Critical point symmetries in boson-fermion systems. The case of shape transition in odd nuclei in a multi-orbit model

0.12 MB·English

by  C. E. Alonso

Critical point symmetries in boson-fermion systems. The case of shape transition in
odd nuclei in a multi-orbit model
C.E. Alonso1, J.M. Arias1, and A. Vitturi1,2
1
Departamento de F´ısica At´omica, Molecular y Nuclear, Facultad de F´ısica
Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain
2
Dipartimento di Fisica Galileo Galilei and INFN, Via Marzolo 8, 35131 Padova, Italy
We investigate phase transitions in boson-fermion systems. We propose an analytically solvable
model (E(5/12)) to describe odd nuclei at the critical point in the transition from the spherical to
γ-unstablebehaviour. In the model, a boson core described within theBohr Hamiltonian interacts
with anunpairedparticle assumed tobemovingin thethreesingleparticle orbitals j=1/2,3/2,5/2.
Energy spectra and electromagnetic transitions at the critical point compare well with the results
obtained within the Interacting Boson Fermion Model, with a boson-fermion Hamiltonian that
describes thesame physical situation.
7
0 PACSnumbers: 21.60.-n,21.60.Fw,21.60.Ev
0
2
The occurrence of shape transitional behaviour char- tive orbits 2d ,2d and 3s , and also in the region
n 5/2 3/2 1/2
acterizes both even-even and odd nuclei. Because of the aroundA=130,inodd-neutronBaisotopeswithrelevant
a
J further complexity embodied in the description of the orbits 2d5/2,2d3/2 and 3s1/2, just to cite a few of them.
8 coupling of the odd particle to the evencore,formalisms At the critical point we extend the E(5/4) model to the
describing phase transitions have been initially mainly multi-j case, obtaining a new analytic solution, which

1 devised for even systems. Only recently the study of we will call E(5/12). Knowing the analytic form of the
v phasetransitionsinodd-nucleihasbeenrevitalized. The eigenfunctions, it is then easy to calculate spectroscopic
6
first case considered by Iachello[1–3] has been the case observables,such as electromagnetictransitionprobabil-
1
of a j=3/2 fermion coupled to a boson core that under- ities. To check the validity of the model, we will also
0
goes a transition from spherical to γ-unstable. At the present more microscopic calculations based on the In-
1
0 criticalpoint,anelegantanalyticsolution,calledE(5/4), teracting Boson Fermion Model[5], with a choice of the
7 has been obtained starting from the Bohr Hamiltonian. boson-fermionHamiltonianthatdescribesthesamephys-
0 The possible occurrence of this symmetry in 135Ba has ical situation. We will show that the E(5/12)and IBFM
/ been recently suggested[4]. However, the restriction of models give comparable behaviours.
h
t the fermion space to a single-j orbit makes difficult the We will thenconsiderthe couplingofa collectivecore,
-
l identification of a particular nucleus as critical. described within the quadrupole collective Bohr Hamil-
c
In this letter we considerthe richercase ofa collective tonian, with an odd particle moving in the 1/2,3/2,5/2
u
core coupled to a particle moving in the j=1/2, 3/2 and orbitals. Onecanusetheseparationofthesingle-particle
n
: 5/2 single-particle orbits, with the boson part undergo- angular momentum into a pseudo-spin and pseudo-
v
ingatransitionfromsphericitytodeformedγ-instability. orbitalangularmomentum(0and2)andassumenocou-
i
X Thissituationislikelytooccurinthemassregionaround pling due to the pseudo-spin. The pseudo-orbital part
A=190, for example in odd-neutron Pt where the rel- transforms as the representations (τF,0) of OF(5), with
r 1
a evant active orbitals around the Fermi surface are the τF being either 0 or 1[6]. One can therefore use a model
1
3p ,3p and2f ,inodd-protonIrisotopeswithac- Hamiltonian of the form
1/2 3/2 5/2
¯h2 1 ∂ ∂ 1 ∂ ∂ 1 Q2
H = − β4 + sin3γ − κ
2B "β4∂β ∂β β2sin3γ∂γ ∂γ 4β4 sin2 γ− 2πκ #
κ 3
X
+ u(β)+k g(β)[2Lˆ ◦Lˆ ]+k′ g(β)Lˆ2, (cid:0) (cid:1)
B F F
(1)
with where Lˆ and Lˆ are the five-dimensional boson and
B F
fermionorbitalangularmomenta,andthedot◦indicates
¯h2
u(β)=0,β <β ; u(β)=∞,β ≥β ; g(β)= , the five-dimensionalscalarproduct. This Hamiltonianis
w w 2Bβ2 the generalizationofthe Hamiltonianusedinthe E(5/4)
(2)
2
model. It should be noted that the lastterm was not in- dimensions E(5)and the algebrafor the fermion spaceis
cludedinE(5/4). Inthatcaseasingle-jwastreatedand, U(12), we will name the model as E(5/12).
consequently,asingleOF(5)representationwasinvolved. Withthischoicethetotalwavefunctioncanbefactor-
For the E(5/12) multi-j case two OF(5) representations ized in the form
are present and the last term in (1) is needed in order
to include the energy difference between j = 1/2 and
Ψ = F(β) [Φ(γ,θ ,η )⊗χ(η )](JBF) , (3)
j =3/2,5/2 single particle orbits. The choice of an infi- i orb spin
nitesquarewellintheβ variableshouldmimictheactual
situation at the critical point of the transition from the where η and η represent the particle coordinates
orb spin
spherical situation (potential with a minimum in β=0) associated with the pseudo-orbital and the pseudo-spin
to the deformedγ-instability (potentialwith a minimum spaces. The wave function Φ is characterized by good
forafinitevalueofβ). Inthiscase,sincethegroupchar- quantum numbers (τ ,τ ,τB,τF,L ,M ) and is given
1 2 1 1 BF L
acterizingthe coretransitionis the Euclideanone in five by
(τB,0) (τF,0) | (τ ,τ ) ℓ ℓ | L
Φ(τ1,τ2),τ1B,τ1F,LBF,ML(γ,θi,ηorb) = ℓB,mℓBX,ℓF,mℓF (cid:28) ℓ1B ℓ1F | L1BF2 (cid:29)(cid:28)mBℓB mFℓF | MBLF (cid:29)
× φ(τ1B,0),ℓB,mℓB(γ,θi) X(τ1F,0),ℓF,mℓF(ηorb) . (4)
The function X(η ) describes the five-dimensional level couplets. Most of the states are therefore degener-
orb
pseudo-orbital part, while χ(η ) is the pseudo-spin ate, starting from the first excited 3/2-5/2doublet. The
spin
wave function, which does not contribute to the energy. values of the angular momenta (twice) for each degen-
The equation for F(β) then acquires the familiar form erate state characterized by the OBF(5) boson-fermion
(τ ,τ ) labels are given in the lowest inset panel. The
1 2
¯h2 1 ∂ ∂ ¯h2 Λ lowest band with ξ = 1 comes from the coupling of the
− β4 + +u(β) F(β) = EF(β) ,
2Bβ4∂β ∂β 2Bβ2 boson states (τ1B,0) with the fermion states (τ1F =0,0).
(cid:20) (cid:21) (5) It is worth mentioning that all these states with τ1F=0
with havepreciselythesameenergiesasthe statesinthe even
E(5) model. The set of three bands associated to the
Λ =τB(τB +3)+k[τ (τ +3)+τ (τ +1) first excited ξ = 1 structure comes from the coupling of
1 1 1 1 2 2
− τB(τB +3)−τF(τF +3) +k′τF(τF +3). (6) (τ1B,0) with (τ1F = 1,0) which produces states (τ1,τ2)
1 1 1 1 1 1 of the full boson-fermion system labelled as (τB +1,0),
1
(cid:3) (τB,1) and (τB − 1,0)[6]. In addition to these ξ = 1
The solutions of this equation, as in the case of the 1 1
bands, excited ξ =2,3,... structures appear.
E(5/4) model, are given in terms of Bessel functions of
We can now calculate quadrupole transitions. In this
order ν = Λ+9/4, in the form
caseonlythe collectivepartofthe quadrupoletransition
p operator has been used, as in Refs. [1–3]. The calcula-
Fξ,{(τ1,τ2),τ1B,τ1F}(β) = cξ,{(τ1,τ2),τ1B,τ1F} tionreducestoasinglenumericalintegralovertheβ vari-
×β−3/2Jν(xξ,{(τ1,τ2),τ1B,τ1F}β/βw) , (7) able. In Fig. 1 we have chosen to represent the B(E2)’s
between the highest spins of initial and final multiplets
where cξ,{(τ1,τ2),τ1B,τ1F} is a normalization constant and (the B(E2)associatedwith the lowest5/2-1/2transition
ixnξg,{(eτi1g,eτ2n)v,τa1Blu,τe1Fs}atrheegiξv-ethn azesro of Jν(z). The correspond- igsenneorramtea.liNzeodtetoth1a0t0s)t,astienscewmithosdtiffleevreelnstaτr1Fe marueltnipotlycodne--
nected between them by E2 transitions.
Eξ,{(τ1,τ2),τ1B,τ1F} = 2¯hB2 xξ,{(τ1,βτ2),τ1B,τ1F} 2. (8) simTiolacrhpechkystihcaelvsailtiudaittyionofwthitehimnotdheel wfreamcaenwodrekscorifbtehae
(cid:18) w (cid:19) Interacting Boson Fermion Model (IBFM)[5], where the
The corresponding spectrum is given in the upper coupling of a single fermion to the even-even boson core
frame of Fig. 1 for the choice of the parameters k = is described by the Hamiltonian
−1/4 and k′ = 5/2. Each state is characterized by the
H = H +H +V , (9)
ξ,τB,τF(τ ,τ )quantumnumbersandarrangedinbands B F BF
1 1 1 2
according to the counting quantum number ξ. The fac- where the term V couples the boson and fermion de-
BF
torization of the pseudo-spin part give rise to repeated grees of freedom. We remind that critical point symme-
3
quadratic Casimir operators of the OB(6) and OB(5) al-
E(5/12) 2,0(2,0) 0,1(1,0)
gebras. By using the intrinsic frame formalism (see for
4
12 3,1(3,1) 9 instance[10]),onecanshowthatthecriticalpointoccurs
23,,00((23,,00))217 2431 0.07.5527510,,00((10,,00)21),,11((321,,1008)) 179254511114521112,,11((12,,7114))110411012,,11((01,,00))ξ=2112 bateIexnncraeitxnt=aelno54gsNNiyv−−ew88ly.itshtutdhiiesdbiossRonefsH.am[7,ilt1o1n,ia1n2,],wohniechcohualds
7 0,1(1,0) choose[13] in our case a boson-fermion Hamiltonian of
10,,00((10,,00))1600 ξ=2 (τ1B+1,0) (τ1B,1) (τ1B-1,0) κ=−1/4 the form
1ξ=1 ξ=1 κ′=5/2 1−x
H = x nˆ +nˆ +nˆ − Qˆ ·Qˆ
IBFM 3,1(3,1) d 3/2 5/2 N BF BF
2,0(2,0) = xC(cid:0) (UBF(5)) (cid:1)
23,,00((23,,00))5161 1217 0.070.229886310,,00((10,,0102),)1,01(,2(113,0(0,100),)0)145 1385 9014310821,,11((21,,6110))8130812,1,1((01,0,0))12 016,1(21,0) wopheircahtoimr−apnl1ide2−sNq1auxac(cid:2)seCipr2ta(arOitniBcFlceh(o6eni)c)eer−goiCfes2t(h[O5e]B.tFoInt(a5lp))aq(cid:3)ru,taicdurluap(r1,o2liet)
1,0(1,0)14 is worth noting that the j = 3/2 and 5/2 orbits are de-
0
0,0(0,0)10 generate and higher in energy than the j = 1/2 orbit.
Such a Hamiltonian has been used thoroughly in the lit-
(0,0):1 ; (1,0):3,5 ; (2,0):3,5,7,9 ; (3,0):1,5,72,9,11,13 erature and, in particular, in a previous study of shape
(1,1):1,3,5,7 ;(2,1):1,32,52,72,92,11; (3,1):1,32,53,73,93,113,132,15 phase transition in odd-mass nuclei using a supersym-
metric approach[14]. It has the advantage that one re-
covers,intheextremecases,theBose-Fermisymmetry[5]
FIG. 1: Upper frame: Energy levels in the odd system at associated with OBF(6)⊗UF(2) (for x=0) and the vi-
s
the E(5/12) critical point (normalized to the energy of the brationalUBF(5)⊗UF(2)(forx=1). Ontheotherhand,
first excited state). Each (degenerate) state is character- s
suchanIBFMHamiltonianonlyapproximatelypreserves
B F
kize=d −by1/t4h,ekξ′,=τ15,/τ21h,(aτv1e,τb2e)enquuasnetduminntuhmebHearms.ilTtohneiavnal(u1e)s. [13] the τ1B and τ1F quantum numbers, while those are
goodquantum numbers for E(5/12). Although it will be
Thevaluesof (twice) theangularmomenta foreach degener-
possible to establish anyway a correspondence between
atestatecan beextracted from theinset inthelowest panel.
SomeselectedvaluesofB(E2)’s,correspondingtothetransi- E(5/12)andIBFMstates,selectionrulesforelectromag-
tionsbetweenthehighestspinsofinitialandfinalmultiplets, netic and one-particle transfer transitions will be differ-
are given in the figure (the B(E2) associated with the lowest ent.
5/2-1/2 transition is normalized to 100). Lower frame: En- For a better comparison with the E(5/12) results we
ergy levels in the odd system within the IBFM Hamiltonian have therefore preferred to use here at the critical point
(13) at the critical point. A number of bosons N = 7 has a boson-fermion Hamiltonian of the form
been assumed. The valuesof k and k′ are thesame as in the
1−x
E(5/12) Hamiltonian. Forthe BE(2)’s, see theupperframe. H = xC (UB(5))− C (OB(6))−C (OB(5))
1 2 2
2N
k (cid:2) (cid:3)
+ C (OBF(5))−C (OB(5))−C (OF(5))
tries,asE(5),X(5)orE(5/4),althoughinspiredbymod- 2 2 2
2N
els with interacting bosons, can only be approximately k′ (cid:2) (cid:3)
obtained within these models. However the comparison + C2(OF(5)) , (13)
2N
with the IBM and IBFM turned out to be useful, e.g. in
the case of the E(5) and E(5/4)symmetries, for a better whereC2(OF(5)),C2(OBF(5))arethequadraticCasimir
understanding of the characteristics of these models[7– operators of the OF(5) and OBF(5) algebras. This
9]. In these cases, the Hamiltonian used for the bosonic Hamiltonianisdesignedtomimicasmuchaspossiblethe
part was a parametrized one, that produces a transition corresponding Hamiltonian in E(5/12), since the terms
between spherical and γ-unstable shapes, of the form 2Lˆ ◦ Lˆ and Lˆ2 in (1) can be written in terms of
B F F
H =xnˆ −((1−x)/N) Qˆ ·Qˆ , (10) Casimir operators as 2LˆB ◦LˆF = (1/2)[C2(OBF(5))−
B d B B C (OB(5)) − C (OF(5))] and Lˆ2 = (1/2)C (OF(5)).
2 2 F 2
wheretheoperatorsintheHamiltonianaregivenbynˆd = This boson-fermion Hamiltonian preserves by construc-
µd†µdµ and QˆB = (s† ×d˜+d† ×s˜)(2), and N is the tion the τ1B and τ1F quantum numbers and all states are
total number of bosons. This Hamiltonian can be recast therefore characterized by the same quantum numbers
P
into the form and degeneracies as in the E(5/12) case. For a better
1−x clarification, we show in Fig. 2 the lattice of algebras
H =xC (UB(5))− C (OB(6))−C (OB(5)) ,
B 1 2N 2 2 relevant for our problem.
(11) Thecorrespondingspectrumisgiveninthelowerframe
(cid:2) (cid:3)
where C (UB(5)) is the linear Casimir operator of the ofFig.1. AnumberN=7ofbosonsis assumedasatypi-
1
UB(5) algebra, while C (OB(6)) and C (OB(5)) are the calvalue. Other choicesofN do notalterthe qualitative
2 2
4
lection rules valid in the E(5/12) model also apply to
UB(6) ⊗ UF(12) the (τ1B,τ1F,τ1,τ2) quantum numbers of initial and final
states in the IBFM approach. The B(E2) values of se-
lected transitions are explicitly given in the figure. All
UB(5) OB(6) OF (6) ⊗ UF (2) Btr(aEn2si)tivoanlu.eSsimarielanrolyrmtoalwizheadtthoa1p0p0efnosrftohretfihresetn5e/r2gi-es1/o2f
k s
the levels,the overallbehaviourisverysimilarto thatof
OB(5) OF (5) the E(5/12).
k
BF
O (5)
Level Energy Energy
B F
OBF(3) SUF (2) ξ,τ1 ,τ1 (τ1,τ2) E(5/12) IBFM
s
BF
Spin (3)
1,1,0 (1,0) 1.00 1.00
1,2,0 (2,0) 2.20 2.30
2,0,0 (0,0) 3.03 2.86
1,0,1 (1,0) 2.20 2.69
FIG. 2: The lattice of algebras involved in theIBFM Hamil-
1,1,1 (1,1) 3.00 3.82
tonian (13).
behaviour of our results. With the choice N=7 the crit-
TABLE I: Energies of selected states in E(5/12) and IBFM
ical point occurs at x = 4N−8 = 20 ≈ 0.74. In the
crit 5N−8 27 (normalized to the first excited multiplet). The values k =
Hamiltonian the values for k and k′ have been assumed −1/4,k′ =5/2 havebeen used in both Hamiltonians (1) and
to be equal to the corresponding values in the E(5/12) (13). In the IBFM calculation, a numberN=7 of bosons has
case presented in the upper frame. been used.
The spectrum has a structure that clearly resem-
bles the one obtained within the E(5/12) model (upper
frame). Energiesof selectedstates in the two models are
compared in Table I. As noted before, in the E(5/12) In this letter we have presented an analytic model
model allstates with τF=0(i.e. with the oddparticle in (E(5/12)) for the critical point of the phase transition
1
the j=1/2 state) have precisely the same energies as the from spherical to γ-unstable shapes in odd nuclei, i.e.
states in the even E(5) model. Similarly, in the IBFM a mixture of fermion and boson degrees of freedom. In
these states have the same energy of the corresponding the model the fermion is allowed to occupy many single
states in the boson Hamiltonian (10). The agreement particleorbitals. Spectrumandintensitiesofelectromag-
for the energies of these states in E(5/12) and IBFM is netictransitionshavebeencomparedwiththoseobtained
therefore that already discused in ref.[7] for the bosonic within the Interacting Boson Fermion Model showing
part. Excited bands with τF=1 (i.e. with the odd parti- comparablebehaviours. Wethink thatourmodel,which
1
cle in the j=3/2,5/2 states), on the other hand, seem to includes the more realistic many j-orbits case compared
be somewhat compressed in E(5/12) with respect to the withtheE(5/4)model,shouldoffermorechancesforthe
IBFM. occurrence of critical point symmetries in odd nuclei, in
Thesamekindofagreementisalsopresentintheelec- additiontothosealreadyfoundinthecaseofevennuclei.
tromagnetic transitions. Consistently with the choice This work has been partially supported by the Span-
presented in the E(5/12), we assume in the IBFM as E2 ish Ministerio de Educaci´on y Ciencia and by the Euro-
transitionoperatorjusttheboson(collective)quadrupole peanregionaldevelopmentfund(FEDER)underproject
operator, i.e. TˆE2 = Qˆ . The order of magnitude of number FIS2005-01105, and by INFN. Andrea Vitturi
B
the differentB(E2)valuesarereflectedindifferentthick- acknowledgesfinancialsupportfromSEUI(SpanishMin-
ness and styles of the arrows shown in Fig. 1. Given isterio de Educaci´on y Ciencia) for a sabbatical year at
the choice of the IBFM Hamiltonian (13) the same se- University of Sevilla.
[1] F. Iachello, Phys. Rev. Lett. 95, 052503 (2005) versit´a degli StudidiCamerino, ed. G. Lo Bianco
[2] F. Iachello, in Symmetries and low-energy phase transi- [3] M.A. Caprio and F. Iachello, Nucl. Phys. A (2006),
tions in nuclear structure physics, Camerino, 2005, Uni- doi:10.1016/j.nuclphysa.2006.10.032
5
[4] M.S. Fetea et al., Phys. Rev. C 73, 051301(R) (2006) [10] J. Jolie, P. Cejnar, R.F. Casten, S. Heinze, A. Linne-
[5] F. Iachello and P. Van Isacker, The Interacting Boson mann,andV.Werner,Phys.Rev.Lett.89,182502(2002)
Fermion Model, Cambridge University Press, (1991) [11] F. Iachello, Phys. Rev. Lett. 85, 3580 (2000)
[6] P.VanIsacker,A.FrankandH-Z.Sun,Ann.Phys.(N.Y) [12] P.S. Turner and D.J. Rowe, Nucl. Phys. A756, 333
157, 183 (1984) (2005)
[7] J. M. Arias, C. E. Alonso, A. Vitturi, J. E. Garc´ıa- [13] C.E. Alonso, in Proceedings of 3rd Workshop on
Ramos, J. Dukelsky, and A. Frank, Phys. Rev. C 68, shape phase transitions and critical point phenom-
041302(R) (2003) ena in nuclei, Athens, 2006, ed. D. Bonatsos,
[8] C.E. Alonso, J.M. Arias, L. Fortunato and A. Vitturi, http://www.inp.democritos.gr/∼lenis/
Phys. Rev. C 72, 061302(R) (2005) [14] J. Jolie, S. Heinze,, P. Van Isacker, and R.F. Casten,
[9] C. E. Alonso, J. M. Arias, and A. Vitturi, Phys. Rev. C Phys. Rev. C 70, 111305(R) (2004)
74, 027301 (2006)