Heavy-ion fusion cross section formula and barrier height distribution

Methods for obtaining fusion cross section formulas are discussed, especially for those that take on an empirical form. A new expression starting with $\ensuremath{\sigma}(E)=\frac{1}{E}{\ensuremath{\int}}_{{E}_{0}}^{E}\left({\ensuremath{\int}}_{{E}_{0}}^{{E}^{\ensuremath{'}}}B({E}^{\ensuremath{''}})d{E}^{\ensuremath{''}}\right)d{E}^{\ensuremath{'}}$ has been explored. Here, $B(E)$ is a reasonable, assumed function of the barrier height distribution, $\frac{{d}^{2}(\ensuremath{\sigma}E)}{d{E}^{2}}$. The resulting analytic cross section formula reproduces very well the excitation functions for many light and heavy fusion systems across wide energy ranges, when $B(E)$ is assumed to be a multi-Gaussian function. This study offers an improved determination of the fusion barrier height distribution over other numerical techniques.