Theory of electron diffraction from planar ideal crystals

Abstract

Contrary to the case of n = 72, for n = 54 the same locally extremal packing is obtained with tetrahedral {3, 3+}4,2 and octahedral {3, 4q-}3,1 surface lattices. It should be noted that this packing also represents a four-branched spherical helix structure (Sz6kely, 1974; Tarnai, 1985).The applied method did not result in Danzerian rigid packings for n = 114 and 282. The result for n = 114 is not of interest since the circle diameter for n = 114 is less than the circle diameter for n = 120. But, the arrangement of 282 circles is quite good, so it is worth improving it, by giving up the icosahedral symmetry.Terms of the packing sequences {3, q+}c÷~,c, {3, q+}c÷2,~ defined with removal of the vertices of the base polyhedra {3, q} present Danzerian rigid arrangements and quite large densities in all of the investigated cases. On the basis of the results obtained it is expected that Danzerian rigid packings will also be obtained in these sequences for values c > 3. A. (1962 AbstractA theory of electron diffraction from a planar ideal crystal of arbitrary thickness is presented. It is based on Schr6dinger's equation. Both the relativistic corrections in energy and wavelength and the electron 'absorption' due to the presence of inelastic scattering may be incorporated as usual. This theory is constructed in an exact differential-equation approach known as rigorous coupled-wave analysis. This is an exact method of diffraction analysis that has been extensively tested for its numerical calculation scheme. The exact solution for electron wave amplitudes of all diffraction orders is formally presented in terms of a standard eigenvalue problem and explicitly expressed in matrix form. Numerical calculation can be implemented on digital computers in a straightforward manner. An approximate conservation law is given for the transmittance and reflectance, which are then the relevant dynamical quantities to be measured in 0108-7673/87/050616-06501.50 a realistic time-dependent diffraction process and to be calculated in this time-independent diffraction theory for comparison. Two derivations of the well known Bragg law are sketched.