# Modeling penetrating collisions in the standard line broadening impact theory for hydrogen

## TL;DR

This paper derives an analytical expression for penetrating collision integrals in hydrogen line broadening theory, simplifying numerical calculations and enhancing physical understanding of plasma spectral lines at high densities.

## Contribution

It introduces explicit analytical formulas for penetrating collision integrals, improving upon the standard long-range dipole approximation in plasma line broadening models.

## Key findings

- Analytical expression for collision integrals using binomial coefficients and Bessel functions.
- Simplified approximate form for the collision operator.
- Enhanced modeling of spectral lines in high-density plasmas.

## Abstract

Spectral lines emitted by plasmas provide information about the thermodynamic conditions, the degree of randomness or the interactions prevailing in the medium. Collisions by plasma electrons penetrating the extent of bound-electron wavefunctions is important at high density, where short-range interactions become dominant. Such collisions are usually not taken into account properly in the standard lineshape theory, assuming long-range dipole approximation. The formalism of penetrating collisions for hydrogen relies on the introduction of a family of integrals calculated using a recursion relation. In this work, we show that such integrals can be expressed analytically, as a finite sum involving binomial coefficients and modified Bessel functions of the third kind. The explicit expression enabled us to obtain a simple approximate analytical form for the collision operator, making numerical implementation and physical interpretation easier. We also propose simple analytical forms of coefficients and integrals important for the modeling of penetrating collisions.