Calculations of harmonic oscillator brackets

Abstract

To facilitate the use of the extended nuclear shell model (NSM), a Fermi module for calculating some of its basic quantities in the framework of Maple is provided. The Moshinsky brackets, the matrix elements for several central and non-central interactions between nuclear two-particle states as well as their expansion in terms of Talmi integrals are easily given within a symbolic formulation. All of these quantities are available for interactive work.Title of program:FermiCatalogue identifier:ADVOProgram summary URL: http://cpc.cs.qub.ac.uk/summaries/ADVOProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:NoneComputer for which the program is designed and others on which is has been tested:All computers with a licence for the computer algebra package Maple [Maple is a registered trademark of Waterloo Maple Inc., produced by MapleSoft division of Waterloo Maple Inc.]Instalations:GSI-Darmstadt; University of Kassel (Germany)Operating systems or monitors under which the program has beentested: WindowsXP, Linux 2.4Programming language used:Maple 8 and 9.5 from MapleSoft division of Waterloo Maple Inc.Memory required to execute with typical data:30 MBNo. of lines in distributed program including test data etc.:5742No. of bytes in distributed program including test data etc.:288 939Distribution program:tar.gzNature of the physical problem:In order to perform calculations within the nuclear shell model (NSM), a quick and reliable access to the nuclear matrix elements is required. These matrix elements, which arise from various types of forces among the nucleons, can be calculated using Moshinsky's transformation brackets between relative and center-of-mass coordinates [T.A. Brody, M. Moshinsky, Tables of Transformation Brackets, Monografias del Instituto de Fisica, Universidad Nacional Autonoma de Mexico, 1960] and by the proper use of the nuclear states in different coupling notations.Method of solution:Moshinsky's transformation brackets as well as two-nucleon matrix elements are provided within the framework of Maple. The transformation brackets are evaluated recursively for a given number of shells and utilized for the computation of the two-particle matrix elements for different coupling schemes and interactions. Moreover, a simple notation has been introduced to handle the two-particle nuclear states in ll-, LSJ-, and jj-coupling, both in the center-of-well and the relative and center-of-mass coordinates.Restrictions onto the complexity of the problem:The program supports in principle an arbitrary number of shell states with the only limitation given by the computer resources themselves. Typically, the time requirements for the recursive computation of the Moshinsky brackets and matrix elements increase rapidly with the number of the allowed shell states but can be reduced significantly by the pre-calculation of the transformation brackets.Unusual features of the program:Moshinsky brackets are computed and provided in either numeric, algebraic or some symbolic form. In addition, the two-particle matrix elements are calculated for a scalar potential, spin-orbit coupling and tensorial forces, both in floating-point and algebraic notation. All two-particle matrix elements are expressed in terms of the Talmi integrals but can be evaluated also explicitly for several predefined types of the interaction. To simplify the handling of the program, a short but very powerful notation has been introduced which help the user to deal with the two-particle states in various coupling notations. The main commands of the current version of the program are described in detail in Appendix B.Typical running time:The computation of all Moshinsky brackets in floating-point notation, up to ρ=6, takes about 5 s at a 2.26 GHz Intel Pentium IIII processor with 512 MB; in algebraic form, the same computations take about 13 s. Similarly, the computation of these brackets up to ρ=10 requires in numeric and algebraic form about 5 and 15 min, respectively. Once these coefficients have been stored, however, the program replies rather promptly on most further requests.