Angle dependent total cross sections and the optical theorem

Cross sections are either represented by generalized asymptotical partial wave expansions or obtained as a spherical average of an appropriate differential cross section. In these cases it is usually assumed that the total scattering cross section, as a property of a scattering object, does not depend on the incident angles. This viewpoint is supported by common knowledge in connection with low energy scattering. However this unconscious belief is not always correct. In the present paper we will show that a non-spherical scatterer may exhibit strong dependence on the incident direction. To do this we will represent the scattering data of the most general potential, separable in ellipsoidal coordinates, in perturbed ellipsoidal (Lamé) wave functions. These functions arise when variables in the Schrödinger equation are separated in an ellipsoidal coordinate system. The Lamé wave functions are analogous to spherical- and Bessel functions in the case of spherical symmetry. We will expand the total scattering cross section and derive the optical theorem explicitly demonstrating the incident angle dependence for such a class of potentials. As an illustration we will present and display some calculations of the total cross section versus incident direction. Unexpected behavior will be discussed and explained. We also use results from classical acoustic scattering by a triaxial ellipsoid. The general character of the ellipsoidal coordinate system is emphasized.