Journal of Physics B Atomic Molecular and Optical Physics

How a change in the interaction potential affects the p-wave scattering volume

M. J. Jamieson, A. Dalgarno

Published Feb 29, 2012• 3 citations

Abstract

We derive a simple expression for the change in the s-wave scattering length in terms of zero-energy wavefunctions, we generalize it to obtain an expression for the change in the p-wave scattering volume and we use both expressions to derive the first order differential equations of variable phase theory that are satisfied by the closely related accumulated scattering length and volume. We provide numerical demonstrations for the example of a pair of hydrogen atoms interacting via the X1Σ+g molecular state.

DOI: 10.1088/0953-4075/45/6/061001

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Journal of Physics B Atomic Molecular and Optical Physics

How a change in the interaction potential affects the p-wave scattering volume

M. J. Jamieson, A. Dalgarno

Published Feb 29, 2012• 3 citations

Publisher

Abstract

DOI: 10.1088/0953-4075/45/6/061001

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We show how the scattering phase shift, the

s

-wave scattering length, and the

p

-wave scattering volume can be obtained from Riccati equations derived in variable phase theory. We find general expressions that provide upper and lower bounds for the scattering length and the scattering volume. We show how, in the framework of the variable phase method, Levinson's theorem yields the number of bound states supported by a potential. We report results from a study of the heteronuclear alkali-metal dimers NaRb and LiRb. We consider ab initio molecular potentials for the $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ and $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ states of both dimers and compare and discuss results obtained from experimentally based $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ and $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ potentials of NaRb. We explore the mass dependence of the scattering data by considering all isotopomers and we calculate the numbers of bound states supported by the molecular potentials for each isotopomer.

We provide accurate expressions for the

s

-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

s

-wave scattering length is a physically relevant parameter. We first describe a numerical procedure to compute the value of the

s

-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

s

-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.