Calculation of the energy-averaged scattering function from high resolution low-energy neutron scattering data

The elastic scattering neutron cross section at low energy displays many narrow resonances. By definition, the optical-model scattering function is equal to a suitable energy average of the actual scattering function. This average should be independent of the particular representation chosen for parametrizing the actual scattering function and also fairly independent of the averaging weight function. In particular, it should be fairly independent of the width of the weight function, even though a sparsity of resonances might require the width to be comparable to the energy range of the measurements. It is shown by direct numerical calculations of the average that this holds in the example of the scattering of $p_{\frac{3}{2}}$ neutrons by ${}_{}{}^{32}{}_{}{}^{}\mathrm{S}$. Attention is drawn to pitfalls which exist when this average is evaluated analytically from parametrized forms of the scattering function. Several ways of graphically representing or of parametrizing the average scattering function are illustrated and discussed.