Higher-rank discrete symmetries in the IBM. III Tetrahedral shapes
HHigher-rank discrete symmetries in the IBM.III Tetrahedral shapes
P. Van Isacker a , A. Bouldjedri b and S. Zerguine b a Grand Acc´el´erateur National d’Ions Lourds, CEA/DRF–CNRS/IN2P3Bvd Henri Becquerel, F-14076 Caen, France b Department of Physics, PRIMALAB Laboratory, Batna 1 UniversityRoute de Biskra, 05000 Batna, Algeria
Abstract
In the context of the sf -IBM, the interacting boson model with s and f bosons, theconditions are derived for a rotationally invariant and parity-conserving Hamilto-nian with up to two-body interactions to have a minimum with tetrahedral shapein its classical limit. A degenerate minimum that includes a shape with tetrahedralsymmetry can be obtained in the classical limit of a Hamiltonian that is transi-tional between the two limits of the model, U f (7) and SO sf (8). The conditions forthe existence of such a minimum are derived. The system can be driven towards anisolated minimum with tetrahedral shape through a modification of two-body in-teractions between the f bosons. General comments are made on the observationalconsequences of the occurrence of shapes with a higher-rank discrete symmetry inthe context of algebraic models. Key words: discrete tetrahedral symmetry, interacting boson model, f bosons PACS:
This paper is a continuation of Refs. [1,2], henceforth referred to as I and II,as part of a series concerning nuclear shapes with a higher-rank discrete sym-metry in the framework of the interacting boson model (IBM) and its possibleextensions [3]. In I and II we considered the case of hexadecapole deformationgiving rise to shapes with octahedral symmetry and their manifestation in the sdg -IBM. In the present paper we turn our attention to tetrahedral symmetry.Shapes with tetrahedral discrete symmetry occur in lowest order through aparticular kind of octupole deformation, namely Y µ ( θ, φ ) with µ = ±
2, and all
Preprint submitted to Elsevier Science 15 September 2020 a r X i v : . [ nu c l - t h ] S e p ther deformations equal to zero [4,5,6]. Whereas evidence for hexadecapoledeformation in nuclei is circumstantial at best, such is not the case for theoctupole degree of freedom. Octupole excitations in spherical nuclei are welldocumented (see, e.g., the review [7]) and there is even experimental evidencefor nuclei with a permanent octupole deformation [8]. This makes the searchfor nuclear shapes with tetrahedral symmetry all the more compelling.The algebraic description of the octupole degree of freedom requires the intro-duction of an f boson with angular momentum (cid:96) = 3 and negative parity, aswas already suggested in the early papers on the IBM [9,10,11]. In principle,the f boson should be considered in addition to the bosons of the elemen-tary version of the model since for a realistic description of nuclear collectivebehavior the quadrupole degree of freedom, and therefore the d boson, can-not be neglected. Furthermore, an octupole deformation causes a shift in thecenter of mass that must be balanced by a dipole deformation, which ne-cessitates the introduction of a p boson [12]. One concludes therefore thatthe search for tetrahedral deformation should be carried out in the frame-work of the spdf -IBM, the properties of which have been studied in detail inRefs. [13,14,15]. Unfortunately, a catastrophe analysis of this model is a rathercomplicated problem and the following simplification suggests itself based onour experience with the search for octahedral deformation in the context ofthe sdg -IBM. Because quadrupole deformations must vanish for the nucleusto acquire a shape with a higher-rank discrete symmetry, it transpires thatthe d boson is not an essential ingredient in our search, the reason being thatit should not or only weakly couple to the other bosons. In fact, the mostimportant conditions for the realization of a shape with octahedral symmetry,as obtained in the sdg -IBM in I and II, could just as well have been derivedin the context of the sg -IBM. By analogy, we suggest therefore that a searchfor tetrahedral deformation in an algebraic context can be carried out in thesimpler sf -IBM, which is the subject matter of the present paper. It should berecognized however that the absence of a rotational SU(3) limit in the sf -IBMconstitutes a limitation of the present approach.The paper is structured as follows. In Section 2 we recall the parameteriza-tion of octupole shapes and how, within this parameterization, a shape withtetrahedral symmetry can be realized. Section 3 introduces the rotationallyinvariant, parity-conserving Hamiltonian of the sf -IBM with up to two-bodyinteractions, of which the dynamical symmetries are discussed in Section 4 andthe classical limit in Section 5. The main results of this paper are presented inSection 6, where a catastrophe analysis of the classical energy surface is carriedout to unveil the existence of minima at shapes with tetrahedral symmetry.Finally, in Section 7 the conclusions of this work are summarized.2 Octupole and tetrahedral shapes
In case of a pure octupole deformation seven variables α µ are needed to definethe intrinsic shape as well as the orientation of that shape in the laboratoryframe. One is therefore confronted with the problem of the separation of in-trinsic from orientation variables. While this problem has a natural solutionin the case of quadrupole deformation [16,17,18], namely intrinsic axes thatare defined by the mutually perpendicular symmetry planes of the quadrupoleshape, no such solution presents itself in the case of octupole deformation [19].The parameterization of Hamamoto et al. [20] is used in the following and thesurface is written as R o ( θ, φ ) = R (cid:34) a Y ( θ, φ )+ (cid:88) µ =1 a µ Y π µ µ ( θ, φ ) − ı (cid:88) µ =1 b µ Y − π µ µ ( θ, φ ) (cid:35) , (1)with π µ ≡ ( − ) µ and where the combinations Y ± λµ ( θ, φ ) = 1 √ Y λµ ( θ, φ ) ± Y λ − µ ( θ, φ )] , (2)are introduced in terms of the usual spherical harmonics Y λµ ( θ, φ ). The surface R o ( θ, φ ) is determined by the seven (real) variables { a , a µ , b µ , µ = 1 , , } .Hamamoto et al. [20] define the intrinsic shape through the four variables { β , δ , ϑ , ϕ } b = β sin δ ,a = β cos δ sin ϑ cos ϕ , (cid:113) a − (cid:113) a = β cos δ sin ϑ sin ϕ , (cid:113) b + (cid:113) b = β cos δ cos ϑ , (3)while three combinations are set to zero, a = (cid:113) a + (cid:113) a = − (cid:113) b + (cid:113) b = 0 . (4)All possible intrinsic octupole-deformed shapes are covered by the followingthree ranges of parameters:(a) β > , − π < δ < π, tan − √ ≤ ϑ < π, < ϕ ≤ π, β > , if δ = π, (c) β > , ≤ δ < π, ≤ ϕ ≤ π, if ϑ = π, (5)where for range (a) the additional constraint (tan ϑ )(sin ϕ ) ≥ β = 0, and can be realized in lowest order with an octupole deformationwith µ = ± β > δ = π , in which case the nuclear surface (1) reduces to R o ( θ, φ ) R = 1 + ıβ Y − ( θ, φ ) = 1 − (cid:115) π β (sin θ ) cos θ sin 2 φ. (6)A single parameter, β , defines the surface with tetrahedral symmetry. sf interacting boson model In this section the most general rotationally invariant and parity-conserving sf -IBM Hamiltonian with up to two-body interactions is presented. It has thesame formal expression as given in I with the additional constraint that parityis conserved.A Hamiltonian of the sf -IBM conserves the total number of bosons and cantherefore be written in terms of the (1 + 7) = 64 operators b † (cid:96)m b (cid:96) (cid:48) m (cid:48) , where b † (cid:96)m ( b (cid:96)m ) creates (annihilates) a boson with angular momentum (cid:96) and z projection m . A boson-number-conserving Hamiltonian with up to two-body interactionsis of the formˆ H = ˆ H + ˆ H , (7)with a one-body termˆ H = (cid:15) s [ s † × ˜ s ] (0) − (cid:15) f √ f † × ˜ f ] (0) = (cid:15) s s † · ˜ s + (cid:15) f f † · ˜ f = (cid:15) s ˆ n s + (cid:15) f ˆ n f , (8)and a two-body interactionˆ H = (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 ) , (9)4ith ˜ b (cid:96)m ≡ ( − ) (cid:96) − m b (cid:96), − m . The multiplication × refers to coupling in angularmomentum (shown as an upper-index in round brackets), the dot · indicatesa scalar product, b † (cid:96) · ˜ b (cid:96) ≡ (cid:80) m b † (cid:96)m b (cid:96)m , ˆ n (cid:96) is the number operator for the (cid:96) boson and the coefficient (cid:15) (cid:96) is its energy. The coefficients v L(cid:96) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48) are theinteraction matrix elements between normalized two-boson states, v L(cid:96) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48) ≡(cid:104) (cid:96) (cid:96) ; LM L | ˆ H | (cid:96) (cid:48) (cid:96) (cid:48) ; LM L (cid:105) . Conservation of parity implies that this interactionmatrix element vanishes unless ( − ) (cid:96) + (cid:96) = ( − ) (cid:96) (cid:48) + (cid:96) (cid:48) . Also, it will be assumed inthe following that all Hamiltonians are Hermitian so 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) . sf -IBM Although the sf -IBM is a schematic model, it of some interest to study itsdynamical symmetries since these correspond to two possible, basic manifes-tations of octupole collectivity in nuclei.The 64 operators b † (cid:96)m b (cid:96) (cid:48) m (cid:48) with (cid:96), (cid:96) (cid:48) = 0 , sf -IBM.The first limit is obtained by eliminating from the generators of U(8) those thatinvolve the s boson; it is specified by the following chain of nested algebras:(I) U sf (8) ⊃ U f (7) ⊃ SO f (7) ⊃ SO f (3) ↓ ↓ ↓ ↓ [ N ] n f υ f ν f L , (10)where the subscripts ‘ s ’ and/or ‘ f ’ are a reminder of the bosons that make upthe generators of the algebra (see below). Below each algebra the associatedquantum number is given: N is the total number of bosons, n f is the numberof f bosons, υ f is the f -boson seniority (i.e., the number of f bosons not inpairs coupled to angular momentum zero) and L is the angular momentumgenerated by the f bosons. (Since L coincides with the total angular momen-tum, its subscript ‘ f ’ is suppressed.) Additional multiplicity labels, collectivelydenoted as ν f and not associated to an algebra, are needed between SO f (7)and SO f (3). In this limit, which shall be referred to as U f (7) or limit I, theseparate numbers of s and f bosons are conserved, giving rise to a vibrational-like spectrum with a spherical shape of the ground state and oscillations inthe octupole degree of freedom.The second dynamical symmetry corresponds to the following chain of nested5lgebras:(II) U sf (8) ⊃ SO sf (8) ⊃ SO f (7) ⊃ SO f (3) ↓ ↓ ↓ ↓ [ N ] υ sf υ f ν f L . (11)The algebras and quantum numbers are identical to those in the vibrationallimit (10) but for the appearance of SO sf (8) and its associated label υ sf ,resulting from the pairing of s and f bosons. As shown in Section 5, theground state acquires a permanent octupole deformation in this limit, whichshall be referred to as SO sf (8) or limit II.The dynamical symmetries of U sf (8) describe the two basic manifestations ofoctupole collectivity in nuclei: octupole vibrations around a spherical shape(limit I) or a permanent octupole deformation (limit II). The latter limit is ofrelevance in the search for tetrahedral deformation but it has the unrealisticfeature that the energies of the s and f boson are taken to be degenerate. InSections 5 and 6 we investigate to what extent non-degenerate single-bosonenergies can be accommodated while still preserving an octupole-deformedminimum, and whether that minimum can have tetrahedral symmetry.For further reference, we list some of the properties of limits I and II. Theclassification of limits I and II can be summarized with the algebraic latticeU sf (8) (cid:46) (cid:38) U f (7) SO sf (8) (cid:38) (cid:46) SO f (7) ↓ SO f (3) . (12)The generators of the different subalgebras in the lattice (12) areU f (7) : { [ f † × ˜ f ] ( λ ) µ , λ = 0 , . . . , } , SO sf (8) : { [ s † × ˜ f − f † × ˜ s ] (3) µ , [ f † × ˜ f ] ( λ ) µ , λ = 1 , , } , SO f (7) : { [ f † × ˜ f ] ( λ ) µ , λ = 1 , , } , G : { [ f † × ˜ f ] ( λ ) µ , λ = 1 , } , f (3) : { ˆ L µ ≡ √ f † × ˜ f ] (1) µ } . (13)Note the presence of the additional (exceptional) algebra G , which occurs inbetween SO f (7) and SO f (3) [23]. It does not appear in Eqs. (10) and (11)because in symmetric irreducible representations the quadratic Casimir op-erators of SO f (7) and G have identical expectation values. The exceptionalalgebra G is therefore discarded from the classifications (10), (11) and (12)without loss of generality.The explicit expressions of linear and quadratic Casimir operators of the al-gebras appearing in the lattice (12) areˆ C [U sf (8)] = ˆ N = ˆ n s + ˆ n f , ˆ C [U sf (8)] = ˆ N ( ˆ N + 7) , ˆ C [U f (7)] = ˆ n f , ˆ C [U f (7)] = ˆ n f (ˆ n f + 6) , ˆ C [SO sf (8)] = [ s † × ˜ f − f † × ˜ s ] (3) · [ s † × ˜ f − f † × ˜ s ] (3) + ˆ C [SO f (7)] , ˆ C [SO f (7)] = 2 (cid:88) λ odd [ f † × ˜ f ] ( λ ) · [ f † × ˜ f ] ( λ ) , ˆ C [SO f (3)] = ˆ L · ˆ L. (14)The expressions for the quadratic Casimir operators ˆ C [U sf (8)] and ˆ C [U f (7)]are not general but are valid in symmetric irreducible representations of U sf (8)and U f (7). A rotationally invariant and parity-conserving Hamiltonian with upto two-body interactions can be written in terms of the Casimir operators (14),ˆ H sym = (cid:15) s ˆ n s + (cid:15) f ˆ n f + a f ˆ C [U f (7)] + b sf ˆ P † sf ˆ P sf + b f ˆ C [SO f (7)]+ c f ˆ C [SO f (3)] , (15)where (cid:15) (cid:96) , a (cid:96) , b (cid:96) , b (cid:96)(cid:96) (cid:48) and c (cid:96) are parameters. The quadratic Casimir operatorof U sf (8) is omitted for simplicity since it gives a constant contribution for afixed boson number N = n s + n f . The pairing interaction for s and f bosonscan be expressed in terms of Casimir operators,ˆ P † sf ˆ P sf = ˆ C [U sf (8)] − ˆ C [U sf (8)] − ˆ C [SO sf (8)] , (16)where ˆ P † sf ≡ s † s † − f † · f † . Equation (15) is the most general Hamiltonian withup to two-body interactions that can be written in terms of invariant operatorsof the lattice (12). It is intermediate between the limits I and II but has lessparameters than the general Hamiltonian (7). The latter contains seven boson–boson interaction matrix elements whereas the symmetry Hamiltonian (15)has only four two-body parameters. 7he U f (7) limit is attained for b sf = 0 leading to the eigenvalues E I = (cid:15) s n s + (cid:15) f n f + a f n f ( n f + 6) + b f υ f ( υ f + 5) + c f L ( L + 1) . (17)The SO sf (8) limit occurs for (cid:15) s = (cid:15) f ≡ (cid:15) sf and a f = 0, in which case theHamiltonian’s eigenstates have the eigenvalues E II = (cid:15) sf N + b sf [ N ( N + 6) − υ sf ( υ sf + 6)] + b f υ f ( υ f + 5)+ c f L ( L + 1) . (18)The eigenspectra in two limits are then determined with the help of the branch-ing rulesU sf (8) ⊃ U f (7) : [ N ] (cid:55)→ n f = 0 , , . . . , N, U f (7) ⊃ SO f (7) : n f (cid:55)→ υ f = n f , n f − , . . . , , U sf (8) ⊃ SO sf (8) : [ N ] (cid:55)→ υ sf = N, N − , . . . , , SO sf (8) ⊃ SO f (7) : υ sf (cid:55)→ υ f = 0 , , . . . , υ sf . (19)The SO f (7) ⊃ SO f (3) reduction from seniority to angular momentum is morecomplicated due to the multiplicity problem. A closed formula is available forthe number of times the angular momentum L occurs for a given seniority υ f in terms of an integral over characters of the orthogonal algebras SO(7) andSO(3) [24]. This number d ( υ f , L ) is given by complex integral [25] d ( υ f , L ) = i π (cid:73) | z | =1 ( z L +1 − z υ f +5 − (cid:81) k =1 ( z υ f + k − z υ f + L +2 (cid:81) k =1 ( z k +1 − dz, (20)which, due to Cauchy’s theorem, can be evaluated by taking the negative ofthe residue of its integrand. An alternative recursive method to determinethe SO f (7) ⊃ SO f (3) reduction was proposed by Rohozi´nski [26]. Tables ofmultiplicities d ( υ f , L ) can be found in Refs. [26,27].Typical energy spectra in the U f (7) and SO sf (8) limits are shown in Fig. 1.The U f (7) spectrum displays octupole-phonon multiplets characterized by afixed number of f bosons, n f = 0 , , . . . The multiplets are further structuredby the seniority quantum number: the n f = 2 multiplet has υ f = 2 except forthe 0 + level, which has υ f = 0, the n f = 3 multiplet has υ f = 3 except for the3 − level, which has υ f = 1, etc. The SO sf (8) spectrum contains sets of levelswith υ sf = N, N − , . . . and, due the repulsive sf -pairing, υ sf = N levelsare lowest in energy. Multiplets characterized by a seniority quantum number υ f = 0 , , . . . occur within each SO sf (8) multiplet.8 � ( � ) � � = � � � = � � � = � � � = � υ � = � υ � = � � + � - � + � + � + � + � - � - � - � - � - � - � - � - ����� � � � � �� ( � � � ) �� �� ( � ) υ �� = � υ �� = � - � υ � = � υ � = � υ � = � υ � = � υ � = � υ � = � υ � = � � + � - � + � + � + � - � - � - � - � - � - � - � + � - � + � + � + ����� � � � � �� ( � � � ) Fig. 1. Energy spectra in the U f (7) and SO sf (8) limits of the sf -IBM for N = 6bosons. For the U f (7) spectrum the non-zero parameters in the Hamiltonian (15)are (cid:15) f − (cid:15) s = 1000, b f = 25 and c f = 10 keV. For the SO sf (8) spectrum the non-zeroparameters are (cid:15) f − (cid:15) s = 0, b sf = 100, b f = 75 and c f = 10 keV. sf -IBM The classical limit of an arbitrary interacting boson Hamiltonian is its expec-tation value in a coherent state [28], which is a function of the deformationvariables and is to be interpreted as a total-energy surface. The method wasfirst proposed for the sd -IBM [29,30]. The coherent state for the sf -IBM isinspired by the surface (1), | N ; a µ , b µ (cid:105) ∝ Γ( a µ , b µ ) N | o (cid:105) , (21)with [31]Γ( a µ , b µ ) = s † + a f † + (cid:88) µ =1 a µ ( f π µ µ ) † + ı (cid:88) µ =1 b µ ( f − π µ µ ) † , (22)where | o (cid:105) is the boson vacuum and the creation operators are defined as( f ± µ ) † = 1 √ (cid:16) f † µ ± f †− µ (cid:17) . (23)The coefficients a µ and b µ have the interpretation of the shape variablesappearing in the expansion (1). In contrast to the geometric model of Bohrand Mottelson [18] where deformation is associated with the entire nucleus,in the IBM it is generated by the valence nucleons only. As a result, theshape variables in both models are proportional but not identical [32]. Inthe parameterization (3) the radial parameter β in the geometric model andin the IBM are proportional while the angles parameters have an identicalinterpretation. 9he coherent state based on the parameterization (3) reads | N ; β , δ , ϑ , ϕ (cid:105) = (cid:115) N !(1 + β ) N Γ( β , δ , ϑ , ϕ ) N | o (cid:105) , (24)with Γ( β , δ , ϑ , ϕ )= s † + β (cid:104) cos δ sin ϑ cos ϕ f † + ı (cid:113) sin δ ( f †− − f † +2 ) − (cid:113) cos δ (cid:16) sin ϑ sin ϕ ( f †− − f † +1 ) + ı cos ϑ ( f †− + f † +1 ) (cid:17) + (cid:113) cos δ (cid:16) sin ϑ sin ϕ ( f †− − f † +3 ) − ı cos ϑ ( f †− + f † +3 ) (cid:17)(cid:105) . (25)The classical limit of a Hamiltonian of the sf -IBM is its expectation value inthe coherent state, (cid:104) ˆ H (cid:105) ≡ (cid:104) N ; β , δ , ϑ , ϕ | ˆ H | N ; β , δ , ϑ , ϕ (cid:105) , (26)which can be obtained by differentiation [33]. The classical limit of the one-body part (8) is (cid:104) ˆ H (cid:105) = N (cid:15) s + (cid:15) f β β , (27)and that of the two-body part (9) can be written in the generic form (cid:104) ˆ H (cid:105) = N ( N − β ) (cid:88) l =0 , , c l ( β ) l + Φ( δ , ϑ , ϕ )( β ) , (28)whereΦ( δ , ϑ , ϕ ) = (cid:88) ijk ( c ijk + b ijk sin δ sin ϕ )(cos δ ) i (cos ϑ ) j (cos ϕ ) k , (29)with coefficients c l , c ijk and b ijk that can be expressed in terms of the inter-actions v L(cid:96) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48) . The expressions for the coefficients c l are c = v ssss , c = v sfsf − (cid:113) v ssff ,c = v ffff + v ffff + v ffff , (30)10nd those for the non-zero coefficients c ijk and b ijk are c = ¯ v, c = − ¯ v, c = − c = ¯ v,c = c = − c = − c = c = − c = − ¯ v,b = − b = √ ¯ v, (31)in terms of the linear combination¯ v ≡ v ffff − v ffff + 7 v ffff . (32)The classical limit of the total Hamiltonian (7) can therefore be written as (cid:104) ˆ H (cid:105) ≡ E ( β , δ , ϑ , ϕ )= N ( N − β ) (cid:88) l =0 , , c (cid:48) l ( β ) l + Φ( δ , ϑ , ϕ )( β ) , (33)where c (cid:48) l are the modified coefficients c (cid:48) = c + (cid:15) (cid:48) s , c (cid:48) = c + (cid:15) (cid:48) s + (cid:15) (cid:48) f , c (cid:48) = c + (cid:15) (cid:48) f , (34)in terms of the scaled boson energies (cid:15) (cid:48) (cid:96) ≡ (cid:15) (cid:96) / ( N − (cid:15) (cid:96) and seven two-body interactions v L(cid:96) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48) . In the classi-cal limit with the coherent state (21), the number of independent parametersin the energy surface E ( β , δ , ϑ , ϕ ) is reduced to four [three coefficients c (cid:48) l and the single combination ¯ v , which determines completely the functionΦ( δ , ϑ , ϕ )]. sf -IBM The question treated in this section is: What are the conditions on the interac-tions in the sf -IBM for the energy surface E ( β , δ , ϑ , ϕ ) in Eq. (33) to havea minimum with tetrahedral symmetry? Fortunately, a complete catastropheanalysis of the surface is not needed to answer this question.The conditions for E ( β , δ , ϑ , ϕ ) to have an extremum at a point p ∗ in the11our-dimensional space of variables { β , δ , ϑ , ϕ } are ∂E∂β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ = ∂E∂δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ = ∂E∂ϑ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ = ∂E∂ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ = 0 , (35)where p ∗ ≡ ( β ∗ , δ ∗ , ϑ ∗ , ϕ ∗ ) is a short-hand notation for a critical point. Acritical point with tetrahedral symmetry will be denoted as t ∗ , which impliesthat t ∗ satisfies β ∗ > δ ∗ = π . The conditions (35) are necessary for E ( β , δ , ϑ , ϕ ) to have an extremum at p ∗ ; the conditions for a minimum require in addition that the eigenvalues of the stability matrix [ i.e. , the partialderivatives of E ( β , δ , ϑ , ϕ ) of second order at p ∗ ] are all non-negative.Three out of the four conditions (35) are always satisfied for p ∗ = t ∗ . Thefourth, namely the one related to the partial derivative in β , leads to a cubicequation in β with the solutions β ∗ = 0 , β ∗ = ± (cid:115) c (cid:48) − c (cid:48) c (cid:48) − c (cid:48) . (36)Only the last solution with a plus sign corresponds to a tetrahedral extremumand therefore the following condition on the ratio of coefficients is obtained:2 c (cid:48) − c (cid:48) c (cid:48) − c (cid:48) > . (37)The partial derivatives of E ( β , δ , ϑ , ϕ ) of second order are identically zeroat p ∗ = t ∗ , except the double derivatives in β and δ . For the eigenvaluesof the stability matrix to be positive the following two conditions must besatisfied:(2 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) ) c ( c (cid:48) − c (cid:48) + c (cid:48) ) > . (38)The condition (37) for an extremum with tetrahedral symmetry, combinedwith the conditions (38) that the extremum is a minimum, therefore lead to2 c (cid:48) − c (cid:48) > , c (cid:48) − c (cid:48) > , c > , (39)which translate into the following conditions on the single-boson energies andinteraction matrix elements:( N − (cid:16) v ssss − v sfsf + (cid:113) v ssff (cid:17) > (cid:15) f − (cid:15) s , N − (cid:16) v ffff + v ffff + v ffff − v sfsf + (cid:113) v ssff (cid:17) > (cid:15) s − (cid:15) f , v ffff − v ffff + 7 v ffff > . (40)These are the necessary and sufficient conditions for the general Hamiltonianof the sf -IBM, Eqs. (7), (8) and (9), to have a minimum with tetrahedralshape in its classical limit.Can these conditions be fulfilled for “realistic” values of single-boson energiesand boson–boson interaction matrix elements? To answer this question, letus first consider the most general Hamiltonian of the sf -IBM except for onematrix element, namely v ssff , which is assumed to be zero. This Hamiltonianis not analytically solvable but the energies of its 0 + ground state and its yrast3 − state are known in closed form: E (0 +1 ) = N (cid:15) s + N ( N − v ssss ,E (3 − ) = ( N − (cid:15) s + (cid:15) f + ( N − v sfsf + ( N − N − v ssss , (41)resulting in E (3 − ) − E (0 +1 ) = (cid:15) f − (cid:15) s − ( N − v ssss − v sfsf ) . (42)Therefore, unless v ssff >
0, the first of the conditions (40) implies that E (3 − ) < E (0 +1 ), which is clearly unphysical.One concludes therefore that the minimum in the energy surface E ( β , δ , ϑ , ϕ )in Eq. (33) can be of tetrahedral shape only if the mixing matrix element v ssff is non-zero. This brings us to the study of the symmetry Hamiltonian (15),which has the classical limit (cid:104) ˆ H sym (cid:105) = N (cid:15) s + Γ f β β + N ( N − a f β (1 + β ) + b sf (cid:32) − β β (cid:33) . (43)where the combination of parameters Γ f = (cid:15) f + 7 a f + 6 b f + 12 c f is introduced.The parameter b sf is the pairing strength for s and f bosons and is positive,such that the ground-state configuration has υ sf = N , akin to the situationin the SO(6) limit of the sd -IBM [34]. Provided b sf is large enough, the en-ergy surface (43) has an octupole-deformed minimum ( β ∗ ≈ b sf → ∞ )but the shape at minimum is pear-like and not tetrahedral. It can be con-cluded therefore that no isolated minimum with tetrahedral symmetry occursin the classical limit of the symmetry Hamiltonian (15). What still can hap-pen, however, is that a degenerate minimum occurs with non-zero octupoledeformation, which, given the instability in δ , includes a tetrahedral shape.13 � ( � )- �� �� ( � ) � + � - � + � + � + � - � - � - � - � - � - � - � + � - � + � + � + �������� � � � � �� ( � � � ) Fig. 2. Energy spectrum of a U f (7)–SO sf (8) transitional Hamiltonian of the sf -IBM for N = 6 bosons. The non-zero parameters of the Hamiltonian (15) are (cid:15) f − (cid:15) s = 1200, b sf = 100, b f = 50 and c f = 10 keV. The fact that no isolated tetrahedral minimum occurs in the classical limit ofthe symmetry Hamiltonian (15) can be understood from the conditions (40),the first and second of which reduce to4 b sf ( N − − a f − b f − c f > (cid:15) f − (cid:15) s , b sf ( N −
1) + a f (2 N + 5) + 6 b f + 12 c f > (cid:15) s − (cid:15) f . (44)Both inequalities can be satisfied provided b sf is positive and large enough.On the other hand, the last of the conditions (40) is not satisfied because thecombination of f -boson two-body matrix elements vanishes identically for thesymmetry Hamiltonian (15),22( a f + b f − c f ) − a f + b f − c f ) + 14( a f + b f + 9 c f ) = 0 . (45)The absence of a tetrahedral minimum for the symmetry Hamiltonian (15) istherefore entirely due to the specific combination of f -boson two-body matrixelements, of which nothing is known, either empirically or microscopically. If v ffff is taken more repulsive, the energy surface in the classical limit of the sf -IBM Hamiltonian acquires a minimum with tetrahedral symmetry. Indeed,this modification does not alter the conditions (44) since the matrix element v ffff does not appear in them, whereas the third of the conditions (40) isnow satisfied. A possible procedure to construct a Hamiltonian in the sf -IBMwith a minimum with tetrahedral shape in its classical limit is therefore toadd to an octupole-deformed symmetry Hamiltonian (15) a repulsive v ffff interaction.We illustrate this procedure with an example, starting from a U f (7)–SO sf (8)transitional Hamiltonian associated with the lattice (12), giving rise to thespectrum shown in Fig. 2. A reasonable energy difference between the s and f bosons is taken and the strength of the sf pairing is chosen so as to obtain anoctupole-deformed minimum. Other parameters in the Hamiltonian (15) areof lesser importance. 14 � ( � ) � ��� ��� ��� ��� β � ������������ β � δ � ( � ) � ��� ��� ��� ��� β � ������������ β � δ � ( � ) � ��� ��� ��� ��� β � ������������ β � Fig. 3. Three energy surfaces E ( β , δ , ϑ ∗ , ϕ ∗ ) obtained in the classical limit of twodifferent Hamiltonians of the sf -IBM for N = 6 bosons. The values of ϑ and ϕ arefixed and the dependence on β > ≤ δ ≤ π is shown. Black correspondsto low energies and the lines indicate changes by 10 keV. (a) The U f (7)–SO sf (8)transitional Hamiltonian is taken with the parameters given in the caption of Fig. 2.(b) and (c) The Hamiltonian of (a) is modified by taking a repulsive interaction v ffff = 500 keV. The energy surface is shown for (b) ϑ ∗ = π and ϕ ∗ = 0, and for(c) ϑ ∗ = π and ϕ ∗ = π . The parameters quoted in the caption of Fig. 3 satisfy the conditions (44) and,as a result, the energy surface in the classical limit of the corresponding Hamil-tonian displays an octupole-deformed minimum. According to the precedingdiscussion, the energy surface is independent of δ unless the matrix element v ffff is made repulsive, in which case an isolated tetrahedral minimum de-velops. This is indeed confirmed by the surfaces shown in Fig. 3, obtained bytaking the classical limit of two different Hamiltonians of the sf -IBM. Fordisplay purposes the values of ϑ and ϕ are fixed and the dependence on β > ≤ δ ≤ π is shown. The classical limit of the symmetry Hamil-tonian (15), for which v ffff = 2 a f + 2 b f − c f , displays an octupole-deformedminimum at β ∗ ≈ .
32 and no dependence of δ , as shown in Fig. 3(a). Thechange to v ffff = 500 keV introduces a minimum with tetrahedral symmetry( δ ∗ = π ) as shown in Fig. 3(b) for ϑ ∗ = π and ϕ ∗ = 0, and in Fig. 3(c) for ϑ ∗ = π and ϕ ∗ = π . The latter energy surfaces display a second minimumwith axially symmetric octupole deformation ( δ ∗ = 0).Although this proves that shapes with tetrahedral symmetry may occur witha reasonable parameterization in the sf -IBM, it is to be expected that theminimum is rather shallow as it occurs as a result of fine-tuning of little-known f -boson interactions. Even with a value as large as v ffff = 500 keV,the tetrahedral ( δ ∗ = π ) and the axially symmetric ( δ ∗ = 0) minima areseparated by a barrier of only ∼
20 keV. As a result, only minute observableeffects can be expected. This is illustrated in Fig. 4, which shows the spectrumof the U f (8)–SO sf (8) transitional Hamiltonian with the modified v ffff matrixelement. Not much difference from the spectrum shown in Fig. 2 can be seen.15 � - ��� � + � - � + � + � + � - � - � - � - � - � - � - � + � - � + � + � + ��������� � � � � �� ( � � � ) Fig. 4. Energy spectrum of a general Hamiltonian of the sf -IBM for N = 6 bosons.The same Hamiltonian is taken as in Fig. 2 but one f -boson two-body matrixelement is modified to v ffff = 500 keV. On the left- and right-hand sides areshown the shapes at the minima in the energy surface obtained in the classical limitof this Hamiltonian. The shape on the left is axially symmetric, octupole deformedwhile the shape on the right has tetrahedral symmetry. Two dynamical symmetries of the sf -IBM have been established: the U f (7)limit with octupole vibrational characteristics and the SO sf (8) limit where s - and f -boson states are mixed through an sf -pairing interaction, which, ifstrong enough, drives the system towards a permanent octupole deformation.This picture is confirmed by a catastrophe analysis of the energy surface ob-tained in the classical limit of a Hamiltonian transitional between the twolimits, indicating that an octupole-deformed minimum can be obtained withreasonable single-boson energies. However, this minimum is always δ indepen-dent and shapes ranging from pear-like to tetrahedral are degenerate in energy.An isolated minimum with tetrahedral symmetry can be obtained by modify-ing two-body interactions between the f bosons to the transitional symmetryHamiltonian. It is separated from another minimum with axial symmetry bya low-energy barrier, even for fairly strong interactions between the f bosons.There are striking similarities between the search for tetrahedral shapes pre-sented in this paper and the corresponding search for octahedral shapes re-ported in I and II. In both cases it is found that no isolated minimum with ahigher-rank discrete symmetry is possible for a symmetry Hamiltonian of U(8)or U(15) but that a degenerate minimum occurs in the SO sf (8) or SO sg (10)limits of sf - or sg -pairing, respectively. An isolated minimum with tetrahe-dral or octahedral symmetry can be obtained through a modification of thetwo-body interaction between the relevant bosons. However, the minima thusconstructed are rather shallow, even for large repulsive matrix elements be-tween the f or g bosons, and their effects on spectroscopic properties areexpected to be minute.With this series of papers the role of higher-rank discrete symmetries in the16ontext of algebraic nuclear models is clarified and a well-defined procedureis established to find out whether a given Hamiltonian of a particular versionof the interacting boson model displays in its classical limit a minimum witha tetrahedral or octahedral shape. This enables the study of observable con-sequences of higher-rank discrete symmetries in the framework of algebraicmodels.The limitations of this series of papers should nevertheless be recognized be-cause the present analysis is restricted to Hamiltonians with up to two-bodyterms. It is possible that, just as triaxial shapes require higher-order interac-tions in the sd -IBM, shapes with a higher-rank discrete symmetry can be iso-lated with a high barrier by introducing higher-order interactions in sdg -IBMand sf -IBM. Also, as mentioned in the introduction of this paper, the anal-ysis of the tetrahedral case so far has been limited to sf -IBM and should becarried out in the more general spdf -IBM. What can be concluded from theexamples reported in this series of papers is that, unless such more compli-cated Hamiltonians are adopted, it will be difficult to identify clear effects ofhigher-rank discrete symmetries in nuclei. References [1] P. Van Isacker, A. Bouldjedri, S. Zerguine, Nucl. Phys. A 938 (2015) 45.[2] A. Bouldjedri, S. Zerguine, P. Van Isacker, previous paper.[3] F. Iachello, A. Arima, The Interacting Boson Model, Cambridge UniversityPress, Cambridge, 1987.[4] X. Li, J. Dudek, Phys. Rev. C 49 (1994) 1250(R).[5] J. Dudek, A. G´o´zd´z, N. Schunck, M. Mi´skiewicz, Phys. Rev. Lett. 88 (2002)252502.[6] J. Dudek, D. Curien, N. Dubray, J. Dobaczewski, V. Pangon, P. Olbratowski,N. Schunck, Phys. Rev. Lett. 97 (2006) 072501.[7] P.A. Butler and W. Nazarewicz, Rev. Mod. Phys. 68 (1996) 349.[8] L.P. Gaffney et al. , Nature 497 (2013) 199.[9] A. Arima, F. Iachello, Ann. Phys. (NY) 99 (1976) 253.[10] A. Arima, F. Iachello, Ann. Phys. (NY) 111 (1978) 201.[11] O. Scholten, F. Iachello, A. Arima, Ann. Phys. (NY) 115 (1978) 325.[12] J. Engel, F. Iachello, Phys. Rev. Lett. 54 (1985) 1126.[13] J. Engel, F. Iachello, Nucl. Phys. A 472 (1987) 61.
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