tensor formalism for rotational and vibrational nuclear

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International Journal of Modern Physics E Vol. 17, No. 1 (2008) 272–275 c World Scientific Publishing Company

TENSOR FORMALISM FOR ROTATIONAL AND VIBRATIONAL NUCLEAR MOTIONS WITH DISCRETE SYMMETRIES: ROTATIONAL TERMS

Int. J. Mod. Phys. E 2008.17:272-275. Downloaded from www.worldscientific.com by UNIVERSITY LOUIS PASTEUR (ULP) on 10/15/12. For personal use only.

†,¶ and J. DUDEK‡,k ´ ZD ´ Z ´ ∗,§ , M. MISKIEWICZ ´ A. GO ∗ Zaklad

Fizyki Matematycznej, Uniwersytet Marii Curie-Sklodowskiej, pl. Marii Curie-Sklodowskiej 1, PL-20031 Lublin, Poland † II LO im. Hetmana Jana Zamoyskiego, ul. Ogrodowa 16, PL-27075 Lublin ‡ Institut Pluridisciplinaire Hubert Curien, Departement de Recherches Subatomiques and Universit´ e L. Pasteur, Strasbourg I, 23 rue du Loess, F-67037 Strasbourg, France Received October 24, 2007

The method of construction of nuclear collective Hamiltonians with discrete symmetries has been considered. The method allows to construct the collective Hamiltonians using O(3) tensors as building blocks. The formula for general form of the rotational reduced matrix elements is derived.

The studies based on the mean–field approaches predict the existence of atomic nuclei with tetrahedral symmetries. To predict the forms of collective behaviour of tetrahedral-symmetric nuclei (or other nuclei with high discrete symmetries, such as octahedral one) one needs to consider the appropriately adapted collective models. Some progress in relation to the description of the collective nuclear rotation has been presented in Ref. 1. The first investigation of the possible tetrahedral nuclear vibrations using the Generator Coordinate Method can be found in Ref. 2. In this paper we adopt the idea of decomposition of the Hamiltonian into spherical tensors used in molecular physics.3 This method allows to construct the building blocks (spherical tensors) of the Hamiltonian with required symmetry. The combinations of tensors which gives the terms of required symmetry for all 32 point groups can be found in the Ph.D. theses.4,5 Nuclear collective Hamiltonians are usually constructed in the intrinsic frames of reference.3,6 In this way, such Hamiltonians satisfy the fundamental requirement to be rotationally invariant in the laboratory frame. However, they can have a series § The

work partially supported by COPIN 04-113. e-mail: [email protected] [email protected] k The work partially supported by COPIN 04-113. e-mail: [email protected] ¶ e-mail:

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of intrinsic symmetries. According to references mentioned in the introduction the list of possible, traditional symmetries used in nuclear physics should be extended to nearly all known point–groups. In the following we are interested in constructions of collective Hamiltonians with discrete symmetries contained in the orthogonal group O(3) = Ci × SO(3). The orthogonal group is the simple product of the inversion group Ci and the rotation group SO(3). For this purpose we need to construct tensors with respect to the orthogonal group O(3). These tensors are defined through their transformation properties. The transformations g(s, Ω) ∈ O(3) are parameterized by four group parameters: s = ±1 which parameterizes the inversion and by the traditional three Euler angles Ω = (Ω1 , Ω2 , Ω3 ). The irreducible tensor operators T (κ,l) with respect to the orthogonal group O(3) are defined as: X (κ,l) l (κ,l) Dm , (1) g(s, Ω)Tm g(s, Ω)† = χκCi (s) 0 m (Ω)Tm0 m0

χκCi (s)

where stand for the characters of the irreducible representations of the inl version group (κ = ±1) and Dm 0 m (Ω) denote the Wigner functions of the rotation group. These tensors can be used to construct collective Hamiltonians consisting of the vibrational- ,Hvib , rotational- Hrot and the coupling-term Hvibrot : ˆ =H ˆ vib + H ˆ rot + H ˆ vibrot . H

(2)

with required symmetries. Since, in this model, the nucleus is described by two kinds of collective variables the tensors (1) should also reflect this feature and should be (κ,λ ) (λ ) composed of the vibrational Vm2 v (vib) and the rotational Rm1r (rot) parts and written in terms of the tensor product: (κ,λ) Tµ(κ,λ) (λv , λr ; vib, rot) = V (κ,λv ) (vib) ⊗ R(λr ) (rot) = µ X (λr ) v) (rot), (3) (vib) Rm (λv m1 λr m2 |λµ) Vm(κ,λ 2 1 µ1 ,µ2

where (λv m1 λr m2 |λµ) are the usual Clebsch-Gordan coefficients for the group SO(3). These tensor operators, expressed in the intrinsic frame, can be used as building blocks of the collective Hamiltonians (2) with required symmetries: X ˆ = H Tµ(κ,λ) (λv , λr ; vib, rot). (4) κ,λ,µ;λv ,λr

The symmetry of the Hamiltonian (4) is determined by the appropriate selection of (κ,λ) the tensors Tµ . However, the symmetry properties are not dependent directly on the tensor indices λv , λr describing internal tensors. This feature allows to construct the wide class of the Hamiltonians with a given symmetry using the appropriate invariant tensor forms with respect to the required symmetry group. The tables of the invariant forms of all 32 point–groups, for the tensor ranks J ≤ 6, are presented in the Refs. 4 and 5.

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As an example, we write below the most interesting forms of the tetrahedral Hamiltonians. The tetrahedral symmetries are represented by three groups T, T h = T × Ci and Td . The lowest order Hamiltonians for the group T can be written as: X (κ,3) (κ,3) ˆ3 = i H T2 (λv , λr ) − T−2 (λv , λr ) (5) κ;λv ,λr

ˆ4 = H

X


κ;λv ,λr

r

! 14 (κ,4) (κ,4) (κ,4) T (λv , λr ) + T4 (λv , λr ) + T−4 (λv , λr ) . 5 0

(6)

The corresponding Hamiltonians for Th can be obtained from (5) and (6) by restriction of the summation to only positive parity terms – in this case the parity is the good quantum number. The tetrahedral symmetry Td leads to the lowest order Hamiltonians which can be built from the same tensors as (5) and (6): X (κ=−1,3) (κ=−1,3) ˆT = H iT2 (λv , λr ) − iT−2 (λv , λr )) + d λv ,λr

r

14 (κ=+1,4) (κ=+1,4) (κ=+1,4) T0 (λv , λr ) + T4 (λv , λr ) + T−4 (λv , λr ) . 5

(7)

The appropriate vector basis for the model Hamiltonians (4) has already been proposed in the Refs. 8 and 9 and can be also found in the Ref. 3 X (κl) J (lmJK|II3 )(−1)m φ−m (vib)rM (8) h|[κlJ]II3 M i = K (rot), mK

√

J J ? where rM 2J + 1DM K (rot) = K (Ω) . The basis is modified here by adding of the parity κ to the vibrational part. One can calculate the SO(3) reduced matrix elements of tensor operators (3) using this basis (see the Ref. 3): p h[κ0 l0 J 0 ]I 0 ||T (π,λ) (λv , λr ; vib, rot)||[κlJ]Ii = (2I 0 + 1)(2λ + 1)(2I + 1)  0   l l λv  (9) J 0 J λr hκ0 l0 ||V (π,λv ) (vib)||κlivib hJ 0 ||R(λr ) (rot)||Jirot .  0  I I λ

This basis allows to separate the matrix elements of the vibrational < k k > vib and the rotational < k k >rot parts. It simplifies the calculations of the matrix elements of the collective Hamiltonians (4). In fact, the Hamiltonians can be defined by the reduced matrix elements of the tensors V (κ,λv ) and R(λr ) , uniquely. All the remaining factors are already expressed in terms of the Clebsch–Gordan coefficients and other special functions. In this letter we consider only the rotational degrees of freedom which can be treated in the general form. In this case the appropriate tensors have to be constructed from the angular momentum operators Jµ expressed in the intrinsic frame. The vibrational parts of the collective Hamiltonians are model dependent.

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The rotational tensors are building blocks of the so called generalized rotor and can be written as: Rµ(λr =1) (rot) = C(J 2 )Jµ and Rµ(λr ) (rot) = C(J 2 )Sµλr ,

(10)

where C(J 2 ) is an arbitrary function of the square of the total angular momentum operator and λr Sµλr = (. . . ((J ⊗ J)(2) ⊗ J)(3) ⊗ · · · ⊗ J)(λ−1) ⊗ J . (11)


µ

The reduced matrix elements of the operators (11) for λ ≥ 2 can be calculated recursively: p ) J(J + 1)(2J + 1) hJ||J||Jirot = , (12) hJ||S (λr ) ||Jirot = F (J, λr )hJ||S (λr −1) ||Jirot hJ||J||Jirot where the function F (J, λr ) is the following combination of the Clebsch–Gordan coefficients: √ F (J, λr ) = { 2J + 1(Jkλr µ|Jk 0 )}−1 X (λr − 1, µ0 1µ00 |λr µ)(J, k + µ00 , λr − 1, µ0 |Jk 0 )(Jk1µ00 |J, k + µ00 ) (13) µ0 µ00

for any k, k 0 and µ fulfilling the condition (Jkλr µ|Jk 0 ) 6= 0. The formula (12) allows to calculate the matrix elements of the collective Hamiltonian (2). In this way we have derived the general structure which allows to construct a set of collective nuclear models with symmetries. These models will allow to predict the spectral properties and electromagnetic branching ratios for the hypothetical tetrahedral nuclei. References 1. J. Dudek, A. G´ o´zd´z, and D. Rosly, Acta Phys. Polonica B32, 2625 (2001). 2. K. Zberecki, P. Magierski, P.-H. Heenen, and N. Schunck, Phys. Rev. C 74, 051302 (2006). 3. W.G. Harter and D.E. Weeks, J. Chem.Phys. 90, 4727 (1988). 4. D. Rosly, “Rotor uog´ olniony i jego zastosowania”, Ph.-D. thesis (in Polish), University of M. Curie–Sklodowska, Lublin, Poland, 2002. 5. M. Mi´skiewicz, “Semiklasyczny i kwantowy opis ruch´ ow rotora uog´ olnionego”, Ph.-D. thesis (in Polish), University of M. Curie–Sklodowska, Lublin, Poland, 2006. 6. A. G´ o´zd´z, M. Mi´skiewicz and A. Olszewski, Int. J. Mod. Phys. E13, 37 (2004). 7. J. Dudek et al., Int. J. Mod. Phys. E16, 516 (2007). 8. J.D. Louck, Diss. Abstr. 19, 840 (1958). 9. K.T. Hecht, J. Mol. Spectrosc. 5, 355 (1960).