A formalism, previously presented, for describing atom-atom scattering in the adiabatic representation to higher orders in
Mm is generalized to describe the problem of different isotopes and different atoms. This is accomplished by a sequence of nonlinear coordinate transformations which introduce a "switching function" which is defined only in the asymptotic limit of separated atoms. It is found that the effective potential for elastic scattering contains energy-dependent terms in order
Mm and that it also depends upon the switching function in this order so that the potential is not unique. It is also found that in this order the definition of adiabatic states is not a trivial problem. A variational choice is made for the switching function and explicit calculations are presented for the He-He interaction. The coupled equations describing transitions are then derived and it is shown that terms are omitted from the usual coupling operator which are of the same order of magnitude as the one usually kept. This leads to a modification of the usual Landau-Zener transition formula.
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.
The low-energy two-body scattering S- and P-wave parameters, together with a shape-dependent S-wave quantity, which appear in the well-known low-density expansions for the ground-state energy of many-boson and many-fermion systems, are accurately calculated for several central pair potentials for spin-polarized atoms, helium atoms, and nucleons. An integral-equation method allowing accurate determination of the coefficients of the above-mentioned parameters expanded in powers of the attraction strength of the pair potential is presented and applied. The results are needed for perturbation studies of ground-state energies of various quantum many-body systems.
At long distances, interactions between neutral ground-state atoms can be described by the van der Waals potential. In the ultracold regime, atom-atom scattering is dominated by
s-waves phase shifts given by an effective range expansion in terms of the scattering length ${\ensuremath{\alpha}}_{0}$ and the effective range
r0. We show that while the scattering length cannot be predicted for these potentials, the effective range is given by the universal low-energy theorem ${r}_{0}=A+B/{\ensuremath{\alpha}}_{0}+C/{\ensuremath{\alpha}}_{0}^{2}$, where
A,
B, and
C depend on the dispersion coefficients
Cn and the reduced diatom mass. We confront this formula to about 100 determinations of
r0 and ${\ensuremath{\alpha}}_{0}$ and show why the result is dominated by the leading dispersion coefficient
C6. Universality and scaling extend much beyond naive dimensional analysis estimates.
Ultracold atoms are increasingly used for high-precision experiments that can be utilized to extract accurate scattering properties. This results in a stronger need to improve on the accuracy of interatomic potentials, and in particular the usually rather inaccurate inner-range potentials. A boundary condition for this short range can be conveniently given via the accumulated phase method. However, in this approach one should satisfy three conditions, two of which are in principle conflicting, and the validity of these approximations comes under stress when higher precision is required. We show that a better compromise between the two is possible by allowing for an adiabatic change in the hyperfine mixing of singlet and triplet states for interatomic distances smaller than the separation radius. Results we presented previously in a brief publication using this method show a high precision and extend the set of predicted quantities. The purpose of this paper is to describe its background. A mass-scaling approach to relate accumulated phase parameters in a combined analysis of isotopically related atom pairs is described in detail and its accuracy is estimated, taking into account both Born-Oppenheimer and Wentzel-Kramers-Brillouin breakdown. We demonstrate how numbers of singlet and triplet bound states follow from the mass scaling.
The scattering length scaling rules for three-body recombination (TBR) of the 4He4He6Li- system in the zero-energy limit (E → 0) are investigated by considering various post-Born-Oppenheimer (post-BO) contributions to the standard BO potential of He-He interaction. It is found that these post-BO effects on the TBR rates that lead to different dimer products could be well fitted by the scattering length scalings Cab with a being the scattering length of the He-He interaction, and b and C the constants. It is interesting to find that the powers b about both the weakly and deeply bound dimer products are quite different from the well-known powers given by the scattering length scaling laws in the fields of ultracold neutral atomic gases. Such difference and the distinct powers associated with various dimer products are qualitatively accounted for by the properties of the major effective potentials and relatively long-range nonadiabatic couplings. Particularly, from these properties, the scaling powers are expected to be dependent on systems, but the power scattering length scaling behaviors to be universal, which has also been checked to apply to a wide range of a by tuning the He-He interaction.