SD-PAIR SHELL MODEL FOR EVEN-EVEN SYSTEMS

January 9, 2009 9:3 WSPC/INSTRUCTION FILE01189International Journal of Modern Physics EVol. 17, Supplement (2008) 245–255c World Scientific Publishing CompanySD-PAIR SHELL MODEL FOR EVEN-EVEN SYSTEMSYAN-AN LUODepartment of Physics, Nankai University,Tianjin 300071, P. R. Chinaluoya@nankai.edu.cnFENG PANDepartment of Physics, Liaoning Normal University,Dalian 116029, P. R. ChinaJERRY P. DRAAYERDepartment of Physics and Astronomy, Louisiana State University,Baton Rouge, LA 70803, USAPING-ZHI NINGDepartment of Physics, Nankai University,Tianjin 300071, P. R. ChinaA nucleon-pair shell model was proposed in 1993 by J. Q. Chen. Due to the success of theinteracting boson model, the full shell model space in the nucleon-pair shell model wastruncated to SD-pair subspace, called SD-pair shell model. Within this model, the experimental spectra in medium-weight and heavy nuclei, limiting cases in the interactingboson model, phase transitions, etc. can be reproduced approximately.1. IntroductionThe discovery of collective motions, such as collective vibration, collective rotation,giant resonances, etc. in nuclei is of great significance. The shell model containsall the nucleonic degrees of freedom and can, in principle, account for any of thesecollective phenomena. Therefore, description of nuclear collective motions in termsof the shell model is a challenge in nuclear structure theory. But even in relativelysimple cases of the medium weight and heavy nuclei, shell model configurations arearound 1014 -1018 and modern computers fail for all these cases. Therefore, if wewant to use the shell model methods to study the medium weight and heavy nuclei,realistic truncations of the model space must be found.In the past decades, many attempts were made along this direction. Based on thegeneralized Wick theorem for fermion clusters,1 a nucleon-pair shell model(NPSM)has been proposed for nuclear collective motion in which collective nucleon pairswith various angular momenta serve as the building blocks.2,3245

January 9, 2009 9:3 WSPC/INSTRUCTION FILE24601189Y.-A. Luo et al.One advantage of the NPSM is that it allows various truncation schemes rangingfrom the truncation to the SD-pair subspace up to the full shell-model space.The tremendous success of the interacting boson model4 has suggested a possibletruncation, the truncation to the SD-pair subspace with SD collective nucleonpairs as the building blocks.5,6 Therefore we truncated the full shell-model spacefor medium and heavy mass nuclei to the collective SD-pair subspace in the NPSM,called the SD-pair shell model (SDPSM).7Our previous work and Zhao’s work show that the collectivity of low-lying statescan be described reasonably well within the SDPSM.7–13,16–19 Problems, such asthe nuclear phase transition, limiting cases of the IBM, validity of the pseudo-spin,etc. were also discussed. In this paper, we concentrate on what we studied by usingthis model.The model will be described in Sec. 2, the application to study Xe isotopes, thelimiting cases of the IBM and the phase transition will be given in Sec. 3, 4 and 5,respectively. Finally, a brief summary will be presented.2. A Brief Review of the SD-Pair Shell ModelIn the SDPSM, a Hamiltonian we used consists of the single-particle energy termH0 , and a residual interaction containing the multipole pairing between like nucleons and the multipole-multipole interaction between all nucleons,XXH (H0 (σ) V (σ)) κt Qtπ · Qtν ,(1)σ π,νH0 XaAs†ν Xcd a n̂a ,tV (σ) Xsgs As† · As sy0 (cds) Cc† Cd† ,νQt Xkt Qt · Qt ,(2)(ri )t Yt (θi φi ) ,(3)tnXi 1where a and n̂a are the single-particle energy and the number operator, respectively.The E2 and M1 transition operators areT (E2) eπ Q2π eν Q2ν ,M 1 M 1(π) M 1(ν),M 1(σ) glσ L gsσ S ,(4)(5)where eπ and eν are effective charge of the protons and neutrons, glσ and gsσ arethe orbital and spin effective gyro-magnetic ratios.In the SDPSM, the model space is constructed by collective SD pairs. Thecollective SD pair, designated as Asν † with s 0, 2, is built from many non-

January 9, 2009 9:3 WSPC/INSTRUCTION FILE01189SD-Pair Shell Model for Even-Even Systems247collective pairs Asν (cd)† in the single-particle orbits c and d in one major shell,XAsν † y(cds)Asν (cd)† ,Asν (cd)† (Cc† Cd† )sν ,(6)cdwhere y(cds) are structure or distribution coefficients satisfying the symmetryθ(cds) ( )c d s .y(cds) θ(cds)y(dcs),(7)It is assumed that there is only one collective S pair and D pair. The creation operator for N pairs r1 , · · · , rN , coupled successively to the total angular momentumJN and with Ji as the angular momentum for the first i pairs, is designated byJ †J †JAMNN (ri , Ji ) AMNN (· · · ((Ar1 † Ar2 † )J2 Ar3 † )J3 · · · ArN † )MNN A(r1 r2 · · · rN ; J1 J2 · · · JN MN )† .(8)The overlap between two states with N fermion-pair is the key quantity, since allthe one- and two-body matrix elements, or the matrix elements of the pair creationoperator and the multipole operator can be expressed in terms of the overlap. Theoverlap between two states with N fermion-pair isJ0Jh0 AMN (si , Ji )AMN (ri , Ji )† 0iNN0 hs1 · · · sN ; J10 · · · JN 1 JN r1 · · · rN ; J1 · · · JN 1 JN iJN sN JN 10b (JbN 1 /JN )( )01XXHN (sN ) · · · Hk 1 (sN )k N Lk 1 ···LN 1 0 ψk δsN ,rk δLk 1 ,Jk 1 hs1 . . . sN 1 ; J10 . . . JN 1 r1 . . . rk 1 , rk 1 . . . rN ; J1 . . . Jk 1 Lk . . . LN 1 i 1XX 00hs1 . . . sN 1 ; J10 . . . JN 1 r1 . . . ri . . . rk 1 , rk 1 . . . rN ; J1 . . . Ji 1 Li . . . LN 1 i ,i k 1 r 0 Li .Lk 2i(9) where Jˆ 2J 1, Hk (sN ) are essentially Racah coefficients, induced by variousre-coupling procedures, ψk is a constant coming from the annihilation of the pairArk † by AsN , and thus depends on the structure of these two pairs, while ri0 rep0resents a new collective pair B ri † resulting from a double–process. First the pairAsN transforms the pair Ark † into a particle–hole pair P t with angular momentumt, which then propagates forward, crosses over the pairs rk 1 , · · · , ri 1 , and finally0 r 0transforms the pair Ari † into the new pair B ri † Ari † , P t i , with a new distribution function y 0 (ak ai ri0 ) depending on the structure of all the three pairs Ark † , Ari †and AsN † , and the intermediate quantum numbers Li . . . Lk 2 Lk 1 . The right sideof Eq. (9) is a linear combination of the overlap for N -1 pairs, therefore the overlapcan be calculated recursively. The details of the model can be found in Refs. 2, 3,7.

January 9, 2009 9:3 WSPC/INSTRUCTION FILE24801189Y.-A. Luo et al.3. Spectra of Even-Even Xe NucleiTo investigate whether the SDPSM can account for the main feature of the lowlying states, even-even Mo(Kr), Xe(Ba) and Pt isotopes in 28-50, 50-82 and 82-128shells were studied, respectively. The Hamiltonian adopted wasH H0 VSDI (ν) VSDI (π) κQ2 (π) · Q2 (ν),(10)where VSDI (σ), σ π(ν), is the surface delta interaction between like nucleons. The collective pairs were taken from the 0 1 and 21 eigenstates of the Hamiltonian7H0 (σ) VSDI (σ) in two-particle system. As an example, the results for Xe nuclei in50-82 shell are presented here. The single particle energy levels are given in Table 1,and the parameters obtained by fitting the spectra are listed in Table 2.The calculated spectra for low-lying states are given in Fig. 1. It is seen thatthe experiments can be reproduced very well. The results for the other isotopes canbe found in Refs. 8, 14–20.Table 1. The single particle (hole) energies for proton (neutron) for133 Sb (131 Sn ).82 508151lj π (MeV)g7/20d5/20.963d3/22.69h11/22.76s1/22.99lj ν (MeV)d3/20h ven Systems251Fig. 3. The γ-unstable spectra. Some relative B(E2) ratios are also shown with the effective chargesfixed at eπ 1.5e and eν 0.5e.Fig. 4. The rotational spectra for the coupled proton-neutrons system in the SDPSM.like-nucleon pairs, and the D-pair was still determined by commutator (11). Withthe quadrupole-quadrupole interaction strength fixed as κ 0.1MeV/r04 , some lowlying states are shown in Fig. 4, in which the levels are arranged into bands. It is

January 9, 2009 9:3 WSPC/INSTRUCTION FILE25201189Y.-A. Luo et al.clear, as shown in Ref. 23, that the rotational level pattern can be reproduced verywell in the SDPSM.45. Phase Transition in the SDPSMSince the SDPSM is built up from SD pairs, it is expected that the SDPSM canreproduce similar transitional patterns to those of the IBM. Two transitional patterns, vibration-rotation and vibration-γ unstable, were studied in the SDPSM. Asan example, the vibration-rotation transitional pattern is discussed in the following.The details can be found in Refs. 21, 22.In order to study the vibration-rotation transitional patterns, an identical nucleon system with N 4 in the N 9 oscillator shell was considered. A schematicHamiltonian was adopted as follows.H αGS † S (1 α)κQ2 · Q2 ,(12)where α is the control parameter 0 α 1. The parameters G and κ were fixedas 0.1 MeV and 0.02 MeV/r04 .Some low-lying energy levels as a function of α are shown in Fig. 5. We can seethat similar to the results reported in Refs. 24–26 there is indeed a minimum in theexcited 0.10.20.30.40.50.60.70.80.91xFig. 5. Some low-lying energy levels across the transitional region, where α 0 corresponds to theaxially deformed shape with a rotational spectra and α 1 to the spherical shape with vibrationalspectra.

January 9, 2009 9:3 WSPC/INSTRUCTION FILE01189SD-Pair Shell Model for Even-Even Systems2536. Pseudo-Spin Transformation in the SDPSMTo study the effect of the pseudo-spin transformation on the SDPSM, a Hamiltonian as Eq. (12) was considered and the interaction strengths were set to beG 0.14 MeV and κ 0.08 MeV. The case we studied has N 4 pairs in the1h9/2 2f7/2 2f5/2 3p3/2 3p1/2 (denoted by h9/2 fp) valence space, the original space. Itis noted that the 1h11/2 level which is a part of the full hf p shell was taken to bea defector, i.e., the 1h11/2 level was considered to be part of the core.Under the pseudo-spin transformation, the Hamiltonian was transformed toH̃ αGS̃ † S̃ (1 α)κ0 Q̃2 · Q̃2(13)with the modified strength of the quadrupole interaction κ0 0.11 MeV and thepairing interaction remained G 0.14 MeV since the pairing interaction is invariant under pseudo-spin transformation. The original model space was transformedg shell), the pseudo-spininto a 1g̃9/2 1g̃7/2 2d 5/2 2d 3/2 3s̃1/2 valence space (or full gdstransformed space which is the space on which the new SD pairs were constructed.Results for H are shown in Fig. 6(b) and H̃ in Fig. 6(c). Generally speaking,the spectra in the transformed space follows that in the original space with slightlylower values at higher energy towards the quadrupole limit. The details can befound in Ref. 27.In the same model, we also studied other problems, the details can be found inRefs. 9, 11, 28–30.SU(3)8 particlesSD-pair shell modelSU(2)8 particlesSD-pair shell model88(h f f p p )pseudo-(g g d d s )9/2 7/2 5/2 3/2 1/29/2 7/2 5/2 3/2 1/28.08.07.0v .20.40.60.81.0 80.01.0(d)Fig. 6. Energy levels (in MeV): (a) in the SU 3 limit; (b) as they vary with the parameter strengthα in the h9/2 fp-shell; (c) the pseudo-spin-transformed counterpart of (b), namely as they varygwith the parameter strength α in the gds-shell;and (d) in the quasi-spin SU 2 limit.