MI strengths _scissors and twist by fjzhxb



l. 71-76


MI strengths (scissors and twist modes) in heavy deformed nuclei e56-160Cd)
J.P. Draayer and G. Popa
Departmem of PhysiC!i Qlld ASlrollomy. LOllisian{l Srate University Batol1 Rouge, LA 70803-4001, U.SA

J.G. Hirsch Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico ApartadoposraI70-543, 04510 México, D.F, Mexico C. Vargas Departamento de Física, Centro de /lll'estigaci Ofl y de Estudios Avanzados de/Instituto Apartado postall4-l40, 07000 México, D.F, Mexico
Recibido el 28 de febrero de 2(X}();aceptado el 5 de mnyo de 2000

Politecflico Nacional

The Elliott SUD) Model. extended for hcavy nuclei using pscudo-spin. is lISCU lo study low-lying magnctic dipole excitations in dcforroce! lluclei. Thc Hamiltollian includcs single-parliclc encrgics as well as quadrupolc-quadrupole and pairing ¡nteraction terms. Sinec the quadrupole-quadrupole interaction dominates, a strong SU(3)~dictated truncalion 01' the model spnce is used. With interaetion strcngths lixed hy syslematics. good agreemenl with experimental results is found for the excitation and f\11 transition spectra of the well-deformed evel1-cven 15ú~160Gd ¡sotopes.
Kl')'lI'ords: SU(3) Model; pseudo-spin;

magnetic dipole excitations: 156-160Gd

El modelo SU(3) de Elliot. extendido usando el pseudo-spit¡ para núcleos pesados, es empIcado para es!udiar excitaciones magnéticas dipolares de haja energía en núcleos deformados. El hamiltoniano incluye energías de partícula independiente e interacciones de apareamiento y cuadrupolo-cuadrupolo. Dado la dominación de la interacción euadrupolar se emplea un e"pacio fuertemente lmncado basado en la simelria SU(3). Usando intensidades de las interacciones basadas en la sistemática se ohtiene un huen acuerdo con los datos experimentales para las energías de excitación y las transiciones M 1 de los isótopos deformados par-par 15(i-160Gd.

Modelo SU(3); pseudo-spill;

excitaciones magnéticas dipolarcs:



PAes: 21.60.Fw: 23.20.-g: 27.70.+q

I. Introduction
In reeent years considerable experimental [1-5] and theoretieal [6-9J cffort has been foeused on the study 01' low-Iying enhanccd MI transition strengths (2-4 single-particle units) round in rare earth and actinide nuclei. A simple geometrical inlerprctation of this phenomena is that al' a collective magl1etic dipole excitatian resulting from relative rotational ascillations of the proton and neutron distrihutions against each othcr, a scissors-Iike actian that ¡ed to the christening al' this excitation as the "scissors" moJe [10]. It is also clear, howevcr, that non-collective features in the nuclear interaction are necessary for a complete interpretation of the experimcntal results. In particular. the fragmentation uf the MI transition strcngth among severallevels that are closely packed and clustercd aroul1J a few strong transition peaks [11J cannot he rcproduced hy a model that incIudes only colleclÍve degrees of freedom. Early on (see Ref. 12 and referenccs therein), models hased 011 more l11icroscopic principIes. while still CI11ploying the hasic concept ol' the rclativc rotalianal mol ion of lhe proton and neutron distrihutions. wcrc proposed for cxplaining the origin of this collective phenomena. In this artiele, the pseudo SU(3) shell model is used to deserihe Ihe

l1on-collcctive as well as the underlying callective nature of low-Iying MI transitions in strongly deformcd nuclei. The pseudo SU(3) model is a many-partide shell-model thcory that takes full aJvantage of pseudo-spin symmetry [13,141. which in heavy Iluclci is manifested in the near degeneraey ofthe orhila pairs ((I-I)j=I+I/2, (1+ 1)]=I-l/2J. Since Ihe tic algehra 01' the pseudo oscillator is the same as for the norlllal oscillalor, the pseudo SU(3) symmetry can he lIsed to partition the full space into distinct subspaces. Since its introduction in the late sixties [15, 16], the pseudo SU(3) model had been applied to various propert;cs 01'heavy deformed nuele; [17-19]. hut these have been Iimiled to schematic nucleon-nuclcon interactions hecause of technical diflieuItics related to the ealeulation 01'SU(3) matrix elements 01' Illore general interactions. Howcver, a cade is now availahle [201 thal removes this limitations and allows for the introduclion nI' interactions, likc pairing which is important rOl'un adcquatc description 01' experimental results. into pseudo SU(3) model ealeulations [21], Thc pseudo SU(J) model can be used lO give a microscopic shcll-modcl interpretation oí' the "scissors" mode [22, 23J. As noted ahoye, a geometrical interpretation of this phenomena in terms oí' a scissors-like relative motion



o •.lhe proran and nCUlron distrihutions--ealled lhe Two Rolor t\lodcl (TRM)-pararnClcrizcs Ihe mOlian in Icrms of an angle H llctwecn lhe principal axcs ofaxially syrnrnclric proIon and neutron distrihutions. shcll-Illodcl schcl1lcs-hoson

and an intrinsic action,

parl lhal describes

the proton.neutron




as wcll as fCfmion, howcvcr, (his scissors aClion is associ¡¡Ict! with lhe rclativc molion nI' "valcncc" protons ami neutrons only. nol lhe cntirc mass dislrihution. Tllat such all intcrprctatioll is a corrcct (lile can he secn from lhe raer Iha1 only in this case do lhe cncrgctics (1 + state al 2-3 McV) within lhe shcll-modcl pit:(ure une discovers Ihal "{\Vis'" modes are possihle for triaxial match ohservatíans. In adJition, distrihulions hecausc rotations hy cPrr and óv ahout the :-axes nI' Ihe prolon and neulron distributions emerge as addilional dcgrces 01' frcedom. (In atlJilion to Ref, 21, see Refs. 2427). As rOl" lhe scissors modc, it is the rclative differcncc in these rotations Ihat gives rise lo an MI cxcitation. This !lCW Illode, which together wilh the scissors construclion determines lhe overall slructure 01' the M 1 transition spcclfllm, has a very simple interrretalion pseudo SU(3) mode!. wilhin lhe framcwork 01' the where

= =

(./'.! -






by a two-

in such a way lllal l/ilJ! can he approximated dimensional, anisotropic oscillalor, Explicitly,


= c(l~ + I~) + :ld[(.\,

+ 1)(.\" + l)e2

+ (/', + 1)("" + 1)1>2]+ E~.

""'0 (110 + ~)

+ 1''''0

(110 + ~)

+ E:,.


w" are dcfined


is a constant

amI lhe oscillalor


Wo antl




+ 1)(.\" + 1), + 1)(/" + 1).

w. =

2. ShclI-l1Jodel gcol1Jctry
The starting roint for a gcolllclrical inlcrpretation of lhe scissors motle within lhe fralllework 01' lhe pseudo SU(3) shell l1lotlcl is the well.knowll relation 01' the SU(3) sYIllIl1l'lry group to the sYlllllletry grour 01' lhe tri axial rolor, /lot(:\) [29.:10], As has heen shown 123,31). a similar relalion holtls for the case 01' two coupled quanturn rotors, one represenling prolons and the olher nClItrons. Based on a cm. rcspontlcllcc oct\\'een (he Casimir operators el ano C:I 01' SU(J) antl invariants of Ihe rolor group.

Here Ao and Ao• respectively, are [he relalive angular momenta aboul an axis perpendicular to and in lhe symrnc[ry plane of the proton.neulron system. Thc structure 01' the intrinsic Hamillonian aIlows for <In interpretation of the coupled SU(3) irreps (Arr, Itrr) and (A", JI,,) for protons ami neutrons, respectively. in lerms 01' oscillator funclions. According to lhe Littelwood rules 132] for coupling Young diagrams, lhe allowed producl configurations can he expressetl in malhematicallcfms wilh the hclp 01"three quanlUm nllmhers (11I.1.k) [31 J:


EB(.\, + .\,,-211I+1./,,+
111.1.1 •.

I'v - 2/+



Tr (O' (e, 1>,,1>,,)] Tr(o'(e,1>rro1>,,)]


2 = 3C


+ 2.

(I) where the parametcrs l antl 1// are defined in a flxed range given by the vallles nI' the initial SU(3) representations, In this fonnulation (A, JI) tum out to he independent 01' k which serves to dislinguish belween Illultiple occurrences 01' equiv. alent (.\,1') irreps in the tensor produel. The numher 01" k values is equal lo the maximum ouler multiplicity, Prnax (p 1,2, .... PlIlax)' The l and m lahels can he idenlified 123) with excitation t)uanla al' the lwo dimensional os-


= :¡C",


Ihe irrcdu<:ihle fcprescnlalion (irrcp) labels (A, Ji) 01' lhe total SU(3) can he mapped onto the eolieetive shape variahles (f! amI I ) of lhe joint rolor syslelll 130].


" h ¡I- = -;:- (
J I.;Ul ~I =

(.,., •. ,1 ),.\- + .\,,
.'"1 1 flllS

] + 1'" + 3 (.\ + l' + 1) ,

eiliator given hy Eq. (6)


/3(/' + 1) 2.\ + l' + 3 '
also suggests


This corrcsponds (o 1\\'0 t1istinct lypcs 01' 1 + lllo1ion, lhe scissors and twis! Illodes, antl their realization in lerms 01' lhe pseudo SU(3) mode!. The leading product. irrep (11I./)

This correspondcnce SU(3) Hami1tonian

(hat one should rewrile the


(O. O). 01" the SU(3) tensor

in lerrns 01' a rotational part.


= (.\, + A,,,


+ 1"')'


is associatcd wilh a rninimulll in lhe relalive angular tlisplaceIIIclHs uf lhe prolon and nClllron Subsyslems, geIlerating maximllm deform<llion rOl" lile composilc system. AII otller vallles

Re¡'. M('x. I-'í.\. "¡(, SI (20()O) 71-76






for m and 1 provide larger expeelalion values for Ihe angular variahles B ami Ó. rcspeclivcly. which, in turn. are rclated lo internal cxcitation cncrgics. Assuming prolatc shapcs fOf lhe parent distrihutions, A<7 > ¡tu. cach phonon added 10Ihe scissors mode produces a largcr ¡ncrease in cnergy than a phonoo added lo Ihe Iwisting molion. This refleels lhe softness 01'Ihe (wist dcgrce of frecdom relative lo lhe stiffcr scissors modc. Thc conliguration



~ o.,

+ O


0.6 0.4
O., O 2.00 1



(A, 11)

(1 1) FIGURE l. The experimenlal ~11 transilion strenglh spectrum of 160Gd (1J. Note how the transilions are c1usteredaround local cen. Iroids and that sorne of the c1uslersappcar to be more fragrnented then others.

is lhe tlrst scissors-likc configuration. 11is always pan of lhe tensor product dccOInposilion and contains a)1I 1+ statc that is the handhead 01' a l\ 1 hand. lis SU(3) parameters are (m, 1) = (1, O), and therefore eorresponds lo a single cxcitation of Ihe () Illode. This can he intcrprClcd as lhe shcllmodel equivalen! 01' Ihe scissors mode 01' lhe TRM. Note, however, tha! in conlrast with lhe TRM, hcrc the relative angular displJcements are associateJ with the JeformeJ valencc nucleon dislributions only.



3. Shell.rnodel ealculations
To investigate the effect of syrnmetry hreaking terms in the interactian on the fragmentation 01' the geomclrical modes [i, llJ, a generalization 01'lhe Hamillonian (3) introduced ahove was used [22J, HpSU(3)

A gcneralization ofthis picture is possihle fOftriaxial distrihutions. In this casc, with eithcr Jl1f or JLv are non-7.cro, J secanJ scissors statc appcars which is given by

=+ Drr

( U2

+ (ls,VIIl)C2 + U;iC3 + bh]'2 + eJ ,
- GrrHj, - GvHf" (14)

(A, 11)


(Arr +A" -1,

Ilrr +,1" -1)

( 12)

L Ii. + Dv L If.

or (m, 1) = (1, 1J. In lerms 01'an underlying geomelrical pielure, this structure is a superposition af a <p twisting motion on top uf the lowest scissors conflguration. The encrgy separation 01' Ihe (m,l) ~ (1,0) and (1,1) modes is a resull 01' different intrinsic l1lotions. The most general situation is when JI1f and Jlv are both nonzcro. that ¡s. when both the proton and neutron distrihutions are triaxial. Thcn two more 1+ statcs can be identifleJ, one heing Ihe (111.1) (0,1) eonliguration wilh


whcre the single-particle angular momcnta and are one-body terms and the proton and neutro n pairing terms Hp and H'íJ are two-body interactions. Also, a parameter a.1Im was included to allow an energy shift for SU(3) irreps with eilher A or 1I odd as Ihese heloog lo differenl irreps (Bo, n = 1,2,3, ralhe, lhan A) 01'the inlrinsie Vierergruppe (D,) symmetry [33J. To select an appropriate set of SU(3) hasis functions, one f1rstdetennines the proton and neutron occupancies hy filling pair-wise from below the single-particle Icvcls of the gencr. alized Nilsson Hamillonian [341, "o



(A. 11)


(Arr + Av + 1, Ilrr + 11"" - 2).

( 13)


"w +

el .s

+ Dl2
2 ,

It differs fmm the others not only through its intrinsic energy hUI also heeause it helongs lo a f{ = O hand. The fourlh state, (111,1, kJ = (1,1,2) differs from lhe seeond, (m, 1, k) (1, 1, 1), only in its value oflhe ouler mulliplieity paramcter (J. In sUlTlmary.a pure pseudo SU(3) picture givcs rise to a maximum 01"four ] + states: scissors, Iwist. and a Jouhly degcncralc scissors-plus-twisl moJe.


r f310


+ SIIl 'Y, +Y -']2)' fñ (Y2 v2



Thc experimental rcsults [11. Fig. l. suggest a much largcr number oí' 1+ statcs wilh non-zero M l transition probabilitics to the 0+ ground state. This can he undcrslood in terrns 01" fragmentation 01'Ihe pure symmetry states under the the intlucnce of SU(3) hrcaking residual intcractions. which is the topic of Ihe following section.

for values 01' (J and 'Y that give lhe lowest total energy 01' the combined prolon and neutron systcms. Dne then determines the number of valencc-space nuclcons in the normal and unique parity Icvels, the Intter bcing intruder states that are pushed down inlo the valcnee spaee from ¡he nex! higher shell hy the slrong spin.orhit interaclion. An overall simplifying assumption made in mosl pseudo SU(3) model ealeulalions is that the relevaot dynamies can he deserihcd hy laking ¡nto account the nucleons in the normal parity sector only [35]; the nuclcons in intrudcr statcs (unique parity sector) are assumed lo follow in an adiabatic manner the motion of the nucleons in normal parity sector with their effect repre-

HI'I'. Mex. f'ís. 46 SI (2000) 71-76


1' __ 1'




12 S'-_M' 6'__
4' __

~: r~.-~:
6' 2' 2" o' O-






2'--2" 0'--0'



M'-_R' b' __ 4' __ 6' 4'


~: ~: -~: ~.
2' 2" o' O'







~ • hp Th~"ry


6' __
4' __

4' 2"


2" __


e)(p thy e)(p thy eJ{p thy
200 2:.0 1,IlJ

Up thy

Enl."rg)' IMeV]

FlCiUR E 2. Encrgy anJ MI tmnsilion spcClra for the even-cvcn 156-16°(Jd ¡SOIOpcS In l. Experimental c'(cilatioo cnergics and E2 transition slrcngths wcrc lIscd as input in a fitting routinc todetermine paramctcrs oflhc Ilamiltoniall fm cach syslcm. These werc then used to calculatc the thcorctical spcctrum and corresponding :.11 transilion slrengths.

TA BLE l. Dcformation and occupation numbers fOf the Gd isolOpcs lIscd in this study. Thcsc pararncters are also used in a determinalion ofthc SU(3) oasis slalcs.



TAHI.E 11. TOlal B(t\ll) transition strcnglhs [Jt~] from experi. mcnl [1,281 and calculalion (22). Experimental and Iheoretical val. lICSfor H(E2.0i -+ 2i) transition in f?b2 are also givcn. Nucleus

8 8 4

6 6 6

6 6 8


¿ IJ(MI)

[,,'(.] Thcory 2.91 3.02 ~.29

B(E2.0t Exp. 4.66 5.02 5.19

-; 27)



:::; '"'( :::; (jO

Theory 4.79 5.23 5.()()

15HGd 1fioGd

0 :::; '"'( ~




3.40 02


sentcd through a rcparametcrization of the thcory. f-or Ihe nU r1ci investigalcd here. the occupalion numbcrs and lhe corrcsponding dcformations (3 and l' are given in Table I. In the ReL 22 aH SU(3) hasis siates with Cf 2: Cf",,,, were selecteJ with Cfi, set so thal all proton (o = rr) antl neulron (O: = ti) irrcp~~nlyingbelow approximately 6 l\.1cV '\o'ereincluded in the analysis. Thcn all possiblc couplings 01' Ihese prolOn antl neutron SU(3) irreps, again within 6 McV of ¡he preJicted grounJ state, were taken lo give coupleJ SU(3) irreps that form basis slalcs of the model spacc. Also. only slales with J ~ 8 and 5' = O were considcred. The paramcters 1'orthe Harniltonian given in Eq. (14) ano Ihe clTcctive chargcs C1f = 1 + qeff ano el-' = qeff useo in lhe £(2) Iransition operalor,

wcrc detcrmincd through a fitting procedurc Ihat incluoco as input all known levels wilh J S S up through 2 MeV in energy amI sclcctcJ B(E2) transition strcngths. This proccdurc gavc. in general. good agrccment bet\Vccn the experimental and thcorcticalnumhers (Fig. 2 and Tahle 11). Rcccnt calculations \Verc oone lIsiog a rctlncd vcrsion of the pseudo SU(3) formalism. The rcalistic Hamiltonian thal was uscd in thcse calculations too k singlc-particlc cncrgics and the quatlrupolc-quadrupolc anJ Illonopole pairing strengths from systcmalics.

T¡\I(E) = A Ir' "

¿ '"

'" u .'(')VI2Af (- (')) • ¿Cu1 1 Tu 1

( 16)

+ (/f"., + hf2 + asym C':1 + (13C3. \./ .,


Rev. Mex. ,..,:, 46 SI (2000) 71-76 ..




75 1. 5

~.~ 2

7' "



6'-_6' 4'-_4'


,'-,. "-,.

"-,. ,.-,.



Exp Th Exp Th Exp Th Exp Th
,'-l' 0'-0'



FIGURE 3. Thc experimental and theoretical graund-balld cncrgy speclra and the corrcsponding MI transilion spectrum for lMGd. The Ihcorcticnl resuhs were calculated wilh singlc-parlicle cncrgics. quadrupole-quadrupolc. and pairing ¡nreraetion strengths frem systematics. Thc othcr (rOlor) pararnctcrs were fil 10 the experimental encrgics below 2 McV.

TABLE 111. D(MI) transition slrengths [tt~] in (he purc syrnrnc. try [¡mil 01' the pseudo SU(3) model. Thc strong coupleó pseudo SU(3) irrep (..\, p)gs Cm the graund Mate is givcn w¡lh its proton and neulron sub-irreps and the irrcps associated w¡lh the 1 + states. (A', ,1') 1 +. In addition. cach transition is labeled as a scissors (s) or

(wiSl(l) or combination mode.
N"cle"s (.\,,1") (.\",1'") (.\,1')9' (.\'.1")1+ 8(1\11) mode


(10,4) (10,4)

(18,0) (18,4)

(28,4) (28.8)

(26.5) (27.3) (26.9) (27.7)1 (27.7)' (29.6)

1.91 1.61 1.77 s



1.82 s + f 0.083 t + ., 0.56 f

complex transition speclrum which is in much betler agreemenl with experimental results. One finds a number oflransilions thal are close lo the observed ones. Also, the centroid of the experimental and theoretical MI lransition slrength distribUlion are usually founu lo lie al approximately Ihe same encrgy, so good overall agreement wilh experiment is obtained. The lolal MI slrenglh. whieh fm Ihe full Hamillonian is a hillower Ihen for ils pure SU(3) limit due lo destruelive inlerferenee associaled wilh Ihe l11ixing (see Tables 11 and IlI). shows reasonahle agrcement wilh the experimenlal results, underestimating them slightly for lhe Gd isolopes. A possihle reason for lhis discrepancy is missing spin I admixtures in lhe wavefunctions which are known to add MI slrength to Ihe syslem [3GJ.

The olher parameters of the Hamiltonian (17), namely, a, {J, (/.3, anll aS'ylll' were determined by fitting the low-energy speclra to lhe experimental numbers. This refined model was applicd to 156Gd. As with lhe earlier ca1culations, the lowenergy spectra was described wel!. In addition, lhe distrioUlion of B(M 1) transition strengths was determined 10 he cJoser lo the experimental values lhan in lhe previous calculalion (Fig. 3). In all cases Ihe M I lransilion operalm [19J T,: = J3/4/IN with orbit q-faclors 9~ ~ 1,
!J~ ~ 0, ( 19)

4. Conclusion
The pseudo SU(3) model uffers a mieroseopic shell-model inlerpre(ation oí"the "scissors" mode. In uddilion. il reveals an addilional "twist" degree 01' freedom that corrcsponds lo allowcd relalive angular motion of the proton and/or neutron suh-distrihutions when (hey are lriaxial. Each MI modes is assoeialed wilh a well-defined SU(3) irrep.ln Ihe pure SU(3) symrnctry limit oí" the theory lhere are up to four non-zero MI transitions from cxcited 1+ stales lo Ihe ground slate. The summcd strenglh is in good agreement wilh experiment. By adding SU(3) syml11eIrybreaking one-body and Iwo-body pairing intcraclions to lhe Hamiltonian. it is possibIe lo descrihe the expcrimentally ohserved fragmenlation 01'the MI sIrenglh.

¿{!I~L~ +!I~S~},


(i.e., with no cffective q-factors) was used lo determine Iran-

silion slrcnglhs hetween the 0+ ground state and 1+ states. In Ihe pure SU(3) Iimil of the lheory, there are at mosl I"our allowcd MI transitions. These, logelher with lhcir classification as the scissors mocle. twisl mode, or a comhination are listed wilb tbe corresponJing SU(3) irreps in Tahle 111. I3ecause 01' the SU(3) syrnmetry hreaking terrns (H;,;") and H;Y, lhis simple piclure gives way to a more

Supponed by Ihe U.S. National Seienee Foundalion Ihrough a U.S.-Mcxico Coopcrativc Rescarch grant (lNT-9500474). a regular granl (PHY-9970769). and a Cooperative Agreemenl (EPS-9720652) Ihal ineludes malehing frol11 Ihe Louisiana Board of Regenls Suppml Fund; and by Conaeyl (Mexieo).

lL ....

Rel'. Mex. Fís. 46 SI (2000) 71-76



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