Microscopic description of octupole collective excitations near N=56 and N=88
MMicroscopic description of octupole collective excitations around N=56 and N=88
K. Nomura, ∗ L. Lotina, T. Nikˇsi´c, and D. Vretenar
1, 2 Department of Physics, Faculty of Science, University of Zagreb, HR-10000 Zagreb, Croatia State Key Laboratory of Nuclear Physics and Technology,School of Physics, Peking University, Beijing 100871, China (Dated: January 11, 2021)Octupole deformations and related collective excitations are analyzed using the framework of nu-clear density functional theory. Axially-symmetric quadrupole-octupole constrained self-consistentmean-field (SCMF) calculations with a choice of universal energy density functional and a pairinginteraction are performed for Xe, Ba, and Ce isotopes from proton-rich to neutron-rich regions, andneutron-rich Se, Kr, and Sr isotopes, in which enhanced octupole correlations are expected to occur.Low-energy positive- and negative-parity spectra and transition strengths are computed by solvingthe quadrupole-octupole collective Hamiltonian, with the inertia parameters and collective potentialdetermined by the constrained SCMF calculations. Octupole-deformed equilibrium states are foundin the potential energy surfaces of the Ba and Ce isotopes with N ≈
56 and 88. The evolution ofspectroscopic properties indicates enhanced octupole correlations in the regions corresponding to N ≈ Z ≈ Z ≈
88 and Z ≈
56, and N ≈
56 and Z ≈
34. The average β deformation parameterand its fluctuation exhibit signatures of octupole shape phase transition around N = 56 and 88. I. INTRODUCTION
The intrinsic shapes of most medium-heavy andheavy nuclei are characterized by reflection symmetric,quadrupole deformations. Reflection-asymmetric, or oc-tupole deformations occurs in specific mass regions withthe proton Z and neutron numbers N near 34, 56, 88 and134 [1, 2]. Octupole correlations determine the system-atics of low-lying negative-parity states, which form ap-proximate alternating-parity doublets with the positive-parity ground-state bands, and the electric dipole and oc-tupole transition strengths. The exploration of stable oc-tupole deformations is a very active research field in bothexperimental and theoretical low-energy nuclear physics.In recent years, experiments with radioactive ion beamshave identified octupole-deformed nuclei, e.g., in light ac-tinides ( Rn and , , Ra) [3, 4], and lanthanides( , Ba) [5, 6]. Experimental studies of octupole defor-mations have also been reported in lighter mass regions,e.g., the neutron-deficient nuclei with N ≈ Z ≈
56 [7–11], and neutron-rich nuclei with N ≈
56 and Z ≈ Z ≈
88 and N ≈ Z ≈
56 and N ≈
88. However, octupole correlations in nuclei withparticle numbers close to 34 and/or 56 have not been ana-lyzed in much detail. A possible reason is that, especiallybecause the N ≈ Z ≈
56 nuclei are close to the protondrip-line, experimental information is insufficient. Few ∗ [email protected] exceptions are perhaps the Nilsson-Strutinsky calculationbased on the Woods-Saxon potential in Refs. [14, 28],the constrained Hartree-Fock+BCS calculation with theSkyrme force [29] of the light Xe and Ba isotopes inRef. [30], and the global analysis of ground-state octupoledeformation within the nuclear density functional theory(DFT) in Ref. [18]. However, in those studies calcula-tions have been carried out at the mean-field level oronly for restricted spectroscopic properties. Because ofrenewed experimental interest in octupole shapes in ex-tended mass regions, it is meaningful to carry out a newtheoretical analysis of octupole deformations and relatedspectroscopy, that also includes the lighter mass regionwith N/Z ≈
34 and 56.Nuclear density functional theory (DFT) provides anaccurate and economic microscopic approach to nuclearstructure that enables systematic studies [31, 32]. Bothrelativistic [33, 34] and non-relativistic [31, 35] energydensity functionals (EDFs) have successfully been ap-plied in the global description of the ground-state proper-ties and collective excitations. The basic implementationis in terms of self-consistent mean-field (SCMF) calcula-tions that produce energy surfaces as functions of shapeand/or pairing collective variables. To compute spec-troscopic properties, the SCMF framework must be ex-tended to include dynamical correlations that arise fromthe restoration of broken symmetries and fluctuationsaround the mean-field minima. A straightforward ap-proach is the generator coordinate method (GCM) [36]with symmetry projections and configuration mixing in-cluded. The GCM has been employed to study octupolecorrelations with axial quadrupole and octupole deforma-tions as collective coordinates [37–40]. In practical appli-cations to medium-heavy and heavy nuclei, however, theGCM is computationally challenging, especially as thenumber of nucleons or collective coordinates increases.Alternative approaches to GCM have thus been devel-oped, such as the quadrupole-octupole collective Hamil- a r X i v : . [ nu c l - t h ] J a n tonian (QOCH) [15, 41, 42] and the mapped sdf -IBM[22, 23].Based on the fully microscopic framework of nuclearDFT, here we carry out a systematic analysis of octupolecollective excitations in the mass A ≈ −
150 regions:Xe, Ba, and Ce isotopes extending from proton-rich ( N ≈ Z ≈
56) to neutron-rich ( N ≈
88 and Z ≈
56) nuclei,and the neutron-rich Se, Kr, and Sr nuclei with Z ≈ N ≈
56. The starting point are axially-symmetricquadrupole-octupole constrained SCMF calculations us-ing the relativistic Hartree-Bogoliubov model with thedensity-dependent point-coupling (DD-PC1) [43] EDF,and a separable pairing force [44]. The relevant excita-tion spectra and transition rates are computed by solv-ing the collective Schr¨odinger equation with the axially-symmetric quadrupole β and octupole β shape degreesof freedom. The constrained SCMF calculations com-pletely determine the moment of inertia, three mass pa-rameters, and collective potential of the QOCH. Thediagonalization of the QOCH yields the positive- andnegative-parity excitation spectra, as well as the electricquadrupole, octupole, and dipole transition rates. Wenote that a similar SCMF+QOCH spectroscopic calcula-tion, based on the PC-PK1 [45] EDF, was performed fora large number of medium-heavy and heavy nuclei: fromRn to Fm, and from Xe to Gd isotopes [15].This paper is organised as follows. In Sec. II we brieflyreview the formalism of the RHB+QOCH model. TheSCMF β − β potential energy surfaces are discussed inSec. III. In Sec. IV the systematics of spectroscopic prop-erties, including excitation energies of low-lying positive-and negative-parity states, and electromagnetic transi-tion rates, are compared to available experimental data.The results for the N = 56 isotones are presented inSec. V. Signatures of octupole shape phase transitionsare examined in Sec. VI. Finally, a brief summary andconclusion are given in Sec. VII. II. THEORETICAL FRAMEWORKA. Relativistic Hartree-Bogoliubov calculation
The first step of the analysis is a set of constrainedSCMF calculations of potential energy surfaces (PESs),performed using the relativistic Hartree-Bogoliubov(RHB) method [33] with the density-dependent pointcoupling (DD-PC1) [43] functional for the particle-holechannel, and a separable pairing force of finite range [44]in the particle-particle channel. The constraints imposedin the SCMF calculations are the expectation values ofthe axially-symmetric quadrupole Q and octupole Q moments: ˆ Q = 2 z − x − y (1)ˆ Q = 2 z − z ( x + y ) . (2) The corresponding quadrupole and octupole deformationparameters β and β are defined by the relations: β = √ π r A / (cid:104) ˆ Q (cid:105) (3) β = √ π r A (cid:104) ˆ Q (cid:105) , (4)where r = 1 . N f = 10 for the region Z ≈
34 and N ≈ Z ≈
56 and N (cid:62)
56 a larger basiswith N F = 12 is used. B. Quadrupole-Octupole Collective Hamiltonian
Collective states are described using an axially-symmetric QOCH, with deformation-dependent parame-ters determined microscopically by the constrained RHBcalculation. The QOCH contains the vibrational and ro-tational kinetic terms, and the collective potential:ˆ H coll = T vib + T rot + V coll , (5)where the vibrational kinetic energy is parametrized bythe mass parameters B , B , and B , T vib = 12 B ˙ β + B ˙ β ˙ β + 12 B ˙ β , (6)and the three moments of inertia I k determine the rota-tional kinetic energy T rot = 12 (cid:88) k =1 I k ω k . (7)Finally, the collective potential V coll includes zero-pointenergy (ZPE) corrections. After quantisation the collec-tive Hamiltonian reads:ˆ H coll = − (cid:126) √ ω I (cid:34) ∂∂β (cid:114) I ω B ∂∂β − ∂∂β (cid:114) I ω B ∂∂β (8) − ∂∂β (cid:114) I ω B ∂∂β + ∂∂β (cid:114) I ω B ∂∂β (cid:35) (9)+ ˆ J I + V coll ( β , β ) , (10)where ω = B B − B . The mass parameters, mo-ments of inertia, and collective potentials as functionsof the collective coordinates ( β , β ), are specified bythe deformation-constrained self-consistent RHB calcu-lations for a specific choice of the nuclear energy densityfunctional and pairing interaction. In the present versionof the model, the mass parameters defined as the inverseof the mass tensor B ij ( q ) = M − ij ( q ), are calculated inthe perturbative cranking approximation M C p = (cid:126) M − M (3) M − , (11)where (cid:2) M ( k ) (cid:3) ij = (cid:88) µν (cid:68) (cid:12)(cid:12)(cid:12) ˆ Q i (cid:12)(cid:12)(cid:12) µν (cid:69) (cid:68) µν (cid:12)(cid:12)(cid:12) ˆ Q j (cid:12)(cid:12)(cid:12) (cid:69) ( E µ + E ν ) k . (12) | µν (cid:105) are two-quasiparticle wave functions, and E µ and E ν the corresponding quasiparticle energies. ˆ Q i denotesthe multipole operators that correspond to the collectivedegrees of freedom. The collective potential V coll is ob-tained by subtracting the vibrational zero-point energy(ZPE) from the total RHB deformation energy E ZPE = 14 Tr (cid:104) M − M (1) (cid:105) . (13)The microscopic self-consistent solutions of the con-strained RHB equations, that is, the single-quasiparticleenergies and wave functions on the entire energy surfaceas functions of the deformations, provide the microscopicinput for the calculation of both the collective inertia andzero-point energy. The Inglis-Belyaev formula is used forthe rotational moment of inertia. From the diagonaliza-tion of the collective Hamiltonian one obtains the collec-tive energy spectrum. The eigenfunctions are expandedin terms of a complete set of basis functions. For eachvalue of the angular momentum I , the basis is definedas: | n n IM K (cid:105) = ( ω I ) − / φ n ( β ) φ n ( β ) | IM K (cid:105) , (14)where φ n λ denotes the one-dimensional HO functions of β λ . For axially-symmetric shapes, the intrinsic projec-tion of the total angular momentum K = 0. The proba-bility density distribution is defined as: ρ Iπα ( β , β ) = e − µ β / √ ω I| ψ Iπα ( β , β ) | , (15)with the normalization (cid:90) ρ Iπα ( β , β ) dβ dβ = 1 . (16)The reduced transition probabilities B ( Eλ ) are calcu-lated from the relation: B ( Eλ ; I i → I f ) =( I i λ | I f × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) dβ s β √ ω I Ψ i M Eλ ( β , β )Ψ ∗ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (17)where M Eλ ( β , β ) is the electric moment of order λ ,and the factor in parentheses on the right-hand side ofthe above expression is the Clebsch-Gordan coefficient. The electric moment is calculated in microscopic mod-els as (cid:104) Φ( β , β ) |M Eλ ( β , β ) | Φ( β , β ) (cid:105) , with Φ( β , β )representing the nuclear wave functions. The correspond-ing operators for dipole, quadrupole, and octupole tran-sitions read: D = (cid:114) π e (cid:16) NA z p − ZA z n (cid:17) (18) Q p = (cid:114) π e (2 z p − x p − y p ) (19) Q p = (cid:114) π e (2 z p − z p ( x p + y p )) , (20)(21)with bare electric charge e , and this means no effectivecharges need to be introduced to calculate electromag-netic transition rates. III. SCMF RESULTSA. Neutron-deficient Z ≈ nuclei The axially-symmetric ( β , β ) PESs for neutron-deficient nuclei: − Xe, − Ba and − Ce aredepicted in Fig. 1. Already at the most neutron-deficientisotopes with N = 54, the potential is considerably softin β deformation, even though the minimum is on the β = 0 axis. Octupole-deformed equilibrium states with β (cid:54) = 0 occur in the N ≈ Z nuclei , Ba and
Ce.There is no stable octupole deformed minimum for theneighbouring Xe isotopes, but the potential exhibits anarrow valley on the prolate side ( β >
0) that is softover a range of β values. Previous mean-field calcula-tions have also suggested there are a few N ≈ Z ≈ N >
64, not shown in the figure, the potential becomesrather softer in the β direction and the prolate deforma-tion becomes larger around the middle of the major shell N = 66, but no octupole minima are found on the corre-sponding SCMF PESs. When approaching the neutronshell closure at N = 82, nearly spherical global minimaare obtained with both the β and β deformations con-verging to zero. B. Neutron-rich Z ≈ nuclei Figure 2 displays the ( β , β ) PESs for the isotopes − Xe, − Ba, and − Ce beyond the N = 82neutron shell closure. These neutron-rich isotopes areclose to the empirical octupole magic number N = 88,and more extensive experimental and theoretical stud-ies have been reported in this region compared to theneutron-deficient one with N ≈ Z ≈
56. In all threeisotopic chains the potential surfaces shown in Fig. 2 are
FIG. 1. SCMF ( β , β ) PESs for − Xe, − Ba and − Ce. Global minima are identified by the red dots. Contourjoints points on the surface with the same energy, and the difference between neighbouring contours is 1 MeV.FIG. 2. Same as in the caption to Fig. 1 but for − Xe, − Ba, and − Ce. more rigid in β , and pronounced octupole correlationsare predicted. In particular, a number of neutron-rich Baand Ce nuclei exhibit octupole-deformed global minima with non-zero value of β , that is, the isotopes − Baand − Ce. The most pronounced octupole globalminimum is found in nuclei with N ≈
88, in agreement
FIG. 3. Same as in the caption to Fig. 1 but for the neutron-rich nuclei − Se, − Kr, and − Sr.FIG. 4. Same as in the caption to Fig. 1 but for the N = 56isotones from Se ( Z = 34) to Ba ( Z = 56). with experimental findings. The β − β PESs obtained in the present analysis for the neutron-rich lanthanidesare also consistent with many of the recent SCMF cal-culations using both relativistic [15, 16, 23] and non-relativistic EDFs [18, 39]. C. Z ≈ nuclei around N = 56 We will also explore another mass region in which oc-tupole correlations could develop. In Fig. 3 we plot theSCMF β − β PESs for the neutron-rich nuclei − Se, − Kr, and − Sr, close to the proton Z = 34 andneutron N = 56 octupole magic numbers. Even thoughoctupole correlations are empirically expected to occurat proton number Z = 34, the PESs in the figure do notexhibit octupole global minima for these nuclei. In gen-eral, the ( β , β ) PESs for the Z ≈
34 neutron-rich nucleiappear rather soft in β deformation. Taking as example , Kr, one notices two shallow local minima on the pro-late side. For many nuclei in this region a number of bothmicroscopic and empirical studies point to the presenceof shape coexistence and/or γ -soft shapes. The presentcalculation is restricted to only axially-symmetric shapesand, thus, a more realistic analysis should take into ac-count the triaxial degrees of freedom. D. N = 56 isotones To analyze the evolution of the empirical N = 56 oc-tupole magic number, we have performed constrainedSCMF calculations along the isotonic chain N = 56. Fig-ure 4 displays the resulting β − β PESs for the N = 56even-even isotones from Z = 34 ( Se) to Z = 56 ( Ba).In the ( β , β ) PESs, neither an octupole deformed equi-librium state nor octupole-soft potential is observed be-low the proton magic number Z = 50. For N = 56 iso-tones beyond Sn, however, the potentials start to becomemore rigid in β and softer in β . Among the N = 56isotones depicted in the figure, the most pronounced oc-tupole minimum is obtained for the nucleus Ba with N = Z = 56. E ne r g y ( M e V ) + + QOCH Experiment Ba + + - - - - - + + + + + + - - - - -
52 48.44 +1-3 +2-1 +25-34 <106<13985.7(2)54.4(2)768794 163530 95 E ne r g y ( M e V ) + + QOCH Experiment Xe + + - - - -
145 0 + + + + + - - - - FIG. 5. Comparison of the QOCH and experimental low-energy excitation spectra for the positive- and negative-parityyrast states of
Ba and
Xe. Solid and dashed arrows de-note E2 and E3 transitions, respectively, and the correspond-ing B ( E
2) and B ( E
3) values are given in Weisskopf units.Experimental results are from Refs. [5, 46]. FIG. 6. Probability density distributions for the lowestpositive-parity (0 +1 ) and negative-parity (1 − ) states of Ba(upper) and
Xe (lower) in the β − β plane. IV. SPECTROSCOPIC RESULTS
In the following we present QOCH results for the spec-troscopic properties relevant to quadrupole and octupolecollective excitations. Note that the calculation also in-cludes N ≈ Z ≈
56 nuclei that are close to the protondrip-line. For these nuclei only very limited experimen-tal information is available: the lightest known Xe, Ba,and Ce isotopes are
Xe,
Ba [11], and
Ce. Forcompleteness, and considering the signatures of octupolecorrelations on the corresponding ( β , β ) PESs in Fig. 1,we also discuss the spectroscopy of proton drip-line nu-clei. A. Benchmark calculation:
Ba and Xe As a test case, we consider the QOCH results for thelow-energy K π = 0 +1 and 0 − bands of Ba and
Xe.These nuclei are specifically considered here as represen-tative of the regions close to the neutron octupole magicnumbers N ≈
88 and 56. In particular, recent exper-iments performed at the Argonne National Laboratory[5, 6], have indicated that the neutron-rich nucleus
Baand neighbouring Ba isotopes are characterized by pro-nounced ground-state octupole deformations. In Fig. 5we compare the lowest K π = 0 +1 and 0 − QOCH bands of
Ba and
Xe to the available data. Both the positive-parity π = +1 and negative-parity π = − K π = 0 +1 of Ba, and predicts B ( E
3) values fortransitions between the π = +1 and π = −
52 56 60 64 68 72 76 80 84 88 92 96 N E ne r g y ( M e V ) (a ) Xe (QOCH) + + + +
52 56 60 64 68 72 76 80 84 88 92 96 N (a ) Xe (expt.)52 56 60 64 68 72 76 80 84 88 92 96 N E ne r g y ( M e V ) (b ) Ba (QOCH) 52 56 60 64 68 72 76 80 84 88 92 96 N (b ) Ba (expt.)52 56 60 64 68 72 76 80 84 88 92 96 N E ne r g y ( M e V ) (c ) Ce (QOCH) 52 56 60 64 68 72 76 80 84 88 92 96 N (c ) Ce (expt.)
FIG. 7. Evolution of QOCH excitation spectra for the positive-parity yrast states along the chains of Xe, Ba, and Ce isotopes.Experimental values are from the ENSDF database [46].
Ba the calculated B ( E
3; 3 − → +1 ) value of 16 W.u. is within the range of experimental uncertainty. In theneutron-deficient N ≈ Z ≈
56 region,
Xe is the light-est nucleus for which experimental information is avail-able. The theoretical excitation spectrum is in qualita-tive agreement with the data, though we note that thecalculated π = +1 band appears to be somewhat morecompressed than the experimental one.Figure 6 plots the probability density distributions ρ Iπα ( β , β ) of the lowest energy positive (0 +1 ) andnegative-parity (1 − ) states in the ( β , β )-deformationspace. One notices that, for both nuclei, the ground state0 +1 probability density is peaked at β ≈ β , min , wherethe global minimum occurs on the PES, and β ≈ − state are, on theother hand, concentrated at the same values of β as thecorresponding ground states, but at finite values of theoctupole deformation β ≈ . − . B. Low-energy excitation spectra
The calculated excitation spectra for both even-spinpositive- and odd-spin negative-parity yrast states areshown in Figs. 7 and 8, respectively, for the Xe, Ba,and Ce isotopic chains in comparison with available data.The theoretical positive-parity states are in good agree-ment with experimental results (Fig. 7), with the excep-tion of nuclei in the immediate vicinity of the neutronmagic number N = 82. For these nuclei the purely collec-tive states of the QOCH cannot reproduce the empiricalexcitation spectrum on a quantitative level.The results for the negative-parity states, shown inFig. 8, are more interesting. The calculated levels foreach isotopic chain exhibit evident signatures of enhancedoctupole collectivity, that is, a parabolic behavior of exci-tation energies with neutron number, centered at around N ≈
56 and N ≈
88. At these neutron numbers the
52 56 60 64 68 72 76 80 84 88 92 96 N E ne r g y ( M e V ) (a ) Xe (QOCH) − − − − −
52 56 60 64 68 72 76 80 84 88 92 96 N (a ) Xe (expt.)52 56 60 64 68 72 76 80 84 88 92 96 N E ne r g y ( M e V ) (b ) Ba (QOCH) 52 56 60 64 68 72 76 80 84 88 92 96 N (b ) Ba (expt.)52 56 60 64 68 72 76 80 84 88 92 96 N E ne r g y ( M e V ) (c ) Ce (QOCH) 52 56 60 64 68 72 76 80 84 88 92 96 N (c ) Ce (expt.)
FIG. 8. Same as in the caption to Fig. 7 but for the negative-parity states. levels become lowest in energy. This is consistent withthe observed trend of the SCMF ( β , β ) PESs in Figs. 1and 2: in most of the nuclei around N = 56 and 88 thecorresponding PESs display global minima at non-zero β .There are only few tentative assignments of negative-parity states in neutron-rich Z = 34 , ,
38 isotopes. InFig. 9 we display the QOCH results for the excitation en-ergies of the lowest negative-parity states in neutron-richSe, Kr and Sr isotopes. Only few data for Kr isotopes areavailable. Also in this case one notices a kind of parabolicbehavior centered at N = 56, but much less pronouncedthan in the proton-rich Xe, Ba, Ce nuclei. Obviously inthe latter case the Z ≈
56 proton and N = 56 num-bers reinforce octupole correlations, and global minimaat non-zero β are predicted. This does occur for theneutron-rich Se, Kr and Sr isotopes, for which the cor-responding PESs are at most soft in the β collective coordinates, as shown in Fig. 3. In fact, it is well knownthat for these nuclei it is far more important to includethe triaxial degree of freedom in order to describe the ex-citation spectra at a quantitative level, and especially theshape transition at N = 60. The present version of theQOCH model is restricted to axially symmetric shapesand, therefore, the calculated positive-parity spectra canonly qualitatively reproduce the empirical isotopic trend. C. Electromagnetic properties
The results for the B ( E
2; 2 +1 → +1 ), B ( E
3; 3 − → +1 ),and B ( E
1; 1 − → +1 ) reduced transition probabilitiesalong the Xe, Ba, and Ce isotopic chains are shown inFig. 10. We note a very nice agreement with the exper-imental B ( E
2) values, especially considering that barecharges are used in the calculation. Much less informa-
52 54 56 58 60 62 64 66 N E ene r g y ( M e V ) (a) Sr (QOCH) − − − − −
52 54 56 58 60 62 64 66 N (b) Sr (Expt.) − − − − −
52 54 56 58 60 62 64 66 N E ne r g y ( M e V ) (c) Kr 52 54 56 58 60 62 64 66 N (d) Se FIG. 9. Same as in the caption to Fig. 8 but for the negative-parity states in Se, Kr, and Sr isotopes. tion is available on the B ( E
3) values. What is inter-esting is that the theoretical B ( E
3) values exhibit twopeaks, one at N ≈
56 and the other at N ≈
88. Theseneutron numbers, of course, correspond to the ones atwhich octupole collectivity is most enhanced. Consid-ering the results on a more quantitative level, in eachisotopic chain the QOCH results for the B ( E , Ba, for which the experimental valuesare characterized by large uncertainties (see also Fig. 5).The fact that the QOCH cannot quantitatively reproducethe experimental B ( E
3) values is probably related to thefact that the data are only available for nuclei that areclose to the neutron magic number N = 82, for whichthe calculated energies of the 3 − state are also not ina particularly good agreement with experimental results(cf. Fig. 8). As shown in Fig. 2, for those nuclei thatare nearly spherical, there is no octupole deformation oroctupole softness at the SCMF level, so the collectivemodel is not expected to provide a very good descrip-tion of E3 transition strength. There is no experimentalinformation for the E B ( E
1) values exhibits certainpeaks for particular nuclei, but they are not necessarilythe same as for the B ( E
3) values. The E E E
3, hencethe collective model does not necessarily provide accurate predictions for the B ( E
1) values.
V. SYSTEMATICS ALONG THE N = 56 ISOTONIC CHAINS
We have also explored the systematics of excitation en-ergies along the N = 56 isotones. The QOCH results forthe low-energy positive-parity and negative-parity spec-tra are shown in Fig. 11. The predicted positive-paritylevels remain almost constant with proton number Z ,except for the Z = 50 shell closure. The model qualita-tively reproduces the corresponding experimental π = +1spectra, with the exception of a pronounced proton-number dependence observed in the region 36 (cid:54) Z (cid:54) Z = 40 proton sub-shell closure, which is not properly ac-counted for in the present calculation restricted to axialsymmetry. The computed π = − Z ≈
40 for the
Z <
50 region. Beyondthe proton magic number Z = 50, level spacing betweenthe negative-parity states are strongly reduced, and theirenergies display the parabolic trend characteristic of pro-nounced octupole correlations. The calculation also re-produces the empirical B ( E
2) values, but underestimatesthe two known B ( E
3; 3 − → +1 ) at Z = 42 and Z = 44by approximately a factor of two. VI. SIGNATURES OF OCTUPOLE SHAPETRANSITIONS
As signatures of quadrupole and octupole shape tran-sitions, we plot in Fig. 12 the average values of the axialquadrupole β (a) and β (b) deformation parametersin the QOCH ground states 0 +1 , and their fluctuations δβ /β and δβ /β , respectively, for the Ce, Ba, Xe, Sr,Kr, and Se isotopes, as functions of the neutron number.Here the average β λ ( λ = 2 ,
3) is defined as β λ = (cid:112) (cid:104) β λ (cid:105) ,and δβ λ denotes the variance δβ λ = (cid:113) (cid:104) β λ (cid:105) − (cid:104) β λ (cid:105) / β λ [47, 48]. In Fig. 12(a), as expected from both the SCMF( β , β ) PESs and the calculated excitation spectra, theaverage deformation β increases towards the middle ofthe major shell, as the quadrupole collectivity becomeslarger. The octupole deformation β exhibits a parabolicbehavior in two regions, centered at the neutron num-bers N = 56 and 88, at which it reaches maximum valueslarger than β ≈ . β deformation for theSe, Kr, and Sr isotopes change abruptly from N = 58 to60. This reflects the rapid structural evolution in thesenuclei, most noticeably in Kr, the relevant spectroscopicproperties indicating phase-transitional behavior at N =60. The fluctuations of β for the Xe, Ba, and Ce isotopesexhibit only a moderate change. This is consistent withthe SCMF results that the minima are more rigid in β .The large β fluctuations at the N = 82 magic number0
54 60 66 72 78 84 90 96 N B ( E ; + → + ) ( W . u . ) (a ) Xe54 60 66 72 78 84 90 96 N B ( E ; − → + ) ( W . u . ) (a ) Xe54 60 66 72 78 84 90 96 N B ( E ; − → + ) ( W . u . ) (a ) Xe 54 60 66 72 78 84 90 96 N (b ) Ba54 60 66 72 78 84 90 96 N (b ) Ba54 60 66 72 78 84 90 96 N (b ) Ba 54 60 66 72 78 84 90 96 N (c ) Ce54 60 66 72 78 84 90 96 N (c ) Ce54 60 66 72 78 84 90 96 N (c ) Ce FIG. 10. B (E2; 2 +1 → +1 ), B (E3; 3 − → +1 ), and B (E1; 1 − → +1 ) reduced transition probabilities for the Xe, Ba, and Ceisotopes. Filled symbols connected by lines denote the QOCH results. Experimental values (open symbols) are taken from theENSDF database. are due to vanishing values of β in the denominator inspherical nuclei.The fluctuation in octupole deformation β , depictedin Fig. 12(d), presents a measure for octupole softness.Especially for the Sr and Kr nuclei near N = 54 to 56,and for the Ba and Ce nuclei from N = 88 to 90, weobserve a marked discontinuity characteristic of octupoleshape-phase transitions. The isotopic dependence of thefluctuation in Fig. 12(d) correlates with the systematicsof spectroscopic properties. VII. CONCLUSIONS
Octupole collective excitations have been analyzedusing the fully microscopic framework of nuclear den-sity functional theory. Axially-symmetric quadrupole-octupole constrained SCMF calculations based on achoice of universal energy density functional and pairinginteraction have been performed in three mass regionsof the nuclear chart in which enhanced octupole correla-tions are empirically expected to occur: neutron-deficientnuclei with Z ≈
56 and N ≈
56, neutron-rich nuclei with Z ≈
56 and N ≈
88, and the neutron-rich nuclei with Z ≈
34 and N = 56. The resulting potential energysurfaces in the ( β , β ) plane indicate octupole-deformedequilibrium states at the SCMF level in , Ba and
Ce on the neutron-deficient side, and in a number ofneutron-rich Ba and Ce nuclei around N = 88.The SCMF calculations completely determine the in-gredients of the quadrupole-octupole collective Hamil-tonian: the moment of inertia, three mass parameters,and the collective potential. The diagonalization of theQOCH subsequently yields the positive- and negative-parity excitation spectra and the electric quadrupole, oc-tupole, and dipole transition strengths that are relevantto the quadrupole and octupole modes of collective exci-tations. The calculated excitation spectra for both pari-ties, and the B ( E
2) and B ( E
3) values are in very goodagreement with experiment for the neutron-deficient andneutron-rich Z ≈
56 nuclei. These quantities indicatea parabolic systematics around the neutron numbers N = 56 and 88, at which the SCMF ( β , β ) PESs exhibitpronounced octupole deformed minima. The calculatedspectroscopic properties for the neutron-rich nuclei with Z ≈
34 and N ≈
56 also indicate a signature of en-hanced octupole collectivity around N ≈
56, though notas distinct as in the case of the Z ≈
56 isotopes. Wehave further explored spectroscopic properties along the N = 56 isotones, from Z = 34 to 58. The relevant quan-tities, i.e., negative-parity spectra and B ( E
3) transitions,again point to an enhancement of octupole correlationsaround Z = 34 and 56. In general, octupole collectivityappears to be more enhanced in the N ≈
88 region than1
34 38 42 46 50 54 58 Z E ne r g y ( M e V ) (a) QOCH ( π = +1 ) N =56
34 38 42 46 50 54 58 Z (b) Expt. ( π = +1 ) + + + +
34 38 42 46 50 54 58 Z E ne r g y ( M e V ) (c) QOCH ( π = − ) 34 38 42 46 50 54 58 Z (d) Expt. ( π = − ) − − − − − FIG. 11. Excitation spectra of the low-lying positive- (upper)and negative-parity (lower) yrast states of the N = 56 iso-tones as functions of the proton number Z . The excitationspectra computed with the QOCH are plotted on the left-hand side of the figure, and are compared with the availableexperimental data [46] on the right. around N ≈
56. The present fully-microscopic spectro-scopic calculation has predicted several nuclei with stableoctupole deformation in the neutron-deficient Z ≈
56 nu-clei, that have not been investigated so far. The average β and β deformations calculated in the QOCH groundstates, as well as their fluctuations, exhibit signatures ofquadrupole- and octupole shape phase transitions.The current implementation of the QOCH method isrestricted to axially-symmetric shapes. Hence some dis-crepancies with the experimental spectroscopic proper-ties could be traced back to this limitation. In par-ticular, the fact that the positive-parity states for theneutron-rich Z ≈
34 and N ≈
56 nuclei have not beenreproduced quantitatively indicates that triaxial shapedegrees of freedom need to be included as additional col-lective coordinates. This requires the inclusion of severalnew terms in the collective Schr¨odinger equation, but inpractical applications such an extension would be verycomplicated. Thus, a method that consists in mappingthe SCMF solutions onto the interacting-boson Hamilto-nian [49] could be more feasible for the inclusion of tri-axial degrees of freedom. Work in this direction presentsan interesting future study.
52 56 60 64 68 72 76 80 84 88 92 960.00.10.20.30.40.5 β (a)52 56 60 64 68 72 76 80 84 88 92 960.060.080.100.120.140.16 β (b)52 56 60 64 68 72 76 80 84 88 92 960.00.10.20.30.40.50.60.70.8 δ β / β (c)52 56 60 64 68 72 76 80 84 88 92 96 N δ β / β (d) XeBaCeSeKrSr
FIG. 12. Average values of the β (a) and β (b) deformationparameters in the 0 +1 ground state, β λ = (cid:112) (cid:104) β λ (cid:105) , and thefluctuations δβ /β (c) and δβ /β (d), for the Xe, Ba, Ce,Se, Kr, and Sr isotopes as functions of the neutron number N . The variance δβ λ is defined as δβ λ = (cid:113) (cid:104) β λ (cid:105) − (cid:104) β λ (cid:105) / β λ . ACKNOWLEDGMENTS
This work has been supported by the Tenure TrackPilot Programme of the Croatian Science Foundationand the ´Ecole Polytechnique F´ed´erale de Lausanne, andthe Project TTP-2018-07-3554 Exotic Nuclear Struc-ture and Dynamics, with funds of the Croatian-SwissResearch Programme. It has also been supported inpart by the QuantiXLie Centre of Excellence, a projectco-financed by the Croatian Government and Euro-2pean Union through the European Regional Development Fund - the Competitiveness and Cohesion OperationalProgramme (KK.01.1.1.01). [1] P. A. Butler and W. Nazarewicz, Rev. Mod. Phys. ,349 (1996).[2] P. A. Butler, J. Phys. G: Nucl. Part. Phys. , 073002(2016).[3] L. P. Gaffney, P. A. Butler, M. Scheck, A. B.Hayes, F. Wenander, M. Albers, B. Bastin, C. Bauer,A. Blazhev, S. B¨onig, N. Bree, J. Cederk¨all, T. Chupp,D. Cline, T. E. Cocolios, T. Davinson, H. D. Witte,J. Diriken, T. Grahn, A. Herzan, M. Huyse, D. G. Jenk-ins, D. T. Joss, N. Kesteloot, J. Konki, M. Kowal-czyk, T. Kr¨oll, E. Kwan, R. Lutter, K. Moschner,P. Napiorkowski, J. Pakarinen, M. Pfeiffer, D. Radeck,P. Reiter, K. Reynders, S. V. Rigby, L. M. Robledo,M. Rudigier, S. Sambi, M. Seidlitz, B. Siebeck, T. Stora,P. Thoele, P. V. Duppen, M. J. Vermeulen, M. vonSchmid, D. Voulot, N. Warr, K. Wimmer, K. Wrzosek-Lipska, C. Y. Wu, and M. Zielinska, Nature (London) , 199 (2013).[4] P. A. Butler, L. P. Gaffney, P. Spagnoletti, K. Abrahams,M. Bowry, J. Cederk¨all, G. de Angelis, H. De Witte,P. E. Garrett, A. Goldkuhle, C. Henrich, A. Illana,K. Johnston, D. T. Joss, J. M. Keatings, N. A. Kelly,M. Komorowska, J. Konki, T. Kr¨oll, M. Lozano, B. S.Nara Singh, D. O’Donnell, J. Ojala, R. D. Page,L. G. Pedersen, C. Raison, P. Reiter, J. A. Rodriguez,D. Rosiak, S. Rothe, M. Scheck, M. Seidlitz, T. M. Shnei-dman, B. Siebeck, J. Sinclair, J. F. Smith, M. Stryjczyk,P. Van Duppen, S. Vinals, V. Virtanen, N. Warr,K. Wrzosek-Lipska, and M. Zieli´nska, Phys. Rev. Lett. , 042503 (2020).[5] B. Bucher, S. Zhu, C. Y. Wu, R. V. F. Janssens, D. Cline,A. B. Hayes, M. Albers, A. D. Ayangeakaa, P. A. But-ler, C. M. Campbell, M. P. Carpenter, C. J. Chiara, J. A.Clark, H. L. Crawford, M. Cromaz, H. M. David, C. Dick-erson, E. T. Gregor, J. Harker, C. R. Hoffman, B. P. Kay,F. G. Kondev, A. Korichi, T. Lauritsen, A. O. Macchi-avelli, R. C. Pardo, A. Richard, M. A. Riley, G. Savard,M. Scheck, D. Seweryniak, M. K. Smith, R. Vondrasek,and A. Wiens, Phys. Rev. Lett. , 112503 (2016).[6] B. Bucher, S. Zhu, C. Y. Wu, R. V. F. Janssens, R. N.Bernard, L. M. Robledo, T. R. Rodr´ıguez, D. Cline, A. B.Hayes, A. D. Ayangeakaa, M. Q. Buckner, C. M. Camp-bell, M. P. Carpenter, J. A. Clark, H. L. Crawford, H. M.David, C. Dickerson, J. Harker, C. R. Hoffman, B. P.Kay, F. G. Kondev, T. Lauritsen, A. O. Macchiavelli,R. C. Pardo, G. Savard, D. Seweryniak, and R. Von-drasek, Phys. Rev. Lett. , 152504 (2017).[7] S. L. Rugari, R. H. France, B. J. Lund, Z. Zhao, M. Gai,P. A. Butler, V. A. Holliday, A. N. James, G. D. Jones,R. J. Poynter, R. J. Tanner, K. L. Ying, and J. Simpson,Phys. Rev. C , 2078 (1993).[8] C. Fahlander, D. Seweryniak, J. Nyberg, Z. Dombr´adi,G. Perez, M. J´ozsa, B. Nyak´o, A. Atac, B. Ceder-wall, A. Johnson, A. Kerek, J. Kownacki, L.-O. Norlin,R. Wyss, E. Adamides, E. Ideguchi, R. Julin, S. Juu-tinen, W. Karczmarczyk, S. Mitarai, M. Piiparinen,R. Schubart, G. Sletten, S. T¨orm¨anen, and A. Virtanen, Nucl. Phys. A , 773 (1994).[9] J. F. Smith, C. J. Chiara, D. B. Fossan, G. J. Lane, J. M.Sears, I. Thorslund, H. Amro, C. N. Davids, R. V. F.Janssens, D. Seweryniak, I. M. Hibbert, R. Wadsworth,I. Y. Lee, and A. O. Macchiavelli, Phys. Rev. C , R1037(1998).[10] J. Smith, C. Chiara, D. Fossan, D. LaFosse, G. Lane,J. Sears, K. Starosta, M. Devlin, F. Lerma, D. Sarantites,S. Freeman, M. Leddy, J. Durell, A. Boston, E. Paul,A. Semple, I. Lee, A. Macchiavelli, and P. Heenen, Phys.Lett. B , 13 (2001).[11] L. Capponi, J. F. Smith, P. Ruotsalainen, C. Scho-ley, P. Rahkila, K. Auranen, L. Bianco, A. J. Boston,H. C. Boston, D. M. Cullen, X. Derkx, M. C. Drum-mond, T. Grahn, P. T. Greenlees, L. Grocutt, B. Ha-dinia, U. Jakobsson, D. T. Joss, R. Julin, S. Juutinen,M. Labiche, M. Leino, K. G. Leach, C. McPeake, K. F.Mulholland, P. Nieminen, D. O’Donnell, E. S. Paul,P. Peura, M. Sandzelius, J. Sar´en, B. Saygi, J. Sorri,S. Stolze, A. Thornthwaite, M. J. Taylor, and J. Uusi-talo, Phys. Rev. C , 024314 (2016).[12] T. Rzaca-Urban, W. Urban, A. Kaczor, J. L. Durell,M. J. Leddy, M. A. Jones, W. R. Phillips, A. G. Smith,B. J. Varley, I. Ahmad, L. R. Morss, M. Bentaleb,E. Lubkiewicz, and N. Schulz, Eur. Phys. J. A , 165(2000).[13] E. T. Gregor, M. Scheck, R. Chapman, L. P. Gaffney,J. Keatings, K. R. Mashtakov, D. O’Donnell, J. F. Smith,P. Spagnoletti, M. Th¨urauf, V. Werner, and C. Wiseman,Eur. Phys. J. A , 50 (2017).[14] W. Nazarewicz, P. Olanders, I. Ragnarsson, J. Dudek,G. A. Leander, P. M¨oller, and E. Ruchowsa, Nucl. Phys.A , 269 (1984).[15] S. Y. Xia, H. Tao, Y. Lu, Z. P. Li, T. Nikˇsi´c, and D. Vrete-nar, Phys. Rev. C , 054303 (2017).[16] S. E. Agbemava, A. V. Afanasjev, and P. Ring, Phys.Rev. C , 044304 (2016).[17] S. E. Agbemava and A. V. Afanasjev, Phys. Rev. C ,024301 (2017).[18] Y. Cao, S. E. Agbemava, A. V. Afanasjev,W. Nazarewicz, and E. Olsen, Phys. Rev. C ,024311 (2020).[19] J. Engel and F. Iachello, Nucl. Phys. A , 61 (1987).[20] N. V. Zamfir and D. Kusnezov, Phys. Rev. C , 054306(2001).[21] N. V. Zamfir and D. Kusnezov, Phys. Rev. C , 014305(2003).[22] K. Nomura, D. Vretenar, and B.-N. Lu, Phys. Rev. C ,021303 (2013).[23] K. Nomura, D. Vretenar, T. Nikˇsi´c, and B.-N. Lu, Phys.Rev. C , 024312 (2014).[24] P. G. Bizzeti and A. M. Bizzeti-Sona, Phys. Rev. C ,011305 (2013).[25] D. Bonatsos, A. Martinou, N. Minkov, S. Karampagia,and D. Petrellis, Phys. Rev. C , 054315 (2015).[26] T. M. Shneidman, G. G. Adamian, N. V. Antonenko,R. V. Jolos, and W. Scheid, Phys. Lett. B , 322 (2002).[27] T. M. Shneidman, G. G. Adamian, N. V. Antonenko,R. V. Jolos, and W. Scheid, Phys. Rev. C , 014313(2003).[28] J. Skalski, Phys. Lett. B , 6 (1990).[29] T. H. R. Skyrme, Nucl. Phys. , 615 (1958).[30] P.-H. Heenen, J. Skalski, P. Bonche, and H. Flocard,Phys. Rev. C , 802 (1994).[31] M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod.Phys. , 121 (2003).[32] J. Erler, P. Kl¨upfel, and P.-G. Reinhard, J. Phys. G: Nucl.Part. Phys. , 033101 (2011).[33] D. Vretenar, A. V. Afanasjev, G. A. Lalazissis, andP. Ring, Phys. Rep. , 101 (2005).[34] T. Nikˇsi´c, D. Vretenar, and P. Ring, Prog. Part. Nucl.Phys. , 519 (2011).[35] L. M. Robledo, T. R. Rodr´ıguez, and R. R. Rodr´ıguez-Guzm´an, J. Phys. G: Nucl. Part. Phys. , 013001(2019).[36] P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer-Verlag, Berlin, 1980).[37] L. M. Robledo and P. A. Butler, Phys. Rev. C , 051302(2013).[38] J. M. Yao, E. F. Zhou, and Z. P. Li, Phys. Rev. C , 041304 (2015).[39] R. N. Bernard, L. M. Robledo, and T. R. Rodr´ıguez,Phys. Rev. C , 061302 (2016).[40] R. Rodr´ıguez-Guzm´an, Y. M. Humadi, and L. M. Rob-ledo, J. Phys. G: Nucl. Part. Phys. , 015103 (2020).[41] Z. P. Li, B. Y. Song, J. M. Yao, D. Vretenar, and J. Meng,Phys. Lett. B , 866 (2013).[42] Z. Xu and Z.-P. Li, Chin. Phys. C , 124107 (2017).[43] T. Nikˇsi´c, D. Vretenar, and P. Ring, Phys. Rev. C ,034318 (2008).[44] Y. Tian, Z. Y. Ma, and P. Ring, Phys. Lett. B , 44(2009).[45] P. W. Zhao, Z. P. Li, J. M. Yao, and J. Meng, Phys. Rev.C , 034316 (2010).[48] J. Srebrny, T. Czosnyka, C. Droste, S. Rohozi´nski,L. Pr´ochniak, K. Zaj¸ac, K. Pomorski, D. Cline, C. Wu,A. B¨acklin, L. Hasselgren, R. Diamond, D. Habs,H. K¨orner, F. Stephens, C. Baktash, and R. Kostecki,Nucl. Phys. A , 25 (2006).[49] K. Nomura, N. Shimizu, and T. Otsuka, Phys. Rev. Lett.101