Octupole correlations in light actinides from the interacting boson model based on the Gogny energy density functional
K. Nomura, R. Rodríguez-Guzmán, Y. M. Humadi, L. M. Robledo, J. E. García-Ramos
OOctupole correlations in light actinides from the interacting boson model based on theGogny energy density functional
K. Nomura, ∗ R. Rodr´ıguez-Guzm´an, Y. M. Humadi, L. M. Robledo,
3, 4 and J. E. Garc´ıa-Ramos Department of Physics, Faculty of Science, University of Zagreb, HR-10000, Croatia Physics Department, Kuwait University, 13060 Kuwait, Kuwait Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid and CIAFF, E-28049 Madrid, Spain Center for Computational Simulation, Universidad Polit´nica de Madrid,Campus de Montegancedo, Bohadilla del Monte, E-28660-Madrid, Spain Departamento de Ciencias Integradas y Centro de Estudios Avanzados en F´ısica,Matem´atica y Computaci´on, Universidad de Huelva, E-21071 Huelva, Spain (Dated: August 21, 2020)The quadrupole-octupole coupling and the related spectroscopic properties have been studiedfor the even-even light actinides − Ra and − Th. The Hartree-Fock-Bogoliubov approxi-mation, based on the Gogny-D1M energy density functional, has been employed as a microscopicinput, i.e., to obtain (axially symmetric) mean-field potential energy surfaces as functions of thequadrupole and octupole deformation parameters. The mean-field potential energy surfaces havebeen mapped onto the corresponding bosonic potential energy surfaces using the expectation valueof the sdf
Interacting Boson Model (IBM) Hamiltonian in the boson condensate state. The strengthparameters of the sdf -IBM Hamiltonian have been determined via this mapping procedure. Thediagonalization of the mapped IBM Hamiltonian provides energies for positive- and negative-paritystates as well as wave functions which are employed to obtain transitional strengths. The results ofthe calculations compare well with available data from Coulomb excitation experiments and pointtowards a pronounced octupole collectivity around
Ra and
Th.
I. INTRODUCTION
It is a well known fact that just a handful of nuclei ex-hibit reflection asymmetric ground states with non zerooctupole deformation. Reflection asymmetric shapes arefavored in some very specific regions of the nuclear chartwith neutron N and/or proton Z numbers around 34,56, 88, 136, . . . [1, 2]. However, dynamical octupole cor-relations have attracted considerable attention in recentyears as they play a relevant role in the description ofmany negative parity collective states like the low-lying1 − states in the spectra of even-even nuclei that are usu-ally considered fingerprints of octupole correlations [3, 4].In the common situation where the ground state of thosenuclei is quadrupole deformed, there exists a 3 − state,member of the corresponding negative-parity rotationalband, which decay through fast E + ground state. On the other hand, the decay of the 1 − tothe ground state proceeds via E Rn,
Raand , Ra [5, 6]) and lanthanides ( , Ba [7, 8]).The study of octupole correlations also has a potentialimpact on other research fields. Indeed, the presence of ∗ [email protected] static (and dynamic) nuclear octupole correlations en-hance the fingerprints of the existence of a non-zero elec-tric dipole moment of elementary particles. The existenceof such an effect would imply the violation of the CP sym-metry implying the existence of new physics beyond theStandard Model of particle physics [9].From a theoretical point of view, both relativistic[10, 11] and non-relativistic [12, 13] approaches rootedin the nuclear energy density functional (EDF) frame-work [12] have been extensively employed to describeintrinsic nuclear shapes and the related spectroscopicproperties. In particular, the static and dynamic as-pects associated with the spontaneous breaking of reflec-tion symmetry have been studied using the self-consistentmean-field (SCMF) approximation based on a given non-relativistic or relativistic EDF [3, 4, 14–38]. Dynamicalbeyond-mean-field correlations, stemming from symme-try restoration and/or fluctuations in the relevant collec-tive deformations, have been considered within configu-ration mixing approaches in the spirit of the GeneratorCoordinate Method (GCM) [12, 13, 39].On the one hand, beyond-mean-field configuration-mixing approaches are required to access spectroscopicproperties such as, the excitation energies of negative-parity states as well as B ( E
1) and B ( E
3) reduced tran-sition probabilities. On the other hand, beyond-mean-field approaches become computationally expensive inmedium and heavy nuclei, specially when several collec-tive coordinates are to be included in the GCM ansatz.This drawback of the GCM justifies the introduction ofcomputationally less expensive approaches like the in-teracting boson model (IBM) mapping procedures intro-duced in Refs. [40, 41] In this approach, the SCMF po- a r X i v : . [ nu c l - t h ] A ug tential energy surfaces (SCMF-PESs) are mapped ontothe corresponding (bosonic) IBM-PESs as to determinesome of the strength parameters of the correspondingIBM Hamiltonian, which is subsequently used to com-pute excitation spectra and transition probabilities. Themethod has been employed to study octupole related ef-fects like the surveys of octupole related properties in therare-earth and actinide regions [42–44] or the descriptionof octupole bands in neutron-rich odd-mass nuclei [45].The SCMF-PESs have been computed using the rela-tivistic DD-PC1 [46] or the non-relativistic Gogny-D1M[47, 48] EDFs.Due to the renewed experimental interest in the lightactinide region, we consider in this work the evolutionof the octupole shapes and the resulting spectroscopicproperties in a wide range of actinide nuclei including − Ra and − Th. To this end, the quadrupole-octupole SCMF-PESs, obtained within the (axially sym-metric) Hartree-Fock-Bogoliubov (HFB) approximationbased on the parametrization D1M [48] of the Gogny-EDF [47], are mapped onto the expectation value of theinteracting-boson Hamiltonian in the condensate stateconsisting of the monopole L = 0 + ( s ), quadrupole 2 + ( d ), and octupole 3 − ( f ) bosons [49, 50]. The map-ping procedure, employed to obtain the IBM-PESs fromthe SCMF-PESs, completely determines the consideredquadrupole-octupole sdf -IBM Hamiltonian and its di-agonalization provides wave functions which are subse-quently used to compute positive- and negative-parityspectra as well as transition strengths.The paper is organized as follows. The Gogny-D1Mquadrupole-octupole SCMF-PESs, i.e., the microscopicbuilding blocks of the calculations, are discussed inSec. II. The mapping procedure to obtain the IBM Hamil-tonian is illustrated in Sec. III. The results obtainedfor low-energy excitation spectra, electric quadrupole,octupole and dipole transition strengths as well as forthe transition quadrupole and octupole moments are dis-cussed in Sec. IV. Finally, Sec. V is devoted to the con-cluding remarks and work perspectives. II. SCMF GOGNY-D1M CALCULATIONS
To obtain the quadrupole-octupole SCMF-PESs, theHFB equation has been solved with constrains on theaxially symmetric quadrupole ˆ Q and octupole ˆ Q op-erators [29, 44]: ˆ Q = z − (cid:0) x + y (cid:1) ˆ Q = z − z (cid:0) x + y (cid:1) (1)The mean values (cid:104) Φ HF B | ˆ Q | Φ HF B (cid:105) = Q and (cid:104) Φ HF B | ˆ Q | Φ HF B (cid:105) = Q also define the quadrupole and octupole deformation parameters β and β : β λ = (cid:112) π (2 λ + 1)3 R λ A Q λ (2)where R = 1 . A / fm. In the following, the subscriptzero in β λ ’s and Q λ ’s ( λ = 2 ,
3) is omitted, unless oth-erwise specified. The center of mass is fixed at the ori-gin to avoid spurious effects associated with its motion[29, 30]. The HFB quasiparticle operators [39] have beenexpanded in a deformed (axially symmetric) harmonicoscillator (HO) basis containing 17 major shells to grantconvergence for the studied physical quantities.The constrained calculations provide a set of HFBstates | Φ HF B ( β , β ) (cid:105) labeled by their static deformationparameters β and β . The HFB energies E HF B ( β , β )associated with those HFB states define the contourplots referred to as SCMF-PESs in this work. Asthe HFB energies satisfy the property E HF B ( β , β ) = E HF B ( β , − β ) only positive β values are consideredwhen plotting the SCMF-PESs.The SCMF-PESs obtained for − Ra and − Th are depicted in Fig. 1. Along the β -directionthere is a shape/phase transition from spherical orweakly deformed ground states in the lightest isotopes( Ra and
Th) to well quadrupole deformed groundstates in heavier nuclei. On the other hand, the SCMF-PESs are rather soft along the β -direction. A globaloctupole deformed minimum with β ≈ . N ≈
132 (
Ra and
Th). This minimumbecomes deeper as one approaches the neutron number N = 136 ( Ra and
Th). In our calculations themost pronounced octupole deformation effects are foundaround this neutron number with β ≈ .
15 for
Raand
Th, in good agreement with the experiment [1].Beyond this neutron number, as one moves towards N = 150, the corresponding β values decrease andreflection symmetric HFB ground states are obtainedfor the heaviest isotopes in both chains.Previous SCMF calculations including the quadrupoleand octupole constrains simultaneously can be found inthe literature for nuclei in this region of the nuclearchart. For example, calculations have been carried outin Ref. [43] for − Ra and − Th using the rela-tivistic DD-PC1 EDF [46]. The overall systematic of thequadrupole and octupole deformations associated withthe DD-PC1 SCMF-PESs is similar to the one obtainedin the present study with the Gogny-D1M EDF. How-ever, in the case of the DD-PC1 EDF, the N = 132isotopes ( Ra and
Th) exhibit a reflection symmet-ric SCMF ground state while those nuclei are predictedto be octupole deformed in the Gogny-D1M calculations.Pronounced octupole deformation effects are predictedby both EDFs for
Ra and
Th though deeper globalminima are found in the relativistic approach. Thequadrupole-octupole coupling has been studied for Rn,Ra and Th nuclei in Ref. [30]. A comparison of severalrelativistic EDFs in a survey of octupole correlations canbe found in Ref. [32]. A thorough account over a large set
FIG. 1. (Color online) SCMF-PESs computed with the Gogny-D1M EDF for the nuclei − Ra and − Th. The colorcode indicates the total HFB energies plotted with respect to the global minimum. For more details, see the main text. of even-even nuclei of observables associated to octupolecorrelations was presented in Refs. [3, 4] using the Gogny-HFB approach, parity projection and octupole configura-tion mixing. Octupole deformations have also been stud-ied for Ra isotopes [26] using the HFB approach based onthe Barcelona-Catania-Paris (BCP) [26] and Gogny-D1S[51] EDFs.
III. MAPPING ONTO THE BOSON SYSTEM
Having the (fermionic) Gogny-D1M SCMF-PESs athand, we map them onto the corresponding (bosonic)IBM-PESs using the methods developed in Refs. [40–42]. In order to account for negative-parity states theIBM space includes, in addition to the positive-paritymonopole s ( L = 0 + ) and quadrupole d ( L = 2 + ) bosons,the negative-parity f ( L = 3 − ) boson. Within the IBM FIG. 2. (Color online) IBM-PESs computed with the mapped sdf -IBM Hamiltonian Eq. (5) for the nuclei − Ra and − Th. For more details, see the main text. framework, bosons represent collective pairs of valencenucleons [52]. In particular, the f boson can be viewed asformed by coupling the normal and unique parity orbitals π ( i / ⊗ f / ) (3 − ) and ν ( j / ⊗ g / ) (3 − ) in the light ac-tinides with Z ≈
88 and N ≈ sdf -IBMphenomenology, the number of f bosons involved in theIBM space is limited to one or, at most, three. In thepresent work, we do not assume any such truncation forthe f -boson number. Thus, the numbers n s , n d and n f of s , d , and f bosons are arbitrary and satisfy the condi-tion that the total boson number N B = n s + n d + n f isconserved for a given nucleus.The mapping of the Gogny-D1M ( β , β )-PESs ontothe IBM ones is achieved by introducing the intrinsicstate for the boson system [53]: | φ (cid:105) = 1 √ N B ! ( b † c ) N B | (cid:105) , (3) . . . . . . (cid:15) d ( M e V ) (a) RaTh . . . . . . − (cid:15) f ( M e V ) (b) − κ s d ( k e V ) (c) − . − . − . − . − . − . − . . χ dd (d) . . . . . . . χ ff (e) − − − − κ s d ( k e V ) (f) κ s d f ( k e V ) (g) − . − . − . − . − . − . . χ d f (h) C (i) C (j) N ν . . . . . . . e ( ) B ( √ W . u . ) (k) N ν e ( ) B ( √ W . u . ) (l) FIG. 3. (Color online) The strength parameters of the sdf -IBM Hamiltonian Eqs. (6) to (9) (panels (a) to (h)), the co-efficients C λ ’s for the deformation parameters (panels (i) and(j)), and the boson effective charges e ( λ ) B Eqs. (14) and (15)(panels (k) and (l)) are plotted as functions of the neutronboson number N ν for − Ra and − Th. where N B and | (cid:105) denote the number of bosons and theboson vacuum, respectively. The condensate boson op-erator b c is given by b c = (1 + α + α ) − / ( s + α d + α f ) , (4)with amplitudes α and α . The doubly-magic nucleus Pb is taken as boson vacuum. Therefore, N B runsfrom 5 to 15 (6 to 16) for − Ra ( − Th). The amplitudes α and α can be related to the deformationparameters β and β as α = C β and α = C β [43, 44, 53] where, C and C represent dimensionlessparameters.The IBM-PES is obtained analytically, by taking theexpectation value of the sdf -IBM Hamiltonian in the bo-son condensate state Eq. (3). The sdf -IBM Hamiltonianis the sum of the Hamiltonians for the sd and f bosonspaces plus a coupling ˆ H sdf between them:ˆ H = ˆ H sd + ˆ H f + ˆ H sdf . (5)The sd -boson Hamiltonian readsˆ H sd = (cid:15) d ˆ n d + κ sd ˆ Q sd · ˆ Q sd + κ (cid:48) sd ˆ L d · ˆ L d , (6)where the first term represents the number operator forthe d bosons with (cid:15) d being the single d boson energyrelative to the s boson one. The second term representsthe quadrupole-quadrupole interaction with strength κ sd and the quadrupole operator ˆ Q sd = s † ˜ d + d † ˜ s + χ dd [ d † × ˜ d ] (2) . The third term in Eq. (6) is the rotational termwith the angular momentum operator ˆ L d = √ d † × ˜ d ] (1) .The Hamiltonian for the f -boson space readsˆ H f = (cid:15) f ˆ n f + κ f ˆ Q f · ˆ Q f + κ (cid:48) f ˆ L f · ˆ L f , (7)with the f -boson quadrupole operator ˆ Q f = [ f † × ˜ f ] (2) and the angular momentum operator ˆ L f = √ d † × ˜ d ] (1) .The sdf Hamiltonian employed here takes the followingform: ˆ H sdf = κ (cid:48) sdf ˆ Q sd · ˆ Q f + κ sdf ˆ O · ˆ O, (8)The last term in Eq. (8) is the octupole-octupole inter-action with the strength parameter κ sdf . The octupoleoperator takes the formˆ O = s † ˜ f + f † ˜ s + χ df [ d † × ˜ f + f † × ˜ d ] (3) , (9)with χ df being a parameter.For simplicity, we assume κ (cid:48) sdf = 2 κ sd χ ff , κ f = κ sd χ ff , and κ (cid:48) f = κ (cid:48) sd . The independent parameters ofthe Hamiltonian are, therefore, (cid:15) d , (cid:15) f , κ sd , κ (cid:48) sd , χ dd , χ ff , κ sdf , and χ df as well as the coefficients C and C forthe β and β deformations. These parameters are de-termined via the mapping procedure. The Hamiltonianin Eq. (5) is similar to the one employed in our previousstudy in the rare-earth region [44], except for the ˆ L f · ˆ L f term considered in this work. This rotational correctionterm is considered because a good amount of f -bosoncomponents is present in the calculated yrast states forboth parities and the inclusion of this term has a sizableeffect on the moments of inertia obtained for the positiveand negative-parity yrast bands. A more detailed ac-count of the other terms as well as the analytical form ofthe IBM-PES as a function of the β and β deformationscan be found in Ref. [44].The strength parameters of the sdf -IBM Hamiltonianin Eq. (5) are determined so that the IBM-PES re-produces the topology of the Gogny-D1M SCMF-PESaround the global minimum. Only the parameter for theˆ L d · ˆ L d term in Eq. (6) is determined independently insuch a way that the bosonic cranking moment of inertia(see Ref. [54] for details) at the global minimum is equalto the Thouless-Valatin [55] moment of inertia for the 2 +1 state.The mapped sdf -IBM-PESs are depicted in Fig. 2 forthe studied nuclei. As expected, the original Gogny-D1M( β , β )-PESs are nicely reproduced around the globalminimum. The IBM-PESs are, however, much flatterfar away from this minimum. This is a common featurefound in previous IBM studies and can be attributed tothe size of the IBM model space [40, 53]. The boson con-figuration space consists of only valence nucleons whileall the nucleons are involved in the Gogny-HFB calcu-lation. The resulting sdf -IBM Hamiltonian, with thestrength parameters determined via the mapping proce-dure, is then diagonalized to obtain excitation energiesand transition strengths for a given nucleus.The strength parameters obtained for the sdf -IBMHamiltonian are plotted in panels (a) to (j) of Fig. 3 asfunctions of the neutron boson number N ν (= N B − ( Z − / N ν . However, some of the parameters for theinteraction terms involving f bosons, e.g., (cid:15) f , and χ df ,display abrupt changes around N = 136. This resultsfrom the difference in the topology of the SCMF-PESscorresponding to neighboring isotopes in this transitionalregion (see, Fig. 1). IV. SPECTROSCOPIC PROPERTIESA. Systematic of excitation spectra
The low-energy excitation spectra corresponding toeven-spin positive and odd-spin negative-parity yraststates are plotted in Fig. 4 as functions of the massnumber A . Those states are assumed to be membersof the K π = 0 +1 and 0 − bands. The excitation ener-gies of the positive-parity states decrease with increasingneutron number. This reflects the onset of pronouncedquadrupole deformation effects with increasing neutronnumber (see, Figs. 1 and 2) and the corresponding transi-tion from vibrational to well-developed rotational bands.For both isotopic chains, the predicted positive-parityspectra agree reasonably well with the experimental onesalso included in the figure.The excitation energies of the negative-parity statesexhibit a parabolic behavior as functions of the neutronnumber. The lowest excitation energies correspond to N ≈
136 isotopes. Around this neutron number thepredicted negative-parity band lies quite close in energy to the positive-parity band. This situation correspondsto an alternating-parity rotational band (see, Sec. IV B)that is a neat fingerprint of permanent octupole deforma-tion [1]. For larger neutron numbers, the negative-parityband is higher in energy and completely decoupled fromthe positive-parity band, i.e., the octupole vibrationalregime, associated with the β -softness of the potential,sets in. The predicted excitation energies of the negative-parity states are also in good agreement with the exper-imental data though the former somewhat overestimatethe latter, in particular around N = 136. In the case ofthe lightest isotopes Ra and
Th, the predicted ex-citation energies for both parities are too high. This maybe a consequence of the reduced IBM space employedin the calculations, which is is not large enough to ac-count for the low-lying structures of those nuclei close tothe N = 126 neutron shell closure. Note also that for , Ra and , Th the 1 − energy level is predictedabove the 3 − level. In the case of Ra this contradictsthe experiment. This inversion could be, once more, theresult of the limited IBM space employed in the calcula-tions.The probability amplitudes of the f -boson componentsin the IBM wave functions corresponding to even-spinpositive-parity and odd-spin negative-parity yrast statesin − Ra and − Th, are plotted in Fig. 5 as func-tions of the spin I . The amplitudes are computed asexpectation values (cid:104) ˆ n f (cid:105) of the f -boson number opera-tor ˆ n f Eq. (7) in the IBM wave functions. For all thestudied isotopes, at low spins I π (cid:54) + , the fraction ofthe f -bosons in the positive-parity states is rather low.However, for spins I π (cid:62) + the contribution from the f -boson components increases in nuclei with neutron num-bers 130 (cid:54) N (cid:54) f -boson contributions becomesignificant for I π > − . For both parities and isotopicchains, the f bosons play a major role up to N ≈ (cid:104) ˆ n f (cid:105) tends tobe larger for lighter isotopes and becomes much smallerwithout significant changes for heavier isotopes. For thelighter isotopes the mixing of different configurations inthe sdf -IBM states is pronounced. B. Possible alternating-parity band structure
As a more quantitative measure of the extent to whichthe predicted positive- and negative-parity bands resem-ble alternating parity bands, we have considered thequantity S ( I ) = E ( I + 1) + E ( I − − E ( I ) , (10)where E ( I ) represents the excitation energy of the I =0 + , 1 − , 2 + , . . . yrast states. In the limit of an ideal al-ternating parity band, this quantity goes to zero. Thequantity S ( I ) is depicted in Fig. 6 as a function of thespin I . For most of the isotopes in both chains, the S ( I )
218 220 222 224 226 228 230 232 234 236 238
Mass number . . . . . . E xc i t a t i onene r g y ( M e V ) (a) Ra (theo.)
218 220 222 224 226 228 230 232 234 236 238
Mass number(b) Ra (expt.) + + + + +
220 222 224 226 228 230 232 234 236 238 240
Mass number . . . . . . E xc i t a t i onene r g y ( M e V ) (c) Th (theo.)
220 222 224 226 228 230 232 234 236 238 240
Mass number(d) Th (expt.)
218 220 222 224 226 228 230 232 234 236 238
Mass number . . . . . . E xc i t a t i onene r g y ( M e V ) (a) Ra (theo.)
218 220 222 224 226 228 230 232 234 236 238
Mass number(b) Ra (expt.) − − − − −
220 222 224 226 228 230 232 234 236 238 240
Mass number . . . . . . E xc i t a t i onene r g y ( M e V ) (c) Th (theo.)
220 222 224 226 228 230 232 234 236 238 240
Mass number(d) Th (expt.)
FIG. 4. (Color online) Low-energy even-spin positive and odd-spin negative-parity excitation spectra of yrast states for − Ra and − Th computed with the sdf -IBM Hamiltonian Eq. (5). Experimental data are taken from Ref. [56]. values exhibit an odd-even staggering pattern. This stag-gering pattern is less pronounced for N ≈
136 reflectingthat the negative-parity band becomes particularly lowin energy and forms an approximate alternating-paritystructure with the positive-parity ground-state band. For N (cid:62) β -softness of the potential. C. Transition strength properties
For the computation of the reduced transition proba-bilities, we have employed the quadrupole and octupoletransition operators:ˆ T E2 = e (2)B ˆ Q , ˆ T E3 = e (3)B ˆ Q (11) I ( ¯ h ) . . . . . . . h ˆ n f i (a) Ra ( π = +1 ) I ( ¯ h )(b) Th ( π = +1 ) I ( ¯ h ) . . . . . . . h ˆ n f i (c) Ra ( π = − ) I ( ¯ h )(d) Th ( π = − ) FIG. 5. (Color online) The f -boson contents in the wavefunctions of the even-spin positive-parity (a,b) and odd-spinnegative-parity yrast states (c,d) in − Ra and − Th, obtained as the expectation value (cid:104) ˆ n f (cid:105) in a given state, areplotted as functions of the spin I . I ( ¯ h ) − − S ( I ) ( M e V ) (a) Ra (theo.) I ( ¯ h ) (b) Ra (expt.) I ( ¯ h ) − − S ( I ) ( M e V ) (c) Th (theo.) I ( ¯ h ) (d) Th (expt.)
FIG. 6. (Color online) The relative energy splitting betweenpositive- and negative-parity yrast bands S ( I ) Eq.(10), ob-tained for − Ra and − Th, is plotted as a functionof the spin I . where e ( λ )B ’s are effective charges andˆ Q = s † ˜ d + d † s + χ (cid:48) dd [ d † × ˜ d ] (2) + χ (cid:48) ff [ f † × ˜ f ] (2) (12)ˆ Q = s † ˜ f + f † s + χ (cid:48) df [ d † × ˜ f + f † × ˜ d ] (3) . (13) Mass number B ( E ; + → + )( W . u . ) (a) Ra Mass number B ( E ; − → + )( W . u . ) (c) Ra Mass number − − − − − − B ( E ; − → + )( e · b ) (e) Ra Mass number(b) Th Mass number(d) Th Mass number(f) Th FIG. 7. (Color online) Reduced transition probabilities B (E2;2 +1 → +1 ) (a,b), B (E3; 3 − → +1 ) (c,d), and B (E1; 1 − → +1 )(e,f) for − Ra and − Th. Theoretical values are rep-resented by filled symbols connected by lines. Experimentaldata have been taken from Refs. [5, 6, 56]. They are rep-resented by open symbols with error bars. The B (E2) and B (E3) rates are in Weisskopf units while the B (E1) rates in e · b units are plotted using a logarithmic scale. The quadrupole and octupole transition operatorsEq. (11) have the same form as the ones in the Hamilto-nian Eqs. (6) to (8) but with new parameters χ (cid:48) dd , χ (cid:48) ff ,and χ (cid:48) df . The effective charges e ( λ )B ’s are determined sothat the intrinsic quadrupole (octupole) moment in theIBM, obtained as the expectation value of the operatorˆ T E λ in the coherent state at the minimum of the PES [49]is equal to the Gogny-HFB one. Introducing the bosonicdeformation parameters ¯ β λ = C λ β λ ), corresponding tothe minimum of the PES, we obtain the following equa-tions e (2)B N B (2 ¯ β − (cid:113) χ (cid:48) dd ¯ β − √ χ (cid:48) ff ¯ β )1 + ¯ β + ¯ β = Q min20 (14) I ( ¯ h ) | Q ( I → I − ) | ( e · f m ) (a) Ra I ( ¯ h ) (b) Th I ( ¯ h ) | Q ( I → I − ) | ( e · f m ) (c) Ra I ( ¯ h ) (d) Th I ( ¯ h ) | Q ( I → I − ) | ( e · f m ) (e) Ra I ( ¯ h ) (f) Th FIG. 8. (Color online) The transition quadrupole and octupole moments (in e · fm λ units) obtained for − Ra and − Thare plotted as functions of the spin I . For more details, see the main text. e (3)B N B ¯ β (1 − √ χ (cid:48) df ¯ β )1 + ¯ β + ¯ β = Q min30 . (15)For the parameter χ (cid:48) dd we have adopted the value χ (cid:48) dd = −√ / χ (cid:48) ff = 1 . χ (cid:48) df = − . χ ff and χ df values employed for the Hamil-tonian, respectively. The effective charges e (2)B and e (3)B have been further multiplied by the scale factors s and s , respectively. The scale factor s is assumed to takethe form s = 1 . / (9 . − . N B ), in order to reproducethe experimental systematic of the B (E2; 2 +1 → +1 ) val-ues. The boson-number dependence in the denominatorof s has been introduced so that the computed B (E2;2 +1 → +1 ) is not too large for N = 150 isotopes (close tothe neutron mid-shell N = 154). On the other hand, wehave considered s = 0 .
33 so that an overall agreementwith the systematic of the experimental B (E3; 3 − → +1 )values is obtained. The effective charges e (2)B and e (3)B (in √ W . u . units)Eqs. (14) and (15), are plotted in panels (k) and (l) ofFig. 3 as functions of the neutron boson number N ν . Theeffective charge e (2)B increases smoothly with the neutronnumber while e (3)B exhibits a parabolic behavior with amaximum at N ν ≈ N ≈
136 at which the most pronounced octupoledeformations are found.The electric dipole (E1) mode is yet another charac-teristic property of pear-shaped nuclei. In the sdf -IBMframework, the E1 operator readsˆ T E1 = e (1)B ( d † × ˜ f + f † × ˜ d ) (1) , (16)with the E1 effective charge e (1)B . We have taken e (1)B =0 . e · b / in order to reproduce the experimental B (E1; 1 − → +1 ) value for Ra.The predicted B (E2; 2 +1 → +1 ), B (E3; 3 − → +1 ), and B (E1; 1 − → +1 ) transition rates are compared in Fig. 70 E xc i t a t i on ene r g y ( M e V ) - + - + + + + - - + - - + + + + + + + Ra + + Theo.Expt. - - - - E xc i t a t i on ene r g y ( M e V ) - + - + + + + - - + - - + + + + + + + Ra + + Theo.Expt. - - - - + - + FIG. 9. (Color online) The energy spectra obtained for
Ra(top panel) and
Ra (bottom panel) are compared with theexperimental ones [5]. with the available experimental data. The increase in the B (E2) values (panels (a) and (b)) correlates well withthe increase in quadrupole collectivity along the studiedisotopic chains. The B (E3) strengths (panels (c) and (d))display a parabolic behavior, similar to the one obtainedfor the excitation energies of negative-parity states, witha maximum around the neutron number N = 136.The B (E1; 1 − → +1 ) strengths (panels (e) and (f)) in-crease smoothly. The predicted B (E1) values reproducethe reasonably well the experimental ones for − Raand − Th. However, the calculations are not able toaccount for the experimental B (E1) values in Ra and , Th. Here, one should keep in mind that E1 tran-sitions are less collective in nature and very sensitive tothe occupancy of high- j orbitals around the Fermi sur-face [20, 22]. Due to this sensitivity to single particleproperties, specific details of E1 transitions may be, atleast for some nuclear systems, out of reach for the IBMdescription (based on collective nucleon pairs) employedin this study. Phenomenological IBM studies (see, forexample, Refs. [57–60]) have often considered the dipole L = 1 − ( p ) boson to effectively describe E1 transitions. However, such a boson has not been included in this worksince its microscopic origin is less clear than for the s , d ,and f bosons. D. Transition quadrupole and octupole moments
The quadrupole Q ( I → I −
2) as well as the oc-tupole Q ( I → I −
3) and Q ( I → I −
1) moments, ob-tained from the reduced matrix elements (cid:104) I − (cid:107) ˆ T E2 (cid:107) I (cid:105) , (cid:104) I − (cid:107) ˆ T E3 (cid:107) I (cid:105) , and (cid:104) I − (cid:107) ˆ T E3 (cid:107) I (cid:105) , are often consideredas signatures of quadrupole and octupole collectivity.Those transition multipole ( λ = 2 ,
3) moments can beexpressed as: √ I + 1 (cid:114) λ + 116 π ( Iλ | I (cid:48) Q λ ( I → I (cid:48) ) = (cid:104) I (cid:48) (cid:107) ˆ T E λ (cid:107) I (cid:105) , (17)where ( Iλ | I (cid:48)
0) denotes a Clebsch-Gordan coefficient.These quantities have been computed for the in-band E2transitions within K π = 0 + and K π = 0 − bands with | I − I (cid:48) | = ∆ I = 2, and for the ∆ I = 3 and ∆ I = 1 E3transitions between the K π = 0 + and K π = 0 − states.They have been computed up to the spin I π = 8 + . TABLE I. Theoretical and experimental B (E2), B (E3), and B (E1) transition rates (in Weisskopf units) for Ra. Ex-perimental values are taken from Ref. [5]. For comparison,results based on the relativistic DD-PC1 EDF [43] have alsobeen included in the table. All transitions, exception made of B (E2; 2 +2 → + ), are between yrast states.Experiment Theory Ref. [43] B (E2; 2 + → + ) 98 ± B (E2; 3 − → − ) 93 ± B (E2; 4 + → + ) 137 ± B (E2; 5 − → − ) 190 ±
60 103 97 B (E2; 6 + → + ) 156 ±
12 154 159 B (E2; 8 + → + ) 180 ±
60 138 153 B (E2; 2 +2 → + ) 1.3 ± B (E3; 3 − → + ) 42 ± B (E3; 1 − → + ) 210 ±
40 86 85 B (E3; 3 − → + ) <
600 57 46 B (E3; 5 − → + ) 61 ±
17 85 61 B (E1; 1 − → + ) < × − × − × − B (E1; 1 − → + ) < . × − × − . × − B (E1; 3 − → + ) 3 . +1 . − . × − × − × − B (E1; 5 − → + ) 4 +3 − × − × − × − B (E1; 7 − → + ) < × − × − . × − The transition quadrupole and octupole moments, ob-tained for − Ra and − Th, are shown in Fig. 8as functions of the spin I . The quadrupole moments(panels (a) and (b)) remain rather constant with spin al-though a certain staggering pattern is observed. In thecase of the octupole moments, depicted in panels (c) to1(f) of the figure, the lightest isotopes display rather ir-regular patterns with spin. However, the amplitudes ofthe oscillations become smaller for 130 (cid:54) N (cid:54) Q moments, for both the ∆ I = 1and ∆ I = 3 transitions, are around 2000 e · fm . E. Low-energy excitation spectra, reducedtransition probabilities and reduced matrix elementsfor selected Ra isotopes
In what follows, the low-energy excitation spectra pre-dicted for , Ra are discussed in detail to further ex-amine the predicted power of the employed IBM frame-work based on the Gogny-D1M EDF. The energy spec-trum obtained for
Ra is compared with the experi-mental one [5] in the top panel of Fig. 9. The ground-state K π = 0 + band is reproduced reasonably well bythe calculations up to I π = 6 + . However, for I π (cid:62) + the predicted band looks stretched as compared with theexperiment. As can be seen from Fig. 5, in the case of Ra, the f -boson content of states with I π (cid:54) + is (cid:104) ˆ n f (cid:105) ≈ . I π (cid:62) + the f -boson content turnsout to be (cid:104) ˆ n f (cid:105) ≈ .
5. For the K π = 0 − band, the 1 − (bandhead) state is higher in energy than the experimen-tal one, although features such as the moment of inertiaand energy spacing agree well with the experiment. Upto I π = 7 − the f -boson content of the band is (cid:104) ˆ n f (cid:105) ≈ . I π = 9 − more f -bosons start to play a role,i.e., (cid:104) ˆ n f (cid:105) ≈ .
0. Alternating parity doublets are visible,in both the theoretical and experimental spectra, from I π = 5 − . The predicted non-yrast 0 +2 and 2 +2 states(above 1 MeV) have also been included in the figure.These states have a double octupole phonon nature with (cid:104) ˆ n f (cid:105) ≈
2. In the bottom panel of Fig. 9, we have alsoplotted the energy spectrum obtained for
Ra. Thisspectrum compares slightly better with the experimentthan in the case of
Ra. Here, the change in the struc-ture of states in the K π = 0 + ( K π = 0 − ) band is lesspronounced with (cid:104) ˆ n f (cid:105) ≈ . − . (cid:104) ˆ n f (cid:105) ≈ . − . I π = 16 + ( I π = 11 − ). Similar results are found for , Th.The B (E2) and B (E3) transition rates obtained for Ra are shown in Table I. We observe a very reason-able agreement with the corresponding experimental val-ues. The only exceptions are the B (E2; 5 − → − ) and B (E3; 1 − → + ) transitions which are underestimatedby a factor of two to three. In addition, we have alsoincluded in the table results from previous IBM calcula-tions based on the relativistic DD-PC1 EDF [43]. As canbe seen, both (mapped) IBM calculations provide rathersimilar predictions for the B (E2) and B (E3) rates. The B (E1; 1 − → + ) values obtained in the present studycompare slightly better with the experiment. However,other E1 transition strengths are larger than the ones ob-tained in Ref. [43] typically by one order of magnitudeand overestimate the experiment [5] by a factor from 10 to 10 .Finally, let us have a look on the reduced ma-trix elements | (cid:104) I − (cid:107) ˆ T E2 (cid:107) I (cid:105) | , | (cid:104) I − (cid:107) ˆ T E3 (cid:107) I (cid:105) | , and | (cid:104) I − (cid:107) ˆ T E3 (cid:107) I (cid:105) | in the case of − Ra for which exper-imental data are available [5, 6, 61]. They are depictedin Fig. 10 as functions of I . The predicted E2 matrix el-ements (panels (a ) to (a )) increase with spin and agreereasonably well with the experimental ones. For someof the studied nuclei, the E2 matrix elements are almostzero at high spins (for example, at I = 12 + for Ra andat I = 15 − for Ra). This is probably due to band mix-ing effects occurring in the high-spin regime, as can beexpected from the structural changes in the correspond-ing wave functions (see, Fig. 5). The | (cid:104) I − (cid:107) ˆ T E3 (cid:107) I (cid:105) | (panels (b ) to (b )) and | (cid:104) I − (cid:107) ˆ T E3 (cid:107) I (cid:105) | (panels (c )to (c )) matrix elements also increase as functions of I .However, they exhibit a pronounced staggering even atlow spin that contradicts the pattern observed in theavailable experimental data. A similar staggering hasalso been obtained in previous IBM studies [43, 58]. Ithas been concluded, within the framework of the phe-nomenological spdf -IBM model [58], that at least 3 pf bosons ( n p + n f = 3) are required to account for the ex-perimental systematic of the reduced E1 matrix elementsthat linearly increase with spin. It would be interestingto examine whether the inclusion of the p -boson degree offreedom can also improve the systematic of the E3 transi-tions in the (mapped) IBM framework. Another possibleremedy for the staggering problem observed in the E3 andE1 transition matrix elements within the sdf -IBM frame-work could be to consider higher-order terms in the cor-responding transition operators [62] (see, Eqs. (13) and(16)). V. SUMMARY
In this paper, we have considered the quadrupole-octupole coupling and collective excitations in the even-even actinides − Ra and − Th due to the re-newed experimental interest in the region. The con-strained Gogny-D1M HFB approach has been employedto obtain (axially symmetric) quadrupole-octupoleSCMF-PESs. The SCMF-PESs have been mapped ontothe corresponding IBM-PESs using the expectation valueof the sdf -IBM Hamiltonian in the boson condensatestate. The strength parameters of the bosonic Hamilto-nian have been determined via this mapping procedure.The wave functions resulting from the diagonalization ofthe (mapped) sdf -IBM Hamiltonian have been used tocompute octupole-related quantities such as, for exam-ple, both positive- and negative-parity excitation spectraand transition strengths.The SCMF-PESs are rather soft along the β -direction.A global mean-field reflection-asymmetric minimumemerges at N = 132 (i.e., for Ra and
Th). Forboth isotopic chains, the most pronounced octupole de-formation effects are found at N = 136 (i.e., for Ra2 I ( ¯ h ) | h I || ˆ T E || I − i | ( e · f m ) (a ) Ra I ( ¯ h )(a ) Ra I ( ¯ h )(a ) Ra I ( ¯ h )(a ) Ra I ( ¯ h ) | h I || ˆ T E || I − i | ( e · f m ) (b ) I ( ¯ h )(b ) I ( ¯ h )(b ) I ( ¯ h )(b ) I ( ¯ h ) | h I || ˆ T E || I − i | ( e · f m ) (c ) I ( ¯ h )(c ) I ( ¯ h )(c ) I ( ¯ h )(c ) FIG. 10. (Color online) Reduced matrix elements | (cid:104) I − (cid:107) ˆ T E2 (cid:107) I (cid:105) | , | (cid:104) I − (cid:107) ˆ T E3 (cid:107) I (cid:105) | , and | (cid:104) I − (cid:107) ˆ T E3 (cid:107) I (cid:105) | for , , , Ra.Experimental data are taken from Refs. [6] ( , Ra), [5] (
Ra), and [61] (
Ra). Theoretical values are represented byfilled symbols connected by lines. Experimental data are shown as open symbols with error bars. Experimental values withouterror bars represent upper limits [5, 61] . and
Th). This agrees well with the experimental find-ings of stable pear-like shapes for this particular neutronnumber. The octupole deformed minimum becomes lessprominent with increasing neutron number and disap-pears from N = 142 (i.e., for Ra and
Th) onward.These features are also found in the mapped sdf -IBM-PESs which nicely reproduce the basic topology of thefermionic PESs around the global minima.The spectroscopic properties, resulting from the diag-onalization of the sdf -IBM Hamiltonian, have been stud-ied in detail. Within this context a parabolic behavior,centered around the nuclei
Ra and
Th, has beenfound for the low-lying negative-parity spectra and the B (E3; 3 +1 → +1 ) reduced transition probabilities. Forisotopes in the neighborhood of N = 136, an approxi-mate alternating-parity band structure has been found.Octupole-related properties have been analyzed in detail for , , , Ra. The calculations reproduce reason-ably well the trends observed in the data available fromCoulomb excitation experiments. However, the fact thatthe calculations cannot account for the correct system-atic of the B (E1; 1 +1 → +1 ) rates and/or the E3 tran-sition matrix elements suggests that improvements, suchas the inclusion of dipole p bosons, are still required inthe employed mapping procedure.From the comparison of the results obtained in thiswork with the available experimental data as well aswith previous (mapped) IBM calculations based on therelativistic mean-field approximation [42, 43], we con-clude that the trends predicted for the studied nucleiare independent of the underlying microscopic input, i.e.,they are robust. Given the predictive power and compu-tational advantages of the mapping procedure togetherwith the IBM, studies of octupolarity in odd-mass ac-3tinides and heavier nuclear systems appear as our nextplausible steps. ACKNOWLEDGMENTS
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