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.
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 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, which showed that it has 75% preference for positive values for
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, ${r}_{e}{=F}_{n}\ensuremath{-}{G}_{n}{/a+H}_{n}{/a}^{2},$ where
Fn, Gn, and
Hn depend only on \ensuremath{\gamma}. Thus, we know the
s-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
s-wave atomic scattering in a wide range of momenta, as well as the positions of upper bound states of the diatomic molecule.
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.
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, (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
aB has been obtained, where
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.
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.
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.
For elastic scattering of 133 Cs atoms by 85 Rb and 87 Rb atoms, interacting via the X 1 Σ + and a 3 Σ + 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 Schrödinger equation at very large interatomic separations. The sensitivity of the results to the choice of potentials, to the van der Waals dispersion coefficient C 6 and to the reduced mass of the colliding atoms is explored, and is found to be great.