M J Jamieson

The variable phase method used to calculate and correct scattering lengths

It is shown that the scattering length can be obtained by solving a Riccati equation derived from variable phase theory. Two methods of solving it are presented. The equation is used to predict how long-range interactions influence the scattering length, and upper and lower bounds on the scattering length are determined. The predictions are compared with others and it is shown how they may be obtained from secular perturbation theory.

Elastic scattering of cold caesium and rubidium atoms

For elastic scattering of 133Cs atoms by 85Rb and 87Rb atoms, interacting via the X 1 Σ+ and a 3Σ+ molecular states of RbCs, calculations are presented of the scattering length and the effective range and also of two volume parameters, one for low energy p-wave scattering and one for the energy of an ensemble of fermions. The calculations are performed by various methods, with several representations of the interatomic potentials, including new ab initio potentials presented here. It is shown how the calculated data may be corrected to include the influence of the long range interactions, thus removing the need for numerical solutions of the radial Schrodinger equation at very large interatomic separations. The sensitivity of the results to the choice of potentials, to the van der Waals dispersion coefficient C6 and to the reduced mass of the colliding atoms is explored, and is found to be great.

Scattering parameters for cold Li–Rb and Na–Rb collisions derived from variable phase theory

We show how the scattering phase shift, the s-wave scattering length, and the p-wave scattering volume can be obtained from Riccati equations derived in variable phase theory. We find general expressions that provide upper and lower bounds for the scattering length and the scattering volume. We show how, in the framework of the variable phase method, Levinsons theorem yields the number of bound states supported by a potential. We report results from a study of the heteronuclear alkali-metal dimers NaRb and LiRb. We consider ab initio molecular potentials for the X {sup 1}{sigma}{sup +} and a {sup 3}{sigma}{sup +} states of both dimers and compare and discuss results obtained from experimentally based X {sup 1}{sigma}{sup +} and a {sup 3}{sigma}{sup +} potentials of NaRb. We explore the mass dependence of the scattering data by considering all isotopomers and we calculate the numbers of bound states supported by the molecular potentials for each isotopomer.

A comment on the use and accuracy of the Richardson deferred approach to the limit

Abstract Some higher order methods for solving eigenvalue differential equations are contrasted with simpler second order methods combined with extrapolation. Particular attention is given to the Coulomb and centrifugal singularities in the radial part of the Schrodinger equation.

On finding eigenvalues in the presence of singularities

The difficulty of finding eigenvalues in the presence of singularities is illustrated for the harmonic oscillator perturbed by an essentially singular potential and for the perturbed rigid rotator where the singularity is removable. The source of the difficulty is investigated in each case. For the oscillator a comment is made on the use of asymptotic matching conditions.

s-wave and p-wave scattering in a cold gas of Na and Rb atoms

Using improved experimentally based X1Σ+ and a3Σ+ molecular potentials of NaRb, we apply the variable phase method to compute new data for low energy scattering of 23Na atoms by 85Rb atoms and 87Rb atoms. These are the scattering lengths and volumes, numbers of bound states and effective ranges, which we use to obtain the low energy spin-change cross section as functions of the system temperature and the isotope masses. From an analysis of the contributions of s-wave and p-wave scatterings to the elastic cross section we estimate temperatures below which only s-wave scattering is dominant. We compare our quantal results to data obtained from the semiclassical approximation. We supply evidence for the existence of a near zero energy p-wave bound state supported by the singlet molecular potential.

S-wave scattering lengths and effective ranges for collisions of ground state Be atoms

The scattering lengths and effective ranges for collisions of ground state Be atoms interacting via the X1Σ+g molecular state of the Be2 dimer are calculated for two published X1Σ+g potentials. A new ab initio X1Σ+g potential is presented with its vibrational energies and rotational constants. The scattering parameters calculated with it and with the existing potentials are compared. Simple formulae showing how the scattering parameters are influenced by adiabatic coupling to electronic motion are derived within the semi-classical formulation and evaluated for the potentials considered. The formulae imply that a semi-classical account of adiabatic coupling can be made by increasing each atomic mass by one electron mass.

A note on the generalisation of Aitken's δ 2 transformation

Shanks generalisation (1955) of Aitkens delta 2 transformation is presented as a ratio of persymmetric determinants, and an algorithm for calculating these is given. The superiority of the generalised transformation over repeated delta 2 transformations for summing perturbation series is shown.

The Accurate Calculation of Atomic Properties by Numerical Methods

Publisher Summary This chapter reviews some applications of numerical procedures to obtain accurate values for atomic properties. The chapter has cited applications in more diverse areas, as one-center expansions for molecules, the calculation of excitation cross sections by the impact parameter method, or a simple solution of continuum and semicontinuum models for excess electrons in liquids. Yet even greater possibilities lie with time-dependent theories. Recently, a number of extensions of the time-dependent Hartree–Fock (TDHF) method have been proposed to handle the general open-shell system. This development probably offers the most promising line of advance in the subject, particularly for the study of photoionization. Even for closed-shell atoms, such as barium, complicated effects are apparent in df/dЄ at high energies, and though there is broad agreement with experiment substantial discrepancies exist between different RPA computations. Possibly only a direct numerical solution of the equations will be sufficiently accurate for successful applications of the TDHF approximation to complex systems.

Parameters for Cold Collisions of Lithium and Caesium Atoms

We calculate the s-wave scattering length and effective range and the p-wave scattering volume for 7Li atoms interacting with 133Cs atoms via the X1Σ+ molecular potential. The length and volume are found by fitting the log-derivative of the zero energy wave function evaluated at short range to a long range expression that accounts for the leading van der Waals dispersion potential and then incorporating the remaining long range dispersion contributions to first order. The effective range is evaluated from a quadrature formula. The calculated parameters are checked from the zero energy limits of the scattering phase shifts. We comment on ill-conditioning in the calculated s-wave scattering length.