Quantum-Size Effect of Bloch Wave Functions of Ultra-High Energy Electrons in a Thin Single-Crystal Film

The reflection coefficient of ultra-high-energy electrons (~1 MeV) at their normal incidence on a thin single-crystal film is calculated. It is shown that even at such high particle energies, the quantum-size effect of the Bloch waves formed in the film is noticeably manifested. Narrow Bragg reflection peaks are found to appear at certain electron energies. A formula is given that determines their position and intensity on the reflection curve. A comparison of reflection coefficients at intermediate, high, and ultra-high particle energies is made.