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STUDIES OF LI = 4 BIFC-RC:\TION

IS .4 - 150 Sl-PERDEFOR.\IED B.\SDS

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Abstract

This s?.st,miatic- si ir~cl\ o f 1 f - -! t) if i i ïc*ii t i o i i I i i i ~ I ~ v l t i iihtlci t O i.igoroiis1~ t wt

the riiodd of Pavlich~nko\.. itiid t h a t ut' iiarrianiotc~ arid h t t oison. I t 1s chriorist ï i i t ~ f l

that rieither of ttirse niodals is capable of esplainine; the resiilts prw(~ritod.

Firially. a st,atistical analysis o f t tio st aggwiri y pa t t cms psiw~iit t d i r i t his t tirhis

is praposcd and carried o i i t . This arialysis iintlrrlirirs t l i c x t i i g h l ~ iiuri-star 1st ical m t urtl

of t lic stsggeririg ptieiiomvnon. but also oniptiasizes the iinportaiicLc of indep~ridrnt

vcrificat ion of t hcse rcsdts .

iii

Acknowledgement s

1 would frst lilw to t h i k t i i y tvifil Dwiiiir fo r livr l o v ~ :iiid q q ~ o i . t a i i d for

kivy~iiig riic iii die nianner to rdiicli I ' w I - i t ~ ~ i ~ i c acxustoriicd.

111 tlly t\VO >'f'il.LS ikS 21 gYilt!ll>ltC U [ ~ ( l l l t i111[\ t hS0P \'O>lrS AS i l S l l l ~ l l ~ ~ t ~ ~ ' $1 ll(!Otlt

Contents

1 Introduction to Superdeforniation

2 Tlieory of Superdeformed Bands

. . . . . . . . . . . . . . . . . . . . . . . . 3.1 Thc Particle-R.otor 1Iodt~l

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 .A ligniiients

. . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Effective Xlignrrivrit

. . . . . . . . . . . . . . . . . . . . . . 2 . 2 . Incrernental Alignnitwt

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Routhian Diagranis

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Ideritical Bands

. . . . . . . . . . . . . . . . . . . . . 2.4. I Theoretical Explanations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 A i = 4 Bifurcation

2.3. l Experiiiit~iital E1;i[lciiïci . . . . . . . . . . . . . . . . . . . . . . 28

2 . 5 . Theoretical Efforts . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Staggering in Identical Bands

4 . 1 1 [O t i\.;tt.ic,ri LN t hi) Espr11.i i ~ i o r i r . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . -4.2 Tlir Esp(~ririi(liir

. . . . . . . . . . . . . . . . . . . . . . . . . 4 . 3 Espcrirncntiil Rosii l~s

4 . 3 1 Y B i . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . 4 . 3 2 Transit ioii Eritlrgy Dtwrriiiriat ioii

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . 3 . 3 S taggcring

4 [ntcrpretation of t h Stiiggclririg Rcsults . . . . . . . . . . . . . . . . .

4.4.1 Sinusoidid Staggeritig . . . . . . . . . . . . . . . . . . . . . . .

4.4.2 A 'ilore Conipletr Descripriori . . . . . . . . . . . . . . . . . .

4 . 3 Implications for Pavlichen kov . . . . . . . . . . . . . . . . . .

5 Further Studies of Staggering

5.1 Staggcriiig in "'"11(2) . . . . . . . . . . . . . . . . . . . . . . . . .

5.2 Staggeririg iri 1.'7.1.'XC y [ 1 . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Llotiv;lt.ioris . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 . 2 . Bands Sut -4ii;ilyzrd . . . . . . . . . . . . . . . . . . . . . . .

5 . 2 3 Rcsults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 . 3 S t;lggcririg iii "'Eii . . . . . . . . . . . _ . . . . . . . . . . . . . . .

.J..[ Si ;ixal~riIla i i i 'XI,'j' Gd . . . . . . . . . . . . . . . . . . . . . . . . .

- - . t t t 1 i t J I . . . . . . . . . . . . . . . . . . . . . . .

6 Conclusions

6. I 1 I = -1 Bit'iirrat ioii i i i I ( l i b ~ i t ic7;il Exicls . . . . . . . . . . . . . . .

2 Syitvtiiat ic: S i i r w , ~ of' 1 1 = L U i i ' i i r w t ioii . . . . . . . , . . , . . ,

6.2.1 Ttlsts i)t' t hi) . \ Io[ f i l l : ; . . . . . . . . . . . . . . . . . . . . . . .

2 . 2 Sti-itist i t d Siyiitic.;i~iw . . . . . . . . . . . . . . . . , . .

6 .3 Fut iirr P r o s ~ w t s . . . . . . A . . . . . . . . . . , , . ,

A Oscillation of the Tunnelirig Amplitude

B Calculation of the Staggering Amplitude

C Glossary of Symbols

Bibliography

List of Figures

4.1 Single-partick Roiir liiiiiis at dt>fortriat iotis appropriitto t.o t l i i b .. I - L j O

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . region a.2

4.2 Partial 3-ray spectra uf " 'Eu(l) arid 1"YGd(6) . . . . . . . . . . . . . 60

4.3 Staggering patterns for l'l%u( 1) arid "'%Gd(6) . . . . . . . . . . . . . . 62

4 4 Systrniatic uii(.t~rtaintii~s i i i a staggoriiig plut . . . . . . . . . . 66

4.5 Ideiitical baiid st.aagcririg par r t ~ i i s plot t o d as n f i i r i ( ~ t i o i i of q i i i i . . . . 69

4.6 Coocl fits to ttic idriitical band staggeriiig data . . . . . . . . . . . . . 12

4.7 Poor fits to thp idrritiçal baiid staggering data. . . . . . . . . . . . Ï:J

List of Tables

5-10 Staggering sigtiiticaricil for t l i ~ bitiirls .;tiiiticd i r i tliis n w k . . . . . . . 102

6.1 Tests of Pav1ic:hikov'c; riod del. . . . . . . . . . . . . . . . . . . . 110

Chapter 1

Introduction to Superdeformation

-4s discussed in Section 1.1. the propertics of these Irvels nia! soriietinies be

understood by supposirig t h the levels arc excitations of single protons or neutrons.

T l i i s <bIi i i l ) t .c '~ ~iro\*i(ii.s ;i I~ r io f i i i t r o i ! i i ( ~ r i ~ ~ i i 1 0 i i i i i . I i l i i r physirs. u i i l i i u i (.iii-

p l i ; i s i s 0 1 1 s i i p o r d r f t - m i i ; i r ion i i i t lit. . I l i i i i y i o r i ' . ( - ' I l i i ~ ) i (XI, 2 t jxplor(ls i l l o r t b ( lo(~pl>-

st,rilo OF t.li(1 r p l t . v a r i t l ~ ; i ~ . k ~ i . ( > i i i i , i t ( 1 r I l i 5 K, > r k . i i i ( h i i i i i i g ( l ~ t ailod i x p l ; i i i ; i ~ i o i i s O[ ~ v l i i i t

is i i i ih; i i i t t " 1 1 = -1 I ) i f i l i . r - ; i t i o i i " ;111(1 . . i ( l ~ ~ i i ! i t - ; i l t ) i i t i ( i ~ " . S O I I ~ ( I of t t ~ x p o r i 1 1 1 ( ~ 1 1 -

ro(- / l r1i(!1los t I lki t l i ; ~ v ~ ~ I ( V I I I I S ~ Y ~ 111 r Ili,< \\.I , I . I ~ 1 0 51 i ~ t i i . SIIIJ('!'I l(>!i ) t . t t i ~ ~ I 1 ):-111i IS : I I . ( )

c l i s ( . i i s s o ~ l i l ) C ' l i a p t c l i . :j . ' r i I ( l i ~ s i - i i i ~ o ~ , ; i l \ q w i i i i m i ~ l i i d i :i.;t-. ~ ) o i ~ f i ) i - i i i t v l i r i

o r t l ~ l r t o stii{i\- A1 = -1 I i i l i i r i s ; i r i i i i i i i i : ' ' ( ; ( l ; i i i < l ! '"Eii. i i i i i l ~ t i o kt.. i ~ ~ s i i l t s w l i i d i

i - t r i t Ir i C ' 1 i ; i p t i . i -5. (Iiit i i t ' r o i i i ;i i i i i i d w r 01 ' ~ ~ q i w i i i i o i i i s i i r o i i i i : ~ l \ - z c ~ i h i t i i -

y . o i i ~ t h r s s t r t i i s r I 1 = l i i t o ~ . T l i o i w r i < . l i i s i o i i s

d s a w i i f ro r~ i t l i i s work am1 p w s o 1 1 t ( ) c i i i i C ' l i q 1 r 1 1 r (i.

1.1 Nuclear Models

As nientioned above. the esatbr solut i o r i 01' t h t l n i i c h r svst iw is h ~ o i i c l t t i ~ wacli

of today's coniputers. As a result. the tlirorct i c x l ( 4 ' ~ t s r» iiridi.rstard r tir niic-leus

niust rely on rnodels of the riuclctis. This s ~ c t i o n d i s c i i s s r s t h r two hasic families of

nuclear rnodels. the single-particle and collrctive rnodi4s.

'-4 is the nuclear m a s . See Appendix C for a glossary of synibols useri 111 t h tliesis.

1.1.1 The Single-Particle Mode1

This approacii beçonics ver? conipiitat iorially i n t m s i w rsr r t ~ i i i r l l ! . qiiicbklv. .-\t

present. the most powerful siiptircotnpiitcis kiaw orily hwri i i t ) l v r o (10 stic.11-riiotlrl

calculatioiis for nuclei wi tli niasses iip to -4 - 50 !Caii9-4. lLar97'. Tiiiis. for tit~;1vi~r

nuclei one is forced to abandon these tecliniqiies and niake rriorr :issiiriipt ions about

t lie iiuc4cus before beginning.

.A poor-man's version of t l i ~ shell niodel is a rriean-field rtiotlrl. .A niean-

field mode1 assumes that each nucleoii rnows in an average putential which is the

WOODS-SAXON POTENTIAL I

QUADRUPOLE DEFORMATION

Figure 1. i : Slicll gaps in a Roods-Saxon potentiiil[Diid871. Particlch ~iunibws at varioils gaps i r e shown. At nlro (kforrriation. thcl orbitals arr lahel1cc.i w-ith t l ~ o spectroscopie notation 1,. w h m I ( j 1 is tlic orbital (total) arigular nionientuni of t h particle. At large deformation. t h t y arr 1at)rllcd by t h projectioii of t h ~ i r angiilar rnoinentum on the syrnnietry axis of the niirleiis. and hy tlieii parity. Shaded regioris dcnote regions of liigher level density.

1.1.2 Collective Models

This collection of states is callcti a rotational bi~nci . The statw i r i a rotational

band have angular mornenta 1. I + 2 . I t 4 . . . .. as a iesult of sv i i in i r i ry (.onsicl-

erations [PreZ] . The quantity ,7 is the moment of inertia of the iiuclciis. In niiclear

t t h t i r . i t i I o s i t I - t ~ t i t i o ~ l I . Tliis

Tlius. the rotational frequericy is g i w i 1~

1.1.3 Deformation Parameters

in t his section.

The surface of t lie nucleus may be describctl [RiriSO/ hy t hc funcr i o r i R(0. o 1 -

'-4 triaxial sliape is an ellipsoid wi th rio two u e s equal. like a kiwi.

1.2 Superdeformation: P hysics in the Second Well

1.2.1 Superdeformed Fission Isomers

s.4n isoiner is a long-lived excited nuclear state. Xlthough the focus here is on ttie short fission lifetime of these particular states. the lifetime is still tens of microseconds. inaking it a rnillion times longer than is typical of rnost nudear excited states.

t l i i fission isonier mj*stery. \\lirii escited t u ~ i w g i e s abow r liat of t . h ~ siipor(lc1l;)rriied

miniiiiurii (which could happen i i i dit. rpactioiis st uilirs). r kir iiiiclo~is ( ~ 1 i i 1 d tirsr t ~ i n n c l

froni the near-spherical mininiurn to the superdeforrned rninimuni. aritl t lien t urine1

into a fissioned system. Because the barriers are so miicli narrowrr iri tliis case thc

t liat. as long as t lie motlierit of incrt ia rriiiaiiis constant. t lie1 encrgirls of tlic -. rays

are proportional to I . so that the spectriini of -O rit!. iwrrgies slioiiltl look l i k ~ a

20 transitions. giving them a very disririctirc signature. I t shoiild tw tioted ttiat

- 130 superdefornied baiids arr never populat t ~ l wi t li iiiorci t han t i\.o p~rwrit of

tlir iiitctisity o f the cha~iriel. niaking thrni iriipossiblc t,o set) witliuut iisirig i i iiiiiiit,rr

of cmlr\-t1r clsp~ririieiital trv:tiiiiqiirs. Tlicsr techiiiqiios arc tiisc*ussrd i i i C ' l i i t l > t i l ~ 3 .

The .4 .- 150 bands arc stabilized by the proton s t i~ l i +ips ;it (52 66 ancl tht.

neutron shell gaps a t 84-86 (sec Figure 1.1). The intruder orbitals stabilizing these

gaps are the proton l13 /? and the neutron jlji2 orbitals with Nilsson numbers .Y = 6

Chapter 2

Theory of Superdeformed Bands

2.1 The Particle-Rotor Mode1

I n the particlc-rotor rriodrl. one in1agint.s tliat t tir riiichs ia; i i i r r m t id il3

a fow partirlcs couplecl to a dcforrried mrr . The Hartiiltoiiiiiii H t~ t l i i b s?-stim i?;

niade up of a cullective part HCoii. a t t r i b u t d t o t h clore. ami a siriglcl-i);irt,icltb pari

r . gireii by tlic energ!. of the particles oiitside the cor^ ( i n t ht. fi&l o f t lw dchmied

c o ~ e ):

into c . This means that the total HarriiIturiian cari \)c w i t trn as

k 2

THEORI.' OF SIrPERDEF0R:IIED B.4 .VDS

2.2.1 Effective Alignment

'This assuniption that ( j l l ' ) is iiist:iisi tivt. ti, t lie ri-iïlriis is n p;irt.ii*ularly guod ont3 when the m u s - A nucleus is a rigid core. like '"'Dy. Hotvever. it is foiinri t.liai. this assitnipt.ion r~rna ins good tliroughout the -4 - 150 superdciornitici nuclpi. ~vfiere niosr d t h riilr.ltv)ris c.:tii still btl itssurried to be involved in rotation r d the core.

Gamma-ray Energy Ey (keV) 1000 1100

0.50 0.55 Rotational Frequcncy ho (%le\')

Th~is. Ai, is approsiniately eqiial to the aligntltl spiri of tlio l;i$t i i i i h ) r i . I l o w

generally. Ai, For two baiids is a ineasurr of tlir suiri uf t lir aligiiid spins of r i i ~

nuclcons wliich makc the structures of the two bands c i i f f i w w .

Calculating csperiniental effective aligrinients is siiriple. Corisidrr ;i rrfrrrrirr

band whose mth transition energy is ~ ~ ' ( r n ) . Coiisidrr also ano t t i r r harid \vit11

transition energies E,(n). The effective alignnient of tlir second batid witli rcspect

tvhrre tlir notation is thc siinic. as i i i tlic prc\-ioiis swt ion . T h o rthii(l(lr n-il1 rwogriizc.

t liis as t lie t icgat iw of t tic srriiriil t vrni i i i t i i ~ i ~ f f ~ r t ive :iligriiwiit Tho inrrcriii~iital

a l ignnie~i t thus ignores t liil iwiit s i h t ion t O t 110 o f f w t i n l ;iligririiimt wtiiiiig frorti t lio

difference in spins of the rr~iit t iiig lowls. and rcrains t lu. . h t ( l r p k m d a\ig11111(>1it'*

cornirig from the differcnce in -!-ray energies.

Although the increniental alignnient has no convenicnt pliysical interpreta-

tioii. as does the effectivr d i g r m e n t . it is ras' to cal[-ulatr and finds cornrrion use

-1. . d - h - . d

k 'J - - C

u - d

'7 - - d - x 'A . d - . - 3- H .-. . - 3 - CI

,L

> - - c: V . d 'n -n z d

'L

$ i-. d H Y

5 4 C œ

C a 'Y? >, 'J2

5s 4 Y . - Ci

c C s - r;: C I - - x . +

u -4 - d r œ .d

.A

THEORY OF SUPERDEFORMED B A V D S

F i 2 . : R.eprescntativr1 Roiithiiiii iliagrarii for r i ~ i i t n ~ ~ i s n t w .Y = Sri (.Y is th r neutron nurnber: Z is t tic proton tiiiriibtlr). Sorrir of t h irit riidilr o r h i t als iirc latwll~tl. The Yilsson niimbers of soriit. others a r ~ also shown.

a plot of o' as a fuiiction of frrqiicm~y.

.4 represeritative Routliian diagrarn is stiown i r i Figure. 2 . 7 . Ea(1ti linr is t h r

R,outhian of a different orbital. with thc linc types indicating thv signature. o. and

parity, n. of the orbital ( t o be defined). The label "BY means that 55 orbitals lie

below that gap.

lccording to Equation 2.17. the dope of a Routhian is equal to the negative

of the aligned spin j, of that orbital. The orbitals labelled "71" and have large

slopes and are tlius higlily aligned. These are the N = T intruder orbitals. which help

stabilize the superdeformed shapes in the -4 150 region.

THEOR).' OF SCTPERDEFORltlED B A N D S '23

Routhians with coristant slopes are of special interest. Thcse orbitsls have

coiistant alignrnents as a Fiiriction of rotatioiial frequency. which is a necessary (bu t

riot sufficient) condition for the grneratiori of idriitical bands. as disciissed in tlie

followi~ig section.

2.4 Identical Bands

Consider two superdeformed bands in neighbouring nuclei in the .-1 - 150 region.

Since the masses of the nuclei differ by one part in 150. and since the niornent of

THEORY OF SUPERDEFORMED BAlVDS

inertia scales like .4'i3. one sliould espect the moments of inertia of these two nuclei

to differ bu approxirnatcly one percent. Siiice i-ray energies are inverscly proportional

to the moment of inertia. one r n q expect one percent sliifts in gamma-ray energies

between these two bands. and since the -/-rai enrrgies are approsiniatcly 1000 ke\'.

these shifts should be around 10 keY. The superdeforni~ct bands in ii~iglibuiiriiig

nurlt4 should, t liereforc, hear lit t lc reserriblance to onc a n o t l w .

This expectatioii is fiilfillcd iri niost pairs uf t)arids in rieighl)i~iiririg nurlri. A

typic-al pair is showii i i i tlw top t ~ v o 5pecti.a of Figiirc 2.3. Horwvri, t l i ~ top aricl

bottoiri spectra i r i this Figiircl arc. higlily corrclated. This pair of t~arids is iiri ~s ; l r r ip l~

of rv hat ml calleci iderit ical t~aricis.

Sirice the - p r q eriergies of isospectral bands are the same. their relative in-

cremental alignments are zero. and their relative effective alignments arp a constant.

the spin difference of the levels de-excited by the identical 7 rays. L h r ~ gerierally.

bands with identical moments of inertia have constant incremental and effective align-

ments. Plots of these alignments are often used in the nuclear physics literature to

demonstrate that two bands are identical.

TNEORY OF SUPERDEFORMED BANDS

1200 1350 1500

Energy (keV) Figure 2.3: Exampies of identical and non-identical bands. The upperniost spectrum is that of the SD band 1"9Gd(l) [Flig'i], the rniddle one 148Gd(l) . and the bottom spectrum '"Eu(1). It is clear iiow little correlation there is betweeri the top two spectra. but how much there is between the top and bottom spectra.

THEORY OF SUPERDEFORklED BANDS '26

The identical bands L51Tb(2) and 15'Dy(1) diRer i r i structure by a single par-

ticle. a proton in the negative-signature [301]1/2 Nilsson orbital. This is also true of

the identical pair I5%d(2) and 15'Tb(l). and of other SD bands in the -4 - 150 rnüss

region. It thus appears that there is sorriething special about this orbital tliat makes

it responsible for the generation of identical barids. Tlic sarrie appriir.; to be triitl of

. . tiir. positive-signature i4 i 1 j 1 j j neutron ori>itai. ait iiougii tiiil oq)rririieiirai rlvidr~irr

for t tiis is riot as abiindant.

2.4.1 Theoret ical Explanations

SOI-OI-:I~ at tchrripts have t~eeii riiacic to esplairi t l i i . idiw iciil Ixmls pli(liii>iiieni~n. Ijilt

oril- tlic t,wo best-knoari explanations will b r disciisstd tioro.

Tlie Rrst explanatiori. proposd by Ragni i rs~~i i jRiig90. Rii<!l:3!. is t hat t iic

siriiilarity betweeri the 7-ray rriergicls i r i t h r:iV(, I~aiids is i1ssontiall\- ;icx9iileiit al. HP

lias prewcted somc eïidence that t iir rffwtiw ;iligriiiit~rit of t h rii1g;it iw-signat iiro

[XI Lj 112 protori orbital (which niust bcx - 1/2 t o gciicwtil i s o s p . t 1-;il t~;iiiils) (#;in I N )

rcproduced in his cranked Xilssoii-Strutinsky calciilai iixis. Iri t liis riiodol. t lit. rviii~~-;\l

of this proton from the iiucleus causcs a dccwtw in t~uc-lcw rriass. ; ~ i i c i iiri iri('ri3asr

in iiiii4wr defoririatiori: thesc two etfects carir~l one aiiotli~i-. iiiakirig t tio iiiunients of

incrtia identical iri the two cases.

This approach is soniewliat unsatisfying. iii that it takes what appears to b~

an incredible. cross-nucleus degeneracy and relegates it to the status of a coinridence.

Another probieni is that the calculations of Ragnarsson [Rag931 reprodure the oh-

served -1!2 alignment to no better than twenty percent. Thus. niariy people have

sought alternate explanations.

.A very attractive o p ~ i o n invokes the concept of pseudospin. The notion of

pseudospin [HecGS, hi691 arises from the observation that as the deformation be-

THEORY O F SUPERDEFORMED BANDS

conies large. orbitals in a g iwn oscillator shell with quantuni numbers (1. j = 1 + 112)

and (! +' . l+3 / '2 ) approacli one another. I t has becri faund coriveriierit to r(hbe1 these

orbitals witli pseudo-orbital angular mornerituni i and psciitlo-spiri 3. su that tliese

orbitals form t h e pseudo-spin doublct ( 1 = I + 1. j = I i 112) with srriiill pscudo-spin-

orbit coupling. This tricl; of ( h i i g i n g t h I-valut) iri o rdor t o dr tmuo t.liv spin-orbit

splittiiig torids to work I~ec.ausc> r l i t . tiiiclear Haniiltonlari depc~itis r r i i ~ s t l y ori j. and is

That is. the energies are the sanie as those iri ! " D y ( l ) (which has spins I + 112).

Thus. l5lTb(2) should be identical to 15'Du(l) in the limit t h a t the deformations are

large. By the same reasoning. the neutron orbital with Nilsson numbers [U l ] 1 / 2 and

pseudo-Nilsson nurnbers [jiO]1/2 should also generate identical bands.

THEORY OF SUPERDEFORMED BANDS 23

Tliis is an intriguing theory. but it has a major fiaw wtiich was glossrd olver in

the above disciission. This flaw is that the tl1eoi.y ignores the differentrrs in niornerit

of inertia between the tivo nuclei. The pseudospin mode1 c m shed no light on this

aspect of the phenomenoii. and t herefore remains iricomplrt~.

2.5 LI - 4 Bifurcation

2.5.1 Experimental Evidence

In 1993. Flibottc et al. jFli9Sj pr~sentcd evidenrr for ail iiricq)r(wd i i t i i ) t i i ; i l ~ ( c . ; i l l d

AI = -1 t~ifiircation. or staggvriiig) ir i tlic dynariiic rrioriicwr of iiicwia of [tic yrasr SD

h r i d of l%d. This niorii~iit of irirrtia. plotrrd iri Figiirv 2.4 is riot a sriioor li funrtiori

of rotatioiial frcqurncy. h r whibi ts a regular "zig-zag" I1i4iiivioiir. TIiv cictiiils of

t h ii~iiisual oscillation c w i Iwst [ ) t h s r rn iri il "st;tggcriiig ilo or". r l i t l ( - i ) i i ~ t r u c t i i i n of

whicli is csplaiiied helow.

Sincc the dynamic riionicnt of inertia is giren hy r t w rrcipro(lii1 o f A E , . one

can also see the staggering ~ f f e c t in a plot of AE,. shown in t lie upper panel of

Figure 2.5. Since one is interested not in the gross characteristics of this graph. but

rather i n the subtle deviations from its mean behaviour. a smooth reference curve

AEff is constructed from the data. also shown in the iipper panel of Figure 2.5. For

the s r n o d i reference. the well-known four-point smoothing formula IBev92I is used:

0.35 0.55 0.75 Rotational Frequency ( M e V )

Oric. t l ien subtracts frorii i w l i iI;it:i point t h(> wlw of t IN. sriiuot li r t ~ f ( ~ r c ~ r i w ;ir t lw

samc rotational freqiienry. T h i s giws a staggi2ring plot. sIiown iri t tic^ lo~ï.rr piiricl of

Figure 2.5.

This staggering in AE, indicatcs rhat the seciilence of lewls in tlir siipcr-

deformed band has been separated irito t w sequenws ( t h e sc3qiiericil lias "l~ifiircated" ).

one wirh spins ( I . I +4. I +B. . . .). sliifted slightly i r i ericrgy with respect, t a the other.

wi th spins (1 + 2. I + 6. 1 + 10. . . .). Since the niagnitude of the staggering in AE,

is approximately 240 eV at the mid-point of the staggering pattern. the relative shift

THEORY OF SUPERDEFORMED BANDS

Rotational Frequency (MeV)

Figure 2 . 5 : Construction of a staggering plot. In the upper panel. the points represent the AE7 values for '"'Gd(1); the solid line is obtained bu applying a four-point srnoothing formula to the data. In the lower panel is the differencc between the data and the srnooth curve. The staggering effect is clearly seen.

THEORY OF SUPERDEFORMED BANDS 31

of the enrrgy levels of the tivo sequences is only 60 eV. This is quite sniall. especially

when one considers that the energy levels of interest are at excitatiori energies of

about 10 Meir.

Some auttiors have suggested [RevSG. Do11961 that t hç. staggering effect is due

to one or more band erossings. Tliat is. lewls froni i i t l i ~ r SD (~onfigiirarions with the

sanie spin and parity as the ' '"c~L yrast band are rii is~d arit ti t tiostl of t Iii) vrast band.

producing levcl repulsion and t h i s staggwiiig. k i t s s i t 1 1);iiid crossiiigs

('ilri pro<lii(*r irregiiliir sr iiggoriiig c~f f r~ t : : owr ;i rclar iwl!. riiisroa spiri r.iy,iori ( a.s shonm

1)'. Riloy et r d . [RilSGi). i r is difficwlt to iniagirit. ii l);lii(l imsi;iiig swri;irio capat)l(> of

prodiic~irig a attiggcring pat,tern as rpgiilar as t hi i t ~ ) f ''YC;(l( 1 ) t hiit porsists ovrr so

rriany t riirisit ioris.

2.5.2 Theoretical Efforts

Thew Iiai~e b w n nurnrroiis a t r rnipts to espliiiri thil origiri of ll = 4 I>ifiiix-ation iMnrri9-4.

Bur95. llac95. hIag95. Liik95. F; i~95. Stiii95;. \lii~i!. of t litwb 1 I i w r i ~ ~ s ;tssclrt t Iiat t h c h

presrricc of t\vo rcgular I I = 4 ~ i ~ q u i ~ i i < + t ' s ' i i i g g ~ ~ t i l 1 1 t > ~ l ) l i l t l i l t i i > t ~ i~i\.oli-ing il four-

Fold rotational syninietry (an in~ariaricr wi th rcispivbt to t h C', poir i~ syiiirietryT).

The best ktiowri sucli tlieory is tliat of Haniamoto arid .\Iottc?lsun [HariiW!. the basics

of whicli arc describcd below. This approacli is Yrry siriiilar to ttiat proposed bu

Pavlichenkov and Flibotte [Pav95].

Hamamoto and Mot telson begin witli a generic C.i-irivariarit Hamiltoniari.

which they suppose to be a perturbation to the conventional particle-rotor Hamilto-

th general, a C, symmetry is an invariance with respect to a rotation through 360/n degrees.

THEORY OF SUPERDEFORMED BANDS 32

where 1, are the three components of the nuclear angular momentum I witb the 3-axis

coiricident with the long axis of the prolate shape and the 1- arid ?-axes lying in the

equatorial plane. The quantities .4. BI. and B2 are constants. The first terrn in this

Haniiltonian serves to keep I away from the 3 -a i s . as expected for a siip~rdrformed

nticleus. The second term breaks the C, syrnrnetry of the Haniiltuniari and is the

siniplest terni (lowest power in 1 ) which niakes C'., t l ir dominant syriirictry. 1 hc third

tcrrri is asirilly syninictrici. 11ut is iridiidcri so ttiat d l CI-iiiwriiirit tt1rrris of fourth circler

iri I iirtl pr txr i t iri t,hp Haniilti~riia~i. To soriic tlsttmt. it ~uiiiitcr;ic~ts t lir fi rst tcrm

(siriw ( I f + 1:) is proportional to - I ; ) . so it niust be assuriietl tliat 20-I" -4 s«

t h t htl angiilar nioniciiturri lies iii the equat or. Sorne u t hors [lI;ic95! hnw sliown

tliiit B2 inust bc very srnail a o t hat t lit1 rrgulas rotatioriiil striii*iiiril of t lir. SD t ~ r i c t

is iiot dcs t royd . In this thesis. & will bc sct tci zilro.

The Haniiltonian dcscrilm a poterit ial i w q y siiri;iw ivi t 11 foiir i ~ i i i i i i l i i t I ri t l i ~

cvluatori;il planr. Quantuni nici:liariics allows t l i t . ;irigiiiai. nioiiioiit i i r n t i r i i i i r i i ~ l t'rorri

oncl of thcse minima to iinother. producing n lt11.d split tiiig wliic*h is proportioii;il to

t h i iiniiding niatris clrrncnt. I t is shown iii App~r id is .-\ t h for 1' > .4/-!B1. tlir

tiiriiieliiig niat r is elcniciit a q u i r e s a factor wtiicl: cliangm sigri wlirn I changes hy 2h.

This prodiices an effect like that ubservetl in %Gd(l).

This mode1 is not without its critics. Doriau. Frauendorf. and Meng [Don961

have provided the most notable criticism. performing calciilations to estirnatc the

relative magnitudes of the paranieters BI and A. Their <:alculations sliotv tliac it is

extrernely unlikely that the C, syrriinetry-breaking terms i r i ttic a b w e Hamiltoniari

corne from a static hexadecapole deformation of the superdeformed shape, as mas

suggested by Harnanioto and Mottelson. However, this does not negate the validity

of the model; it only establishes that the appearance of C4 terms in the Hamiltonian

THEORY O F SUPERDEFORMED BANDS

is not the result of a static deforniation.

Recently. Pavlichenkov has takeii a new but relatrd appriiach. He has consid-

ered [Pavg'ia] a Routliian corisistirig of single-particle cm.~rgics. tao-body quadrupole

and liexadecapole interactions. and cranking terrris for rotation ü l m i t t lic axes per-

pendiriilar to the long asis of thr niicleus. F'rorii t his. ho h;ls ( l ( b i + i \ - o i l ilri offoctivcl

Harnil t o ~ i i m

THEOR).' OF SUPERDEFORMED BANDS

1 3 5 7 9 11 13 15 Number of nucleons in the j-shell

Number of nucleons in the j-shell

Figure 2.6: Moments Qd4 for j = 1312 (upper panel) and j = 1512 (lower panel) subshells. Calculations are performed for rotational frequencies of 0.1 MeV (circlss). 0.6 Me\,' (diamonds), and 0.8 MeV (triangles). Takeii from [Pav97b. Pav981.

THEOR Y OF SUPERDEFORMED BANDS

tested in this thesis.

The theoretical stage is rlow set for this irivestigatiori of 1 1 = 4 t~ifiircatiori in

superdeformed bands. I t is now iiiiportanr to understand r h r experirrierital îc.ch~iiques

used in this thesis, before the discussion of t h r data begins.

Chapter 3

Experiment al Techniques

3.1 Heavy-Ion Fusion-Evaporation Reactions

Higli-spin riiiclt'ili. staros ;iw iiiost oftcn popula t~d in htmy-ion t'iiioii-o\;\por;iri»n

r~actiims. In r liesc rc;ii1t ioiis. i l I > r ~ i i ~ i i of projectilr nuclci ( r g . '"1 l is writ r tiroiigli

a target cornposecl of sorne otlicr niiiterial ( c g . '"'Sn). 111 ttitl expwirrit1nt.s dcsc.ribctl

iii this tlicsis. the targrts werc atmut oiie niicrometw tliick: t tiest. t hickric1ssrs iiro

noriiially given in microgranis p w square centimeter. If tlie hearii cnrrgy is high

enougli for tlie target and proJwt il<. niiclei to overconic their Coiiloiiili rrpiilsim. t tir!.

can fuse. forming a compound iiucleus (in this case. !j3Gd). The cuiiipuuiid nucleus

formed has a fixed excitation energy. given by the beani energy and the Q-value for

the reaction: however. tlie angular monientum of the compourid nucleus has a wide

distributiori. related to the distribution of impact parameters for the rcactiori. For the

reactions considered in t his t hesis. the compound nuclei were formed at an excitation

energy greater than 50 MeV. with angular rnomenta as high as 70A.

When it is formed, the conipound nucleus is excited to tens of MeV above

EXPERIAdENTA L TECHAWl) U E S 3'7

the yrast line. and must finil a way to cocjl off. This is niost often acconiplislied by

neutron evaporation. altliough fission can cornpete w i t h this procers at the highest

angular inornenta. Neutron cvaporation carries away alrriost 10 SI& [Hac96]. on

averagr. in binding energy aritl ki~ietic. energy: tliis niükes i t i~iuch r r i i m t#iiirnt. than

S - r q rrnission, whicli reniores mily a few Ne\* on averagr. Neutrons takr away oril!

about Lh iri anguiar niornrritiirri. so tiie anguiar rrioriirriruiri ( i i~ t r l t )~ i t i~) r l is i;irgrli!.

unaltcretl. Thc conipouiid riiicl(w'; caii ;ilsu riiiit pr(miis o r i i l ~ j l i i ~ [);II t i ( . l t 'b . \,iit t h w j

c*tiargo(i particles rniist o~- i~ i . (~or~ i i~ i i ( 'o i i lo i t i t i i);iri.i~i. i i i o i - i I ~ i . to o \ ( . . i ~ ) ( ~ Tliiis. t ticbir

oitiissioti is i~ftcri siippross~vl. iiiiloss t tic rorripoiirid riiii*l[~iis is iitliit i.011 iloli(*itliit.

.As t hc decay pat ti gt3ts (*loscl ro t hi1 vrast liiir. t /ici Ivwl ihisit!- (li~çrcvws.

and t lie population of spccific pat liways bcconics largrl cnoiigli for t h v pat l i w y to

be observed through their individual -; rays. The supcrdefornicd bands analyzed in

this work are al1 yrast or near-yrast decay pathways at angular niunirrita a t~ovr 5Oh.

whose 5-ray decays can be sern in discrete pcalis.

Below 50h, the superdeformed structures are no longer yrast. Howcvcr. tlic

transition matrix elements between adjacent superdeformed states arc so large that

"'Level density" refers to the number of energy levels present at a giveri excitation energy. per unit excitation energy. It increases exponentially as one rrioves away from the yrast lirw.

E X P E R M E N T A L TECHNlQ UES 38

in-band decays remnin favoured until spins as low as 25 - 3 X . At thesr lower spins.

the nucleus is again several MeLV in excitation energy above the (nornial-drforniedj

yrast linc. t he normal-deformed level density becomes quite large. aiid n continuum

decay takes o w r again. Experimenters have not been able to tind discrrtc-line decay

patliways betwetw the superdefornied bands and the normal-defornird states in the

.-I - 150 regioii: this has rriadi' it impossible to niake spin and t~srit;ttioii rriwgy

assigririieirts for t!w supwddoriried harids in t his regiori.

3.2 Gamma-Ray and C harged-Part icle Detectors

Bot h -/-ray aiid diargrd-particle det cct'ors operat r on t lit. sarrir prilici plr. T h

ray or cliarged particlr eriters somc detection niedium. and crcatcs f+ctrcri-holr

pairs. With charged particles. this is done through direct ionization of atoiris t)? the

cliarged particles. Gamma rays first create eiiergetic elrctrons. t lirougli r he photo-

electric effect. Comptori scattering. or pair production: t hese r1t.c troris t h n ionize t hr

atoms in the detection mediuni. These electrons and holes are then dctct- td to infer

some of the properties of t h r incident particle or ray.

The detectors in use today fa11 into two categories. solid-state detectors and

scint illators. These are discussed in the following sections.

ESPERIMENTA L TECHiVIQUES

3.2.1 Solid-State Detectors

Today's high-spin 7-ray spectroscopy is done wi th a r r q s of Conipton-siipprcssed

gerrrianium detectors. The experirnents described in this thesis were perforrned wi th

Gamrnasphere[Lee90]. an American arrav consisting of up to 110 detectors. This de-

trctor can be run on its own. or iri conjunction mith a variety of ausiliary detectors.

One of these auxiliary detectors is the Microbal1 [Sar96]. a charged-particle detector

which will be described briefly in the following section.

Channel

Figurr 3.1 : Illustration of t lit. C'imiptoii t~ackgroiiiid. TIw sprrt riirn skiowri ix of ;i

"Co soiirce. collected rvith uriv o f t l i v C;ariiniaspkierii geriiianiiirri t i ( i toc torh . Cimptori siippr~ssioii has been turneri off. Data is taken fmrn [Svr98].

3.2.2 Scintillators

Scintillators do not directly detect the electrons and holes. but rather the visible light

that is emitted wtien they recombine. Scintillators made frorn bismuth germanate

(BGO) are used as Compton suppressors for the germanium detectors in Gammas-

phere. The Microball uses an array of 95 cesium iodide scintillators to detect protons

and alpha particles emitted in fusion-evaporation reactions.

EXPERIMENTA L TECHNIQUES 4 1

3.3 Coincidence Spectroscopy

Tliv top p ; ~ ~ i ~ l of Figlirtl :3.2 is c t hackgroiirid-siil~t rwt rd pilrt id -.-r;iy sptlctruni oh-

taincd frorii the GS- 113 mp~ri~iic:rit discussrd in t liis T ticsis. T l i ~ spwtriirri shoivs al1

of the A. rays obscrved in ttiis espr r immt . and is ofteri ca l ld t h sutotal projection".

Duc to the nature of h c a ~ y - i o n fusion-evapiiratiori rwctioris. niany niirlci arc pro-

duced in this experirnetit. and the total projection is kriowri to coiitain *( rays frorri

14s- 1-is Eu and L"5-147Sni. Thus . if one wants to assign r q s to a particiilar nucleus.

or if one aants to study ver- weak structures. like superdefornicd barids. one has to

look beyond t h e total projection. T o d q ' s 7-ray spectrometers detect in germanium

detectors as many a s 6 or 7 of the y rays emitted by a given rompound nucleus. so

one can make great strides by asking which rays are seen in coincidence ivith other

EXPERIMENTA L TECHNIQUES

1 . :il 1 1 al, n , II , i p

vu * S I , b ~ 4 ' ~ , , ' rlJJ 1 ~+k",+/

y-ray Energy (keV)

Figure 3.2: A dernonstration of coincidelice spectroscopy. In the top panel is the total projection. In the middle panel is the spectrum in coincidence with the 721 keV *, ray in l'"Eu. whiie the bottom panel shows the spectrum in coincidence with the 1439 keV y ray in ""u.

E.XPERIIZIENT.4 L TECHNIQUES

7 rays. A simple csaniple of this t t x h i q u e follows.

Partial level scliemes of 1"6*L47Eii arc sliown iii Figure 3.3 [E~css . Fle77. ZlioYii.

Because the *, r q s on the left are in the sanie cascade. they should be seen in coin-

cidencc with one anottier. For esample. if the 607 keV -{ ray is observed. the ones

below it in the cascade arerc also eniicted, and may have been detected. The same is

true of the y rays on the right. However one should not expect to see the 721 keV 7

ray in coiricidence with any of the ones on the right: the! belong to different nuclei.

EXPERIMENTA L TECHNIQ C'ES 44

and should not be emitted iri the sanie evenc ( that is. by the sariir residual nurleiis).

This is demonstrated in Figure 3.2. The middle paiiel is the spectruni of -4

rays in coincidence with the 721 ke\' 7 ray. while the bottorn panel is the spectrum

in coiricidence with tlir 1439 kei' r q f ( in the language of s-ray spectroscopy. the

721 kt.\' or 1439 ke\. -( say are calleti '*gates". arid the spectra in coincidence with

tlieiii are calleci "gated spectra" ) . As expected the 7'21-gatcd spectriiiri has peaks at

366 k ~ \ . . 350 kt., and 60'7 kc!.. wtiilc the 1439-gated spectriini has pealis at 285 kei'.

434 kib\-. arid 5 18 kr\.: w i t lirr spiJct rurii 1 i i l ~ pcaks t>rloiigirig to t hr ot lirr casmiic.

I i i addit iori. tlio gated s p c t r a oadi Iiaw Fcwr pt'aks tliari tlir total pn).jwtiori. as a

rc~siilt of th[) cw*liisiciri of ot1itlr h r i r i t k This ;illo~vs rriorr3 clorrtilotf s l~c~ctr l iscop to

C ~ O I ~ P . This td i i i iq l i t~ c:lIl 110t o~il)* h? 1 1 ~ d t O Wpir:lttl r ~ ; ~ . t ~ O I I ( ' I l i i t~~li~k: i l (.:lti

also tw i i s i~ i tlistinguish hctwwri d i f fc~mt decil! paths iri t h sarric r i i i i h i : ; . arid morr

goriclrall!. t u drtcrniiiic wliiit t hi) dota!. pathux!.s a w ir i a gi~.vri riii(,l(~iis.

3.3.1 klultiple Gating

Analyis is o f tm pcrforriied mi spwt ra in coiricidvnw wit h t w or nior(' I rays. This

allows ovrri more sc4cc*ti\ity iri ctioixirig a givm der- patli. alid it rediices further

the iincorreiated backgruurids. siicti as the Compton background. .An esample of t t i ~

p o w r of multiple gatirig in t h spwtroscopy of suprrdefornitd txinds (*an be srrn in

Figure 3.4. describeci in the folloniiig paragraphs.

The uppermost panel of Figure 3.4 shows the total projection from ari ex-

perirnent on l"%d. This riucleus has a superdeformed band with nineteen knonn

transitions. However. i t is populated in only one percent of' al1 reactions popiilating

LG18Gd. and so it cannot be seen in the total pro,jection.

One can improve things by setting gates. The other panels in Figure 3.1 show

sums of spectra gated orice. twice. three times. four times. five times. and sis times

E.YPERIi\IENTA L TECHNIQUES

y-ray Energy (keV)

Figure 3.4: Gated spectra of the superdeformed band "'%Gd( 1 ) .

on the band. That is. the serond panel sliows the suni of thr spectrum gated on the

first transition. plus the spectruni gated or1 the second transition. and so on up to

the nineteerith transition. I t is clear that one can enhance the signal f r m the band.

arid siniultaneoiisly dccrease the background by setting large numbers of gates.

T'lie biggest draaback to sctting niultiple gates is the loss iri stntistics. Re-

qiiiring ari eveiit t40 contain four 7 rays from th! barid is a stririgcnt coiiditiori. One is

fortiiiiatc wit h superdefornieci bands that tliey coiitaiii si> tiiilnu traiisitioiis: with nine-

trcii tmrisitiuiis. the qiiadriiply-gated slwtsutii i r i the Rgiirr is a siiiii of' (':) = 3876

iiiilividiial qiiaririiplr-gatid spc.c6t ril. Ho~vr\.t~r. t lit. ligiircb shows Iio\i. O I ~ P iuurinot srt

arl~itriirily Iiigli giit,iiig cuiitlitioris.

3.3.2 Elliptical Gatirig

.-\ siiiglo-gatcl cririditiori is ;i stateni~ri t of t hcl forrri. ..if ttir -1-riq vriclrg!. lirs hetwwn n

r i t . . . ." . This can hl goiirritlizi~d to tivo dimrrisions. t o hrcmicl. - i f t ho Hrst -,-r;iy

m q y . El . 1ic.s hrtivcrn r i ancl b. ancl the s~corid -?-r;iy rri(ygy. 6. lif's I ) ~ t i w t ~ n r . arid

ri. . . .". This kiiid of gaw is ciillrd a rectangular gatc (iriiagiric a grapli of E l \.ersils E2.

and pictiire t h regiori passirig the gating condition ) . Tlic spwtra in Figure 3.4 used

rectangular gates. and t tieir higlier-dimensiorial analogs: t iiey are o bviousl y tiigh ly

effective. but they could bc iriiproved. In El E2 space. tlir pcaks c l 0 not liai-e a

rectangular geometry. but ari elliprical one with ases in thc ratio ( b - a ) : ( d - C)

(see Figure 3.5) . This nieans that the corriers of the rectangular gatr (:ont ain rnostly

background.

This situation can be aineliorated with a so-called "elliptical gate" [CroSL

Wil97bl. Tha t is, in place of the rectangular gating statement above. one uses the

coridition, "if the point ( E l , E2) lies within the ellipse centred at (9. 2) with

E.YPERliCfENTAL TECHNIQUES

Figurcl 3.5: Contour plot of ii t wo-~1iiiii~iisiori;il C;i\ii~~iiiii di5t i - i l i i i t i i , r i . S o t t t r /IO distribut ion has an rllipriixl givjrliilr I-y iii El E2 spactl.

ascs 1 and . . ." . .4lt lioiigli t h iniplcrrir~itat ion of siich a stilt.(\ri,iilit is riioro - i:otiipiitationally i n t c ~ i s i \ ~ tliii~i th;it of a rcctangular gatci. it can bi3 iisofiil in rwliicing

backgrounds.

3.3.3 Spike-Free Incrementation

The reader will likely have noticed that the sestuply-gated spcctrum in Figure 3.4

contains some decidedly non-statistical fluctuations. These anomalies. calied spikes.

arise as a result of the algorithrn used to create these spectra. as discussed below.

EXPERIMENTA L TECHNIQUES 48

For many years. t l i ~ efficiency of -~-ray spectrometers was sucli that thev

could collect only two rays per r ~ ~ e n t (called H o l d da ta) . -4s a result. da ta arialysis

techniques developed around the analysis of "doubles". Evmts with t h rw detected

7 rays were too infrrquont to be arialyzed by double-gating: as a result. the! were

"iinpacked" irito doubles. Ttiac is. ari evcnt w i t h -.-ray rriergicls ( E l . E2. El) was

coiiwrted into tiiree doubic-wincidcricc rvents. ( E l . C2 j . ( E l . El 1. and ( t2. 1 . This

cwiscd rio problrriis ;\ri( l twt+airiil st iiridard.

C'utisitl(~r ovcqit i l l i t 7 S . S I f i d f i ~ l l irisid(> gates of a

supcrdrforriit~tl harici. I f oiio w n t s t o rtiakc a s ~ s t iiply-gated spclc-triini. then this

ciglit-folcl tv.c~rit is i inpackd iri t O oight se\-cri-fold ~ \ ~ i i t S. SCYPII of t hoso ~ v e r i t , ~ ro~i ta in

the ; ray ttiat does not fiil1 iliside t he gates. and sis ttiat do. .As a resuit. the ;-Tay

eriergy falling oiitsidc ttw gütes will bc increnierited seven times in tlic spectrum.

These incrrmentations are tiot iritlrpcndcrit of one ariother. and so the spertrurn is no

longer statistically correct. in that t lie spectrurn no longer ubeys Poisson statistics.

Beausang et al. (Bea951 have suggest,ed an incrementation scheme in whicli

these spikes are not created. Their algorithm takes each ;. ray in the event in t u rn . and

asks whether the gating conditions arc satisfi ed by the rest of the î rays in the event.

If so. that energy is incremented once in the spectrum; if not, then no incrementation

is made. Obviously~ in this scheme a given energy is incremented only once and spikes

EXPERIMERiTrl L TECHNIQUES

grounds inhererit to tliat spectrurii. This is a less offensive forni i j f I)arkground. but

cari I>e distracting. S l i p prirnary goal of background suhtracltioii is to remove the

first component to the background; the second is a smootli curve whirh can be re-

nioved bv hand. It should also be noted that for multiply-gated spectra. the first

coniponerit becornes niuch riiore complicated. wherms tdhe second cvmpuncnt retains

its s in ip i ic i t~

Palameta and L\iddingtori [PalB5] siiggested t tiat t titi n l ~ o w i i~s i i inpt ion ( t hat

the \)ackground spectrurn is independent of the position of the brickgroiincl gate) is

so good that one can make a single background spectruni gated on al1 hockground

channels in the spectriirn. This background spectrum c m be iised as the hackgrotind

for al1 single-gated spectra. Elloreover. since the total nimber of background channels

is large. this spectrrini cari be made with great statistical accuracy. The generalization

of this method to higher folds is also considerablv simpler t h a i the earlier alternative.

This is furtiier explored in the follorving section.

3.4.1 Background Subtraction in Higher Folds

This section will give the reader a serise of how background subtraction is perfornied

on multiply-gated spectra. For a coniplete discussion of tliis topic. t h ) rcadcr is

directed to the article b>- Hackniaii and Naddington [Hac95].

Consider a spectriini triplc-gatcd on thrw -. rays. This spwtriirii raii tw

tliouglit of as cornprisirig foiis styaratcl cori!potii!rits:

Oril?. tuiiipont'nt I is tlosiri~d. C'orripoiirrits '2. 3. ;iiici 4 shoiild t~r si1 l ~ t r;u1t cd.

In analogy \vit l i t h discsiission iri t h 1 p s w d i n g sectiori. a spccBtriini of corii-

ponerit 2 ni- be protliiced by triplr-gat ing on two praks of intcvst and a s ~ t of

background cliannels. Similasly. coniponents 3 and 4 m- be estractcd. Even il

spectrum employing a sum of triplr gatcs. as in Figure 3.4. is only rnarginally rnorr

complicated. sincc the set of backgrouiid cliannels is taken to be univcrsal. and not

affiliated with any particular gatr.

There are some coniplications in this scheme. For example. the peak-peak-

background gate used to create the spectrum for cornponent 2 contains coniporients

3 and 4 as backgrounds. .As a result. when this spectrum is subtracted from the

EXPERil\lENTAL TECHNIQUES 52

original triple-gated sprctrutii. corriponent 3 is oversubtracted. and must be added

back. in addition. while the use of a single set of background channels simplifies

enormously the background subtmction of a spectrum niade from a suni of multiple

gates. there a re still fairly cietailed comhinatorics argumerits whicti determine how

much of each backgrourid spectrurii to subtract from. or add to. t h original in order

to rccovcr cornporient 1. H o w t v r . t tie so-cailed .'operator riietlioti oi Hackinari and

\ lkddington 1.s out al1 of thest) dotails.

This chiiptrr lias ( l w r i i w t l rl i t l irriportaiit priiicipicls u~iderlyiiig thil i:olloc:tion

of -# -r- spwt roscopy t l i i t a. ;ml its arialysis. Coiiplrd wit h tlir kriowlrdge of nudear

st ruct iircl tlirory glo;iiioil froiii t lit) prilceding ckiaptcrs. t his iriforrriat,iori sliould have

rlir r d { l r p r o p d y priirii~d i o iiiitlcrstand and nppreriatc t h rw i l t s prcwiircd iri t tie

diiip t r rs t O ronilx.

Chapter 4

St aggering in Ident ical Bands

Tliis cliapter preseiits t l i r rrsiilts uf iiii i~sp?riirit!rit pcdbrrricd in .July 1996 to stiidy

tlio staggcri~is plicnoiiimuri. I t cori<ilutles with a disriissiori c)f the iriiportariw 01' ttiis

w o r k tu tlic r:urreiit iindrwt.ari(liiig of tlic i+fmrt.

4.1 Motivation for the Experiment

Soct ion 2.3 cicscribed 1 I = -4 t ) i f ~ i r ( x t I O H :mi r ho w c i t ( w ~ t i t i t q w r a t t i r ~ T 110

riuclciir st ruct u r ~ conirriiiriit~.. T h f in t t ~ i d m w of t \lis plwnortitwm i x r r i ~ [rorn

' % G d ( 1 ) [FliD3]. Tlie staggclririg pat t ~ r i i n t liis l ) i~r i ( i rmiairis t lw c*loartlst .jiriglc

csnnipl(1 of flic effect. siric~. i r sliows a regiilar oscillation t h persists owr 2Oh in

arigular nionientuni. .4ltliougli srwral groiips tiaw clairned tu sep s t agg~r ing in othor

superdeformed bands [SernS~. de.496. Fis96. Ccd9-l. Iiru961. doubt remains regarding

t lie statistical significance and reproducibility of t tiese results.

An understanding of this phenonienon clearly demands investigation of stag-

gering in nuclei other than 14'Gd. Iinowing which bands eshibit A I = 4 bifurcation

could be valuable in determining which aspects of the structure of thesc bands rnight

be responsible for the efFect . Pavlichenkov's model. for iristance. makes de finite pre-

dictions about certain orbitals and their contribut ion to the staggering phenornenon.

STAGGERING I N IDENTICAL B . W D S 5-4

It c m , therefore. be rigoroiisly tested if it is known which bands in the n i a s region

exhibit the staggering effeçt. .As a rcsult. a systematic survey of the rnass region

could be very useful iii tracking down the cause of 2.1 = 4 bifurcation. Hoivever.

with the modern 7-ray spectronieters in high demand. a more fociissed and l r ss tinie-

coiisuming experiment niiist be proposeci.

As disciissed in .-ippc~ndis A. Haniaiiioto and I lu t t~ l son l i i i \ . i b s!iuwri i hat, iri

tlieir rnod~ l . the staggclriiig ;iri!pli t iidc is approxiniatrly proport iorial t o (.os( ? ( I - A,) ).

wtitw Io is a pliase fiictur iiiiiqiii4y &terniinc.cl II- the paranit1tc3rs .4 aiid BI of rlicir

Hairiiltoriiaii (Hani94j. This riiiglit Iiaw sonie prcdictive p o i w if ono kiiow t t i ~ origin

of' t t i i w paranieters. For itistitri(.o. this i ~ s p r ~ w i o n woiild hc us th1 if oiir ki i tw- rliat.

t , l i c b paranieters w i w t tir siiiiith i r i t iw diffwmt t~aricls. If i har wort~ t ho t livri ~ r i v

coiild bcgiri to cotistrain t,tir iiiodd I)iirirnit.tcrs.

1s tliere a possibilit! bit p;iii.s of baritis cari I)e tUiiri(1 in whidi r l i o p;ii;iiiivt tLrs

o l tlic rriodel Haniiltoiiiaii ;irr r l iv si~iiio? .-\ri iritriguing a\.cwic3 of itiwsi igat i o i i is t o

study pairs of t den t l ca l txiiids. .As tlisciissed iri Section 2.4. idontical I)iiiids I i i t ~ . ~ :-rav

riicrgics that are corrclatcd t o a riiiirh higher degree than niiglit h r ospwtrd . Tlir

origiri of tliis phenonieiiori is riot y t understood. and so the t>xterit of tlw siniilarity

rnay run deeper thari is gencrally rralized. If an? bands have a c h a n w of sharirig CI

Hatniltonian parametcrs. idcntical bands do.

It has previously been notcci that the removal of a proton fror~i the negative-

signature [301]1/2 Nilsson orbital in 15'Dy and other -4 - 130 nuclei generates an

isospectral identical band. -4s shown in Figure 4.1. this orbital is oclc*upi~d and at

the Fermi surface of 1'9Gd(l). suggesting that the yrast superdeformed tmid in lqaEu

should be identical to "'Gd(1). Being yrast . this band should be easy to populate

and identify.

STAGGERING I N IDENTICAL BANDS

0.4 0.6 ho (MeV)

ho (MeV)

Figure 4.1: Single-particle Routliians at deforniations (J2. JI. - i ) = (0.57.0-07.3.8'). 1 1 1 Orbitals with parity and signature (ii. a ) = (+. +:). (+, - - ) . (-. +:). and (-. -? )

are indicated by solid. dotted. dash-dotted, and dashed lines. respectively.

STA GGERING IN IDENTICA L BANDS 56

Before proceeding. it should be noted that . wen if the paranieters of the CI

Harniltonian do not remain the same in the two bands. ihis is an interesting exper-

iiiient to perform. Sup r rde fo rmd coiifiguratioris have not previously heen observed

in any europium nuclei with -4 > 114. The experiment thus probes nuclei wliich have

not ber11 stiidied at higli spins. In addition. becausc '."EU and '."Gd diffa- by oiily

uric procori. the configuratiuiis o i ciir siiper(ieforriied striirtiires ir i t i i c ~ ~ iiucic4 are PX-

pecttxl to I>e quite similar. The presciicc or absciii:i) of staggrring i i i "%1i bands ma?

still. t litwfore. provitir iiiforriiat ion rrgarcling t lit. i i i i d t w sr riirt ilri] origiri of I I = 4

t)ifiirc3;it inti. a s disçussivl at t hcl I ) t y j r i r i i tig 01' t his wct rori.

4.2 The Experiment

Cy t l i ~ staritliirds of IiigIi-spin nuclcar physirists. ""11 is n:wt rori-ridi. This rrimns

t tiat riibiit rori rvaporat ion \ rd1 ilurriiriatr wr r cliiirgtd-part icIi. t ~ a p o r ; l t ioii ir i a hcavy-

ioii fiisioii-i1\.aporatio1i r i w t iori prodiiiirig "'Eu. Idtwll!.. t Iwrohrc. ;i t w i r r i ;irirl targrt

combination is desirrd ttiat î a n populatc ""Eu i n a ncwtron ompvration cliannel.

Unfortunately. iiu cornbiiiatioris csist ttiat cari poptilatr ' '"11 w i t h the spin

aiid escit atiori energ? appropriatcl to a st utly of siiptwirfornicd h i i t l s . On r riiiist ,

tlirreforr. resort t o a reactioii in wliich a charged particle 1s cirnittcci. A proton

emission ctiannel is prcfrrred. since alpha particles terid to reniovc more angular

mornrntuni.

There are several suitable reactions pnpulating '"'Eu. a t the riglit spin and

excitation energy. in which a proton is emitted. .A beam of "Si with an energy of 158

MeY and a 700 pg/cm' target of ""Sn was chosen. This produces 14%u following the

evaporation of one proton and four neutrons. This esperiment was carried out a t the

Ernest Orlando Lawrence Berkeley National Laboratory, with the 88-Inch Cyclotron

-4s nieritiorieil i i l ~ o ~ < ) . ir i.i,iis(~cliiori(-tl of iisiiig t tiis r i w t ion is t t i t j pi.o(tu(-tiuii of

;i h r p d~ i ta SC:^ for i . ' 7 , 1 " y ~ ~ i . i ~ i t 11 ' "G(1 ~ I ' U ( \ U C O C \ 011t i ~ ~ i i ~ l l \ * ~ O I ' ~ ~ l [ ) t ~ ~ . ~ ~ t l ! ' t ~ ~ . t ~ i i ~ t i ~ ~ ~ ~ .

A l t tioiigti these riiiclvi I i : i w I)i.rii iwll stiidicd b!. ot l i c m [Th%. i 1 ( 4 ' 3 G . ~ic+'!J5. BvrS8j.

a t lioroiigli aiiiilysis id r l i v st aggrririg i r i t. l i t w riiicloi bas nor Iwtw p r f o r r i i d . Of

sprrial iriterrst is '"'Gd(C)'. This I~arirl is i d m t ical ti, "'Gd( 1 ) ;riid lias t h ' % t l ( l )

configuration [deF95]. witli a iiiiiit.ron tiole in t ht. positive-signature [4 1 11 1 / 2 orbital

(see Figure 4.1 for the neiitxori Routhians in this mass region). .As described in

Section 2.4.1. this orbi ta1 is also significarit in the pseudospin explanation of identical

bands; this makes it another iriteresting candidate for study. This band and lq8Eu(1)

'.Alpha particles were also detected witti the llicroball. leading to the formatiori of siipcrdeforrned States in '"Sm. .A description of the two obscrved bands does not appear in this thcsis, but can be found in [Has98a].

'In this thesis, the superdelormed bands in L48Gd are nurnbered wi th the scheme suggested in Reference [deF95], and employed in [SinYG. Hw97, Byr981. An alternate nurnbering scheme exists. suggested in Reference (de.1961 and used in [The96. Pav97b. Has98bI. In this latter scheme, bands nunibered 1-6 in this thesis are numbered 1.3.63.3. and 4. respectively.

STAGGERING IN IDENTICAL BANDS 58

are the focus of this chapter: analyses of the 0 t h bands in 1.L7-1%cl are left to

Cliapter 5.

4.3 Experimental Results

4.3.1 New Bands

4.3.2 Transition Energy Determination

Al1 aiialysis in tliis thesis was pcdh-rried oii triple-gatd sportra. I t is possiblr that

quadruple-gating is optimal for ttie rriost intense t~aiids in tlic gadoliniiirii data sets

analyzed in Chapter 5. However. because these bands arr so intense. these consid-

erations are not important. In most cases. the spectra werc created wi th spike-free

incrementation? wit.h the background subt.racted according tci the operator approach.

For the piirposes of this thesis. the most important quantities measured in

these bands are their transition energies, since these permit the extraction of stagger- - .-. -

§The intensities are extracted as explained in [Hasgdb]; for more detail on intensity rneasurernents in gated spectra, the reader is directed to [Hac96].

ing data. To ertract these eiiergics. the ?-ray peaks werp fitted with Gaussian shapes

with the code G F ~ [RadSS]. The background was fixed to be flat. and in most fits it

was fixed to be zero. The transition energies are. liowcver. rclatively inscrisitive to

the height of the backgroilnri.

In cases where a peak riras deemed to be "contaniiriated" by a pclak frorii t hc

normal-deformed level sctir~ritl. tliis riorriial-deLiriid p a k aas f i r i ~ i l i i i t h i l back

groiirid spectrurri. a n d its positiori arid ivici th ot)tiiint.cl Frorii tliis t i r w r o tistd iii thci

ti t of tlio siipcrdef~)siiicd baiid spwt i ~ i i i i .

nlicw F . G. and H arc piiranict.crs. and r is t h i-liaiiiid riiinihvr rli\.i(ld II!- orw

tlioiisancl. The widths of tlicl supcml~~forrrird p a l i s ~ w r c tlien fixed acvmiing to ttiis

Fuiiction and the spcctruni l i as rcl-fit. Ln cases n l i c r ~ sewral baiids w r r fit iri t h

sarne nucleus, the parameters (F. G. H) extractcd from the fit of the most intense band

were used to fis the widths in al1 of the bands. This is desirable because the width

parameters are determined by Doppler broadening and detector characteristics. and

so should be the same for al1 bands in a given niicleus produced in a given esperiment.

Figure 4.2 shows partial -,-ray spectra of the bands Eu( 1) arid lJ8Gd(6).

The transition energies of these bands are given in Table 4.1. The transithm energies

for lq8Eu(2) will be presented in Chapter 5. when the staggering of this band is

ST'4 GGERING I N lDENTICi.4 L BANDS

y-ray Energy (keV)

Figure 4.2: Partial -,-ray spectra of ld8Eu(l) and L4%d(6).

STA GGERING I L ' V IDENTIC.4 L B44NDS 61

Table 4.1: Gamma-ray transition energies. in keY. of the superdeformed bands "'Eu(l) and ld8Gd(6).

4.3.3 Staggering

The staggering results arr. of roiirsc3. t l i ~ rriost iriiportaiit r r su l t s of this chapter.

Figure 4.3 shows the staggcring patterns e s t racted for '-IX E u ( 1) and l'l%Gd ( 6 ) . us-

ing the energies from Table 4.1. Also shown is the staggc~ring pat terii for l'''Gd( 1).

froni [FliSS]. One is imrnediately struck t)y the presctice of LI = 4 bifurcation in

both of the identical bands. This is reniarkable becaiise staggeririg is a rcliitively rare

phenomenon. Chapter 5 will show many esamples of bands t ha t rxhibit no staggering

at all. This makes the occurrence of the phenomenon in three identical bands quite

surprising. Although neither of these new examples persists over as large a range in

Rotational Frequenc y (MeV)

Figure 4.3: Staggering patterns for '""E(l) and '"'Gd(6). That of '%d(i) is also shown, for reference [Fli93].

SZ4GGERING IN IDENTICA L BANDS 63

T h s e roncrriis art3 largcly uiifourided. I t slioiilti tirst II( ! riotrd tliat staggtlring

di?pcrids uii difftwnccs of w-ay cncrgics. I t is tliiih iiiiii<wssiiry to Li;i~t) ;(II al~soliitcb

calibratioii acciiratc to 0.1 kt>\' . L r y i ~ i g t tic calihrilt io i i cwffkiwts ( i v i t hi11 roasoriatilc

bounds) has virtually rio efftlct on a staggering plot.

Non-linearities in the hDC's cannot b<i resporisiblt. for t h v s t iiggering plicl-

iiorricriori cither. .-\ given peak in a siiperdcforni~tf biiri(1 ocnirs i l [ a diffcrrrit set of

chaniirls for each of the detc~ctors. Th i s occiirs hccausr tlir : rays irrc Doppler-stiifted

due tu the velocity of the nucleus. and because the signals from the detectors are

not aligned anuway? As a result. a systematic non-linearity in the ADC's would be

applied to a different energy for each detector. K i t h one hundred detectors. these

effects are completely washed out . I t is also important to note ttiat. as stated above.

Corrections for these effects are made in software.

many bands have been s h o w in t h work to exhibit no staggering at d l . suggesting

that there are no systeniatic problenis wi th the detector. The '%d( l ) staggering

tias also been measured wi th three different detector arrays [HaaSO. Fli93. LFiv96].

I t is worth dernonstrating at this time that the reported uncertainty in the

is q i i a l to the pcak art1a. arid griicralizr). Tliiis. t hr i qm- tc t i iiric*i.rtainty iti t tir

crritruici is 0.14 keL'. This is sniiillvr t.lian t tic rq~ortrvi iiriwrtairity of O. 17 k tT . Lhr

iilloivs the area of the pcdi to var'.. Nevtirtheless. the roportrd iiriwrtaiiit i ~ s ;in. no

srrialler t han t hose t hat rnight be expected from sini plo statist ical considerat.ions.

The most likely source wf systeniatic uncertairities is the backgroiirid siibtrac-

tion. This, more than an. stage in the analysis. depends on the judgernent of the

esperimeiiter. The experimenter must decide on the shape of the sniooth Compton

and continuum backgrounds. determine wliich superdeformed peaks are contaminated

and how to fit these contaminating peaks. Even the ratio of smooth background to

gated background in the subtraction is subject to the experimenter's discretion: it is

felt that this is the dominant source of systematic uncertainty.

STA GGERING IN IDENTICAL B,4,VDS

Iii order to make an estimate of the size of tliis uriccrtainty. a spectrurn was

prepared in wliich approximately five percent more gated background tvas subtracted

than iii the optimal spectrum. .A spectrum was also prepared in which five percent l e s

gated background was subtracted. In both cases. the amount of sniootti background

subtracted was adjustcd to make the counts in the background cliarinels surn to Lero

in tlie tinal spectrum. The tive percent deviatioti \vas riot c~oriiplrtc~ly arhitrary: iit t hi5

l e~e l . the spectra no longer look properly bilrkgroiiii(l -;iihtr;i~,:c~l. Tlitw ohviiiiisli.

oi-tly- aiiïl iiiidcr-~ubtïii(~t etl spcl(*t ra ~ T S P t l i i ~ i i f i t t i v l i ri t t i ~ SiLiriil iv;i\. t los(.ri l w f l

; i l~ow. ;iiid a staggeririg plot axs iiiadr. Figurtl 4.4 ~Iitm-s t l i o wsi11t~ of t h 5 stu(1-y.

Sliown is a sragg~ring plot of '%d(6): tlic tillwl r i s c h ;istl t h s;iiiitb d;it;i as p l r ~ t t d

iri Figiisr. 4.3 and t h ) twur bars arr t lie stat k t icA twors ori t liosil piiiirs. T l i v iipnarcl-

iiiid do~vti\viird-poiritirig trimglrs .irtl t lir rvstilts of t l i ~ i~riiilyws i ~ f t l i t b m-ils- aiid iiiiiicr-

sul)tr;ictrd spectra. resprrtivrly. This imiilysis is miwiiragirig. iii t l i i i t i t s h o w tliat

iii riiuiy cmascbs. t h r twi t s arc3 vi1~tual1-y t l i o siiiii(l Ii)r dl t Ili-IV iL;isos. ~ i ( l i t is r;ircl tliar

tliis s>.stcriiatic iiiicrrtaiiity is (xjriqxirablc tu t h stat kt ;(.id t~ars . Tl iorc~f~~ri~. oritl

caii sah.1. ncglcct tlic syste~iiatic uricrrtaiiitirs froiii tinc4qy-oiiri(l si1 I N i ; i iht ion.

The staggering results presented in t his rhiiptcr wrt iiinly look rion-st aiist icül.

Hoaever. it is iiiiportant to ask what t h prol~ability is tliat a nicxsiir~nicrit of ü

perfrrtly regular band would yield a staggering plut liktl tlio ones sliowii twre. simply

as a result of statistical fluctuations. This question is tiot as easy to answer as one

inight imagine because the points on the staggering plot are so h ~ a v i l y correlatecl*.

However. the analpsis described below gives an answer t O t tiis question.

A computer prograni was writ ten aliich takw as input tcn roiisecutive transi-

tion energies (with their statistical uncertainties) from one of ttie bands. The energies - -

'Recall h m Equation 2.21 that each point on a staggering plot depends on three AE, 's . and therefore four y r a y energies.

Rotational Frequency ( MeV)

F igiiw 4.-l: Systeniatic uiiwrtairit i(.s in a staggclririg plot. Tlic f i l l ~ t l cirrlcs g i w the stiiggrririg d a t a for 1 4 % d ( ~ ) and t hc errur bars show thcl statistical uncwtainties in tliese points. The upward (do~vii1vard)-poir1tirig triangles stiow t lic rosults ~f the samc analysis. but with an ov~r(iiritli~r)--siibtracted spcctriirri.

are selected from a region centered at E, = 1200 keY. The prograrii theri prodiic~s

100000 copies of this band. wit ti tlie transition energies in rach copy randomized with

Gaussian distributions with means and standard deviations qua1 t u the rneasured

transition energies and their uncertainties. respectively. For each copy. the seven

staggering values X i are calculated and the rnean staggering 1. is calculated according

STAGGERING IN IDENTICA L B44 NDS 67

Table 4.2: hlean staggering (-Y / and staggering significaim 1 - for '%d( 1 ) . 1'1%11( 1 ). and ' '%Gd(G). Also given is the probability P as defined in the test.

B a d Mean Staggering ldy1 (keV) Significance 1' Probability P ll%d( 1) 0.2 l (9) 2.3 2.1% iA'8Eu(l) 0.30( 13) 2.3 2.1% l%d (6) 0.37( 12) 3.1 O. 19%

-Hic. ratio Y = j.Y/ /qy is a mcasurc: of t hr sigiiificaiiw of tlitl r~s i i l t . i r i t tiat

t l i ~ rcsiilt diîFw by Y staritlard dct.iat,ioris Frorii tliv i i i i l l r t w i l t i r i o sraggc3riiig). Orir

caii rc3port cithcr tfiis 1 ~ ~ 1 of sigiiifiwiw o r d i t . prolxildit!- t l i i i t ;i riiridorii siiiiiplc

frorii a staritlard norrrial clist r i h t iori tvould >-irlil ii rwiilt u-it h ;tl)solii t (i i*aIiio grwtilr

tliaii 1- (tliis caii be fourid in tab1c.s in Bevingtori [Btii.!I2] or virtuiilly ariy statistics

t e s t ) . Tlie probability. P. is esactly equal to the prohability t h a nicl;isiircrricm o f

a barid with no AI = -1 bifurcation would prodiiw a valuv of i.yI as large as that

iibscrved. answering the initial question.

Tlie results of this analysis for the three bands in Figure 4.3 are shown in Ta-

ble -4.2. In each case. the mean staggering IS(. thc significance 1- and the probability

P are given. The table shows h o a unlikely i t is that a measurernent of a regular band

would produce these results by chance. This issue is revisited in Section 5.5 after a

more systematic survey of staggering in this mass region is performed.

4.4 Interpretation of the Staggering Results

4.4.1 Sinusoidal Staggering

The goal in perforrning this experiment was to see if there is an>- elidence that

the niodel pararneters of Hanianloto and hlottelson rerriaiii constant iii the identical

baii~is. This section exa~riines tliis question.

In Mariianioto arid \lot telson's niodel. t lic band staggcring is prupcirt ional to

spiii. sliotild ;il1 lie ori i l single cusi~ic c i i rw . Onc slioiil(l. thrrcforo. plot thcwb data

its it hiiictiori of spin. H ~ ~ T v I ? ~ . as p~ i l l t ed 011t. 111 Section 1.3. t h e l spins of -4 - 130

is = - 112. That is. the spiris of the levels arr 2 n - 112. wliere r i is an integer.

Now consider the structure of "l%(l). As stated a t tlic beginning (.if this diapter.

the s t r ~ ~ c t u r e s of these two bands differ only b:: a proton in the negatiw-signature

[301]1/2 proton orbital. This orbital contributes -112 to the riiiclear arigular mo-

rnentuni. Thus the corresponding ' IgGd(l) and '%u( l ) levels obey the relation

That is. the spins of the levels in 148Eu(l) are 2n. Similarly, t,he structure of lJaGd(6)

' I t is highly unlikely that tliis is wrong.

Assumed Spin (Ti)

Figure -1.5: Staggeririg pat ttlrns foi. t l i ~ t h r w itirritic:al txiric!s ;is ;i f i i i i c b t ion of spiri. -4 cosine is also plotted.

differs from "%Gd(l) hy a neutron in the positive-signariirc. 1-1113 L/2 orlital. Thiis.

the spins of the 1.18Gd(~) levels ohe-

and so I(1.t"Gd(6)) = 2n - 1.

The staggering patterns of the three bands are plotted as a function of spin in

Figure 4.5. The spin assignnients of Ragnarsson [Rag93] for IigGd(l) have been used

instead of the generic 2n - 112; t his h a no effect on this resul t . Over the spin range

where data from al1 three data sets are available. t h cosine does a remarkablr job

of describing the data. At büth higher and lower freqiiencirs. the fit is (vrisidt~riihly

poorer: however, the results in Figure 4.5 suggest that the pira~rieters arc sirnilar in

the t tiree bands.

Tlic fit of this cosiric to ttic data permits a cu~istrairit of c;(iïiic of the niodel

4.4.2 A More Complete Description

These discussions have ignored the imaginaru part of tlir action. This is

definitel- a concern. however. sincc the imaginary part coritrols the amplitude of the

staggering (apart From a cuiistant factor). It is therefore interestiiig tu ask whr the r

one can learn angthing by lookirig at the amplitude.

In Appendix B. the suthor examines the iniaginary part of the action. and

its effect on the amplitude of the staggering. Semi-classical arguments are used to

derive an analytic expression for the staggering amplitude as a function of angular

moment uni. Tliis expression is

If the mode1 parameters of Hamainoto and Mottelson are the saine for al1 idenrical

bands (as the resiilts of tlir prtl~ioiis section siiggest) and if thes? pariinietcrs art'

spiii-independent (wtiich also srciiis to be t r u c ar least in the spin r;iiig(> plottid i r i

Figiirc 4.5). tlieii this <qwss io i i (:RH t r fitted to estract valiit's for t l i t l j)ilriirii(.ttbrs.

Tliis At lias t)cori ptlrforiiifvi o n tlitj data. ;\II data I)ct~vctbrl I = 37.3 : i~ id

1 = 61.5 has t)ec?ii iiic*liitlocl. This oriiits tliil four lowwt- ariti tno Iiiglicst-spiri poiiita iri

tlicl ! "Gd(1) data set. aiid r l i ~ lowst-spiri point iii tlitl 1 4 8 E ~ i ( 1). Thtir<> is iio ph~,si(~al

rpnsori to tasclude t hrse poiiits. h i r tlie data dv riot tbsiiibit r ~ g i i l a r st aggvririg i r i t l i i w

ri.gioris. aritl so forcc t hi. airipli t iido . I r o br t oo m a l l . Tticstl rrgioris iiro tlihc*iissivl

lat (Y i r i t his sectio~i.

Tlic fit took thcl fcwiii of i i grid scarcli ii i Ili. rvith iiri c x u ~ soltir io i i f o r . I ;it

t w h value of Io . .Ali valuils of 1,) froin 0.01 to 10.00. i r i s t ~ p s u t ' 0 . 0 1 . w r o r ~ s r i d .

For ciich. -4 mas dctcrrniiic.tl witli ;in mor-wtiighwd avcragv. It is appmlrit fr«rii tlio

graplis in this cliaptrr rhor t l i ~ mw- bars or1 t hc ataggcririg data ;irr too large t,r>

strongly constrain tlir paraiiic3tc1rs. That is. the rpduced k' values for tliescl fits Iverr

always rriiich smaller ttian unit!.. I-Iowever. much can still be learned from this fit.

Some of the best fits are shown in Figure 4.6. The analysis of the previoiis

section revealed that the data arc1 best fit b>. Io 2 6.75 mod 2 . This arialysis agrces

in some sense, but also shows ttiat only a sniall subset of the possiblc values of Io

adequately fit the data. Figure 4.6 shows the staggering data as filled circles. Each

panel also contains a solid curve. showing the best-fit ( A E , - 1~:') for Io (?qua1 to

0.75. 2.75, and 4.75. Both Io = 0.75 and la = 2.75 do an excellent job of fitting the

STAGGERING IN IDENTICA L BANDS

Assumed Spin (fi)

Figure 4.6: Good fits to the identical band staggering data. All t h e e panels show the data from Figure 4.5. The upper, middle, and !ower panels also show the best fit to these data with Io equal to 0.73. 2.75. and 4 . X 1 respectively.

STAGGERING IN IDELVTICAL B'4NDS

40 50 60 Assumed Spin (Ti)

Figiirc 4.7: A s i~ni lar set c ~ f graphs t u tliuscl i i i Figiirv 4.6. H t w . t l i ~ iippvr i i r i c l loiwr pancls sliow the best fits to the dara with Io cqual t» 6.75 aiid 8.75.

data. The Io = 4.75 fit is also acccpt,able. but t h t ~ clear decrime in airiplit,uclc as a

function of spin is opposite to the trend of the data. This is everi more truc of larger

Io values. Figure 4.7 shows exarnples for Io = 6.75 and 8.75. demonstrating that 4.75

is the largest value of Io which can replicate the data.

One is notv left with the inevitable question: what do these three possible

values mean? Table 4.3 gives the values of -4 and BI corresponding to the three

acceptable values of Io. Also given is the approximate height of the barrier tbrough

Sm GGERING IN IDENTICA L BAIVDS I I

Table 4.3: .4 and BI values for the three possible Io solutions of tt ici Harnamoto and >lottelson Hamiltoniaii. Also given is the height of the t~arrier separating the classically allowed regioris at I = 5Oh.

where the Coriolis terni has bccn neglected. The trrrris u p tu f rriay 1w foiinti in

Equation 2.8. and the last two are siiggested Ir>? Haniamoto aritl Mot tolson.

For superdeformed bands. h2/2,7 is a p p r o s i m a t ~ l y 7 Lx\*. I t lias becri sug-

gested [HaniS-l] that A shoiild he -*large" and positive cornparcd to r i 2 / ' 2 ~ JO that

the arigular rnomentum remains perpendicular to the 3-asis. since a small value of .4

would rriake rotation arourid the 3-axis energetically favoiirahle. contrary to what is

observed. The Io = 0.75 solution jwith .4 = 2.6 eV) does not satisfy this criterion. 111

other words, the single-particle part of the nuclear Hamiltonian must surely contain

terms proportional to 1: with coefficients much Larger than 2.6 eV.

STAGGERING IN IDENTICAL BANDS 9-

1 il

The solution Io = 2.75 is intriguing, since the corresponding .4 is approxi-

mately equal to h " 2 ~ . This rnakcs it sirnilar in size to the existing 1: terrn in the

particle-rotor model. The significance of this is not clear. I t should again be rioted

that this -4 does not satisf'y th^ requirement of Hamamoto and Mottelson that it be

"large".

rhere rs littie to sa? about t h e la = 4.75 soiutiori. lt swnis iiic'i)rri3c*t. in

tlic seiisc t h 1.6 hIeiV is a large prefactor for a supposedly perturbativc. atlditiciri

tci t lit. Haniiltoniari. How\-c~r. i i i w I,, reiiiaiiis as srriall as 0.5. t liis (wri(*rrri rriay bo

ll~lf'Oi11l~~t~d.

Bcfore this scctiori c.oric.liiclcs. tlw issiic o f data cx4usiori frorii t tw lits skioiilci

be rp-acidrcssed. Rclatiwly l i t t l e data has been csîluckd at liigli spiiis. aritl Fli-

11ot t r [ F l K ] lias cxprcssc~tl rcwrvat ioris a lmit t lie fits to t lic highost-oricrgy pwks.

As ;i rcsul t . t h escliisioii it i i iv iiot h so iiiiporr ;mr. 'The t?srlusioii ;kt l o n ~ r spins

is niorr of a conccrii. Sc\.rrd da ta points Iiaw I w n loft out of t iiil f i t h . ;in(l i t is

clcar t liat tlic staggcring effwt (Ioils riot rsist ;it t h t w rotnt ional f i q t i i w ~ i t l s ( s w Fig-

iirt. 4 . 3 ) . This mq- iridimto t hat m c or rnc~rc of t t i ~ parmirtors of t l i t ) I-I:uiiilt oriim

are spiri-clepciideiit. I t Ilas t ~ w i i iiorrtl [Fli93] ~ t i a t thcl alignrnrnr of t h .Y = (i pro-

roiis iit LI = 0.4 hleY occiirs r i c w the onset of the M = 4 bifurcation i n '"%i(l).

This could be an extrenicly important chie. altliough lictle has bren rriaric of it in the

theorctical literature,

4.4.3 Implications for Pavlichenkov

Pavlichenkov tias discussed (Pav97bl the significance of these results to his niodel.

This discussion is summarized below.

In Pavlichenkov's modei. one can determine she ther a band will eshibit stag-

gering by determining the sign of a quantity proportional to the proton and neutron

STAGGERING IN IDENTICAL BANDS 76

rnultipole nionients. Q14 ( r r ) and QA4 (v). If both Q's are positive. thr niicleiis exliibits

staggering: if botli are ricgative. the nucleus does not exhibit staggrring. I f thc two

have different signs. the11 the relative sizes of the terms rnatter.

For ""Gd(l), Q.,.\(a) < O and Q4&) > O (sec Figure 2.61. Pavliclienkov's

cal<:ulatioii iiidicatcs ttiat this t~aritl sliould riot staggpr. but uncertainties iri this

caiciiiatiori riiake trie predicrioii aeak. However. Pa~iichenkov pimirs i ~ u t that tlir

contributions of low-j orhitals (slich ils t h e (j = 112) [301]lj:! or th^ ( J = 3 / 2 )

[411]1/2) to the rriultipuh. riioiiicliit m i small. Tlierrforc.. i f ""Gd( 1 ) staggors. tlirii

o r i ~ sliould also c x p ~ c t i t h t w ~ ic1t)ritiral l~ands to stagger. siiiw t h r s ~ i~arids diffcr frorii

"'C;d( 1) oril? in thc occ*iip;itioii d low-1 orbitals. This issiio will I ) P rol.isit i d i r i t

r i m t d ~ p t er.

1 t is also iritclrclsi iiig r O riot icbc t lit1 frcqiiciicy rirprritltiriw o f () , , . I':i~li(-tiori1;ov's

c.;rl(biilat ions slioiv Q,, (h;tiigiiig sigii rirar 2 0.8 .\ lc\* [Pav97al. (-;iiiiiiig t l i i l oiisclt

of staggoririg. Sigri c - h i g c ~ s likc tliis could causc t l i ~ AI = 4 hifiirtwiuii to t~ogin

at~riiptly at 1 2 3 X . as ol)sc~rvc.d (it sliould bc riotcd tliat I = :1X (mr~spon t l s to

frcqiiericirs rlosc to Li; 2 0.4 !JcY).

The ohservatioii of iwrrchtcvi staggering patt3tms in tliesc id~~ntic;il baiitls

is interesting and provides a conipelling reasori to continue this aiial?.sis in other

pairs of identical bands. and in otlier siipertleforrned bands in gerierai. Th nest

chapter presents sucti ari analysis for the other superdefornied bands cibsc~rved in tliis

esperirnent. and in two other ~spe r imen t s performed with thc Ganiniasphrre array.

Chapter 5

Furt her St udies of Staggering

5.1 Staggering in 14 '~u (2 )

.-\~iid!~is of t hr GS-6-1 data rcwalcd ari rscitcd siipercleb~rriitd t~aiid i r i "%. This

band. called 14%~1(2). is assigried the l i g G d ( l ) configuration witli a M e in the

positive-signature si30 11 1 / 2 orhit al' (see Table 5.1 for 1 4 % ~ 1 band coiifigiirations).

In light of the results obtaineri for l JnEu( l ) . this band is an interesting case for study.

In particular. the levels in this band have odd spins. so the staggeririg pattern should

be the same as for 1'1aGd(6). if it fits into the existing pir ture. Pavliclienko~ would

also expect this band to stagger. since it differs structurally from 14'Gd(l) only bu a -- --

'Note that in this chapter the signature of an orbital will often tie iridicatcd by a subscript (+) or (-) following the Nilsson nurnbers, for positive and negative signatures, respectively.

FURTHER STUDIES O F STAGGERING 78

Table 5.1: Single-particlr corifigurations for superdeformed bands in lq8Eu. The con- figurations are given relative to 149Gd(l) . which is ~ 6 ~ ~ 7 ' and filled up to the proton and neutron shell gaps at particle nurnbers 64 and 83 (see Figure 4. I ) .

Barid Configuration relative to "%d( 1 ) Protoris Xeutrons

1 1 3 1 1 2 , - . . . L.L"~i(2) { [ O 1 , - . . .

lw- j M P .

Crifortiiniitrl!-, ;is t 110 s p t ~ ~ t rurii of t his barid iri t ho tippcr p;~riol of' Figure 5.1

iridicatcs, tlic statistiïs ori this h t i d arc low. This rrlakos i t ciiffi(-iilt t o cletwtnine

wliich pcaks art. coritaiiiiriatctl II>* 7 rays in t h e norrtial-<lc~fi)rrri(~(\ lcwl s<.lic~riio. ; \ n d

altliougti tlic oversubtr;ictrd po;k at 1 k r i s i t i t h i t his s p t ~ * r riini is

i~i~rita~iiiriatrtl. i t c:oiilil tiot Iw (lrtimiincvl frorii t tic) t ) i i ( .k~~i>ii~i( l spcvmt riiiii cviiitt tlir

soiirc.ta of t l io c*oiitariiiiiiitioii is. Tlicwforc~. riu r-orrwtiori t'or c~orititlriiriiitir~g lintls lias

bwii mule iri tlir staggoririg iirialysis of this band.

It should also Iw iiotcd that the background subtractiori for tliis case was not

perfornied witti the optlriitor k~rriialisrn. Instead a fraction of a sppirt riirri gated twice

oii the band and once oii a. set of background channels was iisrd. hl thoiigh this is no:

strictly correct. it is a decent approximation and reduces tlir statistical fluctuations

in the final spectrurn. rnaking it preferable for weak bands.

The low statistirs also produce huge error l~ars in the staggering plot. also

shown in Figure 5.1 (the 7-ray eriergies are given in Table 5.2) with the '""Gd(6) stag-

gering pattern overlaid as a comparison. Although there is no evidence for staggering

in this graph. the error bars are much too Large to make a definitive statement at this

time. The clear evidence in the staggering plot for a contaminated peak or a barid

FURTHER STUDIES OF ST.4GGERIiVG

1200 y-ray Energy (keV)

Figure 5.1: Upper panel: Partial 7-ray spectrum of ""~(2). Lower panel: staggering plot for this band (fiiled circles). and IJ8Gd(6) (open circles), for reference.

FURTHER STUDIES OF ST-AGGERING

Table 5.2: Ganinia-ray transition energies. in k e L of "%1(2).

Staggering in

5.2.1 Motivations

.As nientioned in the prtbvioiis <:li;ipter. the GS-64 espcrimcwt pro(hicw1 1.3 hillion

events populating ""Gd a r i t l '.''Gd. Bot h of t hese nuclci haw hrcn wll-st iidird:

l%d has six knowri superdeformed bands [The96]. and 1''8Gd has nine' [Byr98).

However. prior to this work. LI = 4 biwcat ion has only been studied in one of

these bands. l"Gd(1) [deh96]. .\s a result , this experiment represents an excellent,

opportiinity to do the first extensive study of the staggering in these bands.

The single-particle configurations assigned to the bands in 147q148 Gd are given

tOnly the first six are çtudied in this thesis. The last three were reported near the completion of this work, and are likely too weak to annlyze in this way with the GS-64 da ta set.

FURTHER STUDIES OF STAGGERING 81

Table 5.3: Single-particle configurations for superdefornied bands in l'"pl'l%Gd. The configurations are given relative to 14%d(l). which is n6'~7l and filled up to the proton and neutron shell gaps at particle numbers 64 and 85.

Band Configuration relative to L . 1 9 ~ d (1)

arialysis of Cliiipter -4 siiggws t liat holes iii tlio 130 11 1 / 2 < - , or [-Il 1 j l / 2 ( +, orbitals

producc no change in th(. H a n i a n i o ~ o and Slut tclson pararneters. Par lichen kov agrees

that liolrs in these orbitals should rieither produïe nor destroy staggering. I t would

be interesting, therefore. to further cssrnine the role of these orbitals witli respect to

the staggering phenornenon. Bands 4 and 3 in ""Gd provide such an opportunity.

wheti compared to '''Gd(l.2). Band 6 in l ' "Gd(~) is similarly interesting. if it could

be compared to the band with the [3Ol]l/?, - orbital filled. Honever. tliis would be a

band in ld8Tb, a nucleus in which no bands are knorn. The final example of th is type

is L4aGd(5), which has the :5'Dy(l) structure with two holes in the [301]1/2 orbitals

and two holes in the [411]1/2 orbitals.

FURTHER STUDIES OF STAGGERING 82

The other bands differ from each otlier and from '"%Gd(l) in the occupation

of = 6 and iV = 7 neutron orbitals. It would be interesting to discover whether any

of these orbitals have any kind of systematic effect on the staggering. Such results

could be very important in testing Pavlichenkov's model.

5.2.2 Bands Not Analyzed

Staggering is a subtlc perturbation of the î-ray eriergies of a siipcrtirforrncd hand.

TIiat oiic caii observe this p~r t i i rha t ion ar al1 is (lue to the sriioot liness of t lie triorrimm

uf iriertia of tliesc barids. Coriwrs(il>*. if thc rnoriitArit of inertiii of a siipi~r<lrforiiit.d

ierits of iritlrt ia. rl'liis

is diic to the crossing of thr ri<.gativc-signatiirc [ G X ] l/'? a n d [6-12:3/'2 Silssori l>r-

Iiitals [Tlir96]. a topic not disriisscd in this ivork. \ \Ir11 thi.; r*i-ossiiig ih r i o r I>lor*krvl.

thil riiorrierit of incrtia rliarigcs so ~iipidl\. t hnt i t 1s iinpossi l i l ~ t O t'strii(*t a s ~ l i a h l ~

staggoring plot froni ttic data. Tliis is tlir castl for 11"~l(2). ' "Gd( 1 ) . ' " 'G~ l (5 ) . and

li7Gd(6). In al1 other cases. thr crossing is t11ockt.d. the morriciit of inrrtiii is sniouthor

and the analysis cari be done.

5.2.3 Results

The y-ray energies of the siiperdeforrned bands in i'"*l"%d analyzed in this work

are given in Table 5.4 (erccpt for i L ' 8 G d ( ~ ) , the energies of which werr giwn in th^

previous chapter). The discussion will be broken up into sections according to the

experimental motivations presented in Section 5.2.1

X primary motivation is to examine the effect of the [301]1/2 and [Jll] 112

Nilsson orbitals. The first case in which this can be done consists of I4'Gd(l) and

FURTHER STUDIES OF STAGGERING

Table 5.4: G a m m a - r o transition energies of the superdeforrried bands in 1 4 i * 1 4 8 ~ d .

G d ) . ident ical b;mds wliicli ditfer in striirt iirv hy a iic.iit,roii iri t ,hr posit ivv-

sigriatiirc [-Ill] 112 orbital. A disciissioii of t l i t w (iatil is dtlfor.rcil t u Scctiori 5.3.

whcrr the results for (1 ttiirti idt~ritical I~aritl art1 pswntivl .

The secorid case is '""Gd(5). ari ideritiral I~arid t o "'Dy(1). -4s skiown in Fig-

ure 5 .2 . this band exhibits a regular sraggering pattern. with a stat istical significance

of 2.8 standard deviations lrorn the niil1 resiilt. Biintl 1 in '''Dy shows no e~ idence

of 1 1 = 4 bifurcation [Fli33]. in accordance with Padichrnkov's predictioris. How-

ever. in this model. bands differing only by particles in lon=-j orbitals should have

similar staggering behaviour: the differing results for "'Dy(1) and 148Gd(5) cannot

be reconciled with this model. The model of Ha~nanioto and Mottelson is difficult

Band 2 Band 3 Band 4 729.9(1) 856.6(2) 931.5(4)

778.34(7) 910.6(2) 988.3(2)

to test, since it makes few predictions. However. an idea developed in Chapter 4

Band 1 Band 3 Band -4 Barid a 897.87(3) 873.1 ( 2 ) 890.2(1) 899.7(1) 950.33(3) 924.9( 1) 938.6(1) 944,7(1)

2 . 1 966.3(2\ 1 0 4 1 . 9 3 1 0 3 . 1 9 . 1 988.411'i Wl .n ( 1) S77.46(6) 1022.1(3) 1096.3(2) 1 1035.39(1'j O . 1 1039.-4(1) 103S.2(1) ( 5 ) O . 1) 1 . ) 1 l l ( 1 lOgl.-l( 1 ) 1084.Y(l) 981.31(5) L137.3/2) l l O ( 4 ) 1 1 1 . 6 4 ) 3 . L l-l-l.-L(l) 1131.9(1)

1035.32(5) 1194.9(2) ( 2 ) 1 0 ( ) 1 . 2 13'>3.9(2) 1 1 4 6 . 2 ) 1 3 1 0 4 2 135-LÏ(-1)

4 ( ) 1LSG.II 1) 1195.0(1) 11?9.0(1) 1335.68(4) 1 1 ) 1 . ) 1226.S( 1 ) 1:3-44.OG(4) 1292.8121 1308.2(2) 1274.3(1)

120:3.-14(6) 1361.6(2) 1 - ( 4 ) 1 2 ( ) 1 - S 1 1:364.3(2) l;3??.?( 1) 1261 .37(6) 1423.3(3) 1 1 - 4 1 . O 2 - 1 . 2 2 ) 1369.3( 1) 1320.08(7) 1476.9i-4) I 1220.9'2(8) 1-i79.1(2) 1417.4(2) 1379.33(8) i lX0 .2 (1 ) lqX7.2(3 j 146-1.S(?)

1439.G(l) / LG39.?('2) Lv3!17.3(G) lX3.U(3) 1.300.2(2) I 1560.7(7)

-- 1609( 1 )

0.6 0.7 Rotational Frequency (MeV)

Figiirc 5.2: The cffcct of tlir [.Wl1l/:! arid (41 l 1 l / 2 Nilssoii orbitwls o n staggrring. S h o w is the staggering pattern of '""Cd(5). a I~antl ident i d t u li2D!.( 1 i.

t hat the modcl parameters arc ~inclianged 11'. t lit. [3O 11 1 /:! aiid (4 1 l ] 1 /:! orhit;ils. a

"constarit-parameter extension" of the mode! is more testable. It also fails the test.

since both 148Gd(5) and "'Dy(1) rnust have the same staggering plot i f both bands

have the same spins. as the configuration assignment predicts. Perhaps the constant-

parameter estensiori should only apply to the a[301]1/2(-; orbital and not to the

other signature.

Another area of interest is the effect of i'V = 7 neutron orbitals on 31 = 4

bifurcation. Figure 5.3 shows the staggering plot of 14'Gd(3). which differs from

FURTHER STUDIES Or" STAGGERING

Rotational Frequency (MeV)

l"%Gd(l) only in tlic occupation of t h e il i n t r ~ i d ~ r . Th plot s h o ~ clearly that

'"'Gd(3) does not st,agges. This r w d t is in accord i v i t h Pav1ic:hrrrkov's rriodrl. \\-itli-

out any :V = 7 orbitals occupied. the Q.14 moment of "'Gd(3) should be dominated

by the contribution froni its two .Y = 6 protons. Parlichcnkov shows rhis contribiition

to be negative, indicating t h a t '%d(3) should exhil i t rio staggcring.

Finally, there is the sole of Y = 6 neutron orbitals in t h c staggering phe-

nomenon. Figure 5.4 shows staggering plots for four superdeformed bands in '47j148Gd.

Al1 of these bands differ from l'19Gd(l) by one or two X = 6 neutron orbitals. and

FCJRTHER STUDIES OF STAGGERING

Rotational Frequency (MeV)

Figure 5.4: The effect of N = 6 neutron orbitals on staggering. -411 of the bands s h o w differ in structure from 14'Gd(l) by one or ttvo N = 6 neutrons.

FURTHER STUDIES O F ST.4GGERING 87

none of them exhibits statistically sipnificant AI = 4 bifurcation. Ttie absence of

staggering in '""djl) is in contradiction with t h e tentative result of [deA96]. These

four cases provide a ( someaha t coniplicated) test of thc Pavlichenkov niodel. The

[651]1/2(+) orbi ta l is the second occupied orbital in the 1 1 ~ ~ ~ ~ subshcll. Thc [64'2]5/2,+,

orbital is the s i s th occupirti orbital in the I r l . l ,2 subslirll. -4s sud i . bot l i st3rvo t o do-

trrciase QI.I at frequencitis above 0.5 Me\' anci holcs in tti~scl orbitiils stioiil(i irirmmc

VI4'. Orie tlius espects il11 of tlir bands in Figurrl 5.-1 t u ilstiit~it I I = 4 ~~ifui-catiuiiq

- (witli thil possiblr. cswpt ior i of ' %d13). wliidi also lias a IioIil i r i t titi i iritr~&r.

t i t G d ( 1 j ) Figiirr 5.4 dcriioiistrat<~s t hat nimo 01' t l i c w t>iiii<ls stiiggclr. in

coiitratlictioii with thc prcvlict ioiis.

5.3 Staggering in l - ' ' ~u

Two bands identical to %d( 1)

t iirrs t o "'!'Gd( 1) . but "%( 1)

" " 4 6 ) has a hole in tlic v[-Il i

Th is band is i ~ i '"'Eu and should bc at ari rstxitatioii c1ric>rgy. rdat . iw t u the

size to t ha t obtairied for '''Gd should be required to stiitl!- t,his band. 111 practical

terms, this means tha t the riucleus must be populated in a neutrons-only evaporation

channel. The only reaction employing stable beams and targets a l i ich can produce

1'!7Eu at a spin and escitatiuri energ' suitable for a stiidy of çuperdeforniation is a

beam of 23Na on a target of 130Te. The Slchlaster group proposed th i s c,perinicnt. - --

I I ~ is important for this argument that both orbi ta l~ produce the same effect. since t h e ~ becoine so heavily mixed in the superdeformed bands

(.lote that the conclusions drawn in this discussion are at variance with those drawli by Pavlichenkov in Reference [Pavg'ib].

FURTHER STUDIES O F ST.4 GGERINC: 88

and perforrned it in Jiily 1997 at the Lawrence Berkeley Natioiial Laborator? with the

Gammasphere array. For tliis experiment. named GS-118. Ganimasphere comprised

103 germanium detectois. Tlie beam energy was 145 Me\'. and two stacked 500

pg/crn2 foils of '''Te were iised as targets. each evaporated on a siniilar thickness of

goltl for support.

Ttie siriglp-partic10 coiifigiirations of the five obserwd baiicis arc g iwn in Ta-

ble 5.5 (the corifiguratiun for barid 5 is trntative). Tlie justiticatims for t h ~ w con-

figurations are summarized iri Rcfererice [HasSShj. .\ niirriber of t tiesc bards are

interesting cases for study. Bands 1 and 5 differ h m the ''l'Gd yrast band by a

hole i r i one of the [SOI] 1/2 orbitals. thus providing another opportunit? to study the

effect of these orbitals on the staggeririg. The other bands have different intruder

configurations and provide additional tests of Pmliclienkov's tnodel.

Most of these bands have pronounced irregularities in their moments of inertia.

Bands 2 and 3 show a dramatic increase in J ( ~ ) at high frequency, while band 4 shows

- L L-2

+ - + '. 1. '.- IG IL LG -

: + l ..- - - - Cs- -

I I I

FURTHER STUDIES O F ST.4 GGERING

Table 5.6: Gamma-ray transition energies. in ke\ '. for barids

(xtii\ptcr. and t1ict-c. is l i t tlv point iri goiiig iiito aiiy hir t ii(.i. i i (1 t ail ; i I )o \ l t t l i t . Siiiliii .r) of

t I I 1 1 1 t e S C of ' 1 1 1 ) Hwevcr. tlw nieilslirwicwt o l r hl sr aggoriiig in t iiis

baiid perrriits an intrrestirig wniparison. Tho i ippflr parid of Figurtl 3.6 s h o ~ v s ~ k i ~

farniliar staggrriiig patterns of ' "'C;tl(l). ' ' X E u ( 1 ) . ancl ' %i( lj 1 . i ) lo r t i v l ;is ii f i i n i s -

tion of angu1a.r munieritum. as was illiistrated i r i Figure 4.5. 'Ttir) I ; i t t ( ~ i . t ~ v o 11;inds

arc rrlated striicturally to tlie first by haies in the proton [301]1/2, - , m d neutron

[-Il 11 112, + orbitals. respect ivrly. The corrrlatiori hecawn t h r hrer staggering pat-

terns is remarkable. and inspires furthcr investigatiori into the rolr of these orbitals

as regards the staggering plienornenon.

The lower panel of Figure 5.6 is the analogous plot for '-"Gd( 1 ) anci its two

identical bands, 147Eu( l ) and ldiGd(-k). Each band differs from one in the uppw

panel only by a neutron iri the [651]1/2 Nilsson orbital (the pairs are plotted with

the same symbols). It is clear that none of the bands in the lower panel exhibit any

FURTHER STUDIES OF ST.4GGERING 9 1

Rotational Frequency (MeV)

F i g i i r ~ 5 . k Staggrbring plots for bands in "'Eii.

AI = 4 bifurcation. This fi gurr dernonstrates tno vcry irnportaiit roqiiirrriirnts of

a rriodel attcmptiiig to explain the plienomenon of AI = 4 bifurcation. The [riodel

niust explain

1. why the 1~[30ljl/2(-, and v[411]1/2~+l holes preserve the staggering ef fec t in

the niiclei depicted in the iipper panel. arid

2. why so many orbitais, including the v[651]1/2~+, orbital illustrated here. com-

pletely destroy the effect .

Assumed Spin (fi)

Stiiggeriiig plots ;is ii furictioii of spiri. In t h upprr pud. thc circles. upward-poiiiting tnianglrs. iirld dotvriwartl pointirig triarigles reprtwnt t. tir (lata for the well-kriowri cases '"%d( 1 ) . "%i( L 1. and l " r G d ( ~ ) . r i yxc t iv~ ly . In t h r l o w r panel arc tlic analogous cases. lacking the positive-signat use [G 11 1 / 2 orbit al. These are "%d(l). "'Eu(1). and '"'Gd(4). respectively.

Neither of the rnodels described in this thesis are up to this challeiige. The failure of

Pavlichenkov's mode1 for N = 6 neutrons has already been explored in Section 5 . 2 . 3 .

Hamamoto and Mottelson's niodel fares even worse. The lack of a physical basis for

their mode1 gives them no predictive power regarding an' orbitals.

FURTHER STUDIES O F STAGGERING

5.4 Staggering in i 5 o . i ~ ~ d

'Originally labelled bands 3 and 4 in Reference [Bea93]. The notation used in ttiis thesis is that of (Ert981. th this thesis. only ten are studied in '"'Gd and four in '"'Gd. The others are either extreniely

weak or without a configuration assigriment.

FURTWER STUDIES OF STAGGERlh'C 94

Table 5.7: Single-particle configurations for superdeforrned bands in Ij0.15'C;d. The configurations are given relative to '%d(l) , which is T G ' Y ~ ~ and filleci up to the pro- ton and neutron sheil gaps at particle numbers 64 and 85. The signature assignnicnts of '50Gd(6a.~b) are not certain.

Band Corifiguration relative to 14%d( 1) Pro turis X P U t rorls

' " O ~ d ( 1 ) . . . {'h } 15'Gd(2) ( 8 : I } { ~ . L } {[30l]l /?} -' {7?} 15"Gd(3) {fi:{} {[301]1/2( - ) } - ' { G ) l"OG<l(-l; i) . . . {[40215/2; - ) }

1ioGd(4b) . . . ( [-102!2/2, + , } ljUG<l(r>) . . . { G 1 '"'Gd(Gn) . . . {ix W'?( - ) } 1 5 0 ~ < i ( ( i t ) ) . . . ( [ 2 m / 2 , . . -il l"OGci(Sa) {(il} { [301] 1 / 2 ( - i } - ' { \ -102~5/2 , -., ]

"'Gd( 1). Thr orlicr b;iritis i n '"''Cd Iiaw a i iuml~cr o f diKmwit priit ori ;rrid ricliit ron

iritrutier confi giirations. whicli w u l t i pnividc tests of Pa\.lic.lit'nki>\'s pi.idi<+tions. In

'"Gd. al1 of the bands tiaw the T G ~ U ; ~ intruder configiiratiori. wi th tlic 87th npiitron

occupying one of several strongly-coupled orbitals. Thus. al1 of tliesc harids arc ver!.

sirnilar to 150Cd( 1). and niight provide interesting compnrisons.

C'nfortunateiy. iiot al1 of the bands in these data cniild bp an;ilyzed in this

investigation. Bands 3. -la. and 8b in l5'C;d have large irregularitics in their nioments

of inertia which prevent the calculation of a smooth reference. Band 5 in l4j0Gd. and

band 3b in 'jlGd are too contaminated by peaks from the normal-deformed level

scheme to permit this kind of analysis.

FURTHER STUDIES OF STAGGERING 9 5

Table 5.8: Ganiriia-rw transition energies. in ke\'. for bands in lXGd.

Band 1 Band 2 Band 4b Band 6a Band Gb Band Sa 849.50(2) O ( ) Y60.77(6) 8 9 801.26(8) S75.3(2) 888.26(2) 1097.90(8) 91 l.'L3(6) 930.5(1) 835.83(8) S'Z-I.S('2 j 929.23(2) 11-15.84(8) 963.20(7) 983.2(1) 906.12(7) 974.8(2) 971.26(2) O 1 - 4 9 1 0 1 . 5 1 1034.7 11 ! K . O J 121 lOX.7( 2.1

10L3.78(2) 125 1. lW) 1068.76(8) 1055.0(3) 1008.61 (8) 1077.3(2) 1oaû.78(3) 1301.4q 101 1 i 4 . : ~ O C > I . ~ G ( S I i 1 2 9 . w 1

1100.35(3) 3 . ( 1 0 ) 1 L1i .3 l ( l l ) 1 l!I5.8(2) 11 l - l . ( ~ - l ( 1 1 ~ 1 lSZA(2) 1 4 4 ( l O l ( 1 ) 1 3 7 0 0 ) 1 1 0 ) 11G8.Olj(%O) 1236.-1(2) 1 l9O.ÏG(3) 1-450.34(L7) ( 1 1 1306.5(2) l ? E . 6 l ( l l ) 1290.72 1 1235.09(3) 1-198.Y9(2SSI ( l ) 1363.0(2) l?X.3!J( 13) 13-13.-1(2) ( 3 ) 1548.:36(-13) 1-4Ol.OO( 14) l-ll!l. l ( 2 ) 13;3:Je91( 17) 1-!00.3(3) ~ :~ :~G.sG(- I ) ~ ~ ï , r . i , ç ( 1s 1 1 . o . 1-.~:,fi.:3(.~) 1388.10(5) llj14. 14(25) 1 3 . ( 4 ) 144G.90(75) 1440.31 (7) 136(3.66(49) 1 9 7 ( 1 0 1503.52(47)

l-iW.07( 10) 133).27(93) 1547.-F(2-L)

The first arw of intcrcst is the bands i r i IioCd ttiat arc identi(.;il to '"'Gcl(1).

Figure 5.7 shows the staggering patterns of these bands. Neittier baiid -4b nor barid Ga

cxhibit A I = 4 bifurcation. This implies that the constant-paraineter estension to the

mode1 of Hamamoto and ' r lo t t th i i does not apply to the [402]5/2,,, or (521]3/7,.+,

orbitals. The staggering results disagree with the predictions of Pavlichenkov. The

neutrons that have been added to the lq9Gd(l) system in these two bands arc iri

small-j (dJ12 and fTI2) orbitals. -4s a result. they should have litt'le cffect on the

staggering and one would expect that both of these bands should stagger.

Rotational Frequency (MeV)

Figure 5.7: Staggering plots for bands in 150Gd identical to 14'Gd(l).

FURTHER STUDIES OF ST.4 GGERING 91

Table 5.9: Gamma-ray transition energies. in keV. for bands in '"'Gd.

Thc resiilts for band 6b arc also interesting as a test of Padichrnkov's rrio(lr1.

If this band is a bonafide exaniple of a staggering band, then it is intrigiiing to ask why

the [521]3/2(-) orbital shoiild preserve LI = 4 bifurcation. but the positive-signature

orbital should destroy it ( c f . band 64. hccording to Pavlictienkov's model. tlicre

is no distinction between these two orbitals, so a different rwi l t . for the two cases is

troubling.

Figure 5.8 shows the staggering plots for 15%Gd(l) and "'Gd(la.lb.3a). ;\II

FURTHER STUDIES OF ST.4 GGERIXG

0.5 0.6 0.7 Rotational Frequency (MeV)

Figure 5.8: Staggering plots for bands identical to 150Gd(l).

FURTHER STUDIES OF ST4GGERING Y9

of these bands are built on ri6'vï" iritruder configurations. as discussed abovi.. Tlicir

structure is. t hrrefore. sufficieiitly different from " % G d ( 1) for the mode1 of Hamarrioto

and Mottelson to have no prcdictiw p o w r . Honrver. Pavlichenkov's p rd i c t i on for

this iritriider configuration is dcfiiiite. The niornent Q14 is negative for both the proton

iiriti nciitrori subsystenis. and tliiis no staggering is expected.

Fiiially. F i g u r ~ 5.9 slioas spwral bands wi t h niorc csot ic s t ru r t i i r c s Batid 2

iri '%d lias a ;r6"u'i2 corif gwat i i m and is idmt iral t o 1r12Dy( 1 ) . Tliat i liis haricl d w s

not stagger is in accord witli Pavliclirril<w's prrdict ion for tliis i r ~ t riitlor rwiifigiirat i o n .

I t caii also be recoiiciled ni t l i the roristaiit-parariicttlr estorisiori of t tw Hiiniamoto i i r i< i

Slottclson model: sirice iieither band staggers. one only has to assurnc t liai B I = 0 for

these bards. Hiiwever. one should remerriber the results of Section 5.2.3 for ld8Gd(5).

a band ideritical to '"Dy(1) that does stagger. Th results for thesc t h r w identical

bands caniiot be understood togetki~r i r i eitlier rnodel.

Band 8a in I5OGd does not stagger, either. Since this band has a 7r6"vi1

intruder configuration. it is unlike any of the bands studied so far. The model of

Mamamoto and Mottelson is. therefore. not useful here: Pavlichenkov's model is also

FURTHER STUDIES OF STA G'GERING

0.5 0.6 0.7 Rotational Frequency (MeV)

1 - . .

Figure 5.9: St,aggoririg plots for nioit3 I~arids iri "'"Gd.

anihiguous for ttiis casr .

5.5 Stat ist ical Considerations

In Chapter 4. the statistiîai significances of the staggering patterns in "%d(i j.

1 . ' 8 E ~ ( l ) , and '"Gd(6) were discussed. Procedures were deceloped for caiciilating

the mean staggering S, its signifiçance Y, and the probability P that a measurement

of a perfectly regular band would produce a mean staggering of JXJ or larger as a

FURTHER STUDIES OF STAGGERING

result of statistical fluctuations. I t i ~ a s shown that the sigiiificance of cach o f the

patterns in Chapter 4 is two to tliree standard deviatioris frorn the nul1 result. and

that the probability that sucti patterns would arise tlirough chance is no largrr tlian

about two percent. I t was arguetl that the staggering patterns in tlicsc. bands are

rion-statisticai in nature. Honever. ~ i o ~ tliat staggcririg pat toms liavc. tio(w rlstractcd

froni tweiity-two bands. it is tvortli looking ;tt t h proptlrtiw of t lit. 1 - (list sibiitioii.

Tlic riiust sigiiific-ant i'~ii~iil,ltls of r htl st;iggiiririg ~ f f w t ;irrl. riot siirprisingly.

tli(~sit tliscbiissd i r i C h p t o r 4 ; i r i t l ""Gd(5 i . T~ikfm iridividuiill>.. t lir staggoriiig plots

for ttitw bards riiakv tlio rffoct look ver? sigriificaiit. Tlic ligiirc i i l~o shows tliat

tliew clxamplcs starid rorisià~rably tiigher t hari the Chussiari tiist ribiitioii. Honcver.

the area under t l i ~ Gaiissiari ri irw is non-negligil~l~ nt t,htw values O F 1 -. and i t is

instructive to ask how manu cases with this level of sigriificarice rriight bo cspected

froni a random sample of twenty-two bands.

The chance that the measiirement of a non-staggeriiip Iland will yicld stag-

gering with a sigiiificarice greater than 2.3 is 2.1%. Thus. for a saniple of twenty-two

bands. 0.46 cases of this magnitude are expected. This rnakes the four observed cases

look very significant; using a Poisson distribution with a mean of 0.46. the proba-

FURTHER STUDIES OF STAGGERING 1 O2

Table 5.10: Mean staggeriiig ).YI and the staggering significance 1.' for the bands studied in this work. Also given is the probability P as defined in Chapter 1.

Band Xlean Staggering 1x1 (keiT) Significarice Y Probability P 147Eu(1) 0.074(78) 0.95 i3 4 Y?

bility of observing four or niow cxscs is 0.2%. Howewr. it niust bc notcd that ttirl

probability of observing at leasc one case is 3776, and therefore it cannot be rulcd out

that one of these cases is only a statistical fluctuation!

Slightly less significant are those th suggestive" cases noted enrlicr in tliis cliap-

ter. lS0Gd(6b) and 15'Gd(la). If these are bonu fide exaniples of staggering. the?

SThe original "9Gd(l) example is almost certainly not a fluctuation. The significance for this band quoted in this chapter is based on the results published by Flibotte et al.. However, the independent confirmation of this result in an earlier experiment by the 8n group. and in a later experiment at Eurogam II makes it much niore significant.

FURTHER STlJDIES O F STAGGERING 103

1 .O 2 .O Significance Y

ivoiild be interesting for ttw r t w o n s disrusseci rarlier in this chapter. H o w r v ~ r . at

this level of sigriificancc. t t ~ y rrriiaiii at the thresholci of twli~vabilit~y.

The probability that, a rneasiirement of a regular band yields staggrring witli

significance greater than 1.8 and sinaller than 2.3 ( to separate this discussion froni

the previous orle) is 5.0%. Thcrefore. a sample of twenty-two barids stioiild yield 1.1

exaniples of tliis kind. The obserwd yield of two bands is riot significant in this light.

and in tact the probability of drawing two or more cases from a Poisson distribution

with a mean of 1.1 is 30.4%. Thus. it is not proven that either of these bands exhibits

FURTHER STUDIES OF STA GGERING

LI = 4 bifurcation.

Before leaving tliis discussion. it is worth making sorne observations about the

distribution as a rvliolc. The rnean of the squares of the significances is the reduced y'

statistic for this distribution ( untier the assumption that no bands eshibi t staggering).

Tlie value of this statistic is 1 .;S. For a saniple wit ti twenty-two deg re~s of freedom.

ttiis is an iiiisrasond~ly largc valiio of i" the prohability of swing a k 2 this large.

is appnxiinately 1 - 2% . This is ;in iridic-at ioii t tiat t l i ~ assiiiiird distribiitioii (nu

st agp!riug) is i ~ i i i p p r o p ~ i i ~ ~ t ) .

Srcorid. t h niav iridicat c. a systcriiatic bias ori t h part ut' t l i v qwririioiitor.

The arialyses of sei.~riil hancis wrrc ahortcd brcauso it sccnicd clcar diiriiig thil analysis

that the band was hriiig ci-ossd os coritarniriated in sunic way t h could not be

corrected. Pertiaps these deviations are genuine statistical fluctuations. and t heir

exclusion froni t his distribution biases the sample toward small significances. This

would result in a smallcr valiir of i'. as observed.

Third. this may indicate that it is inappropriate to esclude al1 four of the

staggering bands from the distribution. That is. it should be conceded that one of

these cases is nothing more than a fluctuation. Adding. for example. the 14'Eu(l)

Chapter 6

Conclusions

6.1 il I = 4 Bifurcation in Identical Bands

Stiiggcriiig p l t tvriis (*;III 1)o stwi i r i t w of t I io itiwticiil h i i l s o t ~ s ~ r ~ ~ i ~ i i r i ttiis

rsp~rir i i rnt . %ii( 1) a n d '"C;d(Ci i . Givrn thcl rrkitiw rarity o f 1 I = 4 I)ifiircat ioii in

superdeforrricd bands (ti(mioristsated 111 this n o r k ) . t h occurrerire of t tiis plicmonitqon

in thsee identical bands is niust ilnusual. E w n rriorc striking is t titi graph of staggrring

versus arigular moment i i r i i for t t i i w thrre baritls. stiowri iri Figiiro 4.;. T tiis grapli

shows a rcmarkable rorrcllation bet~vwri the s t a g g ~ r i n g patterns i r i rhtl threc bands.

and raises a number of interestirig issues. First. it suggests that there is some vaiidity

to the mode1 of Hanianioto and Mottelson m ha mg.^]. nhich clairns that the staggering

in the superdeformed bands is proportional to c o s ( ? ( I - I o ) ) . 'iloreover. it suggests

that the parameters -4 and Bi that characterizc this mode1 are the sanie in al1 three

identical bands, a totally unexpected finding. This vindication of Hamamoto and

blottelson's mode1 is even niore surprising since there is no known physical origiri for

CONCL USIONS

their Hamiltonian.

An effort Lias beeri riiade tu e ~ d u a t e the full staggering amplitude (iiot just the

oscillatory cosine terni) scriii-classically for the Haniiltonian put lorward by Hamanioto

and hlottelson. The rcsults of ttiis w r k . sti~nrriarized in Appendis B. have been iised

to coristrain the mode1 para~iictei-s -4 aiid BI by fitting the observetl staggrririg par-

terris in the tliree ideriticsa1 \)aritls t,o tlicl espression drrivcd iri tlw A p p w t l i s . Tho

possihlc values of tlir. pariiiiiotors ;irt1 giwii in Table 4.3: it is iiot clear wliic4i of thcl

t,lirw c~lioi(*cs is corrcvst. ( l (~sp i t (~ i l i ~ f i u . ~ t h 0iii:Ii optioii is t i i f f c v r i t h i n i th(> iithcrs

t)y s ~ w r a l orders of rr i ; ip , i i i t i i ( l ib . I t is cspcxmtt~d that tlicsc! pararrictcrs wil l put s r w r r

rcstrictions on future clioicw hr r tic. pliysical origiii of tliis Haniiltoiiiari.

tlirr (31-irirnce of correliitotl staggoririg pat twns iri iiloiitical I);iri(ls. Of t l i ~ or her 11;iriils

idrrit i c d to 1'1%~1( 1) (tiarriol!.. ' '' Eiii 2 ) and '"OC;di - h . ~ a . ~ b ) I . rioric. ixliil~it staggoririg

patt,criis witti an aniplitiidt. or Irwl of significancr approacliing that of t l i ~ f rs t thrrc

cases. ancl none fit on the ciirvr of Figure 4.5. For ttir '"'Gtl(5) -"OGcil2) idcritical

band pair. the former eshibits signiticarit staggering whilr the latter does not. This

is a damning result for two bands which should have the same spins. The I5OGd(l)

famil- (also including 151 Gd(1a. 1 b.3a) ) are riiostly rion-staggering. althoiigh the rrsult

for lSIGd(ia) is ambiguous. Finally. the triplet ('. '%d(l). L " E ~ ( ~ ) . 1%d(4)) are al1

consistent with zero. as s h o w in Figure 5.6. Thiis. the only correlation observed in

identical band staggering patterns beyond the first three cases is in families where

none of the members exhibit - I I = 4 bifurcation. a rather weak result.

CONCL USIONS

6.2 Systematic Survey of AI = 4 Bifurcation

The search for staggering in bands identical to 14YGd(1) has perrnitted ii niiich widcr

investigation of AI = 4 bifurcatiori. In the course of this work. staggering plots

have been produced for twrity-one superdeformerl bands in ' Li.''18 EU m d 14i-151 Gd .

Howéccr. only threc of t l i w ! twcrity-one bands eshibit staggeriiig wi th statisticnl

sigiiificancc grwter th;iri 2.3.

This s w w y ~ i r i ( i d i ~ i ( ~ ~ (~suvial c p w im at)o~it the stagpsirig ~ ~ l l o r l o r ~ ~ ( ~ i ~ o r l :

what is it about '''%ci( 1 ) tliat. hriiigs illioiit 1 I = -4 I~ifiirciuiori? OF t l i v t'uiir kiiowii

stiiggeritig I~arids. oric is "''G(l\ l ) . ariti t\vu ot lirrï ;rra i(iontic.;il r O t liat Iiiiii(1. Tliis

srcriis to iridicatr that '"!'G(i( i l i i i ~ il vrlry ptlciiliiir pi.opcq-t!- t li;it hriiigs i i h ~ i i t t lit>

c‘ff~c-t . altliotigli it. is i i i ik r i~u~i l ~ \ . l i i i t r liis proport\. is. ( I f t*o~irstl. ariv t ~ ~ l ) i ; ~ ~ l i ~ t ioti

sliotiltl also t q d a i n th^ staggcv-iiig in ' " G d ( 5 ) . a Imri t l witli a w r y tliti~rriit strii(.tiiril.

6.2.1 Tests o f t h e Models

The model of Harrianioto and llottclson is not a goud physical riidcl. s i n w

it riiakes very few predictioris. Ccrtaiiily. i t niakes rio predictioris abuiit what t lir

parameters of the Hamiltoniari siiould be for various bands. or evcn which bands

shoiild eshibit the effect. Howewr. as disciissed above. the results for the 149Gd(l)-

like bands observed in GS-64 are encouraging for proponents of tliis model. since

the staggering patterns seeni to lie along a common cosiiie curve. This suggests not

only that the staggeriiig is described by a cosine. but that the parameters of the

Hamiltonian are the saine for the three identical bands. This result has spawned

CO N C L USIOlVS L 09

the so-called "constant-paranieter esterision" of the niodel. in wliich it is assumed

that the parameters of the Harniltoniari are the sanie for pairs of identical bands.

This extension of the niode1 is testable. However. the lack of correlatiori between

stnggering patterns in al1 of t h at lier idcntical bands (discussed earlier ) iridicatrs

tliat tliis mode1 is insiifficimt to ciescrit~t. this phenornenon.

Table 6.1 siininiarizt.b t licl prodirt ioiis of t l i v P;i~li(~Iivriko~ i i i o d d t'or t lio l);iiids

stucliecl in this thesis. The I~ands arc urgi~niz~(1 acwrdirig t.o thvir i r i t riidcr iuitigii-

ratioris. the dominant factor i r i ~vhctiier a h i d will staggpr. Tlir a6'v7' bands art1

furtticr brokeri dowri i n t o two siihcts. Thcl first s r~t~sct i~icliidcs bztrids whicti ciiffer

froni '"%d(l) by one or riiorc holcs in .Y = 6 iirutrori orbital?;. Thclse bands shodd

stagger. by the arguments given in Section 5.2.3. The second siibset differs from

ih'SGd(t) only by holes in low-j orbitals. These bands should also stagger. hecause

low-j particles have little effect on A I = 4 bifurcation. iri Pavlichenkov's niodel. The

predictions in the table use the generous assumption that Pavlichenkov's calculations

can be fixed up to produce staggering in ldgGd(l) and no staggcririg in 152Dy(1) .

another ambiguous case.

Table 6.1: -1 summary of the tests of Pavliclienkoï's mode1 niade in this work. C' ~1ven are Pavliclienkov's predictions (as "stag." or "no" for a prediction of staggering or rio staggering, respectivelu). the actual results for these bands (using the samp shorthand). and whether these rpsults agree with the prediçtions (as "!.es" or "no" for agreement or no agreement. r e s p ~ c t k l y ). The only bands omitted froiri this table are 14'Gd(3) and lSoGd(8a) (for wliich the predictions are not clear). and 150Cd(6b) and lSIGd(la) (for which the csperiniental cvidenw is not rlcar).

Coiifigurat iori (stag./rio) IO ( y s , h o ) , Ï - G * U Ï ~ !.17E~i(:3) ~ t i q . 110 110

Table 6.1 also sliows whether staggering was observcd in these bands. and

whether the theoretical prediction agrees with the experirnental rcsult. I t is appar-

ent that this theory has as man- failures as it has successes. This indicates that

Pavlichenkov haç not correctly identiîied the origin of A I = 4 bifurcation. or that his

calculations of the QJ4 moments are not accurate enough to make these predictions.

CONCL USIONS

6.2.2 Statistical Significance

Yoiv tthat so many bands have been nieasured. it is alsu possible to conduct a statistical

esarriiriation of the staggering cfkct. Scctiori 4.3.3 describeci a method by which the

statistical sigiiificance Y of the st,aggering in a giww \>and rnay be evaluated. The

siçiiificarices for al1 of thc barids stiitlied iri tliis w r k Iiiivr bcen calciilatrd and appear

in Table 5.10: a liistograiii of this tlistribiitiori can bc. sceri i r i Figure 5.10.

The biiiitls G d 1). "'Eii( 1). "C;d(B). ~ i r i t l l '"C(5) al1 liaw staggi~riiig

par t(.riis with I > 2.3. TIicl ~ t ; i t i s h x l aiiaiysis priwiito(l iri tliis tlicsis lias slion-ri

tli;it rnost, if riot all, of t i i c w c1asc3s arc! riot thcl rihsiilt of s t iitisticA tlilcbt iiat ions. Tho

probal~ility of obsr r~i i ig four staggcriiig pattrriis w i t h t h . ; l i v l 01' sig~iifirari(~c~ iii ;i

sariipltl of trvcrity-two I)aiids is i).?C/i. arid t h c l protiiibility i d i hwrviiig rio sr aggvririg

pattrmis tliis sigriifiraiit, iri tliis wiiiplo is 63%. This . i t ii: likoly tliat al1 four of tli~scl

barids arc1 borl(i fidr rs;iiiipltis uf A I = 1 bifiircatiori.

Tlic sariic cariiiot br said of t,lic stagg~ritig p;ittcbriis i11 ' i U C ~ I ( ~ I ) ) i t ~ i c l lilGd( l a ) .

Tlirsc patterns liavc sigiiitii:ancrs of 1.8. Ln t,his snnipltl of twrity-two biinils. on(> cs-

pects 1.1 i~arids witli staggrlririg patt(wis iv i t l i 1 .Y < 1' < 2 .3 pimly by diaiiro. and

tlic probability of obserl-irig twu or rnore bands at tliis lrwl of sigriifirancr is 37%.

These statistics iridicattl ttiat rieittwr of thcsc bands sht~uld hc assi~nierl to eshibit

A I = 4 bifurcation a i t hout an independent wrihcat ion froni a second esperinient.

The distribution of eighteen non-staggering bands (not including the four

cases witli 1' > 2 .3) is reasonably well described by the assumption that none of

these bands staggers. III fact. the y' statistic for this distribution seenis too srnall.

This niay be a chance occurrence: it rnay also indicale a systematic bias on the part

of the experimenter, or that one of the four staggering bands is really nothing more

than a statistical fluctuation. The choice between these options cannot be made now

CONCL USIONS

on the b a i s of the facts at hand.

6.3 Future Prospects

This thesis lias certainly cstended the c u r e n t knowledgc of AI = 4 bifurcation in

thc -4 -+ 150 region of su!)t~r<i~foriiiation. Howevrr. t1ic.r~ is consiclcral~lr work v ~ t tn

do.

It woiild also probably t ~ r wurttin-hile to rorifirrri o r i.(~futc soriio d thr rtlsiilts

prescnted iii ttiis ths is . 111 tlir prcceciing sectioii. it was iiotrd t t i i i t i t is possiblr

that one of the four best esaniples of staggering pr~srnt i ld iri tliis thrsis r r i q hc a

çtatistical fluctuation. I t is also vcry likely that one or botli of thr Y = 1.9 cases

is a fluctuation. A re-nieasurenierit of the staggering patterris in ttiese bands would

almost certainly settle tliese issues. For example. bp merel? repeating any of the

experiments described in this thesis. the statistical error bars on the st aggering plots

could be reduced by a factor of a, thus changing a 1' = 1.8 result into a Y = 2.5

result, if the mean staggering .Y remains constant.

CONCL USIONS 113

Firially, it is certain that Inore theoretical nork or1 ttiis ptirnomc~noii is rc-

qiiired. This thesis has denionstrated that neither of the approacbes wns i< l e rd in

this ivork is capable of niaking accurate predictions of wherr the staggering will occur.

or weii of convincingly denioiistrating what the origin of tlic phenonienon is. It is also

fair to say tha t none of tlir iriaiiv other proposed tisplanatioiis tias gcwratcd niucli

iriterest. although tha t does not riecessarily mean tha t thosc esplaiiations are without

riiclrit. Tlic large qiiaiitity of da t a presentrd in tliis t h i s s1ioii1(1 pro\.itiv ;miplr~ trsrs

for tlicoretical rriocirls. ; i i i t l t h i l rirrio is iIt+iiitol!, iipv 6 ~ . mi ry)l;iiiatiori o f A I = -4

bifiirîatiori wliicli cari c'~piiiiii tlio par t c w i of ocBriirwiico ol)soi*\.(vl i i i t l i i h n.tisk.

Appendix A

Oscillation of the Tunneling Amplitude

At this tirne! the notation is also chariged slightly. with I3 + h-. and BI + B. This

will elirninate the need for many subscripts and should make the following easier to

read.

I l 4

OSCILLATION OF THE TLNiVELIiVG A M P L I T U D E 115

In tliese new cco-ordinates and with these substitutions. tlic Ha:iiiltciriian he-

This clearly has four niininia in t h Ii = O plane. at Q = T/-1. 3;7/4. 3;7/4. alid 7 7 ~ / 4 .

as discosscd in Cliapter 2 . At t hcs r rriininia. the enrrgy is eqiial to zcro.

The qiiantity of iiit(wst is the tiirineling iimplit udc.. t . In t lw \ \ ' I d 3 approsi-

mat ion. t his is giwn I>y

wiiclrcl S is thc action foi. travcllitig froni one mininiiirii to ; i r i i d i t l i . . For i ~ i s t ; t ~ i w . 5'

iv\.litlri. Ii(0) is givcri hy t h soliir.ioii of H = O. Th t . is.

Tliere are two choires to riiakr Iirrcl. The first is a choici) hrtwwn positiw niid rivgativc

K . and is arbitrary. Both (:kioi<w &scribe equivaletit paths b r t w r r i t l i ~ mininia of

the Haniiltonian. The secorid clioice is important. but relat ivelu simplr. Consider

the limit cos(-@) -+ O. In this limit Ji' must also approach zero. since cos(20) + O

describes the minima of the Hamiltonian. However. only the positive root yields

h' = O. Therefore. for tliis discussiori the followirig solution will be used:

OSCILLA TIQN OF THE TUNNELING AVlPLITUDE 116

Qualitatively, this systeni has two types of behaviour. depending on the value

of 1. Consider frst the case 1" 4 4 B . In this regime. the quantity in the inrierniost

square root. tha t is A' - 4.4B12 cos2(2d), is always positive. In turn. the argument of

the outermost square mot is aIways ncgative. and so K is aiway piirely iiiiaginary.

'Thus. the action is purrly iriiagiriary. and the tunnelirig amplitude is a rra1 i~poncnt i ; i l

wi th no oscillatory tieliarwiir.

Conrersely. if I 2 > .-I/-IB. tlicti thc arguriirnt of rhc iririt.rriiost s i i i i i r o root

i~la;i!.s twcoiries negativii for si1111~ O valiles bc t i~wr i t tie riii1iiiii;l. T l i i ih . t t i o i i i .il t i i i l r i t

O F t tir oiitcrrtiost scp1ai.c. mot is wniplos. ancl so A' is cimiplos. T!ius. I lw w t ioii is

(-oriiplcs. and thc tiiiiridiiig iiiriplit i i d ~ obtiiiiis a Fact»r of cos(%( S) 1 r v i i i d i i i i ; iy (.tiiiiigti

sigti as ;l funct ion of spiri. I i i t l i ~ rclst of t his appc.ii(lis, R(S) ri-il1 Iw itvtoriiiiiitv l. ; i r i t l

t h s t tic^ oscil1;tti)ïv 11at us(> of' / .

T h act icxi i i i t cyy-;il is

Now. factor out and cal1 I J ~ = (note that the abo\!e condition on 1'

makes 112). Tliis lcads ro

For II, < sec-' ( - fiO), the integrand is purely imaginary. For Q: > sec- ' ( - fiq).

the integrand has both real and imaginary parts. Since the goal of tliis hppendix is

OSCIL LATION OF THE TUNNELING AMPLITC'DE 117

the calculation of R(S). i t is sufficient to consider only this latter part of the integral.

In tliis regioii. thc integrand is of the form d m . where S = r12 - sec' LI. and

1- = 1 sec J2rr? - sec:! IL!. -4 little algebra shows that the real part of rhis intcgrarid

i\.h~lix~ t l i ~ quar i t i t~ . Io = J . 4 1 - l ~ 1 i i i ~ twrri introdiiud.

Tliiis. the tiiniiding aiiiplitiitlt> is of thc f{jriii t = 1 1 1 clos($([ - I , , ) ) . ~rliii./i ii!jr

o i i l ~ ~ cliaiigcs sign. but do ci^ so twr!. two units of spin. prodiicirig il st;iggrriiig p;ir torr1

likr thlit observeci iri i'L%Gd(l!.

Appendix B

Calculat ion of the Staggering Amplitude

[ t was denionstrated ttiat the rcal part of this expression Imds to staggering propor-

tiorial to c o s ( : ( l - - In)). In ttiis appcndix the i~iiaginary part of thv ~sprcssiori w i l l

be evaluated to deterrrii~ie t lit? amplitude of the oscillation.

The condition c7 > sec-' ( - fiO) has been identified as necessary and sufficient

for ttic integrand to have both real and imaginary parts. Again. in this region. the inte-

grand is of the form ,/(A- + i l r ) . with S = q2 -sec2 @, and Y = 1 sec zc1 J2$ - sec' v.

118

CA LCULATION OF THE STA CGERING A M P L I T U D E 119

This niakes the irnaginary part of this integrand. in the region cq > s r ~ - ~ ( - & r ~ ) .

cqual to the surprisingly siniplc resiilt. sec u / d . The integral of this term. called SI.

is given by

-

The two terms of this integral are tackled separatelv. The first term. hereafter

called is evaluated through t lie substitution u = sec <. This yields

lini t+'%

Ln . r. Pc E-'

CALCULATION OF THE ST44GGERING A M P L I T U D E 121

tliis rotational systein, rn is replaced by &. the inertial "mass" for movernent in the

direction of oscillation 0. Thus. k = and since .4 = 1/(2Jl) . one obtairis

This. coiiibiricd wi th t h r rcsult of Griffiths. gives the splitting of t h cnrrgy lewls

t~roi igl i t aboiit t ~ y tiiririrli~ig betwecm rriiriima i r i the Hamiltaniari of Harrianioto arid

Ilottt~lsori. .Arid. siiicc ( J E , - LE:'') is four timcs as

This i~sprrwioii is îit tcvl t o t lir st i ig~wing diitii in Ctiiipt or 4.

Appendix C

Glossary of Symbols

is firsr. i i s t~ I is d s o irictimt cd.

BI Pararrietcr in the Harriamoto and ,110ttelson modcl. (Section 1 . 5 2 )

B2 Parameter in the Hamamoto and blottelson niodcl. ISwt ion 2 . 5 . 2 )

f3 Parameter in the Pavlichenkov modcl. (Sectioii 2.5 2 )

c, .\ symmetry groiip iri which the menibers are invariant under rotations of 360/n degrees. (Section 2 . 5 . 2 )

c Paranieter in the Pavlichenkov Hamiltonian. (Section 2 . 5 . 2 )

d Parameter in the Pavlichenkov Hamiltonian. (Section 2.5.2)

E Energy of a nuclear state. (Section 1.1.2)

GLOSSARY OF SYMBOLS

Incsemrntill alignrnciiit.. (Section 2.2 )

Eflective alignnierit. (Section 2.2)

Ilorrierit of incrtia. Subscripts nia? be used to denote t h r principal moments. (Sect,iori 1.1.2)

Kinernatical monient of inertia. (Section 1. i 2 )

Dynaniical moment of inertia. (Section 1.1.2)

Total angular momenturn of a particle. (Section 1.1.1)

GLOSSA Rk' O F SYhlBOLS

Action for tuiineling between minima. (Appendix A )

Tuniielirig arriplitiidc. (Appendix .A)

Peak width. (Stlctio~i 1.3.2)

The mean staggcring of a band. given by the niean of the distribution of 2's. (Section 4.:3.:3)

Sirigle-part idr ronrribiit ion t» the niiclear Haniilturiiaii. (Scct iori 2 .1 )

Parameter iiscd iri t hc solution of the Hamamoto ancl httt31~011 Haniil- tonian. qiiai to I Jm. (j lppendis .A)

Projection of t h single-particle orbital aiigular monientum on t lie pro- late axis. (Section 1.1 .?)

Counterpart of .\ in the pseudospin framework. (Section 2.4.1)

Prefix indicating tliat the following orbital is a neutron orbital. (Sec- tion 1.3)

T Prefis indicating tliat the folloming orbital is a proton orbital. (Sec- tion 1.3)

0 .Y Standard cit?viatioii of the distribution of .Ys. (Section -4.3.3)

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