# 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.

## Key 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.