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arXiv:2302.08331 · American Journal of Physics

Low-energy scattering parameters: A theoretical derivation of the effective range and scattering length for arbitrary angular momentum

Jordi Pera, Jordi Boronat

Published Jan 23, 2023• 5 citations
arXivHTMLPDF

Abstract

The most important parameters in the study of low-energy scattering are the s-wave and p-wave scattering lengths and the s-wave effective range. We solve the scattering problem and find two useful formulas for the scattering length and the effective range for any angular momentum, as long as the Wigner threshold law holds. Using that formalism, we obtain a set of useful formulas for the angular-momentum scattering parameters of four different model potentials: hard-sphere, soft-sphere, spherical well, and well-barrier potentials. The behavior of the scattering parameters close to Feshbach resonances is also analyzed. Our derivations can be useful as hands-on activities for learning scattering theory.

DOI
10.1119/5.0079744

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The most important parameters in the study of low-energy scattering are the s-wave and p-wave scattering lengths and the s-wave effective range. We solve the scattering problem and find two useful formulas for the scattering length and the effective range for any angular momentum, as long as the Wigner threshold law holds. Using that formalism, we obtain a set of useful formulas for the angular-momentum scattering parameters of four different model potentials: hard-sphere, soft-sphere, spherical well, and well-barrier potentials. The behavior of the scattering parameters close to Feshbach resonances is also analyzed. Our derivations can be useful as hands-on activities for learning scattering theory.
Elastic cross-sections for electron scattering on neon from 0 energy up to 16 eV are analyzed by an analytical approach to the modified effective range theory (MERT). It is shown that energy and angular variations of elastic differential, integral and momentum transfer cross-sections can be accurately parameterized by six MERT coefficients up to the energy threshold for the first Feshbach resonance. MERT parameters are determined empirically by numerical comparison with large collection of available experimental data of elastic total (integral) cross-sections. The present analysis is validated against numerous electron beams and swarm experiments. The comparison of derived MERT parameters with those found for other noble gases, helium, argon and krypton, is done. The derived scattering length (for the s-partial wave) in neon, 0.227 a 0, agrees well with recent theories; it is small but, differently from Ar and Kr, still positive. Analogue parameters for the p-wave and the d-wave are negative and positive respectively for all the four gases compared.
Scattering processes are a fundamental way of experimentally probing distributions and properties of systems in several areas of physics. Considering two-body scattering at low energies, when the de Broglie wavelength is larger than the range of the potential, partial waves with high angular momentum are typically unimportant. The dominant contribution comes from l=0l=0l=0 partial waves, commonly known as sss-wave scattering. This situation is very relevant in atomic physics, e.g. cold atomic gases, and nuclear physics, e.g. nuclear structure and matter. This manuscript is intended as a pedagogical introduction to the topic while covering a numerical approach to compute the desired quantities. We introduce low-energy scattering with particular attention to the concepts of scattering length and effective range. These two quantities appear in the effective-range approximation, which universally describes low-energy processes. We outline a numerical procedure for calculating the scattering length and effective range of spherically symmetric two-body potentials. As examples, we apply the method to the spherical well, modified Pöschl-Teller, Gaussian, and Lennard-Jones potentials. We hope to provide the tools so students can implement similar calculations and extend them to other potentials.
The concepts of scattering length a and effective range re{r}_{e}re​ previously introduced for low-energy scattering from a potential V(r) in a plane and in higher dimensions are extended to include a 1/r potential (strength parameter \ensuremath{\gamma}). Both a and re{r}_{e}re​ have the physical significance of being equal to the radius of an equivalent hard sphere giving rise to the same O(k0{k}^{0}k0) and O(k2{k}^{2}k2) terms in the expression for the phase shift. The method used is based on the properties of the ``local scattering length'' a(r,\ensuremath{\gamma}) for the potential V(r) cut off at radius r and an ``equivalent hard-sphere radius'' a(r,k,\ensuremath{\gamma}) for wave number k\ensuremath{\ne}0. It is shown that these quantities have a smooth behavior for \ensuremath{\gamma}\ensuremath{\rightarrow}0 and for dimension n\ensuremath{\rightarrow}2.
It is shown how the concepts of scattering length and effective range, previously introduced for low-energy scattering from a potential V(r) in a plane, correspond to the well-known parameters in three dimensions. This is done by considering low-energy scattering in a general dimension n\ensuremath{\ge}2 and subsequently showing that both the n=2 and n=3 cases fit naturally in such a generalized treatment. Furthermore, our previous work is extended to long-range potentials, decreasing faster than 1/rn+1{r}^{n+1}rn+1. The method used is based on the properties of a local scattering length a(r) for the potential V(r) cut off at radius r and an equivalent hard-sphere radius a(r,k) for k\ensuremath{\ne}0. Some applications and illustrative examples are given.
We investigate the scattering theory of two particles in a generic DDD-dimensional space. For the sss-wave problem, by adopting an on-shell approximation for the TTT-matrix equation, we derive analytical formulas which connect the Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)$ of the interaction potential to the sss-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 sss-wave scattering length as{a}_{s}as​ and the sss-wave effective range rs{r}_{s}rs​ for D=3D=3D=3, D=2D=2D=2, and D=1D=1D=1. Our results, which are strongly dependent on the spatial dimension DDD, 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 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.