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Physical Review

Upper Bounds on Electron-Atomic Hydrogen Scattering Lengths

Leonard Rosenberg, Larry Spruch, Thomas F. O'Malley

Published Jul 1, 1960• 54 citations
Publisher

Abstract

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, ${A}_{T}$ and ${A}_{S}$ respectively, satisfy the inequalities ${A}_{T}\ensuremath{\leqq}1.91{a}_{0}$ and ${A}_{S}\ensuremath{\leqq}6.23{a}_{0}$, where ${a}_{0}$ 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, ${r}_{0S}$, of about $2.7{a}_{0}$. The triplet trial functions which were obtained were not sufficiently accurate to be useful in a determination of the triplet effective range, ${r}_{0T}$.

DOI
10.1103/physrev.119.164

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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 AT\ensuremath≦1.91a0{A}_{T}\ensuremath{\leqq}1.91{a}_{0}AT​\ensuremath≦1.91a0​ and AS\ensuremath≦6.23a0{A}_{S}\ensuremath{\leqq}6.23{a}_{0}AS​\ensuremath≦6.23a0​, 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 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 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.
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.
The scattering length AAA characterizes the zero-energy scattering of one system by another. It was shown some time ago that a variational upper bound on AAA could be obtained using methods, of the Rayleigh-Ritz type, which are commonly employed to obtain upper bounds on energy eigenvalues. Here we formulate a method for obtaining a variational lower bound on AAA. Once again the essential idea is to express the scattering length as a variational estimate plus an error term and then to reduce the problem of bounding the error term to one involving bounds on energy eigenvalues. In particular, the variational lower bound on AAA is rigorously established provided a certain modified Hamiltonian can be shown to have no discrete states lying below the level of the continuum threshold. It is unfortunately true that necessary conditions for the existence of bound states are not available for multiparticle systems in general. However, in the case of positron-atom scattering the adiabatic approximation can be introduced as an (essentially) solvable comparison problem to rigorously establish the nonexistence of bound states of the modified Hamiltonian. It has recently been shown how the validity of the variational upper bound on AAA can be maintained when the target ground-state wave function is imprecisely known. Similar methods can be used to maintain the variational lower bound on AAA. Since the bound is variational, the error in the calculated scattering length will be of second order in the error in the wave function. The use of the adiabatic approximation in the present context places no limitation in principle on the accuracy achievable.
We present a convergent variational basis-set calculational scheme for elastic scattering of positronium atom by hydrogen atom in S wave. Highly correlated trial functions with appropriate symmetry are needed for achieving convergence. We report convergent results for scattering lengths in atomic units for both singlet (=3.49pm0.20=3.49\\pm 0.20=3.49pm0.20) and triplet (=2.46pm0.10=2.46\\pm 0.10=2.46pm0.10) states.