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

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Abstract

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 is made of reflection coefficients at medium, high and ultra-high particle energies.

About the authors

S. M. Shkornyakov

Shubnikov Institute of Crystallography of the FSRC “Crystallography and Photonics” of the RAS

Author for correspondence.
Email: shkornyakov@mail.ru
Russian Federation, Moscow

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2. Fig. 1. Schematic of the potential energy of an electron in a film: V0 - average internal energy; N - number of monolayers parallel to the surface in the film; c - period of one-dimensional lattice; t - amplitude of the passed wave; r - amplitude of the reflected wave

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3. Fig. 2. Plot of the dependence of the reflection coefficient R on the incident electron energy E in the range of about 1 MeV. Parameters: c = 5 Å, y = -2, V0 = 0, N = 300

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4. Fig. 3. Dependence of the reflection coefficient R on the incident electron energy E in the range of about 1 MeV in the enlarged scale on the ordinate axis

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5. Fig. 4. Dependence of the reflection coefficient R on the incident electron energy E in the range of about 1 MeV at even larger scaling up both on the x-axis and y-axis

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6. Fig. 5. R(E) curves for three ranges of incident electron energy: a - medium; b - high; c - ultrahigh

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