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Journal of Physics A Mathematical and General

Scattering length and effective range in two dimensions: application to adsorbed hydrogen atoms

BOUDEWIJN J. VERHAAR, J.P.H.W. van den Eijnde, M. A. J. Voermans, M Schaffrath

Published Feb 21, 1984• 74 citations
Publisher

Abstract

The concepts of scattering length (a) and effective range (r e ) are introduced in two dimensions for an isotropic finite-range potential. Application to mutual scattering of pairs of hydrogen atoms in the triplet state adsorbed at a liquid helium surface shows the usefulness of a description in terms of two parameters. The values derived for this case are a=2.468a 0 and r e =11.62a 0 .

DOI
10.1088/0305-4470/17/3/020

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The concepts of scattering length (a) and effective range (r e ) are introduced in two dimensions for an isotropic finite-range potential. Application to mutual scattering of pairs of hydrogen atoms in the triplet state adsorbed at a liquid helium surface shows the usefulness of a description in terms of two parameters. The values derived for this case are a=2.468a 0 and r e =11.62a 0 .
Accurate scattering lengths and effective ranges for hydrogen atomic collisions are determined from effective range theory. The precise ab initio b3Σ+u potential energy curve values of H2 calculated by Kolos and Wolniewicz and C6, C8, C10 long-range expansion coefficients are used in obtaining accurate representation of the two-body interaction potential. The s-wave scattering length obtained for this potential (A =1/33455 ao) is used to calculate the zero temperature energy as a function of density of electron spin-polarized ("spin-aligned") atomic hydrogen (H+) using previous many-body perturbation theory results for bosons. Neglecting small potential shape dependent terms, one obtains E/N (energy per particle ) as a function of density, which may be compared with similar results obtained from Monte-Carlo or variational calculations. The scattering lengths and effective ranges for the other hydrogen isotopic systems are also presented. It is shown that the first bound state for the b3Σ+u potential (within the Born-Oppenheimer approximation) only exists for a hypothetical isotopic species with mass larger than a value m = 3.258176 g/mole (where the scattering length becomes infinite). In terms of the quantum parameter, η ≡ [MATH]2/ (2µεσ2), two-body binding occurs in the b3Σ+u potential only for η < 0.17094. The second order liquid-to-gas transition (corresponding to the onset of infinite-body binding) occurs at the accurately determined critical value of the quantum parameter ηCB = 0.43891. This corresponds to zero scattering length.
Recently developed variational techniques for determining upper bounds on scattering lengths are applied to singlet and triplet scattering of zero-energy electrons by atomic hydrogen. The results obtained are not only rigorous but are in fact somewhat lower and therefore somewhat better than those previously obtained by variational methods. We find that the triplet and singlet scattering lengths, AT{A}_{T}AT​ and AS{A}_{S}AS​ respectively, satisfy the inequalities ${A}_{T}\ensuremath{\leqq}1.91{a}_{0}$ and ${A}_{S}\ensuremath{\leqq}6.23{a}_{0}$, where a0{a}_{0}a0​ is the Bohr radius. The only assumptions involved in the deduction of these results are that there be no bound triplet state and one and only one bound singlet state. The singlet trial function determined during the course of the calculation generates a singlet effective range, r0S{r}_{0S}r0S​, of about 2.7a02.7{a}_{0}2.7a0​. The triplet trial functions which were obtained were not sufficiently accurate to be useful in a determination of the triplet effective range, r0T{r}_{0T}r0T​.
A parametrization of low-energy two-dimensional quantum scattering is given which is in the tradition of familiar three-dimensional effective-range theory. A scattering length differing from previous treatments is introduced whose behavior reflects the repulsive or attractive nature of the interaction, and whether or not bound states are present. However its definition is subject to an unknown length parameter. An effective-range parameter is also defined and together with the scattering length explains the nonexistence of the Efimov effect in two dimensions. It is also shown that the Ramsauer–Townsend effect exists in 2D. Application to the case of a square-well potential reveals direct relationships between the unknown length parameter and the effective range to the actual range of the potential.
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.
The elastic scattering of two hydrogen atoms interacting at very small energies is explored numerically using the most accurate potential-energy curves available for the X $^{1}\mathrm{\ensuremath{\Sigma}}_{\mathrm{g}}^{+}$ and $^{3}\mathrm{\ensuremath{\Sigma}}_{\mathrm{u}}^{+}$ states of H2{\mathrm{H}}_{2}H2​. The scattering lengths and low-temperature-averaged cross sections are determined and compared with previous calculations. The scattering lengths are 0.41a0{\mathit{a}}_{0}a0​ for the singlet case and 1.91a0{\mathit{a}}_{0}a0​ for the triplet. The predicted total triplet cross section at 2 K is 5.5\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}15}$ cm2{\mathrm{cm}}^{2}cm2 but it could be lower, at 4.8\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}15}$ cm2{\mathrm{cm}}^{2}cm2, with a different fit to the triplet potential between 12a0{\mathit{a}}_{0}a0​ and 15a0{\mathit{a}}_{0}a0​. A more accurate triplet-state potential is needed at these separations.
Abstract The textbook effective-range expansion of scattering theory is useful in the analysis of low-energy scattering phenomenology when the scattering length ∣ a ∣ is much larger than the range R of the scattering potential: ∣ a ∣ ≫ R . Nevertheless, the same has been used for systems where the scattering length is much smaller than the range of the potential. We suggest and numerically study an effective-range expansion for ∣ a ∣ ≪ R as well as improved two-parameter effective-range expansions for the cases ∣ a ∣ > R and ∣ a ∣ < R . The improved effective-range expansions for ∣ a ∣ > R and ∣ a ∣ < R reduce to the expansions for ∣ a ∣ / R ≫ 1 and ∣ a ∣ / R ≪ 1 , respectively, in appropriate limits.