Microscopic description of large-amplitude shape-mixing dynamics with inertial functions derived in local quasiparticle random-phase approximation
Nobuo Hinohara, Koichi Sato, Takashi Nakatsukasa, Masayuki Matsuo, Kenichi Matsuyanagi
aa r X i v : . [ nu c l - t h ] J a n Microscopic description of large-amplitude shape-mixing dynamics with inertialfunctions derived in local quasiparticle random-phase approximation
Nobuo Hinohara, Koichi Sato,
2, 1
Takashi Nakatsukasa, Masayuki Matsuo, and Kenichi Matsuyanagi
1, 4 Theoretical Nuclear Physics Laboratory, RIKEN Nishina Center, Wako 351-0198, Japan Department of Physics, Graduate School of Science, Kyoto University, 606-8502 Kyoto, Japan Department of Physics, Faculty of Science, Niigata University, Niigata 950-2181, Japan Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: November 17, 2018)On the basis of the adiabatic self-consistent collective coordinate method, we develop an effi-cient microscopic method of deriving the five-dimensional quadrupole collective Hamiltonian andillustrate its usefulness by applying it to the oblate-prolate shape coexistence/mixing phenomena inproton-rich , , Se. In this method, the vibrational and rotational collective masses (inertial func-tions) are determined by local normal modes built on constrained Hartree-Fock-Bogoliubov states.Numerical calculations are carried out using the pairing-plus-quadrupole Hamiltonian including thequadrupole-pairing interaction within the two major-shell active model spaces both for neutrons andprotons. It is shown that the time-odd components of the moving mean-field significantly increasethe vibrational and rotational collective masses in comparison with the Inglis-Belyaev crankingmasses. Solving the collective Schr¨odinger equation, we evaluate excitation spectra, quadrupoletransitions and moments. Results of the numerical calculation are in excellent agreement with re-cent experimental data and indicate that the low-lying states of these nuclei are characterized asan intermediate situation between the oblate-prolate shape coexistence and the so-called γ unstablesituation where large-amplitude triaxial-shape fluctuations play a dominant role. PACS numbers: 21.60.Ev, 21.10.Re, 27.50.+e
I. INTRODUCTION
The major purpose of this paper is to develop an effi-cient microscopic method of deriving the five-dimensional(5D) quadrupole collective Hamiltonian [1–4] and illus-trate its usefulness by applying it to the oblate-prolateshape coexistence/mixing phenomena in proton-rich Seisotopes [5–8]. As is well known, the quadrupole collec-tive Hamiltonian, also called the general Bohr-MottelsonHamiltonian, contains six collective inertia masses (threevibrational masses and three rotational moments of in-ertia) as well as the collective potential. These sevenquantities are functions of the quadrupole deformationvariables β and γ , which represent the magnitude andtriaxiality of the quadrupole deformation, respectively.Therefore, we also call the collective inertial masses ‘in-ertial functions.’ They are usually calculated by meansof the adiabatic perturbation treatment of the movingmean field [9], and the version taking into account nu-clear superfluidity [10] is called the Inglis-Belyaev (IB)cranking mass or the IB inertial function. Its insuffi-ciency has been repeatedly emphasized, however (see e.g.,Refs. [11–14]). The most serious shortcoming is that thetime-odd terms induced by the moving mean field areignored, which breaks the self-consistency of the theory[15, 16]. In fact, one of the most important motives ofconstructing microscopic theory of large-amplitude col-lective motion was to overcome such a shortcoming ofthe IB cranking mass [15].As fruits of long-term efforts, advanced microscopictheories of inertial functions are now available (see Refs.[15–26] for original papers and Refs. [27, 28] for reviews). These theories of large-amplitude collective motion havebeen tested for schematic solvable models and appliedto heavy-ion collisions and giant resonances [18, 26]. Fornuclei with pairing correlations, Dobaczewski and Skalskistudied the quadrupole vibrational mass with use of theadiabatic time-dependent Hartree-Fock-Bogoliubov (AT-DHFB) theory and concluded that the contributions fromthe time-odd components of the moving mean-field sig-nificantly increase the vibrational mass compared to theIB cranking mass [16]. Somewhat surprisingly, however,to the best of our knowledge, the ATDHFB vibrationalmasses have never been used in realistic calculations forlow-lying quadrupole spectra of nuclei with superfluid-ity. For instance, in recent microscopic studies [29–34] bymeans of the 5D quadrupole Hamiltonian, the IB crank-ing formula are still used in actual numerical calcula-tion for vibrational masses. This situation concerning thetreatment of the collective kinetic energies is in markedcontrast with the remarkable progress in microscopic cal-culation of the collective potential using modern effectiveinteractions or energy density functionals (see Ref. [35]for a review).In this paper, on the basis of the adiabatic self-consistent collective coordinate (ASCC) method [36],we formulate a practical method of deriving the 5Dquadrupole collective Hamiltonian. The central con-cept of this approach is local normal modes built onconstrained Hartree-Fock-Bogoliubov (CHFB) states [37]defined at every point of the ( β, γ ) deformation space.These local normal modes are determined by the lo-cal QRPA (LQRPA) equation that is an extension ofthe well-known quasiparticle random-phase approxima-tion (QRPA) to non-equilibrium HFB states determinedby the CHFB equations. We therefore use an abbrevia-tion ‘CHFB+LQRPA method’ for this approach. Thismethod may be used in conjunction with any effec-tive interaction or energy density functional. In thispaper, however, we use, for simplicity, the pairing-plus-quadrupole (P+Q) force [38, 39] including thequadrupole-pairing force. Inclusion of the quadrupole-pairing force is essential because it produces the time-oddcomponent of the moving field [40].To examine the feasibility of the CHFB+LQRPAmethod, we apply it to the oblate-prolate shape coex-istence/mixing phenomena in proton-rich , , Se [5–8, 41, 42]. These phenomena are taken up because weobviously need to go beyond the traditional frameworkof describing small-amplitude vibrations around a singleHFB equilibrium point to describe them; that is, theyare very suitable targets for our purpose. We shall showin this paper that this approach successfully describeslarge-amplitude collective vibrations extending from theoblate to the prolate HFB equilibrium points (and viceversa). In particular, it will be demonstrated that wecan describe very well the transitional region between theoblate-prolate shape coexistence and the γ unstable sit-uation where large-amplitude triaxial-shape fluctuationsplay a dominant role.This paper is organized as follows. In Sec. II, weformulate the CHFB+LQRPA as an approximation ofthe ASCC method and derive the 5D quadrupole col-lective Hamiltonian. In Sec. III, we calculate the vibra-tional and rotational masses by solving the LQRPA equa-tions, and discuss their properties in comparison withthose calculated by using the IB cranking formula. InSec. IV, we calculate excitation spectra, B ( E , , Se and discuss properties of the oblate-prolateshape coexistence/mixing in these nuclei. Conclusionsare given in Sec. V.
II. MICROSCOPIC DERIVATION OF THE 5DQUADRUPOLE COLLECTIVE HAMILTONIANA. The 5D quadrupole collective Hamiltonian
Our aim in this section is to formulate a practicalmethod of microscopically deriving the 5D quadrupolecollective Hamiltonian [1–4] H coll = T vib + T rot + V ( β, γ ) , (1) T vib = 12 D ββ ( β, γ ) ˙ β + D βγ ( β, γ ) ˙ β ˙ γ + 12 D γγ ( β, γ ) ˙ γ , (2) T rot = 12 X k =1 J k ( β, γ ) ω k , (3)starting from an effective Hamiltonian for finite many-nucleon systems. Here, T vib and T rot denote the ki- netic energies of vibrational and rotational motions, while V ( β, γ ) represents the collective potential. The veloci-ties of the vibrational motion are described in terms ofthe time-derivatives ( ˙ β , ˙ γ ) of the quadrupole deforma-tion variables ( β , γ ) representing the magnitude and thetriaxiality of the quadrupole deformation, respectively.The three components ω k of the rotational angular ve-locity are defined with respect to the intrinsic axes asso-ciated with the rotating nucleus. The inertial functionsfor vibrational motions (vibrational masses), D ββ , D βγ ,and D γγ , and the rotational moments of inertia J k arefunctions of β and γ .As seen in the recent review by Pr´ochniak and Ro-hozi´nski [4], there are numerous papers on microscopicapproaches to the 5D quadrupole collective Hamiltonian;among them, we should quote at least early papers byBelyaev [2], Baranger-Kumar [43, 44], Pomorski et al.[12, 13], and recent papers by Girod et al. [33], Nikˇsi´cet al. [29, 30], and Li et al. [31, 32]. In all these works,the IB cranking formula is used for the vibrational iner-tial functions. Below, we outline the procedure of deriv-ing the vibrational and rotational inertial functions onthe basis of the ASCC method. B. Basic equations of the ASCC method
To derive the 5D quadrupole collective Hamiltonian H coll starting from a microscopic Hamiltonian ˆ H , we usethe ASCC method [36, 45]. This method enables us to de-termine a collective submanifold embedded in the large-dimensional TDHFB configuration space. We can usethis method in conjunction with any effective interactionor energy density functional to microscopically derive thecollective masses taking into account time-odd mean-fieldeffects. For our present purpose, we here recapitulatea two-dimensional (2D) version of the ASCC method.We suppose existence of a set of two collective coordi-nates ( q , q ) that has a one-to-one correspondence to thequadrupole deformation variable set ( β, γ ) and try to de-termine a 2D collective hypersurface associated with thelarge-amplitude quadrupole shape vibrations. We thusassume that the TDHFB states can be written on thehypersurface in the following form; | φ ( q , p , ϕ , n ) i = e − i P τ ϕ ( τ ) e N ( τ ) | φ ( q , p , n ) i = e − i P τ ϕ ( τ ) e N ( τ ) e i ˆ G ( q , p , n ) | φ ( q ) i , (4)withˆ G ( q , p , n ) = X i =1 , p i ˆ Q i ( q ) + X τ = n,p n ( τ ) ˆΘ ( τ ) ( q ) , (5)ˆ Q i ( q ) = ˆ Q A ( q ) + ˆ Q B ( q )= X αβ [ Q Aαβ ( q ) a † α a † β + Q A ∗ αβ ( q ) a β a α + Q Bαβ ( q ) a † α a β ] , (6)ˆΘ ( τ ) ( q ) = X αβ [Θ ( τ ) Aαβ ( q ) a † α a † β + Θ ( τ ) A ∗ αβ ( q ) a β a α ] . (7)For a gauge-invariant description of nuclei with super-fluidity, we need to parametrize the TDHFB state vec-tors, as above, not only by the collective coordinates q = ( q , q ) and conjugate momenta p = ( p , p ) butalso by the gauge angles ϕ = ( ϕ ( n ) , ϕ ( p ) ) conjugate tothe number variables n = ( n ( n ) , n ( p ) ) representing thepairing-rotational degrees of freedom (for both neutronsand protons). In the above equations, ˆ Q i ( q ) and ˆΘ ( τ ) ( q )are infinitesimal generators which are written in termsof the quasiparticle creation and annihilation operators( a † α , a α ) locally defined with respect to the moving-frameHFB states | φ ( q ) i . Note that the number operators aredefined as e N ( τ ) ≡ ˆ N ( τ ) − N ( τ )0 subtracting the expecta-tion values ( N ( n )0 , N ( p )0 ) of the neutron and proton num-bers at | φ ( q ) i . In this paper, we use units with ~ = 1.The moving-frame HFB states | φ ( q ) i and the infinites-imal generators ˆ Q i ( q ) are determined as solutions of themoving-frame HFB equation, δ h φ ( q ) | ˆ H M ( q ) | φ ( q ) i = 0 , (8)and the moving-frame QRPA equations, δ h φ ( q ) | [ ˆ H M ( q ) , ˆ Q i ( q )] − i X k B ik ( q ) ˆ P k ( q )+ 12 "X k ∂V∂q k ˆ Q k ( q ) , ˆ Q i ( q ) | φ ( q ) i = 0 , (9) δ h φ ( q ) | [ ˆ H M ( q ) , i ˆ P i ( q )] − X j C ij ( q ) ˆ Q j ( q ) − " ˆ H M ( q ) , X k ∂V∂q k ˆ Q k ( q ) , X j B ij ( q ) ˆ Q j ( q ) − X τ ∂λ ( τ ) ∂q i e N ( τ ) | φ ( q ) i = 0 , (10)which are derived from the time-dependent variationalprinciple. Here, ˆ H M ( q ) is the moving-frame Hamiltoniangiven byˆ H M ( q ) = ˆ H − X τ λ ( τ ) ( q ) e N ( τ ) − X i ∂V∂q i ˆ Q i ( q ) (11) and C ij ( q ) = ∂ V∂q i ∂q j − X k Γ kij ∂V∂q k (12)with Γ kij ( q ) = 12 X l B kl ( ∂B li ∂q j + ∂B lj ∂q i − ∂B ij ∂q l ) . (13)The infinitesimal generators ˆ P i ( q ) are defined byˆ P i ( q ) | φ ( q ) i = i ∂∂q i | φ ( q ) i , (14)with ˆ P i ( q ) = i X αβ [ P iαβ ( q ) a † α a † β − P ∗ iαβ ( q ) a β a α ] , (15)and determined as solutions of the moving-frame QRPAequations.The collective Hamiltonian is given as the expectationvalue of the microscopic Hamiltonian with respect to theTDHFB state: H ( q , p , n ) = h φ ( q , p , n ) | ˆ H | φ ( q , p , n ) i = V ( q ) + X ij B ij ( q ) p i p j + X τ λ ( τ ) ( q ) n ( τ ) , (16)where V ( q ) = H ( q , p , n ) p = , n = , (17) B ij ( q ) = ∂ H ∂p i ∂p j p = , n = , (18) λ ( τ ) ( q ) = ∂ H ∂n ( τ ) p = , n = , (19)represent the collective potential, inverse of the collectivemass, and the chemical potential, respectively. Note thatthe last term in Eq. (10) can be set to zero adopting theQRPA gauge-fixing condition, dλ ( τ ) /dq i = 0 [45].The basic equations of the ASCC method are invari-ant against point transformations of the collective coordi-nates ( q , q ). The B ij ( q ) and C ij ( q ) can be diagonalizedsimultaneously by a linear coordinate transformation ateach point of q = ( q , q ). We assume that we can intro-duce the collective coordinate system in which the diag-onal form is kept globally. Then, we can choose, withoutlosing generality and for simplicity, the scale of the col-lective coordinates q = ( q , q ) such that the vibrationalmasses become unity. Consequently, the vibrational ki-netic energy in the collective Hamiltonian (16) is writtenas T vib = 12 X i =1 , ( p i ) = 12 X i =1 , ( ˙ q i ) . (20) C. CHFB+LQRPA equations
The basic equations of the ASCC method can be solvedwith an iterative procedure. This task was successfullycarried out for extracting a one-dimensional (1D) collec-tive path embedded in the TDHFB configuration space[46, 47]. To determine a 2D hypersurface, however,the numerical calculation becomes too demanding at thepresent time. We therefore introduce practical approxi-mations as follows: First, we ignore the curvature terms(the third terms in Eqs. (9) and (10)), which vanish atthe HFB equilibrium points where dV /dq i = 0, assum-ing that their effects are numerically small. Second, wereplace the moving-frame HFB Hamiltonian ˆ H M ( q ) andthe moving-frame HFB state (cid:12)(cid:12) φ ( q , q ) (cid:11) with a CHFBHamiltonian ˆ H CHFB ( β, γ ) and a CHFB state | φ ( β, γ ) i ,respectively, on the assumption that the latters are goodapproximations to the formers.The CHFB equations are given by δ h φ ( β, γ ) | ˆ H CHFB ( β, γ ) | φ ( β, γ ) i = 0 , (21)ˆ H CHFB ( β, γ ) = ˆ H − X τ λ ( τ ) ( β, γ ) e N ( τ ) − X m =0 , µ m ( β, γ ) ˆ D (+)2 m (22)with four constraints h φ ( β, γ ) | ˆ N ( τ ) | φ ( β, γ ) i = N ( τ )0 , ( τ = n, p ) (23) h φ ( β, γ ) | ˆ D (+)2 m | φ ( β, γ ) i = D (+)2 m , ( m = 0 ,
2) (24)where ˆ D (+)2 m denotes Hermitian quadrupole operators,ˆ D and ( ˆ D + ˆ D − ) / m = 0 and 2, respectively(see Ref. [46] for their explicit expressions). We definethe quadrupole deformation variables ( β, γ ) in terms ofthe expectation values of the quadrupole operators: β cos γ = ηD (+)20 = η h φ ( β, γ ) | ˆ D (+)20 | φ ( β, γ ) i , (25)1 √ β sin γ = ηD (+)22 = η h φ ( β, γ ) | ˆ D (+)22 | φ ( β, γ ) i , (26)where η is a scaling factor (to be discussed in subsectionIII A).The moving frame QRPA equations, (9) and (10), thenreduce to δ h φ ( β, γ ) | [ ˆ H CHFB ( β, γ ) , ˆ Q i ( β, γ )] − i ˆ P i ( β, γ ) | φ ( β, γ ) i = 0 , ( i = 1 ,
2) (27)and δ h φ ( β, γ ) | [ ˆ H CHFB ( β, γ ) , i ˆ P i ( β, γ )] − C i ( β, γ ) ˆ Q i ( β, γ ) | φ ( β, γ ) i = 0 . ( i = 1 ,
2) (28) Here the infinitesimal generators, ˆ Q i ( β, γ ) and ˆ P i ( β, γ ),are local operators defined at ( β, γ ) with respect to theCHFB state | φ ( β, γ ) i . These equations are solved ateach point of ( β, γ ) to determine ˆ Q i ( β, γ ), ˆ P i ( β, γ ), and C i ( β, γ ) = ω i ( β, γ ). Note that these equations are validalso for regions with negative curvature ( C i ( β, γ ) < ω i ( β, γ ) takes an imagi-nary value. We call the above equations ‘local QRPA(LQRPA) equations’. There exist more than two solu-tions of LQRPA equations (27) and (28), and we need toselect relevant solutions. A useful criterion for selectingtwo collective modes among many LQRPA modes will begiven in subsection III C with numerical examples. Con-cerning the accuracy of the CHFB+LQRPA approxima-tion, some arguments will be given in subsection III F. D. Derivation of the vibrational masses
Once the infinitesimal generators ˆ Q i ( β, γ ) and ˆ P i ( β, γ )are obtained, we can derive the vibrational masses ap-pearing in the 5D quadrupole collective Hamiltonian (1).We rewrite the vibrational kinetic energy T vib given byEq. (20) in terms of the time-derivatives, ˙ β and ˙ γ , of thequadrupole deformation variables in the following way.We first note that an infinitesimal displacement of thecollective coordinates ( q , q ) brings about a correspond-ing change, dD (+)2 m = X i =1 , ∂D (+)2 m ∂q i dq i , ( m = 0 , , (29)in the expectation values of the quadrupole operators.The partial derivatives can be easily evaluated as ∂D (+)20 ∂q i = ∂∂q i h φ ( β, γ ) | ˆ D (+)20 | φ ( β, γ ) i = h φ ( β, γ ) | [ ˆ D (+)20 , i ˆ P i ( β, γ )] | φ ( β, γ ) i , (30) ∂D (+)22 ∂q i = ∂∂q i h φ ( β, γ ) | ˆ D (+)22 | φ ( β, γ ) i = h φ ( β, γ ) | [ ˆ D (+)22 , i ˆ P i ( β, γ )] | φ ( β, γ ) i , (31)without need of numerical derivatives. Accordingly, thevibrational kinetic energy can be written T vib = 12 M [ ˙ D (+)20 ] + M ˙ D (+)20 ˙ D (+)22 + 12 M [ ˙ D (+)22 ] , (32)with M mm ′ ( β, γ ) = X i =1 , ∂q i ∂D (+)2 m ∂q i ∂D (+)2 m ′ . (33)Taking time-derivative of the definitional equations of( β, γ ), Eqs. (25) and (26), we can straightforwardly trans-form the above expression (32) to the form in terms of( ˙ β , ˙ γ ). The vibrational masses ( D ββ , D βγ , D γγ ) are thenobtained from ( M , M , M ) through the following re-lations: D ββ = η − (cid:16) M cos γ + √ M sin γ cos γ + 12 M sin γ (cid:19) , (34) D βγ = βη − [ − M sin γ cos γ + 1 √ M (cos γ − sin γ ) + 12 M sin γ cos γ (cid:21) , (35) D γγ = β η − (cid:16) M sin γ − √ M sin γ cos γ + 12 M cos γ (cid:19) . (36) E. Calculation of the rotational moments of inertia
We calculate the rotational moments of inertia J k ( β, γ )using the LQRPA equation for the collective rotation [46]at each CHFB state, δ h φ ( β, γ ) | [ ˆ H CHFB , ˆΨ k ] − i ( J k ) − ˆ I k | φ ( β, γ ) i = 0 , (37) h φ ( β, γ ) | [ ˆΨ k ( β, γ ) , ˆ I k ′ ] | φ ( β, γ ) i = iδ kk ′ , (38)where ˆΨ k ( β, γ ) and ˆ I k represent the rotational angle andthe angular momentum operators with respect to theprincipal axes associated with the CHFB state | φ ( β, γ ) i .This is an extension of the Thouless-Valatin equation [48]for the HFB equilibrium state to non-equilibrium CHFBstates. The three moments of inertia can be written as J k ( β, γ ) = 4 β D k ( β, γ ) sin γ k ( k = 1 , ,
3) (39)with γ k = γ − (2 πk/ D k ( β, γ )above are replaced with a constant, then J k ( β, γ ) re-duce to the well-known irrotational moments of inertia.In fact, however, we shall see that their ( β, γ ) depen-dence is very important. We call J k ( β, γ ) and D k ( β, γ )determined by the above equation ‘LQRPA moments ofinertia’ and ‘LQRPA rotational masses’, respectively. F. Collective Schr¨odinger equation
Quantizing the collective Hamiltonian (1) with thePauli prescription, we obtain the collective Schr¨odingerequation [2] { ˆ T vib + ˆ T rot + V } Ψ αIM ( β, γ, Ω) = E αI Ψ αIM ( β, γ, Ω) , (40) whereˆ T vib = − √ W R ( β " ∂ β β r RW D γγ ∂ β ! − ∂ β β r RW D βγ ∂ γ ! + 1 β sin 3 γ " − ∂ γ r RW sin 3 γD βγ ∂ β ! + ∂ γ r RW sin 3 γD ββ ∂ γ ! , (41)ˆ T rot = X k =1 ˆ I k J k (42)with R ( β, γ ) = D ( β, γ ) D ( β, γ ) D ( β, γ ) , (43) W ( β, γ ) = (cid:8) D ββ ( β, γ ) D γγ ( β, γ ) − [ D βγ ( β, γ )] (cid:9) β − . (44)The collective wave function in the laboratory frame,Ψ αIM ( β, γ, Ω), is a function of β , γ , and a set of threeEuler angles Ω. It is specified by the total angular mo-mentum I , its projection onto the z -axis in the laboratoryframe M , and α that distinguishes the eigenstates pos-sessing the same values of I and M . With the rotationalwave function D IMK (Ω), it is written asΨ αIM ( β, γ, Ω) = X K =even Φ αIK ( β, γ ) h Ω | IM K i , (45)where h Ω | IM K i = s I + 116 π (1 + δ k ) [ D IMK (Ω) + ( − ) I D IM − K (Ω)] . (46)The vibrational wave functions in the body-fixed frame,Φ αIK ( β, γ ), are normalized as Z dβdγ | Φ αI ( β, γ ) | | G ( β, γ ) | = 1 , (47)where | Φ αI ( β, γ ) | ≡ X K =even | Φ αIK ( β, γ ) | , (48)and the volume element | G ( β, γ ) | dβdγ is given by | G ( β, γ ) | dβdγ = 2 β p W ( β, γ ) R ( β, γ ) sin 3 γdβdγ. (49)Thorough discussions of their symmetries and the bound-ary conditions for solving the collective Schr¨odinger equa-tion are given in Refs. [1–3]. III. CALCULATION OF THE COLLECTIVEPOTENTIAL AND THE COLLECTIVE MASSESA. Details of numerical calculation
The CHFB+LQRPA method outlined in the preced-ing section may be used in conjunction with any effec-tive interaction, e.g., density-dependent effective inter-actions like Skyrme forces, or modern nuclear densityfunctionals. In this paper, as a first step toward suchcalculations, we use a version of the P+Q force model[38, 39] that includes the quadrupole-pairing in addi-tion to the monopole-pairing interaction. Inclusion ofthe quadrupole-pairing is essential, because neither themonopole-pairing nor the quadrupole particle-hole inter-action contributes to the time-odd mean-field effects onthe collective masses [16]; that is, only the quadrupole-pairing induces the time-odd contribution in the presentmodel. Note that the quadrupole-pairing effects werenot considered in Ref. [16]. In the numerical calculationfor , , Se presented below, we use the same notationsand parameters as in our previous work [47]. The shellmodel space consists of two major shells ( N sh = 3 ,
4) forneutrons and protons and the spherical single-particle en-ergies are calculated using the modified oscillator poten-tial [49, 50]. The monopole-pairing interaction strengths(for neutrons and protons), G ( τ )0 , and the quadrupole-particle-hole interaction strength, χ , are determined suchthat the magnitudes of the quadrupole deformation β andthe monopole-pairing gaps (for neutrons and protons)at the oblate and prolate local minima in Se approxi-mately reproduce those obtained in the Skyrme-HFB cal-culations [51]. The interaction strengths for Se and Se are then determined assuming simple mass-numberdependence [39]; G ( τ )0 ∼ A − and χ ′ ≡ χb ∼ A − ( b denotes the oscillator-length parameter). For thequadrupole-pairing interaction strengths (for neutronsand protons), we use the Sakamoto-Kishimoto prescrip-tion [52] to derive the self-consistent values. Followingthe conventional treatment of the P+Q model [53], weignore the Fock term, so that we use the abbreviationHB (Hartree-Bogoliubov) in place of HFB in the follow-ing. In the case of the conventional P+Q model, the HBequation reduces to a simple Nilsson + BCS equation(see, e.g., Ref. [37]). The presence of the quadrupole-pairing interaction in our case does not allow such a re-duction, however, and we directly solve the HB equation.In the P+Q model, the scaling factor η in Eqs. (25) and(26) is given by η = χ ′ / ~ ω b , where ω denotes thefrequency of the harmonic-oscillator potential. Effectivecharges, ( e n , e p ) = (0 . , . β, γ )plane, we employ a two-dimensional mesh consisting of3600 points in the region 0 < β < . ◦ < γ < ◦ . Each mesh point ( β i , γ j ) is represented as β i = ( i − . × . , ( i = 1 , · · · , (50) γ j = ( j − . × ◦ , ( j = 1 , · · · . (51)One of the advantages of the present approach is that wecan solve the CHB + LQRPA equations independentlyat each mesh point on the ( β, γ ) plane, so that it is suitedto parallel computation.Finally, we summarize the most important differencesbetween the present approach and the Baranger-Kumarapproach [43]. First, as repeatedly emphasized, we intro-duce the LQRPA collective massess in place of the crank-ing masses. Second, we take into account the quadrupole-pairing force (in addition to the monopole-pairing force),which brings about the time-odd effects on the collectivemasses. Third, we exactly solve the CHB self-consistentproblem, Eq. (21), at every point on the ( β, γ ) plane us-ing the gradient method, while in the Baranger-Kumarworks the CHB Hamiltonian is replaced with a Nilsson-like single-particle model Hamiltonian. Fourth, we donot introduce the so-called core contributions to the col-lective masses, although we use the effective charges torenormalize the core polarization effects (outside of themodel space consisting of two major shells) into thequadrupole operators, We shall see that we can well re-produce the major characteristics of the experimentaldata without introducing such core contributions to thecollective masses. Fifth, most importantly, the theoret-ical framework developed in this paper is quite general,that is, it can be used in conjunction with modern densityfunctionals going far beyond the P+Q force model. B. Collective potentials and pairing gaps
We show in Fig. 1 the collective potentials V ( β, γ ) cal-culated for , , Se. It is seen that two local minimaalways appear both at the oblate ( γ = 60 ◦ ) and prolate( γ = 0 ◦ ) shapes, and, in all these nuclei, the oblate mini-mum is lower than the prolate minimum. The energy dif-ference between them is, however, only several hundredkeV and the potential barrier is low in the direction oftriaxial shape (with respect to γ ) indicating γ -soft char-acter of these nuclei. In Fig. 1 we also show the collectivepaths (connecting the oblate and prolate minima) deter-mined by using the 1D version of the ASCC method [47].It is seen that they always run through the triaxial valleyand never go through the spherical shape.In Fig. 2, the monopole- and quadrupole-pairing gapscalculated for Se are displayed. They show a signifi-cant ( β, γ ) dependence. Broadly speaking, the monopolepairing decreases while the quadrupole pairing increasesas β increases. FIG. 1: (Color online) Collective potential V ( β, γ ) for , , Se. The regions higher than 3 MeV (measured fromthe oblate HB minima) are drawn by rosybrown color. One-dimensional collective paths connecting the oblate and prolatelocal minima are determined by using the ASCC method anddepicted with bold red lines.
C. Properties of the LQRPA modes
In Fig. 3 the frequencies squared, ω i ( β, γ ), of variousLQRPA modes calculated for Se are plotted as func-tions of β and γ . In the region of the ( β, γ ) plane wherethe collective potential energy is less than about 5 MeV,we can easily identify two collective modes among manyLQRPA modes, whose ω i ( β, γ ) are much lower than FIG. 2: (Color online) Monopole- and quadrupole-pairinggaps for neutrons of Se are plotted in the ( β, γ ) deforma-tion plane. ( upper left ) Monopole pairing gap ∆ ( n )0 . ( lowerleft ) Quadrupole pairing gap ∆ ( n )20 . ( lower right ) Quadrupolepairing gap ∆ ( n )22 . See Ref. [46] for definitions of ∆ ( n )0 , ∆ ( n )20 ,and ∆ ( n )22 . those of other modes. Therefore we adopt the two low-est frequency modes to derive the collective Hamiltonian.This result of numerical calculation supports our assump-tion that there exists a 2D hypersurface associated withlarge-amplitude quadrupole shape vibrations, which isapproximately decoupled from other degrees of freedom.The situation changes when the collective potential en-ergy exceeds about 5 MeV and/or the monopole-pairinggap becomes small. A typical example is presented in thebottom panel of Fig. 3. It becomes hard to identify twocollective modes well-separated from other modes when β > .
4, where the collective potential energy is high (seeFig. 1) and the monopole-pairing gap becomes small (seeFig. 2). In this example, the second-lowest LQRPA modein the 0 . < β < . β > .
5. In fact,many non-collective two-quasiparticle modes appear inits neighborhood. This region in the ( β, γ ) plane is notimportant, however, because only tails of the collectivewave function enter into this region.It may be useful to set up a prescription that workseven in a difficult situation where it is not apparent howto choose two collective LQRPA modes. We find that thefollowing prescription always works well for selecting twocollective modes among many LQRPA modes. This maybe called a minimal metric criterion. At each point onthe ( β, γ ) plane, we evaluate the vibrational part of themetric W ( β, γ ) given by Eq. (44) for all combinations oftwo LQRPA modes, and find the pair that gives the mini-mum value. We show in Fig. 4 how this prescription actu-ally works. In this figure, the W ( β, γ ) values are plottedas functions of β and γ for many pairs of the LQRPA -2 0 2 4 6 8 10 12 14 0 10 20 30 40 50 60 ω ( β , γ ) ( M e V ) γ (deg) [ β =0.305]-4-2 0 2 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5 0.6 ω ( β , γ ) ( M e V ) β [ γ =0.5(deg)] -4-2 0 2 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5 0.6 ω ( β , γ ) ( M e V ) β [ γ =30.5(deg)] FIG. 3: Frequencies squared ω of the LQRPA modes calcu-lated for Se are plotted as functions of β or γ . The LQRPAmodes adopted for calculation of the vibrational masses areconnected with solid lines. ( top ) Dependence on γ at β = 0 . middle ) Dependence on β along the γ = 0 . ◦ line. ( bottom )Dependence on β along the γ = 30 . ◦ line. modes. In the situations where the two lowest-frequencyLQRPA modes are well separated from other modes, thisprescription gives the same results as choosing the twolowest-frequency modes (see the top and middle panels).On the other hand, a pair of the LQRPA modes differentfrom the lowest two modes is chosen by this prescrip-tion in the region mentioned above (the bottom panel).This choice may be better than that using the lowest-frequency criterion, because we often find that a nor-mal mode of pairing vibrational character becomes thesecond lowest LQRPA mode when the monopole-pairinggap significantly decreases in the region of large β . Thesmall values of the vibrational metric implies that the di-
0 10 20 30 40 50 60 W ( β , γ ) / β ( M e V - ) γ (deg) [ β =0.305] LQRPAIB10
0 0.1 0.2 0.3 0.4 0.5 0.6 W ( β , γ ) / β ( M e V - ) β [ γ =0.5(deg)] LQRPAIB10
0 0.1 0.2 0.3 0.4 0.5 0.6 W ( β , γ ) / β ( M e V - ) β [ γ =30.5(deg)] LQRPAIB FIG. 4: Dependence on β and γ of the vibrational part of themetric W ( β, γ ) calculated for Se. ( top ) Dependence on γ at β = 0 .
3. ( middle ) Dependence on β along the γ = 0 . ◦ line.( bottom ) Dependence on β along the γ = 30 . ◦ line. The crosssymbols indicate values of the vibrational metric calculatedfor various choices of two LQRPA modes from among thelowest 40 LQRPA modes; the lowest mode is always chosenand the other is from the remaining 39 modes. The smallestvibrational metric is shown by solid line. For reference, thevibrational metric calculated using the IB vibrational mass isindicated by broken lines. rection of the infinitesimal displacement associated withthe pair of the LQRPA modes has a large projection ontothe ( β, γ ) plane. Therefore, this prescription may be wellsuited to our purpose of deriving the collective Hamilto-nian for the ( β, γ ) variables. It remains as an interestingopen question for future to examine whether or not theexplicit inclusion of the pairing vibrational degree of free-dom as another collective variable will give us a betterdescription in such situations. FIG. 5: (Color online) Vibrational masses, D ββ ( β, γ ), D βγ ( β, γ ) /β , and D γγ ( β, γ ) /β , in unit of MeV − calculatedfor Se.
D. Vibrational masses
In Fig. 5 the vibrational masses calculated for Se aredisplayed. We see that their values exhibit a significantvariation in the ( β, γ ) plane. In particular, the increasein the large β region is remarkable.Figure 6 shows how the ratios of the LQRPA vibra-tional masses to the IB vibrational masses vary on the( β, γ ) plane. It is clearly seen that the LQRPA vibra-tional masses are considerably larger than the IB vibra-tional masses and their ratios change depending on β and γ . In this calculation, the IB vibrational masses are FIG. 6: (Color online) Ratios of the LQRPA vibra-tional masses to the IB vibrational masses, D ββ /D (IB) ββ and D γγ /D (IB) γγ , calculated for Se. evaluated using the well-known formula: D (IB) ξ i ξ j ( β, γ ) = 2 X µ ¯ ν h µ ¯ ν | ∂ ˆ H CHB ∂ξ i | i h | ∂ ˆ H CHB ∂ξ j | µ ¯ ν i [ E µ ( β, γ ) + E ¯ ν ( β, γ )] , (52)( ξ i = β or γ )where E µ ( β, γ ), | i , and | µ ¯ ν i denote the quasiparticleenergy, the CHB state | φ ( β, γ ) i and the two-quasiparticlestate a † µ a † ¯ ν | φ ( β, γ ) i , respectively (see Ref. [46] for themeaning of the indices µ and ¯ ν ).The vibrational masses calculated for , Se exhibitbehaviors similar to those for Se.
E. Rotational masses
In Fig. 7, the LQRPA rotational masses D k ( β, γ ) cal-culated for Se are displayed. Similarly to the vibra-tional masses discussed above, the LQRPA rotationalmasses also exhibit a remarkable variation over the ( β, γ )plane, indicating a significant deviation from the irrota-tional property.0
FIG. 7: (Color online) Rotational masses D k ( β, γ ) in unit ofMeV − , calculated for Se. See Eq. (39) for the relation withthe rotational moments of inertia J k ( β, γ ). Figure 8 shows how the ratios of the LQRPA rotationalmasses D k ( β, γ ) to the IB cranking masses D (IB) k ( β, γ )vary on the ( β, γ ) plane. The rotational masses calcu-lated for , Se exhibit behaviors similar to those for Se.As we have seen in Figs. 5–8, not only the vibra-tional and rotational masses but also their ratios to theIB cranking masses exhibit an intricate dependence on β and γ . For instance, it is clearly seen that the ra-tios, D k ( β, γ ) /D (IB) k ( β, γ ), gradually increase as β de-creases. This result is consistent with the calcula-tion by Hamamoto and Nazarewicz [54], where it is FIG. 8: (Color online) Ratios of the LQRPA rotational massesto the IB rotational masses, D k ( β, γ ) /D (IB) k ( β, γ ), calculatedfor Se. shown that the ratio of the Migdal term to the crank-ing term in the rotational moment of inertia (about the1st axis) increases as β decreases. Needless to say, theMigdal term (also called the Thouless-Valation correc-tion) corresponds to the time-odd mean-field contribu-tion taken into account in the LQRPA rotational masses,so that the result of Ref. [54] implies that the ratio D ( β, γ ) /D (IB)1 ( β, γ ), increases as β decreases, in agree-ment with our result. To understand this behavior, it isimportant to note that, in the present calculation, the dy-namical effect of the time-odd mean-field on D ( β, γ ) isassociated with the K = 1 component of the quadrupole-1pairing interaction and it always works and increase therotational masses, in contrast to the behavior of the staticquantities like the magnitude of the quadrupole-pairinggaps, ∆ and ∆ , which diminish in the spherical shapelimit. Obviously, this qualitative feature holds true ir-respective of details of our choice of the monopole andquadrupole pairing interaction strengths.The above results of calculation obviously indicate theneed to take into account the time-odd contributionsto the vibrational and rotational masses by going be-yond the IB cranking approximation. In Refs. [29–32],a phenomenological prescription is adopted to remedythe shortcoming of the IB cranking masses; that is, aconstant factor in the range 1.40-1.45 is multiplied tothe IB rotational masses. This prescription is, however,insufficient in the following points. First, the scalingonly of the rotational masses (leaving the vibrationalmasses aside) violates the symmetry requirement for the5D collective quadrupole Hamiltonian [1–3] (a similarcomment is made in Ref. [4]). Second, the ratios takedifferent values for different LQRPA collective masses( D ββ , D βγ , D γγ , D , D , and D ). Third, for every col-lective mass, the ratio exhibits an intricate dependenceon β and γ . Thus, it may be quite insufficient to sim-ulate the time-odd mean-field contributions to the col-lective masses by scaling the IB cranking masses with acommon multiplicative factor. F. Check of self-consistency along the collectivepath
As discussed in Sec. II, the CHB+LQRPA method is apractical approximation to the ASCC method. It is cer-tainly desirable to examine the accuracy of this approxi-mation by carrying out a fully self-consistent calculation.Although, at the present time, such a calculation is toodemanding to carry out for a whole region of the ( β, γ )plane, we can check the accuracy at least along the 1Dcollective path. This is because the 1D collective path isdetermined by carrying out a fully self-consistent ASCCcalculation for a single set of collective coordinate andmomentum. The 1D collective paths projected onto the( β, γ ) plane are displayed in Fig. 1. Let us use a nota-tion | φ ( q ) i for the moving-frame HB state obtained byself-consistently solving the ASCC equations for a singlecollective coordinate q [46, 47]. To distinguish from it,we write the CHB state as | φ ( β ( q ) , γ ( q )) i . This notationmeans that the values of β and γ are specified by the col-lective coordinate q along the collective path. In otherwords, | φ ( β ( q ) , γ ( q )) i has the same expectation values ofthe quadrupole operator as those of | φ ( q ) i . It is impor-tant to note, however, that they are different from eachother, because | φ ( β ( q ) , γ ( q )) i is a solution of the CHBequation which is an approximation of the moving-frameHB equation. Let us evaluate various physical quantitiesusing the two state vectors and compare the results.In Fig. 9 various physical quantities (the pairing gaps, V ( q ) ( M e V ) (1+3)D ASCCCHB+LQRPA 1 1.1 1.2 1.3 1.4 1.5 ∆ ( τ ) ( q ) ( M e V ) np-2-1012345 ω ( q ) ( M e V ) D ( q ) ( M e V - ) D ββ D βγ / β D γγ / β D k ( q ) ( M e V - ) γ (q) D D D FIG. 9: (Color online) Comparison of physical quantities eval-uated with the CHB + LQRPA approximation and those withthe ASCC method. Both calculations are carried out alongthe 1D collective path for Se and the results are plotted asfunction of γ ( q ). From the top to the bottom: 1) the col-lective potential, 2) monopole-pairing gaps, ∆ ( n ))0 and ∆ ( p )0 ,for neutrons and protons, 3) frequencies squared ω of thelowest and the second-lowest modes obtained by solving themoving-frame QRPA and the LQRPA equations, 4) vibra-tional masses, D ββ , D βγ /β , and D γγ /β , 5) rotational masses D k . In almost all cases, results of the two calculations are in-distinguishable, because they agree within the widths of theline. the collective potential, the frequencies of the localnormal modes, the rotational masses, and vibrationalmasses) calculated using the moving-frame HB state | φ ( q ) i and the CHB state | φ ( β ( q ) , γ ( q )) i are presentedand compared. These calculations are carried out alongthe 1D collective path for Se. Apparently, the resultsof the two calculations are indistinguishable in almost allcases, because they agree within the widths of the line.This good agreement implies that the CHB+LQRPA isan excellent approximation to the ASCC method alongthe collective path on the ( β, γ ) plane. As we shall seein the next section, collective wave functions distributearound the collective path. Therefore, it may be reason-able to expect that the CHB+LQRPA method is a good2approximation to the ASCC method and suited, at least,for describing the oblate-prolate shape mixing dynamicsin , , Se.
IV. LARGE-AMPLITUDE SHAPE-MIXINGPROPERTIES OF , , SE We have calculated collective wave functions solvingthe collective Schr¨odinger equation (40) and evaluatedexcitation spectra, quadrupole transition probabilities,and spectroscopic quadrupole moments. The results forlow-lying states in , , Se are presented in Figs. 10–15.In Figs. 10, 12, and 14, excitation spectra and B ( E Se, Se, and Se, calculated with theCHB+LQRPA method, are displayed together with ex-perimental data. The eigenstates are labeled with I π =0 + , + , + , and 6 + . In these figures, results obtained us-ing the IB cranking masses are also shown for the sakeof comparison. Furthermore, the results calculated withthe (1+3)D version of the ASCC method reported in ourprevious paper [47] are shown also for comparison withthe 5D calculations. We use the abbreviation (1+3)Dto indicate that a single collective coordinate along thecollective path describing large-amplitude vibration andthree rotational angles associated with the rotational mo-tion are taken into account in these calculations. Theclassification of the calculated low-lying states into fami-lies of two or three rotational bands is made according tothe properties of their vibrational wave functions. Thesevibrational wave functions are displayed in Figs. 11, 13,and 15. In these figures, only the β factor in the vol-ume element (49) are multiplied to the vibrational wavefunctions squared leaving the sin 3 γ factor aside. Thisis because all vibrational wave functions look like triax-ial and the probability at the oblate and prolate shapesvanish if the sin 3 γ factor is multiplied by them.Let us first summarize the results of the CHB+LQRPAcalculation. The most conspicuous feature of the low-lying states in these proton-rich Se isotopes is the dom-inance of the large-amplitude vibrational motion in thetriaxial shape degree of freedom. In general, the vibra-tional wave function extends over the triaxial region be-tween the oblate ( γ = 60 ◦ ) and the prolate ( γ = 0 ◦ )shapes. In particular, this is the case for the 0 + statescausing their peculiar behaviors; for instance, we obtaintwo excited 0 + states located slightly below or above the2 +2 state. Relative positions between these excited statesare quite sensitive to the interplay of large-amplitude γ -vibrational modes and the β -vibrational modes. Thisresult of calculation is consistent with the available ex-perimental data where the excited 0 + state has not yetbeen found, but more experimental data are needed toexamine the validity of the theoretical prediction. Below,let us examine characteristic features of the theoreticalspectra more closely for individual nuclei.For Se, we obtain the third band in low energy. The 0 +2 and 2 +3 states belonging to this band are also shown inFig. 10. Their vibrational wave functions exhibit nodesin the β direction (see Fig. 11) indicating that a β -vibrational mode is excited on top of the large-amplitude γ vibrations. As a matter of course, this kind of state isoutside of the scope of the (1+3)D calculation. The vi-brational wave functions of the yrast 2 +1 and 4 +1 statesexhibit localization in a region around the oblate shape,while the yrare 2 +2 , +2 , and 6 +2 states localize around theprolate shape. It is apparent, however, that all the wavefunctions significantly extend from γ = 0 ◦ to 60 ◦ overthe triaxial region, indicating γ -soft character of thesestates. In particular, the yrare 4 +2 and 6 +2 wave functionsexhibit two-peak structure consisting of the prolate andoblate peaks. The peaks of the vibrational wave functiongradually shift toward a region of larger β as the angu-lar momentum increases. This is a centrifugal effect de-creasing the rotational energy by increasing the momentof inertia. In the (1+3)D calculation, this effect is absentbecause the collective path is fixed at the ground state.Thus, the 5D calculation yields, for example, a muchlarger value for B ( E
2; 6 +1 → +1 ) in comparison with the(1+3)D calculation. Actually, in the 5D CHB+LQRPAcalculation, the wave function of the yrast 6 +1 state lo-calizes in the triaxial region (see Fig. 11) where the mo-ment of inertia takes a maximum value. This leads toa small value for the spectroscopic quadrupole moment(see Fig. 16) because of the cancellation between the con-tributions from the oblate-like and prolate-like regions.This cancellation mechanism due to the large-amplitude γ fluctuation is effective also in other states; although thespectroscopic quadrupole moments of the yrast 2 +1 and4 +1 (yrare 2 +2 , +2 , and 6 +2 ) states are positive (negative)indicating their oblate-like (prolate-like) character, theirabsolute magnitudes are rather small.The E γ -unstable situation; for instance, B ( E
2; 6 +2 → +1 ), B ( E
2; 4 +2 → +1 ), and B ( E
2; 2 +2 → +1 )are much larger than B ( E
2; 6 +2 → +1 ), B ( E
2; 4 +2 → +1 ),and B ( E
2; 2 +2 → +1 ); see Fig. 10. Thus, the low-lyingstates in Se may be characterized as an intermedi-ate situation between the oblate-prolate shape coexis-tence and the Wilets-Jean γ -unstable model [55]. Usingthe phenomenological Bohr-Mottelson collective Hamil-tonian, we have shown in Ref. [56] that it is possible todescribe the oblate-prolate shape coexistence and the γ -unstable situation in a unified way varying a few param-eters controlling the degree of oblate-prolate asymmetryin the collective potential and the collective masses. Thetwo-peak structure seen in the 4 +2 and 6 +2 states maybe considered as one of the characteristics of the inter-mediate situation. It thus appears that the excitationspectrum for Se (Fig. 10) serves as a typical exampleof the transitional phenomena from the γ -unstable to theoblate-prolate shape coexistence situations.Let us make a comparison between the spectra inFig. 10 obtained with the LQRPA collective masses andthat with the IB cranking masses. It is obvious that3 E x c i t a t i o n E n e r g y ( M e V ) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Se (1+3)D ASCC CHB+IB CHB+LQRPA EXP (432)(146) (457)(329)(580)(500)(343) (207)(36) (374)(785)(2) (533)(933)(1289)(400) (1043) (563)(568)(247)(676)(30)(550)(892)(1030)(42) (1191)(857)(492) (1026)(657)(783)(395)(508) (30)(12) (557)(1179) (517)(43) FIG. 10: Excitation spectra and B ( E
2) values calculated for Se by means of the CHB+LQRPA method (denotedCHB+LQRPA) and experimental data [5–7]. For comparison, results calculated using the IB cranking masses (denotedCHB+IB) and those obtained using the (1+3)D version of the ASCC method (denoted (1+3)D ASCC) are also shown. Only B ( E e fm . FIG. 11: (Color online) Vibrational wave functions squared β | Φ Ik ( β, γ ) | calculated for Se. the excitation energies are appreciably overestimated inthe latter. This result is as expected from the too lowvalues of the IB cranking masses. The result of our cal-culation is in qualitative agreement with the HFB-basedconfiguration-mixing calculation reported by Ljungvall etal. [8] in that both calculations indicate the oblate (pro-late) dominance for the yrast (yrare) band in Se. Quiterecently, the B ( E
2; 2 +1 → +1 ) value has been measured in experiment [7]. The calculated value (492 e fm ) is infair agreement with the experimental data (432 e fm ).The result of calculation for Se (Figs. 12 and 13) issimilar to that for Se. The vibrational wave functions ofthe yrast 2 +1 , +1 , and 6 +1 states localize in a region aroundthe oblate shape, exhibiting, at the same time, long tailsin the triaxial direction. We note here that, differentlyfrom the Se case, the 6 +1 wave function keeps the oblate-4 E x c i t a t i o n E n e r g y ( M e V ) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Se (1+3)D ASCC CHB+IB CHB+LQRPA EXP (2) (632)(405)(61)(126)(426)(639)(18) (561)(390)(472)(1198) (0.00) (1313)(565)(35) (909)(2)(1296)(988)(568) (856)(105) (1843)(13) (541)(521)(600) (497) (423)(1184)(921) (21) (1376)(923)(404)(0.4)(102) (619)(530) (1738)(25) (320)(478)(158) (342)(370)(530) FIG. 12: Same as Fig. 10 but for Se. Experimental data is taken from Refs. [8, 41].FIG. 13: (Color online) Same as Fig. 11 but for Se. like structure. On the other hand, the yrare 2 +2 , +2 , and6 +2 states localize around the prolate shape, exhibiting, atthe same time, small secondary bumps around the oblateshape. For the yrare 2 +2 state, we obtain a strong oblate-prolate shape mixing in the (1+3)D calculation [47]. Thismixing becomes weaker in the present 5D calculation, re-sulting in the reduction of the B ( E
2; 4 +1 → +2 ) value.Similarly to Se, we obtain two excited 0 + states in lowenergy. We see considerable oblate-prolate shape mix-ings in their vibrational wave functions, but, somewhatdifferently from those in Se, the second and third 0 + states in Se exhibit clear peaks at the oblate and prolateshapes, respectively, Their energy ordering is quite sen-sitive to the interplay of the large-amplitude γ vibrationand the β vibrational modes. The calculated spectrumfor Se is in fair agreement with the recent experimentaldata [41] , although the B ( E
2) values between the yraststates are overestimated.The result of calculation for Se (Figs. 14 and 15)presents a feature somewhat different from those for Seand Se; that is, the yrast 2 +1 , +1 , and 6 +1 states local-ize around the prolate shape instead of the oblate shape.5 E x c i t a t i o n E n e r g y ( M e V ) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Se (1+3)D ASCC CHB+IB CHB+LQRPA EXP (29) (543)(443)(88)(837) (74)(297) (337)(639)(390) (329)(48)(19)(9) (1169)(742)(371)(1519)(1077)(593) (119)(93)(25) (977) (763)(58) (320)(15) (467)(799)(558) (1454)(993)(547) (286) (1097) (397)(775)(760)(902)(358)(733)(93) (27)(488) (93)(141) (40) (1220)(882)(405) FIG. 14: Same as Fig. 10 but for Se. Experimental data is taken from Refs. [8, 42].FIG. 15: (Color online) Same as Fig. 11 but for Se.
The localization develops with increasing angular mo-mentum. On the other hand, similarly to the , Secases, the yrare 2 +2 , +2 , and 6 +2 states exhibit the two-peak structure. The spectroscopic quadrupole momentsof the 2 +1 , +1 , and 6 +1 states are negative, and their abso-lute magnitude increases with increasing angular momen-tum (see Fig. 16) reflecting the developing prolate char-acter in the yrast band, while those of the yrare statesare small because of the two-peak structure of their vi-brational wave functions, that is, due to the cancellationof the contributions from the prolate-like and oblate-like regions. Also for Se, we obtain two excited 0 + statesin low energy, but they show features somewhat differ-ent from the corresponding excited 0 + states in , Se.Specifically, the vibrational wave functions of the sec-ond and third 0 + states exhibit peaks at the prolate andoblate shape, respectively. As seen in Fig. 14, our re-sults of calculation for the excitation energies and B ( E +1 , +1 , and 6 +1 states in Se.Experimental E -100-80-60-40-20 0 20 40 60 80 100 2 4 6 Q ( I ) ( e f m ) I Se -100-80-60-40-20 0 20 40 60 80 100 2 4 6 Q ( I ) ( e f m ) I Se )) ) 2 4 6I Se
2 4 6I Se (1+3)D ASCC(gs(1+3)D ASCC(ex)CHB+IB(gsCHB+IB(ex)CHB+LQRPA(gsCHB+LQRPA(ex) FIG. 16: Spectroscopic quadrupole moments for , , Se.Values calculated with the LQRPA collective masses areshown with the triangles. For comparison, values calculatedwith the IB collective masses and those obtained with the(1+3)D version of the ASCC method are also shown with thesquares and the circles, respectively. The filled symbols showthe values for the yrast states, while the open symbols thosefor the yrare states.
V. CONCLUSIONS
On the basis of the ASCC method, we have developed apractical microscopic approach, called CHFB+LQRPA,of deriving the 5D quadrupole collective Hamiltonianand confirmed its efficiency by applying it to the oblate-prolate shape coexistence/mixing phenomena in proton-rich , , Se. The results of numerical calculation forthe excitation energies and B ( E
2) values are in goodagreement with the recent experimental data [7, 8] for the yrast 2 +1 , +1 , and 6 +1 states in these nuclei. It is shownthat the time-odd components of the moving mean-fieldsignificantly increase the vibrational and rotational col-lective masses and make the theoretical spectra in muchbetter agreement with the experimental data than cal-culations using the IB cranking masses. Our analysisclearly indicates that low-lying states in these nuclei pos-sess a transitional character between the oblate-prolateshape coexistence and the so-called γ unstable situationwhere large-amplitude triaxial-shape fluctuations play adominant role.Finally, we would like to list a few issues for the fu-ture that seems particularly interesting. First, fully self-consistent solution of the ASCC equations for determin-ing the two-dimensional collective hypersurface and ex-amination of the validity of the approximations adoptedin this paper in the derivation of the CHFB+LQRPAscheme. Second, application to various kind of collec-tive spectra associated with large-amplitude collectivemotions near the yrast lines (as listed in Ref. [28]).Third, possible extension of the quadrupole collectiveHamiltonian by explicitly treating the pairing vibrationaldegrees of freedom as additional collective coordinates.Fourth, use of the Skyrme energy functionals + density-dependent contact pairing interaction in place of theP+Q force, and then modern density functionals cur-rently under active development. Fifth, application ofthe CHFB+LQRPA scheme to fission dynamics. TheLQRPA approach enables us to evaluate, without needof numerical derivatives, the collective inertia masses in-cluding the time-odd mean-field effects. Acknowledgments
Two of the authors (K. S. and N. H.) are supportedby the Junior Research Associate Program and the Spe-cial Postdoctoral Researcher Program of RIKEN, re-spectively. The numerical calculations were carried outon Altix3700 BX2 at Yukawa Institute for TheoreticalPhysics in Kyoto University and RIKEN Cluster of Clus-ters (RICC) facility. This work is supported by Grants-in-Aid for Scientific Research (Nos. 20105003, 20540259,and 21340073) from the Japan Society for the Promotionof Science and the JSPS Core-to-Core Program “Inter-national Research Network for Exotic Femto Systems.” [1] A. Bohr and B. R. Mottelson,
Nuclear Structure , vol. II(W-A. Benjamin Inc., 1975; World Scientific, 1998).[2] S. T. Belyaev, Nucl. Phys. , 17 (1965).[3] K. Kumar and M. Baranger, Nucl. Phys. A , 608(1967).[4] L. Pr´ochniak and S. G. Rohozi´nski, J. Phys. G , 123101(2009). [5] S. M. Fischer, C. J. Lister, and D. P. Balamuth, Phys.Rev. C , 064318 (2003).[6] S. M. Fischer, D. P. Balamuth, P. A. Hausladen, C. J.Lister, M. P. Carpenter, D. Seweryniak, and J. Schwartz,Phys. Rev. Lett. , 4064 (2000).[7] A. Obertelli, T. Baugher, D. Bazin, J. P. Delaroche,F. Flavigny, A. Gade, M. Girod, T. Glasmacher, A. G¨orgen, G. F. Grinyer, et al., Phys. Rev. C , 031304(2009).[8] J. Ljungvall, A. G¨orgen, M. Girod, J.-P. Delaroche,A. Dewald, C. Dossat, E. Farnea, W. Korten, B. Melon,R. Menegazzo, et al., Phys. Rev. Lett. , 102502(2008).[9] D. R. Inglis, Phys. Rev. , 1059 (1954).[10] S. T. Beliaev, Nucl. Phys. , 322 (1961).[11] K. Kumar, Nucl. Phys. A , 189 (1974).[12] K. Pomorski, T. Kaniowska, A. Sobiczewski, and S. G.Rohozi´nski, Nucl. Phys. A , 394 (1977).[13] S. G. Rohozi´nski, J. Dobaczewski, B. Nerlo-Pomorska,K. Pomorski, and J. Srebrny, Nucl. Phys. A , 66(1977).[14] J. Dudek, W. Dudek, E. Ruchowska, and J. Skalski, Z.Phys. A , 341 (1980).[15] M. Baranger and M. V´en´eroni, Ann. Phys. , 123(1978).[16] J. Dobaczewski and J. Skalski, Nucl. Phys. A , 123(1981).[17] F. Villars, Nucl. Phys. A , 269 (1977).[18] K. Goeke and P.-G. Reinhard, Ann. Phys. , 328(1978).[19] D. J. Rowe and R. Bassermann, Can. J. Phys. , 1941(1976).[20] T. Marumori, Prog. Theor. Phys. , 112 (1977).[21] J. Libert, M. Girod, and J.-P. Delaroche, Phys. Rev. C , 054301 (1999).[22] M. K. Pal, D. Zawischa, and J. Speth, Z. Phys. A ,387 (1975).[23] N. R. Walet, G. Do Dang, and A. Klein, Phys. Rev. C , 2254 (1991).[24] D. Almehed and N. R. Walet, Phys. Lett. B , 163(2004).[25] T. Marumori, T. Maskawa, F. Sakata, and A. Kuriyama,Prog. Theor. Phys. , 1294 (1980).[26] M. J. Giannoni and P. Quentin, Phys. Rev. C , 2060(1980).[27] G. D. Dang, A. Klein, and N. R. Walet, Phys. Rep. ,93 (2000).[28] K. Matsuyanagi, M. Matsuo, T. Nakatsukasa, N. Hino-hara, and K. Sato, J. Phys. G , 064018 (2010).[29] T. Nikˇsi´c, D. Vretenar, G. A. Lalazissis, and P. Ring,Phys. Rev. Lett. , 092502 (2007).[30] T. Nikˇsi´c, Z. P. Li, D. Vretenar, L. Pr´ochniak, J. Meng,and P. Ring, Phys. Rev. C , 034303 (2009).[31] Z. P. Li, T. Nikˇsi´c, D. Vretenar, J. Meng, G. A. Lalazissis,and P. Ring, Phys. Rev. C , 054301 (2009).[32] Z. P. Li, T. Nikˇsi´c, D. Vretenar, and J. Meng, Phys. Rev.C , 034316 (2010). [33] M. Girod, J.-P. Delaroche, A. G¨orgen, and A. Obertelli,Phys. Lett. B , 39 (2009).[34] J. P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hi-laire, S. P´eru, N. Pillet, and G. F. Bertsch, Phys. Rev. C , 014303 (2010).[35] M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod.Phys. , 121 (2003).[36] M. Matsuo, T. Nakatsukasa, and K. Matsuyanagi, Prog.Theor. Phys. , 959 (2000).[37] P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer-Verlag, 1980).[38] D. R. Bes and R. A. Sorensen,
Advances in NuclearPhysics , vol. 2 (Prenum Press, 1969).[39] M. Baranger and K. Kumar, Nucl. Phys. A , 490(1968).[40] N. Hinohara, T. Nakatsukasa, M. Matsuo, and K. Mat-suyanagi, Prog. Theor. Phys. , 567 (2006).[41] G. Rainovski, H. Schnare, R. Schwengner, C. Plettner,L. K¨aubler, F. D¨onau, I. Ragnarsson, J. Eberth, T. Stein-hardt, O. Thelen, et al., J. Phys. G , 2617 (2002).[42] R. Palit, H. C. Jain, P. K. Joshi, J. A. Sheikh, and Y. Sun,Phys. Rev. C , 024313 (2001).[43] M. Baranger and K. Kumar, Nucl. Phys. A , 241(1968).[44] K. Kumar and M. Baranger, Nucl. Phys. A , 273(1968).[45] N. Hinohara, T. Nakatsukasa, M. Matsuo, and K. Mat-suyanagi, Prog. Theor. Phys. , 451 (2007).[46] N. Hinohara, T. Nakatsukasa, M. Matsuo, and K. Mat-suyanagi, Prog. Theor. Phys. , 59 (2008).[47] N. Hinohara, T. Nakatsukasa, M. Matsuo, and K. Mat-suyanagi, Phys. Rev. C , 014305 (2009).[48] D. J. Thouless and J. G. Valatin, Nucl. Phys. , 211(1962).[49] T. Bengtsson and I. Ragnarsson, Nucl. Phys. A , 14(1985).[50] S. G. Nilsson and I. Ragnarsson, Shapes and Shells inNuclear Structure (Cambridge University Press, 1995).[51] M. Yamagami, K. Matsuyanagi, and M. Matsuo, Nucl.Phys. A , 579 (2001).[52] H. Sakamoto and T. Kishimoto, Phys. Lett. B , 321(1990).[53] M. Baranger and K. Kumar, Nucl. Phys. , 113 (1965).[54] I. Hamamoto and W. Nazarewicz, Phys. Rev. C , 2489(1994).[55] L. Wilets and M. Jean, Phys. Rev. , 788 (1956).[56] K. Sato, N. Hinohara, T. Nakatsukasa, M. Matsuo, andK. Matsuyanagi, Prog. Theor. Phys.123