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Physical Review A

Addendum to “ s -wave scattering for deep potentials with attractive tails falling off faster than −1/r2 ”

Tim-Oliver Müller, Alexander Kaiser, Harald Friedrich

Published Nov 15, 2011• 6 citations
Publisher

Abstract

In a recent work, we derived a formula for the elastic $s$-wave phase shifts for deep potentials with attractive tails vanishing faster than $1/{r}^{2}$ asymptotically [Phys. Rev. A 84, 032701 (2011)]. As an addendum to that work, we reformulate its key result, i.e., the formula for the scattering phase shift ${\ensuremath{\delta}}_{0}$, in such a way that the contributions independent of the short-range influence of the full potential are explicitly expressed in terms of the amplitudes for transmission through and quantum reflection by the nonclassical region of the potential tail.

DOI
10.1103/physreva.84.054702

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In a recent work, we derived a formula for the elastic sss-wave phase shifts for deep potentials with attractive tails vanishing faster than 1/r21/{r}^{2}1/r2 asymptotically [Phys. Rev. A 84, 032701 (2011)]. As an addendum to that work, we reformulate its key result, i.e., the formula for the scattering phase shift \ensuremathδ0{\ensuremath{\delta}}_{0}\ensuremathδ0​, in such a way that the contributions independent of the short-range influence of the full potential are explicitly expressed in terms of the amplitudes for transmission through and quantum reflection by the nonclassical region of the potential tail.
For potentials with attractive tails, as occur in typical atomic interactions, we present a simple formula for the sss-wave phase shift ${\ensuremath{\delta}}_{0}$. It exposes a universal dependence of ${\ensuremath{\delta}}_{0}(E)$ on the potential tail and the influence of effects specific to a given potential, which enter via the scattering length aaa, or equivalently, the noninteger part ${\ensuremath{\Delta}}_{\mathrm{th}}$ of the threshold quantum number nth{n}_{\mathrm{th}}nth​. The formula accurately reproduces ${\ensuremath{\delta}}_{0}(E)$ from threshold up to the semiclassical regime, far beyond the validity of the effective-range expansion. We derive the tail functions occurring in the formula for ${\ensuremath{\delta}}_{0}(E)$ and demonstrate the validity of the formula for attractive potential tails proportional to 1/r61/{r}^{6}1/r6 or to 1/r41/{r}^{4}1/r4, and also for a mixed potential tail consisting of a 1/r41/{r}^{4}1/r4 term together with a non-negligible 1/r61/{r}^{6}1/r6 contribution.
We study the phases of the amplitudes for transmission through and near-side quantum reflection by the nonclassical regions of singular attractive potential tails, as occur in the interaction of atoms and molecules with each other and with surfaces. The derivatives of these phases with respect to energy or momentum are interpreted as time and space shifts according to the prescription of Eisenbud and Wigner. The results for transmission through the nonclassical part of the potential follow a straightforward pattern: Near threshold, the time and space shifts are close to those of a free particle and only marginally resolve the influence of the attractive potential; with increasing energy they rapidly approach the expectations for a classical particle accelerated in the potential. The phase of the amplitude for near-side quantum reflection leads to a negative time shift. In a regularized potential obtained by smoothly continuing the attractive tail to a deep constant at negative distances, this can be understood as a small boost for a wave packet approaching the origin from the left---relative to a free wave totally reflected at the origin.
The interactions between atoms and molecules with each other and with surfaces are typically deep potential wells with attractive tails behaving asymptotically as an inverse power of the distance. In such potential wells, bound state energies En{E}_{n}En​ are determined by a quantization function F(E)F(E)F(E) according to ${n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})$, and F(E)F(E)F(E) is dominantly determined by the singular potential tail for near-threshold states. In this paper we formulate a general theory for the contribution Ftail(E){F}_{\text{tail}}(E)Ftail​(E) of the singular potential tail to the quantization function. The general expression for Ftail(E){F}_{\text{tail}}(E)Ftail​(E) contains two terms: a difference of action integrals and a difference of outer reflection phases, taken at threshold and at energy EEE. Close to threshold, E=0E=0E=0, strongly energy dependent and nonanalytic contributions of both terms cancel, so Ftail(E){F}_{\text{tail}}(E)Ftail​(E) acquires a universal form determined by a threshold length and an effective length which is related to a subthreshold effective range. For a homogeneous potential tail proportional to $\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}$, one universal expression for Ftail(E){F}_{\text{tail}}(E)Ftail​(E) caters for all potential strengths. We give an explicit analytical expression for the important case $\ensuremath{\alpha}=6$ and present applications involving the derivation of atom-atom scattering lengths from the binding energies of high-lying bound states of the associated diatomic molecule. We also demonstrate how the dissociation energy of a diatomic molecule can be determined from spectroscopic energies of high-lying states, and we make a quantitative comparison with the performance of the LeRoy-Bernstein formula, which fails near threshold, because the strongly energy dependent and nonanalytic contribution from the action integrals is not, as it should be, compensated by terms coming from the corresponding energy dependence of the outer reflection phase.
We formulate an effective-range theory for the near-threshold behavior of the amplitudes describing quantum reflection in attractive potential tails by adapting the effective-range theory of ordinary elastic scattering to the case of incoming boundary conditions at small distances. For homogeneous attractive potentials proportional to $\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>5$, the effective range turns out to be a real multiple (with known coefficient) of the complex scattering length which defines the leading, linear momentum dependence of phase and modulus of the quantum reflection amplitude. Analytical expressions are also given for the leading and next-to-leading terms in the near-threshold behavior of the quantum reflection amplitudes for homogeneous attractive potentials proportional to $\ensuremath{-}1∕{r}^{3}$, $\ensuremath{-}1∕{r}^{4}$, and $\ensuremath{-}1∕{r}^{5}$.
We study the near-threshold behavior of scattering phase shifts, quantum reflection amplitudes, and quantization functions for systems described by a circularly symmetric potential in two spatial dimensions. In contrast to the three-dimensional case, the centrifugal potential $({m}^{2}\ensuremath{-}\frac{1}{4}){\ensuremath{\hbar}}^{2}∕(2M{r}^{2})$ is nonvanishing and even attractive for sss waves, m=0m=0m=0, and the leading near-threshold energy dependence of phase shifts and amplitudes for scattering and quantum reflection is logarithmic in this case. The near-threshold behavior of the sss-wave phase shifts and amplitudes can nevertheless be characterized by a well-defined scattering length (for potentials falling off faster than 1∕r21∕{r}^{2}1∕r2) and an effective range (for potentials falling off faster than 1∕r41∕{r}^{4}1∕r4). For a potential with a bound state at energy ${E}_{n}=\ensuremath{-}{\ensuremath{\hbar}}^{2}{\ensuremath{\kappa}}_{n}^{2}∕(2M)$ very near threshold, the scattering length obeys $a\stackrel{{\ensuremath{\kappa}}_{n}\ensuremath{\rightarrow}0}{\ensuremath{\sim}}2{e}^{\ensuremath{-}{\ensuremath{\gamma}}_{\mathrm{E}}}∕{\ensuremath{\kappa}}_{n}+O({\ensuremath{\kappa}}_{n})$, with no term of order ${\ensuremath{\kappa}}_{n}^{0}$---in contrast to the three-dimensional case. Analytical results are derived for homogeneous potentials, and the necessary modification of the effective-range expansion is given for potentials proportional to 1∕r41∕{r}^{4}1∕r4. For $m\ensuremath{\ne}0$ we give analytical expressions for the near-threshold behavior of phase shifts as well as scattering and quantum reflection amplitudes, which are generally valid, even when mmm is neither integer nor half-integer.
We formulate the quantization rule ${n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})$ for bound-state energies En{E}_{n}En​ in a deep potential well with a long-range attractive tail for an extended range of near-threshold energies. This makes it possible to accurately read off the scattering length aaa from the energy of a high-lying (not necessarily the highest-lying) bound state via $a=\overline{a}+b∕\mathrm{tan}[\ensuremath{\pi}F({E}_{n})]$. For a potential tail proportional to $\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}$, $\ensuremath{\alpha}>3$, the constants a‾\overline{a}a and bbb are known and an explicit and universally valid expression for the function F(E)F(E)F(E) is given analytically.