Positron Annihilation Experiments and the Band Structure of V₃Si

Abstract

In the positron annihilation method¹ the angular correlation between the two γ-rays emitted as a result of a two-photon positron-electron annihilation is measured. The angular correlation is directly connected to the momentum distribution of the annihilating electron-positron pair and depends mainly on the momenta of the electrons in the solid to be studied, since the positron thermalizes prior to annihilation. The probability of the annihilation γ-rays carrying away momentum p is given, in the independent particle model, by

$${\rm{\rho }}\left( {\underline {\rm{p}} } \right) = {\sum\limits_{\underline {\rm{k}} {\rm{,}}\ell } {\left| {\int {{{\rm{d}}^3}\underline {\rm{r}} \,\exp \left( { - {\rm{i}}\underline {\rm{p}} \cdot \underline {\rm{r}} } \right)} {{\rm{\psi }}_ + }\left( {\underline {\rm{r}} } \right){{\rm{\psi }}_{\underline {\rm{K}} {\rm{,}}\ell }}\left( {\underline {\rm{r}} } \right)} \right|} ^2}$$

(1)

where Ψ₊ (r) is the ground state position wavefunction and Ψ_{k, ℓ} is the electron wavefunction with wave number k and band index ℓ; the summation is over all occupied states k,ℓ In the actual experiment, the angular distribution N(⦵) is obtained using long slits (fig. 1), so that only one component of the momentum is being measured

$${\rm{N}}\left( {{{\rm{p}}_{\rm{z}}}} \right) = \smallint \smallint {\rm{\rho }}\left( {{{\rm{p}}_{\rm{x}}}{{\rm{p}}_{\rm{y}}}{{\rm{p}}_{\rm{z}}}} \right){\rm{d}}{{\rm{p}}_{\rm{x}}}{\rm{d}}{{\rm{p}}_{\rm{y}}}\,;\,{{\rm{p}}_{\rm{z}}}{\rm{ = mc\theta }}$$

(2)

The experimental work was supported at Brandeis University by the National Science Foundation and the U. S. Army Research Office, Durham, N. C. The theoretical computations were performed at the Hebrew University.