Correcting the Input Data for Calculating the Asymmetry of Hydrogenic Spectral Lines in Plasmas

Abstract

We provide corrections to the data in Sholin’s tables from his paper in Optics and Spectroscopy 26 (1969) 27. Since his data was used numerous times by various authors to calculate the asymmetry of hydrogenic spectral lines in plasmas, our corrections should motivate revisions of the previous calculations of the asymmetry and its comparison with the experimental asymmetry, and thus should have a practical importance.

The input data presented in Sholin’s tables from paper [1] was used numerous times by various authors to calculate the asymmetry of hydrogenic spectral lines in plasmas. (For the latest advances in the theory of the asymmetry we refer to papers [2,3] and references therein). However, we found that there are incorrect entries tabulated in paper [1] for the the Ly-γ, Ly-$ϵ$, and H-$\alpha$ lines, in both the intensity corrections and the quadrupole frequency corrections.

The dipole and quadrupole frequency corrections are given in paper [1] as

$${\Delta }_k^{dipole}=nq-n^{\prime }{q}^{\prime },$$

(1)

and

$${\Delta }_k^{quadrup}=\frac{1}{3}[n^4-n^2-6n^2{q}^2-{n^{\prime }}^4+{n^{\prime }}^2+6{n^{\prime }}^2{q^{\prime }}^2],$$

(2)

where n and $n^{\prime }$ are the principal quantum numbers of the upper and lower energy levels, respectively; $q=n_1-n_2$ and $q^{\prime }={n_1}^{\prime }-{n_2}^{\prime }$ are the combinations of the corresponding parabolic quantum numbers.

Frequency Corrections

For Ly-gamma (n = 4), Equation (2) becomes:

$${\Delta }_k^{quadrup}\left(q\right)=80-32q^2$$

(3)

It yields ${\Delta }_k^{quadrup}\left(0\right)=80$, ${\Delta }_k^{quadrup}\left(\pm 1\right)=48$, ${\Delta }_k^{quadrup}\left(\pm 2\right)=-48$, ${\Delta }_k^{quadrup}\left(\pm 3\right)=-208$. The comparison shows that in Sholin’s table there are typographic errors in ${\Delta }_k^{quadrup}\left(0\right)$ entered as 60 (instead of 80) and in ${\Delta }_k^{quadrup}\left(\pm 3\right)$ entered as −206 (instead of −208).

For Ly-epsilon (n = 4), Equation (2) becomes:

$${\Delta }_k^{quadrup}\left(q\right)=420-72q^2$$

(4)

It yields ${\Delta }_k^{quadrup}\left(0\right)=420$, ${\Delta }_k^{quadrup}\left(\pm 1\right)=348$, ${\Delta }_k^{quadrup}\left(\pm 2\right)=132$, ${\Delta }_k^{quadrup}\left(\pm 3\right)=-228$, ${\Delta }_k^{quadrup}\left(\pm 4\right)=-732$, ${\Delta }_k^{quadrup}\left(\pm 3\right)=-1380$. The comparison shows that in Sholin’s table there are typographic errors in ${\Delta }_k^{quadrup}\left(\pm 2\right)$ entered as 108 (instead of 132).

Intensity Corrections

The intensity corrections are calculated from the corresponding corrections to the wave functions. The latter are given, e.g., in the Appendix of paper [4].

For H-alpha (n = 3 to n = 2 transition), the comparison shows that in Sholin’s table there are typographic errors in ${ϵ}_k{}^{\left(1\right)}$ corresponding to ${\Delta }_k^{dipole}=2$ entered as −62 (instead of −62/9) and ${\Delta }_k^{dipole}=-2$ entered as 62 (instead of 62/9), as shown in detail below.

For ${\Delta }_k^{dipole}=2$:

$$\begin{array}{l}{I}_k=<{110}^{″}\left|\mathrm{z}\right|{010}^{″}{>}^2\\ =(<110\left|z\right|010>-3\frac{a_o}{R}<200\left|z\right|010>+3\frac{a_o}{R}<020\left|z\right|010\\ >-\frac{a_o}{R}<110\left|z\right|100>+3\frac{a_o{}^2}{R^2}<200\left|z\right|100>-3\frac{a_o{}^2}{R^2}<020|z|100\\ >{)}^2\approx <110\left|z\right|010{>}^2-6\frac{a_o}{R}<200\left|z\right|010>+6\frac{a_o}{R}<020\left|z\right|010\\ >-2\frac{a_o}{R}<110\left|z\right|100>=I_k{}^{\left(0\right)}(1-\frac{a_o}{R}\frac{62}{9}).\hfill \end{array}$$

(5)

For ${\Delta }_k^{dipole}=-2$:

$$\begin{array}{l}{I}_k=<{110}^{″}|\mathrm{z}|{100}^{″}{>}^2\\ =(<110|z|100>-3\frac{a_o}{R}<200|z|100>+3\frac{a_o}{R}<020|z|100\\ >+\frac{a_o}{R}<110|z|010>-3\frac{a_o{}^2}{R^2}<200|z|010>+3\frac{a_o{}^2}{R^2}<020|z|010\\ >{)}^2\approx <110|z|100{>}^2-6\frac{a_o}{R}<200|z|100>+6\frac{a_o}{R}<020|z|100\\ >+2\frac{a_o}{R}<110|z|010>=I_k{}^{\left(0\right)}(1+\frac{a_o}{R}\frac{62}{9}).\end{array}$$

(6)

We note in passing that the robust perturbation theory, as developed by Oks and Uzer [5], allows for analytically calculating corrections to the eigenfunctions due to the quadrupole interaction in a much simpler way than in Sholin paper [1]. Details are presented in Appendix.

For completeness, we list below also previously known (for a long time) corrections to the tabulated entries from paper [1] for the H-beta line.

For the Stark components corresponding to the radiative transitions between the parabolic states 210 and 010 or between 120 and 100, the unperturbed intensity should be 81, instead of 16.

For the Stark component corresponding to the radiative transition between the parabolic states 210 and 001, the intensity correction ${ϵ}_k{}^{\left(1\right)}$ should be −20 (instead of −16).

For the Stark component corresponding to the radiative transition between the parabolic states 120 and 001, the intensity correction ${ϵ}_k{}^{\left(1\right)}$ should be 20 (instead of 16).

There are also two corrections (known for a long time) to the following typographic errors from paper [1].

In Table 2 from [1] for the H-alpha line, in the header of the last column, the scaling factor should be 10⁶ instead of 10⁵.

In Equation (21) from [1], in its 2nd term in the right hand side, the coefficient should be (3/8) instead of (3/16). We note that after this correction, Equation (21) from [1] coincides with the corresponding term (proportional to 1/R⁴) in Equation (4.59) from book [5] after setting in the latter Z₁ = 1, Z₂ = Z. Equation (4.59) from book [5] was derived from the exact expression for the energy in elliptical coordinates for the two Coulomb center problem by expanding the latter in powers of 1/R up to (including) the term ~1/R⁶. Therefore, Equation (4.59) from book [5] can be considered, in particular, as the benchmark for testing Equation (21) from [1]. Such a test also confirms that the 2nd term in the right hand side of Equation (21) from [1] correctly contains the first power of Z (while there were incorrect suggestions that this term should contain Z²).

In summary, since Sholin’s input data from paper [1] was used numerous times by various authors to calculate the asymmetry of hydrogenic spectral lines in plasmas, our corrections should motivate revisions of the previous calculations of the asymmetry and its comparison with the experimental asymmetry, and thus should have a practical importance.