Higher-rank discrete symmetries in the IBM. II Octahedral shapes: Dynamical symmetries
HHigher-rank discrete symmetries in the IBM.II Octahedral shapes: Dynamical symmetries
A. Bouldjedri a , S. Zerguine a and P. Van Isacker b , a Department of Physics, PRIMALAB Laboratory, Batna 1 UniversityRoute de Biskra, 05000 Batna, Algeria b Grand Acc´el´erateur National d’Ions Lourds, CEA/DRF-CNRS/IN2P3Bvd Henri Becquerel, F-14076 Caen, France
Abstract
The symmetries of the sdg -IBM, the interacting boson model with s , d and g bosons,are studied as regards the occurrence of shapes with octahedral symmetry. It isshown that no sdg -IBM Hamiltonian with a dynamical symmetry displays in itsclassical limit an isolated minimum with octahedral shape. However, a degenerateminimum that includes a shape with octahedral symmetry can be obtained from aHamiltonian that is transitional between two limits, U g (9) ⊗ U d (5) and SO sg (10) ⊗ U d (5), and the conditions for its existence are derived. An isolated minimum withoctahedral shape, either an octahedron or a cube, may arise through a modificationof two-body interactions between the g bosons. Comments on the observationalconsequences of this construction are made. Key words: discrete octahedral symmetry, interacting boson model, g bosons PACS:
This paper is the second in the series initiated with Ref. [1], henceforth referredto as I. The overall purpose of this series is the study of nuclear shapes with ahigher-rank discrete symmetry, in particular of the tetrahedral or octahedraltype, in the framework of a variety of interacting boson models. For the generalcontext of this study we refer the reader to the introduction in I, of which onlythe essential points and references are mentioned here.The paper by Li and Dudek [2] pointed out the possibility of intrinsic nuclearshapes with a higher-rank discrete symmetry. Subsequent publications by the
Preprint submitted to Elsevier Science 1 September 2020 a r X i v : . [ nu c l - t h ] A ug trasbourg group and their collaborators [3,4,5,6] showed the possible occur-rence of tetrahedral, octahedral and icosahedral symmetries through combina-tions of deformations of specific multipolarity. In parallel with these theoreticaldevelopments, a search was initiated for experimental manifestations of suchsymmetries in nuclei [7,8,9]. In addition to these experimental searches cited inI, a more recent study of this type appeared in 2018 [10], formulating criteriafor the identification of tetrahedral and/or octahedral symmetries in nuclei.From the theory side most work related to higher-rank discrete symmetrieshas been carried out in the context of mean-field models. The aim of thisseries of papers is to address the question of the possible occurrence of higher-rank discrete symmetries from a different theoretical perspective. Our studyis carried out in the context of algebraic collective models inspired by theinteracting boson model (IBM) of Arima and Iachello [11,12,13], which pro-poses a description of quadrupole collective nuclear states in terms of s and d bosons with angular momentum (cid:96) = 0 and (cid:96) = 2. The application of this ideato collective states of different nature, notably of octupole and hexadecapolecharacter, requires the introduction of other bosons [14], in particular f and g bosons with angular momentum (cid:96) = 3 and (cid:96) = 4. Since shapes with tetrahe-dral symmetry arise in lowest order through a particular octupole deformationand those with octahedral symmetry emanate from a combination of hexade-capole deformations, the study of such shapes in an algebraic context requiresthe introduction of f and g bosons, respectively.We initiated this program in I with the study of octahedral shapes in the sdg -IBM. We considered the most general rotationally invariant Hamiltonianwith up to two-body interactions between the bosons and derived the condi-tions for this Hamiltonian to have in its classical limit a minimum with anintrinsic shape with octahedral symmetry. Owing to the general nature of theanalysis in I, only qualitative conclusions could be drawn with regard to the(non-)existence of such minima in the sdg -IBM. In the present paper we arriveat more concrete conclusions by considering a subset of all possible Hamilto-nians of the sdg -IBM, namely those that have a dynamical symmetry and areanalytically solvable. Although none of the symmetry Hamiltonians leads toa stable shape with octahedral symmetry, a subset can be used to proposea generalization with additional interactions between the g bosons that drivethe system toward such a shape.The paper is structured as follows. To avoid repeatedly referring to equationsin I, we list in Section 2 formulas from I that are needed in this paper. InSection 3 we focus on two dynamical symmetries of the sdg -IBM of particularrelevance in our quest for octahedral shapes and in Section 4 the classicallimit is derived for the Hamiltonian that is transitional between these twolimits. The central results of this paper are presented in Section 5, with acatastrophe analysis of the energy surface associated with the transitional2ymmetry Hamiltonian, and suitable generalizations thereof, to uncover theexistence of minima at stable shapes with octahedral symmetry. Finally, inSection 6 the conclusions of the second paper in this series are summarized. sdg -IBM and its classical limit A boson-number-conserving, rotationally invariant Hamiltonian of the sdg -IBMwith up to two-body interactions is of the formˆ H = (cid:15) s ˆ n s + (cid:15) d ˆ n d + (cid:15) g ˆ n g + (cid:88) (cid:96) ≤ (cid:96) ,(cid:96) (cid:48) ≤ (cid:96) (cid:48) ,L ( − ) L v L(cid:96) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48) (cid:113) (1 + δ (cid:96) (cid:96) )(1 + δ (cid:96) (cid:48) (cid:96) (cid:48) ) [ b † (cid:96) × b † (cid:96) ] ( L ) · [˜ b (cid:96) (cid:48) × ˜ b (cid:96) (cid:48) ] ( L ) , (1)where ˆ n (cid:96) is the number operator for the (cid:96) boson, (cid:15) (cid:96) is the energy of the (cid:96) bosonand v L(cid:96) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48) is a boson–boson interaction matrix element.A geometric understanding of this quantum-mechanical Hamiltonian is ob-tained by considering its expectation value in a coherent state. As discussedin I, a study of shapes with octahedral symmetry in sdg -IBM requires a co-herent state of the form | N ; β , β , γ , γ , δ (cid:105) = (cid:115) N !(1 + β + β ) N Γ( β , β , γ , γ , δ ) N | o (cid:105) , (2)withΓ( β , β , γ , γ , δ ) = s † + β (cid:104) cos γ d † + (cid:113) sin γ ( d †− + d † +2 ) (cid:105) (3)+ β (cid:104)(cid:16)(cid:113) cos δ + (cid:113) sin δ cos γ (cid:17) g † − (cid:113) sin δ sin γ ( g †− + g † +2 )+ (cid:16)(cid:113) cos δ − (cid:113) sin δ cos γ (cid:17) ( g †− + g † +4 ) (cid:105) , in terms of the deformation parameters β , β , γ , γ and δ , following theconvention of Rohozi´nski and Sobiczewski [15]. A shape with octahedral sym-metry is obtained for β = 0, β (cid:54) = 0 and δ = 0 (octahedron) or δ = π (cube) with arbitrary γ . Another solution with octahedral symmetry existsfor γ = 0 and δ = arccos(1 / δ = 0. The parameterization introducedin Ref. [15] therefore does not define a unique intrinsic state.3he expectation value of the Hamiltonian (1) in the coherent state (2) (or theclassical limit of ˆ H ) leads to the energy surface (cid:104) ˆ H (cid:105) ≡ E ( β , β , γ , γ , δ )= N ( N − β + β ) (cid:88) kl β k β l c (cid:48) kl + (cid:88) ij c ijkl cos( iγ + jγ ) φ ijkl ( δ ) , (4)where φ ijkl ( δ ) are trigonometric functions defined in I. The coefficients c (cid:48) kl areknown in terms of the scaled single-boson energies (cid:15) (cid:48) (cid:96) ≡ (cid:15) (cid:96) / ( N −
1) and theinteraction matrix elements v L(cid:96) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48) , c (cid:48) = v ssss + (cid:15) (cid:48) s ,c (cid:48) = (cid:113) v ssdd + v sdsd + (cid:15) (cid:48) s + (cid:15) (cid:48) d ,c (cid:48) = v ssgg + v sgsg + (cid:15) (cid:48) s + (cid:15) (cid:48) g ,c (cid:48) = v dddd + v dddd + v dddd + (cid:15) (cid:48) d ,c (cid:48) = √ v ddgg + √ v ddgg + v dgdg + v dgdg + v dgdg + v dgdg + (cid:15) (cid:48) d + (cid:15) (cid:48) g ,c (cid:48) = v gggg + v gggg + v gggg + v gggg + v gggg + (cid:15) (cid:48) g . (5)Only a single coefficient c ijkl is needed in the subsequent analysis, viz. c = − v gggg + v gggg + v gggg − v gggg , (6)introducing a δ dependence in the energy surface (4) since φ ( δ ) = 2 cos 2 δ + 17 cos 4 δ . (7)It is assumed in the following that the boson Hamiltonian is Hermitian andtherefore that v L(cid:96) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48) = v L(cid:96) (cid:48) (cid:96) (cid:48) (cid:96) (cid:96) . With this assumption the expressions forthe coefficients c kl and c ijkl , given in Eqs. (20) and (21) of I, are valid with v L(cid:96) (cid:96) · (cid:96) (cid:48) (cid:96) (cid:48) = v L(cid:96) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48) . Note that for a general Hamiltonian one has v L(cid:96) (cid:96) · (cid:96) (cid:48) (cid:96) (cid:48) =( v L(cid:96) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48) + v L(cid:96) (cid:48) (cid:96) (cid:48) (cid:96) (cid:96) ) /
2, which corrects by a factor 2 the expression given in I. g (9) and SO sg (10) limits Since the Hamiltonian of the sdg -IBM conserves the total number of bosons,it can be written in terms of the (1 + 5 + 9) = 225 operators b † (cid:96)m b (cid:96) (cid:48) m (cid:48) . The 225operators generate the Lie algebra U(15) with a substructure that determinesthe dynamical symmetries of the sdg -IBM.4 comprehensive list of the dynamical symmetries of the sdg -IBM is given byDe Meyer et al. [16], and their group-theoretical properties are extensively dis-cussed by Kota et al. [17]. It is found that the model has seven major dynami-cal symmetries, four of strong coupling, SU(3), SU(6), SU(5) and SO(15), andthree of weak coupling, U s (1) ⊗ U dg (14), U sd (6) ⊗ U g (9) and U sg (10) ⊗ U d (5).The question treated in this paper is whether any of the dynamical symme-tries of the sdg -IBM corresponds to a shape with octahedral symmetry. In thissection a choice of the limits that possibly have such property is made on thebasis of intuitive arguments. Subsequently, in Sections 4 and 5, the conditionsfor the existence of a shape with octahedral symmetry are derived rigorouslyon the basis of the results obtained in I.A minimum with octahedral shape requires mixing of s and g bosons, so asto induce hexadecapole deformation, and no or weak mixing of these withthe d boson to ensure zero quadrupole deformation. These conditions rule outall limits where s , d and g bosons are strongly mixed on an equal footing,that is, they discard the SU(3), SU(6), SU(5) and SO(15) limits [18]. A strictdecoupling of the s and g from the d bosons is obtained by the reductionU(15) ⊃ U d (5) ⊗ U sg (10) ↓ ↓ ↓ [ N ] n d n sg , (8)and, furthermore, zero quadrupole deformation follows from a U(5) classifica-tion for the d bosons,U d (5) ⊃ SO d (5) ⊃ SO d (3) ↓ ↓ ↓ n d υ d ν d L d , (9)where underneath each algebra the associated quantum number is given.Specifically, N is the total number of bosons while n (cid:96) is the number of (cid:96) bosons and n (cid:96)(cid:96) (cid:48) is the number of (cid:96) plus (cid:96) (cid:48) bosons. The seniority label asso-ciated with an (cid:96) boson is denoted as υ (cid:96) and corresponds to the number of (cid:96) bosons not in pairs coupled to angular momentum zero. Additional (or miss-ing) labels, not associated with any algebra, are indicated with ν (cid:96) . Finally, theangular momentum generated by the (cid:96) bosons is denoted as L (cid:96) .The U sg (10) algebra in Eq. (8) allows two classifications of interest. The firstis obtained by eliminating from the generators of U sg (10) those that involve5he s boson, leading to(I) U sg (10) ⊃ U g (9) ⊃ SO g (9) ⊃ SO g (3) ↓ ↓ ↓ ↓ n sg n g υ g ν g L g . (10)In this limit, which for brevity shall be referred to as U g (9) or limit I, theseparate boson numbers n s and n g are conserved. The resulting spectrum isvibrational-like with a spherical ground state and excited states that corre-spond to oscillations in the hexadecapole degree of freedom.The second classification of U sg (10) is specified by the following chain of nestedalgebras:(II) U sg (10) ⊃ SO sg (10) ⊃ SO g (9) ⊃ SO g (3) ↓ ↓ ↓ ↓ n sg υ sg υ g ν g L g , (11)which for brevity shall be referred to as SO sg (10) or limit II. The definingfeature of the reduction (11) is the appearance of the algebra SO sg (10) andits label υ sg , associated with the pairing of s and g bosons. As is shown inSection 5, the ground state in this limit acquires a permanent hexadecapoledeformation and the limit is therefore of interest in our quest for octahedralshapes. On its own, however, limit II implies degenerate energies of the s and g boson, and as such it is not realistic. It is therefore necessary to study acombination of the two limits I and II.In Sections 4 and 5 we investigate to what extent non-degenerate energies canbe taken for the s and g boson that still lead to a hexadecapole-deformedminimum and whether that minimum can have octahedral symmetry. In theremainder of this section we list some of the properties of limits I and II thatare necessary to carry out this analysis.The classification of limits I and II can be summarized with the algebraic6atticeU(15) ⊃ U d (5) ⊗ U sg (10) | (cid:46) (cid:38)↓ U g (9) SO sg (10)SO d (5) (cid:38) (cid:46)| SO g (9) ↓ ↓ SO d (3) SO g (3) (cid:38) (cid:46) SO(3) , (12)where SO(3) is associated with the total angular momentum L , which resultsfrom the coupling of L d and L g . The generators of the different algebras inthe lattice (12) are as follows:U d (5) : { [ d † × ˜ d ] ( λ ) µ , λ = 0 , . . . , } , SO d (5) : { [ d † × ˜ d ] ( λ ) µ , λ = 1 , } , SO d (3) : { ˆ L d,µ ≡ √ d † × ˜ d ] (1) µ } , U sg (10) : { [ s † × ˜ s ] (0)0 , [ s † × ˜ g ] (4) µ , [ g † × ˜ s ] (4) µ , [ g † × ˜ g ] ( λ ) µ , λ = 0 , . . . , } , U g (9) : { [ g † × ˜ g ] ( λ ) µ , λ = 0 , . . . , } , SO sg (10) : { [ s † × ˜ g + g † × ˜ s ] (4) µ , [ g † × ˜ g ] ( λ ) µ , λ = 1 , , , } , SO g (9) : { [ g † × ˜ g ] ( λ ) µ , λ = 1 , , , } , SO g (3) : { ˆ L g,µ ≡ √ g † × ˜ g ] (1) µ } , SO(3) : { ˆ L µ ≡ ˆ L d,µ + ˆ L g,µ } . (13)The linear and quadratic Casimir operators of the algebras appearing in thelattice (12) can be expressed as follows in terms of the generators (13):ˆ C [U(15)] = ˆ N = ˆ n s + ˆ n d + ˆ n g , ˆ C [U(15)] = ˆ N ( ˆ N + 14) , ˆ C [U d (5)] = ˆ n d , ˆ C [U d (5)] = ˆ n d (ˆ n d + 4) , ˆ C [SO d (5)] = 2 (cid:88) λ odd [ d † × ˜ d ] ( λ ) · [ d † × ˜ d ] ( λ ) , C [SO d (3)] = ˆ L d · ˆ L d , ˆ C [U sg (10)] = ˆ n s + ˆ n g , ˆ C [U sg (10)] = (ˆ n s + ˆ n g )(ˆ n s + ˆ n g + 9) , ˆ C [U g (9)] = ˆ n g , ˆ C [U g (9)] = ˆ n g (ˆ n g + 8) , ˆ C [SO sg (10)] = [ s † × ˜ g + g † × ˜ s ] (4) · [ s † × ˜ g + g † × ˜ s ] (4) + ˆ C [SO g (9)] , ˆ C [SO g (9)] = 2 (cid:88) λ odd [ g † × ˜ g ] ( λ ) · [ g † × ˜ g ] ( λ ) , ˆ C [SO g (3)] = ˆ L g · ˆ L g , ˆ C [SO(3)] = ˆ L · ˆ L. (14)The expressions for the quadratic Casimir operators of unitary algebras are notgeneral but are valid in a symmetric irreducible representation. A rotationallyinvariant Hamiltonian with up to two-body interactions can be written interms of the Casimir operators (14):ˆ H sym = (cid:15) d ˆ n d + a d ˆ C [U d (5)] + b d ˆ C [SO d (5)] + c d ˆ C [SO d (3)]+ (cid:15) s ˆ n s + (cid:15) g ˆ n g + a sg ˆ C [U sg (10)] + a g ˆ C [U g (9)] + b sg ˆ P † sg ˆ P sg + b g ˆ C [SO g (9)] + c g ˆ C [SO g (3)] + c ˆ C [SO(3)] , (15)where (cid:15) (cid:96) , a (cid:96) , a (cid:96)(cid:96) (cid:48) , b (cid:96) , b (cid:96)(cid:96) (cid:48) and c (cid:96) are parameters. The quadratic Casimir operatorof U(15) is omitted for simplicity since it gives a constant contribution for afixed boson number N = n s + n d + n g . Furthermore, it is convenient to define,instead of the quadratic Casimir operator ˆ C [SO sg (10)], the combinationˆ P † sg ˆ P sg = ˆ C [U sg (10)] − ˆ C [U sg (10)] − ˆ C [SO sg (10)] , (16)where ˆ P † sg ≡ s † s † − g † · g † is the pairing operator for s and g bosons. The sym-metry Hamiltonian (15) is less general than Eq. (1) but it is the most generalone that can be written in terms of invariant operators of the lattice (12) andas such it is intermediate between the limits I and II.The U g (9) limit occurs for b sg = 0, leading to the eigenvalues E I = (cid:15) d n d + a d n d ( n d + 4) + b d υ d ( υ d + 3) + c d L d ( L d + 1)+ (cid:15) s n s + (cid:15) g n g + a sg n sg ( n sg + 9) + a g n g ( n g + 8) ++ b g υ g ( υ g + 7) + c g L g ( L g + 1) + c L ( L + 1) . (17)The SO sg (10) limit is attained for (cid:15) s = (cid:15) g ≡ (cid:15) sg and a g = 0, in which case theHamiltonian’s eigenstates have the eigenvalues8 � ( � ) � � = � � � = � � � = �� � � = � � + � + � + � + � + � + � + � + � + � + � + � + � + � + � + � + ���� � � � � �� ( � � � ) �� �� ( �� ) υ �� = � υ �� = � - � υ �� = � - � � + � + � + � + � + � + � + � + � + � + � + � + � + � + � + � + � + � + ���� � � � � �� ( � � � ) Fig. 1. Energy spectra in the U g (9) and SO sg (10) limits of the sdg -IBM for N = 5bosons. For the U g (9) spectrum the non-zero parameters in the Hamiltonian (15)are (cid:15) d − (cid:15) s = 800, (cid:15) g − (cid:15) s = 1000, b d = 40, c d = 10, b g = 25 and c g = c = 5 keV.For the SO sg (10) spectrum the non-zero parameters are (cid:15) d − (cid:15) s = 800, (cid:15) g − (cid:15) s = 0, b d = 40, c d = 10, b sg = 60, b g = 50 and c g = c = 5 keV. E II = (cid:15) d n d + a d n d ( n d + 4) + b d υ d ( υ d + 3) + c d L d ( L d + 1)+ (cid:15) sg n sg + a sg n sg ( n sg + 9) + b sg [ n sg ( n sg + 8) − υ sg ( υ sg + 8)]+ b g υ g ( υ g + 7) + c g L g ( L g + 1) + c L ( L + 1) . (18)The eigenspectra are then determined with the help of the necessary branchingrules. The reduction U(15) ⊃ U d (5) ⊗ U sg (10) implies the relation N = n d + n sg or the branching rule[ N ] (cid:55)→ ( n d , n sg ) = (0 , N ) , (1 , N − , . . . , ( N, . (19)The branching rules for the classification (9) are known from the U(5) limitof the sd -IBM [11] and those for the classifications (10) and (11) can be foundin Ref. [17].Typical energy spectra in the U g (9) and SO sg (10) limits are shown in Fig. 1.The U g (9) spectrum displays quadrupole- and hexadecapole-phonon multi-plets characterized by a fixed number of d and g bosons. The multiplets arefurther structured by a seniority quantum number: for example, the n d = 2multiplet has υ d = 2 except for the 0 + level, which has υ d = 0, and similarlyfor the n g = 2 multiplet and the υ g seniority. Also combined quadrupole–hexadecapole multiplets occur in the spectrum. The SO sg (10) spectrum con-tains sets of levels with υ sg = N, N − , . . . (for n d = 0) with υ sg = N − , N − , . . . (for n d = 1) etc., and υ sg = N levels are lowest in energy duethe repulsive sg -pairing. Multiplets characterized by the seniority quantumnumber υ g = 0 , , . . . occur within each SO sg (10) multiplet. Note that in thislimit, unless b sg is small, the first-excited state has J π = 4 + . This is a conse-quence of the unrealistic condition that the energies of the s and g bosons aredegenerate, (cid:15) s = (cid:15) g . 9 Classical limit of the symmetry Hamiltonian
The classical limit of the symmetry Hamiltonian (15) can be obtained withthe general procedure outlined in I. First, a conversion to the standard rep-resentation (1) is carried out. A quadratic Casimir operator ˆ C ( G ) is givengenerically as an expansion over the generators,ˆ C ( G ) = (cid:88) λr a λr (cid:88) (cid:96) (cid:96) α λr(cid:96) (cid:96) (cid:16) b † (cid:96) × ˜ b (cid:96) (cid:17) ( λ ) · (cid:88) (cid:96) (cid:48) (cid:96) (cid:48) α λr(cid:96) (cid:48) (cid:96) (cid:48) (cid:16) b † (cid:96) (cid:48) × ˜ b (cid:96) (cid:48) (cid:17) ( λ ) , (20)with coefficients α λr(cid:96) (cid:96) and a λr that are specific to each algebra G , as given inEq. (14). The overall sum in Eq. (20) is over the multipolarity λ of the gener-ators and over an additional index r to distinguish different generators withthe same λ . With use of the expansion (20) one finds the following expressionfor the matrix element of ˆ C ( G ) between two-boson states [19]: (cid:104) (cid:96) (cid:96) ; L | ˆ C ( G ) | (cid:96) (cid:48) (cid:96) (cid:48) ; L (cid:105) = [ f (cid:96) ( G ) + f (cid:96) ( G )] δ (cid:96) (cid:96) δ (cid:96) (cid:96) + 2( − ) (cid:96) + (cid:96) (cid:113) (1 + δ (cid:96) (cid:96) )(1 + δ (cid:96) (cid:96) ) (cid:88) λr a λr (2 λ + 1) × (cid:34) α λr(cid:96) (cid:96) α λr(cid:96) (cid:96) (cid:40) (cid:96) (cid:96) λ(cid:96) (cid:96) L (cid:41) + ( − ) L α λr(cid:96) (cid:96) α λr(cid:96) (cid:96) (cid:40) (cid:96) (cid:96) λ(cid:96) (cid:96) L (cid:41)(cid:35) , (21)where the symbol between curly brackets is a Racah coefficient [20]. The quan-tity f (cid:96) ( G ) is the expectation value of the operator ˆ C ( G ) between single-bosonstates, f (cid:96) ( G ) ≡ (cid:104) (cid:96) | ˆ C ( G ) | (cid:96) (cid:105) = (cid:88) λr λ + 12 (cid:96) + 1 a λr (cid:88) (cid:96) (cid:48) ( − ) (cid:96) + (cid:96) (cid:48) α λr(cid:96)(cid:96) (cid:48) α λr(cid:96) (cid:48) (cid:96) . (22)The Hamiltonian (15) can, with use of Eqs. (21) and (22), be converted intoits standard representation on which the classical-limit expression (4) can beapplied. The procedure results in the energy surface E ( β , β ) = N ( N − β + β ) (cid:88) kl c (cid:48) kl β k β l , (23)where the non-zero coefficients c (cid:48) kl are c (cid:48) = a sg + b sg + Γ s , c (cid:48) = Γ s + Γ d , c (cid:48) = 2 a sg − b sg + Γ s + Γ g ,c (cid:48) = a d + Γ d , c (cid:48) = Γ d + Γ g , c (cid:48) = a sg + b sg + a g + Γ g , (24)10n terms of the combinationsΓ s ≡ N − (cid:15) s + 10 a sg ) , Γ d ≡ N − (cid:15) d + 5 a d + 4 b d + 6 c d + 6 c ) , Γ g ≡ N − (cid:15) g + 10 a sg + 9 a g + 8 b g + 20 c g + 20 c ) . (25)All coefficients c ijkl of Eq. (4) vanish identically in the classical limit of thesymmetry Hamiltonian (15). From the outset it should be clear that energy surface (23) cannot have anisolated minimum with octahedral shape since it is independent of γ , γ and δ . What can still happen, however, is the occurrence of a minimum with zeroquadrupole and non-zero hexadecapole deformation ( β ∗ = 0 and β ∗ (cid:54) = 0),which, given the instability in γ and δ , includes a shape with octahedralsymmetry. Therefore, the goal of this section is to establish the conditions onthe parameters in the symmetry Hamiltonian (15) such that its classical limitdisplays a minimum with β ∗ = 0 and β ∗ (cid:54) = 0, and, subsequently, to identifythe interactions in the general Hamiltonian (1) that generate a dependenceon δ , enabling the formation of an isolated minimum with octahedral shape.This problem can be investigated with the procedure outlined in I.According to the analysis of the previous section, the classical energy (23) is atwo-variable function E ( β , β ). The conditions for this energy surface to havean extremum are ∂E∂β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ = ∂E∂β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ = 0 , (26)where p ∗ ≡ ( β ∗ , β ∗ ) is a short-hand notation for an arbitrary critical point.Furthermore, a critical point at an extremum with β ∗ = 0 and β ∗ (cid:54) = 0 shallbe denoted as h ∗ . The condition (26) in β is identically satisfied for p ∗ = h ∗ and does not lead to any constraints on the coefficients c (cid:48) kl . The condition in β leads to a cubic equation with the solutions β ∗ = 0 , β ∗ = ± (cid:115) c (cid:48) − c (cid:48) c (cid:48) − c (cid:48) . (27)11nly the last solution corresponds to an extremum h ∗ and implies the followingcondition on the ratio of coefficients:2 c (cid:48) − c (cid:48) c (cid:48) − c (cid:48) > . (28)While the condition (28) is necessary and sufficient to have an extremum at β ∗ = 0 and β ∗ (cid:54) = 0, a minimum at these values implies further constraints.They are obtained by requiring that the eigenvalues of the stability matrix [ i.e. ,the partial derivatives of E ( β , β ) of second order] are all positive. Since theoff-diagonal element of the stability matrix vanishes for the energy surface (23),the existence of a minimum follows from the uncoupled conditions ∂ E∂β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ∗ > , ∂ E∂β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ∗ > , (29)or, in terms of the coefficients c (cid:48) kl in Eq. (23),(2 c (cid:48) − c (cid:48) )[2 c (cid:48) ( c (cid:48) − c (cid:48) ) + c (cid:48) ( c (cid:48) − c (cid:48) − c (cid:48) ) + 2 c (cid:48) c (cid:48) ]( c (cid:48) − c (cid:48) + c (cid:48) ) > , (2 c (cid:48) − c (cid:48) )(2 c (cid:48) − c (cid:48) ) ( c (cid:48) − c (cid:48) + c (cid:48) ) > . (30)If we write Eq. (28) as A/B >
0, the second inequality in Eq. (30) becomes AB / ( A + B ) > A and B should be positive, A ≡ c (cid:48) − c (cid:48) > B ≡ c (cid:48) − c (cid:48) >
0, leading to the constraint − b sg − a g < Γ g − Γ s < b sg . (31)The first inequality in Eq. (30) can be reduced to2 b sg (2Γ d − Γ s − Γ g − a sg − a g ) + a g (Γ d − Γ s − a sg ) > . (32)The conditions (31) and (32) are necessary and sufficient for the energy sur-face E ( β , β ) to have a minimum at zero quadrupole and non-zero hexade-capole deformation. To obtain an intuitive understanding of them, we notethat Γ g − Γ s , for a reasonable choice of parameters, is positive. The upperpart of the inequality (31) therefore expresses the need for b sg to be positiveand sufficiently large, corresponding to a repulsive sg -pairing interaction thatputs the configuration with maximal sg seniority υ sg = n sg at lowest energy.For b sg > a g >
0, the lower part of the inequality (31) is automaticallysatisfied. The condition (32) is easier to appreciate if it is assumed that thecoefficients in front of the quadratic Casimir operators of the unitary algebras12 d (5), U g (9) and U sg (10) vanish, a d = a sg = a g = 0. This assumption is justi-fied if anharmonicities are neglected in the various limits. Given that b sg > d − Γ s − Γ g > . (33)In terms of the original parameters in the symmetry Hamiltonian (15) (as-suming a d = a sg = a g = 0) the conditions to have a minimum at β ∗ = 0 and β ∗ (cid:54) = 0 can be summarized as − N − b sg < (cid:15) g − (cid:15) s + 8 b g + 20( c g + c ) < N − b sg , (cid:15) d − (cid:15) s − (cid:15) g + 8( b d − b g ) + 4(3 c d − c g − c ) > . (34)We now ask the question whether two-body interactions can be added to thesymmetry Hamiltonian (15), which lift the ( γ , δ ) instability and create aminimum at δ = 0, δ = arccos(1 /
6) or δ = π . To achieve this goal, we recallthe result from I that for a general Hamiltonian of the sdg -IBM an extremumwith β ∗ = 0 and β ∗ (cid:54) = 0 occurs for β ∗ = ± (cid:115) c (cid:48) − c (cid:48) c (cid:48) − c (cid:48) + 38 c . (35)The term in c introduces a dependence in δ that may lead to an isolated min-imum with octahedral symmetry. Given the expression (6) for c , this argu-ment suggests adding g -boson interactions v Lgggg to the Hamiltonian (15). Theclassical energy (23) then becomes a three-variable function E ( β , β , δ ), forwhich the above catastrophe analysis can be repeated. With these additionaltwo-body interactions the extremum and stability conditions (26) and (29)become2 c (cid:48) − c (cid:48) > , c (cid:48) − c (cid:48) + 38 c > , c < , (36)2 c (cid:48) ( c (cid:48) − c (cid:48) ) + c (cid:48) ( c (cid:48) − c (cid:48) − c (cid:48) ) + 2 c (cid:48) c (cid:48) − c (2 c (cid:48) − c (cid:48) ) > . For a d = a sg = a g = 0, these conditions imply the inequalities − b sg − ˜ v < Γ g − Γ s < b sg , ¯ v ≡ v gggg − v gggg − v gggg + 56 v gggg > , b sg (2Γ d − Γ s − Γ g ) + (Γ d − Γ s − b sg )(˜ v − b sg ) > , (37)where ˜ v is the following linear combination of g -boson interaction matrix ele-13 � ( � )- �� �� ( �� ) � + � + � + � + � + � + � + � + � + � + � + � + � + � + � + � + ���� � � � � �� ( � � � ) Fig. 2. Energy spectrum of a U g (9)–SO sg (10) transitional Hamiltonian of the sdg -IBM for N = 5 bosons. The non-zero parameters of the Hamiltonian (15)are (cid:15) d − (cid:15) s = 1200, (cid:15) g − (cid:15) s = 1500, b d = 40, c d = 10, b sg = 150, b g = 25 and c g = c = 5 keV. ments:˜ v = v gggg + v gggg + v gggg + v gggg . (38)For the Hamiltonian (15) the linear combination ¯ v vanishes identically and thesecond inequality in Eq. (37) is not fulfilled. This expresses the δ independenceof the symmetry Hamiltonian and the fact that its classical limit does notacquire an isolated minimum with octahedral shape. Furthermore, for thesymmetry Hamiltonian one has ˜ v = 2 b sg and the conditions (37) reduce toEq. (34).There are clearly many ways to find matrix elements v Lgggg that satisfy allconditions (37) but one way is particularly simple. Note that the quadrupolematrix element v gggg does not appear in the combination ˜ v . By making thismatrix element more repulsive, the second inequality in Eq. (37) is satisfiedwhile the other two conditions are not modified with respect those in Eq. (34)valid for the symmetry Hamiltonian (15). A possible procedure to constructan sdg -IBM Hamiltonian whose classical energy displays a minimum withoctahedral shape is therefore to add to a hexadecapole-deformed symmetryHamiltonian (15) a repulsive v gggg interaction.Let us illustrate this procedure with an example. The starting point is a U g (9)–SO sg (10) transitional Hamiltonian associated with the lattice (12), giving riseto the spectrum shown in Fig. 2. Note that the choice of the single-bosonenergies (cid:15) (cid:96) for this figure is realistic in the sense that the g -boson energyis higher than that of the d boson. The sg -pairing strength b sg , of whichlittle is known either empirically or microscopically, is chosen such that ahexadecapole-deformed minimum occurs in the classical limit. Other parame-ters in the Hamiltonian (15) are of lesser importance and are chosen as to liftdegeneracies in the spectrum. Note that with this choice of parameters theresulting spectrum, as shown in Fig. 2, is rather closer to the U g (9) and thanto the SO sg (10) limit. 14 �� ��� ��� β � ������������ β � Fig. 3. The energy surface E ( β , β ) obtained in the classical limit of aU g (9)–SO sg (10) transitional Hamiltonian of the sdg -IBM. Parameters of the Hamil-tonian (15) are given in the caption of Fig. 2. Black corresponds to low energies andthe lines indicate changes by 10 keV. δ � δ � ( � ) ������������ ��� ��� ��� ��� β � β � δ � δ � ( � ) ������������ ��� ��� ��� ��� β � β � Fig. 4. Energy surfaces E ( β , δ ) obtained in the classical limit of two differentHamiltonians of the sdg -IBM for N = 5 bosons. The dependence on β > ≤ δ ≤ π is shown for β ∗ = 0. Black corresponds to low energies and the linesindicate changes by 10 keV. (a) The U g (9)–SO sg (10) transitional Hamiltonian istaken with the parameters given in the caption of Fig. 2. (b) The Hamiltonian of(a) is modified by taking a repulsive interaction v gggg = 500 keV. The parameters quoted in the caption of Fig. 2 satisfy both conditions (34).As a result, the energy surface in the classical limit of the correspondingHamiltonian has a minimum for β ∗ = 0 and β ∗ ≈ .
34, as shown in Fig. 3.According to the preceding discussion, the surface is independent of δ , whichis indeed confirmed by Fig. 4(a). If the v gggg matrix element is modified, adependence in δ is introduced, as illustrated in Fig. 4(b) for the value v gggg =500 keV. It is seen that the energy surface displays three isolated minima thatare exactly degenerate. The three minima all have an octahedral symmetry,corresponding to either an octahedron [ δ ∗ = 0 and δ ∗ = arccos(1 / ≈ . o ]or a cube ( δ ∗ = π ).Although this analysis shows that isolated minima with octahedral symmetrycan be obtained in the classical limit of an sdg -IBM Hamiltonian with rea-sonable parameters, it can be expected that such minima are rather shallow.15 �� - ��� � + � + � + � + � + � + � + � + � + � + � + � + � + � + � + � + ���� � � � � �� ( � � � ) Fig. 5. Energy spectrum of a general Hamiltonian of the sdg -IBM for N = 5 bosons.The same Hamiltonian is taken as in Fig. 2 but one g -boson two-body matrixelement is modified to v gggg = 500 keV. On the left- and right-hand sides are shownthe shapes at the minima in the energy surface obtained in the classical limit ofthis Hamiltonian. They have octahedral symmetry and correspond to either anoctahedron or a cube. Even for the fairly large value of the interaction matrix element in the aboveexample, v gggg = 500 keV, the three minima are separated by a barrier of ∼
20 keV, inducing only very weak observable effects. This point is illustratedwith Fig. 5, which shows the spectrum of the U g (9)–SO sg (10) transitionalHamiltonian with the modified v gggg matrix element. Except for some minutechanges the spectrum is essentially the same as that shown in Fig. 2.One subtle point made clear by the current study is that it is not sufficientto carry out a catastrophe analysis of the generic surface (4) obtained in theclassical limit of the most general sdg -IBM Hamiltonian (1) with up to two-body interactions between the bosons. The coefficients c (cid:48) kl and c ijkl cannot betreated as free parameters but their expressions in terms of the single-bosonsenergies and boson–boson interactions are an essential part of the analysis.To illustrate this point consider the energy surface shown in Fig. 4(b). Theminima at δ ∗ = 0 and δ ∗ ≈ . o correspond to the same intrinsic shape (anoctahedron, shown on the left-hand side of Fig. 5) and, as a consequence, theminima must be exactly degenerate. This behavior is generally valid. There-fore, whatever single-boson energies and boson–boson interactions one adoptsin the Hamiltonian (1), the energy surface in its classical limit must satisfy theconstraint that points on the surface with same intrinsic shape (e.g., δ ∗ = 0and δ ∗ ≈ . o ) are at the same energy. The main conclusion of this paper is that no sdg -IBM Hamiltonian with adynamical symmetry that includes Casimir operators of up to second orderdisplays in its classical limit an isolated minimum with octahedral shape.Nevertheless, a degenerate minimum that includes a shape with octahedral16ymmetry can be obtained from a Hamiltonian transitional between two lim-its. In the limits in question the d boson is decoupled from s and g bosons.Furthermore, limit I, U g (9), has hexadecapole vibrational characteristics whilein limit II, SO sg (10), s - and g -boson states are mixed through an sg -pairinginteraction. A catastrophe analysis of the energy surface obtained in the classi-cal limit of this transitional symmetry Hamiltonian indicates that a minimumwith zero quadrupole and non-zero hexadecapole deformation can be obtainedwith reasonable parameters. However, this minimum is always δ independent,meaning that it ranges from an octahedron to a cube and includes interme-diate shapes without octahedral symmetry. Isolated minima with octahedralsymmetry can be obtained by adding two-body interactions between the g bosons to the transitional symmetry Hamiltonian. The resulting energy sur-face displays in this case minima with octahedral symmetry, with the shapeof either an octahedron or a cube, separated by a barrier with low energyeven for fairly strong interactions between the g bosons. The conclusion ofthis analysis in the context of the sdg -IBM is therefore that it will be difficultto find experimental manifestations of octahedral symmetry in nuclei. Acknowledgements
This work has been carried out in the framework of a CNRS/DEF agreement,project N 13760.
References [1] P. Van Isacker, A. Bouldjedri, S. Zerguine, Nucl. Phys. A 938 (2015) 45.[2] X. Li and J. Dudek, Phys. Rev. C 94 (1994) 1250(R).[3] J. Dudek, A. G´o´zd´z, N. Schunck, M. Mi´skiewicz, Phys. Rev. Lett. 88 (2002)252502.[4] J. Dudek, A. G´o´zd´z, N. Schunck, Acta Phys. Pol. B 34 (2003) 2491.[5] J. Dudek, D. Curien, N. Dubray, J. Dobaczewski, V. Pangon, P. Olbratowski,N. Schunck, Phys. Rev. Lett. 97 (2006) 072501.[6] D. Rouvel, Essai sur les Sym´etries G´eom´etriques et les Transitions de Forme duNoyau de l’Atome, PhD thesis, University of Strasbourg, 2014, unpublished.[7] D. Curien, J. Dudek, K. Mazurek, J. Phys.: Conf. Ser. 205 (2010) 012034.[8] D. Curien, Recherche Exp´erimentale des Sym´etries de Haut-Rang en StructureNucl´eaire, Habilitation `a Diriger des Recherches, University of Strasbourg, 2011,unpublished.
9] M. Jentschel, W. Urban, J. Krempel, D. Tonev, J. Dudek, D. Curien, B. Lauss,G. de Angelis, P. Petkov, Phys. Rev. Lett. 104 (2010) 222502.[10] J. Dudek, D. Curien, I. Dedes, K. Mazurek, S. Tagami, Y.R. Shimizu,T. Bhattacharjee, Phys. Rev. C 97 (2018) 021302(R).[11] A. Arima, F. Iachello, Ann. Phys. (NY) 99 (1976) 253.[12] A. Arima, F. Iachello, Ann. Phys. (NY) 111 (1978) 201.[13] A. Arima, F. Iachello, Ann. Phys. (NY) 123 (1979) 568.[14] F. Iachello, A. Arima, The Interacting Boson Model, Cambridge UniversityPress, Cambridge, 1987.[15] S.G. Rohozi´nski, A. Sobiczewski, Acta Phys. Pol. B 12 (1981) 1001.[16] H. De Meyer, J. Van der Jeugt, G. Vanden Berghe, V.K.B. Kota, J. Phys. A19 (1986) L565.[17] V.K.B. Kota, J. Van der Jeugt, H. De Meyer, G. Vanden Berghe, J. Math. Phys.28 (1987) 1644.[18] A. Bouldjedri, P. Van Isacker, S. Zerguine, J. Phys. G 31 (2005) 1329.[19] Note the correction with respect to Eq. (2) of Ref. [18].[20] I. Talmi,
Simple Models of Complex Nuclei. The Shell Model and the InteractingBoson Model (Harwood, Chur, 1993).(Harwood, Chur, 1993).