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Manuscript Title: Fast computer evaluation of radiative properties of hydrogenic systems.
Authors: P.J. Storey, D.G. Hummer
Program title: RADZ1
Catalogue identifier: ABZS_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 66(1991)129
Programming language: Fortran.
Computer: VAX 3900, VAX 6400.
Operating system: VMS.
Word size: 60
Keywords: Astrophysics, Radiative transfer, Scattering, Photon, Hydrogenic radiative Transitions, Einstein coefficients, Photoionization, Free-free, Gaunt factors.
Classification: 1.3, 21.2.

Nature of problem:
Although the cross sections for the absorption and emission of radiation by hydrogenic systems are known exactly in the nonrelativistic approximation, the analytic expressions are not easy to evaluate, especially for large values of the principal quantum number. However, for bound-bound and bound-free transitions, recursion techniques have proved to be accurate and stable for values of the principal quantum number as large as 500. For free-free transitions, recursion techniques are not applicable, and the analytic expressions must be evaluated in other ways. In most applications the free-free cross section must be thermally arranged to account for the Maxwellian distribution of electron speeds.

Solution method:
The bound-bound and bound-free cross sections are evaluated by recurrence on the radial dipole matrix elements. The free-free Gaunt factors have been expressed in terms of hypergeometric functions, which we have evaluated by direct summation combined with a variety of approximate analytic techniques. These results have been thermally averaged and fitted to two-dimensional Chebyshev polynomials, which can be evaluated with very few operations.

Running time:
The running times of the recurrence procedures in the bound-bound and bound-free subroutines are independent of the principal quantum numbers involved, but there is a fixed overhead. Thus the 2->1 transition probability requires 124 mu s to calculate on a VAX 6400, whereas 21 mu s is required to compute each of the 100->50 probabilities. The average running time for the bound-free subroutine is about 40 mu s per cross section. The free-free subroutine requires 88 multiplications and the evaluation of two logarithms for each combination of temperature and charge, and then 8 multiplications for each frequency. Evaluation of the free-free Gaunt factor at 100 frequencies required 15 mu s per frequency.