On the Li-Rosmej analytical formula for energy level shifts in dense plasmas

TL;DR

This paper revisits the analytical formula for energy level shifts in dense plasmas, correcting previous inaccuracies by deriving a new, consistent expression based on existing mathematical formulas.

Contribution

It provides a simple, accurate analytical expression for the expectation value r^{3/2} and refines the formula for plasma-induced energy level shifts within Rosmej's framework.

Findings

Derived a simple expression for r^{3/2} using Shertzer's formula.

Provided a corrected analytical formula for energy level shifts in dense plasmas.

Clarified inconsistencies in previous models of plasma screening effects.

Abstract

Li and Rosmej derived analytical fits for the energy level shifts due to plasma screening on the basis of a free-electron potential published by Rosmej et al. one year earlier. The derivation of the fits, which were shown by Iglesias to be inconsistent with the fundamental premise of the ion-sphere model, was motivated by the belief that no analytical expression exists for the expectation value $\langle r^{3/2}\rangle$, an assertion that was also contradicted by Iglesias. In this short note, I point out that a simple expression for the latter quantity can be obtained as a particular case of a formula published by Shertzer, and I provide a corresponding compact analytical expression for the level shifts in the framework of Rosmej's formalism.

Equations

$$V_{\mathrm{f}}(r)=4\pi n_e\left(R_{\mathrm{ws}}\right)\left\{\frac{R_{\mathrm{ws}}^2}{2}-\frac{r^2}{6}+\frac{4}{3\sqrt{\pi }}{\left[\frac{\left(Z-N_{\mathrm{b}}\right)}{k_B{T}_e}\right]}^{1/2}{R}_{\mathrm{ws}}^{3/2}-\frac{8}{15\sqrt{\pi }}{\left[\frac{\left(Z-N_{\mathrm{b}}\right)}{k_B{T}_e}\right]}^{1/2}{r}^{3/2}\right\},$$

$$R_{\mathrm{ws}}={\left[\frac{3\left(Z-N_{\mathrm{b}}\right)}{4\pi n_e\left(R_{\mathrm{ws}}\right)}\right]}^{1/3}.$$

$$\Delta {ϵ}_{n\ell }=⟨n\ell |V_{\mathrm{f}}(r)|n\ell ⟩={\int }_0^{\infty }{V}_{\mathrm{f}}(r)R_{n\ell }^2\left(r;Z_{\mathrm{eff}}\right)r^{2}dr,$$

$$⟨r^2⟩=\frac{n^2}{2Z_{\mathrm{eff}}^2}\left[5n^2+1-3\ell (\ell +1)\right]$$

$$⟨r⟩=\frac{1}{2Z_{\mathrm{eff}}}\left[3n^2-\ell (\ell +1)\right].$$

$$⟨n\ell |r^{\beta }|n{\ell }^{\prime }⟩=A_{n,\ell ,{\ell }^{\prime }}\sum _{i=0}^{n-\ell -1}\frac{(-1{)}^i\Gamma (\ell +{\ell }^{\prime }+3+i+\beta )}{i!(2\ell +1+i)!(n-\ell -i-1)!}\frac{\Gamma (\ell -{\ell }^{\prime }+2+i+\beta )}{\Gamma (\ell +3-n+i+\beta )}$$

$$A_{n,\ell ,{\ell }^{\prime }}=\frac{(-1{)}^{n-{\ell }^{\prime }-1}}{2n}{\left(\frac{n}{2Z_{\mathrm{eff}}}\right)}^{\beta }{\left[\frac{(n+\ell )!(n-\ell -1)!}{(n+{\ell }^{\prime })!(n-{\ell }^{\prime }-1)!}\right]}^{1/2},$$

$$⟨r^{3/2}⟩=\frac{(-1{)}^{n-\ell -1}}{2n}{\left(\frac{n}{2Z_{\mathrm{eff}}}\right)}^{3/2}\sum _{i=0}^{n-\ell -1}\frac{(-1{)}^i\Gamma (2\ell +9/2+i)}{i!(2\ell +1+i)!(n-\ell -i-1)!}\frac{\Gamma (7/2+i)}{\Gamma (\ell +9/2-n+i)}.$$

$$\Delta {ϵ}_{n\ell }$$

Full text

1 Introduction

In order to take into account plasma density effects on bound energy levels, several analytical formulas were obtained (see for instance the non-exhaustive list of references [1, 2, 3, 4]) to approach ion-sphere potentials. Applying first-order perturbation theory to them together with hydrogenic-scaled mean ionization yields analytical formulas to estimate level shifts. In 2011, Rosmej et al. proposed an asymptotic expansion to express the free-electron screening potential in finite-temperature plasmas in a closed analytical expression [5] (in atomic units):

[TABLE]

where $n_{e}\left(R_{\mathrm{ws}}\right)$ is the free-electron density at the Wigner-Seitz radius (radius of the ion-sphere), $N_{\mathrm{b}}$ the number of bound electrons, $k_{B}$ the Boltzmann constant and $T_{e}$ the electron temperature. We have

[TABLE]

The energy shift of $n\ell$ subshell is then obtained by

[TABLE]

where $R_{n\ell}\left(r;Z_{\mathrm{eff}}\right)$ represents the radial part of the hydrogenic wavefunction of the subshell with effective nuclear charge $Z_{\mathrm{eff}}$. The formula involves quantities such as

[TABLE]

and $\langle r^{3/2}\rangle$ (I use the simplified notation $\langle n\ell|f(r)|n\ell\rangle=\langle f(r)\rangle$). In 2012, Li and Rosmej, convinced that no formula exists for $\langle r^{3/2}\rangle$, derived an alternative analytical fit for $V_{\mathrm{f}}(r)$, depending only on $\langle r^{2}\rangle$ and on $\langle r\rangle$, which is known to be:

[TABLE]

It was shown very recently by Iglesias that the fit published by Li and Rosmej was inconsistent with the ion-sphere model [6], on the contrary to the “original” potential of Eq. (1). In addition, the remark of Li and Rosmej, which led to the work of Ref. [2], is not correct: $\langle r^{3/2}\rangle$ can definitely be expressed analytically. This was also pointed out in Ref. [6], where the author indicates that such a quantity can be evaluated analytically, following the procedure given in Appendix E of Ref. [7] using the generating-function formalism (see for instance the textbook by Bransden and Joachain [8]) and yielding to a complicated expression (in the same paper, a table is provided with particular values displayed in the form of rational fractions). I would like to mention here that a simple expression for $\langle r^{3/2}\rangle$ exists. It is a particular case of a relation published by Shertzer in 1991 [9], who provided an expression for $\langle n\ell|r^{\beta}|n\ell^{\prime}\rangle$ for arbitrary $\beta$:

[TABLE]

and

[TABLE]

applying therefore also for off-diagonal terms ($\ell\neq\ell^{\prime}$). $\Gamma$ is the usual Gamma function, which evaluation was widely treated in the literature (see for instance the recent simple and efficient approximation given by Chen [10]). In the present case, we have $\ell=\ell^{\prime}$ and $\beta=3/2$, and we get

[TABLE]

Is is worth mentioning that a relativistic equivalent of Eq. (6) for $\ell=\ell^{\prime}$ was published by Salamin in 1995 [11]. Inserting Eqs. (4) and (8) in Eq. (3) gives

[TABLE]

The potential first published by Rosmej et al. in 2011 [5] and which is consistent with the fundamental neutrality requirement of the ion-sphere model as shown by Iglesias [6], can therefore be directly used to derive simple analytical formulas to estimate energy level shifts in dense plasmas. The alternative fit by Li and Rosmej [2], which is not consistent with the ion-sphere model, was motivated by the belief that no analytical expression exists for $\langle r^{3/2}\rangle$, a statement that was invalidated by Iglesias as well. In this short note, I pointed out that a compact analytical expression can be obtained for that quantity, as a particular case of a relation published by Shertzer, and the resulting formula for the energy level shift due do plasma screening effects was given, for a direct use in atomic-structure codes.