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Manuscript Title: Recurrence relations for Coulomb excitation electric multipole radial matrix elements.
Authors: L.D. Tolsma
Program title: RECREM
Catalogue identifier: AALF_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 41(1986)41
Programming language: Fortran.
Computer: BURROUGHS 7900.
Operating system: MCP.
Word size: 48
Keywords: Atomic physics, Heavy ions, Electron, Radial matrix element, Molecular physics, Coulomb excitation, Scattering, Recurrence relation, Inelastic, Low energy.
Classification: 2.4.

Subprograms used:
Cat Id Title Reference
ACMI_v2_0 CLMINT CPC 32(1984)421

Nature of problem:
The radial Schrodinger equation which has to be solved for the quantum mechanical description of inelastic collisions between charged particles can be rewritten as an equivalent set of coupled integral equations. The partial wave radial function is written as a linear combination of two linearly independent basic functions with more or less slowly varying amplitudes. For large r values of the integration region or for high l values of the orbital angular momentum these amplitudes consist of electric multipole radial matrix elements, i.e., integrals Il,l'(lambda), over a finite interval (R1, R2) and with an integrand containing a product of the Coulomb wave functions Xl(eta,kr) and Yl'(n',k'r) and a form factor r-lambda-1 where lambda > 1. Such integrals have to be determined for one or more radial intervals when solving the set of integral equations. The calculation of the excitation probabilities for analysing experimental data needs the solution of the Schrodinger equation and, thus, the knowledge of these integrals for a few hundred or even thousand partial waves, especially, for heavy ion collisions.

Solution method:
Radial matrix elements Il,l'(lambda) of any multipolarity satisfy recurrence relations. Diagonal (l'=l) and upper-diagonal (l'=l+1) matrix elements are calculated using an upward recursion, starting with four initial integrals. Each of these four initial values is obtained by a call to the subprogram CLMINT. Using the diagonal and upper- diagonal matrix elements in their turn as initial values, the remaining lower- and upper-diagonal matrix elements are calculated by a sidewards recursion with l,l' values for which |l-l'| <= lambda. The diagonal and upper-diagonal matrix elements can also be calculated by solving a pentadiagonal system of linear equations obtained by combining and rearranging two recurrence relations. Four boundary values of the radial matrix elements are required: two low l values and two for high l values. Each of these four boundary values is also obtained by a call to CLMINT.

If the recurrence relations of the radial matrix elements are used in an upward or downward recursion, then they are susceptible to error growth. This growth depends largely on the ratio of k and k'. The more this ratio differs from unity, the more the recurrence relations will lose their accuracy due to the cancellation of terms. This loss of accuracy is not encountered when two recurrence relations are combined and rearranged into a pentadiagonal system of linear equations which can be solved by standard methods.

Running time:
The running time is mainly determined by the computation time for the initial or boundary radial matrix elements required by CLMINT, i.e., it depends on whether the radial matrix elements are calculated by an upward recursion or by solving a system of linear equations. The output of the test runs gives the processor time of both alternatives. The computation time for the initial or boundary radial matrix elements depends largely on the parameters and the lower limit R1. These integrals are generated efficiently by CLMINT for parameters encountered in heavy ion scattering processes.