By M. Schlüter (auth.), T. J. Wieting, M. Schlüter (eds.)

This quantity is dedicated to the electron and phonon strength states of inorganic layered crystals. The virtue of those low-dimensional fabrics is their effortless mechanical cleavage alongside planes parallel to the layers. this selection signifies that the chemical binding inside of every one layer is way more suitable than the binding among layers and that a few, yet no longer inevitably all, actual houses of layered crystals have two-dimensional personality. In Wyckoff's Crystal constructions, SiC and similar com­ kilos are considered as layered constructions, simply because their atomic layers are alternately stacked in keeping with the necessities of cubic and hexagonal close-packing. How­ ever, the uniform (tetrahedral) coordination of the atoms in those compounds excludes the type of structural anisotropy that's basic to the fabrics dis­ stubborn during this quantity. somebody layer of a layered crystal might be composed of both a unmarried sheet of atoms, as in graphite, or a suite of as much as 5 atomic sheets, as in Bi2 Te3' A layer can also have extra complex preparations of the atoms, as we discover for instance in Sb S . however the targeted characteristic universal to these kinds of fabrics is two three the structural anisotropy, which at once impacts their digital and vibrational homes. the character of the vulnerable interlayer coupling is not good understood, regardless of the widespread attribution of the coupling within the literature to van der Waals forces. major proof, although, have emerged from all studies.

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Thus each factor group contains some representations appropriate for k and some representations appropriate for k' = 2nk. These 'pairs' of symmetry points, lines or planes are the following: (A, (H, K), (L, M), (S, T), (R, 2') and (G, N). Let us now explicitly construct the enlarged factor group Gk/Tk for the line S which contains representations appropriate for both lines Sand T. The four symmetry elements for S (see Table 7) are {B {C2, {lTh 't}, {lTv 't}. From these elements we construct the following eight 'complexes' which span the factor group Gk/Tk for S and which can be decomposed into five classes: I n, IO}, I C1 C2 C3 C4 Cs IO}, I I t eyen} {B I todd} {C 2 , I t eyen}, {C 2 , I todd} {lTh I 't + t even}, {lTh I 't + todd} {lTv I 't + t even}, {lTv I 't + tOdd}' {B (A2S) The abstract group of eight elements which form five classes is either D4 , C 4v or D2d which are all isomorphic with each other.

The table corresponds to Table lOc. Time reversal symmetry causes L3 to be degenerate with L 4 • L E T o o 2 2 T o o 2i -2i o o o o o o T Time inversion o o b b TABLE 131 Character table for the extra representations of the double factor group Gk/Tk for the line R. The table corresponds to Table lOd. Time reversal symmetry causes Rs to be degenerate with itself. R E Time inversion T 2 o o o a 36 CHAPTER A TABLE 13m Character table for the extra representations of the double factor group Gk/Tk for the plane G.

We note that, instead of diagonalizing a matrix of order 233 using simple plane waves, we have now to diagonalize twelve different submatrices with dimensions ranging from four to nineteen, which results in a considerable reduction of computing time. TABLE 17 Number of symmetrized plane waves obtained from the first 24 stars or 233 G-vectors of D~h at (ideal cia ratio). g. listed in Slater's book [14]. The final symmetry coefficients multiplying the simple plane waves are given in the Tables 18a to 18f.

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