Microscopic origin of reflection-asymmetric nuclear shapes
MMicroscopic origin of reflection-asymmetric nuclear shapes
Mengzhi Chen ( 陈 孟 之 ),
1, 2
Tong Li ( 李 通 ),
1, 2
Jacek Dobaczewski,
3, 4 and Witold Nazarewicz
5, 1 Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA Department of Physics, University of York, York Y010 5DD, UK Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, 02-093 Warsaw, Poland Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA (Dated: March 5, 2021)
Background:
The presence of nuclear ground states with stable reflection-asymmetric shapes is supported by richexperimental evidence. Theoretical surveys of odd-multipolarity deformations predict the existence of pear-shapedisotopes in several fairly localized regions of the nuclear landscape in the vicinity of near-lying single-particle shellswith ∆ (cid:96) = ∆ j = 3. Purpose:
We analyze the role of isoscalar, isovector, neutron-proton, neutron-neutron, and proton-proton mul-tipole interaction energies in inducing the onset of reflection-asymmetric ground-state deformations.
Methods:
The calculations are performed in the framework of axial reflection-asymmetric Hartree-Fock-Bogoliubov theory using two Skyrme energy density functionals and density-dependent pairing force.
Results:
We show that reflection-asymmetric ground-state shapes of atomic nuclei are driven by the odd-multipolarity neutron-proton (or isoscalar) part of the nuclear interaction energy. This result is consistent withthe particle-vibration picture, in which the main driver of octupole instability is the isoscalar octupole-octupoleinteraction giving rise to large E Conclusions:
The necessary condition for the appearance of localized regions of pear-shaped nuclei in the nuclearlandscape is the presence of parity doublets involving ∆ (cid:96) = ∆ j = 3 proton or neutron single-particle shells. Thiscondition alone is, however, not sufficient to determine whether pear shapes actually appear, and – if so – whatthe corresponding reflection-asymmetric deformation energies are. The predicted small reflection-asymmetricdeformation energies result from dramatic cancellations between even- and odd-multipolarity components of thenuclear binding energy. I. INTRODUCTION
While the vast majority of atomic nuclei have eitherspherical or ellipsoidal (prolate or oblate) ground-state(g.s.) shapes, some isotopes exhibit pear-like shape de-formations that intrinsically break reflection symmetry.Experimental evidence for such shapes comes from char-acteristic properties of nuclear spectra, nuclear moments,and electromagnetic matrix elements [1, 2]. Pear-shapedeven-even nuclei display low-energy negative-parity exci-tations that are usually attributed to octupole collectivemodes. For that reason, pear-shaped nuclei are often re-ferred to as “octupole-deformed.”There are two regions of g.s. reflection-asymmetricshapes that have been experimentally established overthe years: the neutron-deficient actinides around
Raand the neutron-rich lanthanides around
Ba. Nucleartheory systematically predicts these nuclei to be pear-shaped (see Ref. [3] for a recent survey of theoreticalresults). Other regions of pear-shaped nuclei predictedby theory, i.e., lanthanide nuclei around
Gd as well asactinide and superheavy nuclei with 184 < N <
206 aretoo neutron rich to be accessible by experiment [3–7]. Ingeneral, deformation energies associated with reflection-symmetry breaking shapes are much smaller than thoserelated to stable ellipsoidal shapes [8, 9]. Consequently,for octupole-deformed nuclei, beyond mean-field meth-ods are needed for a quantitative description, see, e.g.,Refs. [10–13]. According to the single-particle (s.p.) picture, the ap-pearance of pear-shaped deformations can be attributedto the mixing of opposite-parity s.p. shells [14, 15]. Inthe macroscopic-microscopic (MM) approach, the macro-scopic energy favors spherical shapes. Therefore, stablerefection-asymmetric shape deformations obtained in theMM method [9, 16] can be traced back to the shape po-larization originating from proton and neutron s.p. lev-els interacting via parity-breaking fields. Since shell cor-rections are computed separately for protons and neu-trons, the results are usually interpreted in terms ofdeformation-driving proton or neutron shell effects. Theproton-neutron interactions are indirectly considered inthe macroscopic energy with the assumption of identi-cal proton and neutron shape deformation parameters,which follow those of the macroscopic term.In general, in the description based on the mean-fieldapproach, nuclear shape deformations result from a cou-pling between collective surface vibrations of the nu-cleus and valence nucleons. Such a particle-vibrationcoupling [17] mechanism can be understood in terms ofthe nuclear Jahn-Teller effect [18, 19]. The tendencytowards deformation is particularly strong if the Fermilevel lies just between close-lying s.p. states. In sucha case, the system can become unstable with respectto the mode that couples these states. Simple esti-mates of the particle-vibration coupling (Jahn-Teller vi-bronic coupling) for the quadrupole mode (multipolar-ity λ = 2) [20, 21] demonstrate that its contribution to a r X i v : . [ nu c l - t h ] M a r the mass quadrupole moment at low energies doubles thequadrupole moment of valence nucleons. The Hartree-Fock (HF) analysis [22, 23] confirmed this estimate. Itshowed that the main contribution to the quadrupoledeformation energy comes from the attractive isoscalarquadrupole-quadrupole term, which can be well approx-imated by the neutron-proton quadrupole interaction.When it comes to reflection-asymmetric deformations,the leading particle-vibration coupling is the one due tothe octupole mode (multipolarity λ = 3). This cou-pling generates a vibronic Jahn-Teller interaction be-tween close-lying opposite-parity s.p. orbits that may re-sult in a static reflection-asymmetric shape. For g.s. con-figurations of atomic nuclei, such pairs of states can befound just above closed shells and involve a unique-parityintruder shell ( (cid:96), j ) and a normal-parity shell ( (cid:96) − , j − N oct = 34, 56, 88, and, 134 [1].Indeed self-consistent calculations systematically predictpear shapes for nuclei having proton and neutron num-bers close to N oct .To understand the origin of reflection-asymmetric g.s.deformations, in this study we extend the quadrupole-energy analysis of Refs. [22, 23] to odd-multipolarityshapes. To this end we decompose the total Hartree-Fock-Bogoliubov (HFB) energy into isoscalar, isovector,neutron-neutron ( nn ), proton-proton ( pp ), and neutron-proton ( np ) contributions of different multipolarities.This paper is organized as follows. In Sec. II we esti-mate the octupole polarizability and coupling strengthsof the octupole-octupole interaction. Section III de-scribes the multipole decomposition of one-body HFBdensities and the HFB energy. The results of our analy-sis calculations and an analysis of trends are presented inSec. IV. Finally, Sec. V contains the conclusions of thiswork. II. SIMPLE ESTIMATE OF LOW-ENERGYOCTUPOLE COUPLING
In this section, we follow Refs. [20, 21], which used aschematic particle-vibration coupling Hamiltonian con-sisting of a spherical harmonic-oscillator one-body termand a multipole-multipole residual interaction. Thismodel was used in the early paper [22] in the context ofquadrupole deformations. The model Hamiltonian withthe octupole-octupole interaction isˆ H = ˆ H + 12 κ ˆ Q ˆ Q + 12 κ ˆ Q ˆ Q , (1)where ˆ Q = ˆ Q n + ˆ Q p and ˆ Q = ˆ Q n − ˆ Q p are single-particle octupole isoscalar and isovector operators, re-spectively, and ˆ H is a spherical one-body harmonic-oscillator Hamiltonian. For the case of high-frequency oc-tupole oscillations (giant octupole resonances), the cou-pling constants of the isoscalar and isovector octupole-octupole interactions, κ and κ , respectively, can be written as: κ = − π M ω A (cid:104) r (cid:105) , κ = πV sym A (cid:104) r (cid:105) , (2)where ω is the oscillator frequency, V sym is the repulsivesymmetry potential ( ∼
130 MeV), and M is the nucleonmass. Since the isovector coupling constant κ is positive,the g.s. neutron and proton deformations are expectedto be similar, as assumed in the MM approaches.Within the Hamiltonian (1), the g.s. octupole polariz-ability of the nucleus is given by [20] χ ,τ = − κ τ κ τ + C (0)3 , (3)where τ = 0 or 1 and C (0)3 is the restoring force pa-rameter. There are two types of octupole modes involv-ing s.p. transitions with ∆ N = 1 or 3, where N is theprincipal oscillator quantum number. The correspondingrestoring-force parameters are: C (0)3 (∆ N = 1) = 16 π M ω A (cid:104) r (cid:105) , (4) C (0)3 (∆ N = 3) = 3 C (0)3 (∆ N = 1) . (5)By using the estimate in Ref. [20] V sym M ω ≈ . (cid:104) r (cid:105)(cid:104) r (cid:105) , (6)one obtains: χ , (∆ N = 1) = 3 , χ , (∆ N = 3) = 1 / . (7)The isovector octupole polarizabilities are obtained in asimilar way by assuming a uniform density distribution: χ , (∆ N = 1) = − . , χ , (∆ N = 3) = − . . (8)While the collective octupole modes couple the ∆ N =1 and 3 transitions, the low-frequency mode is primarilyassociated with the ∆ N = 1 excitations. At low ener-gies, associated with nuclear ground states, the strengthcoefficients in Eq. (2) should be renormalized by factors(1 + χ ,τ ) to account for the coupling to high-energyoctupole collective vibrations. We indicate them by˜ κ τ = (1 + χ ,τ ) κ τ . Following Ref. [22], we rearrange theoctupole-octupole Hamiltonian into nn , pp , and np partswith the coupling constants˜ κ nn = ˜ κ pp = ˜ κ + ˜ κ , ˜ κ np = ˜ κ − ˜ κ . (9)By assuming the average values of octupole polarizabili-ties χ , ≈ χ , ≈ − .
4, the ratio of the couplingconstants becomes:˜ κ nn ˜ κ np = ˜ κ pp ˜ κ np ≈ . . (10)We can thus conclude that the octupole-octupole np in-teraction may indeed be viewed as being responsible forthe development of the octupole deformation. III. MULTIPOLE EXPANSION OF DENSITIESAND HFB ENERGY
In self-consistent mean-field approaches [24–26] withenergy-density functionals (EDFs) based on two-bodyfunctional generators, the total energy of a nucleus isexpressed as: E = Tr( T ρ ) + Tr(Γ ρ ) + Tr(˜Γ˜ ρ ) . (11)Here T is the kinetic energy operator, Γ and ˜Γ are meanfields in particle-hole (p-h) and particle-particle (p-p)channels, respectively, and ρ and ˜ ρ are one-body p-h andp-p density matrices, respectively. (Instead of using thestandard pairing tensor [24], here we use the “tilde” rep-resentation of the p-p density matrix [27].) The meanfields Γ and ˜Γ are defined as T + Γ = δEδ (cid:48) ρ , (12)˜Γ = δEδ (cid:48) ˜ ρ , (13)where δ (cid:48) denotes the variation of the total energy thatneglects the dependence of the functional generators ondensity, that is, the mean fields (12) and (13) do notcontain so-called rearrangement terms [25]. A. Multipole decomposition
As observed in Ref. [22], the density matrices and meanfields can be split into different multipole components as ρ = ρ [0] + ρ [1] + ρ [2] + ρ [3] + . . . (14a)˜ ρ = ˜ ρ [0] + ˜ ρ [1] + ˜ ρ [2] + ˜ ρ [3] + . . . , (14b)Γ = Γ [0] + Γ [1] + Γ [2] + Γ [3] + . . . (14c)˜Γ = ˜Γ [0] + ˜Γ [1] + ˜Γ [2] + ˜Γ [3] + . . . , (14d)where ρ [ λ ] , ˜ ρ [ λ ] , Γ [ λ ] , and ˜Γ [ λ ] are rank- λ rotational com-ponents of ρ , ˜ ρ , Γ, and ˜Γ, respectively. Traces appearingin Eq. (11) are invariant with respect to unitary transfor-mations, and, in particular, with respect to spatial rota-tions. Therefore, the traces act like multipolarity filtersprojecting the total energy on a rotational invariant. Inthis way, when the multipole expansions (14) are insertedin the expression for the total energy (11), only diagonalterms remain: E = E [0] + E [1] + E [2] + E [3] + . . . , (15)where E [ λ ] = Tr(Γ [ λ ] ρ [ λ ] ) + Tr(˜Γ [ λ ] ˜ ρ [ λ ] ) . (16)In the above equation, we add the kinetic energy to themonopole energy E [0] since T is a scalar operator whichimplies E kin = Tr( T ρ ) ≡ Tr(
T ρ [0] ). Therefore we define E [0] = E kin + Tr(Γ [0] ρ [0] ) + Tr(˜Γ [0] ˜ ρ [0] ) . (17) When parity symmetry is conserved, only even- λ mul-tipolarities appear in Eqs. (14) and (15). In Refs. [22,23], this allowed for analyzing the monopole ( λ = 0),quadrupole ( λ = 2), and higher even- λ components.In the present work, we analyze broken-parity self-consistent states and focus on the reflection-asymmetric(odd- λ ) components of the expansion. As our multipoleexpansion is defined with respect to the center of massof the nucleus, the integral of the isoscalar dipole density ρ [1] , namely, the total isoscalar dipole moment, vanishesby construction. Nevertheless, the dipole density ρ [1] anddipole energy E [1] can still be nonzero.In the spherical s.p. basis, the expansions (14) can berealized by the angular-momentum coupling of basis wavefunctions. Since the HFB equation is usually solved in adeformed basis, an explicit basis transformation is thenneeded. Moreover, the direct angular-momentum cou-pling does not benefit from the fact that Skyrme EDFsonly depend on (quasi)local densities, which is the prop-erty that greatly simplifies the HFB problem. Inspiredby the latter observation, in this work, we determinethe multipole expansions of (quasi)local densities and(quasi)local mean fields directly in the coordinate space.With axial symmetry assumed, particle density ρ ( r )can be decomposed as [28] ρ ( r ) = (cid:88) J ρ [ λ ] ( r ) Y J,M =0 (Ω) , (18)where ρ [ λ ] ( r ) = (cid:90) d Ω ρ ( r ) Y ∗ J,M =0 (Ω) . (19)An identical decomposition can be carried out for allisoscalar ( t = 0) and isovector ( t = 1) (quasi)local p-h densities [29] (cid:37) t ≡ { ρ t , τ t , ∆ ρ t , J t , ∇ · J t } , plus localneutron ( q = n ) and proton ( q = p ) pairing densities ˜ ρ q .The p-h densities depend on neutron and proton densitiesin the usual way: (cid:37) = (cid:37) n + (cid:37) p , (cid:37) = (cid:37) n − (cid:37) p . (20)Our strategy is to use the energy-density expression forthe time-even total energy (11), E = (cid:90) d r (cid:26) (cid:126) m τ ( r ) + H ( r ) + ˜ H ( r ) (cid:27) , (21)where the standard Skyrme energy densities read [29, 30]: H ( r ) = (cid:88) t =0 , H t ( r ) , (22a)˜ H ( r ) = (cid:88) q = p,n ˜ H q ( r ) , (22b)and where H t ( r ) = C ρt ρ t ( r ) + C ∆ ρt ρ t ( r )∆ ρ t ( r )+ C τt ρ t ( r ) τ t ( r ) + C J t J t ( r ) (23a)+ C ∇ Jt ρ t ( r ) ∇ · J t ( r ) , ˜ H q ( r ) = V q (cid:20) − V (cid:18) ρ ( r ) ρ (cid:19) γ (cid:21) ˜ ρ q ( r ) . (23b)For simplicity, the Coulomb energy is not included inEq. (21); it will be discussed later.It is convenient to rewrite the energy densities (23) in terms of local p-h and p-p potentials as H t ( r ) = V t ( r ) ρ t ( r ) + (cid:88) ij V tij ( r ) J tij ( r ) , (24a)˜ H q ( r ) = ˜ V q ( r )˜ ρ q ( r ) , (24b)where V t ( r ) = C ρt ρ t ( r ) + C ∆ ρt ∆ ρ t ( r )+ C τt τ t ( r ) + C ∇ Jt ∇ · J t ( r ) , (25a) V tij ( r ) = C J t J tij ( r ) , (25b)˜ V q ( r ) = V q (cid:20) − V (cid:18) ρ ( r ) ρ (cid:19) γ (cid:21) ˜ ρ q ( r ) , (25c)with indices i, j denoting the components of the spin-current tensor density J tij ( r ) in three dimensions. Inanalogy to Eqs. (18) and (19), we then determine themultipole expansions of the local potentials (25). In thisway, the total energy (15) can be decomposed into mul-tipole components: E [ λ ] = (cid:90) d r (cid:88) t =0 , V t [ λ ] ( r ) ρ t [ λ ] ( r ) + (cid:88) ij V tij [ λ ] ( r ) J tij [ λ ] ( r ) + (cid:88) q = p,n ˜ V q [ λ ] ( r )˜ ρ q [ λ ] ( r ) . (26)Finally, the same strategy can be applied to theCoulomb energy, which contributes to the multipoleterms of Eq. (15) through the multipole expansions ofdirect and exchange potentials: E Coul[ λ ] = (cid:90) d r (cid:104) V dir[ λ ] ( r ) + V exc[ λ ] ( r ) (cid:105) ρ p [ λ ] ( r ) , (27)where V dir ( r ) = e (cid:90) d r (cid:48) ρ p ( r (cid:48) ) | r − r (cid:48) | , (28) V exc ( r ) = − e (cid:2) π ρ p ( r ) (cid:3) . (29) B. Isospin and neutron-proton energydecomposition
In the isospin scheme, the total energy can be writtenas E = E t =0 + E t =1 + E Coul + E pair , (30)where E t = E kin δ t + (cid:90) d r H t ( r ) , (31a) E pair = (cid:88) q = p,n (cid:90) d r ˜ H q ( r ) . (31b) Note that the kinetic energy E kin is included in theisoscalar energy E t =0 . The Coulomb energy E Coul isseparated out because the Coulomb interaction breaksthe isospin symmetry. The pairing functional is notisospin invariant either as the neutron and proton pairingstrengths differ.By decomposing the isoscalar and isovector p-h den-sities (cid:37) t into the neutron and proton components (20),the total energy can be expressed in the neutron-protonscheme [22]: E = E kin + E nn + E pp + E np . (32)In Eq. (32), the individual E qq (cid:48) components ( q, q (cid:48) = n or p ): E qq (cid:48) = (cid:90) d r (cid:104) H qq (cid:48) ( r ) + δ qq (cid:48) ˜ H q ( r ) (cid:105) , (33)are defined through the energy densities H qq (cid:48) and ˜ H q ,which are bilinear in the densities (cid:37) q or ˜ ρ q . Note that theCoulomb energy E Coul is included in the proton energy E pp . As discussed earlier, all the energy terms enteringthe isospin and neutron-proton decompositions can beexpanded into multipoles. IV. RESULTS
The systems we studied are even-even barium, radiumand uranium isotopes. They are predicted to have stablepear shapes at certain neutron numbers [3]. For com-parison, we also calculate ytterbium isotopes which havestable quadrupole but no reflection-asymmetric deforma-tions. We performed axial HFB calculations using thecode hfbtho (v3.00) [31] for two Skyrme EDFs givenby SLy4 [32] and UNEDF2 [33] parametrizations. Weused the mixed-pairing strengths of V n = − .
25 MeVand V p = − .
06 MeV (SLy4) and V n = − .
30 MeVand V p = − .
04 MeV (UNEDF2). For UNEDF2, wedid not apply the Lipkin-Nogami treatment of pairing;instead, we took the neutron pairing strength V n to re-produce the average experimental neutron pairing gap for Sn, ∆ n = 1.245 MeV. The proton pairing strength V p was adjusted proportionally based on the default valuesof V n and V p .In the first step, we performed parity-conservingcalculations by constraining the octupole deformationto zero and determined the corresponding equilibriumquadrupole deformation β (0)2 . At the fixed value of β (0)2 ,we varied β from 0.0 to 0.25. In the hfbtho code,multipole constraints are actually applied to quadrupole( Q ) and octupole ( Q ) moments related to β and β through β = Q / (cid:32)(cid:114) π π AR (cid:33) ,β = Q / (cid:32)(cid:114) π π AR (cid:33) , (34)where A is the mass number, R = 1 . × A / , and Q = (cid:10) z − x − y (cid:11) ,Q = (cid:10) z (cid:0) z − x − y (cid:1)(cid:11) . (35)Figure 1 shows reflection-asymmetric deformation en-ergies ∆ E ( β ) = E ( β ) − E ( β = 0) determined for Raand
Ba obtained in this way. We see that UNEDF2gives a higher octupole deformability than SLy4 in bothnuclei. This is consistent with the results of Ref. [3].
A. Multipole expansion of the deformation energy
The convergence of the multipole expansion (15) pro-vides a check on the accuracy of our results. In Fig. 2,we show the energy difference, E diff ( λ ) = λ (cid:88) λ (cid:48) =0 E [ λ (cid:48) ] − E (36)for Ra at two values of the octupole deformation, β = 0 .
05 and 0.15. We see that at β = 0 .
15, the multi-pole components decrease exponentially with λ , with the β Δ E ( M e V ) UNEDF2SLy4 Ba Ra FIG. 1. The deformation energies, ∆ E ( β ) = E ( β ) − E ( β =0), as functions of β for Ra (dashed lines) and
Ba (solidlines) calculated at β (0)2 with the SLy4 (circles) and UNEDF2(triangles) EDFs. λ E d i ff ( M e V ) Ra, SLy4 β =0.15β =0.05 −3 −2 −1 FIG. 2. Convergence of E diff ( λ ) (36) for Ra computed withSLy4 at β =0.05 (dashed line) and 0.15 (solid line). monopole component off by about 150 MeV and the sumup to λ = 9 exhausted up to about 20 keV. At a smalloctupole deformation of β = 0 .
05, high-order contri-butions decrease. As expected, the octupole componentbrings now less energy as compared to the quadrupoleone. The results displayed in Fig. 2 convince us that cut-ting the multipole expansion of energy at λ = 9 providessufficient accuracy.Figure 3 shows how the reflection-asymmetric deforma-tion energy builds up. It presents the four leading mul-tipole components ∆ E [ λ ] ( β ) = E [ λ ] ( β ) − E [ λ ] ( β = 0),for λ = 0 −
3, of the deformation energies shown in Fig. 1.We can see that the pattern of contributions of differentmultipolarities is fairly generic: it weakly depends on thechoice of the nucleus or EDF. Figure 3 clearly demon-strates that the main driver of reflection-asymmetricshapes is a strong attractive octupole energy ∆ E [3] . Theattractive dipole energy ∆ E [1] is much weaker. The SLy4 UNEDF2 Ba Ra Δ E [ λ ] ( M e V ) β [ ] [ ] [ ] [ ] (a) (b)(c) (d) FIG. 3. Multipole components, ∆ E [ λ ] ( β ) = E [ λ ] ( β ) − E [ λ ] ( β = 0), of the total deformation energy shown in Fig. 1,plotted for λ = 0 − β at β (0)2 . Upper (lower) panels show results for Ra(
Ba) obtained with the SLy4 (left) and UNEDF2 (right)EDFs. monopole and quadrupole energies are repulsive alongthe trajectory of β (with a fixed quadrupole deforma-tion β (0)2 ) and essentially cancel the octupole contribu-tion. Indeed, one can note that while individual multi-pole components can be of the order of tens of MeV, thetotal reflection-asymmetric deformation energy shown inFig. 1 is an order of magnitude smaller. Therefore, the fi-nal reflection-asymmetric correlation results from a largecancellation between individual multipole components,and even a relatively small variation of any given com-ponent can significantly shift the net result. In addition,as discussed in Sec. IV C below, higher-order multipolecomponents ( λ >
3) can be important for the total en-ergy balance.
B. Isospin and neutron-proton structure of theoctupole deformation energy
To analyze the origin of the octupole energy ∆ E [3] , inFig. 4 we show its isospin and neutron-proton compo-nents as defined in Eqs. (31a) and (33). Again, a genericpattern emerges. In all cases, the octupole energy is al-most equal to its isoscalar part ∆ E t =0[3] . The isovectorenergy ∆ E t =1[3] is indeed very small, even if the studiednuclei have a significant neutron excess; this is consis-tent with the simple estimates of Sec. II. The contribu-tion from the pairing energy ∆ E pair[3] is also practicallynegligible. In the neutron-proton scheme, the np com-ponent always clearly dominates the nn and pp terms.The latter two are very small for UNEDF2 and hence∆ E [3] ≈ ∆ E t =0[3] ≈ ∆ E np [3] for this EDF. For SLy4, the nn and pp terms provide larger contributions to the oc-tupole deformation energy, accompanied by a reductionof the np term. Regardless of these minor differences be- Δ E [ ] ( M e V ) β (a) (b)(c) (d) SLy4 UNEDF2 Ba Ra FIG. 4. Similar to Fig. 3 but for different isospin and neutron-proton components of the octupole energy ∆ E [3] . tween the EDFs, we can safely conclude that it is theisoscalar octupole component (or the np octupole energycomponent) that plays the dominant role in building upthe nuclear octupole deformation. C. Reflection-asymmetric deformability alongIsotopic chains
At this point, we are ready to study structuralchanges that dictate the appearance of nuclear reflection-asymmetric deformations. The results shown in Figs. 3and 4 tell us that a mutual cancellation of near-parabolicshapes of different components of the deformation en-ergy results in a clearly non-parabolic dependence of thetotal deformation energy, as seen in Fig. 1. Therefore,to track back the positions and energies of the equilib-rium reflection-asymmetric deformations to the proper-ties of specific interaction components is not easy. To thisend, we analyze the properties of reflection-asymmetricdeformabilities of nuclei, that is, we concentrate on thecurvature of reflection-asymmetric deformation energiesat β = 0. To investigate the variation of the reflection-asymmetric deformability with neutron number, we per-formed SLy4-HFB calculations for the isotopic chains ofeven-even − Ba, − Ra, and − U isotopes,which are in the region of reflection-asymmetric insta-bility, as well as − Yb, which are expected to bereflection-symmetric [3]. In Fig. 5 we show the baselinequadrupole deformations β (0)2 . For the Ba, Ra, and Uisotopic chains, spherical-to-deformed shape transitionsare predicted slightly above the neutron magic numbers.The considered open-shell Yb isotopes are all predictedto be well deformed.As a quantitative measure of the octupole deformabil-ity, we analyze the deformation energy ∆ E = E ( β =0 . − E ( β = 0) calculated at a small octupole defor-mation of β = 0 .
05, with the quadrupole deformation β ( β = ) Neutron number (a) (b)(c) (d)
UBa RaYb
FIG. 5. Equilibrium quadrupole deformations β (0)2 as func-tions of N for the isotopic chains of (a) Ba, (b) Ra, (c) U,and (d) Yb computed with the SLy4 EDF. (a) (b)(c) (d) Neutron number Δ E ( M e V ) UBa RaYb
FIG. 6. Similar to Fig. 5 but for the deformation energy∆ E = E ( β = 0 . − E ( β = 0) . fixed at β (0)2 . We have checked that for different energycomponents, curvatures ∆ E/β are stable within about1% up to β = 0 .
05, so values of ∆ E taken at β = 0 . E calculated for the fourstudied isotopic chains. We see that the negative val-ues of ∆ E delineate regions of neutron numbers wherereflection-asymmetric deformations set in in Ba, Ra, andU isotopes [3].We now study ∆ E [ λ ] , the multipole components of thetotal deformation energy, for the four isotopic chainsconsidered to see whether they could provide insightsinto the neutron-number dependence of octupole defor-mations. Figure 7 shows that the answer is far fromobvious. Indeed, we observe strong cancellations of con-tributions coming from different multipole components ofthe reflection-asymmetric deformation energy. For exam-ple, both the repulsive monopole and attractive octupolecomponents are an order of magnitude larger than the (a) (b)(c) (d) Neutron number Δ E [ λ ] ( M e V ) UBa RaYb [3][1][2][0]
FIG. 7. Similar to Fig. 5 but for the deformation energies∆ E [ λ ] = E [ λ ] ( β = 0 . − E [ λ ] ( β = 0) for λ = 0 − total deformation energies shown in Fig. 6. Therefore,we can expect that in order to understand the behav-ior of the deformation energies, higher-order multipolecomponents ∆ E [ λ ] should be considered. Indeed, it hasbeen early recognized that higher-order deformations canstrongly influence the octupole collectivity of reflection-asymmetric nuclei [34–41]. (a) (b)(c) (d) Neutron number Δ E [ - λ m a x ] ( M e V ) UBa RaYb
FIG. 8. Similar to Fig. 7 but for the deformation energies∆ E = E ( β = 0 . − E ( β = 0) with multipole compo-nents summed up from λ = 0 to λ max . The values of λ max are listed in the legend. The regions of deformed isotopes ex-hibiting reflection-asymmetric instability in Fig. 6 are markedby shading. To better see accumulation effects with increasing mul-tipolarity and subtle fluctuations at different orders, inFig. 8 we plot multipole components of the octupole de-formability summed up to λ max . Noting dramaticallydifferent scales of Figs. 6 and 8, we see that summationsup to about λ = 5 or 7 are needed for the results to con-verge. Although the octupole component contributes byfar most to the creation of the reflection-asymmetric de-formation energy, its effect is counterbalanced by a verylarge monopole component and, therefore, even highermultipole components are instrumental in determiningthe total reflection-asymmetric deformability. This as-pect is underlined in the results shown in Figs. 9 and 10,where we separately show analogous sums of only odd- λ (odd parity) and even- λ (even parity) components, re-spectively. It is clear that the octupole polarizability isa result of a subtle balance between positive (repulsive)effect of the even-parity multipoles and negative (attrac-tive) effect of the odd-parity multipoles. (a) (b)(c) (d) Neutron number Δ E [ , ,.. λ m a x ] ( M e V ) UBa Ra Yb
FIG. 9. Similar to Fig. 8 but for the cumulative sum involvingodd- λ multipoles only. (a) (b)(c) (d) Neutron number Δ E [ , ,.. λ m a x ] ( M e V ) U Ba Ra Yb
FIG. 10. Similar to Fig. 8 but for the cumulative sum involv-ing even- λ multipoles only. D. Relation to shell structure
To gain some insights into the shell effects behindthe appearance of stable reflection-asymmetric nuclearshapes, Figs. 11 and 12 show, respectively, the s.p. leveldiagrams for
Yb and
Ra as functions of β . While β S i ng l e - p a r ti c l e e n e r gy ( M e V )
82 70566482100126 104 (a) neutrons(b) protons ε F ε F FIG. 11. Single-particle (canonical) neutron (top) and pro-ton (bottom) SLy4-HFB levels as functions of β ( β = 0)for Yb. Solid (dashed) lines indicate positive- (negative-)parity levels. Fermi levels ε F at N = 106 and Z = 70 aremarked by dash-dotted lines. The equilibrium deformation of Yb is indicated by a vertical dotted line. such diagrams cannot predict symmetry breaking effects per se , they can often provide qualitative understanding.The well-deformed nucleus
Yb is characteristic ofa stiff octupole vibrator. Indeed, its nucleon numbers( Z = 70 , N = 106) lie far from the “octupole-driving”numbers N oct . Due to the large deformed Z = 70 gaparound β = 0 .
32, there are no s.p. states of oppositeparity and the same projection Ω of the total s.p. angu-lar momentum on the symmetry axis that could producep-h excitations with appreciable λ = 3 strength acrossthe Fermi level. As for the neutron s.p. levels, the low-Ω positive-parity states originating from the 1 i / shelllie below the Fermi level, which appreciably reduces the1 i / ↔ f / strength. Because of the large quadrupoledeformations of Yb isotopes considered, the s.p. orbitalangular momentum (cid:96) of normal-parity orbitals is fairlyfragmented within the shell [42]. As seen in Figs. 10dand 9d, all multipole components of ∆ E for Yb varyvery smoothly with neutron number. β S i ng l e - p a r ti c l e e n e r gy ( M e V ) (a) neutrons(b) protons
130 132 134 136 138140 142144130 132 134136 138140 142144 FIG. 12. Similar to Fig. 11 but for
Ra. Fermi levels foreven-even Ra isotopes with N = 130 −
144 are marked bycircles. They have been shifted according to the position ofthe spherical 2 g / neutron and 1 h / proton shell. The equi-librium deformation of Ra is indicated by a vertical dottedline.
The Nilsson diagram shown in Fig. 12 is character-istic of transitional neutron-deficient actinides in whichthe octupole instability is expected. The unique-parityshells, 1 i / proton shell and 1 j / neutron shell, areof particle character, which results in an appearance ofclose-lying opposite-parity pairs of Nilsson levels with thesame low Ω-values at intermediate quadrupole deforma-tions. These levels can interact via the octupole field,with the dominant π i / ↔ π f / and ν j / ↔ ν g / couplings.As seen in Figs. 9 and 10, in the regions of octupole in- stability, the monopole and quadrupole deformation en-ergies become locally reduced while the octupole and do-triacontapole ( λ = 5) contributions to ∆ E grow. Ac-cording to our results, the effect of the dotriacontapoleterm is essential for lowering ∆ E around N oct . Thisnot surprising as the main contribution to the dotria-contapole coupling comes from the ∆ (cid:96) = ∆ j = 3 ex-citations [38, 40], i.e., the octupole and dotriacontapolecorrelations are driven by the same shell-model orbits.Interestingly, it is the attractive λ = 5 contribution to∆ E rather than the octupole term that exhibits the lo-cal enhancement in the regions of octupole instability.The shallow octupole minima predicted around Baresult from an interplay between the odd- λ deformationenergies, which gradually increase with N (see Fig. 9a)and the even- λ deformation energies, which gradually de-crease with N (see Fig. 10b). Again, the dotriacontapolemoment is absolutely essential for forming the octupoleinstability. V. CONCLUSIONS
In this work, we used the Skyrme-HFB approach tostudy the multipole expansion of interaction energies inboth isospin and neutron-proton schemes in order to ana-lyze their role in the appearance of reflection-asymmetricg.s. deformations. The main conclusions and results ofour study can be summarized as follows:(i) Based on the self-consistent HFB theory, reflection-asymmetric ground-state shapes of atomic nucleiare driven by the odd-multipolarity isoscalar (or,in neutron-proton scheme, np ) part of the nuclearinteraction energy. In a simple particle-vibrationpicture, this can be explained in terms of the verylarge isoscalar octupole polarizability χ , (∆ N =1) = 3.(ii) The most favorable conditions for reflection-asymmetric shapes are in the regions of transitionalnuclei with neutron and proton numbers just abovemagic numbers. For such systems, the unique-parity shell has a particle character, which createsfavorable conditions for the enhanced ∆ (cid:96) = ∆ j = 3octupole and dotriacontapole couplings.(iii) The presence of high-multipolarity interaction com-ponents, especially λ = 5 are crucial for the emer-gence of stable reflection-asymmetric shapes. Mi-croscopically, dotriacontapole couplings primarilycome from the same ∆ (cid:96) = ∆ j = 3 p-h excitationsthat are responsible for octupole instability. Ac-cording to our calculations, the attractive λ = 5contribution to the octupole stiffness is locally en-hanced in the regions of reflection-asymmetric g.s.shapes.In summary, stable pear-like g.s. shapes of atomic nu-clei result from a dramatic cancellation between even-0and odd-multipolarity components of the nuclear bindingenergy. Small variations in these components, associated,e.g., with the s.p. shell structure, can thus be instrumen-tal for tilting the final energy balance towards or awayfrom the octupole instability. One has to bear in mind,however, that the shell effect responsible for the sponta-neous breaking of intrinsic parity is weak, as it is associ-ated with the appearance of isolated ∆ (cid:96) = ∆ j = 3 pairsof levels (parity doublets) in the reflection-symmetric s.p.spectrum. In this respect, the breaking of the intrinsicspherical symmetry in atomic nuclei (presence of ellip-soidal deformations) is very common as every sphericals.p. shell (except for those with j = 1 /
2) carries an in-trinsic quadrupole moment that can contribute to the vibronic coupling.
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