A Many-Body Theory for Nearly-Free Electrons and Its Application to Annihilaion of Positrons in a Periodic Field

A many-body theory for nearly-free electrons has been constructed by introducing “ Bloch-pair operators,” α k and β k , which are related to free-particle operators through the transformation {aligned} {pmatrix} αₖ\\ βₖ {pmatrix} = {pmatrix} 1/√2 & θₕ/√2\\ -θₕ/√2 & 1/√2 {pmatrix} {pmatrix} aₖ\\ aₖ₊ₕ {pmatrix} , {aligned} where h is 2π times a reciprocal lattice vector, and θ h is the sign of the relevant Fourier component, v h , of the periodic field. This theory has been then applied to the annihilation of positrons in the periodic field. The interaction between a positron and a Bloch electron, has been, for the first time, taken into account through a few simplest diagrams. The result shows that a fairly sharp break of the angular-correlation curve occurs a the hexagonal-zone-face position in copper just as expected from the experiments previously made by the present author and his collaborator.