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Progress of Theoretical Physics

Low Energy Electron-Hydrogen Scattering

Takashi Ohmura, Yasuo Hara, Takahiko Yamanouchi

Published Jul 1, 1958• 27 citations
Publisher

Abstract

The effective range theory is developed for electron-hydrogen scattering. The scattering length as and the effective range r0s of the singlet state are determined by making use of the accurate solution of the H− ion states, as as = 7.03, r0s = 3.37 (in atomic unit). The scattering length at and the effective range r0t of the triplet state are calculated from the zero energy solution by a variation-perturbation method. The results are at = 2.34, r0t = 1.29.

DOI
10.1143/ptp.20.82

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Journal of the Physical Society of Japan•Mar 15, 1986•16 citations
The effective range theory is developed for electron-hydrogen scattering. The scattering length as and the effective range r0s of the singlet state are determined by making use of the accurate solution of the H− ion states, as as = 7.03, r0s = 3.37 (in atomic unit). The scattering length at and the effective range r0t of the triplet state are calculated from the zero energy solution by a variation-perturbation method. The results are at = 2.34, r0t = 1.29.
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​.
The effective range in the singlet electron-hydrogen system has been evaluated as 2.646\ifmmode\pm\else\textpm\fi{}0.004 atomic units by using the asymptotic amplitude of the 202-parameter ${\mathrm{H}}^{\ensuremath{-}}$ wave function of Pekeris. This value of the effective range, together with the value of the electron affinity of ${\mathrm{H}}^{\ensuremath{-}}$, determines the scattering length in the singlet system as 6.167 atomic units. The effective range in the triplet system is calculated to be 1.219 atomic units by a Hartree-Fock approximation. It is shown that the effective-range approximation is very good for all energies at which only elastic scattering is allowed. The photo-ionization of ${\mathrm{H}}^{\ensuremath{-}}$ is briefly discussed on the basis of the effective-range theory.
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.
The non-relativistic Schrödinger equation of electron plus hydrogen-atom system is solved by the Hulthén type variational method at the limit of zero incident electron energy. 8 and 5 parameter trial functions are used for the singlet and triplet states respectively, and the following upper bounds on the scattering length a are obtained, as ≦6.217a0 (singlet), at ≦2.272a0 (triplet). Some discussions are given on the accuracy of the result. It is shown that the relation aOH > aSH > aK > a (true) holds for trial functions of the same form if a(true) is positive.
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.
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 .