arXiv:1802.07063 · Physical Review A

s -wave scattering length of a Gaussian potential

Péter Jeszenszki, Alexander Yu. Cherny, Joachim Brand

Published Apr 23, 2018• 29 citations

Abstract

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.

DOI: 10.1103/physreva.97.042708

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Resonances in the Gaussian potential

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An HNC Approach to Strongly Interacting Ultracold Bose Gases

T. C. F. van Heijst, Henk T. C. Stoof

Jan 1, 20111 citation

On-shell approximation for the s -wave scattering theory

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Excitation spectrum and effective interactions of a highly elongated Fermi gas

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Analytical calculation of cold-atom scattering

V. V. Flambaum, G. F. Gribakin, C. Harabati

Physical Review AMar 1, 1999180 citations

Calculation of the scattering length in atomic collisions using the semiclassical approximation

G. F. Gribakin, V. V. Flambaum

Physical Review AJul 1, 1993473 citations

arXiv:1802.07063 · Physical Review A

s -wave scattering length of a Gaussian potential

Péter Jeszenszki, Alexander Yu. Cherny, Joachim Brand

Published Apr 23, 2018• 29 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physreva.97.042708

Related papers

Resonances in the Gaussian potential

J. Horáček, Roman Čurı́k

Simple analytical expressions are derived for the bound-state energy as well as for the resonance energy and width in the case of the Gaussian potential. These accurate formulas may find applications in nuclear physics, in studies of low-energy electron–atom collisions, and in simulations of few-body properties in Bose–Einstein condensates and Fermi gases. An important feature of the present approach is that the formulas rely solely on the knowledge of a set of real bound-state energies. In addition, an analogous expression is obtained for the s -wave scattering length. This formula is constructed so as to yield highly accurate values of the scattering length in the ground-state region, where the scattering length becomes large.Simple analytical expressions are derived for the bound-state energy as well as for the resonance energy and width in the case of the Gaussian potential. These accurate formulas may find applications in nuclear physics, in studies of low-energy electron–atom collisions, and in simulations of few-body properties in Bose–Einstein condensates and Fermi gases. An important feature of the present approach is that the formulas rely solely on the knowledge of a set of real bound-state energies. In addition, an analogous expression is obtained for the s -wave scattering length. This formula is constructed so as to yield highly accurate values of the scattering length in the ground-state region, where the scattering length becomes large.

Computer Physics CommunicationsJun 1, 20260 citations

An HNC Approach to Strongly Interacting Ultracold Bose Gases

T. C. F. van Heijst, Henk T. C. Stoof

We argue that the energy of a strongly interacting gas in the unitary limit is parametrized by a universal constant β. For fermions, this parameter is known, but not for bosons. Finding β for bosons is the goal of this thesis. We present the basics of scattering theory, including the s-wave scattering length, which is the characteristic length scale for the interactions at low energies and which goes to infinity at unitarity. Shape and Feshbach resonances are explained. The conventional theory for Bose gases, Bogoliubov theory, is presented, which takes into account quantum fluctuations. It is used to derive identities for the groundstate energy, the condensate fraction and the contact. These are given as expansions in the scattering length, which therefore only hold true for weak interactions. To improve on this, we present an effective action from an atom-molecule theory, which also allows bound states. By a mean-field approach we derive universal identities for the associated energy scales, and in the unitary limit an analytical expression is found for the chemical potential, from which the value β = −0.54 is found. However, this mean-field result is qualitatively not reliable, since quantum depletion is neglected. A new approach is introduced, starting with a Jastrow Ansatz, from which the energy is expressed in the Jastrow function f and radial distribution function g. A cluster expansion is made for g, where a diagrammatic analogy is found. The Hyper-Netted Chain approximation is introduced to calculate g systematically by summing infinitely many diagrams. In HNC/0, we minimize the energy by a variational method, which is based on a well-constructed Ansatz for g. Numerical results using the HNC/0 approximation are given not only for the ground-state energy, but also for the condensate density and the contact. For large scattering lengths the calculations became increasingly less accurate, but the results from Bogoliubov theory are correctly included for small a. An extrapolation for the ground-state energy is made to find β ' −0.20, which is expected to be an upper bound. Moreover, quantum depletion is found to be higher when calculated in HNC/0 for large scattering lengths. Also, it is found that the contact at this value for unitarity remains finite and acquires the value of C ' 10.3n4/3, which is remarkably close to the value for fermions, C ' 11n4/3. Cover image courtesy of [18].We argue that the energy of a strongly interacting gas in the unitary limit is parametrized by a universal constant β. For fermions, this parameter is known, but not for bosons. Finding β for bosons is the goal of this thesis. We present the basics of scattering theory, including the s-wave scattering length, which is the characteristic length scale for the interactions at low energies and which goes to infinity at unitarity. Shape and Feshbach resonances are explained. The conventional theory for Bose gases, Bogoliubov theory, is presented, which takes into account quantum fluctuations. It is used to derive identities for the groundstate energy, the condensate fraction and the contact. These are given as expansions in the scattering length, which therefore only hold true for weak interactions. To improve on this, we present an effective action from an atom-molecule theory, which also allows bound states. By a mean-field approach we derive universal identities for the associated energy scales, and in the unitary limit an analytical expression is found for the chemical potential, from which the value β = −0.54 is found. However, this mean-field result is qualitatively not reliable, since quantum depletion is neglected. A new approach is introduced, starting with a Jastrow Ansatz, from which the energy is expressed in the Jastrow function f and radial distribution function g. A cluster expansion is made for g, where a diagrammatic analogy is found. The Hyper-Netted Chain approximation is introduced to calculate g systematically by summing infinitely many diagrams. In HNC/0, we minimize the energy by a variational method, which is based on a well-constructed Ansatz for g. Numerical results using the HNC/0 approximation are given not only for the ground-state energy, but also for the condensate density and the contact. For large scattering lengths the calculations became increasingly less accurate, but the results from Bogoliubov theory are correctly included for small a. An extrapolation for the ground-state energy is made to find β ' −0.20, which is expected to be an upper bound. Moreover, quantum depletion is found to be higher when calculated in HNC/0 for large scattering lengths. Also, it is found that the contact at this value for unitarity remains finite and acquires the value of C ' 10.3n4/3, which is remarkably close to the value for fermions, C ' 11n4/3. Cover image courtesy of [18].

Jan 1, 20111 citation

On-shell approximation for the s -wave scattering theory

Francesco Lorenzi, Andrea Bardin, Luca Salasnich

We investigate the scattering theory of two particles in a generic $D$ -dimensional space. For the $s$ -wave problem, by adopting an on-shell approximation for the $T$ -matrix equation, we derive analytical formulas which connect the Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)$ of the interaction potential to the $s$ -wave phase shift. In this way we obtain explicit expressions of the low-momentum parameters ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}$ of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}{k}^{2}+\ensuremath{\cdots}$ in terms of the $s$ -wave scattering length ${a}_{s}$ and the $s$ -wave effective range ${r}_{s}$ for $D=3$ , $D=2$ , and $D=1$ . Our results, which are strongly dependent on the spatial dimension $D$ , are a useful benchmark for few-body and many-body calculations. As a specific application, we derive the zero-temperature pressure of a two-dimensional uniform interacting Bose gas with a beyond-mean-field correction which includes both scattering length and effective range.We investigate the scattering theory of two particles in a generic $D$-dimensional space. For the $s$-wave problem, by adopting an on-shell approximation for the $T$-matrix equation, we derive analytical formulas which connect the Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)$ of the interaction potential to the $s$-wave phase shift. In this way we obtain explicit expressions of the low-momentum parameters ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}$ of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}{k}^{2}+\ensuremath{\cdots}$ in terms of the $s$-wave scattering length ${a}_{s}$ and the $s$-wave effective range ${r}_{s}$ for $D=3$, $D=2$, and $D=1$. Our results, which are strongly dependent on the spatial dimension $D$, are a useful benchmark for few-body and many-body calculations. As a specific application, we derive the zero-temperature pressure of a two-dimensional uniform interacting Bose gas with a beyond-mean-field correction which includes both scattering length and effective range.

arXiv:2303.02675Physical Review AMar 23, 202310 citations

Excitation spectrum and effective interactions of a highly elongated Fermi gas

D. Blume, Debraj Rakshit

Full three-dimensional (3D) calculations of small two-component Fermi gases under highly elongated confinement, in which unlike fermions interact through short-range potentials with variable atom-atom $s$ -wave scattering length, are performed using the correlated Gaussian approach. In addition, microscopic one-dimensional (1D) calculations are performed for effective ``atomic'' and ``molecular'' 1D model Hamiltonian. Comparisons of the 3D and 1D energies and excitation frequencies establish the validity regimes of the effective 1D Hamiltonian. Our numerical results for three- and four-particle systems suggest that the effective 1D atom-dimer and dimer-dimer interactions are to a good approximation determined by simple analytical expressions. Implications for the description of quasi-1D Fermi gases within strict 1D frameworks are discussed.Full three-dimensional (3D) calculations of small two-component Fermi gases under highly elongated confinement, in which unlike fermions interact through short-range potentials with variable atom-atom $s$-wave scattering length, are performed using the correlated Gaussian approach. In addition, microscopic one-dimensional (1D) calculations are performed for effective ``atomic'' and ``molecular'' 1D model Hamiltonian. Comparisons of the 3D and 1D energies and excitation frequencies establish the validity regimes of the effective 1D Hamiltonian. Our numerical results for three- and four-particle systems suggest that the effective 1D atom-dimer and dimer-dimer interactions are to a good approximation determined by simple analytical expressions. Implications for the description of quasi-1D Fermi gases within strict 1D frameworks are discussed.

arXiv:0901.3862Physical Review AJul 1, 20098 citations

Analytical calculation of cold-atom scattering

V. V. Flambaum, G. F. Gribakin, C. Harabati

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 $\ensuremath{\gamma}=\sqrt{2\ensuremath{\mu}\ensuremath{\alpha}}/\ensuremath{\Elzxh}\ensuremath{\gg}{a}_{0}^{(n\ensuremath{-}2)/2}$ ( ${a}_{0}$ 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 ${F}_{n},$ ${G}_{n},$ and ${H}_{n}$ 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.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 $\ensuremath{\gamma}=\sqrt{2\ensuremath{\mu}\ensuremath{\alpha}}/\ensuremath{\Elzxh}\ensuremath{\gg}{a}_{0}^{(n\ensuremath{-}2)/2}$ (${a}_{0}$ 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 ${F}_{n},$ ${G}_{n},$ and ${H}_{n}$ 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.

Physical Review AMar 1, 1999180 citations

Calculation of the scattering length in atomic collisions using the semiclassical approximation

G. F. Gribakin, V. V. Flambaum

A simple analytical formula, a=a\ifmmode\bar\else\textasciimacron\fi{}[1-tan(\ensuremath{\pi}/n-2)tan{\ensuremath{\Phi}-[\ensuremath{\pi}/2(n-2)]}, is obtained for the scattering length in atomic collisions. Here a\ifmmode\bar\else\textasciimacron\fi{}=cos[\ensuremath{\pi}/(n-2)]{ \ensuremath{\surd}2M\ensuremath{\alpha} /[\ensuremath{\Elzxh}(n-2)]${\mathrm{}}}^{2/(\mathit{n}\mathrm{\ensuremath{-}}2)}$[\ensuremath{\Gamma}(n-3)/(n-2)]/[T(n-1)/(n-2)] is the mean scattering length determined by the asymptotic behavior of the potential U(r)\ensuremath{\sim}-\ensuremath{\alpha}/ ${\mathit{R}}^{\mathit{n}}$ , (n=6 for atom-atom scattering or n=4 for ion-atom scattering), M is the reduced mass of the atoms, and \ensuremath{\Phi} is the semiclassical phase calculated at zero energy from the classical turning point to infinity. The value of a\ifmmode\bar\else\textasciimacron\fi{}, the average scattering length, also determines the slope of the s-wave phase shifts beyond the near-threshold region. The formula is applicable to the collisions of atoms cooled down in traps, where the scattering length determines the character of the atom-atom interaction. Our calculation shows that repulsion between atoms (a>0) is more likely than attraction with a ``probability'' of 75%. For the Cs-Cs scattering in the $^{3}\mathrm{\ensuremath{\Sigma}}_{\mathit{u}}$ state, a\ifmmode\bar\else\textasciimacron\fi{}=95.5 ${\mathit{a}}_{\mathit{B}}$ has been obtained, where ${\mathit{a}}_{\mathit{B}}$ is the Bohr radius. The comparison of the calculated cross-section energy dependence with the experimental data gives evidence for a positive value for the Cs-Cs scattering length, which makes cesium Bose gas stable.A simple analytical formula, a=a\ifmmode\bar\else\textasciimacron\fi{}[1-tan(\ensuremath{\pi}/n-2)tan{\ensuremath{\Phi}-[\ensuremath{\pi}/2(n-2)]}, is obtained for the scattering length in atomic collisions. Here a\ifmmode\bar\else\textasciimacron\fi{}=cos[\ensuremath{\pi}/(n-2)]{ \ensuremath{\surd}2M\ensuremath{\alpha} /[\ensuremath{\Elzxh}(n-2)]${\mathrm{}}}^{2/(\mathit{n}\mathrm{\ensuremath{-}}2)}$[\ensuremath{\Gamma}(n-3)/(n-2)]/[T(n-1)/(n-2)] is the mean scattering length determined by the asymptotic behavior of the potential U(r)\ensuremath{\sim}-\ensuremath{\alpha}/${\mathit{R}}^{\mathit{n}}$, (n=6 for atom-atom scattering or n=4 for ion-atom scattering), M is the reduced mass of the atoms, and \ensuremath{\Phi} is the semiclassical phase calculated at zero energy from the classical turning point to infinity. The value of a\ifmmode\bar\else\textasciimacron\fi{}, the average scattering length, also determines the slope of the s-wave phase shifts beyond the near-threshold region. The formula is applicable to the collisions of atoms cooled down in traps, where the scattering length determines the character of the atom-atom interaction. Our calculation shows that repulsion between atoms (a>0) is more likely than attraction with a ``probability'' of 75%. For the Cs-Cs scattering in the $^{3}\mathrm{\ensuremath{\Sigma}}_{\mathit{u}}$ state, a\ifmmode\bar\else\textasciimacron\fi{}=95.5${\mathit{a}}_{\mathit{B}}$ has been obtained, where ${\mathit{a}}_{\mathit{B}}$ is the Bohr radius. The comparison of the calculated cross-section energy dependence with the experimental data gives evidence for a positive value for the Cs-Cs scattering length, which makes cesium Bose gas stable.

Physical Review AJul 1, 1993473 citations

We provide accurate expressions for the

s

s

-wave scattering length is a physically relevant parameter. We first describe a numerical procedure to compute the value of the

s

s

We investigate the scattering theory of two particles in a generic

D

-dimensional space. For the

s

-wave problem, by adopting an on-shell approximation for the

T

-matrix equation, we derive analytical formulas which connect the Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)$ of the interaction potential to the

s

-wave phase shift. In this way we obtain explicit expressions of the low-momentum parameters ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}$ of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}{k}^{2}+\ensuremath{\cdots}$ in terms of the

s

-wave scattering length

{a}_{s}

and the

s

-wave effective range

{r}_{s}

for

D=3

D=2

, and

D=1

. Our results, which are strongly dependent on the spatial dimension

D

, are a useful benchmark for few-body and many-body calculations. As a specific application, we derive the zero-temperature pressure of a two-dimensional uniform interacting Bose gas with a beyond-mean-field correction which includes both scattering length and effective range.

s

-wave scattering length, are performed using the correlated Gaussian approach. In addition, microscopic one-dimensional (1D) calculations are performed for effective ``atomic'' and ``molecular'' 1D model Hamiltonian. Comparisons of the 3D and 1D energies and excitation frequencies establish the validity regimes of the effective 1D Hamiltonian. Our numerical results for three- and four-particle systems suggest that the effective 1D atom-dimer and dimer-dimer interactions are to a good approximation determined by simple analytical expressions. Implications for the description of quasi-1D Fermi gases within strict 1D frameworks are discussed.

{a}_{0}

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

{F}_{n},

{G}_{n},

and

{H}_{n}

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.

{\mathit{R}}^{\mathit{n}}

, (n=6 for atom-atom scattering or n=4 for ion-atom scattering), M is the reduced mass of the atoms, and \ensuremath{\Phi} is the semiclassical phase calculated at zero energy from the classical turning point to infinity. The value of a\ifmmode\bar\else\textasciimacron\fi{}, the average scattering length, also determines the slope of the s-wave phase shifts beyond the near-threshold region. The formula is applicable to the collisions of atoms cooled down in traps, where the scattering length determines the character of the atom-atom interaction. Our calculation shows that repulsion between atoms (a>0) is more likely than attraction with a ``probability'' of 75%. For the Cs-Cs scattering in the $^{3}\mathrm{\ensuremath{\Sigma}}_{\mathit{u}}$ state, a\ifmmode\bar\else\textasciimacron\fi{}=95.5

{\mathit{a}}_{\mathit{B}}

has been obtained, where

{\mathit{a}}_{\mathit{B}}

is the Bohr radius. The comparison of the calculated cross-section energy dependence with the experimental data gives evidence for a positive value for the Cs-Cs scattering length, which makes cesium Bose gas stable.