The atomic-string approximation in cross-grating high-energy electron diffraction. I. Dispersion surface and Bloch waves

Abstract

A new real-space method of calculating the dispersion surface and Bloch waves in cross-grating HEED is derived. Instead of using a large many-beam matrix to compute the dispersion surface, the equivalent two- dimensional band structure of allowed transverse energies of the Bloch waves is calculated from a small matrix obtained by approximating the KKR equations derived earlier [Ozorio de Almeida (1975). Acta Cryst. A31, 435-442]. For a close-packed array of atomic strings as in Au [111], transverse energy bands in excellent agreement with conventional 91 × 91 many-beam matrix calculations are obtained with only a 7 × 7 matrix. It is also shown how the calculation of the Bloch waves and their Fourier coefficients C(J)G may be further simplified by replacing the unit cell of the projected potential by its Wigner-Seitz circle. Numerical calculations, again for Au[ 111], show that the C(J)0 so obtained are still in excellent agreement with many-beam calculations but that the higher Fourier coefficients C(J)G, being more sensitive to the form of the Bloch waves in the interstitial region, are less accurate. The form of the Bloch waves is investigated and it is shown that near the zone-axis critical voltage, the nearly degenerate Bloch waves are almost entirely sp hybrids so that a 3 × 3 matrix may be used.