Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The variable phase method used to calculate and correct scattering lengths

Henni Ouerdane, Michael J. Jamieson, D. Vrinceanu, M. J. Cavagnero

Published Sep 19, 2005• 21 citations
OpenAlex

Abstract

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.

Related papers

The variable phase method used to calculate and correct scattering lengths

Henni Ouerdane, M J Jamieson, D Vrinceanu +1 more

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.

arXiv:physics/0509155•Journal of Physics B Atomic Molecular and Optical Physics•Sep 26, 2003•29 citations

Numerical determination of the scattering length for atomic collisions

Ondřej Basler

The goal of this bachelor thesis is to compare numerical methods designed for the determination of scattering length for atomic collisions. We compare nume- rical solutions calculated by the method of direct integration of the Schrödinger equation, the variable phase equation, the Riccati equation for the scattering length and the method of exterior complex scaling. Methods are tested on the exponential potential and the empirical potential of the Cs-Cs dimer. Calculated scattering lengths are further compared with results from other autors. 1

•Jan 1, 2017•0 citations

Analytical calculations of scattering lengths in atomic physics

Radosław Szmytkowski

We describe a method for evaluating analytical long-range contributions to scattering lengths for some potentials used in atomic physics. We assume that an interaction potential between colliding particles consists of two parts. The form of a short-range component, vanishing beyond some distance from the origin (a core radius), need not be given. Instead, we assume that a set of short-range scattering lengths due to that part of the interaction is known. A long-range tail of the potential is chosen to be an inverse power potential, a superposition of two inverse power potentials with suitably chosen exponents or the Lent potential. For these three classes of long-range interactions a radial Schrodinger equation at zero energy may be solved analytically with solutions expressed in terms of the Bessel, Whittaker and Legendre functions, respectively. We utilize this fact and derive exact analytical formulae for the scattering lengths. The expressions depend on the short-range scattering lengths, the core radius and parameters characterizing the long-range part of the interaction. Cases when the long-range potential (or its part) may be treated as a perturbation are also discussed and formulae for scattering lengths linear in strengths of the perturbing potentials are given. It is shown that for some combination of the orbital angular momentum quantum number and an exponent of the leading term of the potential the derived formulae, exact or approximate, take very simple forms and contain only polynomial and trigonometric functions. The expressions obtained in this paper are applicable to scattering of charged particles by neutral targets and to collisions between neutrals. The results are illustrated by accelerating convergence of scattering lengths computed for e - -Xe and Cs-Cs systems.

Journal of Physics A Mathematical and General•Dec 21, 1995•41 citations

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

Henni Ouerdane, Michael J. Jamieson

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, Levinson's 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 $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ and $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ states of both dimers and compare and discuss results obtained from experimentally based $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ and $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ 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.

arXiv:physics/0509146•Physical Review A•Aug 27, 2004•22 citations

Variational calculations for scattering lengths

B. L. Burrows, A J Perks

Approximations for scattering lengths are calculated using variational methods. In most cases bounds for the scattering are obtained. The calculations extend the usual theory to deal with indefinite operators and potentials and in all cases good approximations are achieved.

Journal of Physics B Atomic and Molecular Physics•Jun 21, 1977•3 citations

Variable Phase Method Used to Calculate Ultracold Scattering Properties of 7 Li 33 Cs

Du Bing-Ge, Sun Jin-Feng, Zhang Ji-Cai +3 more

Elastic scattering properties of singlet and triplet states of 7 Li 133 Cs at ultralow temperatures are calculated using the constructed potential curves gleaned from high-precision spectroscopy measurement. We show how to reach the scattering length and the number of bound states via the variable phase method. The scattering lengths of the singlet and triplet states of 7 Li 133 Cs are 50.5a.u. and — 135.7a.u., respectively. We derive two corrections, arising from long range interactions, accurately to at least first order, which provide upper and lower computed bounds to the scattering length. Our results are consistent with the recent experimental data and the theoretical calculation.

Chinese Physics Letters•Oct 1, 2008•6 citations
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.
The goal of this bachelor thesis is to compare numerical methods designed for the determination of scattering length for atomic collisions. We compare nume- rical solutions calculated by the method of direct integration of the Schrödinger equation, the variable phase equation, the Riccati equation for the scattering length and the method of exterior complex scaling. Methods are tested on the exponential potential and the empirical potential of the Cs-Cs dimer. Calculated scattering lengths are further compared with results from other autors. 1
We describe a method for evaluating analytical long-range contributions to scattering lengths for some potentials used in atomic physics. We assume that an interaction potential between colliding particles consists of two parts. The form of a short-range component, vanishing beyond some distance from the origin (a core radius), need not be given. Instead, we assume that a set of short-range scattering lengths due to that part of the interaction is known. A long-range tail of the potential is chosen to be an inverse power potential, a superposition of two inverse power potentials with suitably chosen exponents or the Lent potential. For these three classes of long-range interactions a radial Schrodinger equation at zero energy may be solved analytically with solutions expressed in terms of the Bessel, Whittaker and Legendre functions, respectively. We utilize this fact and derive exact analytical formulae for the scattering lengths. The expressions depend on the short-range scattering lengths, the core radius and parameters characterizing the long-range part of the interaction. Cases when the long-range potential (or its part) may be treated as a perturbation are also discussed and formulae for scattering lengths linear in strengths of the perturbing potentials are given. It is shown that for some combination of the orbital angular momentum quantum number and an exponent of the leading term of the potential the derived formulae, exact or approximate, take very simple forms and contain only polynomial and trigonometric functions. The expressions obtained in this paper are applicable to scattering of charged particles by neutral targets and to collisions between neutrals. The results are illustrated by accelerating convergence of scattering lengths computed for e - -Xe and Cs-Cs systems.
We show how the scattering phase shift, the sss-wave scattering length, and the ppp-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, Levinson's 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 $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ and $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ states of both dimers and compare and discuss results obtained from experimentally based $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ and $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ 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.
Elastic scattering properties of singlet and triplet states of 7 Li 133 Cs at ultralow temperatures are calculated using the constructed potential curves gleaned from high-precision spectroscopy measurement. We show how to reach the scattering length and the number of bound states via the variable phase method. The scattering lengths of the singlet and triplet states of 7 Li 133 Cs are 50.5a.u. and — 135.7a.u., respectively. We derive two corrections, arising from long range interactions, accurately to at least first order, which provide upper and lower computed bounds to the scattering length. Our results are consistent with the recent experimental data and the theoretical calculation.