Analytic phase shifts for yukawa potentials

The Yukawa or Debye‐Hückel potential, by a suitable choice of scale factor ( d ) and the use of the corresponding energy unit ( h 2 /2 md 2 ), may be written in the form V ( r:Z ) = –(2 Z/r ) e −r . For Z << 1 or l >> 1 or E = k 2 sufficiently large the partial wave phase shifts are given by the Born approximation which mathematically vanishes at E = 0. On the other hand, Levinson's theorem indicates that a phase shift, in units of π, is equal to n ( Z,l ), the integral number of bound states sustainable by the potential. In this work we first adapt an optical model code written by Relyea for molecular potentials to be suitable for singular attractive potentials such as the Yukawa potential. This modified code is used to obtain the numerical Schrödinger phase shifts for a broad range of Z and l values. By the addition of integers to assure continuity of the phase shifts and the satisfaction of Levinson's theorem we obtain an array of continuous functions which, in effect, interpolate between the Levinson limit and the Born limit. Next we find an accurate analytic characterization of the number of bound states n a ( Z,l ) suggested by the Sommerfeld model. Finally we develop a simple two‐term analytic formula which interpolates between the Levinson limit and the Born limit and fits the rectified set of Schrödinger phase shifts quite well. In effect we arrive at a four‐dimensional function δ( E,l,Z,d ) which should be useful as a convenient approximation in atomic, nuclear, particle, and plasma physics.