# Thermodynamically-consistent semi-classical $\ell$-changing rates

## TL;DR

This paper compares semi-classical and quantum-mechanical methods for calculating angular-momentum changing rates in Rydberg atom collisions, finding good agreement and implications for early universe recombination modeling.

## Contribution

It introduces a revised semi-classical formalism that aligns well with quantum results, addressing divergence issues in impact parameter integration.

## Key findings

- Revised semi-classical formalism agrees with quantum results.
- Quantum rates are suitable for modeling recombination in the early universe.
- Identifies the cause of divergence in impact parameter integration as finite collision time.

## Abstract

We compare the results of the semi-classical (SC) and quantum-mechanical (QM) formalisms for angular-momentum changing transitions in Rydberg atom collisions given by Vrinceanu & Flannery, J. Phys. B 34, L1 (2001), and Vrinceanu, Onofrio & Sadeghpour, ApJ 747, 56 (2012), with those of the SC formalism using a modified Monte Carlo realization. We find that this revised SC formalism agrees well with the QM results. This provides further evidence that the rates derived from the QM treatment are appropriate to be used when modelling recombination through Rydberg cascades, an important process in understanding the state of material in the early universe. The rates for $\Delta\ell=\pm1$ derived from the QM formalism diverge when integrated to sufficiently large impact parameter, $b$. Further to the empirical limits to the $b$ integration suggested by Pengelly & Seaton, MNRAS 127, 165 (1964), we suggest that the fundamental issue causing this divergence in the theory is that it does not fully cater for the finite time taken for such distant collisions to complete.