Construction of [formula omitted] spherical harmonics and Clebsch–Gordan coefficients

Abstract

The SO ( 5 ) ⊃ SO ( 3 ) spherical harmonics form a natural basis for expansion of nuclear collective model angular wave functions. They underlie the recently-proposed algebraic method for diagonalization of the nuclear collective model Hamiltonian in an SU ( 1 , 1 ) × SO ( 5 ) basis. We present a computer code for explicit construction of the SO ( 5 ) ⊃ SO ( 3 ) spherical harmonics and use them to compute the Clebsch–Gordan coefficients needed for collective model calculations in an SO ( 3 ) -coupled basis. With these Clebsch–Gordan coefficients it becomes possible to compute the matrix elements of collective model observables by purely algebraic methods. Program summary Program title: GammaHarmonic Catalogue identifier: AECY_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AECY_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 346 421 No. of bytes in distributed program, including test data, etc.: 16 037 234 Distribution format: tar.gz Programming language: Mathematica 6 Computer: Any which supports Mathematica Operating system: Any which supports Mathematica; tested under Microsoft Windows XP and Linux Classification: 4.2 Nature of problem: Explicit construction of SO(5) ⊃ SO(3) spherical harmonics on S 4 . Evaluation of SO(3)-reduced matrix elements and SO(5) ⊃ SO(3) Clebsch–Gordan coefficients (isoscalar factors). Solution method: Construction of SO(5) ⊃ SO(3) spherical harmonics by orthonormalization, obtained from a generating set of functions, according to the method of Rowe, Turner, and Repka [1]. Matrix elements and Clebsch–Gordan coefficients follow by construction and integration of SO(3) scalar products. Running time: Depends strongly on the maximum SO(5) and SO(3) representation labels involved. A few minutes for the calculation in the Mathematica notebook.