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Manuscript Title: Quantum rotor and its SU(3) realization.
Authors: O. Castanos, J.P. Draayer, Y. Leschber
Program title: ROTXSU3
Catalogue identifier: ABFO_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 52(1988)71
Programming language: Fortran.
Computer: IBM 3084-QX6, IBM PC/XT.
Operating system: MVS/SP 1.3.3, VMS, DOS.
RAM: 86K words
Keywords: Nuclear physics, Fractional parentage, Shell model, Quantum rotor, Su(3) model, Asymmetric top, Elliott model, Collective motion, Triaxial rotor, Harmonic oscillator, Algebraic theory, Dynamical symmetry, Isotropic oscillator, Integrity basis, Enveloping algebra, Group contraction, Su(3) -> so(3), T5 x so(3).
Classification: 17.18, 17.19.

Nature of problem:
Low-lying energy levels of well-deformed, even-even nuclei can be described by the rotational limit of the collective model and by the SU(3) shell model. The parameters of the rotor hamiltonian are the three principal moments of inertia. The corresponding SU(3) theory is obtained by taking the shell-model hamiltonian to be a linear combination of the operators L2, Xa3 and Xa4 which belong to the SU(3) -> SO(3) integrity basis. The coefficients of these three rotational scalars are the parameters of this theory. Both approaches are phenomenological because the parameters of the model hamiltonians are fit to experimental data. The hamiltonians of the quantum rotor and SU(3) models are given by
     HROT = A1I21 + A2I22 + A3I23                                 (1)    
     HSU3 =   aL2 +   bX3 +  cX4,                                 (2)
respectively. In the rotor case, Ialpha is the projection of the total angular momentum on the alpha-th body-fixed symmetry axis. When the rotor hamiltonian is rewritten in terms of lab-frame operators, it has the same form as the SU(3) hamiltonian. It is therefore not surprising that the dynamics of a quantum rotor can be realized in terms of the SU(3) -> SO(3) group structure. Rigorous justification for this follows from the fact that the algebra of the dynamical symmetry group of the quantum top, T5 X SO(3), is a contraction of SU(3). In particular, it has been shown that one can establish a linear mapping between eigenvalues of invariant operators of the two theories and that this in turn allows one to deduce a relationship between the parameters of the two models. In the language of group theory, SU(3) forms a compact group realization of the noncompact symmetry group T5 X SO(3) that characterizes the quantum rotor. The program ROTXSU3 calculates matrix representations of HROT and HSU3 for selected angular momentum values, diagonalizes the matrices, and displays the corresponding eigenvalues in increasing order. The user is required to select whether to input the inertia parameters of the rotor, in which case the parameters of the SU(3) model are calculated using the mapping formulae, or the parameter of the SU(3) model, in which case the inverse of the mapping transformation is used to determine the parameters of the rotor.
Specifically, the following are required input:
1) Option selection: Specify 1 if the input parameters are for the rotor or 2 if they are for the SU(3) hamiltonian. Enter an ENDFILE to terminate processing.
2) SU(3) representation: Select the representation lambda and mu of SU(3). This choice also fixes the D2 symmetry type of the rotor, see Table 1 of the text.
3) Parameter specification: Enter the inertia parameters of the rotor, Aalpha (alpha=1,2,3) of Eq.(1), under Option 1 or (a,b,c) of Eq.(2) in the SU(3) case for Option 2.
4) SO(3) representation: The desired range of angular momentum values, Lmin and Lmax. The program automatically skips unallowed values of the angular momentum.

Solution method:
First a matrix representation of the rotor hamiltonian is obtained using an orthonormal set of basis states which carries a definite angular momentum and D 2 symmetry. Then a matrix representation of the corresponding SU(3) shell-model hamiltonian is constructed with a nonorthonormal set of basis states. Because of the nonorthonormality of the shell-model basis it is necessary to use a diagonalization algorithm for nonsymmetric matrices. However, since only eigenvalues are required and not eigenvectors, this poses no problem because standard routines for determining the eigenvalues of nonsymmetric matrix forms are available. The reason for selecting a nonorthonormal SU(3) basis rather than orthonormal ones is that general analytic results for matrix elements of the X 3 and X 4 operators can then be given. This, in turn, means the code is general because it can be used for arbitrarily large representations of SU(3).

The only restriction is on the size of the hamiltonian matrix which in the current version is set at 40*40 but this can be changed as necessary. The program requires the user to supply an eigenvalue routine for nonsymmetric matrices.

Running time:
Run times are insignificant, even on a microcomputer. The execution time scales with the size of the matrices and not with the size of the SU(3) irrep.