H, He-like recombination spectra I: $l$-changing collisions for hydrogen

Hydrogen and helium emission lines in nebulae form by radiative recombination. This is a simple process which, in principle, can be described to very high precision. Ratios of He I and H I emission lines can be used to measure the He$^+$/H$^+$ abundance ratio to the same precision as the recombination rate coefficients. This paper investigates the controversy over the correct theory to describe dipole $l$-changing collisions ($nl\rightarrow nl'=l\pm 1$) between energy-degenerate states within an $n$-shell. The work of Pengelly & Seaton (1964) has, for half-a-century, been considered the definitive study which "solved" the problem. Recent work by Vrinceanu et al.(2012) recommended the use of rate coefficients from a semi-classical approximation which are nearly an order of magnitude smaller than those of Pengelly & Seaton (1964), with the result that significantly higher densities are needed for the $nl$ populations to come into local thermodynamic equilibrium. Here, we compare predicted H~I emissivities from the two works and find widespread differences, of up to $\approx 10$%. This far exceeds the 1% precision required to obtain the primordial He/H abundance ratio from observations so as to constrain Big Bang cosmologies. We recommend using the rate coefficients of Pengelly & Seaton (1964) for $l$-changing collisions, to describe the H recombination spectrum, based-on their quantum mechanical representation of the long-range dipole interaction.