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Physical Review A

Near-threshold quantization and scattering lengths

Harald Friedrich, Patrick Raab

Published Jan 7, 2008• 9 citations
Publisher

Abstract

We formulate the quantization rule ${n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})$ for bound-state energies ${E}_{n}$ 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 $a$ 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 $\overline{a}$ and $b$ are known and an explicit and universally valid expression for the function $F(E)$ is given analytically.

DOI
10.1103/physreva.77.012703

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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.

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We formulate the quantization rule nth\ensuremath−n=F(En){n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})nth​\ensuremath−n=F(En​) 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=a‾+b∕tan[\ensuremathπF(En)]a=\overline{a}+b∕\mathrm{tan}[\ensuremath{\pi}F({E}_{n})]a=a+b∕tan[\ensuremathπF(En​)]. For a potential tail proportional to \ensuremath−1∕r\ensuremathα\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}\ensuremath−1∕r\ensuremathα, \ensuremathα>3\ensuremath{\alpha}>3\ensuremathα>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.
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.
For potential wells with long-range attractive tails proportional to $\ensuremath{-}1/{r}^{3}$, as occur in the resonant dipole-dipole interaction in homonuclear alkali-metal dimers, we present a highly accurate analytical expression for the tail contribution to the quantization function F(E)F(E)F(E). This quantization function determines the near-threshold bound-state energies via the quantization rule ${n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})$. The performance of the quantization function derived in this paper is demonstrated by applying it to a model Lennard-Jones potential and to vibrational bound-state spectra of sodium dimers (Na2{}_{2}2​). These results are compared with those obtained via the semiclassical LeRoy-Bernstein formula which neglects quantum effects that are important in the near-threshold regime.
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 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 analyse the influence of the attractive tail of the p–H two-body potential on the energy progression of the near-threshold rotationless bound states of the H molecular ion. For the 1sσ g configuration we study different definitions of the reference potential describing the potential tail. In each case, the scattering length consistent with the energies of the highly excited vibrational states, as calculated by Hilico et al. [Eur. Phys. J. D 12, 449 (2000)], is found to lie in the range between −30.5 and −31 au. In the 2pσ u configuration, for which there is an extremely weakly bound v = 1 vibrational state [Carbonell et al., Europhys. Lett. 64, 316 (2003)], we obtain a scattering length near 762.8 au. In contrast to previously conveyed impressions, the weakly bound v = 1 state can be satisfactorily described as the excited state in a distant shallow potential well of the p–H two-body potential, provided that a sufficient number of terms is included in the representation of the large-distance behaviour of this potential.
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 ${\ensuremath{\delta}}_{0}$, 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.