Abstract: In this paper, we use Green's function method to study the first-order ladder graphs which include lattice-electron and lattice-positron interaction lines. We calculate some low order Feynman graphs in taking the static limit for electron-positron effective interaction potential, and obtain the formula of partial annihilation rate of the positron for p>p_F. Furthermore, we make some numerical calculations, and discuss and compare them with the results of Hede-Carbotte's theory.