COLLECTIVE HAMILTONIANS WITH TETRAHEDRAL SYMMETRY: FORMALISM AND GENERAL FEATURES

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1 International Journal of Modern Physics E Vol. 18, No. 4 (2009) c World Scientific Publishing Company COLLECTIVE HAMILTONIANS WITH TETRAHEDRAL SYMMETRY: FORMALISM AND GENERAL FEATURES A. GÓŹDŹ,, M. MIŚKIEWICZ,, J. DUDEK, and A. DOBROWOLSKI Zak lad Fizyki Matematycznej, Uniwersytet Marii Curie-Sk lodowskiej, pl. Marii Curie-Sk lodowskiej 1, PL Lublin, Poland II LO im. Hetmana Jana Zamoyskiego, ul. Ogrodowa 16, PL Lublin Institut Pluridisciplinaire Hubert Curien, Departement de Recherches Subatomiques and Université de Strasbourg, 23 rue du Loess, F Strasbourg, France Received November 3, 2008 Revised November 25, 2008 Collective octupole variables invariant under tetrahedral point group transformations are used to construct tetrahedrally invariant collective Hamiltonians. The so-called intrinsic groups are considered as a tool for symmetry analysis of such Hamiltonians in intrinsic frames of a nuclei. 1. Introduction Realistic nuclear mean-field calculations suggest an existence of atomic nuclei whose shapes and thus underlying mean-field Hamiltonians have tetrahedral symmetry. 1,2 To arrive at the formulation of experimental criteria possibly based on the collective excitations it is interesting to construct collective Hamiltonians invariant under tetrahedral symmetry point group T d. To generate T d -symmetric mean-field Hamiltonian we will need to construct nontrivial T d -invariant surfaces built out of spherical harmonics {Y λµ } (in the case of the phenomenological mean-fields) or, equivalently, T d -symmetric combinations of the multipole moments Q λµ (in the case of the constrained Hartree-Fock approach). Here we would like to examine the collective model Hamiltonians of symmetries compatible with those of the underlying mean-fields. One can show that the lowest order spherical harmonics generating tetrahedral symmetry correspond to λ = 3 and thus represent octupole modes. However, theoretical description of octupole excitations is much more complicated as compared to the quadrupole ones, the latter well known e.g. in conjunction with the Bohr collective Hamiltonian and, The work partially supported by COPIN Andrzej.Gozdz@umcs.lublin.pl. marekm@kft.umcs.lublin.pl. The work partially supported by COPIN Jerzy.Dudek@ires.in2p3.fr. 1028

2 Collective Hamiltonians with Tetrahedral Symmetry 1029 in fact, even the problem of satisfactory parameterization of the corresponding octupole shapes is not solved satisfactorily up to date. 3 Some interesting derivations of collective octupole Hamiltonians are presented in the papers by Lipas and Davidson. 4,5 They follow the traditional way of quantization of the 7-dimensional harmonic oscillator with a restricted number of dynamical variables. However, they are using a series of various approximations what makes the symmetry analysis very difficult if not impossible. Similar considerations are given in a series of papers which can be helpful in construction of the vibrational terms of the tetrahedral-symmetric Hamiltonians. 6 9 Even more general considerations, which allow to introduce anharmonicities in octupole vibrations can be found in Rohozinski and references therein. 10 In this article we employ the group-theoretical methods to construct collective, schematic Hamiltonians with tetrahedral symmetry; they are useful in building-up a qualitative understanding of properties of tetrahedral-mode excitations. 2. Symmetries and Intrinsic Groups Any model Hamiltonian, to be physically acceptable, must satisfy certain general symmetry properties. Here we consider Hamiltonians with symmetries that are modelled using an auxiliary surface parameterized with the help of the the basis of spherical harmonics: R({α}; θ, φ) =R 0 [ 1+ λµ α λµ Y λµ(θ, φ) ]. (1) Let us emphasize that this form of parameterization predefines the geometry of the implied collective excitations. In this article we consider geometrical symmetries of a space-localized nucleus; such symmetries are necessarily equivalent to one of the point-group symmetries - so is the symmetry of the collective Hamiltonian. In the traditional presentation of the formalism of a tri-axial rotor, given in the intrinsic frame, it is essential that the corresponding angular momentum operators commute with the angular momentum operators (more generally: rotation group generators) in the laboratory frame. A generalization of those concepts can be elegantly achieved using the notion of intrinsic groups G associated with the symmetry groups G. Intrinsic groups are defined by their action on the elements of the corresponding groups G in the laboratory frame. 11 Given a symmetry group G of a certain Hamiltonian. Let us construct this group in a laboratory frame and consider the group s linear space L G constructed out of formal sums S, of elements of the group G. For each element g in G, one can define the corresponding operator g as: gs = Sg, S L G. (2) Group G formed by collection of the operators g is called the intrinsic group of G. Generally, intrinsic groups satisfy the most important physical requirement i.e. that the elements of the intrinsic group are scalars with respect to G:

3 1030 A. Góźdź etal. [G, G] = 0. (3) Groups G and G are anti-isomorphic. This property, after small modifications, allows to redefine all notions related to the group G as the corresponding notions with respect to the intrinsic group G. Since the elements of both considered groups commute, one can find common basis { Γmk } for any representation of the groups G and G. The representation matrices D (Γ) m m (g) andd(γ) (g) are defined implicitly by: g Γmk = m k k D (Γ) m m (g) Γm k and g Γmk = k D (Γ) k k (g) Γmk (4) One can show that the matrices of both representations are related D (Γ) mk (g) =D(Γ) km (g) (5) as well as that generators of both groups satisfy the commutation relations: [X m, X k ]=0; [X m,x k ]=C l mk X l, [X m, X k ]= C l mk X l. (6) One of the most fundamental objects needed in the applications are irreducible tensors defined with respect to Lie groups. The corresponding tensors T Γ m are defined as: Traditional Lie groups: [X ρ,t Γ m ]=+ l D Γ lm (X ρ)t Γ l (7) Intrinsic Lie groups: [X ρ, T Γ k ]= Dlm Γ (X ρ)t Γ l. (8) l Tensor products of either irreducible tensors or intrinsic irreducible tensors are defined in the same way by making use of the standard Clebsch-Gordan coefficients of the group G. In the following we focus on the orthogonal group O(3) = SO(3) C i in R 3 where SO(3) denotes the rotation group and C i the spatial inversion group. It is usually assumed that mathematical objects in the intrinsic frame are scalars with respect to the laboratory rotations. There exists a basic theorem which allows to fulfil these properties for tensors: if the physical SO(3) tensors fulfil the condition: [J ρ,tm λ ]=0, (9) where J ρ are angular momenta operators in the intrinsic frame [generators of the intrinsic group SO(3)], the tensor: [T ] λ m = m D λ m m(ω)t λ m (10) is the intrinsic irreducible tensor T λ m. It follows that [T ] λ m expresses operator T λ m in the intrinsic frame.

4 Collective Hamiltonians with Tetrahedral Symmetry Transformation Properties of Collective Variables The collective variables of the model are defined by Eq. (1). It is known that these variables transform as spherical tensors of multipolarity λ, the associated phonons carry the parity π λ =( 1) λ,aswellasthat ĝα λµ = Dµ λ µ (g) α λµ. (11) µ The deformation parameters α in the intrinsics frame are given by: α λµ = µ D λ µ µ(ω) α λµ (Ω). (12) The parameters α λµ are independent of the Euler angles Ω; they can be considered as the expansion coefficients of the functions α λµ (Ω) into complete set of the Wigner functions. This important property can be also seen by rewriting the right hand side of (11) in the intrinsic frame: ĝα λµ = µ D λ µ µ(g) α λµ = µ D λ µ µ[(g 1 Ω) 1 ] α λµ. (13) The laboratory rotation operator ĝ moves orientation of the intrinsic frame together with the nuclear surface (1). In the next step we need to define action of the intrinsic group SO(3) on the deformation parameters. Following definition (2) we express the action of SO(3) on the spherical functions written in the intrinsic frame: Y λµ (θ w,φ w)=ĝy λµ (θ w,φ w )=ĝ µ D λ µ µ(ω)y λµ (θ, φ) = µ D λ µ µ (g 1 )Y λµ (θ w,φ w ). (14) Now we can use invariance of the nuclear surface written in the intrinsic frame: R({α}; θ w,φ w )=R({α }; θ w,φ w), (15) where the directional angles of the rotated surface θ w and φ w are implicitly defined by Eq. (14). The result of a simple calculation is: α λµ ĝ α λµ = µ D λ µ µ(g 1 ) α λµ. (16) This relation shows that intrinsic deformation parameters are intrinsic tensors (8). Using the relation (16) we can check the physically important condition (9). For this purpose we have to calculate action of the intrinsic group elements on the deformation parameters expressed in the laboratory frame: ĝα λµ = ĝ µ D λ µ µ(ω 1 )α λµ = µ [ĝd λ µ µ(ω 1 )][ĝ α λµ ] = µ µ D λ µ µ ((Ωg 1 ) 1 )D λ µ µ (g 1 )α λµ = µ D λ µ µ (Ω 1 )α λµ = α λµ. (17)

5 1032 A. Góźdź etal. The above invariance of α λµ shows no influence of the intrinsic transformations on the laboratory variables, and thus we have shown that the condition (9) is fulfilled. As it is well known the collective variables α λµ defined by equation (1) are irreducible tensors with respect to the SO(3) group. In the intrinsic frame, we are tempted to distinguish the three orientation angles as natural collective variables, among all the others. This implies that in order to have e.g. the Euler angles among collective coordinates one needs to impose three extra constraints on α λµ. The well known and successful Bohr Hamiltonian can be taken as a reference for the analysis of structure of collective Hamiltonians with pre-defined symmetries (α 2µ a µ ): 12 Ĥ B = 1 3 J 2 [ n 2 J n=1 n (a 0,a 2 ) 2 2 2B a ] 0 2 a 2 + (3a2 0 +6a2 2 )2 2 8a 2 2 (3a2 0 + V (a 0,a 2 ). 2a2 2 )2 (18) Let us observe the following: Bohr Hamiltonian is built out of invariants of its original symmetry group as defined in terms of spherical harmonics Y 20, Y 2,±2, i.e. D 2h (for a 2 0). The invariants retained as building blocks are of the lowest order, i.e. the Bohr Hamiltonian is constructed from invariant collective variables (here a 2 and a 0 ) and of the lowest order (quadratic) intrinsic angular momentum operators that are D 2h invariants). In the following we will use this pattern to built the Hamiltonian invariant under the tetrahedral group T d. 4. Structure of Intrinsic Tetrahedral Symmetry Hamiltonians The name tetrahedral symmetry can be, strictly speaking, associated with three point groups: T, T h and T d.groupt consists of discrete rotations. It has four irreducible representations: A 1,A 2,A 3 and T, of dimensions 1, 1, 1 and 3, respectively. The representations A 2 and A 3 are coupled by time reversal operations and give, in fact, a two-dimensional representation for time even Hamiltonians. The structure of representations for the group T h doubles the one of the group T because T h is a direct product of T and the inversion group C i. The group T d, in contrast, has a different structure, with representations A 1,A 2,E,T 1 and T 2 of dimensions 1, 1, 2, 3 and 3, respectively. The dimensions of the representations correspond to degeneracies of the associated energy levels of the tetrahedral Hamiltonians. In the following we restrict our considerations to the octupole modes. To write down the Hamiltonian invariant with respect to the tetrahedral groups, following the strategy used in the quadrupole Hamiltonian of Bohr, we need tetrahedrally invariant collective variables. One may show that there are no tetrahedrally invariant collective variables of the form µ c µα 3µ for groups T and T h. Choosing symmetry axes of a tetrahedron positioned in a Cartesian reference frame according to the

6 Collective Hamiltonians with Tetrahedral Symmetry 1033 conventions of Cornwell, one obtains (the only) invariant octupole-variable that can be constructed for T d,intheform 13 ξ = i (α 32 α 3, 2 )=Imα 32 and Re α 32 =0. (19) Assuming, as a starting point, a 7-dimensional harmonic oscillator realization of the octupole potential, one can show that the corresponding Cartesian components of the moments of inertia can be expressed in the intrinsic frame by the variable ξ only, the inertia tensor is diagonal and that all the diagonal components are equal: 8 J lm =0 for l m, J 11 = J 22 = J 33. (20) This implies that T d invariant collective octupole-hamiltonians can be expressed by operator ˆξ, its canonically conjugated momentum ˆπ ξ and invariants built out of the angular momentum operators. It then follows that the associated lowest order Hamiltonian can be written down as Ĥ T (ˆπ ξ, ˆξ; {Ĵk}) = C n,m;k1,k 2,k 3 ˆπ n ˆξ ξ m (Ĵ 1) k1 (Ĵ 2) k2 (Ĵ 3) k3 (21) n,m k 1 k 2 k 3 and, after developing and making the notation more compact Ĥ T = Ĥξ(ˆπ ξ, ˆξ)+Ĥrot[ˆξ,Ĵ 2, ( ˆT 3 2 ˆT 3 2 )] + ĤξJ(ˆπ ξ, ˆξ,Ĵ k), (22) where ˆT µ 3 =[(Ĵ Ĵ)2 Ĵ ]3 µ are invariants of group T d. The first term describes pure (here: oscillatory) ξ-motion, the second one describes generalized rotor and the last term describes all possible remaining couplings that involve {Ĵk}. In the following we focus on the first two terms. The simplest rotational term can be derived from the seven-dimensional harmonic oscillator. It is proportional to the square of the angular momentum operator. The resulting Hamiltonian is T d invariant, all its eigen-functions belong to the scalar representation of group T d and thus there are no characteristic 2- or 3-fold degeneracies associated with this part of the Hamiltonian. To obtain all the representations among the solutions, construction (21) should provide an appropriate function of T d scalars (built out of the tensors Tµ λ ), like those included in Eq. (22): Ĥ rot (Ĵ)2 J (ξ) + F [i(t 2 3 T 2)]. 3 (23) It is important to note that hermitian operator i( ˆT 2 3 ˆT 2 3 )istime-oddandthe real operator function ˆF has to be an even-function of its argument if one wishes to work with time-even Hamiltonians. The structure of the corresponding Hamiltonian is: Ĥ = Ĥξ(ˆπ ξ, ˆξ)+Ĥ rot. (24) For F 0 and the inertia parameter independent of ξ, the eigen-functions in Eq. (24) are products of eigen-functions of the vibrational Hamiltonian Ĥξ(ˆπ ξ, ˆξ) and the normalized Wigner functions R J MK (Ω) 2J +1D J MK (Ω). Such product

7 1034 A. Góźdź etal. eigen-functions can be used as a convenient basis for calculations related to both Hamiltonians (22) and (24). In general, the eigen-functions of the Hamiltonians in Eqs. (22) and (24) have the following structure: ψ νjm = K v νjk (ξ) R J MK (Ω), (25) where ν denotes a set of quantum numbers describing the tetrahedral quanta (tetrahedral phonons) and simultaneously the irreducible representations of the T d group. Hamiltonians with the structure of Ĥ describe rotational bands, that respect the characteristic tetrahedral degeneracies, built on top of vibrational states. In order that the collective functions ψ νjm are single-valued in the laboratory frame with respect to variable ξ, they have to be invariant under the octahedral group O h generated by the Bohr transformations defined as R 1 = g(0,π,0), R 2 = g(0, 0,π/2) and R 3 = g(π/2,π/2,π). 14 Since R 1,R 3 T d we find that R 1 ξ = ξ and R 3 ξ = ξ. On the other hand, R 2 belongs to the group T and changes only sign of the variable ξ. The Wigner functions have the following transformation properties: R 1 R J MK (Ω) =( 1) J+K R J M, K (Ω) R 2 R J MK (Ω) =( i) K R J MK (Ω) R 3 R J MK (Ω) =( i) K K ( 1) K d J KK ( π 2 )rj MK (Ω). (26) These properties allow to write down the R-invariance conditions for the eigenfunctions (25). The R 1 -operation relates the functions v νjk (ξ) andv νj, K (ξ): R 1 v νjk (ξ) =( 1) J K v νj, K (ξ). (27) The above condition implies that for K = 0 the angular momentum quantum number can only be equal to an even integer, i.e. J =0, 2, 4,... Transformation R 2 requires that functions v νjk (ξ) are either even or odd. In the first case, v νjk (ξ) = v νjk ( ξ), the only allowed K values are 0, ±4, ±8, ±12,... In the second case, the only allowed K values are ±2, ±6, ±10,...TheR 3 -invariance leads to a more complicated condition: R 3 v νjk (ξ) =( 1) K J K = J ( i) K d J K K ( π 2 ) vνjk (ξ) (28) which relates the functions v νjk (ξ) with different K. The R-symmetry is a general requirement imposed on eigen-functions of the Hamiltonian with any symmetry, in particular: tetrahedral. Relations between the R-symmetry and symmetry of the tetrahedral Hamiltonian are to our knowledge not known and are subject of a follow-up study. At the exact T d -symmetry limit considered here, the collective multipole transition operators can be constructed according to Ref.; 12 up to the second order: Q coll 1µ = 0, because the Clebsch-Gordan coefficient ( )=0,

8 Collective Hamiltonians with Tetrahedral Symmetry 1035 Q coll 2µ = 0, because the Clebsch-Gordan coefficient ( )=0. The only non-zero moment is the octupole one. This would suggest that, strictly speaking, no intra-band E2 transitions, as in fact observed in several Actinide nuclei, the only allowed intra-band transitions having octupole character. The latter are expected to be very weak and difficult to observe. However, it is very likely that weak E2 transitions will take place in nature especially at increasing spins because of various types of polarization effects originating, among others from the quadrupole zero-point motion and Coriolis alignment, cf. Ref. 15 Finally let us mention that the traditional pear-shape symmetry (C )isoften D a competing symmetry with respect to the tetrahedral one in several regions of nuclei such as e.g. neighbourhoods of 80 Zr, 96 Zr and 100 Zr nuclei. For the results with the phenomenological methods the reader is refered e.g. to Figs. (3-4) of Ref., 16 while those using Generator Coordinate Method-to Ref.; 17 both approaches predict that tetrahedral symmetry minima lie lower than the axial-octupole ones. In summary: We used the intrinsic group concept to build classes of Hamiltonians with certain pre-defined symmetry properties and derive some of their general features. Acknowledgments Support within the collaboration TetraNuc through the IN 2 P 3, France and through exchange program between IN 2 P 3 and COPIN, Poland, is acknowledged. References 1. X. Li and J. Dudek, Phys. Rev. C 49 (1994) R J. Dudek et al., Phys. Rev. Lett. 97 (2006) S. G. Rohoziński, Phys.Rev. C 56 (1997) P. O. Lipas and J. P. Davidson, Nucl. Phys. 26 (1961) J.P. Davidson, Nucl. Phys. 33 (1962) 664; ibid. 69 (1965) W. Donner and W. Greiner, Z. Phys. 197 (1966) J. Blocki and W. Kurcewicz, Phys. Lett. B 30 (1969) C. Wexler and G. Dussel Phys. Rev. C 60(1999) D. Bonatsos et al., Phys. Rev. C 71 (2005) ; D. Bonatsos et al., Rom. J. Phys 52 (2007) S. G. Rohoziński, J. Phys. (Nucl. Phys.) G, 4 (1978) Jin-Quan Chen, Jialun Ping and Fan Wang, Group Representation Theory for Physicists (World Scientific, 2002). 12. J. M. Eisenberg and W. Greiner, Nuclear theory, Vol. 1 (North Holland, 1970). 13. J. F. Cornwell, Group theory in physics, Vol. 1 (Academic Press, London, 1994). 14. M. K. Pal,Nuclear Structure (Van Norstrand Reinhold Company, N.Y., 1983). 15. J. Dudek et al., Acta Phys. Polonica, B 38 (2007) J. Dudek, K. Mazurek, D. Curien, A. Dobrowolski, A. Góźdź, D. Hartley, A. Maj, L. Riedinger and N. Schunck, Proc. Int. Zakopane Conference on Nuclear Physics, September 1-7, K. Zberecki, P. -H. Heenen and P. Magierski, arxiv:nucl-th/ v1 (2008); also Int. J. mod.phys E 16 (2007) 533.