# Soft Bremsstrahlung

## TL;DR

This paper examines simple formulas for low-frequency bremsstrahlung energy emission in ionized gases, highlighting limitations of existing formulas and proposing a more accurate, generalized formula based on the soft photon theorem and Born approximation.

## Contribution

It introduces a generalized formula for soft bremsstrahlung emission that accounts for the validity range of the Born approximation and soft photon theorem.

## Key findings

- Existing formulas have limited validity range.
- A generalized formula improves accuracy for soft photon emissions.
- The Born approximation remains valid for sufficiently soft photons.

## Abstract

Simple analytic formulas are considered for the energy radiated in low frequency bremsstrahlung from fully ionized gases. A formula that has been frequently cited over many years turns out to have only a limited range of validity, more narrow than for a formula derived using the Born approximation. In an attempt to find a more widely valid simple formula, a soft photon theorem is employed, which in this context implies that the differential rate of photon emission in an electron-ion collision with definite initial and final electron momenta is correctly given for sufficiently soft photons by the Born approximation, to all orders in the Coulomb potential. Corrections to the Born approximation arise because the upper limit on photon energy for this theorem to apply to a given collision becomes increasingly stringent as the scattering approaches the forward direction. A general formula is suggested that takes this into account.