Journal of Physics B Atomic and Molecular Physics

Simplified model for the scattering of zero energy electrons by helium atoms

S. K. Houston

Published Jan 1, 1973• 3 citations

Abstract

A local effective potential, H ex has been used to account for electron exchange, together with a model hamiltonian for the helium atom in Kohn's variational approximation to the electron helium scattering length, a. H ex was chosen to give agreement with the Born-exchange and static-exchange approximations. Correlation was introduced in the 84 term that trial function by using interelectronic coordinates. The method provides an upper bound to the 'model' scattering length and the value of 1.16a 0 was obtained in good agreement with more elaborate calculations and with experiment.

DOI: 10.1088/0022-3700/6/1/014

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Journal of Physics B Atomic and Molecular Physics

Simplified model for the scattering of zero energy electrons by helium atoms

S. K. Houston

Published Jan 1, 1973• 3 citations

Publisher

Abstract

DOI: 10.1088/0022-3700/6/1/014

Related papers

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An electron in helium vapor at 4\ifmmode^\circ\else\textdegree\fi{}K is characterized by its

s

-wave scattering length off helium atoms. This scattering length is small compared with the average interparticle spacing in the helium vapor. Consequently, we have taken as a model a particle interacting via hard-core repulsion with an ideal gas. This establishes a connection with the hard-sphere Bose problem of more general interest. The model described above is simpler (a) because it has only Boltzmann statistics and (b) because the electron is very light compared with the helium atoms. For this model, we have calculated the interaction free energy of the electron assuming that it is in statistical equilibrium with the helium gas. In the

s

-wave approximation, it is shown that this interaction free energy is rigorously $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ due to single scattering, all higher order multiple-scattering effects being zero. Here, $\ensuremath{\rho}$ is the average density of helium atoms,

d

is the scattering length, and

m

, is the electron mass. Since the

p

-wave approximation contributes a term of order $\ensuremath{\rho}{a}^{3}$, it is evident that the term $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ is good to higher densities than might previously have been supposed. This provides partial justification for the "bubble" model of the electron mobility since the term $2\ensuremath{\pi}\ensuremath{\rho}a(\frac{{\ensuremath{\hbar}}^{2}}{m})$ is certainly good up to densities at which the free energy of the "bubble" configuration becomes smaller.

{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}

s

p

d

, and

f

waves are presented. The width of the

^{2}S

closed-channel resonance below the

n=2

threshold is found to be 0.015 eV, in comparison with the 0.015-0.020-eV estimates of Andrick and Ehrhardt from experiment and the 0.01-eV estimate of Simpson and Fano.