2 Adopted energy levels and "reference'' wavelengths

For H I and He II energy levels we adopt the values given by Erickson (1977). These are based on quantum electrodynamic calculations including the Lamb shift. This work makes a careful comparison between experimental and calculated energy-level differences, and demonstrates good agreement. In general the theoretical QED uncertainty is smaller than the experimental error.

For each transition of H I and He II considered, we adopted a reference wavelength, $\lambda_\mathrm {ref}$, for use as a zero-point in the presentation of the shifts of individual component lines in velocity space. These were computed from the Rydberg formula  

$$\lambda_\mathrm{ref} = \left[ \mu R_{\infty} \left( n_\mathrm{l}^{-2} - n_\mathrm{u}^{-2} \right) \right]^{-1},$$ (1)

where $R_{\infty}$=109737.3153 cm^-1 is the Rydberg constant for infinite nuclear mass (Ferguson 1986), and $\mu$ is the reduced mass of the electron-nucleus system in electron masses. The finite mass Rydberg constants, $\mu R_{\infty}$, that were used to calculate the reference wavelengths are 109677.6155, 109707.4343, 109717.3590 and 109722.2772 cm^-1 for ¹H, ²D, ³He and ⁴He respectively.

The difference in energy, $\Delta_{nj}$, between the energy of a hydrogenic state E_n, given by the Rydberg formula and the energy E_nj obtained from the Dirac equation is approximately  

$$\Delta_{nj} = E_{nj} - E_n = - {{Z^4 \alpha^4 }\over{n^4}} \... ...ty} \left( {{n}\over{j+{{1}\over{2}}}} - {{3}\over{4}} \right),$$ (2)

where $\alpha$ is the fine-structure constant and terms in higher powers of $(Z\alpha)$ have been neglected. This equation describes the relativistic energy shifts of the hydrogenic levels (the mass correction and the Darwin term) as well as the fine-structure due to the spin-orbit interaction. The Lamb shift is not included but can be neglected for the argument made here. For a given n, ($j+{{1}\over{2}}$) runs from one to n, so that the bracket in Eq. (2) is always positive. Thus the levels $n\ell j$ always lie energetically lower than the corresponding non-relativistic energies given from the Rydberg formula. In addition, the energy shift $\Delta_{nj}$ falls rapidly as n increases. As a consequence, the fine-structure components in any transition $n_\mathrm{u} \rightarrow n_\mathrm{l}$ will generally be displaced to wavelengths shorter than $\lambda_0$ as given by Eq. (1), and the corresponding velocity shifts will usually be negative.

We ignore hyperfine structure in this work. Its effect is to double the energy levels of ¹H and ³He, which have finite nuclear spin. The typical splitting is only $0.001\; \mathrm{cm}^{-1}$, which corresponds to a velocity splitting of $0.02\; \mathrm{km}\,\mathrm{s}^{-1}$ for optical and UV lines, and is thus neglected.

  

Table 1: Summary of components, and reference wavelengths of hydrogenic transitions

Table 1 lists, for some commonly observed transitions of H and He, the number of components $N_\mathrm{c}$, the total range in velocity space of the components ${\Delta}v$, and the reference wavelength, defined as above, for the four isotopes ¹H, ²D, ³He and ⁴He. We stress that the reference wavelength does not represent any mean wavelength for the line in question, but is simply a reference point to anchor our chosen scale of velocity shifts. Here, and throughout this paper, wavelengths are given in air for $\lambda\gt 2000$ Å and in vacuum otherwise. The variation of the velocity range of the components as a function of nuclear mass (e.g. between ¹H and ²H, or ³He and ⁴He) is too small to be recorded in Table 1.