Physical Review A

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

Tim-Oliver Müller, Alexander Kaiser, Harald Friedrich

Published Sep 6, 2011• 16 citations

Abstract

For potentials with attractive tails, as occur in typical atomic interactions, we present a simple formula for the $s$-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 $a$, or equivalently, the noninteger part ${\ensuremath{\Delta}}_{\mathrm{th}}$ of the threshold quantum number ${n}_{\mathrm{th}}$. 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 $1/{r}^{6}$ or to $1/{r}^{4}$, and also for a mixed potential tail consisting of a $1/{r}^{4}$ term together with a non-negligible $1/{r}^{6}$ contribution.

DOI: 10.1103/physreva.84.032701

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Near-threshold quantization and scattering lengths

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Physical Review A

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

Tim-Oliver Müller, Alexander Kaiser, Harald Friedrich

Published Sep 6, 2011• 16 citations

Publisher

Abstract

For potentials with attractive tails, as occur in typical atomic interactions, we present a simple formula for the

s

-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

a

, or equivalently, the noninteger part

{\ensuremath{\Delta}}_{\mathrm{th}}

of the threshold quantum number

{n}_{\mathrm{th}}

. 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/{r}^{6}

or to

1/{r}^{4}

, and also for a mixed potential tail consisting of a

1/{r}^{4}

term together with a non-negligible

1/{r}^{6}

contribution.

DOI: 10.1103/physreva.84.032701

Related papers

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

Tim-Oliver Müller, Alexander Kaiser, Harald Friedrich

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.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.

Physical Review ANov 15, 20116 citations

Quantization function for deep potentials with attractive tails

Patrick Raab, Harald Friedrich

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 ${E}_{n}$ are determined by a quantization function $F(E)$ according to ${n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})$, and $F(E)$ is dominantly determined by the singular potential tail for near-threshold states. In this paper we formulate a general theory for the contribution ${F}_{\text{tail}}(E)$ of the singular potential tail to the quantization function. The general expression for ${F}_{\text{tail}}(E)$ contains two terms: a difference of action integrals and a difference of outer reflection phases, taken at threshold and at energy $E$ . Close to threshold, $E=0$ , strongly energy dependent and nonanalytic contributions of both terms cancel, so ${F}_{\text{tail}}(E)$ 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 ${F}_{\text{tail}}(E)$ 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.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 ${E}_{n}$ are determined by a quantization function $F(E)$ according to ${n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})$, and $F(E)$ is dominantly determined by the singular potential tail for near-threshold states. In this paper we formulate a general theory for the contribution ${F}_{\text{tail}}(E)$ of the singular potential tail to the quantization function. The general expression for ${F}_{\text{tail}}(E)$ contains two terms: a difference of action integrals and a difference of outer reflection phases, taken at threshold and at energy $E$. Close to threshold, $E=0$, strongly energy dependent and nonanalytic contributions of both terms cancel, so ${F}_{\text{tail}}(E)$ 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 ${F}_{\text{tail}}(E)$ 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.

Physical Review AAug 8, 200818 citations

Near-threshold quantization and scattering lengths

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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.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.

Physical Review AJan 7, 20089 citations

Phases of the amplitudes for transmission and near-side quantum reflection in attractive potential tails

Eskender Mesfin, Harald Friedrich

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.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.

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Universality and tails of long-range interactions in one dimension

Manuel Valiente, Patrik Öhberg

Long-range interactions and, in particular, two-body potentials with power-law long-distance tails are ubiquitous in nature. For two bosons or fermions in one spatial dimension, the latter case being formally equivalent to three-dimensional $s$ -wave scattering, we show how generic asymptotic interaction tails can be accounted for in the long-distance limit of scattering wave functions. This is made possible by introducing a generalization of the collisional phase shifts to include space dependence. We show that this distance dependence is universal, in that it does not depend on short-distance details of the interaction. The energy dependence is also universal, and is fully determined by the asymptotic tails of the two-body potential. As an important application of our findings, we describe how to eliminate finite-size effects with long-range potentials in the calculation of scattering phase shifts from exact diagonalization. We show that even with moderately small system sizes it is possible to accurately extract phase shifts that would otherwise be plagued with finite-size errors. We also consider multichannel scattering, focusing on the estimation of open channel asymptotic interaction strengths via finite-size analysis.Long-range interactions and, in particular, two-body potentials with power-law long-distance tails are ubiquitous in nature. For two bosons or fermions in one spatial dimension, the latter case being formally equivalent to three-dimensional $s$-wave scattering, we show how generic asymptotic interaction tails can be accounted for in the long-distance limit of scattering wave functions. This is made possible by introducing a generalization of the collisional phase shifts to include space dependence. We show that this distance dependence is universal, in that it does not depend on short-distance details of the interaction. The energy dependence is also universal, and is fully determined by the asymptotic tails of the two-body potential. As an important application of our findings, we describe how to eliminate finite-size effects with long-range potentials in the calculation of scattering phase shifts from exact diagonalization. We show that even with moderately small system sizes it is possible to accurately extract phase shifts that would otherwise be plagued with finite-size errors. We also consider multichannel scattering, focusing on the estimation of open channel asymptotic interaction strengths via finite-size analysis.

arXiv:1705.04225Physical Review AJul 7, 20172 citations

Near-threshold quantization for potentials with inverse-cube tails

Tim-Oliver Müller, Harald Friedrich

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 $F(E)$ . 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 (Na ${}_{2}$ ). 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 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 $F(E)$. 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 (Na${}_{2}$). 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.

Physical Review AFeb 2, 201112 citations

For potentials with attractive tails, as occur in typical atomic interactions, we present a simple formula for the

s

-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

a

, or equivalently, the noninteger part

{\ensuremath{\Delta}}_{\mathrm{th}}

of the threshold quantum number

{n}_{\mathrm{th}}

. 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/{r}^{6}

or to

1/{r}^{4}

, and also for a mixed potential tail consisting of a

1/{r}^{4}

term together with a non-negligible

1/{r}^{6}

contribution.

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.

{E}_{n}

are determined by a quantization function

F(E)

according to ${n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})$, and

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

{F}_{\text{tail}}(E)

of the singular potential tail to the quantization function. The general expression for

{F}_{\text{tail}}(E)

contains two terms: a difference of action integrals and a difference of outer reflection phases, taken at threshold and at energy

E

. Close to threshold,

E=0

, strongly energy dependent and nonanalytic contributions of both terms cancel, so

{F}_{\text{tail}}(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

{F}_{\text{tail}}(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 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.

s

-wave scattering, we show how generic asymptotic interaction tails can be accounted for in the long-distance limit of scattering wave functions. This is made possible by introducing a generalization of the collisional phase shifts to include space dependence. We show that this distance dependence is universal, in that it does not depend on short-distance details of the interaction. The energy dependence is also universal, and is fully determined by the asymptotic tails of the two-body potential. As an important application of our findings, we describe how to eliminate finite-size effects with long-range potentials in the calculation of scattering phase shifts from exact diagonalization. We show that even with moderately small system sizes it is possible to accurately extract phase shifts that would otherwise be plagued with finite-size errors. We also consider multichannel scattering, focusing on the estimation of open channel asymptotic interaction strengths via finite-size analysis.

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 (Na

{}_{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.