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Physical Review A

Elastic scattering of hydrogen atoms at low temperatures

Michael J. Jamieson, A. Dalgarno, John N. Yukich

Published Dec 1, 1992• 53 citations
Publisher

Abstract

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 ${\mathrm{H}}_{2}$. The scattering lengths and low-temperature-averaged cross sections are determined and compared with previous calculations. The scattering lengths are 0.41${\mathit{a}}_{0}$ for the singlet case and 1.91${\mathit{a}}_{0}$ for the triplet. The predicted total triplet cross section at 2 K is 5.5\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}15}$ ${\mathrm{cm}}^{2}$ but it could be lower, at 4.8\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}15}$ ${\mathrm{cm}}^{2}$, with a different fit to the triplet potential between 12${\mathit{a}}_{0}$ and 15${\mathit{a}}_{0}$. A more accurate triplet-state potential is needed at these separations.

DOI
10.1103/physreva.46.6956

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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\ensuremathΣg+^{1}\mathrm{\ensuremath{\Sigma}}_{\mathrm{g}}^{+}1\ensuremathΣg+​ and 3\ensuremathΣu+^{3}\mathrm{\ensuremath{\Sigma}}_{\mathrm{u}}^{+}3\ensuremathΣu+​ 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\ensuremath−15{10}^{\mathrm{\ensuremath{-}}15}10\ensuremath−15 cm2{\mathrm{cm}}^{2}cm2 but it could be lower, at 4.8\ifmmode\times\else\texttimes\fi{}10\ensuremath−15{10}^{\mathrm{\ensuremath{-}}15}10\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.
In a recent paper Jamieson, Dalgarno, and Yukich [Phys. Rev. A 46, 6956 (1992)] discuss the elastic scattering of two hydrogen atoms at very low collision energies. They use as an input to their numerical calculations the potential-energy curve of the $^{3}\mathrm{\ensuremath{\Sigma}}_{\mathrm{u}}^{+}$ state obtained from a configuration-interaction (CI) calculation and report a triplet scattering length that is much larger than found previously. We show that the difference is a consequence of the neglect of a systematic shift in the b $^{3}\mathrm{\ensuremath{\Sigma}}_{\mathrm{u}}^{+}$ curve. Furthermore, we show that by taking the systematic shift into account, the scattering length of the CI potential reduces to a value of (1.2\ifmmode\pm\else\textpm\fi{}0.2)a0{\mathit{a}}_{0}a0​, which is in agreement with previous results.
Potential scattering and spin-exchange scattering cross sections in the energy range from 10 -3 cm -1 to 100 cm -1 have been calculated numerically for hydrogen atoms in collision based on the most accurate Kołos-Woniewicz adiabatic potentials available at present. The scattering lengths for the singlet and triplet states are calculated to be a 0 s =0.595 a.u. and a 0 t =1.34 a.u., respectively. The quasi-bound state with ν=14 and l =4 is found to be at E =1.675 cm -1 above the dissociation limit. The thermally averaged spin-exchange cross section has the low temperature limit \barσ ex (0 K)=0.982 A 2 with a minimum \barσ ex (2.5 K)=0.329 A 2 .
A non-adiabatic model is employed to investigate the H-H scattering at low temperatures. Unlike molecular-orbital calculation technique, no separate calculations are required to obtain the potential energies of electrons in the field of hadrons. This model generates the non-adiabatic effective potential automatically. The coefficients of van der Waals and higher-order potentials are estimated in the present model and are found to be in very good agreement with the corresponding standard values. The s -wave singlet and triplet scattering lengths and thermal-average total triplet cross-section at 2 K are reported and compared with the existing theoretical and experimental predictions.
Interaction potentials for the X $^{1}\mathrm{\ensuremath{\Sigma}}_{\mathrm{g}}^{+}$ and a $^{3}\mathrm{\ensuremath{\Sigma}}_{\mathrm{u}}^{+}$ states of Li2{\mathrm{Li}}_{2}Li2​ are constructed and used in calculations of the elastic scattering of two 7Li^{7}\mathrm{Li}7Li atoms at ultralow temperatures. The extreme sensitivity of the cross sections to the details of the interaction potentials is demonstrated. The calculated elastic and spin-change cross sections are very large, of the order of ${10}^{\mathrm{\ensuremath{-}}13}$ cm2{\mathrm{cm}}^{2}cm2 at zero temperature. The predicted scattering length for the triplet state is negative with a value of -17.2a0{\mathit{a}}_{0}a0​. Pronounced shape resonances appear at energies of 4.9\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}8}$ a.u. in the singlet scattering and 5.5\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}7}$ a.u. in the triplet scattering.
Interaction potentials for the X $^{1}\mathrm{\ensuremath{\Sigma}}_{\mathrm{g}}^{+}$ and a $^{3}\mathrm{\ensuremath{\Sigma}}_{\mathrm{u}}^{+}$ states of Na2{\mathrm{Na}}_{2}Na2​ are constructed and used in calculations of the elastic scattering of two Na atoms at ultralow temperatures. The sensitivity to retardation effects (or Casimir corrections) is explored. The calculated elastic and spin-change cross sections are very large, of the order of ${10}^{\mathrm{\ensuremath{-}}12}$--${10}^{\mathrm{\ensuremath{-}}13}$ cm2{\mathrm{cm}}^{2}cm2 at zero temperature. The predicted scattering lengths are both positive, 34.9 a0{\mathit{a}}_{0}a0​ for the X $^{1}\mathrm{\ensuremath{\Sigma}}_{\mathrm{g}}^{+}$ state and 77.3 a0{\mathit{a}}_{0}a0​ for the a $^{3}\mathrm{\ensuremath{\Sigma}}_{\mathrm{u}}^{+}$ state. Pronounced shape resonances appear for the l=3 and 7 partial waves for the singlet and l=6 for the triplet states.
We have continued our investigation into the elastic scattering of positronium atoms by hydrogen atoms, extending the work to include the spin-one (triplet) state of the two electrons. Once again we have extracted the SSS-wave phase shifts from normalizable trial functions by examining the spatial behavior of the center-of-mass wave functions. An effective-range expansion linear in the positronium energy fits the results very well and leads to a scattering length ${a}^{(\ensuremath{-})}\ensuremath{\simeq}2.36{a}_{0}$, compared with the static result ${a}^{(\ensuremath{-})}=2.476{a}_{0}$ of Hara and Fraser. The SSS-wave total and ortho-para conversion cross sections are calculated by making use of both 1S^{1}S1S and 3S^{3}S3S results. In the 3S^{3}S3S case, we see no indication of any resonances below the excitation threshold at 5.1 eV.