Physical Review A

Calculation of the electron-scattering lengths for the rare-gas atoms

Radosław Szmytkowski

Published Jan 1, 1995• 12 citations

Abstract

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.

DOI: 10.1103/physreva.51.853

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In a recent paper, Reisfeld and Asaf [Phys. Rev. A 49, 348 (1994)] have undertaken an attempt to explain the linear dependence of the electron-atom scattering length on the atomic polarizability. They used the Born approximation, which is valid only for small atomic or molecular polarizabilities. Using the exact formula for the scattering length, we offer a simple expression for estimation of the scattering length by known values of the polarizability and ionization potential.

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Analytical calculations of scattering lengths in atomic physics

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A simple potential for low-energy s-wave scattering of electrons from closed-shell atoms

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

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Physical Review A

Calculation of the electron-scattering lengths for the rare-gas atoms

Radosław Szmytkowski

Published Jan 1, 1995• 12 citations

Publisher

Abstract

DOI: 10.1103/physreva.51.853

Related papers

Comment on ‘‘Relation between the electron scattering length and the van der Waals approximation to the equation of state’’

Igor T. Iakubov, Alexey G. Khrapak

Physical Review AJun 1, 199510 citations

Relation between the electron scattering length and the van der Waals approximation to the equation of state

G. Reisfeld, Uri Asaf

A theoretical relation between the electron scattering length and the van der Waals approximation to the equation of state is described. The theory provides a simple calculation for the scattering length of neutral atomic scatterers such as rare gases, using two experimental parameters: the hard-core radius (calculated from the van der Waals equation of state, using critical data) and the atomic polarizability. The results for the calculated electron scattering length agree with measured values derived either from the energy shifts of highly excited molecular Rydberg states, perturbed by He, Ne, Ar, Kr, and Xe, or from swarm electron cross-section data for the same gases.A theoretical relation between the electron scattering length and the van der Waals approximation to the equation of state is described. The theory provides a simple calculation for the scattering length of neutral atomic scatterers such as rare gases, using two experimental parameters: the hard-core radius (calculated from the van der Waals equation of state, using critical data) and the atomic polarizability. The results for the calculated electron scattering length agree with measured values derived either from the energy shifts of highly excited molecular Rydberg states, perturbed by He, Ne, Ar, Kr, and Xe, or from swarm electron cross-section data for the same gases.

Physical Review AJan 1, 199413 citations

Analytical calculations of scattering lengths in atomic physics

Radosław Szmytkowski

We describe a method for evaluating analytical long-range contributions to scattering lengths for some potentials used in atomic physics. We assume that an interaction potential between colliding particles consists of two parts. The form of a short-range component, vanishing beyond some distance from the origin (a core radius), need not be given. Instead, we assume that a set of short-range scattering lengths due to that part of the interaction is known. A long-range tail of the potential is chosen to be an inverse power potential, a superposition of two inverse power potentials with suitably chosen exponents or the Lent potential. For these three classes of long-range interactions a radial Schrodinger equation at zero energy may be solved analytically with solutions expressed in terms of the Bessel, Whittaker and Legendre functions, respectively. We utilize this fact and derive exact analytical formulae for the scattering lengths. The expressions depend on the short-range scattering lengths, the core radius and parameters characterizing the long-range part of the interaction. Cases when the long-range potential (or its part) may be treated as a perturbation are also discussed and formulae for scattering lengths linear in strengths of the perturbing potentials are given. It is shown that for some combination of the orbital angular momentum quantum number and an exponent of the leading term of the potential the derived formulae, exact or approximate, take very simple forms and contain only polynomial and trigonometric functions. The expressions obtained in this paper are applicable to scattering of charged particles by neutral targets and to collisions between neutrals. The results are illustrated by accelerating convergence of scattering lengths computed for e - -Xe and Cs-Cs systems.We describe a method for evaluating analytical long-range contributions to scattering lengths for some potentials used in atomic physics. We assume that an interaction potential between colliding particles consists of two parts. The form of a short-range component, vanishing beyond some distance from the origin (a core radius), need not be given. Instead, we assume that a set of short-range scattering lengths due to that part of the interaction is known. A long-range tail of the potential is chosen to be an inverse power potential, a superposition of two inverse power potentials with suitably chosen exponents or the Lent potential. For these three classes of long-range interactions a radial Schrodinger equation at zero energy may be solved analytically with solutions expressed in terms of the Bessel, Whittaker and Legendre functions, respectively. We utilize this fact and derive exact analytical formulae for the scattering lengths. The expressions depend on the short-range scattering lengths, the core radius and parameters characterizing the long-range part of the interaction. Cases when the long-range potential (or its part) may be treated as a perturbation are also discussed and formulae for scattering lengths linear in strengths of the perturbing potentials are given. It is shown that for some combination of the orbital angular momentum quantum number and an exponent of the leading term of the potential the derived formulae, exact or approximate, take very simple forms and contain only polynomial and trigonometric functions. The expressions obtained in this paper are applicable to scattering of charged particles by neutral targets and to collisions between neutrals. The results are illustrated by accelerating convergence of scattering lengths computed for e - -Xe and Cs-Cs systems.

Journal of Physics A Mathematical and GeneralDec 21, 199541 citations

A simple potential for low-energy s-wave scattering of electrons from closed-shell atoms

S. H. Patil

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.

Journal of Physics B Atomic Molecular and Optical PhysicsMay 28, 19961 citation

Multiple Scattering and Many-Body Theory: Free Energy of Electrons in Helium

Michael H. Coopersmith

An electron in helium vapor at 4\ifmmode^\circ\else\textdegree\fi{}K is characterized by its $s$ -wave scattering length off helium atoms. This scattering length is small compared with the average interparticle spacing in the helium vapor. Consequently, we have taken as a model a particle interacting via hard-core repulsion with an ideal gas. This establishes a connection with the hard-sphere Bose problem of more general interest. The model described above is simpler (a) because it has only Boltzmann statistics and (b) because the electron is very light compared with the helium atoms. For this model, we have calculated the interaction free energy of the electron assuming that it is in statistical equilibrium with the helium gas. In the $s$ -wave approximation, it is shown that this interaction free energy is rigorously $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ due to single scattering, all higher order multiple-scattering effects being zero. Here, $\ensuremath{\rho}$ is the average density of helium atoms, $d$ is the scattering length, and $m$ , is the electron mass. Since the $p$ -wave approximation contributes a term of order $\ensuremath{\rho}{a}^{3}$, it is evident that the term $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ is good to higher densities than might previously have been supposed. This provides partial justification for the "bubble" model of the electron mobility since the term $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ is certainly good up to densities at which the free energy of the "bubble" configuration becomes smaller.An electron in helium vapor at 4\ifmmode^\circ\else\textdegree\fi{}K is characterized by its $s$-wave scattering length off helium atoms. This scattering length is small compared with the average interparticle spacing in the helium vapor. Consequently, we have taken as a model a particle interacting via hard-core repulsion with an ideal gas. This establishes a connection with the hard-sphere Bose problem of more general interest. The model described above is simpler (a) because it has only Boltzmann statistics and (b) because the electron is very light compared with the helium atoms. For this model, we have calculated the interaction free energy of the electron assuming that it is in statistical equilibrium with the helium gas. In the $s$-wave approximation, it is shown that this interaction free energy is rigorously $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ due to single scattering, all higher order multiple-scattering effects being zero. Here, $\ensuremath{\rho}$ is the average density of helium atoms, $d$ is the scattering length, and $m$, is the electron mass. Since the $p$-wave approximation contributes a term of order $\ensuremath{\rho}{a}^{3}$, it is evident that the term $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ is good to higher densities than might previously have been supposed. This provides partial justification for the "bubble" model of the electron mobility since the term $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ is certainly good up to densities at which the free energy of the "bubble" configuration becomes smaller.

Physical ReviewAug 30, 196519 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

An electron in helium vapor at 4\ifmmode^\circ\else\textdegree\fi{}K is characterized by its

s

-wave scattering length off helium atoms. This scattering length is small compared with the average interparticle spacing in the helium vapor. Consequently, we have taken as a model a particle interacting via hard-core repulsion with an ideal gas. This establishes a connection with the hard-sphere Bose problem of more general interest. The model described above is simpler (a) because it has only Boltzmann statistics and (b) because the electron is very light compared with the helium atoms. For this model, we have calculated the interaction free energy of the electron assuming that it is in statistical equilibrium with the helium gas. In the

s

-wave approximation, it is shown that this interaction free energy is rigorously $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ due to single scattering, all higher order multiple-scattering effects being zero. Here, $\ensuremath{\rho}$ is the average density of helium atoms,

d

is the scattering length, and

m

, is the electron mass. Since the

p

-wave approximation contributes a term of order $\ensuremath{\rho}{a}^{3}$, it is evident that the term $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ is good to higher densities than might previously have been supposed. This provides partial justification for the "bubble" model of the electron mobility since the term $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ is certainly good up to densities at which the free energy of the "bubble" configuration becomes smaller.

{\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.