Microscopic description of quadrupole-octupole coupling in actinides with the Gogny-D1M energy density functional
MMicroscopic description of quadrupole-octupole coupling in actinides with theGogny-D1M energy density functional
R. Rodr´ıguez-Guzm´an ∗ and Y. M. Humadi Department of Physics, Kuwait University, Kuwait
L.M. Robledo
Center for Computational Simulation, Universidad Polit´ecnica de Madrid,Campus Montegancedo, 28660 Boadilla del Monte, Madrid, Spain andDepartamento de F´ısica Te´orica and CIAFF, Universidad Aut´onoma de Madrid, 28049-Madrid, Spain † (Dated: August 24, 2020)The interplay between quadrupole and octupole degrees of freedom is discussed in a series of U,Pu, Cm and Cf isotopes both at the mean-field level and beyond. In addition to the static Hartree-Fock-Bogoliubov approach, dynamical beyond-mean-field correlations are taken into account viaboth parity restoration and symmetry-conserving Generator Coordinate Method calculations basedon the parametrization D1M of the Gogny energy density functional. Physical properties such ascorrelation energies, negative-parity excitation energies as well as reduced transition probabilities B ( E
1) and B ( E
3) are discussed in detail and compared with the available experimental data. It isshown that, for the studied nuclei, the quadrupole-octupole coupling is weak and to a large extentthe properties of negative parity states can be reasonably well described in terms of the octupoledegree of freedom alone.
PACS numbers: 21.60.Jz, 27.70.+q, 27.80.+w
I. INTRODUCTION.
Fingerprints of octupole collectivity in even-even nu-clei are usually associated with the presence of 1 − statesin the low-lying spectra. As the ground state of those nu-clei is usually quadrupole deformed, there is a 3 − state,member of the corresponding negative-parity rotationalbands, which decay through fast E + ground state. On the other hand, the 1 − state de-cays via E E Rn and
Ra unambiguously estab-lished the octupole deformed character of the later nu-cleus. This represents the first unambiguous experimen-tal evidence of permanent octupole deformed even-evennucleus. In multi-step Coulomb excitation experimentsperformed at the ATLAS-CARIBU facility with γ -rayand charged-particle detectors [9] also large E Ba was found pointing to a permanent oc-tupole deformed ground-state. Evidence for permanentoctupole deformation in
Ba has subsequently been ob-tained [10]. Recent experiments [11] have also establishedthe octupole deformed character of
Ra, or measuredthe E Th [12].From a theoretical point of view, various techniquesand models have been employed to study the dynamics of ∗ Electronic address: [email protected] † Electronic address: [email protected] octupole collectivity [4, 13–18]. Some of the approachesuse potential energy surfaces (PESs) obtained within rel-ativistic and nonrelativistic mean-field approximations toobtain the parameters of the Interacting Boson Model[19–22]. Some others rely on microscopic frameworks,both at the mean-field level and beyond, based on thenonrelativistic Skyrme and Gogny as well as relativisticenergy density functionals (EDFs) [23–43].Octupole deformation properties of several even-evenactinides were discussed in Ref. [44] with the helpof octupole-constrained Hartree-Fock-Bogoliubov (HFB)calculations based on the parametrizations D1S [45], D1N[46] and D1M [47] of the Gogny [48] and the BCP [49–52] EDFs. A one-dimensional (1D) collective Hamilto-nian was also built to have access to properties such asthe excitation energies of 1 − states as well as B ( E
1) and B ( E
3) transition probabilities. A thorough account overa large set of even-even nuclei of observables associated tooctupole correlations was presented in Refs. [53, 54] us-ing the octupole-constrained Gogny-HFB approach, par-ity projection and octupole configuration mixing. Fromthe results of these studies it is clear that not only staticoctupole deformation plays a role but also dynamical oc-tupole correlations have a sizable impact on observables.The interplay between quadrupole transitional prop-erties and octupole deformation manifestations in a se-lected set of Sm and Gd nuclei was discussed in Ref. [55]using the D1S and D1M Gogny-EDFs. Both quadrupoleand octupole constrains were considered simultaneously.The mean-field potential energy surfaces (MFPESs) ob-tained for − Sm and − Gd exhibited a very softbehavior along the octupole direction indicating, that dy-namical beyond-mean-field effects should be taken intoaccount. Those beyond-mean-field effects were consid- a r X i v : . [ nu c l - t h ] A ug
16 8 0 8 16 24 32 40Q (b)02.557.51012.515 Q ( b / ) U
16 8 0 8 16 24 32 40Q (b) U
16 8 0 8 16 24 32 40Q (b) U Q ( b / ) U U U Q ( b / ) U U U Q ( b / ) U U FIG. 1: (Color online) MFPESs computed with the Gogny-D1M EDF for the isotopes − U. Taking the lowest mean-fieldenergy as a reference, solid and dashed contour lines extend from 0.25 MeV up to 1 MeV in steps of 0.25 MeV. Solid anddashed contours are then drawn in steps of 0.5 MeV up to 3 MeV and from there up dotted lines are drawn in steps of 1 MeV.The intrinsic HFB energies are symmetric under the exchange Q → − Q . For A = 230, the conversion factor from barn to β values is 0.0212 and the one from b / to β values is 0.0342. For additional details, see the main text. ered via both parity projection of the intrinsic states andsymmetry-conserving quadrupole-octupole configurationmixing calculations, in the spirit of the two-dimensional(2D) Generator Coordinate Method (GCM) [56]. In ad-dition to the systematic of the 1 − excitation energies, cor-relation energies, B ( E
1) and B ( E
3) transition probabili- ties, the results of Ref. [55] suggested a shape/phase tran-sition from weakly to well quadrupole deformed groundstates as well as a transition to an octupole vibrationalregime in the studied nuclei. The quadrupole-octupolecoupling has also been studied for Rn, Ra and Th nucleiwithin the 2D-GCM framework [57]. Let us also mentiona recent state-of-the-art quadrupole-octupole symmetry-projected configuration mixing study for
Ba [43].Given the experimental interest in studying octupoleproperties of nuclei heavier than Th, we considerin the present work the dynamical interplay betweenquadrupole and octupole degrees of freedom in a selectedset of even-even actinides, i.e., − U, − Pu, − Cm and − Cf. These nuclei have Z valuesaway from Z = 88 (Ra) which is considered to be a“magic number” for the existence of permanent octupoledeformation [4]. The study of the dynamical quadrupole-octupole coupling in the selected actinide nuclei allowsus to examine the role of the corresponding zero-pointquantum fluctuations on the systematic of the 1 − exci-tation energies, transition strengths and correlation en-ergies around the N = 134 (a neutron octupole magicnumber) isotones U, Pu,
Cm and
Cf.As in our previous study [55], we consider three levelsof approximation for each of the studied nuclei. The con-strained Gogny-HFB scheme is used to obtain MFPESsas functions of both the quadrupole and octupole mo-ments. As discussed later, those MFPESs can be rathersoft along the octupole direction. Some of the consid-ered nuclei also exhibit transitional features along thequadrupole direction. In this case the HFB approxi-mation can only be considered as a starting point andbeyond-mean-field correlations should be taken into ac-count. First, parity projection is carried out in orderto build the corresponding parity-projected potential en-ergy surfaces (PPPESs). Next, both symmetry restora-tion as well as fluctuations in the collective quadrupoleand octupole coordinates are taken into account withinthe 2D-GCM framework. Although reflection symmetryis also restored by our GCM ansatz (see, Sec. II C), theparity-projected results allow us to disentangle the rela-tive contribution to the total correlation that has to beassociated with the restoration of the reflection symme-try.All the results discussed in this paper have been ob-tained with the Gogny-D1M EDF [47]. Among themembers of the D1 family of parametrizations of theGogny-EDF, D1S [45] has already built a strong rep-utation among practitioners, given its ability to repro-duce a wealth of low-energy nuclear data all over thenuclear chart both at the mean-field level and beyond(see, for example, Ref. [58] and references therein). Nev-ertheless, the parametrization D1M, specially tailored tobetter describe nuclear masses, has already provided areasonable description of nuclear properties in differentregions of the nuclear chart (see, for example, Refs. [59–62] and references therein). In particular, previous stud-ies [44, 53, 55, 57] have shown that the parametrizationD1M essentially keeps the same predictive power as D1Swhen applied to the description of octupole properties.The paper is organized as follows. The different ap-proaches employed in this work are briefly outlined inSecs. II A, II B and II C. In each section the results ob-tained with the corresponding approaches are discussed.
124 128 132 136 140 144 148 D E C O RR , H F B ( M e V ) UPuCmCf
FIG. 2: (Color online) The mean-field octupole correlationenergies Eq.(3) are plotted as functions of the neutron num-ber. Results have been obtained with the Gogny-D1M EDF.For more details, see the main text.
Mean-field results are presented in Sec. II A. We thenturn our attention to beyond-mean-field properties, i.e.,parity restoration and configuration mixing in Secs. II Band II C. Special attention is paid in Sec. II C to 1 − en-ergy splittings, reduced transition probabilities, correla-tion energies and their comparison with the available ex-perimental data [63]. Finally, Sec. III is devoted to theconcluding remarks. II. RESULTS
The aim of this work is to study the quadrupole-octupole dynamics in a selected set of actinide nuclei.Three levels of approximation have been considered: theHFB approach [56] with constrains on the (axially sym-metric) quadrupole and octupole operators, parity pro-jection and the 2D-GCM. In what follows, we outlinethose approaches [55, 57], based on the Gogny-D1MEDF, and discuss the results obtained with each of them.
A. Mean-field
To obtain the MFPESs, the HFB equation with con-strains on the axially symmetric quadrupoleˆ Q = z − (cid:16) x + y (cid:17) (1)and octupole operatorˆ Q = z − (cid:16) x + y (cid:17) z (2)is solved. The mean value with the HFB intrinsic state | Φ (cid:105) of the two operators define the quadrupole and oc-tupole deformation parameters Q and Q . From them
16 8 0 8 16 24 32 40Q (b)02.557.51012.515 Q ( b / ) U p = +
16 8 0 8 16 24 32 40Q (b) U p = +
16 8 0 8 16 24 32 40Q (b) U p = + Q ( b / ) U p = + U p = + U p = + Q ( b / ) U p = + U p = + U p = + Q ( b / ) U p = + U p = + FIG. 3: (Color online) Positive π = +1 parity-projected potential energy surfaces (PPPESs) computed with the Gogny-D1MEDF for the isotopes − U. See, caption of Fig. 1 for the contour-line patterns. one can compute [35] the standard deformation param-eters β l = (cid:112) π (2 l + 1) / (3 R l A ) Q l with R = 1 . A / In order to alleviate the already substantial com- For A = 230 a value of Q = 1000 fm is equivalent to β =0 .
212 and a value of Q = 1000 fm is equivalent to β = 0 . putational effort, both axial and time-reversal symme-tries have been kept as self-consistent symmetries. TheHFB equation is solved using a performing, approximatesecond-order gradient method [64]. The center of massis fixed at the origin to avoid spurious effects associatedwith its motion [55, 57]. The HFB quasiparticle operators[56] have been expanded in a deformed (axially symmet-ric) harmonic oscillator (HO) basis containing 16 majorshells to grant convergence for the studied physical quan-tities.The ( Q , Q )-constrained Gogny-HFB calculationsprovide a set of states | Φ( Q ) (cid:105) labeled by their corre-sponding static deformations Q = ( Q , Q ). The HFBenergies E HF B ( Q ) associated with those states define thecontour plots referred to as MFPESs in this work. Asthe Gogny-EDF is invariant under parity transformation[65, 66] the associated HFB energies satisfy the prop-erty E HF B ( Q , Q ) = E HF B ( Q , − Q ). For this rea-son, only positive octupole moments are considered whenplotting PESs.The MFPESs obtained for the isotopes − U areshown in Fig. 1 as illustrative examples. In our calcu-lations, the Q -grid − ≤ Q ≤
40b (with a step δQ = 1b) and the Q -grid 0b / ≤ Q ≤ / (with a step δQ = 0 . / ) have been employed. Alongthe Q -direction there is a shape/phase transition froma spherical ground state in U to a well quadrupole de-formed ground state in
U. A similar structural evolu-tion along the Q -direction have been obtained for thePu, Cm, and Cf isotopic chains. Spherical or weaklydeformed ground states are obtained for isotopes with N ≈
126 while a well quadrupole deformed groundstate emerges with increasing neutron number. In fact,we have obtained (static) HFB quadrupole deformationswithin the range 0b ≤ Q ,GS ≤ Cf for which Q ,GS = − − U, − Pu, − Cmand − Cf with values of the octupole moment in therange 2b / ≤ Q ,GS ≤ / .The MFPESs depicted in Fig. 1, as well as the onesobtained for the Pu, Cm and Cf isotopic chains are rathersoft along the Q -direction. This is further illustratedin Fig. 5 where the HFB energies obtained for U, Uand
U have been plotted, as functions of Q , for fixedvalues of the quadrupole moment corresponding to theabsolute minima of the PESs.The mean-field octupole correlation energies defined asthe energy gained by allowing octupolarity in the groundstate ∆ E CORR,HF B = E HF B,Q =0 − E HF B,GS (3)are plotted in Fig. 2. The largest values (1 .
25, 1 .
04, 0 . . M eV ) correspond to N = 134 isotones. Note,that the relatively small energies E CORR,HF B result fromthe softness observed in the MFPESs of nuclei with oc-tupole deformed ground states [see, for example, panel(b) of Fig. 5].The softness of the MFPESs discussed in this sec-tion already point towards the key role of dynamicalbeyond-mean-field correlations, i.e., symmetry restora-tion and/or quadrupole-octupole configuration mixing inthe studied nuclei. Two spatial symmetries are brokenin this study. One is the rotational symmetry with thequadrupole moment as the relevant parameter and the other is the reflection symmetry with the octupole mo-ment as the relevant parameter. From the previous dis-cussion of mean-field results it is clear that the octupole isthe softest mode. Therefore, parity is the most importantsymmetry to be restored. It would be desirable to restorealso both the rotational and particle number symmetries.This kind of simultaneous symmetry restoration is fea-sible in lighter nuclear systems. However, when com-bined with the quadrupole-octupole configuration mix-ing of Sec. II C, it becomes a highly demanding compu-tational task [43] out of the scope of an exhaustive surveylike the one discussed in this paper.
B. Parity symmetry restoration
Parity symmetry is broken by intrinsic HFB stateswith a non-zero value of the octupole moment. To re-store the symmetry [23, 32, 55] we build parity-projectedstates | Φ π ( Q ) (cid:105) from the intrinsic HFB states | Φ( Q ) (cid:105) byacting on them with the parity projectorˆ P π = 12 (cid:16) π ˆΠ (cid:17) , (4)where π = ± π one cancompute the projected energy E π ( Q ) = (cid:104) Φ( Q ) | ˆ H [ ρ ( (cid:126)r )] | Φ( Q ) (cid:105)(cid:104) Φ( Q ) | Φ( Q ) (cid:105) + π (cid:104) Φ( Q ) | ˆΠ | Φ( Q ) (cid:105) + π (cid:104) Φ( Q ) | ˆ H [ θ ( (cid:126)r )] ˆΠ | Φ( Q ) (cid:105)(cid:104) Φ( Q ) | Φ( Q ) (cid:105) + π (cid:104) Φ( Q ) | ˆΠ | Φ( Q ) (cid:105) (5)The evaluation of the Hamiltonian overlaps (cid:104) Φ( Q ) | ˆ H [ ρ ( (cid:126)r )] | Φ( Q ) (cid:105) and (cid:104) Φ( Q ) | ˆ H [ θ ( (cid:126)r )] ˆΠ | Φ( Q ) (cid:105) inEq.(5) requires a prescription for the density-dependentpart of the Gogny-EDF. As in previous studies [55, 57],we use the mixed density prescription that amounts toconsider the densities ρ ( (cid:126)r ) = (cid:104) Φ( Q ) | ˆ ρ ( (cid:126)r ) | Φ( Q ) (cid:105)(cid:104) Φ( Q ) | Φ( Q ) (cid:105) , (6)and θ ( (cid:126)r ) = (cid:104) Φ( Q ) | ˆ ρ ( (cid:126)r ) ˆΠ | Φ( Q ) (cid:105)(cid:104) Φ( Q ) | ˆΠ | Φ( Q ) (cid:105) (7)Such a prescription guarantees various consistency re-quirements within the EDF framework and avoidspathologies associated with the restoration of spatialsymmetries [65–68]. The parity-projected proton andneutron numbers, usually differ from the nucleus’ proton Z and neutron N numbers. To correct the energy forthis deviation we have replaced ˆ H by ˆ H − λ Z (cid:16) ˆ Z − Z (cid:17) − λ N (cid:16) ˆ N − N (cid:17) , where λ Z and λ N are chemical potentialsfor protons and neutrons, respectively [55, 57, 69, 70].
16 8 0 8 16 24 32 40Q (b)02.557.51012.515 Q ( b / ) U p =
16 8 0 8 16 24 32 40Q (b) U p =
16 8 0 8 16 24 32 40Q (b) U p = Q ( b / ) U p = U p = U p = Q ( b / ) U p = U p = U p = Q ( b / ) U p = U p = FIG. 4: (Color online) Negative π = − − U. See, caption of Fig. 1 for the contour-line patterns.
The π = +1 and π = − − U are depicted in Figs. 3 and 4 as illus-trative examples. Along the Q = 0 axis, the projectiononto positive parity is unnecessary as the correspond-ing quadrupole deformed even-even intrinsic states arealready pure π = +1 states. On the other hand, inthe case of negative parity, the evaluation of the pro-jected energy along the Q = 0 axis requires to resolve a ”zero-over-zero” indeterminacy [32, 55]. However, the π = − Q = 0 (see, Fig. 5) and its limiting valuedoes not play a significant role in the discussion of thePPPESs. We have then omitted this quantity along the Q = 0 axis in Fig. 4.The absolute minima of the π = +1 and π = − Q ( b ) 1678 1676 1674 1672 1670 1668 1666 1664 E ( M e V ) p = +1 p = UQ =0 b(a) E ( M e V ) UQ =8 b(b) E ( M e V ) UQ =12 b(c) FIG. 5: (Color online) The π = +1 (red) and π = − Q for fixed values of the quadrupole moment Q in the nuclei U, U and
U. The correspondingHFB energies are also included in the plots. Results havebeen obtained with the Gogny-D1M EDF. to the HFB values discussed in Sec. II A. In the case ofthe π = +1 PPPESs, depicted in Fig. 3, a characteristicpocket develops with a minimum at Q = 1 . − . b / .In the case of nuclei with a reflection-symmetric HFBground state, such a minimum is the global one. Thisis illustrated in panels (a) and (c) of Fig. 5 where the π = +1 parity-projected energies obtained for U and
124 128 132 136 140 144 148 N D E C O RR , PP ( M e V ) U PuCm Cf
FIG. 6: (Color online) The correlation energies stemmingfrom parity restoration Eq.(8) are plotted as functions of theneutron number. Results have been obtained with the Gogny-D1M EDF. For more details, see the main text.
U are plotted, as functions of Q , for fixed values ofthe quadrupole moment corresponding to the absoluteminima of the PESs. On the other hand, for nuclei witha reflection-asymmetric mean-field ground state, there isa pronounced competition with a second minimum at Q = 3 . − . b / as illustrated in panel (b) of Fig. 5for U. In the case of
U, the global π = +1 mini-mum at Q = 4 . b / is only 500 KeV deeper than theone at Q = 1 . b / . Similar results have been obtainedfor Pu, Cm and Cf isotopes. For example, the global π = +1 minima correspond to Q = 3 . b / and 4 . b / in Pu and
Pu, respectively, while for other Pu iso-topes as well as for Cm and Cf nuclei they are located at Q = 1 . − . b / . As can be seen from Figs. 1, 3 and 5not only the MFPESs but also the π = +1 PPPESs arerather soft along the Q -direction.The π = − Q = 2 . − . b / . In thecase of nuclei with a reflection-symmetric HFB groundstate, such as U and
U, the absolute π = − π = +1ones [see, panels (a) and (c) of Fig. 5]. On the other hand,for some nuclei with a reflection-asymmetric HFB groundstate, such as U, the (almost degenerate) π = − π = +1 absolute minima have similar octupole deforma-tions [see, panel (b) of Fig. 5]. Similar features have beenfound for the other isotopic chains. Let us mention, thatthe complex topography along the Q -direction as wellas the transition to an octupole-deformed regime foundin our Gogny-D1M calculations has also been studied, asa function of the strength of the two-body interaction,in Ref. [71] using the parity-projected Lipkin-Meshkov-Glick (LMG) model.As a measure of the correlations induced by paritysymmetry restoration one can use the correlation en-ergy, defined in terms of the difference between the HFB E HF B,GS and parity projected E π =+1 ,GS ground state
16 8 0 8 16 24 32 40Q (b)02.557.51012.5 Q ( b / ) U p = +1
16 8 0 8 16 24 32 40Q (b) U p = +1
16 8 0 8 16 24 32 40Q (b) U p = +1 Q ( b / ) U p = +1 U p = +1 U p = +1 Q ( b / ) U p = +1 U p = +1 U p = +1 Q ( b / ) U p = +1 U p = +1 FIG. 7: Collective wave functions Eq.(13) squared for the ground states of the nuclei − U. The contour lines (a successionof solid, long dashed and short dashed lines) start at 90% of the maximum value up 10% of it. The two dotted-line contourscorrespond to the tail of the amplitude (15% and 1% of the maximum value). Results have been obtained with the Gogny-D1MEDF. For more details, see the main text. energies ∆ E CORR,P P = E HF B,GS − E π =+1 ,GS . (8)In Fig. 6, we show this quantity for the different iso-topes considered. The correlation energy shows a mini-mum around N = 132 −
134 corresponding to stronglyoctupole-deformed intrinsic states. As shown later on inSec. II C, the comparison between the correlation energies E CORR,P P and the ones obtained within the symmetry-conserving 2D-GCM framework (see, Fig. 9) reveals thekey role played by quantum fluctuations around thoseneutron numbers.
C. Generator Coordinate Method
We include quantum fluctuations in the quadrupoleand octupole degrees of freedom by considering a linearsuperposition of the HFB states | Φ( Q ) (cid:105)| Ψ πσ (cid:105) = (cid:90) d Q f πσ ( Q ) | Φ( Q ) (cid:105) (9)where, both positive and negative octupole moments Q are included in the integration domain. In this way theparity of the collective amplitude under the change of sign
16 8 0 8 16 24 32 40Q (b)02.557.51012.5 Q ( b / ) U p =
16 8 0 8 16 24 32 40Q (b) U p =
16 8 0 8 16 24 32 40Q (b) U p = Q ( b / ) U p = U p = U p = Q ( b / ) U p = U p = U p = Q ( b / ) U p = U p = FIG. 8: Collective wave functions Eq.(13) squared for the lowest negative-parity states of the nuclei − U. See, caption ofFig. 7 for contour-line patterns. Results have been obtained with the Gogny-D1M EDF. For more details, see the main text. of Q , namely f πσ ( Q , − Q ) = πf πσ ( Q , Q ), deter-mines the parity of | Ψ πσ (cid:105) . The property f πσ ( Q , − Q ) = πf πσ ( Q , Q ) is a direct consequence of the invarianceof the interaction under the parity symmetry operation.The index σ in Eq.(9) labels the different GCM solutions.The amplitudes f πσ ( Q ) are solutions of the Griffin-Hill-Wheeler (GHW) equation [56] (cid:90) d Q (cid:48) (cid:16) H ( Q , Q (cid:48) ) − E πσ N ( Q , Q (cid:48) ) (cid:17) f πσ ( Q (cid:48) ) = 0 . (10)with the Hamiltonian and norm kernels defined in thestandard way H ( Q , Q (cid:48) ) = (cid:104) Φ( Q ) | ˆ H [ ρ GCM ( (cid:126)r )] | Φ( Q (cid:48) ) (cid:105) , N ( Q , Q (cid:48) ) = (cid:104) Φ( Q ) | Φ( Q (cid:48) ) (cid:105) (11)In the evaluation of the Hamiltonian kernel H ( Q , Q (cid:48) )for the Gogny-EDF, we have employed the mixed density prescription ρ GCM ( (cid:126)r ) = (cid:104) Φ( Q ) | ˆ ρ ( (cid:126)r ) | Φ( Q (cid:48) ) (cid:105)(cid:104) Φ( Q ) | Φ( Q (cid:48) ) (cid:105) . (12)As in the parity projection case, first-order correctionsto take into account deviations in both the proton andneutron numbers [55, 57, 69, 70] are included.The HFB basis states | Φ( Q ) (cid:105) are not orthonormal.Therefore, the amplitudes f πσ ( Q ) cannot be interpreted0
124 128 132 136 140 144 148 N D E C O RR , D G C M ( M e V ) U PuCm Cf
FIG. 9: (Color online) The correlation energies obtainedwithin the 2D-GCM framework Eq.(20) are plotted as func-tions of the neutron number. Results have been obtained withthe Gogny-D1M EDF. For more details, see the main text. as probability amplitudes. Instead, one considers the so-called collective wave functions G πσ ( Q ) = (cid:90) d Q (cid:48) N ( Q , Q (cid:48) ) f πσ ( Q (cid:48) ) , (13)written in terms of the square root operator N ( Q , Q (cid:48) )of the norm kernel [55, 56, 65] defined by the property N ( Q ; Q (cid:48) ) = (cid:90) d Q (cid:48)(cid:48) N ( Q ; Q (cid:48)(cid:48) ) N ( Q (cid:48)(cid:48) ; Q (cid:48) ) (14)The overlap (cid:104) Ψ πσ | ˆ O | Ψ π (cid:48) σ (cid:48) (cid:105) of an operator ˆ O between twodifferent GCM states Eq.(9) is required in the compu-tation of physical quantities such as, for example, theelectromagnetic transition probabilities. It reads (cid:104) Ψ πσ | ˆ O | Ψ π (cid:48) σ (cid:48) (cid:105) = (cid:90) d Q d Q (cid:48) G π ∗ σ ( Q ) O ( Q , Q (cid:48) ) G π (cid:48) σ (cid:48) ( Q (cid:48) ) (15)where O ( Q , Q (cid:48) ) = (cid:90) d Q (cid:48)(cid:48) d Q (cid:48)(cid:48)(cid:48) N − ( Q ; Q (cid:48)(cid:48) ) (cid:104) Q (cid:48)(cid:48) | ˆ O | Q (cid:48)(cid:48)(cid:48) (cid:105) ×× N − ( Q (cid:48)(cid:48)(cid:48) ; Q (cid:48) ) (16)For the reduced transition probabilities B ( E , − → + ) and B ( E , − → + ) the rotational formula for K=0bands have been used B ( Eλ, λ − → + ) = e π (cid:12)(cid:12)(cid:12) (cid:104) Ψ π = − σ | ˆ O λ | Ψ π (cid:48) =+1 σ (cid:48) =1 (cid:105) (cid:12)(cid:12)(cid:12) . (17)For B ( E
1) and B ( E
3) transitions σ corresponds to thefirst excited GCM state with negative parity. The elec-tromagnetic transition operators ˆ O and ˆ O are thedipole moment operator and the proton component ofthe octupole operator, respectively [55].Some comments are in order here regarding the use ofEq.(17). Previous studies [72, 73] have revealed that the use of proper angular momentum projected (AMP) wavefunctions concurs in an enhancement of the E E E N ≈
126 nuclei via Eq.(17), should be viewedas lower bounds.The collective wave functions Eq.(13) squared corre-sponding to the ground and lowest negative parity 2D-GCM states in − U are plotted in Figs. 7 and 8, re-spectively. As can be seen from Fig. 7, the ground statecollective amplitudes | G π =+1 σ =1 ( Q , Q ) | reach globalmaxima for octupole moments different from zero onlyin − U. The same holds for − Pu and , Cmwhile for other U, Pu and Cf nuclei, the peaks are locatedaround Q = 0. As illustrated in Fig. 7, the spread-ing of the amplitudes | G π =+1 σ =1 ( Q , Q ) | along the Q -direction is large, indicating the octupole-soft characterof the π = +1 2D-GCM ground states. In the case of the π = − π = − Q ) πσ = (cid:104) Ψ πσ | ˆ Q | Ψ πσ (cid:105) . (18)In the case of a negative-parity operator like ˆ Q thequantity (cid:104) Ψ πσ | ˆ Q | Ψ πσ (cid:105) is zero by construction. Therefore,a meaningful averaged quantity has to be defined [55] byrestricting the integration domain D to positive values of Q and Q (cid:48) ( ¯ Q ) πσ = 4 (cid:90) D d Q d Q (cid:48) G π ∗ σ ( Q ) Q ( Q , Q (cid:48) ) G πσ ( Q (cid:48) ) (19)In the case of a strongly peaked collective inertia, theaverage octupole moment ¯ Q is a good estimator of thelocation of the peak.The ground-state dynamical quadrupole moments( ¯ Q ) π =+1 σ =1 increase as more neutrons are added along agiven isotopic chain and their values remain close to theones predicted at the HFB level. On the other hand, atvariance with the HFB results, once both π = +1 sym-metry restoration and ( Q , Q )-fluctuations are consid-ered at the 2D-GCM level, dynamical octupole deforma-tions 0 . b / ≤ ( ¯ Q ) π =+1 σ =1 ≤ . b / are found in theground states of all the studied nuclei with the largestvalues corresponding to isotopes with neutron numbers N = 132 − Q ) π = − σ corresponding to the lowest negative-parity states alsoincrease their values with increasing N . Moreover, thecorresponding average octupole moments lie within therange 1 . b / ≤ ( ¯ Q ) π = − σ ≤ . b / with their largestvalues being reached once more for N = 132 −
138 iso-topes.1
124 128 132 136 140 144 148 152N00.511.52 E − ( M e V ) U
124 128 132 136 140 144 148 152N10 B ( E ) ( W . u . )
124 128 132 136 140 144 148 152N20406080 B ( E ) ( W . u . ) E − ( M e V ) Pu B ( E ) ( W . u . ) B ( E ) ( W . u . ) E − ( M e V ) Cm B ( E ) ( W . u . ) B ( E ) ( W . u . ) E − ( M e V ) Cf B ( E ) ( W . u . ) B ( E ) ( W . u . ) FIG. 10: (Color online) The 2D-GCM E − energy splittings (left panels) and the reduced transition probabilities B ( E
1) (middlepanels) and B ( E
3) (right panels) are plotted (in black) as functions of the neutron number for the studied U, Pu, Cm and Cfisotopic chains. The available experimental data (in red) have been taken from Ref. [63]. The E − , B ( E
1) and B ( E
3) valuesobtained in the framework of the 1D-GCM [53], with the octupole moment as single generating coordinate, have also beenincluded (in blue) in each of the plots. Results have been obtained with the Gogny-D1M EDF. For more details, see the maintext. E CORR, D − GCM = E HF B,GS − E π =+1 , D − GCM (20)are depicted in Fig. 9. They exhibit a weaker de-pendence with neutron number than the ∆ E CORR,P P values stemming from parity restoration (see, Fig. 6).The inclusion of beyond-mean-field correlations, via the2D-GCM ansatz Eq.(9), substantially modifies the be-havior observed in Fig. 6 around the neutron numbers N = 132 −
134 providing a smoother trend. Further-more, the variation of the correlation energies (withinthe range 1 .
76 MeV ≤ ∆ E CORR, D − GCM ≤ .
46 MeV) isof the same order of magnitude as the rms for the bindingenergy in Gogny-like nuclear mass tables [47] and, there-fore, those correlation energies should be considered inimproved versions of the Gogny-EDF.The energy difference E − between the positive parityground state and the lowest 1 − excited state, obtainedin the 2D-GCM calculations, is shown in the left pan-els of Fig. 10 as a function of the neutron number. Theenergies are very small for − U in agreement withtheir large (dynamical) octupole deformation. Other Uisotopes, with less pronounced dynamical octupole de-formation effects, display larger E − values and the firstnegative parity excited state can be interpreted as an oc-tupole vibrational state. In the same panels, we havealso included the energy differences E − obtained withinthe framework of the 1D-GCM with the octupole mo-ment as single generating coordinate [53]. As can beseen, the trend with neutron number is similar in bothcalculations. However, for heavier isotopes the 2D-GCM E − energies tend to be smaller than the 1D-GCM ones.Regarding the comparison with the the available exper-imental data, we are able to reproduce the increase ofthe excitation energies with increasing neutron number.However, exception made of the N = 138 −
140 isotopes,the predicted E − energies are larger than the experi-mental ones, a feature found in many GCM calculations(see, for example, [53, 55]). Similar results are found forthe other isotopic chains.In the case of the B ( E
1) reduced transition probabil-ities, depicted in the middle panels of Fig. 10, no ex-perimental data are available. Exception made of thenucleus
Pu, the 1D-GCM and 2D-GCM calculationsdisplay a similar pattern with the largest B ( E
1) valuescorresponding to the neutron numbers N = 132 − B ( E
1) strength strongly depend onhow the dipole moment evolves with octupole deforma-tion in the region where the positive and negative paritywave functions overlap. In the
Pu case the dipole mo-ment changes sign in the region of interest and there isa strong cancellation depending upon subtle details ofthe collective wave functions. For other nuclei, however,the sign of the dipole moment does not change with oc-tupole deformation and the dependency with the detailsof the collective wave functions is much weaker. Althoughthe 1D and 2D GCM collective wave functions look very similar, the tiny differences can easily explain the differ-ences in the results of the two calculations. Note, thatboth approaches predict a pronounced minimum for
Ualso consequence of a dipole moment changing its signas the octupole moment increases. The B ( E
3) reducedtransition probabilities are plotted in the right panels ofFig. 10. They show marked maxima for N = 132 − E − energies and the B ( E
1) strengths. Though essentiallythe same trend is obtained, for heavier nuclei the 2D-GCM B ( E
3) values are larger than the 1D-GCM ones.As can be seen from the panels, the predicted B ( E − U and − Pu compare reason-ably well with the available experimental data.Finally, let us mention that the comparison betweenthe 2D-GCM and 1D-GCM results in Fig. 10 revealsthat, to a large extent, there is a decoupling betweenthe quadrupole and octupole degrees of freedom in thestudied nuclei and confirms that the 1D-GCM approach[53] represents a valuable computational tool to accountfor the systematic of the 1 − energy splittings and reducedtransition probabilities in this region of the nuclear chart.In order to explore the robustness of the results witha change of the parametrization of the interaction, wehave carried out in the uranium chain the same kindof 2D GCM calculations but with the D1S and D1M*parametrizations of the Gogny force. The later is a newlyproposed re-parametrization of D1M with the goal of im-proving the slope of the symmetry energy [74] while pre-serving as much as possible other properties of D1M. Theresults are shown in Fig 11 along with the experimentaldata.The trend with neutron number is similar in the threecalculations confirming the consistency of the results.There are some quantitative differences at N = 130where a transition from octupole soft to octupole de-formed ground state takes place. Those differences arelarger for D1S as expected, because D1M* was fitted tobe as close as possible to D1M. From the comparisonwe conclude that the trend of the results with neutronnumber is rather insensitive to the interaction used. III. CONCLUSIONS
In this paper we have studied the interplay between thequadrupole and octupole degrees of freedom in a selectedset of even-even actinides both at the mean-field leveland beyond. To this end, we have resorted to the staticGogny-HFB approach, parity projection as well as 2D-GCM calculations with the multipole moments Q and Q as generating coordinates. At the mean-field levelonly nuclei with neutron numbers 130 ≤ N ≤
138 exhibitoctupole deformed HFB ground states. However, for allthe studied nuclei, the MFPESs and PPPESs are rathersoft along the Q -direction. As a result, once correla-tions associated with parity restoration and quadrupole-octupole configuration mixing are included simultane-3
124 128 132 136 140 144 148 152N00.511.52 E − ( M e V ) D1MD1SD1M* U
124 128 132 136 140 144 148 152N10 B ( E ) ( W . u . )
124 128 132 136 140 144 148 152N20406080 B ( E ) ( W . u . ) FIG. 11: (Color online) Same as Fig 10 but for different parametrizations of the Gogny force (D1M full line, D1M* dotted lineand D1S dashed line). ously within the 2D-GCM approach, their ground statesturn out to be (dynamically) octupole deformed, albeitwith the largest octupole deformation effects still corre-sponding to N = 132 −
138 isotopes. Moreover, withinthe 2D-GCM approach, the correlation energies displaya weaker dependence on the neutron number. Given therange of variation of those 2D-GCM correlation energies,they should be included in the fitting protocol of im-proved versions of the Gogny-EDF. Using the correlated2D-GCM states, we have studied the systematic of the1 − energy splittings as well as B ( E
1) and B ( E
3) reducedtransition probabilities in the considered isotopic chains.The predicted values compare reasonably well with theavailable experimental data. They point towards a (dy-namically) enhanced octupolarity for N = 132 −
138 iso-topes while octupole-vibrational states have been found for other nuclei. The comparison with 1D-GCM results[53] reveals that, for the studied nuclei, the quadrupole-octupole coupling is weak and to a large extent the prop-erties of negative parity states (i.e., energy splittings andreduced transition probabilities) can be reasonably welldescribed in terms of the octupole degree of freedomalone.
Acknowledgments
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