Accurate momentum-space method for scattering by nuclear and Coulomb potentials

Abstract

A coordinate-space boundaxy-matching method is used to solve the problem of including the long-range Coulomb interaction in momentum-space calculations of elastic scattering.There has been considerable interest in momentum-space methods for solving the LippmannSchwinger equation for scattering problems. Although traditionally one solves the Schrodinger equation in coordinate space, and obtains phase shifts and differential cross sections from the solution of the Schrodinger equation, it is often advantageous to solve the scattering problem in momentum space. Particularly when one has to deal with nonlocal potentials, it may be easier to work in momentum space. One example is pionnucleus scattering, where the optical potential derived from a multiple scattering theory is nonlocal. ' Another example is the treatment of heavyion collisions, ' in which the Pauli principle requires a strongly nonlocal optical potential.The difficulty of incorporating the Coulomb interaction in momentum-space methods has discouraged their use, in spite of the attempts of several previous authors to solve this problem. One approach is to include the Coulomb potential in the Green's function that appears in the LippmannSchwinger equation, so that only the short-ranged part of the interaction need be represented in momentum space. The required momentum-space Coulomb Green's function is easily calculated'; however, its singularity is not a simple pole, which could be treated by a subtraction method, ' but a complex power with negative real part, whose infinitely rapid oscillations demand the development of entirely new methods. A second approach uses a free Green's function, but adds the momentum-space representation of the Coulomb interaction to the nuclear interaction. Unfortunately, standard momentum-space methods then become invalid, because the Coulomb matrix elements (k( Vc (k') are singular at k =%, '. A mathematically correct (but numerically inefficient) method of overcoming this difficulty is to introduce an exponential screening factor in the Coulomb potential, and extrapolate calculated cross sections to the limit of infinite range of screening. ' A third approach is to expand the scattering wave function in terms of regular Coulomb functions. The complete matrix of the short-ranged interaction between regular Coulomb functions is then required. Although this method is valid in principle, in practice it would require a very time-consuming calculation of large numbers of Coulomb functions.Several quick approximate methods have been suggested. In one of these methods, ' the nuclear phase shift is approximated by adding the nuclear phase shift produced by the deviation of the Coulomb potential from the point-charge form to the phase shift obtained without Coulomb forces. This is accurate if the phase shifts are small. In another method' the phase shift is calculated for the potential truncated at a distance R. Subtracting an asymptotic term of order (kR) ' then gives an approximation to the nuclear phase shift in the presence of t...

Hyperon–nucleon interactions—a chiral effective field theory approach

“…Since the calculations are done in momentum space, the Coulomb interaction is taken into account according to the method originally introduced by Vincent and Phatak [20] (see also [21]). We have used relativistic kinematics for relating the laboratory energy T lab of the hyperons to the c.m.…”

The Hyperon-Nucleon Interaction: Conventional Versus Effective Field Theory Approach

“…Moreover, for a sensible comparison of the results with experiments it is preferable to solve the scattering equation in the particle basis because then the Coulomb interaction in the charged channels can be incorporated. Here we use the method originally introduced by Vincent and Phatak [27] that was e.g. also applied in the EFT studies of the N N interaction [28].…”

StrangenessS=−1hyperon-nucleon scattering in covariant chiral effective field theory

“…Relativistic kinematics is used throughout to relate the laboratory momenta to the center-of-mass momenta. The Coulomb interaction for charged channels is treated with the Vincent-Phatak method [8,76,77].…”