George H. Rawitscher

Ingoing wave boundary condition analysis of alpha and deuteron elastic scattering cross sections

Abstract An ingoing wave boundary condition calculation (IWB) is used to obtain elastic alpha-Ni scattering cross sections at the lab energies of 25, 33, 43, 50 and 85 MeV and the deuteron-Cu cross section at 11.8 MeV. The calculation is done numerically on an IBM 7090 and is similar to optical model calculations (OM) with the exception that at small distances an ingoing wave boundary condition is imposed on the partial waves. The method describes diffraction by an absorbing sphere and can be considered as providing an extension valid up to large angles of Frauenhofer diffraction formulae in which surface potentials are rigorously included. The experimental alpha scattering data at 33, 50 and 85 MeV are limited to the forward hemisphere and are found to be consistent with the IWB calculations. The diffuseness of the surface potentials as well as the amount of imaginary potential present at the surface is found to increase with energy. The 25 MeV data are not consistent with the calculation. The 43 MeV data extend to 140° and good agreement is found to about the fifth or sixth diffraction minimum. Comparison between optical model and IWB calculations are carried out for this case, and good agreement is found up to large angles (the eleventh diffraction peak) although the optical model result gives a better fit to the data. The deuteron scattering IWB calculations gives poor agreement with both an optical model calculation and the experiment. Nevertheless, the IWB and OM phase shifts have the same average behaviour as a function of angular momentum number.

Nyström-Clenshaw-Curtis quadrature for integral equations with discontinuous kernels

A new highly accurate numerical approximation scheme based on a Gauss type Clenshaw-Curtis quadrature for Fredholm integral equations of the second kind x(t) + ∫ab k(t, s)x(s)ds = y(t), whose kernel k(t,s) is either discontinuous or not smooth along the main diagonal, is presented. This scheme is of spectral accuracy when k(t,s) is infinitely differentiable away from the diagonal t = s. Relation to the singular value decomposition is indicated. Application to integro-differential Schrodinger equations with nonlocal potentials is given.

Resonances and quantum scattering for the Morse potential as a barrier

Quantum scattering in the presence of a potential valley followed by a barrier is examined for a Morse potential, for which exact analytical solutions are known. For our application the sign of the potential is reversed, and the wave function is required to vanish at the origin. This condition requires a special combination of hypergeometric functions, and can lead to resonances for incident energies below the top of the barrier. Numerical values for the analytical phase shifts are presented in and outside the resonant regions, and the corresponding properties of the scattering S matrix are examined in the complex momentum plane. The validity of the Breit–Wigner approximation to the resonant part of the phase shifts is tested, and a new method for finding the location of narrow resonances is described. The time decay of a resonant wave packet slowly leaking out of the valley region (on a time scale proportional to the inverse of the width of the resonance) is compared with theoretical predictions, and com...

Comparison of numerical methods for the calculation of cold atom collisions

Comparison between three different numerical techniques for solving a coupled channel Schrodinger equation is presented. The 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.

A novel method for the solution of the Schrödinger equation in the presence of exchange terms

In the Hartree–Fock approximation the Pauli exclusion principle leads to a Schrodinger equation of an integro-differential form. We describe the extension of a new spectral noniterative method (S-IEM), previously developed for solving the Lippmann–Schwinger integral equation with local potentials, 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 two other methods. One is a noniterative solution 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 S-IEM method turns out to be more accurate than the two comparison methods by many orders of magnitude for the ...

Relation between nuclear matter and nuclear potential in inelastic alpha—nucleus scattering☆

Abstract Inelastic transition form factors for alpha—nucleus collective excitations are derived from deformed matter distributions by averaging an effective alpha nucleus interaction over the nucleons in the nucleus. The method is an extension of the procedure of calculating the real part of elastic alpha—nucleus potentials from nuclear matter distributions. Numerical examples for the nuclei of 42Ca and 142Nd are given. The resulting inelastic form factors depend on the multipolarity of the inelastic excitation and differ from those conventionally derived from a deformed optical potential.

The microscopic feshbach optical potential for a schematic coupled channel example

Abstract By means of a method of expanding the solutions of a system of coupled equations into a set of positive-energy sturmian basis functions, a separable approximation to the microscopic (Feshbach) elastic optical potential is obtained. The properties of this nonlocal potential are analysed in terms of the moments of the potential and of the wronskian of two independent solutions of the single-channel nonlocal Schroedinger equation. A numerical application is given for a schematic coupled channel example, representing a 60 MeV neutron incident on a schematic nucleus with up to 10 channels, coupled by phenomenological surface-peaked potentials. The nature of the nonlocality of this schematic potential is very different from the nonlocality due to exchange effects in that the wronskian is strongly angular momentum dependent and can be larger as well as smaller than unity. The range of the nonlocality is found to be larger than the wavelength of the elastic projectile.

Second-order breakup corrections to elastic deuteron-nickel scattering between 13 and 80 MeV☆

Abstract Breakup corrections to the elastic scattering matrix elements are calculated in the second-order distorted-wave Born approximation at deuteron incident energies of 45 and 85 MeV. The effects of spin are included. The size of the corrections are found to be generally as large as those obtained in a previous study at 13 and 21.6 MeV. The breakup cross section is calculated to first order in the breakup matrix elements by a distorted-wave Born treatment. Comparison with fully coupled calculations shows that the DWBA method overestimates the breakup cross section by a factor of about three. The continuum of breakup states up to a n-p relative momentum 1 fm −1 is included in the calculations. This continuum is discretized by subdividing it first into two bins, and then into four bins. The finer discretization does not make a large difference in either the elastic cross section or the breakup cross sections. The higher bins give only a small contribution to either quantity.

Effect of breakup on the deuteron nucleus spin orbit and tensor potentials

Abstract A previous treatment of breakup in terms of a coupled channel formulation is generalized so as to include the spin of the nucleus. The break up spectrum is described in terms of the eigenstates of the free neutron proton Hamiltonian, based on the Reid soft core potential. Nearly spherically symmetric as well as nearly quadrupole deformed break up states are included. The latter give large contributions to the correction to the spin orbit and tensor parts of the deuteron-nucleus elastic optical model. A numerical application for 13 MeV and 21.6 MeV deuterons incident on Ni is presented, in which virtual breakup is included to second order in the breakup transition matrix element. The breakup correction to the deuteron potential is strongly dependent on the orbital angular momentum and the energy of the deuteron-nucleus motion. The spin orbit potential is rendered less diffuse and acquires an imaginary part. The tensor potential correction is comparable in radial shape to the Watanabe static folding model result, and tends to increase its magnitude by nearly 100%.

Spin dependent deuteron-nucleus interaction caused by the coupling to stripping channels

Abstract Previously employed coupled equations which describe the dissociation and (recombination) of the incident deuteron into (and from) the stripping channels have been reformulated in order to include the spins of the proton, deuteron and the final nucleus involved in the reaction. The effects of nonorthogonality of the incident and the reaction channel are still ignored in the present formulation, as well as the effects due to deuteron breakup. Attention is focused on the spin orbit character of the nonlocal deuteronnucleus interaction which results from the presence of the coupling to the spin dependent stripping channels. A numerical application to the case of 11-MeV deuterons incident on 40 Ca is presented. It is found that roughly half of the vector polarization in the deuteron channel arises from the coupling to the stripping channels. The experimental j dependence of the stripping transition to the 2p 1 2 and 2p 3 2 states in 41 Ca near 100° is not reproduced, and the numerical study suggests that inclusion of deuteron-breakup channels into the coupled equations is called for.