1 Introduction

Recombination lines of hydrogen and helium in the optical, IR and UV spectral regions are frequently used in astronomical spectroscopy for measurements of radial velocities and velocity fields. The levels of hydrogenic ions are not quite degenerate, and lines such as H$\alpha$ ($n=3\rightarrow2$) and He II 4686 Å ($n=4\rightarrow3$) consists of many components; for example for H$\alpha$there are seven components, with transitions between the 3d²D_5/2,3/2, 3p²P_3/2,1/2 and 3s²S_1/2 in n=3 and 2p²P_3/2,1/2 and 2s²S_1/2 in n=2, subject to the usual selection rules for dipole allowed transitions. He II $\lambda$4686 Å has thirteen components.

The precise energies of these levels in hydrogenic ions depend on relativistic shifts as well as fine-structure effects derived from the spin-orbit interaction and quantum electrodynamic effects such as the Lamb shift. For example, the 2s²S_1/2 and 2p²P_1/2 states of H I are separated by 0.0353cm^-1 due to QED effects, while the 2p²P_1/2 and 2p²P_3/2 states are separated by 0.3659cm^-1 due to the spin-orbit interaction.

Unless the line components are resolved, their effect is to broaden the spectral lines and to produce small shifts in the effective position of the line centre, depending on the distribution of intensities between the components. As we show below, the range covered by the line components is 9.1 kms^-1 for hydrogen and 36.6 kms^-1 for helium lines. In an atom of mass m at temperature $T_\mathrm{e}$, the full line width (FWHM) of a single component in velocity units due to thermal broadening is $(8kT_\mathrm{e}\,\ln 2/m)^{1/2}$. For H I and He II, this width has the values of 21.4 and 10.7 kms^-1 at 10⁴K, and 6.8 and 3.4 kms^-1 at 10³K, respectively. Hence the fine-structure broadening can affect line widths of hydrogen and especially helium recombination lines in cool ($T_\mathrm{e}\,\sim$10³K) plasmas. Although many planetary nebulae are somewhat warmer, with $T_\mathrm{e}\,\sim$10⁴K, H II regions often have $T_\mathrm{e}\,\sim$5000K and some old novae have electron temperatures as low as 500K (e.g., DQ Her, Williams et al. 1978).

Because of the combined effects of thermal, turbulent and expansion broadening, we do not expect these components to be completely resolved in observations of ionized plasmas in space. However, they are resolved in laboratory experiments using laser techniques (e.g., Hänsch et al. 1975).

We give several examples of the significance of these splittings. Dyson & Meaburn (1971) showed that the fine-structure of the H$\alpha$ line profile makes a significant difference to derived electron temperatures in nebulae when these are obtained from the ratio of observed widths of the [N II]6584 Å and H$\alpha$ lines. In a typical example, the temperature derived for the Orion region changed from 6200 to 7350K, which would alter the derived O/H ratio (from [O II]3727 Å) by a factor 2.6. In the present paper we update the H$\alpha$ - [N II] correction scheme.

A second application is provided by supernova 1987A. Cumming & Meikle (1993) detected a remarkable short-lived narrow component in H$\alpha$ and H$\beta$ emission from the circumstellar medium of the supernova. In some spatial positions the narrow component's FWHM was as low as $5.3\,\pm\,1.3 \;\mathrm{km}\,\mathrm{s}^{-1}$, which after allowance for the H$\alpha$ fine-structure suggests emission from extremely cool recombining gas. In fact, the formal best fit at some positions was narrower than the spread of the fine-structure components. Of several models discussed, one involved illumination of a circumstellar H I cloud by the H Ly$\beta$ emission line (absorption of Ly$\beta$ would only populate the 3p level, and thus the number of components making up the H$\alpha$ line would be reduced).

Other studies have revealed narrow emission lines from ionized regions. Onello & Phillips (1993) observed emission components with FWHM as low as $3.6\;\mathrm{km}\,\mathrm{s}^{-1}$ in the H168$\alpha$ recombination line towards galactic sources; such components if seen in the Balmer lines would be affected by the fine-structure we discuss here. Gallagher & Hunter (1983) studied widths of the H$\alpha$ line in extragalactic H II regions. Observed widths were corrected for instrumental and thermal broadening, and the "excess widths'' due to the bulk gas velocity dispersion were as small as $1.6 - 13.0\;\mathrm{km}\,\mathrm{s}^{-1}$ (FWHM) for some positions in NGC4214. Correction for fine-structure broadening would reduce these excesses still further.

Lastly, we show below that the He II line at 1640 Å has seven components spanning a range of $36.6 \; \mathrm{km}\,\mathrm{s}^{-1}$ (with the two strongest components $26 \; \mathrm{km}\,\mathrm{s}^{-1}$ apart). Laming & Feldman (1993) analysed SKYLAB spectra of a Solar Prominence which partially resolved the 7 components, and were able to derive information on the process populating the He II n=3 levels in the hot plasma.

Calculations of the recombination spectra of hydrogenic ions in Case B of Baker & Menzel (1938) have been described by Hummer & Storey (1987), Storey & Hummer (1988) and Storey & Hummer (1995) (hereafter HS, SH and SH95). Their calculations take full account of all radiative and collisional processes, and they tabulated emissivities at a range of electron temperatures and densities for principal quantum numbers up to n=50.

Martin (1988) presented effective recombination coefficients for H I and He II states as a function of n and $\ell$, from which the relative intensities of fine-structure components can be derived, but only for the "zero density'' case, in which all collisional processes are ignored. Effective recombination coefficients for hydrogen have also been calculated by Smits (1991) as a function of both electron temperature and density. His calculations extend the calculations of Hummer & Storey (1987) to low electron temperatures, $312.5\, \mathrm{K} \leq T_{\mathrm e} \leq 2500 \, \mathrm{K}$, using essentially the same methods. He does not, however, treat He II, or give the $\ell$-dependent information necessary to calculate the intensities of the fine-structure components.

In this paper we summarize the effects of the fine-structure components and present new calculations of their relative intensities, as a function of electron temperature and density, and in Cases A and B. We also include synthetic spectra showing the components in velocity space, and tabulate the shifts of the line centres for different densities, temperatures and Cases.