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Effective-range expansion for doublet nd scattering

I.V. Simenog, A.I. Sitnichenko, D.V. Shapoval

Published Jan 1, 1985• 1 citation
OpenAlex

Abstract

We derived an expression for the effective range r of doublet nd scattering and perform systematic calculations of r for a broad class of separable interaction profiles. We predict the value (1/2)..cap alpha../sup 3/a/sup 2/r = 1.25 for the slope parameter of the amplitude and establish the quadratic correlation between this parameter and the scattering length a. This correlation and the well-known relation between the triton binding energy and a allow the model-independent explanation of the low-energy behavior of the nd scattering amplitude.

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Physical Review A•Mar 1, 1999•180 citations
We derived an expression for the effective range r of doublet nd scattering and perform systematic calculations of r for a broad class of separable interaction profiles. We predict the value (1/2)..cap alpha../sup 3/a/sup 2/r = 1.25 for the slope parameter of the amplitude and establish the quadratic correlation between this parameter and the scattering length a. This correlation and the well-known relation between the triton binding energy and a allow the model-independent explanation of the low-energy behavior of the nd scattering amplitude.
Relativistic effects on the relation between the root-mean-square radius rD{\mathit{r}}_{\mathit{D}}rD​ of the bound state and the scattering length a are examined for models which simulate the triplet S state of the two-nucleon system. Two models are considered; one is based on the one-body Dirac equation and the other on the two-body Dirac equation. The relation is expressed as (rD{\mathit{r}}_{\mathit{D}}rD​/a)2{)}^{2})2=(c0{\mathit{c}}_{0}c0​+c1{\mathit{c}}_{1}c1​x1{\mathit{x}}^{1}x1+c2{\mathit{c}}_{2}c2​x2{\mathit{x}}^{2}x2+...)/8, where x==\ensuremath{\alpha}r0{\mathit{r}}_{0}r0​; \ensuremath{\alpha} is related to the binding energy and r0{\mathit{r}}_{0}r0​ is the effective range. This expansion is compared with the corresponding expansion of the nonrelativistic Schr\"odinger case. The first three terms are shape independent. The c0{\mathit{c}}_{0}c0​ (=1) and c1{\mathit{c}}_{1}c1​ (=0) are the same as those of the nonrelativistic case but a relativistic correction appears in c2{\mathit{c}}_{2}c2​.
The modifications of effective-range theory, necessitated by the long-range nature of interatomic dispersion and relativistic retardation forces, are discussed. A numerical comparison is given of parameters describing $^{3}\mathrm{\ensuremath{-}}^{3}$He, $^{3}\mathrm{\ensuremath{-}}^{4}$He, and $^{4}\mathrm{\ensuremath{-}}^{4}$He scattering at very low energies calculated with several recently published interaction potentials. The effect of relativistic retardation on these parameters and on the calculated binding energies of the $^{4}\mathrm{\ensuremath{-}}^{4}$He dimer is shown. It is demonstrated that all of these quantities are extremely sensitive to the potential. Scattering lengths and effective ranges for the lowest triplet interactions of H-H and D-D are also reported.
Effective field theories (EFTs) are widely used to study many-body systems by describing two-body interactions using zero-ranged contact potentials. However, when extended to three-body processes, these contact interactions lead to divergences due to the absence of an intrinsic length scale. In EFT, this is typically resolved by introducing a zero-ranged three-body interaction, which can be renormalized to make the low-energy physics independent of the short-distance physics. However, when the two-body potential has a finite range, such as in separable potentials, there is no need for such renormalization. In this work, we derive the integral equation for the three-body scattering amplitude for separable potentials, and solve it in the strongly-interacting regime. With our model, we retrieve the known analytic form of the scattering amplitude for inelastic scattering processes and formulate a new scaling law for elastic three-body scattering processes.
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.