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Journal of Mathematical Physics

Analytical calculations of scattering lengths for a class of long-range potentials of interest for atomic physics

Radosław Szmytkowski

Published Jan 1, 2020• 1 citation
Publisher

Abstract

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.

DOI
10.1063/1.5140726

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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.
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.
Demonstrating the completeness of wave function solutions of the radial Schrödinger equation is a very difficult task. Existing proofs, relying on operator theory, are often very abstract and far from intuitive comprehension. However, it is possible to obtain rigorous proofs amenable to physical insight, if one restricts the considered class of Schrödinger potentials. One can mention, in particular, unbounded potentials yielding a purely discrete spectrum and short-range potentials. However, those possessing a Coulomb tail, very important for physical applications, have remained problematic due to their long-range character. The method proposed in this paper allows to treat them correctly, provided that the non-Coulomb part of potentials vanishes after a finite radius. Nonlocality of potentials can also be handled. The main idea in the proposed demonstration is that regular solutions behave like sine/cosine functions for large momenta, so that their expansions verify Fourier transform properties. The highly singular point at k=0 of long-range potentials is dealt with properly using analytical properties of Coulomb wave functions. Lebesgue measure theory is avoided, rendering the demonstration clear from a physical point of view.
Demonstrating the completeness of wave functions solutions of the radial Schrodinger equation is a very difficult task. Existing proofs, relying on operator theory, are often very abstract and far from intuitive comprehension. However, it is possible to obtain rigorous proofs amenable to physical insight, if one restricts the considered class of Schrodinger potentials. One can mention in particular unbounded potentials yielding a purely discrete spectrum and short-range potentials. However, those possessing a Coulomb tail, very important for physical applications, have remained problematic due to their long-range character. The method proposed in this paper allows to treat them correctly, provided the non-Coulomb part of potentials vanishes after a finite radius. Non-locality of potentials can also be handled. The main idea in the proposed demonstration is that regular solutions behave like sine/cosine functions for large momenta, so that their expansions verify Fourier transform properties. The highly singular point at k = 0 of long-range potentials is dealt with properly using analytical properties of Coulomb wave functions. Lebesgue measure theory is avoided, rendering the demonstration clear from a physical point of view.
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 derive a generalized zero-range pseudopotential applicable to all partial wave solutions to the Schrödinger equation based on a delta-shell potential in the limit that the shell radius approaches zero. This properly models all higher order multipole moments not accounted for with a monopolar delta function at the origin, as used in the familiar Fermi pseudopotential for s-wave scattering. By making the strength of the potential energy dependent, we derive self-consistent solutions for the entire energy spectrum of the realistic potential. We apply this to study two particles in an isotropic harmonic trap, interacting through a central potential, and derive analytic expressions for the energy eigenstates and eigenvalues.