We derived an expression for the effective range r of doublet nd scattering and perform systematic calculations of r for a broad class of separable interaction profiles. We predict the value (1/2)..cap alpha../sup 3/a/sup 2/r = 1.25 for the slope parameter of the amplitude and establish the quadratic correlation between this parameter and the scattering length a. This correlation and the well-known relation between the triton binding energy and a allow the model-independent explanation of the low-energy behavior of the nd scattering amplitude.
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.
Abstract The textbook effective-range expansion of scattering theory is useful in the analysis of low-energy scattering phenomenology when the scattering length ∣ a ∣ is much larger than the range R of the scattering potential: ∣ a ∣ ≫ R . Nevertheless, the same has been used for systems where the scattering length is much smaller than the range of the potential. We suggest and numerically study an effective-range expansion for ∣ a ∣ ≪ R as well as improved two-parameter effective-range expansions for the cases ∣ a ∣ > R and ∣ a ∣ < R . The improved effective-range expansions for ∣ a ∣ > R and ∣ a ∣ < R reduce to the expansions for ∣ a ∣ / R ≫ 1 and ∣ a ∣ / R ≪ 1 , respectively, in appropriate limits.
Instead of calculating phase shifts at small scattering energies for a given two-body central potential and from these deduce by extrapolation to zero energy the associated scattering length and effective range, we write those parameters as integrals which are evaluated numerically to high accuracy for the Lennard-Jones and Aziz et al. interatomic potentials for both
3He and
4He, as well as for the Kolos-Wolniewicz potential between hydrogen isotopes.
The present theory of low-energy scattering by a long-range potential has been examined by omitting the assumptions about the wavefunctions and taking the phase to be a function of the position of the scattered particle. The resulting integro-differential equation for the phase shift has been simplified for a particular case and it is found that the magnitude of the correction terms so obtained is quite large. It is shown that the two-potential formalism can be applied only to light pairs of scattering atoms.
The scattering length is commonly used to characterize the strength of ultracold atomic interactions, since it is the leading parameter in the low-energy expansion of the scattering phase shift. Its value can be modified via a magnetic field, by using a Feshbach resonance. However, the effective range term, which is the second parameter in the phase shift expansion, determines the width of the resonance and gives rise to important properties of ultracold gases. Independent control over this parameter is not possible by using a magnetic field only. We demonstrate that a combination of magnetic and electric fields can be used to get independent control over both parameters, which leads to full control over elastic ultracold interactions.