The Hydrogen Recombination Spectrum

Abstract

The radiative capture-cascade equations for hydrogen are solved, taking explicit account of the orbital angular momentum degeneracy of the atomic levels. All (n, l) levels with n < 13 are considered, together with the continuum. The method of steepest descents is used to obtain asymptotic expansions of hydrogen dipole matrix elements involving the quantum number l explicitly, and other methods are also employed to obtain useful approximate expressions. Hydrogen photo-ionization cross-sections are tabulated for all (n, l) levels with n < 13, and recombination coefficients are given for electron temperatures |${T}_{e}={10}^{4}\,\text{and}\,2\times{10}^{4}\,{^\circ}\text{K}$|⁠. The cascade equations are solved for both case A and case B and |${T}_{e}={10}^{4}\,\text{and}\,2\times{10}^{4}{^\circ}\text{K}$|⁠. The separate l levels for a given n are very far from being populated in proportion to their statistical weights. The present calculations give, for both case A and case B, Balmer decrements similar to those obtained by Baker and Menzel for case B. The observed Balmer decrements of six planetary nebulae, when corrected for reddening, are in good agreement with the calculated decrements.