Angle dependent total cross sections and the optical theorem

Cross sections are either represented by generalized asymptotical partial w ave expansions or obtained as a spherical average of an appropriate differe ntial cross section. In these cases it is usually assumed that the total sc attering cross section, as a property of a scattering object, does not depe nd on the incident angles. This viewpoint is supported by common knowledge in connection with low energy scattering. However this unconscious belief i s not always correct. In the present paper we will show that a non-spherica l scatterer may exhibit strong dependence on the incident direction. To do this we will represent the scattering data of the most general potential, s eparable in ellipsoidal coordinates, in perturbed ellipsoidal (Lame) wave f unctions. These functions arise when variables in the Schrodinger equation are separated in an ellipsoidal coordinate system. The Lame wave functions are analogous to spherical- and Bessel functions in the case of spherical s ymmetry. We will expand the total scattering cross section and derive the o ptical theorem explicitly demonstrating the incident angle dependence for s uch a class of potentials. As an illustration we will present and display s ome calculations of the total cross section versus incident direction. Unex pected behavior will be discussed and explained. We also use results from c lassical acoustic scattering by a triaxial ellipsoid. The general character of the ellipsoidal coordinate system is emphasized. (C) 2001 Elsevier Scie nce Ltd. All rights reserved.