A simple potential is proposed to describe low-energy s-wave scattering of electrons from closed-shell atoms. It has the correct structure at large and small distances from the nucleus and depends on the target polarizability and a dimensionless parameter. This dimensionless parameter is determined by extrapolation from the potential for the bound states in isoelectronic sequences. Apart from providing a satisfactory description of low-energy scattering of electrons from inert-gas atoms, the potential predicts a negative scattering length for electron scattering from Be, Mg, Zn, Cd, Hg, and zeros in the s-wave phase shin at a momentum k in the range 0.05-0.10 au.
A simple potential is proposed to describe low-energy s-wave scattering of electrons from closed-shell atoms. It has the correct structure at large and small distances from the nucleus and depends on the target polarizability and a dimensionless parameter. This dimensionless parameter is determined by extrapolation from the potential for the bound states in isoelectronic sequences. Apart from providing a satisfactory description of low-energy scattering of electrons from inert-gas atoms, the potential predicts a negative scattering length for electron scattering from Be, Mg, Zn, Cd, Hg, and zeros in the s-wave phase shin at a momentum k in the range 0.05-0.10 au.
We have developed a model, l -dependent potential to describe the interaction of an electron with inert gas atoms, and Be, Mg, Zn, Cd, Hg, and their isoelectronic sequences. This potential is adjusted so as to produce bound states with experimental binding energies in isoelectronic sequences of Li I, Na I, K I, Rb I, Cs I and B I, Al I, Ga I, In I, Tl I, and extrapolated to get a potential for the interaction of an electron with inert gas atoms, and with Be, Mg, Zn, Cd, Hg. It provides reliable phase shifts and cross sections for e-inert gas atom, e-Be, e-Mg, e-Zn, e-Cd, e-Hg scattering. Particularly interesting are the signs of the scattering lengths, the changes in the signs of the s-wave phase shifts, and p-wave shape resonances at low energies.
Abstract More than 65 years ago, Jost and Kohn (1952 Phys. Rev. 87 , 977) derived an explicit expression for a class of short-range model potentials from a given effective range expansion with the s -wave scattering length a s being negative. For a s > 0, they calculated another class of short-range model potentials (Jost and Kohn 1953 Dan. Mat. Fys. Medd 27 , no. 9) using a method based on an adaptation from Gel’fand−Levitan theory (Gel’fand and Levitan 1951 Dokl. Akad. Nauk. USSR 77 , 557-60) of inverse scattering. We here revisit the methods of Jost and Kohn in order to explore the possibility of modeling resonant finite-range interactions at low energy. We show that the Jost−Kohn potentials can account for zero-energy resonances. The s -wave phase shift for positive scattering length is expressed in an analytical form as a function of the binding energy of a bound state. We show that, for small binding energy, both the scattering length and the effective range are strongly influenced by the binding energy; below a critical binding energy the effective range becomes negative provided the scattering length is large. As a consistency check we carry out some simple calculations to show that Jost−Kohn potentials can reproduce the standard results of contact interaction in the limit of the effective range going to zero.
In a recent paper, Reisfeld and Asaf [Phys. Rev. A 49, 348 (1994)] suggested a simple model explaining an observed correlation between the electron scattering lengths and electric polarizabilities of rare-gas atoms. They proposed an electron-target interaction potential consisting of a short-range hard-core repulsive part plus a long-range dipole polarization potential, and used the Born approximation to evaluate the scattering length. We show that for this model the scattering length may be evaluated analytically and that theoretical predictions based on an exact expression disagree with experimental data.
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 provide accurate expressions for the s-wave scattering length for a Gaussian potential well in one, two, and three spatial dimensions. The Gaussian potential is widely used as a pseudopotential in the theoretical description of ultracold-atomic gases, where the s-wave scattering length is a physically relevant parameter. We first describe a numerical procedure to compute the value of the s-wave scattering length from the parameters of the Gaussian, but find that its accuracy is limited in the vicinity of singularities that result from the formation of new bound states. We then derive simple analytical expressions that capture the correct asymptotic behavior of the s-wave scattering length near the bound states. Expressions that are increasingly accurate in wide parameter regimes are found by a hierarchy of approximations that capture an increasing number of bound states. The small number of numerical coefficients that enter these expressions is determined from accurate numerical calculations. The approximate formulas combine the advantages of the numerical and approximate expressions, yielding an accurate and simple description from the weakly to the strongly interacting limit.
The method of polarized orbitals is modified to treat low-energy scattering of electrons from highly polarizable systems, specifically alkali-metal atoms. The modification is carried out in the particular context of the e-Li system, but the procedure is general; it consists of modifying the polarized orbital, so that when used in the otherwise orthodox form of the method, it gives (i) the correct electron affinity of the negative ion (in this case ${\mathrm{Li}}^{\ensuremath{-}}$), (ii) the proper (i.e., Levinson-Swan) number of nodes of the associated zero-energy scattering orbital, and (iii) the correct polarizability. A procedure is devised whereby the scattering length can be calculated from the (known) electron affinity without solving the bound-state equation. Using this procedure we adduce a 1S scattering length of 8.69a0. (The 3S scattering length is $\ensuremath{-}9.22{a}_{0}$.) The above modifications can also be carried out in the (lesser) exchange adiabatic approximation. However, they lead to qualitatively incorrect 3S phase shifts. The modified polarized-orbital phase shifts are qualitatively similar to close-coupling and elaborate variational calculations. Quantitative differences from the latter calculations, however, remain; they are manifested most noticeably in the very-low-energy total and differential spin-flip cross sections.