Comparison of numerical methods for the calculation of cold atom collisions

Three different numerical techniques for solving a coupled channel Schroedinger equation are compared. This benchmark equation, which describes the collision between two ultracold atoms, consists of two channels, each containing the same diagonal Lennard-Jones potential, one of positive and the other of negative energy. The coupling potential is of an exponential form. The methods are i) a recently developed spectral type integral equation method based on Chebyshev expansions, ii) a finite element expansion, and iii) a combination of an improved Numerov finite difference method and a Gordon method. The computing time and the accuracy of the resulting phase shift is found to be comparable for methods i) and ii), achieving an accuracy of ten significant figures with a double precision calculation. Method iii) achieves seven significant figures. The scattering length and effective range are also obtained.Comment: 22 pages, 3 figures, submitted to J. Comput. Phys. documentstyle [thmsa,sw20aip]{article} in .te

Inclusion of virtual nuclear excitations in the formulation of the (e,e'N)

A wave-function framework for the theory of the (e,e'N) reaction is presented in order to justify the use of coupled channel equations in the usual Feynman matrix element. The overall wave function containing the electron and nucleon coordinates is expanded in a basis set of eigenstates of the nuclear Hamiltonian, which contain both bound states as well as continuum states.. The latter have an ingoing nucleon with a variable momentum Q incident on the daughter nucleus as a target, with as many outgoing channels as desirable. The Dirac Eqs. for the electron part of the wave function acquire inhomogeneous terms, and require the use of distorted electron Green's functions for their solutions. The condition that the asymptotic wave function contain only the appropriate momentum Q_k for the outgoing nucleon, which corresponds to the electron momentum k through energy conservation, is achieved through the use of the steepest descent saddle point method, commonly used in three-body calculations.Comment: 30 page

A Novel Method for the Solution of the Schroedinger Eq. in the Presence of Exchange Terms

In the Hartree-Fock approximation the Pauli exclusion principle leads to a Schroedinger Eq. of an integro-differential form. We describe a new spectral noniterative method (S-IEM), previously developed for solving the Lippman-Schwinger integral equation with local potentials, which has now been extended so as to include the exchange nonlocality. We apply it to the restricted case of electron-Hydrogen scattering in which the bound electron remains in the ground state and the incident electron has zero angular momentum, and we compare the acuracy and economy of the new method to three other methods. One is a non-iterative solution (NIEM) of the integral equation as described by Sams and Kouri in 1969. Another is an iterative method introduced by Kim and Udagawa in 1990 for nuclear physics applications, which makes an expansion of the solution into an especially favorable basis obtained by a method of moments. The third one is based on the Singular Value Decomposition of the exchange term followed by iterations over the remainder. The S-IEM method turns out to be more accurate by many orders of magnitude than any of the other three methods described above for the same number of mesh points.Comment: 29 pages, 4 figures, submitted to Phys. Rev.

Complex Conjugate Pairs in Stationary Sturmians

Sturmian eigenstates specified by stationary scattering boundary conditions are particularly useful in contexts such as forming simple separable two nucleon t matrices, and are determined via solution of generalised eigenvalue equation using real and symmetric matrices. In general, the spectrum of such an equation may contain complex eigenvalues. But to each complex eigenvalue there is a corresponding conjugate partner. In studies using realistic nucleon--nucleon potentials, and in certain positive energy intervals, these complex conjugated pairs indeed appear in the Sturmian spectrum. However, as we demonstrate herein, it is possible to recombine the complex conjugate pairs and corresponding states into a new, sign--definite pair of real quantities with which to effect separable expansions of the (real) nucleon--nucleon reactance matrices.Comment: (REVTEX) 8 Pages, Padova DFPD 93/TH/78 and University of Melbourn

Coulomb and nuclear breakup of 8^8B

The cross sections for the ($^8$B,$^7$Be-$p$) breakup reaction on $^{58}$Ni and $^{208}$Pb targets at the beam energies of 25.8 MeV and 415 MeV have been calculated within a one-step prior-form distorted-wave Born approximation. The relative contributions of Coulomb and nuclear breakup of dipole and quadrupole multipolarities as well as their interference have been determined. The nuclear breakup contributions are found to be substantial in the angular distributions of the $^7$Be fragment for angles in the range of 30$^\circ$ - 80$^\circ$ at 25.8 MeV beam energy. The Coulomb-nuclear interference terms make the dipole cross section larger than that of quadrupole even at this low beam energy. However, at the incident energy of 415 MeV, these effects are almost negligible in the angular distributions of the ($^7$Be-p) coincidence cross sections at angles below 4$^\circ$.Comment: Revised version, accepted for publication in Phys. Rev.

Investigation of the Coupling Potential by means of S-matrix Inversion

We investigate the inelastic coupling interaction by studying its effect on the elastic scattering potential as determined by inverting the elastic scattering $S$-matrix. We first address the effect upon the real and imaginary elastic potentials of including excited states of the target nucleus. We then investigate the effect of a recently introduced novel coupling potential which has been remarkably successful in reproducing the experimental data for the $^{12}$C+$^{12}$C, $^{12}$C+$^{24}$Mg and $^{16}$O+$^{28}$Si reactions over a wide range of energies. This coupling potential has the effect of deepening the real elastic potential in the surface region, thereby explaining a common feature of many phenomenological potentials. It is suggested that one can relate this deepening to the super-deformed state of the compound nucleus, $^{24}$Mg.Comment: 12 pages with 3 figure