# Approximate transmission coefficients in heavy ion fusion

## TL;DR

This paper evaluates various analytical and semi-analytical methods for estimating transmission coefficients in heavy ion fusion, comparing their accuracy against quantum mechanical results at near-barrier energies.

## Contribution

It introduces an extension of Kemble's approximation to above-barrier energies using complex plane analysis and compares multiple approximation methods for heavy ion tunnelling.

## Key findings

- Kemble's approximation extended to above-barrier energies shows good agreement with quantum results.
- Analytical shape approximations provide useful estimates of transmission coefficients.
- Comparison reveals the validity range of each approximation method.

## Abstract

In this paper we revisit the one-dimensional tunnelling problem. We consider different approximations for the transmission through the Coulomb barrier in heavy ion collisions at near-barrier energies. First, we discuss approximations of the barrier shape by functional forms where the transmission coefficient is known analytically. Then, we consider Kemble's approximation for the transmission coefficient. We show how this approximation can be extended to above-barrier energies by performing the analytical continuation of the radial coordinate to the complex plane. We investigate the validity of the different approximations considered in this paper by comparing their predictions for transmission coefficients and cross sections of three heavy ion systems with the corresponding quantum mechanical results.