Calculation of Gaunt Factors for Free-Free Transitions Near Positive Ions

Abstract

A discussion is given of methods which may be used to compute Gaunt factors for free–free transitions in the neighbourhood of any positive ion. The equivalence of the formulae of Sommerfeld and of Menzel and Pekeris for hydrogen-like systems is demonstrated, and a number of approximations to them discussed. It is shown that the rapid computation of numerical values of monochromatic Gaunt factors for any electron velocity is assured by means of an extensive table of values supplemented by simple asymptotic expansions. The effect of screening on Gaunt factors is estimated for long wave-lengths by a simple formula derived in the Born approximation with the aid of an atomic potential function of the form $${r}^{-1}\Sigma{\alpha }_{n}\,\text{exp}\,(-{\beta }_{n}r)$$ . The method is illustrated by examples in which Thomas-Fermi and self-consistent potential functions are expressed in this form. A simple procedure for calculating Gaunt factors averaged over a Maxwellian velocity distribution is given using a quadrature formula based on evaluation of the integrand at the zeros of Laguerre polynomials. Three integration points are found to be adequate to give accuracy comparable to that attained by more elaborate methods.