Checking the reliability of opacity databases

Abstract

Mathematical inequalities, combined with atomic-physics sum rules, enable one to derive lower and upper bounds for the Rosseland and/or Planck mean opacities. The resulting constraints must be satisfied, either for pure elements or mixtures. The intriguing law of anomalous numbers, also named Benford’s law, is of great interest to detect errors in line-strength collections required for fine-structure calculations. Testing regularities may reveal hidden properties, such as the fractal nature of complex atomic spectra. The aforementioned constraints can also be useful to assess the reliability of experimental measurements. Finally, we recall that it is important to quantify the uncertainties due to interpolations in density-temperature opacity (or more generally atomic-data) tables, and that convergence studies are of course unavoidable in order to address the issue of completeness in terms of levels, configurations or superconfigurations, which is a cornerstone of opacity calculations.

Graphical abstract

Contribution of the Klein–Nishina scattering cross section to the Rosseland mean

We have seen that the Thomson opacity reads

$$\begin{aligned} \kappa _{\textrm{Th}}=\frac{8\pi }{3}\left( \frac{e^2}{4\pi \epsilon _0mc^2}\right) ^2\frac{Z^*}{A}\mathcal {N}_A. \end{aligned}$$

(1)

Let us introduce the reduced parameter

$$\begin{aligned} \gamma =\frac{h\nu }{mc^2}. \end{aligned}$$

(2)

The Klein–Nishina relativistic cross section [67] reads

$$\begin{aligned} \kappa _{\textrm{KN}}={} & {} \kappa _{\textrm{Th}}\left\{ \frac{1+\gamma }{\gamma ^2}\left[ \frac{2(1+\gamma )}{2\gamma +1}-\frac{1}{\gamma }\ln (2\gamma +1)\right] \right. \nonumber \\{} & {} \left. +\frac{1}{2\gamma }\ln (2\gamma +1)-\frac{3\gamma +1}{(2\gamma +1)^2}\right\} . \end{aligned}$$

(3)

If $\gamma \ll 1$, one has

$$\begin{aligned} \kappa _{\textrm{KN}}=\kappa _{\textrm{Th}}\left( 1-2\gamma +\frac{26}{5}\gamma ^2+\cdots \right) \end{aligned}$$

(4)

and if $\gamma \gg 1$:

$$\begin{aligned} \kappa _{\textrm{KN}}\approx \frac{3}{8\gamma }\kappa _{\textrm{Th}}\left[ \ln (2\gamma )+\frac{1}{2}\right] . \end{aligned}$$

(5)

Let us assume that the relativistic effects are negligible. We can then use Eq. (4) yielding

$$\begin{aligned} \frac{1}{\kappa _{\textrm{KN}}}\approx \frac{1}{\kappa _{\textrm{Th}}}\left( 1+2\gamma -\frac{6}{5}\gamma ^2\right) . \end{aligned}$$

(6)

Using the reduced variable $u=h\nu /(k_BT)$, one has for the Rosseland mean in the case of the Klein–Nishina expression of the scattering cross section

$$\begin{aligned} \kappa _{\textrm{R,KN}}= & {} \frac{4\pi ^4}{15}\kappa _{\textrm{Th}}\left[ \int _0^{\infty }\frac{u^4e^{-u}}{(1-e^{-u})^2}\left( 1+\frac{2k_BT}{mc^2}u\right. \right. \nonumber \\{} & {} \left. \left. -\frac{6}{5}\left( \frac{k_BT}{mc^2}\right) ^2u^2\right) \textrm{d}u\right] ^{-1} \end{aligned}$$

(7)

i.e.,

$$\begin{aligned} \kappa _{\textrm{R,KN}}= & {} \frac{4\pi ^4}{15}\kappa _{\textrm{Th}}\left[ \int _0^{\infty }\frac{u^4e^{-u}}{(1-e^{-u})^2}\textrm{d}u\right. \nonumber \\{} & {} \left. +2\frac{k_BT}{mc^2}\int _0^{\infty }\frac{u^5e^{-u}}{(1-e^{-u})^2}\textrm{d}u\right. \nonumber \\{} & {} \left. -\frac{6}{5}\left( \frac{k_BT}{mc^2}\right) ^2\int _0^{\infty }\frac{u^6e^{-u}}{(1-e^{-u})^2}\textrm{d}u\right] ^{-1}. \end{aligned}$$

(8)

Using

$$\begin{aligned} \int _0^{\infty }\frac{u^4e^{-u}}{(1-e^{-u})^2}\textrm{d}u=\frac{4\pi ^4}{15} \end{aligned}$$

(9)

as well as

$$\begin{aligned} \int _0^{\infty }\frac{u^5e^{-u}}{(1-e^{-u})^2}\textrm{d}u=120~\zeta (5) \end{aligned}$$

(10)

and

$$\begin{aligned} \int _0^{\infty }\frac{u^6e^{-u}}{(1-e^{-u})^2}\textrm{d}u=\frac{16\pi ^6}{21}, \end{aligned}$$

(11)

one finally obtains

$$\begin{aligned} \kappa _{\textrm{R,KN}}=\kappa _{\textrm{Th}}\left[ 1+\frac{900~\zeta (5)}{\pi ^4}\left( \frac{k_BT}{mc^2}\right) -\frac{24\pi ^2}{7}\left( \frac{k_BT}{mc^2}\right) ^2\right] \end{aligned}$$

(12)

and using [68, 69]:

$$\begin{aligned} \zeta (5)= & {} \frac{\pi ^5}{294}-\frac{72}{35}\sum _{n=1}^{\infty }\frac{1}{n^5\left( e^{2\pi n}-1\right) }\nonumber \\{} & {} -\frac{2}{35}\sum _{n=1}^{\infty }\frac{1}{n^5\left( e^{2\pi n}+1\right) }\approx 1.03692, \end{aligned}$$

(13)

one has

$$\begin{aligned} \kappa _{\textrm{R,KN}}\approx \kappa _{\textrm{Th}}\left[ 1+9.58057\left( \frac{k_BT}{mc^2}\right) -33.8386\left( \frac{k_BT}{mc^2}\right) ^2\right] . \end{aligned}$$

(14)