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arXiv:1109.6522 · The Journal of Chemical Physics

Rapid, accurate calculation of the s-wave scattering length

Vladimir V. Meshkov, A. V. Stolyarov, Robert J. Le Roy

Published Oct 20, 2011• 27 citations
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Abstract

Transformation of the conventional radial Schrödinger equation defined on the interval r ∈ [0, ∞) into an equivalent form defined on the finite domain y(r) ∈ [a, b] allows the s-wave scattering length a(s) to be exactly expressed in terms of a logarithmic derivative of the transformed wave function φ(y) at the outer boundary point y = b, which corresponds to r = ∞. In particular, for an arbitrary interaction potential that dies off as fast as 1/r(n) for n ≥ 4, the modified wave function φ(y) obtained by using the two-parameter mapping function r(y; ̄r,β) = ̄r[1 + 1/β tan(πy/2)] has no singularities, and a(s) = ̄r[1 + 2/πβ 1/φ(1) dφ(1)/dy]. For a well bound potential with equilibrium distance r(e), the optimal mapping parameters are ̄r ≈ r(e) and β ≈ n/2 - 1. An outward integration procedure based on Johnson's log-derivative algorithm [J. Comp. Phys. 13, 445 (1973)] combined with a Richardson extrapolation procedure is shown to readily yield high precision a(s)-values both for model Lennard-Jones (2n, n) potentials and for realistic published potentials for the Xe-e(-), Cs(2)(aΣ(u)(+)(3)), and (3, 4)He(2)(XΣ(g)(+)(1)) systems. Use of this same transformed Schrödinger equation was previously shown [V. V. Meshkov et al., Phys. Rev. A 78, 052510 (2008)] to ensure the efficient calculation of all bound levels supported by a potential, including those lying extremely close to dissociation.

DOI
10.1063/1.3649946

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Transformation of the conventional radial Schrödinger equation defined on the interval r ∈ [0, ∞) into an equivalent form defined on the finite domain y(r) ∈ [a, b] allows the s-wave scattering length a(s) to be exactly expressed in terms of a logarithmic derivative of the transformed wave function φ(y) at the outer boundary point y = b, which corresponds to r = ∞. In particular, for an arbitrary interaction potential that dies off as fast as 1/r(n) for n ≥ 4, the modified wave function φ(y) obtained by using the two-parameter mapping function r(y; ̄r,β) = ̄r[1 + 1/β tan(πy/2)] has no singularities, and a(s) = ̄r[1 + 2/πβ 1/φ(1) dφ(1)/dy]. For a well bound potential with equilibrium distance r(e), the optimal mapping parameters are ̄r ≈ r(e) and β ≈ n/2 - 1. An outward integration procedure based on Johnson's log-derivative algorithm [J. Comp. Phys. 13, 445 (1973)] combined with a Richardson extrapolation procedure is shown to readily yield high precision a(s)-values both for model Lennard-Jones (2n, n) potentials and for realistic published potentials for the Xe-e(-), Cs(2)(aΣ(u)(+)(3)), and (3, 4)He(2)(XΣ(g)(+)(1)) systems. Use of this same transformed Schrödinger equation was previously shown [V. V. Meshkov et al., Phys. Rev. A 78, 052510 (2008)] to ensure the efficient calculation of all bound levels supported by a potential, including those lying extremely close to dissociation.
We derive two equivalent analytical expressions for an lth partial-wave scattering length al for central potentials with long-range tails of the form V(r)=−ℏ22mBrn−4(rn−2 + Rn−2)2−ℏ22mCr2(rn−2 + Rn−2), (r ⩾ rs, R > 0). For C = 0, this family of potentials reduces to the Lenz potentials discussed in a similar context in our earlier works [R. Szmytkowski, Acta Phys. Pol. A 79, 613 (1991); J. Phys. A: Math. Gen. 28, 7333 (1995)]. The formulas for al that we provide in this paper depend on the parameters B, C, and R characterizing the tail of the potential, on the core radius rs, as well as on the short-range scattering length als, the latter being due to the core part of the potential. The procedure, which may be viewed as an analytical extrapolation from als to al, is relied on the fact that the general solution to the zero-energy radial Schrödinger equation with the potential given above may be expressed analytically in terms of the generalized associated Legendre functions.
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 provide accurate expressions for the sss-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 sss-wave scattering length is a physically relevant parameter. We first describe a numerical procedure to compute the value of the sss-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 sss-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.
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 $\ensuremath{\gamma}=\sqrt{2\ensuremath{\mu}\ensuremath{\alpha}}/\ensuremath{\Elzxh}\ensuremath{\gg}{a}_{0}^{(n\ensuremath{-}2)/2}$ (a0{a}_{0}a0​ 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,a,a, which showed that it has 75% preference for positive values for n=6,n=6,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,a,a, ${r}_{e}{=F}_{n}\ensuremath{-}{G}_{n}{/a+H}_{n}{/a}^{2},$ where Fn,{F}_{n},Fn​, Gn,{G}_{n},Gn​, and Hn{H}_{n}Hn​ depend only on \ensuremath{\gamma}. Thus, we know the sss-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 sss-wave atomic scattering in a wide range of momenta, as well as the positions of upper bound states of the diatomic molecule.
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}/Rn{\mathit{R}}^{\mathit{n}}Rn, (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{}=95.5aB{\mathit{a}}_{\mathit{B}}aB​ has been obtained, where aB{\mathit{a}}_{\mathit{B}}aB​ 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.