N ~ r v s 60 60
(MeV) (fm) (fm) (MeV) (MeV) (fm) (fm) (MeV) (fm) (fm)
Ge
70 54.88 L.14 0.73 2.21 6.03 1.25 0.79 6.30 0.93 0.58 14.54
0.17 0.05 O. L7
72 55.02 1.14 0.73 2.21 6.70 J..ZS 0.79 ';>.74 0.93 0.58 47.34
0.31 0.13 0.21
74 54.41 1.14 0.73 2.21 6.79 1.25 0.79 6.55 0.93 0.513 12.42
0.16 0.07 0.14
76 ~5.30 1.14 0.73 2. .il 6.95 1.25 0.79 5.79 0.93 0.58 LS.08
0.27 0.15 0.29
Se
76 54.60 1.14 0.73 :l.ll 7.03 1.25 0.79 6.07 0.93 0.$8 20.53
0.24 0.11 0.27
78 55.38 1.14 0.73 2.21 7.09 1.25 0.79 5.14 0.93 0.58 35.53
0.25 0.11 0.23
80 ,55.24 1.14 0.73 2.n 7.19 1.25 0.79 5.17 0.93 0.58 18.00
0.15 0.07 O.H)
==========~~~D.W~.~.~~~============~~~~a_ ••• ~~·-~=~==========~===._.~-
78
of theae ~wo triah caO b,;, fouod 10 tables 4.9. As cOl.lld be expected
the x2 values are slightly larger then those or tables 4.8.
In figures 4.4a and 4.4b we have glveo the optical-model
predictions together with the experimcotal points for three
calculations. Ihe Jitterencea between beat fit "earch .. ith W fixed v and th" average. geometry are small and can only be seen in the far
oacil.wan\ augula!:" J:egion /l-no on "ome of the tops and '1alll!Y". Th,;, other
calc~lations (best fit ~Lth W v
free and fixed geometry for S" alld Ge)
gj ve the same k.ind o~ c,'\"ves. In gener".l we "an say that the optical
parameters represent the experimental data very w"ll aod that the
differeoces between best fit anJ fixed geometry ar" negligible.
Whether the fixed-geomet.ry pot.ential should have "Ome significance,
Catl be concl~ded ooJ.y by ~nel.:>sttc 5c.:>ttering calcuiatiOns.
One final remark we have to make: th", predictions of the gl,obal
potential of ll"<:"h,,tti-Gr~enlees do not agree ver:y well with the
experiment. In the figure. une call clearly see that for scattering
angle" larg<!r than 60·-70· the differences in the cr;oss section hecome
larger auo larger. This points to a too small absorptive term In the
potential. The analysing power, however, is represented very well.
This means that the mere presence of a spin-orbit potential is already
sufficient to descr~be tohe data (although "'" might e><:pect that this
observable is dependent on the central strength as well).
4.4.z Volume integrals aoo rme rao~1
I'll! ha'1e "eeo that there is gome scatter among the various
optical-model parameters aod that we can define several sets of
parameters which give near:ly the same v".lues for the observable,;;. The
cxistence of these ambiguities in the optical-model paralllet~rs ¥las
r""o~nized long ago (5SFes). So some new quant~tieg have heen defined
which should flu"tuate less than the individual par"meters in a
cert"in waS" re~ion. these quantities are the voll.lme integral:
2 J - f U(r) r dr (4.1)
(4.2 )
79
Greenlees, Pyle and Tang (~SGre) co~cl~~e~ from a folding model study
that these q~antities are very stable against fl~ct~ation", in the
optical-model parameters. The vol~me integral per nucleon Si ves some
information on the lute-.;action strength, while the rms radills of the
real central ~art of the ~otentlal can be connected to the radius of
the matt.er distribution. We have computed these volume integrals and
root mean square radii and present them in tables 4.10 for the various
sets. From these tables we cau conclude the following:
1. the Sacchetti-Greenlees parameters give values for the volume
integrals which are essentially ~l£ferent from our fit5:
approximately ::>% h:lghe-.; for the central potential, 10% lower for
the aDso-.;pt;lve ~"rt and 10% higher for th" spin-orbit term. This
observ"tion is fully in lil'le with what we al-.;eady have conclUded
from the f1gll--.;es 4.4. It ""n easily be noticed if we look. at the
v«lu",~ of the r~action cross section: these are essentially lower
than the val1l-e6 We found in our fits. This fact is also ~eflected
in the larget" real and in the. smalle.r imaginary volume integral.
For the difference in the spin-orbit part. we. ha~e no e~planation:
it is probably "Onn~et~d with the other two. In t.he. a~aly5ing power
this effect cannot be traced back.
2. th.;: general tren.;! i~ «11 of our ~e.ts 1s tbat they g1 ve very ~table
resllLts for the volume integrals a~(l root: me",n square radii. In
order to have an in~isht in the accuracy of the tabulated number~
we h«ve est.imated the errors 1~ the <;aleulation of ~h,;, \101ume
intesrals and rms radii on basis of tlle dedl,l<;ed en:-or5 in rhe
optic",1-mode1 parameters. We mll-5t then cDnclude from th'" tables
that the pa-.;ame.ter sete obtained give il\ allll<;>S1;: ,,11 "as",,, values
which are the sawe ",i thin the ~rrors. This means thae we "annOt
distinguish between sets sin"e t:he interaction is essentially (with
our experimental accuracy) the SRme. What w@ are not able to see is
h<;>w ac<;urately the final observables (cross section and analysing
pDwer) are described. F-.;om the "alculated curves of the sets we
estimate that the accura<;y of the obser~ables is mo-.;e or less of
the same orde~ as the displayed diffo!orel'lces between the variouS
sets ..
3. Another r~mark concerns the difference~ in the pa~ameter sees
obtained fr<;>w OPTIMO and EGIS79. These are reasM.ably small an.;!
this "o~firms again that both pl'og:r:amme'> deliver e,;s(lntially the
80
Table 4.10a Comparison of volume integrals, reduced rms ~adii and
reaction cross sections (a~eac) fo~ Ge i~otDpeB.
BG a global potential Becchett1-GreenleeB,
SF best-fit pot-en1; i a1,
WVFO be8t~fit potential with W from BG (OPUMO) " WVFE beBt-fit pOtential with W" from BG (BCIS79)
FGGe potential with average geometry for Ge isotopes,
GeS., ~ potential with average geomett'y for Ge and Se
isotop~s.
G" fit J IA J i/A J IA I !3 <R2) <R~>~ <R~ ) 0 1:' so r , so reat:
Mev. fm 3 fm ) (mh)
70 BG 432 US 157 4.66 5.68 4.78 ).250
SF 425(10) 120( 6) 142( 5) 4.52(0.03) ).93(0.06) 4.18(0.09) 1311
'iNfO 418(11) 120( 6) 142( 7) 4.49(0.03) 5.82(0.05) 4.11(0.11) 1304
WVFll 416(47) U8(14 ) 141(17} 4.49(0.\2) ;'.98(0.22) 4.12(0.40) 1295
I:'GGe 422( 1) 120( 1) 145( 4) 4.50 5.83 4.20 1315
GeSe 422( 1) 120( 1) 147( 4) 4.54 5.87 4.24 1341
72 llG 436 121 157 4.69 5.77 4.82 L302
BF 414(19) 127(15) 137(15) 4.43(0.06) 5.96(0.13) 4.30(0.35) 1356
WVFO 420(22) 128 (13) 141(12) 4.44(0.06) 5.91(0.14) 4.30(0.21) 1366
Wvn: 412(62) 125(25) 138(~5) 4.43(0. 18) 5.91(0.41) 4.27(0.56) 1362
FGGe 421( 2) 129 ( 2) 131 ( 6) 4.52 5.88 4.i3 1367
OeSe 4Zl( 2) UOC 2) 134( 6) 4.57 5.92 4.28 1393
74 BG '.39 127 157 4.72 5.85 4.85 1),:)1
llf 4U(12) 128(12) J.48( 7) 4.54(0.04) 6.03(0.10) 4.26(0.12) 1409
WVFO 424(LJ.) L28( 7) 154(10) 4.59(0.03) S.87{0.Ofl) 4.41{0.20) 131<
WVFE 4U(45) 127(18) 147(14) 4.56{0.J.4) 6.02(0.2.'1) 4.19(0.32) 1407
fOC", 414( 1 ) HO( J. ) 150( 4) 4.55 5.92 4.26 1392
GeSe: 41S( 1) 130( 1) 1S3{ 3) 4.59 5.96 4.31 1417
76 BG 442 132. 157 4.74 5.93 4.88 U98
BF 405(24) 135(15 ) H8( 9) 4.59(0.08) 1).43(0.17) 3.91(0.14) 1543
WVFO 396(31) 140(0) lL9( 11) 4.58(0.11) &.45(0.15) 3.78(0.18) L560
WVFE 388(94) 138(28) 118(18) 4.56(0.26) 6.42(0.45) 3.82(0.46) 1548
FGGe 422 ( 2) 1H( 2) 129( 7) 4.:;6 5.97 4.29 142Z
G.,So 421( 1) \31( 2) US( 7) 4.62 6.00 4.:34 1447
======~==================~D_a ____ ~_~~M~~_Dam2_~~~~==~=====~~=2=~~m==a~
Table 4.l0b Comparison of vo!~~e ~~tegral~, reduced rm8 radii and
reaction crOBB 6ectio~s (oreac) for Se isotopes.
BG = global potential Be~chett1-Creen1eeg,
BF ~ beet-fit potential,
WVFO = te~t-fit potenEial with Wv from BG (OPTIMO),
WVFE = test-fit potential with Wv from SG (ECIS79),
FaSe - potential with average geomet~y for Se isotopes,
GeSe = potent~a~ w~th avera*e geome~r1 for Ge and Se
isotope~.
81
Se fit Jr/A J 1/A
Md.~g
o re~c
(mb)
76 sG 433 117 157 4.74 5.84 4.88 1309
SF 418(21) 129(27) 142(11) 4.60(0.08) 6.11(0.15) 4.55(0.24) 1414
WVPO 417(22) 131(15) 140(11) 4.58(0.09) 6.08(0.16) 4.40(0.20) 1428
WVFE 408(54) 12SC 9) 138(17) 4.55(0.16) 6.10(0.09) 4.34(0.43) 1430
FGSe 4l7C 2) 131( 1) 143( 7) 4.67 6.01 4.39 1436
GeSe 41~( 2) 133( 2) 142( 6) 4.62 6.01 4.34 1420
78 SG 436 123 157 4.77 5.92 4.92 1339
SF 398(24) 136(17) L22(LL) 4.63(0.09) 6.23(0.20) ).84(0.18) 1~13
WVFO 405(20) 136(10) LL7(12) 4.64(0.06) 6.33(0.13) 3.94(0.21) 1511
WVfE 399(52) 132(18) 116(15) 4.62(0.15) 6.28(0.31) 3.97(0.42) 1496
raSe 422( 2) 133( 2) ll1( 6) 4.69 6.05 4.42 1466
GeSe 420( 2) 132( 2) 120( 6) 4.65 6.04 4.37 1444
80 BG 438 128 157 4.80 6.00 4.95 1364
Bf 421(10) 133(L1) 146( 5) 4.74(0.03) 6.06(0.08) 4.67(0.14) 1480
WVFO 421(11) 133(10) 146( 8) 4.74(0.06) 6.05(0.10} 4.67(0.17) 1436
WVfE 421(12) 129(11) L44(lL) 4.74(0.07) 6.01(0.L4) 4.61(0.27) 1472
FGSe 420( 1) 134( 1) 136( 3) 4.72 6.09 4.46 1492
GeSe 418( 1) 133( 1) 135( 4) 4.67 6.08 4.40 1469
82
8ame re.~ults. The estimated error!;! in EClS79, however, are in
general about a factor 2 large, than those in OPTIMO. We can trace
this back to the larger quoted errors tor; ttle various radii in
ECIS79. The estimated errors in the volume intesral and in the rills
radius are very aen"icive to the corresponding radius paJ:'a,ueter
through the factor r2 respectively r 4 in the integraJ.s of eql"aticms
(4.1-4.2). Why ECIS19 gives these larger errors we have 00 Idea.
4. Finally we note that the volume integral of the spin-orbit
~otential exhibits rather larse oeviation". The observable, the
analysing power, does not display these differ"nces, as we ha~e
concluded b"fore. So our conclusion is that to, the ob6erVahle it
does not matter what the pre~iS~ shape of the spin-o.b1t potential
looks like. This is not sO surpri8ing since 8ecchettl and Creenlees
could fit a large set of data with a constant spin-orbit potential.
It remains w be seen whether tills holds fo.- inelastic scattering
also.
The values we obtained for the valum" integrals from the be~t-fit
s.::ar(:\les (Br and WVF) c<on be ~ompared with prevloos WOJ;"\<3- for the
imagina.-y volume lntegral there is a study by Agrawel and 800d (75Agr)
and by Hodgson (76Hod.). These authors arrl ve at the ,>aMe <;an~lu5ion~
the imaginary volume int:egral per nocleo1;l. ha~ <0 maaa~ and energy
independent (E=lO-60 MeV) value of (115 ± 15) MeV. fm~. Kallas and
Cupta (77Kai) pe,fo.med a similar analysis and concluded the same with
a slightly diffel"'ent value, (125 .t 21) MeV.fm~. Thi6 condusion is
rathe.- remarkable. One might expect that the abso~pti\1'e ter,u in the
optical poteotial (inco.-porating all open reaction channels) is
dependent on specific nucl""r stroctore 0"1; /it least on "'lergy to
account for opening ot neW reaction channels as the incident nu~leon
energy iuc,eases. Our data confirm ttle findinGS of th,,:;;e authors,
thoogh it Se~m8 that [here is some A(N,Z) dep~nd"nce. Ka1las and Gupta
(77Ka1) /ilSO recognized this and analysed the avaIlable data with the
following empirical formula:
with ~ = (N-Z)/A
and 1 - +1 for proton scattering,
= -1 for neutron scatte~lns.
(4.3 )
... E ... >
II>
:==
~
... E
> .. :f
470
460
450
440
430
420
410
400 390
380
370
360
170
160
150
100
90
SF
Ge
--" . ,-
Til j
70 72 74 76
A
WVf
Sa Ge 51!!
j ... .. ' ....... j .. j .' t
1 I
76 78 80 7072 74 76
A
Figure 4.5 volume integrals per nu~leon of the ~eal and imaginary
part of the op"tcal-model potentials,
BF - best-fic optical potential;
WVF - best-fit optical potential with Wv = 2.21 MeV;
full empirical formula from Kallas and Gupta;
dashed ~ pa~a~trization of the model of Jeukenne,
Lejeune and Mahaux;
dotted - Recchettl-Greenlees global potential;
dot-dashed = ~onBtant empirical values.
83
84
The second part of the expl;eeeion rel?reaenf~ th~ effect of vol"me and
~lJrf?ce contribution.:; to the potential ~ is the ratio of their
strengths. The ag,eement of this ~ml?lriL" fo~mula with our data is
very good. 'rho predicted el;Tor band of 45 MeV. fm 3 ~s pro1;>I;<1;>ly an
overe.,t1mate for [hoe proton scattel:ing volume integr-als because the
parameter~ in the eml?irical formlJla were obtained frow proton as well
.. s from ne\I\:Ton ""atter-ing data, the latter bearing much larger
errors. The result of the theoretical wor\<; of Jeukenne, L"jeune aad
Mahaux (77Jeu), which deals with the constructioa of an optical
potential on basia of nuclear-watt~r calculation5, \5 paTametrized in
the same way. "'"SO tllis Bhows a good agreement with 01.11; data.
The Sl1~e ~ppro~ch waB also u~ed by Kailas and Gup~a (78Ksi) for
the real volume int"gral. A constant value of (465 .t 9) MeV.fm~ ~O1;"
pro too scattering el<periment~ below 25 MeV W(l.6 al~o quoted. Wh~reas
the imaginary volume integral is in good (l.greement, w" fifld for the
real Volume integral a complete disagreement with our data. Our
re~u1t.:; seem to scatter around a constant value of (413 ± 10) MeV.fm 3,
l.ihich i~ 11% lower .. nd outside the 99% confidence ~nterva1. Th" model
of JelJkerme, Lejeune alld Mahallx and the Becchetti-Greenlees g10 1;>", 1
potential give a value 6omew\l",t cloBer to ours, but still off. For
thj,s large disagreement we have ao otllel' e",planation then attribute it
to an energy dependence. Th" data set of K .. 1l(l.s and GUl?ta comprises
mainly low-ene,gy experimentB, which give rise to a laq;e "e81 volume
integral. Greenlees, Pyle and Tang (68Gre) showed tbat SlJeh a l.ieak
e,1ergy dependeace e)l:ists. The reaultiflg vahle f.om the~e author~ (400
± 20 MeV.flu 3) for proton ~c"tteriflg data between i4 and 40 MeV is more
1n aceordance witch our re8ults though their WOl;k i~ appreCiably older.
4.4.3 190spin dependence 1n tAe op~ical potential
The optical-modol potent\a1 fit8 l?erformed W1t\l '" fixed geOm"try
enable us to study of isospin effe"tS ill. the opt~c",l potentials of the
Ge and S" nuclides. This 160spin dep"nd<l:nce is formulated usually in
the f 0 1l0wil.1.g way:
V r
W 8
~ W sO
+
+ T E, W $1
(4.380)
(4.3b)
with s = (N-7.)/A
and T = +1 for protons,
- -1 for n@utrons.
85
(4.3.::)
The third t:erm in expression (4.3a) represents a correction for the
a.verage Coulomb energy, which is only present in the c.ase of proton
scattering (63Per2, 68Gre).
Our experimental data cOver the range from A~70-80, so we should
be able to Say something about the isospin dependence. There are two
approaches:
1. use the optical-model parameters from the individual fits to find
the lsoscala~ and i~ovecto, pa,t~ o~ the potential, or
~. use the observables in 8. direct; Ht of j.eol>t:",lar ",nd isovector
strengths acco~d1ng to rel"'tion", (4.3).
In tables 4.11 some values of these properties are given, with the
"direct fit" indicating the results of method 2. Procee,Hllg (ilong
t:hese lines and using a linear regression on the data of t;ables 4.9
(in table 4.11 labeled ~ith FGGe1, FGSel and G~Sel) we dlscove, rat:her
large dev1ations. In figure 4.6 the Bam~ strength parameters are
pre.~e.n'e.d but now as a function of ~.
We notice.:
~. the l,"e"l and. spin-orbit stre.ngths e>:h1bit some scat;ter (is a
function of 1;,
2. the surface (ibsol,"ption shows a larger r~gulari~y.
The irregularities in \:he. re.al strength mak'" it hard to for!Jl a
definite conclusioll. This is also refl~et .. d in the valLles fo)'; the.
diff~rent analyses when we assume the isoapin dependence of e~press10~
(4.3a). The isoscalar part is reasonably ~ell defined; the lsove'tor
part gives the. large deviations and is coneil>te.nt ~ith no isovector
contribution to the real central term in the optical pot~ntial. This
1s in contra.diction to previous results. In an e",rly review of the
isospin dependenc~. Satchler (69Sat) suggested", value of Vr1 • 25
MeV. The WOl,"~ ot Becchetti and Gr~~nl~es (69Bec) On the global Optical
model potential results in a value of Vrl = 24 MeV. Ev~n when we take
the upper l1mit of our values there 1s not much agreement.
The re.gular behaviour of the surface absorptiOn is in accordance
with the above conclUSion on the volume integral. The empirical
116
Table 4.11 Isoscalar and isovector potential strengths from the
fi"",d-g",om",try Ots (v~l~les in MeV).
Fit
~'GGel
FGG~2
fGSel
FaBe2
fGl
FG2
FG:)
GeSe1
GeSe2
GeBe)
CeSe4
G~S,,5
GeSe6
~GGel - fixed g~omelry Ge isotopes
FGGe2
fGSe~
FGSe<
FGt
FG2
FG3
GeSel
fixed geometry Ge isotopes direct fit
fixed geomet~y Se i~otOp",~
= fixed geomet1:Y Se ~so~ope$ d~rect fit
d8ta f1;om FGGel and FGGel e)<cept lUGe
O;,lt,-; from FCG", 1 "nd FGSel with <;<0.13
data from FGCel and FGSel with <;>0.13
~ fixed geo1ll0try Go and Se isotopes
G",S~2 fi.xed geome.ry Ge and Ge I.sotopes di rec .• fl..
CeSe3 ~ data [rom GeSel wich ~(0.13
GeSe4 data from CeSel with (>0.13
GeSe5 ~ fixed gcolllHry Ce and So;, isotopes direct fit ~<O .13
GeS,,6 fi. xed geometr.y ae "nd: Be igotopes direct fit ;>0.13
V rQ
V rl
W sO
W sl
V soO
V 901
51.8( 1.3) 5.0(10.3) 5.2(0.5) 13.0( 3.8) 6.7(1.2) -5.4( 9.8)
51.8(0.4) J. (>( 1.8) 4.~(0.2) Hi.8( 3.8) 1).1(0.7) -1. 1 ( 0.13)
49.J(l.4) 18.4(HJ.5) 1).1(0.2) 7.6( 1. 7) 6.5(2.6) -7.4(20.1.)
50.J(l.() 1 o. 5( 4.0) 6 . .)(2.0) 5.4( 8.) :>.4(2.2) 0.;>( U ,(,)
6.2(0.1) 6.6( 0.9) 9.2(0.5) -31.S( 4.5)
12.2(2.1) -42.0(L4.L)
51.0«().7) b.9( 5.7) 5.5(0.6) 1().7( 4.8) 6.5(1.0) -4.9( 7.6)
51.10.6) 5.8(l2.~) 4.9 «). 2) 15.9( 5.2) 6.1(0.2) -lo9( 4.9)
H.8(O.6) -27.9( S.7)
11.3(2.1) -35.5(14.1.)
50.8(1.9) 11.2(28.4) 3.7(0.7) 28.0(29.7) 3.5(1.5) -24.3( 8.9)
4:>.0(1.7) 47.3(89.4) 1,.4(0.9) 17.8(14.3) 13.2(2.8) -49.6(24.0)
87
FG GeSe
53.0
52.5
> 52.0 4)
::: .... 51.5 >
51.0
t f
t j j d tf t
T
t t t
7.5
> 7.0 4)
::f
'" 6,5
+. + , + + !, ++ 0
"' + '" Cl DC]
IJ
3: 6.0 ~
IJ DC]
+ []
6_5 > a;
.! 6.0 0
>'" 5.5
5.0
t T +
+
Ij II t t d t t
O.OS 0.10 0.12 D.l4 0,16 0.08 010 0.12 0.14 0.16
E.. ~ Figure 4.6 Potential depths frG~ £1xed-geu~etty optical-~Qdel
potential fits as a function of ~ = (N-Z)/A. The (eai
central depths are corrected fOr the average Coulomb
energy v' = V - 0.4 Z/A1/3. r I;'
Th~ dots co~responds to the Ge isotopes, the triangles to
the Se isotopes, the squares to calculated values derived
from Kailas and Gupta (77Kai)_
FC = fixed geomet~y for One element,
CeSe = f1x@d geometry for Ce and Se.
88
formula of K"ilas a.nd Gupta (77Kd) works well and h"s been used to
calcula~e the a.bsorption strength •• he calculated values of W~ have
been put into figure 4.6 as $<juares. There is "n overall agreement,
when the estima~ed er~or of 2.5 in the calculated points is taken into
aCc.ouIlt .. The figure laheled with GeSe g:J.vea the 1\Ilpre6sion that the
point at ~~O.O86 eOOe) seems to hav~ a too low vall,le 1tl cOmparison
with the other points. With the ca1cul"ted poilus tlcxt to it, th",
point at I;=O.lil ( 765,,) has a too large value.
The large error in the ~~ovector p"rt of the spin-orbit 6trength
is now easily explained. The Vso shows a linear dep",ndence on ~, but
has a jUI~P at " value of I;~O .13. The regol"rity io the data paints
displayed in figor", 4.6 is rewarkable, since our results indicated
thilt the spin-orbit effects are rather independent from the
eorresponding shape of the spin-o~bit potential. ro~sibly the
correlilttDn with th", other parameters masked this effect in earHer
analyses. The jump tn the spin-orbit strength is even more sorprising,
especiilily that it appears for Ce between N=40 and N=42. Some ()ther
effects app",aring at the sam" place - (p, C) and (t, p) r",sult~ for tIle
01 state (77GulZ, 78V,,03) - have been explained as a change irl ovclear
structure, cOnnected to the clORure of the 2P1/
2 neutron subsh~ll. If
this 1s t-cue, [he same. effect shOuld be display"d also by the 5e
isotopes. These. however, ~how th'" jump between ~=44 and N~46. There
m(l",t be a special coupUng of the two exu·" protons to the nelltrOns. A
subshell closvre at Nm40 is anyhow excluded.
In ord~r to be sure that the effects mentioned abQve are reel and
not artificial as " result ~rom the independent analysis of the d",t:a,
we fitted th" isovector and isoscalar strengths directly on all data
for va. 13 and ~)O. 13 with the programme OPTIMO accoJ:ding to the
equations (4.3). The final reso1ts are listed in table 4.11. These "re
ee"entially the same values "5 found J.n the ind"'pendent analyses. so
we may conclude that this e.!'fect is real. A cQnfirmation of this Wo<
find in the recent study of Sen et ",1. (855e05), who found" simllar
change in the elastic vector-"nalysing-power d"u. of polarized
deuterons and protons. They think it not to be an indicatioo of ground
state "hape but more th" result of e£fects ot target ",xcitat~on on
elil6tic scattering.
·the be.st way to obtain the iSQBpln de.pendence is to ev",luate
protOn and neutron B~atter1ng data together. This is difficult in our
89
case, since we did not per£o~1!l neutron scatt~ring eKperirn"nt" and
.tnce data in the literature are scarce for an incident neutron energy
ahe',.,) OJ MeV, From the cOlllp11ation in tabl,,~ 4.3 and 4.6 We see that
there are only t~o expe~iments for Se (76La12), (84Ku09) and nOne for
the Ge isotopes. The work of Kurup et 61. (84Ku09) is a complemellt [0
ano 60 eKtension of the earLier work of Lachkar et 61. (76Lal2), The
latter found anomalous values of the iSovecto~ potential: a too small
value of v r1 (9.3 MeV) and a too large value of Wsl (43 MeV). Kurup et
al. could explain thi~ as due to the negl"c.t of 6tt"01.g-coupli[ll;;
effects in the computations. When optical potentials are deriveo ~rom
coupled-channels calculations with strong cOuplings to excited states,
;hese anomalies are removed. The final values of the lsovec.tor
HLrengths (20 Mell) are mar" in aCCo~daac8 with the predictions. The
p~oblems We ,;,ncOunter in "xt~actiag the isove~tor pa.ts may well have
the 63me cause. Our values are~ however, in all cases too small t ~ith
one remarkable fact: for the Ge isotopes Vrl
is almost zero and Wsl
hal; almost the right value While for the Se isotop1!g it is jUH the
other way around. Whether ehb can be repaJ.~ed with cou!,led-channels
calculations is doubtful.
4.4.4 GeDeralized-optical-mod~l search
As described above we next performed a generall,ed-aptical-mooel
sea.ch with two models (see figure 4.7):
- the harmonic vibrator model (for 0+, 2+ and 3-),
- the symmetri~ rotator model (for 0+, 2+ and 4+)
th" grid search was started with global .;tefe.lIlation param<!:ters
deduced either from a DWBA or ~oupled-channela calculation or from the
literature, Since an optt~al-model Search with three strongly coupled
channels is much more (computer) tlme consuming then with one channel
(approKlmately quadratically with the number of channels), we C,~eQ GO
start with deformation str~ngths that were as oear as possible in the
neighbourlJood of the "'real"' values. So &eneJ:aUy we needed not more
than 9 calculations for the >1ariat:ion of the deformation parameters.
Especially the computations with the symrn~tric-rotator model are very
"xpen",ive; about a factor of two as compared to the vibrator model.
Already in an early stage the values obtained (rom OVIllA analyses
appeared to be considerably larger th~n the final values from couplp.d
channels "alculations an.;t therefore provided no good 6tarti~g val,,"~.
90
Table 4.12a Best-fit generalized-optical-~odel paramete~s for the Ge
isotopes with the vol~me absorption set to 2.21 MeV (frQ~
(69Bec». Coulomb ra~iU8 r~-1.Z5 fm. Each f1rst line
gives the parameters, the second line the errors in theBe
parameters deduced from th~ fit. An asterisk indicates a
fit in which rr and a1
have been handled separately.
Ge V r
(MeV)
VMl £~rBt-order harmonic~vibrator model,
r r
VM2 second-order ha~monic-vibrator model,
SRM ~ymmetrlc-rotator model.
a r
a 1
(fro) (fm)
~s r 1 (MeV) (fm) (fm)
Vso rso aBO XL!N Model
(Mev) (fm) (fm)
70 54.56 1.15 0.75 5.03 1.32 0.69 6.04 0.92 0.63 ll.31 VMl
0.88 0.05 0.02 0.23 0.08 0.02 0.30 0.09 0.06
53.90 1.15 0.68 4.87 1.31 0.~8 5.93 0.95 0.58 17.51 VM2 * 0.35 0.01 0.01 0.19 0.09 0.01 0.11 0.10 0.04
56.00 1.14 0.72 5.92 1.28 0.66 6.64 0.89 0.70 13.92 sRM
0.93 0.05 0.02 0.32 0.06 0.03 0.38 0.11 0.06
72 54.64 l.l6 0.70 4.99 1.28 0.76 5.87 0.95 0.75 32.24 VM~
1.38 0.08 0.03 0.49 0.16 0.05 0.46 0.20 0.09
54.91 1+16 0.62 4+92 1.Z7 Ot74 5.79 1.14 0.56 33.35 VM2 * 0.54 0.01 0.02 0.30 0.13 0.01 0.25 0.15 0.05
54.91 1.15 0.68 5.24 1.22 0.82 6.51 0.97 0.72 ~0.84 SRM
1.34 0.06 0.03 1.03 0.10 0.06 0.42 0.16 0.07
to oe continueQ
91
Table 4.Ua Best-fit generalized~optical-~odel parameters for the Ge
isotopes (continued) •
Ge V r a W r1 at V r a X2/ N MoMl
r r r s sO ~o so (MeV) (fm) (fm) (MeV) (fm) Urn) (Mev) (fm) (fm)
74 52.12 1.18 0.71 5.58 1.28 0.69 6.48 0.92 0.69 16.03 VMl
1.02 0.07 0.02 0.44 0.09 0.04 0.37 0.10 0.08
51.34 1.17 0.64 5.23 1.28 0.69 6.54 0.92 0.65 23.51 VM2 * 0.50 0.01 0.01 0.22 0.10 0.01 0.39 0.10 0.06
:>6.30 1. i.Z 0.73 5.98 1.32 0.04 7.19 0.86 0.65 19.11 SRM
1.21 O.OS 0.02 0.57 0.12 0.03 0.23 0.13 0.08
76 54.38 1.13 0.72 4.73 1.41 0.74 5.49 0.87 0.55 14.53 VMl * 0.61 0.01 0.01 0.28 0.12 0.01 0.40 0.12 0.06
54.37 1.13 0.69 4.63 1.40 0.73 $.76 0.91 0.54 16.83 vM2 * 0.56 0.01 0.0:1. 0.31 0.13 0.01 0.29 0.12 0.08
58.90 1.06 0.78 S.10 1.48 0.6~ 6.33 0.80 0.45 13.00 SRM
2.28 0.14 0.04 0.25 0.18 0.03 0.42 0.l3 0.05 ______ 2D~~3mcmD=~~3~~~=~C~=====~======~~c==n_~~~~-ft __ .~ __ ••• m=~=======
92
Table 4.~2\> Best-fit seneralized-optical-rnodel paramet~rs for the Se
isotopes ~ith the vol\lme absorption aG:t to 2.21 MeV (from
(f)9Bec». Coulomb radius r -1.25 fm. Each first line c
gives the plilrameters, tlte second line the errors in the:ge
Plilrameters deduced from the fit. An asterisk indicates a
fit in which r~ and a1 have been handled separately.
VMl first-order harmoni~-"lbrator model;
VM2 = gecond-or~er harmonic-vibrator model;
SRI'! symmetric-rotator model.
Se V rr a fl r1 a
1 V r a so X2/ N MO~el
J:" r B so so (MeV) (fm) (fm) (MeV) (fm) (frn) (Mev) (tm) (fill)
76 51.04 1.20 0.69 5.86 1.26 0.69 5.54 1.03 0.67 19.98 VM1
,.33 0.09 ().OJ 0.72 0.14 0.06 0.36 0.20 0.08
.51.72 1.19 0.65 5.64 1.23 0.69 6.50 1·00 0.78 22.51 VM2 .. 0.43 0.01 0.01 0.20 0.08 0.01 0.35 0.18 0.07
53.03 1.17 0.69 8.37 1.28 0.55 5.22 1.05 0.50 23.00 SRM
1.~8 C).07 0.03 0.73 0.07 0.03 0.60 0.17 0.15
78 54.16 1.13 0.72 4.59 1.42 0.73 5.86 0.81 0.58 ~4.05 VMl
0.50 0.09 0.02 0.26 0.09 0.02 0.39 0.09 0.07
.54.97 1.13 0.67 4.65 1.36 0.72 5.58 0.83 0.52 29.07 VM2 * 0.56 0.01 0.02 0.32 0.13 0.01 0.41 0.10 0.10
58.H 1.08 0.76 .5.29 1.44 0.65 6.39 0.76 0.50 19.40 SRM
0.84 0.05 0.02 0.28 0.11 0.01 0.36 0.10 0.05
eO 54.3& 1.15 0.78 f).28 1.29 0.74 5.99 0.96 0.74 13.71 VMl
0.75 0.05 O.OZ 0.32 0.07 0.02 0.24 0.13 0.06
54.19 1.15 0.72 6.25 1.26 0.74 6.03 0.96 0.75 17.40 VM2 .. 0.30 0.01 0.01 0.14 0.05 0.01 0.26 O.ll 0.06
56.43 1.12 0.77 6.73 1.28 0.72 6.17 0.94 0.72 13.1) SRM
0.76 0.04 0.02 0.34 0.06 0.02 0.25 0.13 0.06 ==~=~~_~_~_zsa~===z_~_~_~_~~=c~ __ ~_3aD~ __ • __ D~~~ __ • ___ = __ m_. __ ==~_2_._
93
Tap1e 4.l3 Comparison of volume integrals, rme ,adli and r~actiQn
~rQe6 sections (Oreac) of generallzed optical models.
IIMI = fi:tst-order ll'ibrator model,
VM2 • s€cond-o:toer vibrator model,
SRM = ~ymmetric-rotator model.
A fit J/A J itA J /A 1/3 <R2>~ <R2) <a4 >~ 0 60 r i so ,eac
( MeV. fillS (fIll) (mb)
Ge 70 VMl 433{SO) 102(14) 1,40(16) 4,61(0.14) 5.85(0.26) 4.28(0.35) 1298
VM2 413(10) 97(12) 142(14) 4.46(0.03) $.79(0.27) 4.31(0.37) l.294
SRM 432(19) 105(13) 149(21) 4.56(0.06) 5.73(0.23) 4.29(0.41) 1264
72 VM1 431 (7&) 103(31 ) 140(33) 4.56(0.22 ) 5.91 (0. 58) 4.&0(0.82) 1327
VM2 418( 11) 98(21) 166(23) 4.39(0.05) 5.82(0.4:» 5.06(0.60) 1316
SRM 425(68) 106(26) 159(27) 4.54 (0 .20) 5.90(0.36) 4.65(0.55) 1350
74 VM1 431(68) 102(9) l)0(19} 4.66(0.18) 5.82(0.37) 4.43(0.43) LH9
VM2 410(11) 97(1&) 151(18) 4.49(0.03) 5.80(0.34) 4.37(0.39) 1316
SRM 416(80) 108(21) lS,5(Z4) 4.60(0.21) 5.94(0.43) 4.17(0.48) 13S~
76 VM1 402(1l) ll7(22) 120(19) 4.S7(0.05) 6.37(0.41) 4.07(0.47) 1512
VM2 396(11) 112(23) 132(19) 4.5L(0.05) 6.31(0.46) 4.21(U.47) 1478
SRM 386(99) ~Z3(31) 127(20) 4.49(0.37) 6.53(0.61) 3.70(0.52) 1527
Se
76 VMl 437(84) 102(25) 143(32) 4.70(0.25) $.$0(0.49) 4.84(0.85) 1307
VM2 425(10) 94(12) 163(32) 4.59(0.03) 5.69(0.28) 4.89(0.70) 1298
SRM 431(69) 115(18) 138(26) 4.68(0.19) 5.73(0.27) 4.76(0.70) 1279
78 VM1 399(82) 114(17) H9(15) 4.60(0.24) 6.42(0.34) 3.91(0.36) ~457
VM2 395(11 ) 103(19) 116(17) 4.49(0.05) 6.20(0.41) 3.90(0.38) 1426
SRM 394(50) US(;!L) 122(17) 4.60(0.14) 6.44(0.39) 3.63(0.37) 1491
80 VM1 412(13) 119(13) 145(19) 4.76(0.05) 6.09(0.24) 4.74(0.48) 1451
vM2 4L7(LL) 113(10) 146(19) 4.68(0.03) 5.99(0.19) 4.76(0.45) 1429
SR.M 419(40) 121(15) 146(19) 4.74(0.13) 6.07(0.23) 4.66(0.46) 1450 _~~~~~=====~===~~=~3Dm~ __ W_~ft~ __ a=========a=~~_~_._a~============~~~._
94
r 4+
r ~2 ~4 2+
~2 ~3 ~2 0+ 0"
<l. b.
F1Sl.1I:e 4.7 the coupling schemes for the seneralized-optical-model
searches: a. vibrato~ model and b. rot<lCor model.
So we discarded the further u"" of defOJ;m(l.tion parameters deduced fro,"
DwBA analyses.
Th" results of the generallzed-optical-model pa);ameters for the
deformation parameters deduced are siven in tabLes 4.12, 11'1 the
8eneralized-optical-mooel searches we found that for some combinations
of e2 , 6 3 and of 1>2' B4 unphysic<ll v(l.lue" for all. potential par(l.meters
resulted. Thi. is probably caused by the well known ambiguities V .ro r r
;:100 Ws
.a1
, In such case" a fit w;:lS performed fil'st with rr and ai
fixed to reasonable valuee and varying Vr and \Is' wl,!le aft"rwards "1"
and a i were adjuoHd co obt~in the agree(llent with the dat". Paramete~$
obCalneo in ouch a manner are marked w~th all asterisk. in table 4.12.
The e~<o<a as quoted in tabl" 4.12 UIe the combin.ed resultS of the two
fits. It "eems that i[l. such all. analys~e. thE! errors deduc~d are in
general sm<lUer than in the normal fit of all 10 paralJleter~. '['his is
due to missing corr~latlons between fr and ai
lind the other potential
pa);~meters.
n,e grid search with the vibrator [IIooe1 was performed in a first
ord~r approach, 'l'bc par<lmete.rs belong;l,ng to the second-order model
wer", found by fitting the optical potenth.1 with the final values of
the deformation p<lr~mete.rs of the first-order c<llculation. 1he reaSOn
for this was tf,at we found the above.-mentioned ambiguities to be
present especially in the second-order calcul~tions but that only
minor differences showed up in the resulting deformatiOn p,H·<lmeters
between fieBt-order and second-orde~ calculations.
ss
The ",xtl:"action of suitable value;; of Sz and iSs in to.", vibratot
model and of e2 !:'l the. rotator model with the corre.sponding optical
potentials is stl:"aightforward be it with sOme minor difficulties. On
the other hand the extraction of a good value ot ~4 is " notor;lous
problem. As state.d before the 4+ cross section is rather
Structoreless. Moreover it appeared in ou~ analyses that the slopes of
experimental data and caloulated curves are often v",~y moch different.
As e"tra complication we have same doublets of 4+ and 0+ stat",s whlch
had not been r",solved experimentally. Furthe~more the calculated
magnitude already exceeds for 64 -0 the experimental strentths; a
feature that is hard to repair through introduct~on of e1
deformation
since its valu", should be small (e 4 in ehe order of az). One could
r",doce the values of liz by 0.02 or 0.03 {zlO%} to obtain better
agreemene for the 4+ state but then the strength of the 2+ state is
unde~estimat"d completely. In view of our argument.s for the
generalized-optical-model app~oach we th!nk this a methodologically
bad approach. For the determinatton of the 6 4 value We always first
determin~d the ~2 value from the 2+ state (since the influence of the
64
on the 2+ cross se"-tion is only small) and tried ne"t to find a
value for 11 4 , Ou. analysing-powe~ data provld", in this resp"'<:.t some
help since the sign of 6. will influence the results of the
cal,,-ulations. A negativ", sign will modiry the pattern. Connected With
the difficulties of finding a good value of B4 is of "-aura,,, also the
correspondins optioal pOtential,. Fortunately tlle nlue of the B 4 is
only of mino. importan"e and the impact on the resulting potential is
81l1<111" since the potential is mainly determined by the (large) B, deformation.
The values of the potential par"mete.s appear t.o be different
from the va lu", s in tables 4.8. The most dis tinct difference is of
"ourse in the absorptive pOtent~al, where Ws as well as ai
are
small~r. Sut also the real potential i~ $l~ghtly affected, where~s the
spin-orbLt potential exhibits only minor cl1.anges. Also Va1:y distin,,-t
is the difference between the vibrator a.nd the rotato-.: potentials, as
might be expected. From this reault it is cOnfirJDed that explicit
inclusion of strong couplingS lead to a dlff",-.:eut optical potential
an<;l that normal optical-model param"'ters are inco<r",ct in coupled
channels caloul<1tions w1.th strongly excited states 1. e. large
defor~tion parameters.
The volume integrals from these searches, p-.:esented in table 4.13
96
a"tual~y "how the same differences with the previous resl)lts from
section 4.4. i. (table 4.10). The volume integral of the aDsorptive
potential is about 20 MeV.fm 3 lower, i.e. 17%. That is not so
surprising s~nce th"ge cOl)plings e)l;pU,citly account for the loss of
flux. The v(11)me integrals of the real and the spin-orbit parts have,
as expected, about the. Dame value as prev~ouBly and thi", also holds
for the corresponding rrnS radii.
The results we found for the deformation parameters are listed in
table", 4.14. A first comparison of our values Iolith literature show,> a
good agreement. The d1f~erences are small and more or less within the
error intervals. A fl)rther discussion on the details of ou,
deformation paramete~6 6nd other quantities dedl)~ed from these ~ill be
given in the next ~e~tion,
4.5 Inelastic scattering
H6ving obt~ined the deformation parameters together with th~
opti<:.a~ potential we can go on with the analyses of Other Ine~astic:
ch6nnels. ~L rst we perto"med, on the basis of these de.formation
paramet:ers, para\1let:er~fro<e cal~_ulations with the vibrator l~odel and
the asymmetric-rotator model. The vibrator calculations are so-called
pure two-phonon calcula~lons. No new parameters e.nt~r in these.
Since the pu':" two-phonon cha~a"ter is seldom encountered (exce.pt for
the 01 state), we afterwards added a one-phonon cont~!bution for the
21 and the 4t state (see "oupllng scheme 1n tigure 4.8). This results
in tW'e eKt>:a par1ltDeteI;s' the one-phonon ~j and the mixing ang~e 'l'J'
The calcl~latlons Were perforllled according to tlle prescription in
section 2.2.4. The resuLting parameters are listed in table 4.14.
In the asymmetrlc-rot6tar IIlodel we useQ the deeormation
parallleter" and the optical potential dedul'.ed from the ~ymme.tric
rotator model, though this millht not be completely adequate for tne
asymmetric-rotator model. For these calculations one additional
parameter is needed: the a~ymmetry y. For the Se isotopes we obtained
its value from the ratio of the excitatiOn energies of the 2t and 21 states (see. table 4.15). For the Ge 1",otopes this is not possible
since all ratios are ~maller than 2. The vall)e of y for these nI)clides
"'a'l Obtained froIIl the ratio of the 8(1::2) valuM (see table 2.th). It
6eem5 that a value of approximately 27" is appropriate for all
97
n~clides under ~t~dy. lh15 15 in accordance ~ith ~he e~pectatlon 1~30'
for vibrational nuclei as expres~ed in section 2.2.3.
J+
n~ r2 r3 2+
n2 0"
figure 4.8 The coupling ~che~e to~ a m1~ed state of one-phonon and
two-phonon natura.
in figures 4.9-4.15 we present the results of the vibrator
calculat10ns together with the e~perimental data. In each figure four
curvee have been drawn:
- a pure two-phonon calculation with the first-order as well as with
the second-order vibrator model,
- a calculation in wich the 2! and 41 states are mixed with a one
phonon contribut1on, again w1th the first-order as well as With the
second-order vibrator model.
The.se figures give rise to the following r<".mark5:
1. The ground state.
Since we performed generalized-optical-model searches this state
should be r~presented very wel~. ~his 1s indeed the cas~ for the
first-order ~al~ulal;ion",. In the second-order we still notice a
de.crel'~e of the cross section in th<l backward l'n~ull'r region.
This is after all not so surprisin$' In thie kind of calculations
more couplings enter. Especially the d~rect e~citation of a two
phonon component will influence the grOl,lnd etate partic1.l1ary in
the backward angular region.
11. The 21 and the 31 states.
These states, which hsve been included in the generali.ed
optical-model approach ehould be also represented very well. This
i" indeed the case for the first-order calculations. In second~
98
Tabl" 4 .14a Comparison wit~ previous proton-scattering analyses of
the Ge iSotope6.
J)1! B~"L-f1t optical potential,
BG = Becchetti-Green1ees global potential,
CC ~ co,-,p,led~"han<1.els potential fit,
Vlil fir5t-order vibrator model (this work),
VM2 se.cond-"order vibrator model (this work),
SRM - symmetric-rotato~ model (this work).
Harmonic v 1 bra tor (A) symmetric ~otator
Ge OM Bz S3 e2 . 4>2 ~4 ,
q", 62 a4 Y22 Kef.
70 BF 0.Z2 0.25 0.01 0.05 70cu03
CC 0.202 0.202 85Se05
CC 0.22 0.22 -0.013 46 0.080 61 0.22 0.025 28.6 VMJ,-SRM
(0.01)(0.0~)(0.006) (3)(0.027)(6)(0.01) (0.010)
cc o.n 0.22 0.023 54 0.OS5 68 VM2
(0.008) 0)(0.017)(3)
72 ll~' O.V 0.23 0.06 0.01 70Cu03
CC 0.203 0.203 8S5e05
CC 0.25 0.23 0.056 3 -0.037 33 0.2$ -0.010 27 .2 VM1-SRM
(O.01)(0.01)(0.00t)(3) (0.008)(4)(0.01) (0.010)
CC 0.15 0.23 0.0:$5 6 0.042 28 Vt12
(0.001)(2) (0.005)(2)
74 BF 0.23 0.13 0.06 0.02 70Cu03
BO 0.29 0.16 0.07 60 0.02 ~O 82f,,16
cc 0.208 0.208 S$Se05
CC 0.28 0.15 0.011 13 0.037 27 0.27 ~0.015 2:5.6 VM1-SRM
(0.0,)(0.01)(0.002)(3) (0.020)(5)(0.01) (0.0l.'1 )
CG 0.28 0.15 0.052 17 0.028 16 11M2
(0.003)( 5) (0.028)(6)
76 SF 0.22 0.14 0.03 0.00 70CuO}
BG 0.26 0.15 0.085 55 0.020 50 0.26 0.020 24.5 83Ra32
CC o .211 0.2U 855e05
CC 0.25 0.15 0.067 6 0.064 41 0.25 O.ON 26.0 VM1-sRM
(0.10)(0.10)(0.004)(4) (0.011)(3)(0.01) (0.020)
CC 0.25 0.15 0.073 21 0.024 24 VM2
(0.002)(3 ) (0.006)(1 ) ==m_Y~~====2. __ ====~_~~m===== __ ._~==~=~_._~==== ___ -~====~ __ ~3C==~=_ftM=
99
Tablt! 4.141:> Comparison with previous proton-scattering ;l\nalyses of
the. S,,- i"otopes.
BF = B~st-fit optical potential,
CC = coupled-channels potential fit,
VMl • first-order vibrator model (thiB work),
VMZ ~ eecond-order vibrator model (this work),
SRM = symmetric-rotator model (th~s work).
Harmonic. vibrator (A)symm~tric rotator
Se OM 62- ~3 112' h Bit ,
h 6 2 Bit YZ2 Ref·
76 BF 0.28 O. L6 79Ma28
CC 0.281 0.014 65 0.267 0.012 83l1a59
CC 0.26 0.15 0.:)08 0.040 24. 84D,,01
CC 0.28 0.17 0.085 5 0.049 51 0.28 0.01l 24.2 VMl-SRM
(0.01)(0.01)(0.OOZ)(3)(0.OtO)(1) (0.01) (0.010)
CC 0.28 0.17 0.085 16 0.046 46 VM2
(0.001)(2)(0.010)(1)
78 IlF 0.24 0.18 79Ma28
CC 0.256 0.002 60 0.255 0.001 83Ma59
CC 0.24 0.14 0.264 0.027 24. 84DeOl
CC 0.26 O. L7 0.078 13 0.042 28 0.26 0.001 24.5 VMl-SRM
(0.01)(0.01)(0.001)(3)(0.008)(4) (0.01) (0.010)
CC 0.26 0.17 0.094 57 0.015 24 Vlil
(0.012)(2)(0.011)(2)
SO SF 0.21 0.17 79Ma28
CC 0.196 ~0.026 45 0.194 -0.034 83Ma59
CC 0.21 0.14 0.2.29 Q .013 24.5 84DeOl
CC 0.21 0.17 0.082 57 -0.020 79 0.22 -0.025 22.8 VM1-SRM
(0.01)(0.01)(0.020)(5)(0.011)(9) (O.Ol) (0.010)
cc 0.21 0.17 0.06$ 30 -0.051 53 VM2
(0.005)(8)(0.009)(4)
=~W._a==========m~~~~~D~========Z __ ~~_am~=====22~_~_.~~=========~a~~~~= Remark: In the analysis of 84D~01 B~o/arm1.22 and rc~1.21 fm is used
in all calculations.
100
T~l>le 4.15 Calculated values of the parameter y in the Davydov
Filippov model of the asymmetric rotato ••
Ge
70
72
74
76
Exci tat;ion energies f1;om adopted leve15 iii Nuclear Data
Sheets (84Sin, 81Sin, 82S1n).
Reduced transition probabilities 8(~Z) f.om Lecomte et a1.
(80Le24) .
B(E2,2 1..o)
(e 2b 2)
3.57(0.06)
4.18(0.()6)
6.09(0.06)
5.56(0.06)
s"
76
78
80
2.175
2.132
2.115
R(E2, 2 2->l»
(e 2b 2 )
0.026(0.020)
0.019(0.008)
0.13 (0.05
0.17 (0.03
'(
(degree)
26.73
27.16
26.73
)J(E2,2 1->O)
Il(B2,2 2->il)
135(104)
2Z0( 93)
47( 18)
32.7(5.8)
"(
(degr"e)
28.0(~:~ ) (+0.2 )
28.5 -0.5
26.7 (~~:~J 26.0(~:1)
~~~m~~~====~~_a=======~ __ ~_~~===== __ .m====== __ ftU~======
101
order we immediately see the infl\,1ence of the extra coupling
te1;II\S. A clOSeI' look reveal", a contradi.ction. Wh" .. e"" the 2+
state is improved in the backward angular region (clearly visible
in the analysing power) 1;>y the Second-order ~ibrMOt model, the
3- state ia described worae: there is " shift of the pactern
towards larger angles in the analysing-power £ igures. This is
probably due to couplings from the 3- state towards the "two
pllOnon" states which disturb the 8-gr~ement,
iii. The O! states.
If a state exhibits a two-phonon character then it should be the
01 state, since a mixing of one-phonon character is ~~cluded. In
the set Df u\,1clides WOe have studied there are £0\,11; nuclides,
which have 01 states in energy resolved frOm other statesl 70Ge,
72ce, 76ae and 1eSe • ijnfortun~te1y, oowever, the intensity in the
case of 760(0 \i'as not measurable, so we are left with three
reaSOnable data seta. It must be remarked that "lso for these
nuclides the ~ntensities were low (some data points have a cross
Sect~on less then 0.01 mb/sr) and this is reflected in the ratoer
large error bars. Nevertheless these states sho\i' a distinct
pattern, for the croee section as well as for tlle analysing
power. Having this il)- mind we must conclude to 8- remarkable
agreement of the ca1eul8-tions in the second-order v1b~ator model
in view of the faet that in these calculations no ~ree parameter
enters, Especially the minima in the cross sectiOn at about 120 0
and 150· are 8-1so found in the calculations. But also the
analysing poWer oas a good agreement for angles beyond 60·. For
70G" and 72Ge even the absolute magnitude of the cross section is
right. Only in 765e the magn~tude is about a factor 5 too high.
This cannot be repaired by lowe1;ing the value of ~Z. So We must
conclude that some interference terma are missing_ TOe first
order results are in all cases I,lnsatisfactory 1 the diffraction
pattern in the cross seo;tion is no~ described correctly, the
results of the calculation6 even being completely D\,1t of phase
with the data. This .... an'" that the direct-e)(ci~ation component
not present here is re8pons~ ble for the improvement wl;len doing
second~orde, calculations.
102
~ dQ
I mb!zr" I
0.1
10
0.1
o. L
0+ 1
0.000 M,N
BQ 80 l~O t50 leO 0 flom (d1:9 I
90
0·5
0, R
0·5
~o 90 lZ0 ISO 1eo 8<=.rn ',,'.90)
l03
~,.
O. l
01 .. ~
~ "b/.r t O. • 0.01
·O·~
0.00\
o·s
D·
-li-'
'0 .11 ID I~O \50 lao 0 'D 10 .0 ItO IGO 1.0 'fI. Cd.1 J -66. ....... 1
Figure 4.9 Elastic a~d i~elastlc pol~~L~ed-proton scat~erins £ro~
70Ge. The curVes represent ~ibrator-model calculations
full - first-order: pure one- and two-phonon,
da.shed first-order: 2+ ~I\d 4-t states mixed,
dotted second-order: pure one- and two-pho~on.
dot-dashed sec.ond-order: 2+ and 4+ states mixed.
104
o 1
10
d. ~
(lIIIb/ls:r I
O. I
d. di5"
(lI'Ib/:!iif" I
O. I
30
1 i
o· I
~. , O.l3J.iI M.V
3· I
2.515 MfjV
" " "
SO 90 '20 100 180 0 8
ertl [dl:.5l)
o ·s
o· "
-C·s
0·.
o. "
.Q ,:5
30 60 $0 120 '100 180 8" ....... Ides)
105
0-5
"0- O.! di1
! II'Ib/.lJ" l D.
0-0 t -D,S
,. 2
1.464 tMN
0.6
do-;;Q Cd D-t""b/.r J
·0'0
C .01
4+ 1
1.728 MtN
a-s
Jwr'-'" , "-'" _.' dO" \ d2 O. t '-.\+11 _ 0_
I mb/ .. r)
···"'\11t~1~\.~ -O.S \l
,so ISO
Figure 4.10 El~st1c and inelastic polarized-proton scatCe~lng from
72ce. The cu.ves represent vibrator-model calcul~t1ons
wiCh a2-0.25 and 6330.23.
full ~ f1r8~-D~der: pure one- and t~o-phonon,
dashed
dotted
- first-orde.: 2+ and 4+ 8~ates mi~ed,
• se~ond~order: pure One- and tWQ-phonon,
dot-dashed m second-order; 2+ and 4+ S~ates mixed.
~
~
106
0·1
to
da ~
(mb/ .. " )
O.t
o. t
d" ~
(.b/ ... ,. )
O. t
ISO
Figure 4.1l Elastic and inelastic polarized-proton scattering from
7~Ge. The cu~vea rep~esent qibra~Qr-mOdel calculations
with ~2~.28 and 63~O.15. The unresolved doublet of 4+
and OT is shown ~ith the Svm of the contributions.
full .. fi~Bt-o~der: pure one- and two-phononj.
dashed - first-order: 2+ and 4+ states mixed,
dotted ~ second-order: pllJ;"e one- and t~o-phonon.
dot-daBhed = second-order: 2+ and 4+ states mixe<;l.
107
0, >
o. ~
lSO
108
alCTR
0-1
.+ I 10 O_~(>3 MOV
0 ••
do' ~ o· " (lI'Ib/&1"" J
t:~' ;
'" "'0·6
r 1 2.¢'il'1 MeV
0.8
dO' , ,
;rQ O. R
(nlb/ll,.. )
0.1 j,I,,.,+ I'
-O.S
30 60 90 120 150 190 0 30 60 90 120 160 IBO 191;:. (des'
0.01
... d2
(Illb/ ..... ]
0.01
2' 2 1.107 MrN
4+ 1
1.410 MoV
109
~o GO 90 120 150 160 0 lO 60 eo 120 150 leo eCI! (d~s) .9~. (~ICI!;I)
Figure 4.12 Elastic and inelastic polar1~@d-proton scattering from
76Ge. The curves repr€s~nt vibrator-model calculstions
full
dashed
dotted
~ first~order: pure one- and two-phonon,
- first-order: 2+ and 4+ stateB mixed,
~ second-order: pure One- and tWo-phonon,
dot-dashed - ~ecQnd-Qrder: 2+ and 4+ states mi~ed.
110
10
0.1
7OSe (p. p') 76Se ; E - 22.3 MeV p
60
o· 1
0.000 MeV
SO 120 150 IUO a
~.&
o. ~
-0-6
g~ lZ0 Isa 164
0·1
0.01
d" a.l di'
(.b/ .. ,..1
0.01
7SSe (p,p,)76Se ; E • 22.3 MeV p
.' 1
1.:331 Mf!tV
ill
0·8
o.~
O. A
n ••
a· A
-o.s
&0 sn 1:0 I~D 180 0 3D 80 90 120 ISO 180 80 • (4.9 1 ric;:. Id_"
Figure 4.13 Elastic and inela~ttc polarized-proton ~catterin& from
76Se _ The curves represent vibrator-model ~alc~lations
witn 82=0.28 and S3~O.17. full first-order: pure on~- and two-phonon,
dashed = first-order: 2+ and 4+ states mixed,
dotted ~ secend-order: pure one- aud two-phenon,
dot-dashed a se~ond-order: z+ and 4+ states mixed.
112
0-5
o· •
0.1
10
-0.5 0_1
60 ~O 120 150 160 0 150 ISO
113
0.1
0.1
60 9~ 120 150 lao ~ 30 BO SO 120 ISO ~e. (dlll.gl
F1gnr~ 4.14 Elastic and inelastic polari&ed-proton scattering from
l8Se • Tne curves represent vibrator-mode. calculations
with S2~0.26 and 11,"0.17 • The unre60lved doublet of 0+
and 4+ is shown w~th the Sum of the contribut~ons.
full first-order: pure otle- and two-phonon,
dashed - first-o.der: 2+ and 4+ states mixed,
dotted second-order: pur" Oae- and two-photlon.
dot-dashed E1econd-order: 2+ and 4+ states mixed.
~ .. c, •
o. ~
U4
0.\
dO" di
11II'b/u)
0.1
0.1
0+ I
0.00 0 MeV
60 90 120 ISO 160 0 9(lrn C.dfl9)
-0.5
0.5
Q. "
-0·5
30 00 90 Iza 160 leO Belli ~d.9'
ll5
.'.
d~
d6' (.b/.r I
0-1
0-1
O-OC! so 90
8e • Ideg)
2iTO; "J449 MtN
t..479 M~V
.. + I
1.702 MeV
120 150 150 ISO
Figure 4.15 El~~tt~ a~d inelastic polarized-protoa scatterins £~o~
SOSe. The ~~~ves tepresent vibrator-model calculations
with 62~O.21 aud 63~O.17. The unr~~o<ved doublet of 2+
aad 0+ is shown with the Sum of the contributions.
full
dashed
dotted
first-order: pure one- and two-phonon,
~ tlrst-order: 2+ and 4+ states mixed,
~ se~ond-order: pure One- and two-phonon,
dot-dashed ~ se~ond-order: 2+ and 4+ stateB mixed.
0-5
O· A
116
iv. The 21 states.
'l;heee are the fi-r"t ~tat;'8 wh",r", we can test the influem,;, of
ad~l~ing a one-phonon compon;,nt. Th", pure two-phonon calculations
in general do !;lot give a gDOd agreement. By addi!;lg this one
phonon component the description seems to ameliorate, <:I!;lyhow the
magnitude of the calct.tlated C1:"OSS section and also toe phasing
"hows a b~tter agreement. A prefe1:"ence for first- or seco~d-order
vibrator l1Iode~ c"n not be concluded to. Still, the1:"e a"e sOme
discrepancl;,s e.g. the 2+ state in 7UGe is not reproduced Well.
v. The 4t states.
Ibe Co~cl~&ions for the 21 Stat~S are also val~d for the 41 states. The two-phonon calculations usually give the cross
section a too high value. By introducing a one-pho~on component,
which gives a destructive contrib~tion, a better value wil~ b",
obtained. Also he1:"" it is difficult to draw f1~m conclusiun~.
The ~esulcs of th", calculations for the symmetI:ic- and
asymmetric-rotator are presented in figu~es 4.16-4.22. in these
figures only thr",e states are given: .t. 2t and 4t. the groood st8.te
has been omit ted b"c"u8<l it is well represented by the calculat ions,
as might be expected. 'fhe 31 and 1;.1:>", 0t States are not given sin~_e the
(a)symm~tric rotator model does not incorporat'" such states. There ~s
" pr<:scription of Davydov and Chaban (6ClDav), which introduc;,s e vibratiOnS in the asymmet~1.c-1:"otatoI" model, but it t& hard to US" this
iI. il coupled-channels approach. D~laroche et a1. (84DeOl) assumed
instead - as an approximation - that the excited 0+ sCate couples ~o
the lower states in tohe Same way as th,;, ground SI;8.t" does. But the
single case of 76Se in ref"1:"ence 84D<')01 showed a large discrepancy
between the experimen!;",l pointS and the calc~lated curve, the J.a!;to",r
befag about 7 times larger in (ll8gn1t~d" and completely O>.lt of phase.
So we discarded Ulis possiHU ty.
"In th", figures there (He three ~urves, ~epreeenting one
symmetric-rotator calculation ",od two based on an asymmetric rotator.
We started all a8ymmetric-rotator calculat~Qns with a value of y=27°,
since the sensitivity of the 2t and 4t states !;o a small variation of
y (+1' or _1 0) is hardly !;loticeable. In a further calculation we
allowed only the value of y for the 21 state to be different from the
one for the 2t and th'" 4t states. This procedur", was also used by
U7
Ram~~e~u et al. (83Ra32) and by Delarocne et al. (84DeOl) to improve
the agreement between da~a and calculations for the 2t 8ta~e. The
value~ t"esulting from our analyse~ are g1ven in table 4.14 in the
column under Y2Z'
An lnspection of these ~esults gives rise to the following
remarks:
1. The 2t states.
The description of the 2t state in the rotator models is good.
The difference between symmet~1c- and asymmetric-rotator result~
can hardly be Been in the cro~~ sections and ia of minor
significance in the analysing powers. This 16 a confirmation ot our assumption that a generalized optical potential de~1ved with
a symmett"ic rotator can be used for the asymmetric rotator,
whenever the admixing of components, to the walle functions of th",
ground state band is small. The J:esulting cross sections are on
the same level of agreemant as for the vibrato~ or ev~n sligntly
better (e.g. the local ma~imum at 60· is r~produced more
closely). Tbe analysing powars are somewhat worse. Espectally the
mlminum predicted at 120· is ~bsent in mOst cases. The Curves
resulting frOm the symmetric-rotator calculations are als() in
disagreement with the data at other angles.
1i. The 4t states.
The desc~iption of the 4t chann",ls rangee from reasonable to
poor. We encountered the above-mentioned oifficulties of
diffe~eut slopes of the experimental and calcl1lated curves. A
sharp minimum at abo~t 120' for 72Ge, 7bGe, 78S e wae not
reprod~ced by th", calculations. Such an effect was already
observed tn earlier measurements on the Zn isotopes (83Moo). More
important are the diecrepancies between calclilated and measured
analysing power in the backward angular region (~lOO·). The
accurac;y of the data 1s poor in most cases, due to the small
cross Section, but even so there 1s a difference in sign.
Wliereas in ge.neral the e.xperimetlts give a positive. analydng
power, th", reeults of the calculati<;>n~ give an overall negative
Sign. The differences between the three rotator eslcl1lations are
in mo6t cases small, so that an imprOVement cannot be acquired by
applying small cllanges to y. Comparing these w1th t.he v1hator
r~~ult5 we must conclude that there is a prefetence for an
118
0.5 10
o. ;.
D .5
o. l
o. •
0·\
-O,!l,
0·0\
30 £0 90 \ ZO 150 ISO 0 jO 60 90 120 1$0 180 ~'W-m t dlISsl Q~ ... tdcs)
Figure 4.16 In~last1c po~arized-~roton scattering f.o~ 7UGe. The
curves represent rotator-model calculatiOns ~ith eZ=O.22,
and ~4-0.0i5.
ful~ asymmetric rotatDr Y2z~ymZ7·.
dashed ~ asymmetric ,otatDr Yzz-28.o· and y=27·,
dotted symmetric rotator.
10
7ZGe(~,p')72ce; E ~ 22.3 MeV p
lj o.o:u Mf!tV
119
a,s
D, •
0.1 -0 ·5
u·o 0,(
~~
JI ~. " ( .b/ .... 1
0.01
-; 1.728 MeV
O,S
I;···· .. ···············, I
\t'''jf\ o. ~
"
-0.5
30 60 90 ao ISO 180 0 ~O 60 90 120 ISO 180 9t!1. (does) "00. tdol)g!
Figure 4.17 Inelastic pol~r~ze~-proton scattering from 72ce. The
curves represent rota~o~-model calculations with S2=O.25,
an~ ~4~O.OlO.
full = asymmetric rotator Yi2=y=27°,
dashed = asymmetric rotator Y22m27.Z· and y=27°,
dotted - symmet~1c rotator,
120
10 .
0·\
0.01
• I,
4i+ 02 \,464 M.V
1..f.8~ MeV
o. ~
-0·,
o. "
_0.6
0.6
0, "
-0.5
30 60 90 120 160 laO 0 30 SO SO 110 ISO 180 Be"" (deg! 8.c .. (d06{11
Figure 4.18 Inelastic polarized proton scattering from 74Ge • The
curves represent rotator model calculations with ~z-O.27,
and ~~--O.015. The unresolved doublet of 4+ and 0+ is
shown with a 4+ calculation only.
full. asymmetric rotator "Y22~"Y~27·,
dashed - asymmetric rotator "Y22~25.6° and y-Z7',
dotted - symmetric rotator.
10
0.01
0.1
o .Ot
, . •
4+ I
1,';1() M~V
....... -_ ...
121
o. •
O.~
0. ~
-0·,
o '0 $0 so 120 l~O 160 0 ,0 SO 90 120 ISO leo 09011 (d.~) Q". t Iji~~'
Fl~~e 4-19 In~la8tic polarlz~d-proton scattering from 7~Ge. Tne
curves represent rotator-model calculations w(th Sz-O.25,
and a~mO.020 •
.full
dashed ~ Asymmetric rotAtor 122-Z6+O· and ,=27',
dotted ~ symmetric rotator.
122
76Se(p,I") 765e; B - 22.3 MeV p
10
o,~
Q, A
0.1 -0_6
"; 1.216 MeV
0.\
0-01
0,1 0, "
-0,6
0.01 30 ~o 90 120 l$() leO 0 30 SO 90 120 150 160
BIi=I\I {d"'9) ~QIII 'dClgl
Figure 4.20 Inelastic polar1ze~-proton scatte~ing from 7nse. The
CI)r.-eB represent rotator-model c",l~ulatiOIiS with 13 Z=0.28,
and ~4=O.Ol2.
~~11 asymmetric rotator Y22-y~27°,
dashed 3 asymmetric ~ot~tor Y22~24.Zo ",nd y=27°,
dott"d symmetric rotator.
~ d2
(rnb/Dr' )
10
0.1
123
Q. •
O.S
O. A
-o.~
30 BO 90 1~O I~O leo 9~1I ~des'
Figure 4.21 l~elasc1c polarized-proton scat~erin~ from 78Se . The
curvea represent rotator-mooel calculationa wtth 62-0.26,
and 64-0.001. the unresolved ooublet of 0+ and 4+ i5
shown with a 4+ calculation only.
dashed
dotted
asymmetric rotator Y22~y=27·,
a~ymmetr1c rotator Y22~24.2· and y=27°,
symmetric rotator.
1:1.4
10
0·1
0.1
0.01
0.1
2i 0·666 MaV
2i"'"02 1.4 49 M~V
1.479 MeV
.. I
l,ro~ III.V
o.s
-0·.
-0·.
30 SO 90 120 150 ISO 0 30 SO 90 120 150 180 8¢" ('ch:e I I3"c. (.d.9 1
Figure 4.22 Inelastic polar1zed-protQO Bcatt~rio8 from eaSe. Th~
curves repr~sent rotator-model calculationg wich ~2~O.ZZ,
and S4=-O.025. The u~re~ol~ed doublet of 2+ and 0+ is
Bho~n with ~ 2+ calculation only.
full
d~Bhed
dotted
a~1mmetric rotator Y22~y-279,
asymmetric rota"or ~22-ZZ.8° and ~-Z7°, symmetric ~Otator.
in~eJ;"p>:,etation in terms of a vi brat.or. maiIl!y because of the
analysing power.
iii. 2! states.
In eontrast. t.o the 21 and 41 states t.he results of these
calculations were, with the exception o£ the 122 fits, parameter
free. the pure asymmetrie rotat.or results are i.... gO!neral iIl
disagreeme .... t with the measured c.l"OSS section. The diffraetion
pattern is, however, in agreemeat <;;ith the experiments so that
thO! analysing power as a relative quantity also showS th~s sl\me
agreement.. Fitt~ng of 122 gives an imprOVement ip the magnitude
of tile cJ;'oss section, and also the descriptioa of the aaalyaing
power ameliorates. Very good agreement was obtain~d for eoSe , so
sood that we safely may conelude that the contributioa of the 01 "t.ate in this unresolved doublet muSt be very sl1Ia11. In general
the Y22 fits provide a bette~ descriptioa for the 2! states than
the correspoadiag ~ixed-phonon vib~ato~ calculations. The
agreement between the data and the reeults oe the calculations is
for Lhe cross eection as good as or even better than the vibrator
oaes, the analysing powere aJ;e certainly beu"r represented by
the asymmetric rotator model.
4.6 Discussion and eonclusions
In section 2.2.5 we discussed the relations (2.29) between
deformation para~eters resulting from different kinds of transitions.
In tables 4.3 and 4.6 we have .!liven the experiments I<Ihich should
provide the information to test these.
F~rst, lIowever, wc want to make a remark about four experim~nrs
...... mely the work of Matoba et Ill. (79Ma28), Matsuki et a1. (S3Ma5:t),
Tamisier et a1. (82Ta16) and Ra!l\stein et al, (83&.,32). These
experiments have in common that the experimental data were obtain~d by
us~ng 1\ magnetic spectrograph, thus providing well-resolved spectra.
TI;1~s tnstrument is also the cause of O .... e important Shottcoming: the
data are limited to the angular ran),;" or 0·-100·. In contrast to
experiments with semiconductor detectors datti po1~ts could be measured
st ['ather forward angles «15 0) thus making them very sllitable for
extracting deformatioa parameters. The lack ot data in the aagular
range of 100 0 -180 0 inevitably i .... fluences the fin~l results. For
126
instance io. the studies of Tamls:f.l1-r et a1. and RalD"'teirl et a1. the
slobal potentiak of Becchetti and Grel1-nlees (69Bec) wae found suitable
for the analysis of inel(istic scattering. whereaS we h<>d to reject
this potential because of the discrepancies sho1ll1ng uP. at angles
,arser than 70· (see figures 4.4).
In t(ibles 4.14 we h<ove listed our res"ltB together with the
p(irametero published fe>r other proton scattering experiments. ILl. the
p1;eViQ\l$ sec.tion 4.4.5 we already concluded to a good agrl1-ement. A
c.loser inspection of the numbers reveals that the agreement is the
best for the Se isotopes. For the Ge isotopes the present ri!sults
agTee wl1-11 \lith the data o£ the Orsay collaboration (82T<o16, S3Ra32),
but there are SOrrl0 differences between the analyses of Curtis et ,,1.
(70Cu03) and (more recently) of Sen et al. (855e05) and ours. lhis
might b" due to the lower incident energy in both the~e experiments:
14 . .5 MeV and 11. 5 MeV, respec tlvely. <l-t theae energies there might be
"till aome influence af c.ompound reactions. Also saIne effec.t from the
pro>:im1ty of the Coulomb b(irrier (approXimately 8.7 MeV) cannOt be
excluded.
Another dlsagreement ~an be found in the e>:tracted values of the
one-phonO,\ admixtclt"e: the deformation paramG:ter and the corresponoing
milling angle. the agreeml1-nt in the defo~mation parametera "f the o()e
phonon contributiOn is r.ea~onable, but the mixing angles are
coml.'le tely dif ferent. Wher~as the re(lults of the Or8ay colla bora Uon
indicate a small one-phonon admixture (app);<)ximately 30%), our re~ults
tend to a small two-phonon part, especially for the the 2! atate. The
discrepancies we already found between first- and sEc.ond-orde~
vibrator model raise the question of the correc.tness o~ the procedure
for exLractiog these p;orallleters. We thInk. that extractioo ~" toO much
model dependent to <ottribute any 5igni£anc.~ to these parameterB,
anyway for the llIixlIlg "ogle.
We hav8
parameters 10
alre~dy discussed e)Ctensively the extraction of B4
a rotatOr model and the corresponding difficulties.
Delaroche et al. encountered the same problero~. Also their analysing
power data had a sign opposite to that resulting from the
calculations. The values quoted io table 4.14b are given by Del<oroche
et a1. aB supplying the best overall agreement. If only the forward
angular range is congid"r"d a v"lue of ~4-0 was alBo possible. They
concluded from this that also the extraction o~ 8 4 is highly model
1z1
dependent. We, for our part, do not confirlIl this conclusion. The
extraction of the e" parameter is indeed difficult and the final value
will inherently bear a large error, but its value is determ1nable: the
fair 8&reement with Matsuki et al. (83Ma59) and Ram~tein et al.
(83Ra32) confirms this.
The parameter Y22 ~s the la~t of thi~ series. The extractiOn i~
rather easy and it. can be obtained without ambiguities. Our resuHs
are nearly the same a~ thOSE: of Delaroche et al. and of Ramste1t1 et
al. In most cases the value of Y22 we tind ~s smaller than the no,ma1
y for the 2t and ~he 4t Btates, except for 7CGe and 7~e. There is a
g-,;adua1 decJ;ease going from 70Ge. to 76Ge.. Thte indicate.s again the.
change of character between 7%e and 74Ge when we take y~27" as
normal. This &rsdus1 decreaBe is continued in the Sa isotopes. The
question arises what meaning we should actribute to a Value of Y Z2
differing from the normal one. We think that it 1s au indication of
the Q:ott1,1e6~ of th~ nucleus. The mOre the differenc.e between y and
Yzz• the more shape variations will occur. A final conclusion is her~ on in ord~r: what~ver the shape of the Se and Ge nuclides may he, it
is not a~ially symmetric, ceJ;tainly in the excited states.
Whereas We found a reasonable agreement between nur deinrmacin(l
parameters and results from the literature, we shall now also include
other types of transitiol;)s 1n the disCl.Ission. In the table~ 4.16 we
quote the re6ults of relevant earlier e~perimen[s. In some cases these
have been &iven as deformatinn parameters e or in nther cases as
oeformation 1ensth" 6R. In all cases we t.rled to <l~rract the nther
quantity, and also to calcu!;tt@ the corresponding multipnle moment,
when suffi~ient informatinn was available. Especially the data from
(d,d') and of (0,0.') experiments are of lmpoJ;tao.ce, sio.ce these. are
o.ot 1nfl\.lenced by tIle difference in interaction strength between like
and tlnlik@ nucleons. Neutron 6cattering dat", are complementary to
those of proton scattering, but were only avallab~e £or the $e
isotopee. the following empirical relations appear;
Ge: Bdd , < Bern ( Bpp '
s~: aero ~ Bpp' ( B~~, < Bnn'
1'he defo~mation l"-ngth~ (when availablE:) give essentially the same
relations. In the case of the Se isotopes this confirms the
128
TaD~e 4.16~ Quadrupole and octupole deformation para~eters
and defo~mation lengths, and auclear quadrupole and
octupole mOments ill a vU,rliltional context.
VMl = this work: first-order vibrator model,
VM2 = this ~ork: second-order vibrator model,
CE Coulomb excitation.
["eaction
70Ge
E
(MeV)
(p, p') 22.3 0.22 1.043 -52.6 +
(P. p')
(Po p')
(p, p j)
(P. p')
(d,d j)
(d, d')
(d, d')
CE
22.3
11.5
14.$
11
8
16
16
0.22 1.043 -51.0
0.202 0.962 -49.7
0.22 1.124 -57.5
0.18 0.953 -50.1
0.17 0.809 -39.7
0.l7 0.806 -39.0
0.l9 0.822 -39.0
0.204 0.967 -42.4
GE
72Ge
+ (p, p') 22.3 0.25 1.206 -62.8
+ (Po p') 22.3
<p,p') 11.5
(Pop') 14.5
(p.p') 11 +
(d,d') i:l
(d,d') 16
(d.d') 16
( 180, 180') 68.6
CE
CIC
0.25 1.206 -bO.8
0.203 0.976 -52.8
0.22 1.135 -60.8
0.20 \.069 -58.9
(l.17 0.816 -4l.7
0.17 0.8.4 -40.8
0.20 0.882 -43.5
0.99
0.219 1.057 -45.6
0.22 1.04) -322
0.22 1.043 -301
0.25 1.278 -399
0.20 1.059 -349
0.19 0.822 -Z45
0.163 0.773 -20S
0.23 1.110 -348
0.23 1.110 -324
0.23 1.187 -391
0.L8 0.962 -334
0.18 0.794 -240
0.184 0.888 -226
Ref.
VM1
VM2
855,,05
70Cu03
70Pe09 * 8SSe05
85Se05
78S.08
8(lLe16
80L~24
Vt-U
11M2
8~Se05
70Cu03
70Pe09 "
85Se05
855e05
78Sz08
791lg04
80LeH
80Le24
to be: continued
*: DWRA analysis of data obtained by Pe~ey et al. (70Pe09).
~z9
Iable 4.16a Quadrupole and octupole aetormation parameter~
and deformation lengths. and nuclea~ quadrupole and
octupole moments in a vibrational context (continued) •
reaction E /12 /l2R <l.zo ~3 /l,R i[,u Re£.
(MeV) (fIr) (e fmZ) (fm) (e fm 3)
74Ge
(P.P' ) 22.3 0.28 1.387 -76.8 O.lS 0.743 -253 VMl ->
(p,p') 22.3 0.28 1.375 -73.6 0.15 0.737 -233 VM2 +
(P.P') 11.5 0.208 )'.010 -5&.8 85se.05
(P.P') 22.0 0.29 .. 425 -79.6 0.16 0.786 -275 82Ta16
(p,p') 14.5 0.23 1.197 -67.2 0.13 0.677 -235 lOCu03
(P.P' ) 11 0.21 1.133 -65.3 0.14 0.755 -277 70Pe09 * +
(d,d' ) 8 0.197 0.954 -50.6 8SSe05
(d,d ') 16 0.197 0.952 -49.3 8~Se05
(d,d') 16 0.26 1.190 -59.9 0.13 0.595 -182 785z08
CE 0.253 1.253 -55.3 SOLd6
76Ge +
(P. p' ) 22.) 0.25 1.197 -67.4 0.15 0.718 -245 11M 1 .. (p,p') 22.3 0.25 1.197 -66.5 0.15 0.718 -::38 VM2
(;, p') 11.5 0.211 1.033 -60.7 85SeOS
(p,p') :<z.0 0.26 1.289 -75.2 0.-15 0.743 -2n 83Ra32
(p,p') 14.S 0.22 1.156 -67.8 0.14 0.735 -269 70CuO)
(p, p') 11 0.19 1.034 -62.3 0.13 0.708 -273 70Pe09
(d,d' ) * 8 0.197 0.%3 -53.2 85S<!OS
(d,d' ) l6 O.UIl 0.960 -51.6 85SeOS
CE 0.252 1.228 -52.8 80Le16
tim=_~~==========================~~~ ••• ~*W_a~~~=================~~~~~~~ to be continLLcd
*: OWtA analysis of data obtained by Perey et al. (70Pe09).
130
Til-hie 4.1(,,, Quadrupole and occupoLe deformation parameters
and deformation ie~gths. and nuclear quadrupole "nd
octupole moments in a viDrational conte~t (coot1oued).
reacUoo E 1>2 aiR Q20 ~3 /l;JR q 30 Ret.
(MeV) (im) (e fmZ) (frn) (e fro O)
76Se +
(I', p') 22.3 0.28 1.423 -80.6 0.17 0.864 -304 VM1 +
(p,p') n.~ 0.28 1.411 -78.2 0.17 0.1)57 -288 VM2 +
(p,p') Ii> 0.260 1.355 -77 .9 0.150 0.782 -282 84DeOl
(p,p') 51.Y 0.278 1.366 -77 .4 0.164 0.861 -307 79M<128
(o,n') 8,10 0.28 1.483 -77 .1 76LB12
(0, n') 8,10 0.31 1.59 -81.2 0.154 0.79 -244 lH.Ku09
(0:, (X') 42 0.29 0.17 Be~ca'"
Cl! 0.267 1.357 -64.8 0.134 0.681 -200 74:6a80
CE 0.268 1.362 -6S.1 77Lell
78Se .,
(1',1" ) 2.2.3 C).26 1.255 -72.0 C) .17 0.821 -287 VMl .,. (p, p' ) 22.3 C).2(:i 1. 255 -70.5 0.). 7 0.821 -274 vM2
... (1',1") If.> 0.235 1. 235 -74.1 0.140 0.736 -279 84DeOl
(I', 1") 51.9 0.243 1.204 -71.2 0.179 0.948 -355 7',1M",Z8
(n,n') 8,10 0.Z7 1.442 -76.5 761.",12
(a, a') 42 0.25 0.17 Berc.aw
CE 0.253 1.222 -56.7 0.122 0.589 -164 74Ba80
CI': 0.255 1. 231 -57.1 77Ldl
80Se .,.
(1',1") n . .3 0.21 1.041 -1.>4.5 0.17 0.84z -333 VMl +
(P.I" ) 22 . .3 0.21 1.041 -62.9 0.17 0.842 -.31.6 vM2
(p, p') 16 0.210 1. .113 -69.7 0.140 0.742. -295 84DeO 1
(I', p') 51.9 0.210 1.041 -61.5 0.167 0.892 -313 79Ma28
(11,0' ) 8,10 0.25 ),.347 -72.9 76La12
(n, n ') 8,10 0.24 1.25 -66.6 0.15 0.78 -261 84K,,09
(a, a') 42 0.24 0.17 Bercaw
C8 0.208 1.0.31 -49.9 0.063 0.312 -95 74Sa80
GE C).2).0 1.041 -50.4 77L,,1l
====~~_~~m%============~ __ ~_w~am==========~_~~ __ ~_=========~D~~~ __ ~mD==
ilerc.a'W: R. Ber~awt unpublished data ~ited 10 69Ber.
131
Tabl'" 4.1Gb Quadrupole and hexad~capole detorroatlon parameters
an~ deformation lengths, and nuclear quadrupole and
he~decapole mOment~ ~n a rotational ~ontext.
SRM .. thh work+ symmetric-rotator [!lodel.
reaction E 112 e:l qzo 64 B"R q~u l\e£ .
(MeV) (fm) (e £m 2) (fm) (e fm4)
7DGe +
(p,p') 22.3 o.n 1.034 -55.6 0.020 0.094 470 SRM
(p, p') 1>5 390 85Matl
(p,p' ) 11 0.202 0.962 -52.7 85SeOS
(d, d') B 0.1. 7 0.809 -41.8 85SeD5
(d,d ') 16 0.17 0.806 -41.1 8:;Se05
(d,d') 16 -0.18 -0.7789 36.6 0.04 0.173 437 785z08
7%", .. (p, p') 22.3 0.25 1.196 -65.7 -0.010 -0.048 241 Sm
(p, p') 65 460 85Matl .. (p, p') II 0.203 0.976 ~!>(>.O 85Se05
(d,d' ) B O.L7 0.816 -43.9 85Seo~
(d,d' ) 16 0.17 0.814 -43.0 858",05
(1.~· ) 16 -0.12 -0.5292 2!>.9 0.04 0.176 390 78Sz08 74Ce
(p, p') 22.3 0.27 Li70 -73.2 -0.015 -0.071 243 SRM
(p,p' ) 490 8SMatl
(p, p') n 0.208 l.010 -60.4 85SeQ5
(d, d') 8 0.197 0.954 -53.6 85Se())
(;1,.1') 16 0.197 0.952 -52.3 85SeOS
(d,d' ) 16 0.14 0.6407 -34.2 0.02 0.092 ~99 78S,,08
nCe
(p,p' ) 22.3 0.25 1.123 ~66.9 0.001 0.005 351 SRM
(p,p' ) 11 0.211 1.033 -64.7 8SSeOS
(p,p') 22.0 0.26 1.289 -82.8 0.020 0.099 807 83R,,)2
(J.d') a 0.197 0.963 -!>6.3 85Se05 .. (d,d' ) 16 0.197 0.961 -54.8 85SeOS ~.~. ____ ~_D~=Z===========~~_~~.~~ ____ m~~======~=B=.~ •• MW __ =m==========
to b~ continued
132
Table 4.16b Quadrupole and hexadeca~ole deformation paramete~6
and deformation lengths, and nuclear quadrupole and
hexadecapole moments in a rotational context (contin\led).
reaction E GZ B ZR Q20 B" a4R q"u Ref.
(MeV) (fm) (~ fIll2) (fm) (e fm 4 )
76SC
+ (p, p') 22.3 0.28 1.388 -85.0 0.012 Q.069 711 SRM
CP, p') 65 680 8:;Matl
(p, p') 16 0.310 1.589 -103.1 0.040 0.205 1406 84De01
(n,n T) 8,10 0.29 1.49 -83.2 o. o. 584 1l4Ku09
785e
(p, p') 22 .3 0.26 1.200 -73.3 0.001 0.005 414 8RM
+ (p,p') 65 490 85Mat1
+ (p, p') 16 0.26.5 1.370 -90.5 0.027 0.140 1017 84DeOl
5US"
(r, p') 22.3 0.22 1.062 -66·3 -0.025 -0.121 -13 SRM +
(p, p') 65 -90 85Matl
(~, p') 16 0.230 1.199 -80.7 0.013 0.068 670 840e01
(n,n' ) 8,10 0.24 1.25 -71.8 O. O. 426 84Ku09
=====~=~~~~_w~~~m==================~D_~~~ ____ ~============~~_~ ______ 3~
predictions made by Madsen, 6rown and Anderson (7SMadl, 75Mad2). In
the Se isotopes We artl approaching the ,,1o~ure of the nelltron sh,dl at
N-SO. The ratio ~ ,/~ ls, however, close to one as might be pp em
expec.ted for open-shell nuclei. The deformation parameter" [rom «1,,(1,')
also fit w",11 in this ~equen"e. Tn,:, relation appearing for Ge is
somewhat surprising: the (d,d') re"ults are on the other side; j t is
e~pected ~hat these results sho,'l<) give the same reh.ti.on as the
(a,«') data. The smaller ~ for the (d,d') experiments Ill3y al~o be
attributed to the low en"rgy of the incident dcuteJ:o(\s. This may
int lue\"\ce ~he data 1n th,;, same way as we alr~ady COnCl.llded for the low
energy (p,p') data.
The extracted multipole moments fo~ diff"r~nt transit.tons show
larger dIfferences. Witn the ~"c~ption ot the low energy experiment of
Sen et a1. (858,,05) the Q20 moments from (p,p') eKl'e~imentB deriv~d
with a v~oratiOnal interaction are overall the same (Within 10%) and
133
show a gradu,al de.cl:ease 1n the Se isotopes in cOntrast t'1 the Ge
isot'1pes where a miilimum i$ J;'1\,lnd f'1r 1~Ge. l'he same trend is also
exhibited by the Q20 m'1ment~ from (n,n') having about the same value
as those from (p,p'). The differences between Q20 m'1ments derived w~tn
a vibrat~'1nal '1r with a rotati'1nal form factor are small. Oilly for the
data '1e Oelaroche et a1. we find 8isniUcant~y larger values. The
mOlllents in the rotati'1nal COilt"",t have been calc.ulated wi th the
symmet,ic r'1tator, while the parameters of Oelaroche et al. are
obtained with the asymmetric-rotator PK>del. This might be the origin
of the differenCe, though we sh'1wed that the ,:!1fferen(:",~ between tIle
symmetl;ic- and the ,a"ymmetr;-ic-r'1tat'1r model ;in the final results a["e
only 51118.11. Compared to tne moments from Coul'1lllb e2l:cit,aUon, the
moments from (p,p') and (n,il') experiments are somewhat larger. For
these woments also the above ,eLati'1ns hold. To C'1nclude, h'1wevar, to
a dit ferent defo,1Mti'1n of protons and neutrons on bas h o~ these
differences needs llIor", confirmation, particularly by means of (d,d')
or (a,a') experiments at comparable energies.
l'he spread in the Q30 moments from (p,p') reactions 1s l,arger ano
am'1unts up to 20% of the average valu". The moments from (n, n') are
no," also sigilicantly smaller and those from Coulomb e)(cication are
even smaller than the former. Since the differences in the Q30 moments
are m'1re pronounced it seems to bo justifiect to e)(plain these in terlllS
of differ;-ences iil d",formation.
Fo:t: the Q40 mOm",nts we d'1 not have eiloush int'1t"mation to draw a
firm c'1nclusioil. The difficulties in extracting ,the S4 'lalue will,
h'1wever, not lead t'1 the sallie large err'1rs, since these m'1meilts
calculated with a rotatiOo.al f'1rm factor ,are all>O strongly d<i:pendent
on the vslue '1t ~2' Notice in t:ab~e 4.l6b that there is still a
sigilific.ant Q40 moment also whon a4~o. When keeping this in mind, ~e
nave a reasonable agl:eement with Matsuki et sl. (85Matl) but a total
disagreement w~th Delar'1che et ,a1. We tnink that tho values o~ ~4 frOm
the las~ study are t'1'1 large, r~sulting in tO'1 large Q40 moments. The
authors themselves find the extraction of e4 highly m'1del dependent.
~1nally we want to remark t:hat in view of the ab'1ve discussion we
are not yet c'1nvinced that "llIultip'1le m'1ments ,are a better means for
quoting def'1rmati'1ns" (76Mac). All our cooclusions could als'1 be
derived if We had restricted '1urselves to dexQtmation parameter$ Or to
deformatioil lensths. Perhaps multipole moments are a better criterion
134
for permanently defol'meo. nuclei, bu~ at present for the Ge and 5e
nuclides this i~ a decour.
135
Chap~er 5 Nuclear s~ructur~ C&~cuLa~~Qn~
"Door meten tot weten, zOu ~k a),s
zinspreuk boven elk phy8i5~h
laboratoriulU wIllen sehrijllen."
Not only experimenta1ists have studied the Ge-Se ~ss region, but
also theor~ticians have been challenged by the difficulties in
explaining the nuclear structure one encounters here. The theoretical
analyse~ ~tarte4 trom a mixture of microecop1c and collective aSpects
of nuclear 5t~ucture. A comprehensive review of the situation has been
g~ven by Vergnes in 1980 (SOVer). His general concluslons Were:
i. There is considerable experimental evidence for an ob1ate-p.olate
transition between .SCe and 7GGe • lhe Se nuclei exhibit the same
fedtures, but less prono~oced.
ii. No pure ehell-model calculation is capabla to ~aproduce th~ low
lying O! state (see e.g. 76Dev).
iii. Some models seem 1:0 be apt to describe various f"atures of the
spectra, including aome B(E2) values correctly, bu~ theL~
underly;Lns theol:etical assumptions are dHt"rallt. Fo!: illstance
Oidons et 51. (76Did) describe the 01 state as mainly proton
excitations in Collt~S6t to Iwasaki et al. (781"1'5) who attribute
it mainly to neutron excit~tion6.
Toe new analyses since 1980 do not c.ontribute new insights to tha
theoretlc~l descriptions. The.Y' only <:.onfirm the ambiguous vieli" thi"
mass rasion exhibits and wherB confisurat1on mixing, shap~ coexist~nca
and triaxiality sra the keywords. After the introduction of IBA these
apprOaches moved somewhat to the background. As we present here some
schematic lBA-2 calculations, we shall review shortly previous
ealculattollS.
A recent paper dealing with lBA stems from Erokhina, E:fimov,
Lemberg and Mikhallov (85Ero). They established for the S .. i"otopes
the region of admisljible v~lues of the parame.ters E. 5 1 ~Ild X in an
IBA-l type of analysis. Their m<iin o;oncJ.usion on thEt basis ot the
per£o~\Ued analysis of Etn@rgy spectra :and p.obab11ities of quadrupole
traneitioDs "sa that o( the coexistence of .. tates with different
136
deformations in 72. 74S C (sphe~ic~l shape in the ground state ~nd
ax~ally-Bymmetric 6h~pe in the O"! state). For 16.78, 80Se they found
that these h",ve nO stable spherical shape and that the developing
deformations are Y-unstable.
In oreier to c.ompare their experimental result" on 70Se (Ond "leSe
with mo(lel predictIons Matsuzaki and Taketani (82Ma45) performed a
stan(lard IBA-2 analysis tor the Se isotopes with Am 74-80, starling
from an iae,t core of 28 protons and 50 neutrons. A reasonable
agreement ,",as obtilined between the l:'esults and the observed
band stnlctur" (level energies and ll(1l2) ratios). Another IBA-2
analysis wa. perfonned by Kaup, M~nke!lleye); and vOll Brentano (83Ka04 > as an extellsion of earlier similar ana~yse~ of the Kr .... nd. Sr isotopes.
The Se isotopes with N;.42 <!ould be "'ell df!scribed only if one as~umes
th~ 0t .xc.ited state to be an intruder state. Effeets of subshell
cl05ure are most likely respon"ibltIl for the failure ot the model below
N=42.
Duval, Goulte aod Vergnes (830uv) applied the conflg~~ation
mixi[l.g roodel (32Duv, 810uVl, 81Duv2) to the Ge isotOpes. In this model
Olle alia,",,, t\oi'o~part iele two-hole ",,,citations across a closed 8he11.
This means that one tries to explain the nl)o;lear st~l)ctl)r" in terms <;>f
two illteracc1ng configuraClone i.e. that normal in IBA-2 and ao
ao<)itional on~ with two extra bosons. This resembles the coeJ(isu!(lu,
in the eollectlve mooel. Starting XX-OM a simple ansat", Duval, GC/Utte
and Vergnes obtained a qui te goad agreement wi th the e.xpe.rimel~t. The
model predIcte(l, however, a 2+ state in 72ce at 1.36 MeV which has not
been found e){perimentally 8-nd several 3+ ~tat:es at a tou high
excitation energy. The most distioct eiiff"rence l:>etween experiment
and calculatl,ons was that the latt~r actr.ibuted a neg"tive quad!:'''p"le
mOmeCtt to all nuclei consider~d and did not reproduce the oblate
prolate trallsition b~tween 70Ge and ?2Ge. The aCtno'-'nced more complex
<!alculaclons, whlch were a160 to ino;lude quadrl)l'0l~ terms in both
coafiguratiOns in order to obtain triaxial deformation, have not been
published sofar.
The same model has beeD applied to tohe Se nuclei by DeJ.aroche,
Girod and Duval (820el). The results show a general trend of a small
contribution (~10%) of the "excited" uonflguratiOn to a very large.
contrLbution (~YO%) when goinS from 105e to BOSe. They fioally
conclude on the l:>a~ia of this analysis and of a Hartree-Foc~-
137
Bogolyubov approach to the existence of dynamical triaxial
deformat1ons in Se nuclei.
The most recent pape~ with an IBA-2 calculation of G6'68'7D'7~e
is by Yoshida and Arima (85Yos). Since their aim is especially the
description of high-spin etates in these isotopes, they USe an extr~
coupling of two quasipsrt1cles in the g9j2 orbitals. A good agreement
is found with the experimental Spectra, including the gradual lowering
of the 8t state with increasing mass number. The character of the
b~ndS, however, strongly depends on the pa.ametri~ation, necessitating
furthe~ e~perim@ntal tests. Calculations for oth~r nuclide~ 1n this
region, including Se, Kr and Sr, are announced.
When do1ng IRA-:/. calculations one has some advantag<!> of the
mic~oscoplc foundation of the paramete~~ (780ts). e.g, K and X~ and
Xv· Some knowledge of the value5, whi~h a~e ~easonable from' a physical
point of view, is us~ful in confining the parameter space. So it may
be expect~d that ~he va~ues of X are within the int~ryai (-~/7, }/7).
The ex~remes a1:e found for almost filled or almoe t e<1\pty she ill;,
whereas a value of XgO seems suitable for shel!s half filled, Aleo the
number of bosons is an importan~ question to solve. No.mally one takes
the number of <Jalene@ nucleon pairs (hole pairs). In the G@-S@ _»5
region the two YD<IjOl: shells close at N, z=28 and N"SO, bu~ there is
@xperimental evidence for sOme additional shell closure at N=40, be it
wOlak. In that <:.ase the counting of the number of bosons will dUfer.
AnQthe. possibility is the solution proposed by Sch<;>lten (83Sch): when
a subshell cLosure is not very pronounced but nevertheless sllOWS up
one can take this into account by lntroducing some effect1ve number of
bosons (aee table 5.1 and figure 5.1). with a corr@sponding beh3vlour
of the quadrupole parameter x.
In our cal.culations we r"s~rict;e4 oursel"es to the simple IBA.-Z
model, Which mo~eover is treated rather schematically. we think that
thh can be justified since we are interested only in the lowe.t
levele Qf natural parity and only in general trends in deforlndtion
c.q. transitio>1 stre>1sth",. of course SOme caution h3S to be taken in
the interpretation of the low-lying 0+ e~c1ted states in the Ge
nucl@i.
138
6
5
4
3
2
28 30 32 34 36 38 40 42 44 46 48 50
N
Figure 5,1 The possible number of neutrOn bosons (N~) between n~utroa numbors N=28 and N~50.
solid
dotted
dashed
~tendard,
sobshell closas at N=40,
effective.boson ~umber.
Table 5.1 The boson lI11mbers (or Ge and Se ~eotop"s.
Ge
70
72
74
7&
St
76
78
80
N N n V
standard N=40 effective
2
2
2
2
5
4
3
4
2
o
2
3
2
2
2
Ii" started Ollr calcll.Lations from the parameters of l\8.\lp ~t al.
(83Ka04), s10ce only with these We coold obtain a reasonable agreement
with experiment. The other lllA parameter ""'lS COuld noL be "sed due to
incomplete informaClo" and/or to a non-"tandard IBA-2 procedure. Tlte
same paraulete,e wer", also used for the Ge isotDl'e", FUl:thermor« we
l39
varied only o(-~~-~v) ~nd K, for the ij3me reaaone as mentioned by Kaup
et al., viz.:
- restrict the parameter "pace by the requirement that X~ depends only
on the proton and Xv only on the neutron number (then the values of
Xv already deduced for th~ Kr i~otopes can be used and XIT
can be set
to a fixed value for all the isotopes);
- ~ constant value of the Majorana ec.ength (FK) is justified since
it does not vary strongly.
All unknown parame~ers were set to their default values in the
progNmnle NPBOS (HOts). The just:Hication of t-his procedure can be
found in the reasonable a8r~~ment between experimental and calculated
properties of the Se isocopes as obt~ined by Kaup et a1. Bor our
purpos",- We judged oaly the quality of the U t for the lowest exd t~d
stat-es (2t, 2! and 4t) in obta~ning values ot ~ and K for Ge <lad Se.
As we ar~ interested especially in gainin~ ~ome insight in the aatur~
of these states loIe admH that probably this "'ill not give the besc
possible agreement- with the ~ewainder of the spectrulll.
Tablo! 5.2 Values of the ISA-2 parameters (expressed in MeV, except
for t-he boson numbers) as used in the final calculatiofls
tor germanium and selenium.
70Ge 7;'e 74Ge 76Ge 76S e 'I~So 80S e
N 2 2 2 .) 3 3 IT
N\) S 5 4 .) 4 :3 2
'" 1.354 1.129 0.942 0.873 0.963 1.020 1.056
'" -0.112 -0.103 -0.149 -0.209 -0.11:>3 -0.259 -0.272
Xn -1.2- -1.2 -L.2 -1.2 -0.9 -0.9 -0.9
Xv 0.065 0.280 0.49~ 0.710 0.495 0.710 0.925
FK 0.1 0.1 0.1 0.1 0.1 0.1 0.1
The resultins level schemes fo~ the low,,"r excit"d states are
displayed in figure 5.2 and 1n table 5.2 th~ values of th~ parameters
giving this re"",llt are presented. The agree..."nt- between experimeatal
and calculated excitation energies is good, cet:cainly for the Ge
isoto!'e".
> ~
:,;
w
2_
o~ 4~:'--~4
_ O~,
2-,;-2 4-:-~4 ---0
0: 0-_
'~O
1,+2----2
0- ,,_ 0 3 ====<"- 3 2--:--2 4---=4 4----4 4-':--4
2--:-2 2--'-2 2-"-2 --0
:-2 2-'_
0-_
4_-:_4 -0
2--'-2
g==" 4-:,-4
-_0 2_,'-2
'-0 2-'--2 " 2-" 2 ~--'-2 2-"-2 2-'--2 '-
Th Exp Th Exp Th Exp Til hp Th Exp Th Exp
70Ge 72Ge 74Ge 7/iGe 7~Se 7·Se
Figu-re 5.2 Compar lsoa Q f thee re t i<:al (colum marked Th) and
e><perimeatal (colu",n marked Exp) level schemes far the
lQ"eT excited states in 7QGe te 70ee and 7£Se to aoS e _
Th EllP
USe
,.... p o
141
Only the 01 states are poaitione~ ~ncorrectly, but these states a~e
assigned as intruders, the values of " an~ K we find for the Se
isotopes do not differ much from those of Katlp et al. In general the
parameters show a rather regular behaviour. In one respect the values
for Ge are different: the parameter ~ varie~ ~~om 0.9 MeV to 1.4 MeV,
whereas in the 5e and Kr isotopes this parameter i6 abOl,lt 1 MeV, '"
value whi~h is ",160 predicted by microscopic calculations. The $crong
vari",tion of K was also noticed in the calculations of Kal,lp et sl. snd
explained as an effect of sl,lbshell closure, In the Ge isotopes we find
the minimum in K p~edlcted by microscopic calculations at the correct
position between N=38 snd N=48, but its value is about a fsctot of two
smaller.
The wave tU[lctio[ls were used to calculatE;> the >:educed matr1>t
elements of the transition operator l(L) in the standard formulation.
The paramete>:6 Xv and Xn were the same as those in the Hamiltonian and
the boson eff~ctivB charges en and e v were taken as 0,6 and 1.5,
reflecting that t:he interaction strengths for neutrons lind p,otons
differ by a factor of 2,5 when using protons as probe,. the reduced
I I I I I I ISe
0.30 Ge
0.28 t f c , 0.26
IJ
I r32 1 t []
t 0.24
-t c
0,22 ,-0.20
I J I I I I
38 40 42 44 42 44 46
N N
Figare 5.3 Measured !Ii values from this work (dots) compared to
IBA-Z reduced matrix E;>lemel)ts <111 :r (Z ) II f)- (sqtlares)
normalized to the a2 value of eaSe.
142
matdl< elements <fIIT(2) IIi> wer" scaled to th", ~Z value of 80Se for
absolute normalization. The experimental S 2' sand cho"e from IBA-2
calculations are pJ:"es",nt",d in figure 5.:3. A rough "sti'"""te of th",
theoretical Unce~taintiea ia 101. Notice th", raCher good agre,m",ne in
view of the fa~t that this was obtained with only two free parameters.
'this confirms that th", «pproach of Kaup et d. we applied 1s valid
also for the C" isotopes with 1'1>40. The break dOWn for N<42 again
reflects that for these lsotop",. a rliff"r"nt type of nuclear structure
manifesc~ ltself.
Sin~e thO ene1;gy lev",l" as well as the O1+2t transition streugth
are ... ell reproduced by th",se calculations we may have coofid"nce in
conclusiOns aboue the other srates. For that purpose we have given
the probabilities of one and two bosons io these states (see figure
5.4). ,):I'e p\lre twa-phOnOn character (100% two d-bosons) 1s not "resent
in anyone. NeverCheles6 the fraction of two d-bosons !n the O! is
large in the Ge i~o[opcS, which is probably the reasou why our OJ ~t"t"'~ in lOCe and 7%e are SO well represeote-d by a second-oJ:Qer
vibrator calculation. The rather low frdCcioD of 50% io l"Se exp~ainB
the break do ... n of the C(l.lCulation for ~[Iis isotope. For the 21 Stales
we lUusL conchlde to a largely mixed configuration, e,,-cept for 70Ge and
72Ge which a(Q essentially one-d-boson states.
Within the limited ~cope of th16 IBA-2 study we 8lBO looked to
a possible description with other bo",on numbers. Assuming a "hard"
"ub"h"ll closure at N=!+O with c.orrespooding b050rl numbers is clearly
an oversimplification. The remaining bosoo" are too f.ew tG generate
co'~ph>" ~pectra, For instaoce iu 72Coa there are only two proton
bOSOL1G, givj,ng riae to five Levels: che 0+, 2+ dnd 4+ states. The
eff(lcti'le ouoober of b,,~oos does better; ther" eKist, however, strong
con-elations between 0, K' and X. Neverthelese the bOSOn energy £
displays a simi tar behaviOur as was fO\lnd io the standard fit, A study
of the relation bet"een K and X 00 th", one hand aod the effecLi""
b,,"on "umber on the other h/l.nd ia needed, but \01(1" beyond the scope of
thi~ work.
The last item we looked at was the diff"reollce between ~v and ~,,'
In the approach we uoed above ~w WaS set equal to [." There are,
however, indications that theae parameters differ; the excitatloo
eoergies of the first excited 2+ states in che Z~28 isotopes (neutron
leve],s) and in the N=50 1soton0':8 (proton level>;) differ by about 20tl
l43
keV (85Van). We found a large scatt«r ~n the Unal values so a
con~lusion to a systematic behaviour of Ey or e~, or to a differenc~
EV-~W was not pos~1ble. At pr~5eilt an equal value o~ ey and £rr is as
~ood as any other possibility.
In conclusion we ~an say that these calculations, schelllat"lC a~
they are, g1ve a fair account of the structure of the levels
investigated in this study.
Gft 2+ 2 Se
75 '" c 0 ~-- .... -- ...... 0
50 ..c • " ~ 0
25 I I
I I
..... _ _ -Ii
0+ 2
0+ 2 .....
75 -..... '"
... , c "" 0 ... '" ... Q ~'"
I
..c 50 ,
I ...... I -V I ... ~
.... I
'M "
25
N N
figure 5.4 COnt.ributioo.s of one snd two d-bosons to the 01 and 2t eKc:ited st",teS.
dots: one-drboson probabil~ty,
triangles: two-d-boson prob",bility.
The lines are given to guide the eye.
144
145
Cbapt@r 6 F~~al conclusiona and s~ry
Whoever, in the pursuit of science,
seeks at ter immedi",te practical
utility, may gener",lly rest assured
that he will seek i~ vain.
H.L.F. von Helmholtz.
In the previous chapters we have described the e~pe~imental set
up, aspects related to the energy resolution, the eJ(perim~tltal data
obtained in our ~xperiments and tbe theoret1cal calculations cO
iuterpre.t these. In this chapter we shall give a summary of the
conClusions.
A "rHlcal study of the va.lous contributions to the line width
learns the £ollow~ng;
- A large contribution comes from the energy spread of the incident
proton beal)\. A turther de.<::rease ot this spread will inhere"Cly lead
to lower beam intensities with corresponding longer measuring times.
Th~ COOling of semiconductor d~ctectOrs ~esults in a smaller FWHM in
the $peccra. SubSl:antial improvement in this respect requiNs the
use of a magnetic sp~ctrosr8ph.
- The thicker the tar~ets the
especially when the ta~get
scatte~~ng angle.
larger the FWHM
angle deviates much
in
frOm half the
The aualysis of the spectra showed that a 5malle~ FWHM was desirable.
The diLemma of the choice between high intensity beams a.nd short
measuring periods on the one side a~d a small FWHM on the other s1de
is practically unsolvable. The only thin~ we can do is to improve both
beam quality and inteuslty ~t the source and during acceleration.
That ther:e are som", pOSSibilities to incrE!ase I;he inteosity of the
atomic beam sourc~ wi eh simple me~ns \<laS receut ly put forward by
Jaccard (85J8C). Cooling the nozzle of the discharge and splitting the
s"xcupole msgnet into two parts wlll lead to an increase of beam
intensity with a factor of 10. This c~n be partly aac,~t1ceQ tor the
redaction of the ~nergy spread.
The experimental spectta of (in)ela.stic scattered protons from
70'72'74' 76Ge en 76'78' 80Se nuclei were analysed 1n a standard manner
146
with the usual correcti(>ns for isotopic; impurities of the targets. For
each nucli<le we analysed the lowest exci ted stat"" 1. e. the ground
state, the 2t, 31, 0t, 2! and 4t states. ln total we obtained 38 sets
ot the cross section and the analysing power for these nuclides.
The elastic Scattering data "'ere analysed by an opt~cal model fit
by the programme OPTIMO (72Vos). The resulting calculate<l cross
section~ and analysing powers describe the data very well. 'l'hese data
also were used In a Sil)l~la[" fit by the coupled-channels code ECIS79
(82Ray), in ordcr to che<;ok wh"th~r both deliver the "arne results in
spite of the diff.-.rent n"lIlerical methods. This was confirmed by our
findings. The paraw~ter sets found have been used to calculate volume
integrale and rlllS ~adii. The vO).<l11Ie integrals were cOlllpared with
several pJ:"ediction8 from other authors. The imaginary volume integ["al~
a~e we 11 re.pr"sented by phenomenological analyse", and also by the
p1;edi~tion of JeuKe"ne, Lejeune al~d Mahaul< (77Jeu) based 00 a nucl"ar
,natt",< appr0ach. The r;,al vol.ume integra.ls show, however, large
disccepancies between our data and the predlctions of others. An
explanation 1s difflcult, but this might be attributGd to ",ome energy
<lependence nOt used in the predictions. The isoapin d"pendenee in the
opttcal potenti~l also has been looked for. A definite conclusion for
the real strength is not pOS$ible due to the. large seatter ll1 the
valu~s. The imaginary strengths show a larg'H regularity and are more
or less in accordance with prediction(l based on tho! phenomenological
analysis of KaHil'> and Gupta (77Kai). The ,>pin~orb1t strength shows a
jump at (N-Z)(A=Ool3, which is not aD artefact ~ntroduced by the
llletooo. This might point to e~fect6 of excitation ot the target
lIudells in elastj,<;o scattering "'S found by Sen et al (655e05) in «I,d')
experim"nts.
A gene~alized-ol'tical-model seaJ:"ch waS performed to obtain
opti(:al potentials which include the effe(:ts ot couplii\g to strongly
el<Cite<l ~tates (2t and 31 states ill the vibrator mod~l and 2t and 4t stateS in the <otator model). In thia way we obtain a good startiog
poin[ for the calculations of the oth'H Channels st"died. these now
(:ould be calculated nearly par"mo!ter (ree in the vibrator and
asymmetric rotator model. from the IInalysis of the inelastic
5catt~rlng data we learned the following:
t. Th., vi bl':ator model. works well for the Ura t exci te(l 2-+- and 3-
states. Espec1a11y the second-order ~er~ion is alsQ capable in
giving a result that marches the data for the 01 state.
147
ii. The rotator model worke weLl tor the first excited 2+ state. Tne
~econd excitad 2+ state. ~~n be described beat in an asymmetric
rotator context ~~lowlng the asy~et.y parameter YZL
to be
different £ro~ the Y of the 2t sno 4t states.
iii. In both models the 4t 8t~te 1s not de8cribeo very well, thougn
there might be SOme preference for the vibrator model.
Most of our conclueions we.e also given in work publi~hed
e16ewher~. D~lsroche et al. (84DeOl) sssumed beforeh~nd that the
vibrator model W~6 only valid for the first excited 2+ and 3- States.
They analysed the higher lying states with oehe. models (vibrator
rot~tor and extended ~symmetric-rotator) with only limited success.
Our main cOnclusion is, with the words of Vergnes (80Ver), "that
it is oitticult to conclu~el". Our analysis has hrought in some new
scattering data which confirm the ex~etlng problems in this mass
~e3ion. We think, nowever, that the sood descript.ion of the at states
in the vibrator model and of the 2! states in the asymmetri~-rotato~
~odel points to coe~ietence of vibrations and rotations. fu,thermore
we see no evidence for a permanent deformation but our data give
arguments for a 80ft character.
Sche~ati.c theoretical calCl,1lations within the IllA-2 frame snows
that it is very well possible to obtain a reasonable ~escription of
the level schemes for N>40. Using the wavefunctions for extracting
reduced =trix elements and compering theSe with the Il z I S from the
experiments results in a sood agreement fo. all nuclides except 7UGe
and 72Ce. The high fraction of two-d-boeons in the or states is an
explanation for the good description of these state~ in a secol1d
order-vibrator calculatiol1. The highly mixed conf~8u~atiol1 of the 21 states can be the origin of d1tferences between vibrator ea1cuLations
and the data. Moreover in 70Ge ~nd 72 G~ these states app~a~ to h~ve
an ~1most one-d-boaon ~hs.acter.
148
DlRECTE KEKNiEACTIES MET GEPOLARISEERDE PBDTONEN
- Een experimente1e studie van Ge en Se -
Salllenvatting
10 dit proefa~hrift ~ijn meti08en van ve~strooilog van
gepola~tseerd~ protODen aan 70'72'74'76C(:_ en 76'7S'SOSe-kerneo
besehreven met als doel een toetslng van het ~ollectieve model (~owel
vibrator ais rotato); veraie) ell het IllA-model. O~t omdat Kernen in het
massagebied rood Ge en Se in het verleden nogal tegenstrijdige
experimentel~ ge8even~ h~bbell opgeleverd, die een eenvoudig beeld vall
de kernatructuur berooeilijkeo.
Om de eJ(perlmenten optimaa1 te kuone[l \dtvoeren, is gekekell naar
de diverse componenten vao de I1jnbreedte in de spectra. Het bleek d8t
de voorn8arnste compoa.ent a.fkomstig W8S van de energiespreid1rlg vaO de
gepolat:iaeerd" protonenbundel. 13elangrijke bijdrage,~ waren ve.rd~lr een
gevolg van de dikte van he.\: trefplaatje. 001<. <l.I;! ruis 1n de
haHg"leider de\:ec.torell teide me". Een aanz10ill1jke. reduct1e hiervao
k.on .,ardell bereikt door mlddel. Van koeling van deze detectorea.
De verkregen expe~1~entel~ spectra werden ~p een standaard manier
S~.uI"lyae"rd '1t1t\rbij ook conectiea voor d" isotopische aamenste1ling
van het trefpla;;ttje .ijn meesenom(ln. Hierdoor hel:>ben "''' uiteindelijk
38 sets vao differeotl~le ",erkzame doorsneden en a~alyBerende
vermogena v"rkregen VDor de grondtoestand, de 2t. 31- 01, 2'}: el) 4t toes taLloen in deze 7 kernen.
De geg"VellS voor de grondtoe~taod zljo a"nsepast met "en
stand(lard optisch mod"l progr(>mma "OPTIMO". De );eBultaten seven ill
het a.1geme00 de elaStiSche verstroo11ng ze~r goed weer· Tevena ",~rd
ceo.~elfde aanl'<l.ssing uitgevoerd met het g"koppelde-kan<llen prOgramma
EClS79 om te cont~01er"n of et' esseotllHe verschillell opt,aden als
gevolg vao een verschillende Dumerieke a~npak. Deze bleke~ niet
aanwezig te zLjn. Met de g~vollden parameteraets werden '101ume
integraten en rll\~ ~tralen berekend. De volume-integn,len zijn
verseleken met ~nder" 1l.llalyaea. Hierb:l-j bleek een goede ov"reeokoffist
te beataan tusaen "Ide,s gepubllceerd werk e" onze resultaten '100r de.
vo1ume-intcgl:a1en vall de abaorptieterm iO de optlsche potenti",,,,l.
De<elfde integr<lleo eventolel voor het rein" dee1 "arell compleet
149
verschillend met die van andere a~alyses. V~rvolgens is nog he~
1sospin effect bestudeerd. voo-.; de re\!le dieptes waren we ni"'t ~n
staat enige cOncl..,~~", t", t-.;ekken vanwege een grote spreiding in de
ui~eind"'ltjk", parameters. De sterkte van de absorptie vertoonde ",chter
Wat m!i''''-'; r;"'g",lmaat en kwam redelijk overololn met olen waard", afgele;i,d
uit een feoomenologist.he studie Van Ka;i.l-as en Gupta. De spin-ba~HI
sterkte, tenslotte, vertoondol eeO merkwaarde Sprong bij een ~aa~de van
(N-Z)/A-O,13. llier hebheo We well!cht te IMken lIlet effecten van de
excit3t;i.e va~ targetkernen tolruggl!k.oppeld nssr (I.", elastische
ve.r6t~ooing.
De volgende stap 10 de aoalyse was het bepalen van een
gegeoeraliseerd", optisch", potentiaal, waariIl de koppellngen naar
sterk a3ngeslagen niveaus (2t en 3i toeetanden in het vibrator model
en de 2t eo 4t tOestanden in het -.;otator model) ~1jn meegenOmen.
De hierb~j ve,~regen resultaten beschr1jven niet alleen d", aangeslagen
toest3nden goed, maar ook de grondtoestand. De beschr~jvlng van de 4t toestaIld is ~ven*el een problee~. Tevens werden hiermee ook de
deformatieparameters ~2' e3 en 6~ bepaald. De waarden van deza
parameters vertonen in het algemeen een goede ove1;"eenstemming met
eld~rs vertichte experiment en.
Met behcllp van deze gegeoeraliseerde optiB"he potentiaal z1jn
andere inelast~sche toestaodeo berekend zonder g~bruik van veel nt","we
pa-.;ameters. De beschrijv1Ilg vao de ot toestand met het tweede-o-.;oe
vibrator model bl~ak zeer goed te gaan voor 70Ge en "120"" t,;,r",ijl het
dHf.actiepatroon maar niet de sterkte van de d1f£e-renti\!le werkzame
doorsnede van 76S e goed beschreven werd. Daarentegen werd de 2t
toestaod erg £;oed beBchreven in een asymmetrische-rotator context.
waarb1j do!: as),lIIlIli!trieparalIlet.",r Y:<2 verschil1end ruoest zijn van die
voor de 2t en 4t toestanden. De 4t toestandeo ~erden in de mee.te
analyses niet gO bevredigeod beschreven, vooral door de verschil1eode
helltngen van de experim.entele. eO berek.ende "'erkzame doorsnede.n. De
,esclltaten die io di~ ~erk O""r voren l:'ijn gekoruen, djn, voor het
ruere.ndeel, een b",veetiging van elders verrichte e~per1I1lenten. Een
nieu", aspect in dit werk zijn de e.xperimentele resu1tatoln voor de 01 toe.standen - e.en sevolg van de 20rg voor het scheidend vermogen - die
bovend1eo io het twe.ede-orde-vibrator model 80ed beschreven worden.
Als la",tste hebben 'wij eakele ac:hematische IllA-i bereketlingeo
150
I!Hgev<;>erd (uitgaanda van de pa~aUleterB van Kaup et a1.) om enig
J.\~7.J.cht t" krijgen 1n de structuur van deze kernen. Het taagBte deel
van de nive .. us.::.hema I S .... o:>:dt go"d bes"hre'l"'o door de"e b"rekeningen Illet
uitzondering van de 01 taestanJen. Evelleeos eeo goede over"allsteJllm~ng
werd g,,-vonden tussell de experimentele d<
151
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8LDuvl Philip D. Duval and Bru"e R. ]\arrHt, ConflguJ:'atlon miKlns in
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81Duv2 Philip Dewitt Duval, Appl1C(l.tJ.on and description of tl'(!
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8lSin
J.U. Polane, Traltsfer ~e.ctiCnG on 5BNi and 56'e induced by
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Technology, 1981.
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8lThij W.,J.G. ThijsSe1), co11,,(
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81Varl R.L. Varner, J.F. Wilkerson, W.J. Thompson, Y. Tagiahi, E.J.
81var2
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83UhOl C.D. Uhlhorn, B. GonBior, 1). Wegner, K.P, Lie.b, H.H. WolCH
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84Car M. Ca"t~hLdi, H.T. Fortune, C.S.F. Stephans and L.C. Bland,
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179
Tot: be",h,lH
Dit proefsc:hrift zou niet compleet zijn als aiet zOu W'o~d"'n
v,;,rm .. ld d .. genen die. op eS$entil!le wijze hebben bijgedragen aan de
totstandkoming van dit proef5chrift. Het noemen van namen is ulteraard
riskant, omdat 1k nlemand ongenoema wil laten. Zlj die ai@t ,;,xplic:i .. t
l.\enoellld ;:;ijn, mogen lderuit nLet concll.lderea dat hua bijdrage aiet
~ewaardeerd en e.kend is.
De leden van de groep Experimentele Kernfysica ea de FOM
we.kgroep K VIII nemen echter echter eea special(l plaats in. H~t w"rk
dat 1a da proefschrift beschrevea wordt, "'as niet mogelijk ge",eest
aJ.s 1k het alleen had rnoeten doen. Het resultaat is tot stand g"komen
dankzij tea~ork. Dit team, waa~van ik deel moeht uitmaken, bc.taande
l.Ii t:
- de tectmiac:he 5ta£: Leo de Folter, Wim Gudoen, Gerard liamel:5 e.l Rob
Janson
- de ",etenschappelijke staf: piet Van Hall, Siebren Klein, Geriird
Nijgh en Okko Poppema (en tijdelijk voor een jaii~ Chandreshakara en
Pushpalata Bhat),
- de be10e ande~e p~omovendl Kees van Overveld en Rita Petit,
zijn dan in feite ook mede-~uteu(s Van dit proefschrift. Dc afgelopen
jaren war .. n niet een~oudig en ~ol problemen. Maar een ding sto~Q als
een paolI bOl1en water: het werk ~an de promovendi Zo() normaa:!. "'o1:oen
b<!eindigd. En claar h .. bben allen 10yaa1 aan meegewerkt. In ,het byzonder
he,;,fc de inteasiel1e samenwerklag met Rita Petit mijn werk zeer
gestiml.lleerd. De v~eIvuIdige besp~ .. king .. n met promotor Okko Poppem~ en
co-promotor Piet van Hall hebben uit<'lindelijk tOt deze tekst ~eleid>
waarblj Gerard Nljgh en Rita Petit hun best hebben gedaan om de meest:e
typefol.lten en andere onzo~gvuldlgheden t~ winden.
De ell:perimenten waren niet ,nogelijk geweest zonde.r een bruikbaal;"
cyclotron. Hi~rvoor <ocgde de cyclotron bedl;"ijfsgroep. De vele
t"keningen in dit proefschrift zijn het ~e5ultaat van langdurlge
bel;"eken~ngen 0>, de 87700/7900 compute~ vall de Technhche llogeschool.
Ee<l groot deel Van de. f igure.n werden Dok hiermee genlaakt. De
medewerking van het rekencentrum voor h~t b@schikbaar stel1en van de
benodlgde re.kentijd en de. loyale hulp Van de operateu~5 wo.de hie. dan
ook grass vermelcl. In deze wi1 ik de helpende hand noemen vun G,;,(!rt
Jan Vissel;" bij het beSCh1kbaar maken op de B7100/1900 van programma's
180
die in andere kernfysische groepen waren on~w~kkeld. In het bijzonder
ware<l dit het pr08ramllld.-pakke~ PHlNT en zeker niet- te verge ten ECIS79
(met m~dewerkin8 van J. Raynal), zonder welke een groet aantal
re~ultaten 1n dir proetschrift niet hadden kunnen worden berekend.
Tat slot .,11 ik niet Onllermeld laten de steun van mljo. el!""
werkgev.,r irr de afgelopen 4 jaar, de Stichting voor Fundamenteel
O!)derzeek der Mater!e te Utrecht· Zij heeft mij volop gClegenheid
geboden om mijn werk tot ,,<m goed einde te brengen. Ik w~l hier
Spec:.iaal 1.l.oero.en de de~ln.8.ro.e aau d':! volg.ende eveI.1erD.Cnr.en~
- WorkShOp on Direct- Reactions in Nuclear Physics 1981 te 8ad Honnef,
- SlJmmerschool an Nuclear StruetLlre 1,98Z te Dronten,
- International Conf<!rence on Nuclear Structure 1982 te Amst.,rdam,
- International Conee~encc On Nuclea~ Physics 1983 te Florence,
- Int"rnational Conference on General Physics 1984 te 'E'raag (mede
dankdj bijdrage van de European Physical Society), th
- 6 Inte~<Hti()nal Symposium on Polarization Pherlomena in Nuclear
Physics 1985 te 06aka.
l.eveusloop
22 febrl.lari 1956
juni 1974
september 1974
12 maar!: 1980
april 1980-
6 mei 1980
mid 1980~
5 september 1981
Geboren te Kerkrade, Zuid-Limburg.
E~ndexamen Athenaum-B aan het Antonius Doctor
College te Kerkrada,
Start studie Techntsche Natuurkunde aan de
Techrrische Hoges"hoo1 te E1ndhoveLt.
'nseniel.lrsexamen ta~hnis~he Natl.lurkunde aan de
Teehnis"he Hogeschool te Eindhoven
Wecens~happe11jk as.i~t~nt aan de Technische
Hogeschoo1 te Eindhoven.
Vervullins m11itaire dienstplicht,
V!'naf juni 1980 als v;l"Ltdrig ge.detachee.rO bij
Ditectie Materieel Koninklijke Landmacht van het
Ministe.ie van Defensie.
lS1
september 1981- Adjunct wetenschappe1ijk medewerker, in dLenst van
december 1985 oe sticht1ng F.O.M., werkzaam b~j oe Technisch~
HogeachQol te Eindhoven in de werkg~oep K VIII
Experimentele K~rnfy5ica.
1 februari 1986-
hed~n
Sta{~ngen~eur blj Hoosovens B.V. te IJmuiden bij de
Elect~otechnisch8 en !nStrumentaCietechnlscha
Dienst ..
STELLINGEN
behorend~ bij het proefschrift van
W.H.L. Moonen
Eindhoven, II maart 1986.
1. Voor h~t bes.tuderen van 8pecif;l.e~e effec.l.t..::n in ytruc.tuur VJ,(I
~erllell i~ het boekl;ebied "an 90·-180· vall groot b"lanl;' Daarom
zDuden julst in dit gebied hoge reeo<ut1e metillgen mel
magnetische s?ectro8r~feQ uitgevoerd en nlet rand de 90' afgekapt
mOe ten wo~aen.
Dit I'roc£schrift, biz. 125-LZ6.
2. De huidlge concenttatie van n~tuurkundig ondetzoek in Datlonale
inBtit~ten _Ole gunstig zijn vaor de ootwikkoling "an dit onder
zoet maar dreigt ten koste te gaan vart het universttaire ondar
wljs,
F. v.d. Woude, Haalt ·sm~ll· physics het jaar 20DO?, Neclerlands
Tijdschrift voor Natuurkunde B51 (1985) 101 •
. John P. Schiffer, Summary Calk. IIl: l'toc. Int. Conf. on Nucledr
Struct\lre, AllIsterdalil 1932, page 497c. Editors: A. van d"r Woude
and B.J. Verh~~r.
3. Het OlltSLaan van Becond~lre reac[ieprodukten bij het kerllfusie
proces worJt in f,et alg"'l11een onderbelicht en !!eeft aIO\lS eeo ver
keerde indruk van een "schone" eoergiebron+
8. Srandl en C.M. BC(l.affiS, Oe onlsteking van het therillonucl~aire
vuur. NederlanJs 'rljd5chrift \TOO\" Nat"'l'rkund" A49 (1983) 142-145.
A. Robinson en L.C.J.M. de Kock, Pusie onderzoe~ in E\lropa.
Natuur eo Technlek 50 no.12 (1982) 934-953.
4. De tr~nd In de kRrnfysica naar steeds 11lRuWe sDndes, exoClsche
reactieprodukten en hogere ellergi~en dreigt tot gevolg te hebben
dat experiill~nten met l~chte deeltj~s en lage energi~en « 30 MeV)
in de toekoilist (liet me"t moge1ij~ zoUen Zij~l, hetgeen niet io
het belallg van de ~ernfys'ca is.
F.C. KQ~mini~ Heavy 100 accelerators. In~ Ptoc. Irtt. Conf. on
Physics, rlurence 1983. volume 2, page SSl-~78. Rditors:
P. Blast aad I.A. Ricci. Tip08r~f1a Compositori, Bolo8n~ 1~33.
5. De gewQQnte Qm in natuurkundige publicaties bij referentles uit
sluiteud ~uteure en plaats van pub11catie te vermelden, doet te
ko.t aan de eveneenS belangr1jke titel v~n het geretereerde en
getuigt van misplaatste bond~ghe~d.
6. Rapportage door midd~l van voord~~chten wordt terecht ge.ien al~
eeu goed middel van kenn~eOverdracht. Daarom is het d~s te ver
wonde~Lijker, dat binnen het studieprogramma van de a£deling der
Techn1sche Natuurkande van de recnuische Hogeechool Eindhoven
geen eenvoudige instructie "presentatietechniek" be.taat.
7. Studierendement is e~n sleeht criterium voor de bepaLing van de
kwaliteit van net onderw~je.
R.J. in 't Veld. Omderwille van verstandige zelfevaluatie.
VUB&M periodiek 4 (1985) 6-7.
8. De sluiting van de Limburgse kolenmijnen is zo suel uitgevoerd,
dar daardoor volwaardige vervangende lndu8t~i~en zich nlet tijd1g
konden ontwikkelen, Dit is daarmee ~Sn van de belangrijkste oor
zaken van de hu1dige strueturele ~erkeloosheld 1n deze ~e8~o.
B. BrCy, De mijnen gingen open, de mijnen ging~n dicht.
Uitgeverij Anthos/Kosmos, 1l1!.3rn/Amsterd3IU 1980 en referent.i.es
daarin.
W. van den Eelaart. Zeven eeu~en mijnen en mijn~erkers in
Limburg. Uitgeverij Corrie Zee1en, Masebree 1980.
9. Alle moo1e cQmplexe modellen en supercomputers ten spijt. de rea
liteit is onberekenbaar.