A study of the Pais variational phaseshift approximation

Abstract

The accuracy of phaseshifts calculated by means of the Pais variational procedure is investigated and the method is employed to calculate partial derivatives of the phase shifts with respect to a number of potential parameters. One of these, the partial derivative of the phase shift delta L with respect to the orbital angular momentum L, plays an essential role in the semiclassical approach to scattering. Another is the partial derivative delta delta L(k)/ delta E, where E is the energy, which is related to physical quantities like the time delay, virial coefficients and resonance widths. The relative simplicity of the numerical calculations involved in the Pais approach and the fact that it gives accurate results for the range of values of L that are important in semiclassical calculations makes the use of Pais phaseshifts an attractive possibility in such calculations. The formalism that results has possible application to both electron scattering from atoms and molecules and to heavy ion scattering in nuclear physics.

An Extension of the Pais Variational Phase Shift Approximation

“…It was originally applied by Abraham Pais, who was also a very great science biographer in particular of Einstein, in 1946 to the 1 E-mail: william romo@carleton.ca 2 E-mail: valluri@uwo.ca nucleon-nucleon scattering problem with a potential interaction of the Yukawa type, also known as the Debye-Hückel potential. The accuracy of the Pais variational phase shift procedure was earlier investigated [22] by the present authors. The procedure gives reasonably accurate approximations for all phase shifts except those associated with the lowest partial waves.…”

“…This set of parameters provides a crude description of a system in which an 87 keV electron scatters from a Thallium ion with charge plus one. The same set of potential parameters was used for the Yukawa interaction in the numerical examples of one of our earlier papers [22]. However, in that paper the two bodies involved had no long-range Coulomb interaction and the standard Pais method was appropriate.…”

“…For the third set of calculations we chose a Gaussian potential V I (r) = V o exp(−r 2 /R 2 ), with V o = 15.9 keV and R = 0.109Å. This is the same set of parameters used for the Gaussian potential in [22], and as was the case in that paper, the set has no special physical significance other than to produce a set of phase shifts with roughly the same magnitude as in the Yukawa case. For each of the three choices of residual interactions, phase shifts were calculated by three different methods: first the radial Schrödinger differential equations were integrated numerically, for a number of partial waves, out to large values of r where phase shifts were extracted from the asymptotic forms of the resulting radial wave functions, next phase shifts were obtained in first order Born approximation by applying eqs.…”

“…To evaluate a phase shift by the extended Pais method one must be able to evaluate the function S(j, k) for arbitrary values of j. This task was facilitated in our previous paper [22] by the fact that the integral on the right-hand side of eq. ( 10), which defines S, can be expressed as a known analytic function when V I (r) is either a Yukawa potential, an exponential potential, or a Gaussian potential and vanishes, i.e., when there is no long-range Coulomb potential acting between the two particles.…”

“…Eqs. (47) and (49) reduce to expressions in [22] if is set equal to zero. The Hulthén potential which can be expressed as an infinite sum of exponential potentials and is a special case of the Eckart potential [30], can also be treated by the extended Pais method.…”