New Description of the Doublet Bands in Doubly Odd Nuclei

0.41 MB·English

by  H. G. Ganev

New Description of the Doublet Bands in Doubly Odd Nuclei
H. G. Ganev,1 A. I. Georgieva,1 S. Brant,2 and A. Ventura3
1Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,
Sofia 1784, Bulgaria
2Department of Physics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia
3Ente per le Nuove tecnologie, l’Energia e l’Ambiente,
I-40129 Bologna and Istituto Nazionale di Fisica Nucleare,
Sezione di Bologna, Italy
The experimentally observed ∆I =1 doublet bands in some odd-odd nuclei are analyzed within
the orthosymplectic extension of the Interacting Vector Boson Model (IVBM). A new, purely col-
lective interpretation of these bands is given on thebasis of theobtained boson-fermion dynamical
symmetryofthemodel. Itisillustratedbyitsapplicationtothreeodd-oddnucleifromtheA∼130
9 region, namely 126Pr, 134Pr and 132La. The theoretical predictions for the energy levels of the
0 doubletbandsaswellasE2andM1transitionprobabilitiesbetweenthestatesoftheyrastbandin
0 the last two nuclei are compared with experiment and the results of other theoretical approaches.
2
The obtained results reveal theapplicability of theorthosymplectic extension of theIVBM.

n
a
J PACS 21.10.Re,23.20.Lv,21.60.Fw,27.60.+j but they can approach some of them, or at least retain

2 some fingerprints of chirality.
2 Many of the recent experiments and theoretical anal-
I. INTRODUCTION ysis do not support completely the chiral interpretation
]
h [18]-[22]. In particular, in an ideal situation, i.e. per-
t In recent years, extensive experimental evidence for fectly orthogonal angular momentum vectors and stable
-
l the existence of distinct band structures in odd-odd nu- triaxial nuclear shape, a perfect degeneracy between the
c
clei has been obtained. It has created an opportunity identical spin states should be observed. In fact, the at-
u
n for testing the predictions of different theoretical models taintment of degeneracy is one of the key characteristics
[ on the level properties of these nuclei. One such study ofchirality. This feature hasnotbeenobservedinanyof

involvestheobservationofdoublet∆I =1bandsinodd- the chiral structures identified to date. Moreover, states
1
odd N =75 and N =73 isotones in the A 130 region. with different quantum numbers in two nonchiral bands
v ∼
7 A large number of experimental data [1]-[8] have been canalsoshowanaccidentaldegeneracy. Thus,oneofthe
9 accumulated in this mass region, showing that the yrast importanttestofchiralityisthatthedegeneratestatesin
4 and yrare states with the πh νh configuration the two bands should also have similar physical proper-
11/2 11/2
3 form ∆I =1 doublet bands which⊗are nearly degenerate ties,suchasmomentofinertia,quasiparticlealignments,
1. in energy. They are built on the single particle states transition quadrupole moments, and the related B(E2)
0 of a valence neutron and a valence proton in the same values for intraband E2 transitions. Some experimen-
9 unique-parity orbital 0h . Pairs of bands have been tal studies have shown that the two bands have differ-
11/2
0 found also in the A 105 and A 190 mass regions. ent shapes due to the different kinematical moments of
v: Initially, these ∆I =∼1 doublet ba∼nds had been inter- inertia, which suggest a shape coexistence (triaxial and
i preted as a manifestation of “chirality” in the sense of axialshapes). Thisisaninterestingobservationsincethe
X
the angular momentum coupling [9]. Several theoretical quantal nature of chirality automatically demands that
r models have been applied in a number of articles, like a chiral partner band should have identical properties
a
the tilted axis cranking (TAC) model [8],[10]-[12], the to the yrast triaxial rotational band. Similarly, it was
core-quasiparticlecoupling model [13], the particle-rotor also found that the experimental data for the behavior
model(PRM)[14]-[16],twoquasiparticle+triaxialrotor ofotherobservables(equalE2transitions,staggeringbe-
model (TQPTR) [17] , core-particle-holecoupling model havioroftheM1values,thesmoothnessofthesignature
(CPHCM) [6]. All these models have one assumption S(I), etc.) do not support such a chiral structure [18]-
in common, they suppose a rigid triaxial core and hence [22]. These results demand a deeper and more detailed
support the interpretationof the doublet bands of chiral discussion of our understanding of the origin of doublet
structure. On the contrary, all odd-odd nuclei in which bands.
twin bands have been observed have a different charac- Within the frameworkof pair truncated shell model it
teristics in common, they are inregions where even-even was pointed out that the band structure of the doublet
nuclei are γ-soft, i.e., effectively triaxial but not rigid. bands can be explained by the chopsticks-like motion of
Their potential energy surface is rather flat in the γ- two angular momenta of the odd neutron and the odd
direction and the couplings with other core structures, proton [23]-[25]. It was found that the level scheme of
not only the ground state band, are significant. It is ev- ∆I = 1 doublet bands does not arise from the chiral
ident that odd-odd nuclei in these mass regions do not structure, but from different angular momentum config-
satisfy all the requirements for the existence of chirality, urationsofthe unpairedneutronandunpairedprotonin
2
the0h orbitals,weaklycoupledwiththecollectiveex- cleiunderconsiderationisfullyconsistentandstartswith
11/2
citations of the even-even core. The same interpretation the calculation of theirs even-even and odd-even neigh-
wasgivenalsointhequadrupolecouplingmodel[26],[27]. bors. We considerthe simplest physicalpicture in which
An alternative interpretation has been based on two particles (or quasiparticles) with intrinsic spins tak-
the Interacting boson fermion-fermion model (IBFFM) ing a single j value are coupled to an even-even core
−
[28],[29], where the energy degeneracy is obtained but nucleus whose states belong to an Sp(12,R) irreducible
a different nature is attributed to the two bands. A de- representation. Thus, the bands of the odd-mass and
tailedanalysisofthewavefunctionsinIBFFMshowedas odd-odd nuclei arise as collective bands build on a given
wellthat the presence ofconfigurationswith the angular even-even nucleus. So, within the framework of the or-
momenta of the proton, neutron and core in the chiral- thosymplectic extension of the model a purely collective
ity favorable, almost orthogonal geometry, is substantial structure of the doublet bands is obtained.
but far from being dominant. The large fluctuations of The level structure of 126Pr, 134Pr and 132La is an-
the deformation parameters β and γ around the triaxial alyzed in the framework of the orthosymplectic exten-
equilibrium shape enhance the content of achiral config- sion of the IVBM [34]. Thus to describe the structure
urations in the wave functions. The β distribution of of odd-odd nuclei, first a description of the appropriate
the yrast band has its maximum at larg−er deformations even-even cores should be obtained.
than that of the side band. At higher angular momenta,
this difference becomes very pronounced. In addition,
II. THE EVEN-EVEN CORE NUCLEI
the fluctuations of β in the side band become very large
with increasing spin. In both bands the fluctuations of
γ increase with spin, being more pronounced in the side The algebraic structure of the IVBM is realized in
band [30]. The composition of the yrast band, in terms terms of creation and annihilation operators u+(α),
m
of contributions from core states, shows that the yrast u (α) (m = 0, 1). The bilinear products of the cre-
m
±
band is basically built on the ground-state band of the ationandannihilationoperatorsofthetwovectorbosons
even-even core. With increasing spin the admixture of generate the boson representations of the non-compact
the γ band of the core becomes more pronounced. The symplectic group Sp(12,R) [36]:
−
sidebandwavefunctionscontainlargecomponentsofthe
γ band and with increasing spin, of higher-lying collec- FL(α,β) = CLM u+(α)u+(β),
ti−ve structures of the core, which near the band crossing M Xk,m 1k1m k m
become dominant. So, the conclusion of Refs. [21],[30] GLM(α,β) = C1LkM1muk(α)um(β), (1)
was that the existence of twin bands in 134Pr should be Xk,m
attributed to a weak dynamic (fluctuation dominated)
chirality combined with an intrinsic symmetry yet to be AL (α,β)= CLM u+(α)u (β), (2)
revealed. The IBFFM was applied to the doublet bands M k,m 1k1m k m
X
in 134Pr [18],[21],[30]. The B(E2) values of the transi-
where CLM , which are the usual Clebsch-Gordan coef-
tions depopulating the analog states are different from 1k1m
ficients for L = 0,1,2 and M = L, L+ 1,...L, de-
the chiral predictions and the B(M1) staggering is not − −
fine the transformation properties of (1) and (2) under
present [31]. The IBFFM was also applied for the de-
scriptionofthe yrastπh νh bandin126Pr [32]. rotations. The commutation relations between the pair
11/2 11/2
⊗ creationand annihilation operators (1) and the number
The above variety of models and approaches dealing
preserving operators (2) are given in [36].
with the description of the doublet bands in odd-odd
Being a noncompact group, the unitary representa-
nuclei motivated us to consider their properties in the
tions ofSp(12,R)areofinfinite dimension, whichmakes
framework of the boson-fermion extension of the sym-
it impossible to diagonalize the most general Hamilto-
plectic IVBM [33].
nian. When reduced to the group UB(6), each irrep of
Inthepresentworkwecarryoutananalysisofthedou- the groupSpB(12,R) decomposes into irreps of the sub-
blet bands in some doubly odd nuclei from the A 130 group characterized by the partitions [33],[37]:
∼
region within the orthosymplectic extension [34] of the
IVBM. The latter was proposed in order to encompass [N,05] [N] ,
6 6
the treatment of the odd-mass nuclei. Further, the new ≡
version of IVBM was applied for the description of the where N = 0,2,4,... (even irrep) or N = 1,3,5,...
ground and first excited positive and/or negative bands (odd irrep). The subspaces [N] are finite dimensional,
6
of odd-odd nuclei [35]. The spectrum of the positive- which simplifies the problem of diagonalization. There-
parity states in the odd-odd nuclei considered in this fore the complete spectrum of the system can be cal-
paper is based on the odd proton and odd neutron culated through the diagonalization of the Hamiltonian
(both particle-like in contrast to usually considered pro- in the subspaces of all the unitary irreducible repre-
ton particle-like and neutron hole-like nature of the two sentations (UIR) of U(6), belonging to a given UIR of
oddparticles)whichoccupythesamesingleparticlelevel Sp(12,R), which further clarifies its role of a group of
h . The theoretical description of the doubly odd nu- dynamical symmetry.
11/2
3
6 124 6 134
Ce Ce
V] V]
Energy [Me 24 1116482024+++++++ Energy [Me 24 168240+++++ 8756432+++++++Th Exp IBM Th E=xp4+ IBM
0 0+ 0 0+ band
Th Exp IBM Th Exp IBM
GSB
GSB
FIG.1: (Color online) Comparison ofthetheoretical andex- FIG.2: (Color online) Comparison ofthetheoretical and ex-
perimental energies for theground band of 124Ce. perimental energies for the ground and first excited bandsof
134Ce.
The Hamiltonian, corresponding to the unitary limit
of IVBM [33]
4
132
Sp(12,R) U(6) U(3) U(2) O(3) (U(1) U(1)),
Ba
⊃ ⊃ ⊗ ⊃ ⊗ ⊗
(3)
3
expressedintermsofthefirstandsecondorderinvariant V]
e
M
o[3p3e]r:ators of the different subgroups in the chain (3) is gy [ 2 6+ 45++
H =aN +bN2+α3T2+β3L2+α1T02. (4) Ener 1 4+ 23++
H (4) is obviously diagonal in the basis 2+ Th Exp CPHCM
band
0 0+
[N]6;(λ,µ);KLM;T0 (N,T);KLM;T0 , (5) Th Exp CPHCM
| i≡ | i GSB
labeledbythe quantumnumbersofthe subgroupsofthe
chain (3). Its eigenvalues are the energies of the basis
states of the boson representations of Sp(12,R): FIG.3: (Color online) Comparison ofthetheoretical and ex-
perimental energies for the ground and γ bandsof 132Ba.
E((N,T),L,T ) = aN +bN2+α T(T +1)
0 3
+ β L(L+1)+α T2. (6)
3 1 0
frame and is used with the parity (π) to label the differ-
The construction of the symplectic basis for the even ent bands (Kπ) in the energy spectra of the nuclei. For
IR of Sp(12,R) is given in detail in [33]. The Sp(12,R) the even-even nuclei we have defined the parity of the
classificationschemefortheSU(3)bosonrepresentations states as πcore =( 1)T [33]. This allowedus to describe
−
for even value of the number of bosons N is shown on both positive and negative bands.
Table I in Ref. [33] (see also Table I). Further,weusethealgebraicconceptof“yrast”states,
The most important application of the UB(6) introducedin[33]. Accordingtothisconceptweconsider
SpB(12,R)limitofthetheoryisthepossibilityitafford⊂s asyraststatesthestateswithgivenL,thatminimizethe
fordescribingbothevenandoddparitybandsuptovery energy (6) with respect to the number of vector bosons
highangularmomentum[33]. Inordertodothiswefirst N that build them. Thus the states of the ground state
have to identify the experimentally observed bands with band (GSB) were identified with the SU(3) multiplets
the sequences of basis states of the even Sp(12,R) irrep (0,µ) [33]. In terms of (N,T) this choice corresponds
(Table I). As we deal with the symplectic extension we to (N = 2µ,T = 0) and the sequence of states with
areabletoconsideralleveneigenvaluesofthenumberof different numbers of bosons N = 0,4,8,... and T = 0,
vector bosons N with the corresponding set of T spins, T = 0. Hence the minimum values of the energies (6)
0
whichuniquelydefinetheSUB(3)irreps(λ,µ). T−hemul- are obtained at N =2L.
tiplicity index K appearing in the final reduction to the Thepresentedmappingoftheexperimentalstatesonto
SO(3)is relatedtothe projectionofLonthe body fixed theSU(3)basisstates,usingthealgebraicnotionofyrast
4
states, is a particular case of the so called ”stretched” two terms. In analogy it could be shown that the two
states[38]. Thelatteraredefinedasthestateswith(λ + collective modes are mixed in the excited bands as well.
0
2k,µ ) or (λ ,µ +k), where N = λ +2µ and k = Hencewecandescribequitewellinthesamegroupchain
0 0 0 i 0 0
0,1,2,3,.... ofthesymplecticextension,theeven-evencoreswithvar-
Itwasestablished[39]thatthecorrectplacementofthe ious collective properties that need different dynamical
bands in the spectrum strongly depends on their band- symmetries or their mixture in the IBM.
headsconfiguration,andinparticular,ontheminimalor The theoretical predictions for the even-even core nu-
initial number of bosons, N = N , from which they are clei are presented in Figures 1 3. For comparison, the
i
−
built. Thelatterdeterminesthestartingpositionofeach predictions of IBM and CPHCM are also shown. The
excited band. IBMresultsfor124Ceand134CeareextractedfromRefs.
Thus,forthedescriptionofthedifferentexcitedbands, [32],[18]andthoseofCPHCMfor132Bafrom[6],respec-
we first determine the N of the band head structure tively. From the figures one can see that the calculated
i
and develop the corresponding excited band over the energy levels of both ground state and γ bands agree
stretched SU(3) multiplets. This corresponds to the se- rather well up to high angular momenta with the ob-
quence of basis states with N = N ,N +4,N +8,... serveddata. Exceptfor the GSB of 134Ce, for which the
i i i
(∆N =4). ThevaluesofT forthe firsttypeofstretched IVBM and IBM results are almost identical, the IVBM
states (λ changing) are changing by step ∆T = 2, predictions reproduce better the band structures com-
−
whereasforthesecondtype(µ changing) T isfixedso pared to CPHCM and IBM.
− −
thatinbothcasestheparityispreservedevenorodd,re-
spectively. For all presented even-even nuclei, the states
of the described excited bands are associated with the III. FERMION DEGREES OF FREEDOM
stretched states of the first type (λ changing).
−
To describe the structure of odd-mass and odd-odd In order to incorporate the intrinsic spin degrees of
nuclei, first a description of the appropriate even-even freedom into the symplectic IVBM, we extend the dy-
coresshouldbeobtained. Wedeterminethevaluesofthe namical algebra of Sp(12,R) to the orthosymplectic al-
five phenomenological model parameters a,b,α3,β3,α1 gebra of OSp(2Ω/12,R) [34]. For this purpose we intro-
byfittingtheenergiesofthegroundandγ bandsofthe duceaparticle(quasiparticle)withspinj andconsidera
−
even-even nuclei to the experimental data [40], using a simplecoreplusparticlepicture. Thus,inadditiontothe
χ2 procedure. bosoncollectivedegreesoffreedom(describedbydynam-
Numerous IBM studies of even-evennuclei in the A ical symmetry group Sp(12,R)) we introduce creation
∼
130 mass region have shown that these nuclei are well and annihilation operators a† and a (m = j,...,j),
m m −
described by the O(6) symmetry of the IBM, that in the which satisfy the anticommutation relations
classicallimit correspondsto the Wilets-Jeanmodel of a
ltγoi−ownusnitsshtteahbsayltesttrehometyoartai[rc4et1rγ]e,−nadsnodoftft.thhTaethCetheceoisraoectcnoeuppcetlsee.duTsihn12tee4hrCpeearevftoiealr-- {{aam†m,,aa†m†m′′}} == δ{mamm,′.am′}=0, (8)
isotopes are γ−soft (O(6)-like in the IBM terminology), All bilinear combinations of a+m and am′, namely
andthe lighteronesareconsiderablydeformed,butthey
never reach the rigid rotor structure which corresponds fmm′ = a†ma†m′, m6=m′
tthoetsheetwSUo(s3t)rulcimtuirteosfotchceurIsBMfor. T12h6eCter,aannsidtioisnrbeefltewceteend gmm′ = amam′, m6=m′; (9)
in the dynamics of bands in the neighboring odd-even Cmm′ = (a†mam′ −am′a†m)/2 (10)
and odd-odd nuclei. In contrast to the O(6)-like spec-
generate the (Lie) fermion pair algebra of SOF(2Ω).
tra observed in the odd-odd isotopes 130,132Pr [42], the
Theircommutationrelationsaregivenin[34]. Thenum-
structure of 126Pr reflects the transitional SU(3)-O(6)
berpreservingoperators(10)generatemaximalcompact
nature of the core nucleus 124Ce.
subalgebra UF(Ω) of SOF(2Ω). The upper script B or
Here,wemustpointoutthatonlyintheconsidereddy-
F denotes the boson or fermion degrees of freedom, re-
namicalsymmetry(3)oftheIVBM,duetotheemployed
spectively.
“algebraic yrast” condition N = 2L and the reduction
rules connecting the values of the number of bosons N
with their angular momentum L the energies of the col-
A. Fermion dynamical symmetries
lective states of ground state band [33] for example, can
be written as:
As can be seen from (10), the full number conserving
E (L)=(2a 4b)L+(4b+β )L(L+1) , (7) symmetryofafermionofspinj isUF(2j+1). Ingeneral,
g 3
− the full dynamical algebra build from all bilinear com-
where obviously the rotational L(L+1) and vibrational binations (9),(10) of creation and annihilation fermion
L collectivemodes aremixedandthe type ofcollectivity operators is the SO(2Ω) algebra (for a multilevel case
depends on the ratio of the coefficients in front of these Ω = (2j +1)). One can further construct a certain
j
P
5
fermion dynamical symmetry, i.e. the group-subgroup where Sp(2j+1) is the compact symplectic group. The
chain: dynamicalsymmetry (12) remainsvalid also for the case
of two particles occupying the same level j. In this case,
SO(2Ω) G′ G′′ .... (11) the allowedvalues ofthe quantumnumber I ofSU(2)in
⊃ ⊃ ⊃
(12) according to reduction rules are I =0,2,...,2j 1
In particular for one particle occupying a single level j [43]. If the two particlesoccupydifferent levelsj and−j
1 2
we are interested in the following dynamical symmetry:
of the same or different major shell(s), one can consider
the chain
SOF(2Ω) Sp(2j+1) SUF(2), (12)
⊃ ⊃
U(Ω ) Sp(2j +1) SU (2)
ր 1 ⊃ 1 ⊃ I1 ց
SO(2Ω) U(Ω) SUF(2) (13)
⊃
U(Ω ) Sp(2j +1) SU (2)
ց 2 ⊃ 2 ⊃ I2 ր
where Ω = Ω + Ω . We want to point out that al- i.e. G GB GF. Itis ourintentioninthis paper todo
1 2
⊃ ⊗
thoughthefinalgroupSUF(2)thatappearsinthe chain that for chains describing odd-odd nuclei.
(13) is the same as in (12), its content is different.
Here the values of the common fermion angular momen-
tum I are determined by the vector sum of the two MakinguseoftheembeddingSUF(2) SOF(2Ω)and
⊂
individual spins I and I , respectively. Nevertheless, considerations from the preceding section, we make or-
1 2
for simplicity hereafter we will use just the reduction thosymplectic (supersymmetric) extension of the IVBM
SO(2Ω) SUF(2) (i.e. dropping all intermediate sub- which is defined through the chain [34]:
groups be⊃tween SO(2Ω) and SUF(2)) and keep in mind
the proper content of the set of I values for one and/or
two particles cases, respectively.
B. Bose-Fermi symmetry OSp(2Ω/12,R) SOF(2Ω) SpB(12,R)
⊃ ⊗
⇓
Once the fermion dynamical symmetry is determined UB(6)
⇓ ⊗
weproceedwiththeconstructionoftheBose-Fermisym- N
metries. Ifafermioniscoupledtoabosonsystemhaving
⇓
itself adynamicalsymmetry (e.g.,suchasanIBMcore), SUF(2) ⊗ SUB(3)⊗UTB(2)
the full symmetry of the combined system is GB GF. I (λ,µ) (N,T)
⊗ ⇐⇒
Bose-Fermi symmetries occur if at some point the same
ց ⇓
group appears in both chains SOB(3) U(1)
⊗ ⊗
L T
0
GB GF GBF, (14)
⊗ ⊃ ⇓
SpinBF(3) SpinBF(2),
⊃
i.e. the two subgroup chains merge into one. It should J J
0
be notedthat(14)istrueonlyforthe diagonalsubgroup (15)
GB GF, i.e. the one in which the two group elements where below the different subgroups the quantum num-
⊗
multiplieddirectly areparametrizedbythe sameparam- bers characterizing their irreducible representations are
eters. In this way the Bose-Fermi symmetry not only given. SpinBF(n)(n=2,3)denotes the universalcover-
constrains parameters by the choice of particular sub- ing group of SO(n).
group chains in the boson and fermion sectors, but also
specifies the interaction between the two.
In the next section we present the application of the
boson-fermion extension of IVBM, developed for the de-
IV. DYNAMICAL SUPERSYMMETRY scription of the collective bands of even-even [33] and
odd-mass [34] nuclei, in order to include in our consid-
The standard approach to supersymmetry in nuclei erations the positive parity states of the yrast and side
(dynamical supersymmetry) is to embed the Bose-Fermi bands of odd-odd nuclei from A 130 region, build on
subgroup chain of GB GF into a larger supergroup G, πh νh configuration. ∼
11/2 11/2
⊗ ⊗
6
V. THE ENERGY SPECTRA OF ODD-MASS
TABLEI:Classification schemeofbasisstates(16)according
AND ODD-ODD NUCLEI
thedecompositions given by thechain (15).
We can label the basis states according to the chain N T (λ,µ) K L J =L±I
(15) as:
0 0 (0,0) 0 0 I
2 1 (2,0) 0 0,2 I; 2±I
[N] ;(λ,µ);KL;I;JJ ;T
| 6 0 0 i≡ 0 (0,1) 0 1 1±I
[N] ;(N,T);KL;I;JJ ;T , (16)
| 6 0 0 i 2 (4,0) 0 0,2,4 I; 2±I;4±I
where [N] the U(6) labeling quantum number and 4 1 (2,1) 1 1,2,3 1±I; 2±I; 3±I
6
(λ,µ) the −SU(3) quantum numbers characterize the 0 (0,2) 0 0,2 I; 2±I
−
coreexcitations,K isthemultiplicityindexinthereduc- 3 (6,0) 0 0,2,4,6 I; 2±I; 4±I;6±I
tion SU(3) SO(3), L is the core angular momentum, 1±I; 2±I; 3±I;
⊃ 2 (4,1) 1 1,2,3,4,5
I the intrinsic spin of an odd particle (or the common 4±I; 5±I
−
spin of two fermion particles for the case of odd-odd nu- 6 1 (2,2) 2 2,3,4 2±I; 3±I; 4±I
clei), J,J0 arethe total(coupled boson-fermion)angular 0 0,2 I; 2±I
momentum and its third projection, and T,T are the
0 0 (0,3) 0 1,3 1±I; 3±I
T spin and its third projection, respectively. Since the
SO−(2Ω)labelis irrelevantfor ourapplication,we dropit 4 (8,0) 0 0,2,4,6,8 I; 2±I; 4±I;
in the states (16). 6±I; 8±I
The Hamiltonian of the combined boson-fermion sys- 1±I; 2±I; 3±I;
tem can be written as linear combination of the Casimir 3 (6,1) 1 1,2,3,4,5,6,7 4±I; 5±I;6±I;
operators of the different subgroups in (15): 7±I;8±I
2±I; 3±I; 4±I;
2 (4,2) 2 2,3,4,5,6
H = aN +bN2+α3T2+β3′L2+α1T02 5±I; 6±I
+ ηI2+γ′J2+ζJ2 (17) 8 0 0,2,4 I; 2±I; 4±I
0
1 (2,3) 2 2,3,4,5 2±I; 3±I; 4±I;5±I
and it is obviously diagonal in the basis (16) labeled by 0 1,3 1±I; 3±I
the quantumnumbersoftheirrepresentations. Thenthe 0 (0,4) 0 0,2,4 I;2±I; 4±I
eigenvalues of the Hamiltonian (17), that yield the spec- . . . . . .
. . . . . .
trum of the odd-mass and odd-odd systems are: . . . . . .
E(N;T,T ;L,I;J,J )=
0 0
aN +bN2 The basis states (16) can be considered as a result of
+α3T(T +1)+β3′L(L+1)+α1T02 tshpeincφouplingwoafvtehefuonrcbtiiotanls.| (TNh,eTn,);iKf tLhMe p;Tar0iity(5o)fatnhde
+ηI(I +1)+γ′J(J +1)+ζJ2. (18) j≡I,m
0 singleparticleis π ,the parityofthe collectivestatesof
sp
the odd A nuclei will be π = π π [34]. In analogy,
core sp
Wenotethatonlythe lastthreetermsof(17)comefrom −
one can write π =π π (1)π (2) for the case of odd-
core sp sp
the orthosymplectic extension. We choose parameters
odd nuclei. Thus, the description of the positive and/or
β′ = 1β and γ′ = 1γ instead of β and γ in order to
3 2 3 2 3 negative parity bands requires only the proper choice of
obtaintheHamiltonianformofref. [33](settingβ =γ),
3 the core band heads, on which the corresponding single
when for the case I = 0 (hence J = L) we recover the
particle(s)is (are)coupled to,generatingin this waythe
symplectic structure of the IVBM.
different odd A (odd-odd) collective bands.
The infinite set of basis states classified according to −
Furtherinthe presentconsiderations,the ”yrast”con-
thereductionchain(15)areschematicallyshowninTable
ditions yield relations between the number of bosons N
I. The fourth and fifth columns show the SOB(3) con-
and the coupled angular momentum J that character-
tentoftheSUB(3)group,givenbythestandardElliott’s
izes each collective state. For example, the collective
reduction rules [44], while in the next column are given
states of the GSB Kπ = 5+ (125Ce) of the odd-mass
the possible values of the common angular momentum J 2
nuclei are identified with the SU(3) multiplets (0,µ)
J,obtainedbycouplingoftheorbitalmomentumLwith
which yield the sequence N = 2(J I) = 0,2,4,... for
the spin I. The latter is vector coupling and hence all
the corresponding values J = 5,7−,9,.... The T spin
possible values of the total angular momentum J should 2 2 2 −
for the SU(3) multiplets (0,µ) is T = 0 and hence
beconsidered. Forsimplicity,onlythemaximallyaligned
π =( 1)T =(+). Here it is assumed that the single
(J =L+I)andmaximallyantialigned(J =L I)states core
−
− particle has j I = 5/2 and parity π = (+), so that
are illustrated in Table I. sp
≡
the common parity π is also positive.
For the description of the different excited bands,
7
125
Ce 133
4 4 La
V] V]
Me 3 29/2- Me 3 27/2- 25/2-
Energy [ 2 222357/222--- Energy [ 2 23/2- 21/2-
01 11115973157///2/22///2222+++++++ 111111937579///////2222222------- 01 111195///222--- 1173//22-- Th Exp CPHCM
Th Exp IBFM Th Exp IBFM Th Exp CPHCM
GSB:K=5/2 K=7/2 Yrast Band
FIG.4: (Color online) Comparison ofthetheoretical andex- FIG.6: (Color online) Comparison ofthetheoretical and ex-
perimental energies for the ground and first excited bandsof perimental energies for the yrast band of 133La.
125Ce.
Theodd-Aneighboringnuclei125Ce and135Ce canbe
considered as a neutron coupled to the even-even cores
8 124Ce and 134Ce, while the 133La as a proton coupled
135
7 Ce to the 132Ba, respectively. The low−-lying positive parity
29/2+ states of the GSB in odd-A neighbors are based on posi-
eV] 6 27/2+ tive parity protonandpositive parity neutronconfigura-
M
Energy [ 345 221359///222--- 12229153////2222++++ otgthinrooenhufis1pr1(ss/st2S21.aO,Ivdn(a232iol,Ωaud1br52)l,ecagosnn27idns),i/gdolweerrhpaUeatr(ireoΩtani1scs)ltewhwoeoirtstbheaiktoΩefj11inn=e(tggo(ae2antjciev1creo+autpin1ant)rg)io.ttnyhl−ye
2 17/2- The comparison between the experimental spectra for
15/2- Th Exp
1 1131//22-- K=19/2 tmhoedGelSpBaraanmdetfierrsstgeivxecniteindTbaabnldeIuIsIinfogrtthheevnaulculeesi1o2f5Cthee,
0 Th Exp
135Ce and 133La is illustrated in Figures 4 6. One can
Yrast Band
−
see from the figures that the calculated energy levels
agree rather well in general with the experimental data
FIG. 5: (Color online) The same as in the Fig. 4 but for up to very high angular momenta. For comparison, in
135Ce. Figures4and6theIBFMandCPHCMresultsfor125Ce
and133Laarealsoshown. TheyareextractedfromRefs.
[32] and [6], respectively.
we first determine the N of the band head structure Forthecalculationoftheodd-oddnucleispectraasec-
i
and then we map the states of the corresponding band ond particle shouldbe coupled to the core. In our calcu-
onto the sequence of basis states with N = N ,N + lationsaconsistentprocedureisemployedwhichincludes
i i
2,N + 4,... (∆N = 2) and T = even = fixed or the analysis of the even-even and odd-even neighbors of
i
T = odd = fixed, respectively. This choice corresponds the nucleus under consideration. Thus, as a first step an
to the stretchedstates of the secondtype (µ changing). oddparticlewascoupledtothebosoncoreinordertoob-
−
The number of adjustable parameters needed for the tain the spectra of the odd-mass neighbors 125Ce, 135Ce
complete description of the collective spectra of both and 133La. As a second step, we consider an addition of
odd-A and odd-odd nuclei is three, namely γ, ζ and a second particle to the boson-fermion system.
η. The first two are evaluated by a fit to the experi- In our application, the most important point is the
mental data [40] of the GSB of the correspondingodd-A identification of the experimentally observed states with
neighbor, while the last one is introduced in the final a certain subset of basis states from (ortho)symplectic
step of the fitting procedure for the odd-odd nucleus, extension of the model. Here we consider a more gen-
respectively. For the A 130 region where the dou- eral mapping when the states of the GSB of the odd-
∼
blet bands are built on πh νh configuration, odd nuclei are associatedwith a sequence of SU(3) mul-
11/2 11/2
⊗
the two fermions occupy the same single particle level tiplets (0,µ) but the band starts with the multiplet
j = j = j = 11/2 with negative parity (π = ) and (0,µ ) instead of (0,0). Thus, to the states of the yrast
1 2 sp 0
−
the fermion reduction chain (12) can be used. band with J = I,I + 1,I + 2,... of the odd-odd nu-
8
clei we put into correspondence the SU(3) multiplets
(0,µ) (µ changing) of the basis states (16) which in
terms of−(N,T) correspond to (N = 2µ,T = 0) and 8 126Pr
the sequence of states with different numbers of bosons 7
N =N ,N +2,N +4,...(∆N =2). The chosensetof 27+
SU(3)0mult0iplets (00,µ) means that the GSB of the odd- MeV] 6 2256++
ootofddtthhaeeseIwvBeeMlnl-a,esvthethneasctyoomrfeponldeudcct-lmiecuacss.osrWneuesctlrreeuicciastlulbrutehila(ddtoeisnnctrchiobenetGdraSbsByt nergy [ 45 22224321++++
E 3 20+
t”bdiiyavffrneeadrsa’etsln”lostcwtoSaintnUedgs(it3toi)hovenmercuwhtlhthaiinepcghlseettrrseoes(ftu0ctl,hhtµesed)f)rnsowutmmaitthbtehiesnris(tλomhf0eab=IpoVps0Bion,nMµgs0.oi+fsTtahhkcee)- 012 11913+++11111111802849567+++++++++
yields N =2µ0+2L (or k =L). In particular,when the Th Exp IBFFM
bandheadstructureisdeterminedbyN =0bosons,the Yrast
0
yrast condition reduces to N = 2L (or µ = L) [33],[34].
In order to visualize the correspondence under consider-
FIG.7: (Color online) Comparison ofthetheoretical and ex-
ation, we illustrate the selected subset of basis states in perimental energies for the yrast band of 126Pr.
TableII. Henceoneobtainstheobservedgroundstateof
the yrastbandwithKπ =8+ for126Pr,134Pr and132La
J
nuclei simply attributing to it only the angular momen-
tumI =8fromthevectorcouplingoftheprotonI = 11
p 2 134
and neutron In = 121 momenta. 5 Pr
V] 4
TABLEII:Thesubsetofbasisstates(16)associatedwiththe y [Me 3 1176++ 1167++
cstoantfiegsuorfatthioenG.SBofodd-oddnuclei,basedonπh11/2⊗νh11/2 Energ 2 111543+++ 11114532++++
N(λ,µ) N(00,µ0) N(00,µ+02+1) N(00,µ+04+2) N(00,µ+06+3) ...... 1 1119012++++ 118910++++
L 0 1 2 3 ... 0 8+ Th Exp IBFFM
J I I+1 I+2 I+3 ... Th Exp IBFFM Side
Yrast
FIG.8: (Color online) Comparison ofthetheoretical and ex-
perimental energies for the yrast and side bandsof 134Pr.
For the description of the side (yrare) band build also
on the πh νh configuration which can be con-
11/2 11/2
⊗
sidered as an excited band, we first determine the col-
lective structure of the band head N = λ +2µ and
i 0 0
then map the states of this band onto the sequences of 132
4 La
basis states with N = N ,N +2,N +4,... (∆N = 2)
i i i
and T = even = fixed. This choice corresponds to the
V]
ssSutUrgeg(t3ec)shtemdsuimslttiiapltalerestscoof(lλltehc6=etiv0see,cµos)tnradutctttryuipbreuet(efµodr−tctohhtaishnegbiasningdd)e.bcToamnhde- ergy [Me 23 111567+++ 111645+++
pttahaterieoydnratosoftttbhhaeanttdwofoisibtbsaan”sddicsoautlalbykleebtsuppilladarctoenneinrt”ht.heSeGiImSBiBFlaFr(dMien,stcwerrihpbererede- En 1 11113412++++ 111231+++
10+ Th Exp CPHCM
within a single SU(3) multiplet (λ,µ)) of the even-even 0 89++ Side Band
core, while the structure of the side band is that of odd Th Exp CPHCM
proton and odd neutron coupled to the γ band of the Yrast
−
core and in the high spin region contains sizeable com-
ponents of the higher-lying core structures .
Thetheoreticalpredictionsfortheyrastandsidebands FIG. 9: (Color online) The same as Fig. 8, butfor 132La.
basedonπh νh configurationforthethreeodd-
11/2 11/2
⊗
odd nuclei 126Pr, 134Pr and 132La from A 130 region
∼
9
consider the E2 and M1 transitions in the framework of
TABLE III:Values of the model parameters.
the orthosymplectic extension of the IVBM.
Nucl. bands Ni T T0 J χ2 parameters
a=0.02855
Yrast:
126Pr 24 0 0 L+I 0.0017 b=−0.00120 VI. ELECTROMAGNETIC TRANSITIONS
Kπ =8+
α3 =0.00680
I =8 β3 =0.01774 A successful nuclear model must yield a good descrip-
α1 =0.01387 tion not only of the energy spectrum of the nucleus but
η=−0.00906 also of its electromagnetic properties. Calculation of the
latter is a good test of the nuclear model functions. The
γ =0.01691
most important electromagnetic features which manifest
ζ =−0.01132
themselves in doublet bands are the E2 and M1 tran-
a=0.07449 sitions. In this section we discuss the calculation of the
Yrast:
132La 44 0 0 L−I 0.0034 b=0.00690 E2andM1transitionstrengthsbetweenthestatesofthe
Kπ =8+
α3 =0.05709 yrastbandoftheodd-oddnucleibasedonπh11/2 νh11/2
⊗
I =8 sKidπe=: 11+ 50 2 0 L−I 0.0088 αβ31==00..0046804776 ceoxnpfiergiumreantitoanl daantda.compare the results with the available
For a mixed systems of bosons and fermions it is con-
η=0.02360
venient to expand the coupled basis states (16) into the
γ =0.04796
direct product of the boson and fermion states. The lat-
ζ =0.02960
ter significantly simplifies the applicationof the Wigner-
a=0.08190 Eckart theorem in the practical calculations of the tran-
Yrast:
134Pr 10 0 0 L+I 0.0046 b=0.00473 sition rates.
Kπ =8+
α3 =0.03637
I =8 side: 14 4 0 L+I 0.0020 β3 =0.03660
Kπ =8+ α1 =0.04424 A. E2 transitions
η=−0.01876
γ =0.03002 As was mentioned, in the symplectic extension of the
ζ =0.00061 IVBMthecompletespectrumofthesystemisobtainedin
alltheevensubspaceswithfixedN-evenoftheUIR[N]
6
ofU(6),belongingtoagivenevenUIRofSp(12,R). The
arepresentedinFigures7,8and9,respectively. Forcom- classificationscheme of the SU(3) boson representations
parison,the IBFFM(refs.[32],[30])andCPHCM(ref.[6]) forevenvaluesofthe numberofbosonsN waspresented
results are also shown. In Table III, the values of N , in Table I.
i
T, T , J and χ2 for each band under consideration are In the present paper, the states of the yrast band are
0
alsogiven. Fromthe figures one cansee the goodoverall identified with the SU(3) multiplets (0,µ). This yields
agreementbetween the theory and experiment which re- the sequence N = N0,N0 +2,N0+4,... for the corre-
veals the applicability of the boson-fermion extension of sponding values J = I,I +1,I +2,... (see Table II). In
the model. terms of (N,T) this corresponds to (N =2µ,T =0).
To investigate the structure of the doublet bands in a Usingthe tensorialpropertiesoftheSp(12,R)genera-
certainnucleus, it is crucialto determine the B(E2) and torswithrespectto(3)itis easytodefine theproperE2
B(M1) values which are very important for establishing transition operator between the states of the considered
the nature of these bands. So, in the next section we band as [45]:
TE2 =e A[1−1]6 20+θ([F F] [4]6 20 +[G G] [−4]6 20 . (19)
h (1,1)3[0]2 00 × (0,2)[0]2 00 × (2,0)[0]2 00i
The first part of (19) is a SU(3) generator and actually The tensor product
changes only the angular momentum with ∆L=2.
10
[F F] [4]6 20 = C [2]6 [2]6 [4]6 C(2,0)(2,0) (0,2)
× (0,2)[0]2 00 (2,0)[2]2(2,0)[2]2 (0,2)[0]2 (2)3 (2)3 (2)3
X
(20)
C20 C10 F [2]6 20F [2]6 20
× 20 20 11 1−1 (2,0)[2]2 11 (2,0)[2]2 1−1
of the operators (1) that are the pair raising Sp(12,R) operatorTE2 (19) to the experimentfor eachofthe con-
generatorschangesthenumberofbosonsby∆N =4and sideredbands. TheB(E2)strengthsbetweenthepositive
∆L = 2. It is obvious that this term in TE2 (19) comes paritystates ofthe yrastband, aswereattributed to the
from the symplectic extension of the model. In (19) e is SU(3)symmetry-adaptedbasis states (16)of the model,
the effective boson charge. are calculated. For these SU(3) multiplets, the proce-
ThetransitionprobabilitiesarebydefinitionSO(3)re- dure for their calculations actually coincides with that
duced matrix elements of transition operators TE2 (19) given in [45] and modified for the case of odd-odd nuclei
between the i initial and f final collective states in[46]. The theoreticalpredictionsfor the134Pr nucleus
| i− | i−
(16) are compared with the experimental data [30] in Figure
10. For comparison, the IBFFM and TQPTR results
B(E2;J J )= 1 f TE2 i 2 . (21) (ref. [30]) are also shown. From the figure one can see
i f
→ 2Ji+1 |h k k i| thegoodoverallreproductionoftheexperimentalvalues,
which is obviously better than the IBFFM and TQPTR
The basis states (16) can be considered as a result of
ones.
the coupling of the orbital (N,T);KLM;T (5) and
| 0i In Figure 11 the theoretical predictions for the 132La
spin φ wave functions. Since the spin I (I fixed)
Im − nucleus are compared with the experimental data [20].
is simply added to the orbital momentum L, the action
OneseesthattheexperimentalbehaviorofB(E2)values
of the transition operator TE2 concerns only the orbital
of this nucleus is also reproduced quite well.
part of the basis functions (16).
2 b] 000...344505 TETIBQhxFPpeFeoTMrrRiyment Pe a==r a0m.04.e50t 0ee1rbs9: 134Pr W.u.] 112400 EThxepoerryiment Pe a==r a0m.04.e50t0e0rs7:4 132La
2 E2;J J-2) [e0000....12235050 B(E2;J J-2) [18000
B(00..0150 N0=10 60 N0=44
0.00 8 10 12 14 16 18 20 22 24
8 10 12 14 16 18 20 22
J [h/2 ]
J [h/2 ]
FIG. 11: (Color online) Comparison of the theoretical and
FIG. 10: (Color online) Comparison of the theoretical and
experimentalvaluesfortheB(E2)transitionprobabilitiesfor
experimentalvaluesfortheB(E2)transitionprobabilitiesfor the132La.
the 134Pr. The theoretical predictions of the IBFFM and
TQPTR are shown as well.
Inordertoprovethecorrectpredictionsfollowingfrom
B. M1 transitions
our theoretical results we apply the theory to the two
nuclei 134Pr and 132La for which there are available ex-
perimental data for the transition probabilities between The structure of M1 transition operator between the
the states of the yrast bands. The application actually states of the yrast band can be defined in the following
consists of fitting the two parameters of the transition way: