Pairing vibrations in the interacting boson model based on density functional theory
PPairing vibrations in the interacting boson model based on density functional theory
K. Nomura, D. Vretenar,
1, 2
Z. P. Li, and J. Xiang
4, 3 Department of Physics, Faculty of Science, University of Zagreb, HR-10000 Zagreb, Croatia ∗ State Key Laboratory of Nuclear Physics and Technology,School of Physics, Peking University, Beijing 100871, China School of Physical Science and Technology, Southwest University, Chongqing 400715, China School of Physics and Electronic, Qiannan Normal University for Nationalities, Duyun 558000, China (Dated: September 23, 2020)We propose a method to incorporate the coupling between shape and pairing collective degreesof freedom in the framework of the interacting boson model (IBM), based on the nuclear densityfunctional theory. To account for pairing vibrations, a boson-number non-conserving IBM Hamilto-nian is introduced. The Hamiltonian is constructed by using solutions of self-consistent mean-fieldcalculations based on a universal energy density functional and pairing force, with constraints onthe axially-symmetric quadrupole and pairing intrinsic deformations. By mapping the resultingquadrupole-pairing potential energy surface onto the expectation value of the bosonic Hamiltonianin the boson condensate state, the strength parameters of the boson Hamiltonian are determined.An illustrative calculation is performed for
Xe, and the method is further explored in a moresystematic study of rare-earth N = 92 isotones. The inclusion of the dynamical pairing degree offreedom significantly lowers the energies of bands based on excited 0 + states. The results are inquantitative agreement with spectroscopic data, and are consistent with those obtained using thecollective Hamiltonian approach. I. INTRODUCTION
Pairing correlations are among the most prominent fea-tures of the nuclear many-body system [1–4] and, to alarge extent, determine the structure of low-energy nu-clear spectra. Pairing vibrations [4–7], in particular, playan important role in fundamental processes such as neu-trinoless ββ decay [8], and spontaneous fission [9–12].The relevance of pairing vibrations in structure phenom-ena has been investigated using a variety of nuclear mod-els. Here we particularly refer to theoretical studies sincethe early 2000’s, that have used the geometrical collectiveHamiltonian [13–16], the time-dependent Hartree-Fock-Bogoliubov approaches [17], the nuclear shell model [18],the quasiparticle random-phase approximation [19, 20],and the generator coordinate methods (GCM) [8, 21].Nuclear density functional theory (DFT) is at presentthe most reliable framework for the description of low-energy structure of medium-heavy and heavy nuclei.Both the relativistic [22–24] and nonrelativistic [25–27]energy density functionals (EDFs) have been success-fully implemented the self-consistent mean-field (SCMF)studies of static and dynamical properties of finite nu-clei. Within this framework, the calculation of excita-tion spectra requires the restoration of broken symme-tries and configuration mixing, e.g., using the generatorcoordinate method (GCM) [3]. However, when multiplecollective coordinates need to be taken into account, thistype of calculation becomes computationally excessive.In the recent work of Ref. [16], the coupling betweenshape and pairing degrees of freedom has been consid- ∗ [email protected] ered using a quadrupole plus pairing collective Hamilto-nian based on the relativistic mean-field plus Bardeen-Cooper-Schrieffer (RMF+BCS) scheme. It has beenshown that the inclusion of the pairing degree of free-dom significantly improves the description of low-lying0 + states in rare-earth nuclei. The current implementa-tion of this approach is, however, restricted to axially-symmetric shapes.Nuclear spectroscopy is also studied with a theoret-ical method that consists in mapping the solutions ofthe DFT SCMF calculation onto the interacting-bosonHamiltonian [28, 29]. The interacting boson model(IBM) [30, 31], originally introduced by Arima andIachello, is a model in which correlated pairs of valencenucleons with spin and parity 0 + and 2 + are approxi-mated by effective bosonic degrees of freedom ( s and d bosons, respectively) [31, 32]. In the DFT-to-IBM map-ping procedure of Ref. [28], the strength parameters ofthe IBM Hamiltonian are completely determined by map-ping a SCMF potential energy surface (PES), obtainedfrom constrained SCMF calculations with a choice of theEDF and pairing force, onto the expectation value ofthe Hamiltonian in the boson coherent state [33]. Themethod has been successfully applied in studies of a va-riety of interesting nuclear structure phenomena, such asshape coexistence [34, 35], octupole collective excitations[36–39], quantum phase transitions in odd-mass and odd-odd nuclei [40–42], and β decay [43, 44].Considering the microscopic basis of the IBM in whichthe bosons represent valence nucleon pairs [31, 32, 45],one might attempt to implement also pairing vibrationalmodes in the IBM. In Refs. [46–49] additional monopoleboson degrees of freedom, different from the standard s boson, were introduced in the IBM to reproduce low-lying excited 0 + energies. Because of the inclusion of new a r X i v : . [ nu c l - t h ] S e p building blocks, however, the number of free parametersincreases in such an approach. Except for the referencesabove, very little progress has been made in explicitlyincluding pairing vibrations in the IBM framework.In this work, we develop a method to incorporate bothshape and pairing vibrations in the IBM. To accountfor the pairing degree of freedom, we introduce a ver-sion of the IBM (denoted hereafter by p v -IBM) in whichthe number of bosons is not conserved but is allowed tochange by one. Subsequently the boson space consistsof three subspaces that differ in boson number by one.The three subspaces are mixed by a specific monopolepair transfer operator. The strength parameters of the p v -IBM Hamiltonian are completely determined by themapping of the SCMF ( β, α ) potential energy surface,obtained from RMF+BCS calculations, onto the bosoniccounterpart. We demonstrate that the inclusion of dy-namical pairing in the IBM framework significantly low-ers the energies of excited 0 + states, in very good agree-ment with data.The paper is organised as follows. In Sec. II we brieflyreview the underlying SCMF calculations. In Sec. IIIthe p v -IBM model is introduced, and a method for map-ping the SCMF onto bosonic deformation energy surfacesis described. The model is illustrated using as an ex-ample the excitation spectrum of the nucleus Xe inSec. IV. In Sec. V the newly developed method is fur-ther explored in a study of low-energy K π = 0 + bands infour axially-symmetric N = 92 rare-earth isotones. Sec-tion VI presents a summary of the main results and anoutlook for future study. II. QUADRUPOLE-AND-PAIRINGCONSTRAINED SCMF CALCULATION
In a first step, constrained self-consistent mean-field(SCMF) calculations are performed within the frame-work of the relativistic mean-field plus BCS (RMF+BCS)model. In the present study, the particle-hole channelof the effective inter-nucleon interaction is determinedby the universal energy density functional PC-PK1 [50],while the particle-particle channel is modeled in the BCSapproximation using a separable pairing force [51]. Amore detailed description of the RMF+BCS frameworkcombined with the separable pairing force can be foundin Ref. [52]. The constraints imposed in the SCMF calcu-lation are on the expectation values of axial quadrupoleˆ Q and monopole pairing ˆ P operators. The quadrupoleoperator ˆ Q is defined as ˆ Q = 2 z − x − y , and itsexpectation value corresponds to the dimensionless axialdeformation parameter β : β = √ π r A / (cid:104) ˆ Q (cid:105) , (1)with r = 1 . P = 12 (cid:88) k> ( c k c ¯ k + c † ¯ k c † k ) , (2)where k and ¯ k denote the single-nucleon and the corre-sponding time-reversed states, respectively. c † k and c k are the single-nucleon creation and annihilation opera-tors. The expectation value of the pairing operator in aBCS state | α (cid:105) = (cid:89) k> ( u k + v k c † ¯ k c † k ) | (cid:105) , (3)corresponds to the intrinsic pairing deformation param-eter α , α = (cid:104) α | ˆ P | α (cid:105) = (cid:88) τ (cid:88) k> u τk v τk , (4)which can be related to the pairing gap ∆. The sum runsover both proton τ = π and neutron τ = ν single-particlestates. The quadrupole shape deformation Eqs. (1) andpairing deformation (4) represent the collective coordi-nates for constrained SCMF calculations [16]. As an ex-ample, in Fig. 1 we display the SCMF deformation en-ergy surface for the nucleus Xe in the plane of theaxial quadrupole β and pairing α deformation variables.The global minimum is found at β ≈ .
32 and α ≈ α . FIG. 1. (Color online) The potential energy surface (PES)of
Xe in the ( β, α ) plane, calculated by the constrainedRMF+BCS with the PC-PK1 energy density functional andseparable pairing interaction. All energies (in MeV) in thePES are normalized with respect to the binding energy of theabsolute minimum. The contours join points on the surfacewith the same energies, and energy difference between theneighbouring contours is 200 keV.
III. PAIRING VIBRATIONS IN THE IBMA. The Hamiltonian
In the next step we introduce a model that relates theSCMF ( β, α ) potential energy surface (PES) to an equiv-alent system of interacting bosons. The boson space com-prises monopole s and quadrupole d bosons, which repre-sent correlated L = 0 + and 2 + pairs of valence nucleons[31, 32, 45]. In the conventional IBM, the number ofbosons, denoted as n , is conserved for a given nucleus,i.e., n = n s + n d , where n s and n d stand for the s and d boson number, respectively. The boson number is equalto half the number of valence nucleons counted from thenearest closed shells and, in the illustrative case Xe,the boson core nucleus is
Sn and hence n = 9. Wedo not distinguish between neutron and proton degreesfreedom in the boson space. Considering the underlyingmicroscopic structure, the monopole pair transfer opera-tor in the bosonic system should be expressed, to a goodapproximation, in terms of the s boson degree of free-dom, i.e., ˆ P ∝ s † + s . Hence the s boson is expectedto be the most relevant for a description of the pairingvibration mode.To take explicitly into account the pairing vibrationmode, the boson configuration space is extended in sucha way that the total number of bosons is no longer con-served, but is allowed to change in the boson number byone, that is, 8 (cid:54) n (cid:54)
10 for the illustrative case of
Xe.The following IBM Hamiltonian is employed:ˆ H = (cid:88) n ˆ P n ( (cid:15) s ˆ n s + (cid:15) d ˆ n d + κ ˆ Q · ˆ Q + κ (cid:48) ˆ L · ˆ L ) ˆ P n + (cid:88) n (cid:54) = n (cid:48) ˆ P n t s ( s † + s ) ˆ P n (cid:48) (5)where ˆ P n is the projection operator onto the subspace[ n ]. The parameters for the Hamiltonian could differ be-tween different configuration spaces, but here the sameparameters are used for the three configurations. There-fore, for brevity, in the following the operator ˆ P n will beomitted, unless otherwise specified. The first and secondterms in Eq. (5) are the s and d boson-number operatorswith ˆ n s = s † · s and ˆ n d = d † · ˜ d . (cid:15) s and (cid:15) d are absolute val-ues of the single s and d boson energies. The third termis the quadrupole-quadrupole interaction with the bosonquadrupole operator ˆ Q = s † ˜ d + d † s + χ ( d † × ˜ d ) (2) . Thefourth term, with the boson angular momentum operatorˆ L = √ d † × ˜ d ) (1) , makes a significant contribution tothe moments of inertia of the K π = 0 + bands. The lastterm with strength t s in the above Hamiltonian repre-sents the one s -boson (monopole pair) transfer operator.It is the boson-number non-conserving term, and thusmixes the subspaces [ n − n ], and [ n + 1]. For laterconvenience, and since the total boson number operatoris given as ˆ n = ˆ n s + ˆ n d , the above Hamiltonian is rewrit- ten in the form:ˆ H = (cid:15) s ˆ n + (cid:15) d ˆ n d + κ ˆ Q · ˆ Q + κ (cid:48) ˆ L · ˆ L + t s ( s † + s ) (6)where (cid:15) d is the d -boson energy relative to the s boson one,i.e., (cid:15) d = (cid:15) d − (cid:15) s . The first term (cid:15) s ˆ n does not contributeto the relative excitation spectra, and is thus neglected inmost IBM calculations. In the present framework, how-ever, since we allow for the boson number to vary, thisglobal term is expected to play an important role, espe-cially for excitation energies of the 0 + states.The Hamiltonian Eq. (6) is diagonalized in the follow-ing M -scheme basis with M = 0, expressed as a directsum of the bases for the three configurations: | Φ (cid:105) = [ | ( sd ) n − (cid:105) ⊕ | ( sd ) n (cid:105) ⊕ | ( sd ) n +1 (cid:105) ] M =0 , (7)where M denotes the z -projection of the total angularmomentum I . The value of I for a given eigenstate isidentified by calculating the expectation value of the an-gular momentum operator squared, which should give theeigenvalue I ( I + 1).The present computational scheme is formally similarto IBM configuration-mixing calculations that describethe phenomenon of shape coexistence [35]. In the conven-tional configuration-mixing IBM framework, several dif-ferent boson Hamiltonians are allowed to mix [53]. Eachof these independent (unperturbed) Hamiltonians is as-sociated with a 2 m -particle-2 m -hole ( m ∈ Z ) excitationfrom a given major shell to the next and, since in the IBMthere is no distinction between particles and holes, differin boson number by two. The configuration-mixing IBMthus does not conserve the boson number, similar to thepresent case. Here, however, the model space comprisesa single major shell, and the boson number conservationis violated not by the contribution from next major shell(i.e., pair transfer across the shell closure), but by pairingvibrations. B. The boson condensate
The IBM analogue of the ( β, α ) PES is formulated an-alytically by taking the expectation value of the Hamil-tonian of Eq. (5) in the boson coherent state | Ψ( (cid:126)α ) (cid:105) [33, 54, 55]: | Ψ( (cid:126)α ) (cid:105) = | Ψ( n − , (cid:126)α ) (cid:105) ⊕ | Ψ( n, (cid:126)α ) (cid:105) ⊕ | Ψ( n + 1 , (cid:126)α ) (cid:105) , (8)where (cid:126)α represents variational parameters. Since herethe IBM model space comprises three different boson-number configurations, the above trial wave function isexpressed as a direct sum of three independent coherentstates. Each of them is given by | Ψ( n, (cid:126)α ) (cid:105) = 1 √ n ! ( b † c ) n | (cid:105) , (9)and the condensate boson b c is defined as b c = ( α + α ) − / ( α s + α d ) , (10)where the amplitudes α and α should be related tothe pairing deformation α and the axial deformation pa-rameter β in the SCMF calculation, respectively. Thevariable α can be considered as the shape deformationparameter in the collective model: α = ¯ β, (11)where ¯ β is the IBM analog of the axially symmetricSCMF deformation parameter. We propose to performthe following coordinate transformation for the variable α : α = cosh (¯ α − ¯ α min ) . (12)The new coordinate ¯ α is equivalent to the pairing defor-mation α . ¯ α min stands for the ¯ α value corresponding tothe global minimum on the IBM PES. We assume thefollowing relations that relate the amplitudes ¯ β and ¯ α in the boson system to the β and α coordinates of theSCMF model: ¯ β = C β β, ¯ α = C α α. (13)The dimensionless coefficients of proportionality C β and C α are additional scale parameters determined by themapping.Since our model space comprises three subspaces withdifferent number of bosons, the PES of the boson systemis expressed in a matrix form [56]: (cid:32) E n − ,n − (¯ α, ¯ β ) E n − ,n (¯ α, ¯ β ) 0 E n,n − (¯ α, ¯ β ) E n,n (¯ α, ¯ β ) E n,n +1 (¯ α, ¯ β )0 E n +1 ,n (¯ α, ¯ β ) E n +1 ,n +1 (¯ α, ¯ β ) (cid:33) (14)In the limit in which boson number is conserved, onlythe diagonal element E n,n is considered. The energy-surface matrix of Eq. (14) is diagonalized at each pointon the surface ( ¯ β, ¯ α ), resulting in three energy surfaces[56]. The usual procedure in most IBM calculations withconfiguration mixing is to retain only the lowest energyeigenvalue at each deformation.The analytical expressions for the diagonal and non-diagonal elements of the matrix Eq. (14) are obtained bycalculating expectation values of the Hamiltonian Eq. (6)in the coherent state Eq. (10), with the amplitudes de-fined in Eqs. (11) and (12). The right-hand side ofEq. (12) is Taylor expanded: cosh ¯ α (cid:48) = 1+ ¯ α (cid:48) / O (¯ α (cid:48) ),and thus cosh ¯ α (cid:48) = 1+ ¯ α (cid:48) + O (¯ α (cid:48) ), where ¯ α (cid:48) ≡ ¯ α − ¯ α min .Terms of the order of O (¯ α (cid:48) ) and higher are hereafterneglected. The resulting analytical expressions for thematrix elements in Eq. (14) read: E n,n (¯ α, ¯ β ) = (cid:15) s n + n [5 κ + ( (cid:15) d + 6 κ (cid:48) + κ (1 + χ )) ¯ β ]1 + ¯ α (cid:48) + ¯ β + κn ( n − α (cid:48) + ¯ β ) (cid:34) α (cid:48) ) ¯ β − (cid:114) χ ¯ β + 27 χ ¯ β (cid:35) , (15) for the diagonal elements, and E n,n (cid:48) (¯ α, ¯ β ) = E n (cid:48) ,n (¯ α, ¯ β ) = t s √ n + 1 (cid:112) α (cid:48) + ¯ β , (16)for the non-diagonal elements with n > n (cid:48) . The termproportional to ¯ α (cid:48) ¯ β in the numerator of the third termof Eq. (15), and the term quadratic in ¯ α (cid:48) in the numeratorof Eq. (16) are neglected.The functional forms in Eqs. (15) and (16), in par-ticular the norm factor N = 1 + ¯ α (cid:48) + ¯ β that de-pends quadratically on ¯ α , most effectively produce an α -deformed equilibrium state that is consistent with theSCMF PES. The form of the norm factor N ensures thatno divergence occurs at ¯ β ≈ α (cid:48) ≈
0. In the limit¯ α → ¯ α min (= C α α min ), the expression for E n,n (¯ α, ¯ β ) re-duces to the one used in standard sd -IBM calculations[31, 33]. FIG. 2. (Color online) Same as in the caption to Fig. 1 butfor the IBM energy surface.
C. Mapping the boson Hamiltonian
The p v -IBM Hamiltonian in Eq. (6) is constructed inthe following steps:1. The strength parameters that appear in the bosonnumber conserving part of the Hamiltonian: (cid:15) d , κ ,and χ , as well as the scale factor C β , are determinedso that the diagonal matrix element E n,n (¯ α, ¯ β ) re-produces the SCMF PES at α = α min .2. The strength κ (cid:48) of the rotational ˆ L · ˆ L term is de-termined separately so that the bosonic moment ofinertia calculated in the intrinsic frame [57] at theglobal minimum, should equal the Inglis-Belyaev[58, 59] moment of inertia at the corresponding con-figuration on the SCMF energy surface. The detailsof this procedure can be found in Ref. [60].3. The s boson energy (cid:15) s and mixing strength t s , aswell as the scale factor C α , are determined in the( ¯ β, ¯ α ) plane so that the lowest eigenvalue of the en-ergy surface matrix Eq. (14) reproduces the topol-ogy of the SCMF PES in the neighbourhood of theequilibrium minimum. TABLE I. Strength parameters of the boson HamiltonianEq. (6) determined by mapping the SCMF ( β, α ) energy sur-face to the boson space. The parameters χ , C β , and C α aredimensionless, while the others are in units of MeV. (cid:15) s (cid:15) d κ χ κ (cid:48) t s C β C α − . − . − .
029 0.18 2.75 0.045
The values of the resulting parameters of the IBMHamiltonian are listed in Table I, and the correspond-ing IBM PES is shown in Fig. 2. Consistent with theSCMF PES, the equilibrium minimum of the IBM PESis found at β ≈ .
32 and α ≈
10. The potential energysurfaces exhibit a similar topography except for the factthat, away from the global minimum, the IBM surfacetends to be softer than the DFT one obtained using theconstrained SCMF method. This is a common charac-teristic of the IBM [28] that arises because of the morerestricted boson model space as compared to the SCMFapproach based on the Kohn-Sham DFT. The former isbuilt only from the valence nucleons, while the lattermodel space contains all nucleons. Therefore, the bo-son Hamiltonian parameters are determined by the map-ping procedure that is carried out in the neighbourhoodof the global minimum, as this region is most relevantfor low-energy excitations. The Hamiltonian Eq. (6) isdiagonalised in the M -scheme basis of Eq. (7). E x c i t a t i o n e n e r g y ( M e V ) + + + + + + [ n −
1] 0 + + + + + + [ n ] 0 + + + + + + [ n +1] 0 + + + + + + + + + + + + After mixing
FIG. 3. (Color online) Excitation spectra for the unperturbedboson configurations [ n − n ] and [ n + 1], and the final oneobtained after mixing the three configurations. IV. ILLUSTRATIVE EXAMPLE:
XEA. Energy spectra
Figure 3 depicts the calculated excitation spectra cor-responding to the unperturbed boson configurations [ n − n ] and [ n + 1], and the spectrum obtained by mixingthe three different configurations. Without mixing, the0 + ground states for the three configurations cluster to-gether within a small energy range, and the first excited0 + states are also found in a narrow interval around 3MeV. This is, of course, easy to understand because thespaces in which the Hamiltonian is diagonalized only dif-fer by ∆ n = 1 in the boson number. Allowing for configu-ration mixing (boson-number non-conserving term in theHamiltonian Eq. (6)), states with the same spin repel andthe two lowest 0 + excited states are found at excitationenergies E exc ≈ + states. Mixing configurations thatcorrespond to different boson numbers will be essentialfor the description of low-energy 0 + excitations.Figure 4 compares the excitation spectra for Xe cal-culated using the IBM with a single configuration [ n ],where the boson number n = 9 is conserved and the effectof the pairing vibration is not taken into account (left-hand panel, (a)), with those obtained with the IBM thatincludes pairing-vibrations ( p v -IBM), shown in the cen-tral panel (b)). Part of the available experimental energyspectra [61, 62] is also shown in the right-hand panel ofFig. 4. The theoretical states are grouped into bands ac-cording to the sequence of calculated E2 strength values.Since we aim to describe excited 0 + states, only bandsthat are built on a 0 + state and that follow the ∆ I = 2E2 transition systematics are shown in the figure. Theremarkable result is that the K π = 0 + bands built on the0 +2 and 0 +3 states are dramatically lowered in energy bytaking into account configuration mixing, that is, by theinclusion of pairing vibrations. The resulting excitationspectrum is in much better agreement with experiment. B. Structure of wave functions
To shed more light upon the nature of excited statescalculated in the p v -IBM, we show in Fig. 5 the proba-bilities of the three different boson-space configurations[ n − n ], and [ n + 1] in the lowest five 0 + , 2 + , and4 + states of Xe. Let us consider, for example, the 0 + states. Only half the wave function of the ground state0 +1 is accounted for by the [ n ] configuration, while therest is equally shared by the [ n −
1] and [ n + 1] config-urations. The 0 +2 state exhibits a structure that is com-pletely different from the ground state. The dominantcontributions come from the [ n −
1] and [ n + 1] configu-rations, both with probabilities of nearly 50 %, whereasthere is almost no contribution from the [ n ] configura- E x c i t a t i o n e n e r g y ( M e V ) Xe +1 +1 +1 +1 +2 +3 +3 IBM +1 +1 +1 +1 +2 +2 +2 +2 +3 +4 +4 +3 pv -IBM + + + + + + + (2) + Expt
FIG. 4. (Color online) Excitation spectra of
Xe resulting from the IBM calculation with a single boson-number configuration[ n = 9] (left panel), and including configuration mixing between [ n = 8], [ n = 9], and [ n = 10] boson spaces ( p v -IBM).Experimental states are from Refs. [61, 62] (right panel). The 0 + band-head states of the K π = 0 + bands are highlighted withthick lines. P r o b a b ili t y ( % ) (a) + k [ n − n ] [ n +1] P r o b a b ili t y ( % ) (b) + k k P r o b a b ili t y ( % ) (c) + k FIG. 5. (Color online) Probabilities of the [ n − n ], and[ n + 1] components in the wave functions of the five lowest-energy 0 + , 2 + , and 4 + states in Xe. tion space. The structure of the 0 +3 state is very similarto that of the 0 +1 . The state 0 +4 appears to be differ-ent from the lower ones in that the three configurationsare more equally mixed: the [ n ] and [ n −
1] componentsare found with approximately 40 % probability each, andthe remaining 20 % belongs to the [ n + 1] configurationspace. The content of the 0 +5 wave function is similar to | › ˆ H m i x fi | ( M e V ) (a) I +1 [ n − − [ n ][ n ] − [ n +1] | › ˆ H m i x fi | ( M e V ) (b) I +2 I ( )0.00.20.40.60.81.01.2 | › ˆ H m i x fi | ( M e V ) (c) I +3 FIG. 6. (Color online) Matrix elements of the monopole pairtransfer operator ˆ H mix (last term in Eq. (6)) between theunperturbed [ n −
1] and [ n ] configurations, and between theunperturbed [ n ] and [ n +1] configurations, for the lowest threeeven-spin states up to I = 8 + . that of 0 +2 . A corresponding structure is also found forthe 2 + and 4 + states. The only exception is perhaps thefourth lowest state of 2 + and 4 + , nevertheless in eachstate 0 +4 , 2 +4 , and 4 +4 the largest contribution to theirwave function comes from the [ n ] configuration. C. Mixing matrix elements
Figure 6 displays the matrix elements of the mixinginteraction | (cid:104) I + k | ˆ H mix | I + k (cid:105) | ( I even, k = 1 , , H mix = t s ( s † + s ), that couple the unperturbed [ n −
1] and[ n ] configurations, and the unperturbed [ n ] and [ n + 1]configurations. For all of the unperturbed I , , states,the mixing between the [ n −
1] and [ n ] configurationsis almost identical to the coupling between the [ n ] and[ n + 1] configurations. In both cases the mixing is gen-erally stronger between states with lower spin, and grad-ually decreases in magnitude as the angular momentumincreases. D. Electromagnetic transitions
The electric quadrupole (E2) and monopole (E0) tran-sition rates can also be analyzed in the p v -IBM. Thecorresponding operators are defined asˆ T E = e B ˆ Q (17)ˆ T E = ξ ˆ n d + η ˆ n (18)with e B is the E2 boson effective charge, and ξ and η areparameters. The B ( E
2) and ρ ( E
0) transition rates arethen calculated using the relations: B ( E I i → I (cid:48) j ) = 12 I i + 1 | (cid:104) I (cid:48) j (cid:107) ˆ T E (cid:107) I i (cid:105) | (19) ρ ( E I i → I j ) = Z e r A / I i + 1 | (cid:104) I j (cid:107) ˆ T E (cid:107) I i (cid:105) | . (20) TABLE II. B ( E I i → I (cid:48) j ) values in Weisskopf units, calcu-lated in the IBM and p v -IBM. The experimental values aretaken from ENSDF database.IBM p v -IBM Experiment B ( E
2; 2 +1 → +1 ) 80 79 78(4) B ( E
2; 4 +1 → +1 ) 113 114 114(6) B ( E
2; 6 +1 → +1 ) 121 124 1.1 × (4) B ( E
2; 2 + K =0 +2 → + K =0 +2 ) 46 79 B ( E
2; 4 + K =0 +2 → + K =0 +2 ) 57 111 B ( E
2; 6 + K =0 +2 → + K =0 +2 ) 64 119 In Table II we compare the B ( E
2) values calculatedwith ( p v -IBM) and without (IBM) the inclusion of dy-namical pairing. A typical value for the E2 effectivecharge e B = 0 . e · b is used both in the IBM and p v -IBM calculations. The B ( E
2) transitions between theyrast states do not change by the inclusion of the pair-ing degree of freedom. The results of both calculationsare consistent with the experimental values [61]. In the p v -IBM calculation, the E2 transitions in the 0 +2 -based TABLE III. Reduced matrix elements of the d -boson numberoperator ˆ n d and the total boson number operator ˆ n for E0transitions between the lowest three IBM and p v -IBM states0 + and 2 + of Xe. IBM p v -IBM I + i I + j (cid:104) I j (cid:107) ˆ n d (cid:107) I i (cid:105) (cid:104) I j (cid:107) ˆ n d (cid:107) I i (cid:105) (cid:104) I j (cid:107) ˆ n (cid:107) I i (cid:105) +2 +1 − .
166 0.473 0.7210 +3 +1 − .
139 0.006 0.0170 +3 +2 − .
431 0.451 0.6872 +2 +1 − . − . − . +3 +1 − .
482 0.0842 +3 +2 − . − . band display a more pronounced collectivity, comparableto that in the ground state band. As shown in Fig. 5, inthe p v -IBM wave functions we find a rather large contri-bution from the [ n +1] configurations to the 0 +2 band, andthis accounts for the enhanced B ( E
2) strengths withinthis sequence of states.Since there are no data for the E Xe,in Table III we compare the calculated reduced matrixelements of the ˆ n d and ˆ n operators, which constitutethe E0 operator of Eq. (18). Note that in the number-conserving IBM only the ˆ n d term contributes. FromTable III one notices that the reduced matrix elements (cid:104) I j (cid:107) ˆ n d (cid:107) I i (cid:105) in the p v -IBM calculation are systematicallysmaller in magnitude than the corresponding quantity inthe IBM, most notably for the 0 +3 → +1 transition. Thematrix element (cid:104) I j (cid:107) ˆ n (cid:107) I i (cid:105) is generally of equal magnitudeas that of ˆ n d and, therefore, one expects that it will givea sizeable contribution to the ρ ( E
0) values in p v -IBM. V. APPLICATION TO N = 92 ISOTONES
For a more detailed analysis, we apply the p v -IBMtheoretical framework to a study of the structure of theaxially-symmetric N = 92 rare-earth isotones. For nu-clei in this region of the nuclear chart, an unexpect-edly large number of low-energy excited 0 + states havebeen observed [63, 64]. From a theoretical point of view,they have been interpreted in terms of pairing vibrations[16], contributions of intruder orbitals [46], and excita-tions of double octupole phonons [65, 66]. The occur-rence of low-lying excited 0 + states also characterizesthe quantum shape-phase transition from spherical toaxially-deformed nuclear systems [67]. A. ( β, α ) potential energy surfaces In Fig. 7 we plot the SCMF deformation energy sur-faces in the ( β, α ) plane for the N = 92 isotones: Nd,
Sm,
Gd, and
Dy. Note that this is the sameas Fig. 7 in Ref. [16], in which the coupling of shape
FIG. 7. (Color online) Same as in the caption to Fig. 1 butfor the N = 92 isotones Nd,
Sm,
Gd, and
Dy.FIG. 8. (Color online) Same as in the caption to Fig. 7, butfor the IBM energy surfaces. and pairing vibrations was analyzed using a collectiveHamiltonian based on nuclear DFT. Pronounced axiallysymmetric global minima are calculated at β ≈ . α , and the minima extend ina rather large interval 5 (cid:54) α (cid:54)
15. As already noted inRef. [16], this softness is reduced with the increase of theproton number, while simultaneously the energy surfacesbecome more soft in the quadrupole collective deforma-tion.
TABLE IV. Same as the caption Table I, but for the N = 92isotones. (cid:15) s (cid:15) d κ χ κ (cid:48) t s C β C α Nd 1.40 0.478 − . − . − . Sm 1.37 0.626 − . − . − . Gd 1.30 0.530 − . − . − . Dy 1.32 0.533 − . − . − . The corresponding bosonic energy surfaces in the( β, α ) plane are drawn in Fig. 8. They exhibit a non-zero α global minimum, consistent with the microscopicSCMF PESs. As already noted above in the case of Xe, the IBM PESs are considerably softer than theSCMF ones, especially far from the global minimum.This is due to the more restricted boson model space,that is, the restricted space of valence nucleons fromwhich the bosons are built does not contain the high-energy configurations that contribute to the SCMF solu-tions far from the equilibrium minimum. The strengthparameters of the boson Hamiltonian in Eq. (5), deter-mined by mapping the SCMF energy surfaces to the ex-pectation values of the Hamiltonian in the boson con-densate, are listed in Table IV for the N = 92 iso-tones . The large negative values of the derived pa-rameter χ parameter, close to the SU(3) limit of theIBM χ SU(3) = −√ /
2, reflect the pronounced axially-symmetric prolate quadrupole deformation of these nu-clei.
B. Low-energy excitation spectra
Figures 9, 10, 11, and 12 compare the three-lowest K π = 0 + bands of four N = 92 isotones: Nd,
Sm,
Gd, and
Dy, respectively, computed using the IBMand p v -IBM and IBM. In addition to the correspondingdata, we also include the results of our recent study thathas used the newly developed Quadrupole-Pairing Col-lective Hamiltonian (QPCH) to analyze the low-energyspectra of these nuclei [16]. A detailed description of theQPCH model can be found Ref. [16]. All three Hamilto-nians (IBM, p v -IBM, and QPCH) used here are based onthe same energy density functional and pairing interac-tion. The excitation spectra shown in Figs. 9-12 clearlyillustrate the striking effect of the coupling between shapeand pairing degrees of freedom. The inclusion of dynami-cal pairing significantly lowers the bands based on excited0 + states. The bands calculated with p v -IBM and QPCHare in much better agreement with experiment, especiallythe band based on 0 +2 . We note that the overall quality ofthe p v -IBM description of K π = 0 + bands is comparableto that of the fully microscopic QPCH model.Even though we only show the K π = 0 + bands inFigs. 9-12, the K π = 2 + (or γ -) bands are also observedexperimentally for the N = 92 isotones. The IBM models E x c i t a t i o n e n e r g y ( M e V ) Nd +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +3 +4 +4 IBM +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +3 +3 +3 +3 +3 +3 pv -IBM + + + + + + + (2 + )(4 + ) Expt +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +3 +3 +3 +3 +3 +3 QPCH
FIG. 9. (Color online) Low-energy K π = 0 + bands of Nd, calculated using the IBM without and with the dynamical pairingdegree of freedom, in comparison to available data [61]. The corresponding spectrum obtained with the Quadrupole-PairingCollective Hamiltonian (QPCH) model is included for comparison. E x c i t a t i o n e n e r g y ( M e V ) Sm +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +3 +4 +4 IBM +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +3 +3 +3 +3 +3 +3 pv -IBM + + + + + + + + + + + + (4 + ) Expt +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +3 +3 +3 +3 +3 +3 QPCH
FIG. 10. (Color online) Same as in the caption to Fig. 9, but for
Sm. can be used to compute these states but, since this studyis restricted to axial symmetry, the focus is on K π = 0 + bands. For completeness, the K = 2 +1 bandhead is cal-culated to be 2.248 (2.315), 2.330 (2.451), 2.102 (2.114),and 2.085 (2.099) MeV, for Nd,
Sm,
Gd, and
Dy in the p v -IBM (IBM) calculations, respectively.Thus, in the axial case, the energies of the γ band arehardly affected by the inclusion of the pairing degree offreedom. The corresponding experimental 2 + γ energiesfor Sm,
Gd, and
Dy are: 1.440 [68], 1.154 [63],0.946 MeV [64], respectively, whereas no γ band has beenidentified in Nd. Therefore we note that, for a quan-titative comparison with data, the theoretical frameworkshould be extended with the γ degree of freedom (non-axial shapes). C. Structure of the wave functions
In Fig. 13 we plot the probabilities of the three differentboson configurations [ n − n ], and [ n +1] in the p v -IBMwave functions of the four lowest-energy 0 + states. In allfour N = 92 isotones nearly half of the wave function ofthe 0 +1 ground state (a) is accounted for by the [ n ] config-uration. The structure of wave function for the 0 +2 stateis based mainly on the [ n −
1] and [ n + 1] configurations,with almost no contribution from the states of the [ n ]-boson model space. The 0 +3 state is mainly composed of[ n ]-boson configurations, similar to the 0 +1 ground state.The wave function of the 0 +4 state (d) somewhat differsin structure from the lower-energy 0 + states: each of the[ n ] and [ n −
1] configurations takes approximately 40 %of the wave function, and the remaining 20 % consists ofthe [ n + 1] configuration.0 E x c i t a t i o n e n e r g y ( M e V ) Gd +1 +1 +1 +1 +1 +1 +2 +3 +3 +3 +3 +3 +3 +4 +4 IBM +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +3 +3 +3 +3 +3 +3 pv -IBM + + + + + + + + + + + + + + + + (8 + )10 + Expt +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +3 +3 +3 +3 +3 +3 QPCH
FIG. 11. (Color online) Same as in the caption to Fig. 9, but for
Gd. E x c i t a t i o n e n e r g y ( M e V ) Dy +1 +1 +1 +1 +1 +1 +2 +3 +3 +3 +3 +3 +3 +4 +4 IBM +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +3 +3 +3 +3 +3 +3 pv -IBM + + + + + + + + + + + + Expt +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +3 +3 +3 +3 +3 +3 QPCH
FIG. 12. (Color online) Same as in the caption to Fig. 9, but for
Dy.
D. Transition rates
The B ( E
2) and ρ ( E
0) values calculated with the p v -IBM, IBM and QPCH models are compared to availabledata in Tabs. V and VI, respectively, The effective bo-son charge in the E2 operator is e B = 0 . e · b. Theparameters of the E0 operators: ξ = 0 .
095 and η = 0 . for the p v -IBM, and ξ = 0 .
075 fm for the IBM, areadjusted to obtain the best agreement with the experi-mental ρ ( E
0) values for
Gd, and kept unchanged forall four N = 92 isotones. There are no adjustable param-eters for the calculation of transition rates in the QPCHmodel. Note that there are no E2 transitions related tothe γ band with K π = 2 + in the QPCH model since, aspointed out above, the present version of QPCH does notinclude the triaxial degree of freedom.As shown in Table V, the B ( E
2) transition strengthswithin the ground state bands are reproduced very nicelyby all the models. There is no significant difference be- tween the B ( E
2) values calculated with the IBM and p v -IBM. Many experimental results are available for thetransition rates of Sm and, generally, they are well re-produced by all three models. In
Gd the theoreticalresults reproduce the data, except for an overestimate ofthe experimental B ( E
2; 2 + K =0 +2 → + K =0 +2 ) value of 52 ± Dy.The calculated ρ ( E
0) values are generally in satisfac-tory agreement with available data (Table VI), except inthe case of
Sm, in which both the IBM and QPCH ap-proaches considerably overestimate the measured [68] up-per limits of the ρ ( E
0; 2 +2 → +1 ) and ρ ( E
0; 4 +2 → +1 )values. It appears that the IBM and p v -IBM models re-produce the data somewhat better than QPCH, but thiscomes at the expense of additional adjustable parametersin the E P r o b a b ili t y ( % ) N =92 (a) +1
60 62 64 66(b) +2 [ n − n ] [ n +1]
60 62 64 66 Z P r o b a b ili t y ( % ) (c) +3
60 62 64 66 Z (d) +4 FIG. 13. (Color online) Probabilities of the [ n − n ], and[ n + 1] components in the p v -IBM wave functions of the fourlowest 0 + states in the N = 92 isotones. VI. CONCLUSION AND OUTLOOK
We have developed a model that incorporates the cou-pling between nuclear shape and pairing degrees of free-dom in the framework of the IBM, based on nuclear DFT.To account for pairing vibrations, a boson-number non-conserving IBM Hamiltonian is introduced. The bosonmodel space is then extended from the usual one in whichthe boson number equals half the number of valence nu-cleons, to include three subspaces that differ in bosonnumber by one. The three subspaces are mixed by aspecific monopole pair transfer operator. In a first stepof the construction of the IBM Hamiltonian, a set ofconstrained SCMF calculation is performed for a spe-cific choice of the universal EDF and pairing force, andwith the constraints on the expectation values of the ax-ial mass quadrupole operator and monopole pairing op-erator. These calculations produce a potential energysurface (PES) in the plane of the axial quadrupole β and pairing α collective coordinates. The energy surfaceis then mapped onto the expectation value of the IBMHamiltonian in the boson condensate state. The mappingdetermines the strength parameters of the IBM Hamilto-nian, and from the corresponding eigenvalue equation ex-citation energy spectra and transition rates are obtained.As a first application of the newly developed model,this work has focused on the excitation spectrum of Xe. By the inclusion of the dynamical pairing degreeof freedom in the IBM and the resulting boson-numberconfiguration mixing, it has been shown that the excita-tion energies of the 0 +2 and 0 +3 states and the bands builton them, are dramatically lowered by a factor of two orthree, thus bringing the theoretical spectrum in quanti-tative agreement with experiment. The validity of themethod has been further examined in a more system- TABLE V. B ( E I i → I (cid:48) j ) values (in Weisskopf units) for Nd,
Sm,
Gd, and
Dy. Experimental values [61, 63,69] are compared to results of the p v -IBM, IBM and QPCHmodel calculations. Expt p v -IBM IBM QPCH Nd 2 +1 → +1 ±
10 160 162 1624 +1 → +1 ±
11 225 227 2316 +1 → +1 +51 −
240 241 253
Sm 2 +1 → +1 ± +1 → +1 ± +1 → +1 ± +1 → +1 ±
17 283 281 3240 + K =0 +2 → +1 ± . + K =0 +2 → +1 ± .
04 0.7 1.7 1.12 + K =0 +2 → +1 ± .
09 1.4 2.0 1.62 + K =0 +2 → +1 ± .
15 3.7 0.6 3.14 + K =0 +2 → +1 ± .
11 0.7 0.9 1.54 + K =0 +2 → +1 ± .
18 1.2 1.3 1.44 + K =0 +2 → +1 ± .
21 3.7 1.9 2.72 + γ → +1 ± . + γ → +1 ± . + γ → +1 ± .
05 2.4 2.8
Gd 2 +1 → +1 ± +1 → +1 ± +1 → +1 ± +1 → +1 ±
17 304 303 33510 +1 → +1 ±
14 299 296 3420 + K =0 +2 → +1 +4 − + K =0 +2 → +1 ± .
06 0.6 0.8 1.82 + K =0 +2 → + K =0 +2 ±
23 196 136 2362 + K =0 +2 → +1 ± . + K =0 +2 → +1 ± . + K =0 +2 → + K =0 +2 +110 −
276 180 3374 + K =0 +2 → +1 +0 . − . + K =0 +2 → +1 +0 . − . + K =0 +3 → +1 +2 . − . + γ → +1 ± .
16 2.4 2.42 + γ → +1 ± .
25 6.3 5.82 + γ → +1 ± .
04 0.2 0.2
Dy 2 +1 → +1 ± +1 → +1 ±
15 309 311 2846 +1 → +1 × (4) 335 337 3128 +1 → +1 × (7) 342 342 3262 + γ → +1 ± . + γ → +1 ± + γ → +1 ± . + K =0 +2 → +1 ± . + K =0 +2 → +1 ± . + K =0 +2 → +1 ± atic study of the axially-symmetric N = 92 rare-earth2 TABLE VI. Same as the caption to Tab. V but for the ρ ( E I i → I j ) × values. The experimental ρ ( E
0) arefrom Refs. [61, 64, 68, 70–73].Expt p v -IBM IBM QPCH Sm 0 +2 → +1 ±
42 43 39 542 +2 → +1 (cid:54) . ± . +2 → +1 +12 . − .
38 28 53
Gd 0 +2 → +1 ±
21 42 43 730 +3 → +1 +1 . − . +3 → +2 +27 −
41 40 970 +4 → +1 +2 . − .
54 0.07 3.40 +4 → +3 +5 . − . + K =0 +2 → +1 ± + K =0 +3 → +1 +0 . − . + K =0 +2 → +1 +25 −
38 34 724 + K =0 +3 → +1 <
15 4 × − Dy 2 + K =0 +2 → +1 ±
12 40 48 75 isotones. The microscopic coupling between shape andpairing degrees of freedom leads to a boson Hamiltonianthat, when compared to the standard IBM, significantlylowers the K π = 0 + bands based on excited 0 + statesin Nd,
Sm,
Gd, and
Dy. The calculated ex-citation spectra are in an excellent agreement with ex-periment, and are fully consistent with the results of thecorresponding quadrupole-pairing collective Hamiltonianmodel [16]. Both models also reproduce the empirical E2and E0 transition properties with a reasonable accuracy.The present study has shown a new interesting possi-bility for extending the DFT-to-IBM mapping method. By incorporating explicitly the dynamical pairing degreeof freedom in the IBM, this model can be used to de-scribe pairing vibrational modes and quantitatively re-produce the excitations of low-energy 0 + states. Here wehave only considered the coupling of the pairing degree offreedom with the axial shape deformation. A more chal-lenging case, but also more realistic, will be the couplingbetween the pairing and triaxial ( β, γ ) shape degrees offreedom. This will be particularly important in γ -softnuclei and systems that exhibit shape coexistence. Inprinciple, such an extension is also possible in the QPCHapproach, however this generates additional terms in thecollective Schr¨odinger equation that represent the cou-plings of the β − γ and γ − α variables. In contrast,it is rather straightforward to extend the present IBMframework to triaxial nuclei, since there is no need fornew building blocks in the boson Hamiltonian. Work inthis direction is in progress, and will be reported in aforthcoming article. ACKNOWLEDGMENTS
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