Some exact results on the reduced density matrix for neutralized atoms

A theoretical study of positive – ion neutralization scattered from surfaces in low kinetic energies has presented. The simulation of scattering process based on the reduced density matrix element for neutralized atom model (RDMEM). The approximation techniques used to finding an analytical formula for the reduced density matrix element (RDME). The wide band approximation (WBA) and narrow band approximation (NBA) are valid according to the width of the solid surface valence band. Residue theory and WBA applies to contract the integral representation of the RDME, which is resulted form using of the diagrammatic approach to the perturbation expansion. The current RDME compares with another RDME that obtained from different technique. The exact analytic formula and NBA for the RDME has presented when the normalization factor D(t)= in RDMEM . The calculations perform on the real system, H+ - CaO as a wide valence band and H+ - Alkali- Halide as a narrow valence band. Neutralization probability for H+ reaches to the unity in the low kinetic energies only if the atomic energy level for H+ has not embedded in the valence band of the solid.