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Physical Review A

Calculation of the scattering length in atomic collisions using the semiclassical approximation

G. F. Gribakin, V. V. Flambaum

Published Jul 1, 1993• 473 citations
Publisher

Abstract

A simple analytical formula, a=a\ifmmode\bar\else\textasciimacron\fi{}[1-tan(\ensuremath{\pi}/n-2)tan{\ensuremath{\Phi}-[\ensuremath{\pi}/2(n-2)]}, is obtained for the scattering length in atomic collisions. Here a\ifmmode\bar\else\textasciimacron\fi{}=cos[\ensuremath{\pi}/(n-2)]{ \ensuremath{\surd}2M\ensuremath{\alpha} /[\ensuremath{\Elzxh}(n-2)]${\mathrm{}}}^{2/(\mathit{n}\mathrm{\ensuremath{-}}2)}$[\ensuremath{\Gamma}(n-3)/(n-2)]/[T(n-1)/(n-2)] is the mean scattering length determined by the asymptotic behavior of the potential U(r)\ensuremath{\sim}-\ensuremath{\alpha}/${\mathit{R}}^{\mathit{n}}$, (n=6 for atom-atom scattering or n=4 for ion-atom scattering), M is the reduced mass of the atoms, and \ensuremath{\Phi} is the semiclassical phase calculated at zero energy from the classical turning point to infinity. The value of a\ifmmode\bar\else\textasciimacron\fi{}, the average scattering length, also determines the slope of the s-wave phase shifts beyond the near-threshold region. The formula is applicable to the collisions of atoms cooled down in traps, where the scattering length determines the character of the atom-atom interaction. Our calculation shows that repulsion between atoms (a>0) is more likely than attraction with a ``probability'' of 75%. For the Cs-Cs scattering in the $^{3}\mathrm{\ensuremath{\Sigma}}_{\mathit{u}}$ state, a\ifmmode\bar\else\textasciimacron\fi{}=95.5${\mathit{a}}_{\mathit{B}}$ has been obtained, where ${\mathit{a}}_{\mathit{B}}$ is the Bohr radius. The comparison of the calculated cross-section energy dependence with the experimental data gives evidence for a positive value for the Cs-Cs scattering length, which makes cesium Bose gas stable.

DOI
10.1103/physreva.48.546

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A simple analytical formula, a=a\ifmmode\bar\else\textasciimacron\fi{}[1-tan(\ensuremath{\pi}/n-2)tan{\ensuremath{\Phi}-[\ensuremath{\pi}/2(n-2)]}, is obtained for the scattering length in atomic collisions. Here a\ifmmode\bar\else\textasciimacron\fi{}=cos[\ensuremath{\pi}/(n-2)]{ \ensuremath{\surd}2M\ensuremath{\alpha} /[\ensuremath{\Elzxh}(n-2)]{\mathrm{}}}^{2/(\mathit{n}\mathrm{\ensuremath{-}}2)}[\ensuremath{\Gamma}(n-3)/(n-2)]/[T(n-1)/(n-2)] is the mean scattering length determined by the asymptotic behavior of the potential U(r)\ensuremath{\sim}-\ensuremath{\alpha}/Rn{\mathit{R}}^{\mathit{n}}Rn, (n=6 for atom-atom scattering or n=4 for ion-atom scattering), M is the reduced mass of the atoms, and \ensuremath{\Phi} is the semiclassical phase calculated at zero energy from the classical turning point to infinity. The value of a\ifmmode\bar\else\textasciimacron\fi{}, the average scattering length, also determines the slope of the s-wave phase shifts beyond the near-threshold region. The formula is applicable to the collisions of atoms cooled down in traps, where the scattering length determines the character of the atom-atom interaction. Our calculation shows that repulsion between atoms (a>0) is more likely than attraction with a ``probability'' of 75%. For the Cs-Cs scattering in the 3\ensuremathΣu^{3}\mathrm{\ensuremath{\Sigma}}_{\mathit{u}}3\ensuremathΣu​ state, a\ifmmode\bar\else\textasciimacron\fi{}=95.5aB{\mathit{a}}_{\mathit{B}}aB​ has been obtained, where aB{\mathit{a}}_{\mathit{B}}aB​ is the Bohr radius. The comparison of the calculated cross-section energy dependence with the experimental data gives evidence for a positive value for the Cs-Cs scattering length, which makes cesium Bose gas stable.
The interaction between atoms behaves as $\ensuremath{-}\ensuremath{\alpha}{/r}^{n}$ at large distances and, owing to the large reduced mass \ensuremath{\mu} of the collision pair, allows a semiclassical treatment within the potential well. As a result, the low-energy scattering is governed by two large parameters: the asymptotic parameter $\ensuremath{\gamma}=\sqrt{2\ensuremath{\mu}\ensuremath{\alpha}}/\ensuremath{\Elzxh}\ensuremath{\gg}{a}_{0}^{(n\ensuremath{-}2)/2}$ (a0{a}_{0}a0​ is the Bohr radius) and the semiclassical zero-energy phase $\ensuremath{\Phi}\ensuremath{\gg}1.$ In our previous work [Phys. Rev. A 48, 546 (1993)] we obtained an analytical expression for the scattering length a,a,a, which showed that it has 75% preference for positive values for n=6,n=6,n=6, characteristic of collisions between ground-state neutral atoms. In this paper we calculate the effective range and show that it is a function of a,a,a, ${r}_{e}{=F}_{n}\ensuremath{-}{G}_{n}{/a+H}_{n}{/a}^{2},$ where Fn,{F}_{n},Fn​, Gn,{G}_{n},Gn​, and Hn{H}_{n}Hn​ depend only on \ensuremath{\gamma}. Thus, we know the sss-phase shift at low momenta $k\ensuremath{\ll}{\ensuremath{\gamma}}^{\ensuremath{-}2/(n\ensuremath{-}2)}$ from the expansion $k\mathrm{cot}{\ensuremath{\delta}}_{0}\ensuremath{\simeq}\ensuremath{-}1/a+\frac{1}{2}{r}_{e}{k}^{2}.$ At $k\ensuremath{\gg}{\ensuremath{\gamma}}^{\ensuremath{-}2/(n\ensuremath{-}2)}$ the phase shift is obtained semiclassically as ${\ensuremath{\delta}}_{0}=\ensuremath{\Phi}+\ensuremath{\pi}/4\ensuremath{-}{I}_{n}{\ensuremath{\gamma}}^{2/n}{k}^{(n\ensuremath{-}2)/n},$ where ${I}_{n}=[n/(n\ensuremath{-}2)]\ensuremath{\Gamma}((n\ensuremath{-}1)/n)\ensuremath{\Gamma}((n+2)/2n)/\sqrt{\ensuremath{\pi}}.$ Therefore, \ensuremath{\gamma} and \ensuremath{\Phi} determine the sss-wave atomic scattering in a wide range of momenta, as well as the positions of upper bound states of the diatomic molecule.
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.
A semiclassical result is derived for the scattering volume for near-threshold p -wave scattering by a potential falling off as 1/ R n , n ⩾ 6, with R being the interatomic separation. This result depends only on the s -wave zero-energy classical action and properties of the long-range potential and hence can be related to the semiclassical result for the scattering length. The exact result for the scattering volume for the Lennard-Jones (2 n − 2, n ), n ⩾ 6, is derived and it is shown that the corresponding semiclassical result is within 1% for potentials supporting at least two rotationless bound levels. Comparisons are made for the scattering volume with a variety of results in the literature for a range of systems supporting between 0 and 83 rotationless bound levels. The semiclassical result is generally within 5% and was never in error by more than 15%, which occurred for a case supporting no bound states.
Simple analytical expressions are derived for the bound-state energy as well as for the resonance energy and width in the case of the Gaussian potential. These accurate formulas may find applications in nuclear physics, in studies of low-energy electron–atom collisions, and in simulations of few-body properties in Bose–Einstein condensates and Fermi gases. An important feature of the present approach is that the formulas rely solely on the knowledge of a set of real bound-state energies. In addition, an analogous expression is obtained for the s -wave scattering length. This formula is constructed so as to yield highly accurate values of the scattering length in the ground-state region, where the scattering length becomes large.
We calculate the s-wave scattering length and effective range and the p-wave scattering volume for 7 Li atoms interacting with 133 Cs atoms via the X 1 Σ + 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.
Scattering properties of ultracold atoms are sensitive to the interatomic potential. Based on the accurate triplet least-bound state energy, we calculate the triplet s-wave scattering length for 23 Na— 40 K. The scattering length is −814.1 +29 −31.3 a 0 with a 0 being the Bohr radius. By using the mass scaling method, those scattering lengths are also obtained for 23 Na— 41 K and 23 Na— 39 K. The degenerate internal state approximation is used to estimate the scattering data of atoms colliding in different spin states.