1. Introduction

We consider the possibility of zeros in the non-relativistic continuum-continuum dipole matrix element for radiative transitions of an electron in the static field of an atom or ion. In previous work [1,2], the existence of zeros in upward transitions from bound states was predicted by calculating the sign of such transition matrix elements in the soft and hard photon limiting cases. Using arguments for the continuity of the matrix element in energy, the existence of zeros then followed whenever the sign was different in the two limiting cases. The present work can be considered a natural extension of these arguments to the case of continuum-continuum transitions.

In Section 2 we examine the behavior of the dipole radial matrix element, holding one electron energy fixed, in the soft photon limit, demonstrating that the sign of the matrix element in this limit can be determined from a knowledge of the phase shifts for elastic scattering in the potential. Next we again fix one electron energy and consider the limiting case where the other electron energy is taken to infinity. Utilizing the knowledge that in this limit the radial matrix element is determined at small distances [3], the sign of the matrix element may be determined by evaluating the radial integral using Coulomb wavefunctions. From the continuity of the matrix element in energy, the existence of zeros in certain cases can then be predicted. Our results for the soft photon case are also suggestive of the intimate connection to elastic scattering phenomena. In Section 3 we discuss this connection and the relation of free-free zeros to Ramsauer-Townsend minima observed in elastic scattering. Finally, in Section 4 we discuss the issue of observability of the predicted zero crossings in these free-free transition matrix elements.