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Example text

6(e)]. Because each translated string of atoms intersects physical space in the form of an atomic electron density centred on the position of the atom in the modulated structure, the string of atoms defines a string of points (the positions of the atoms) that is identical to the string obtained by the first construction. Comparison of the generalized electron density constructed in this way with a Fourier map in superspace (Fig. 3) shows the equivalence of the reciprocal space and direct space approaches.

All points with the same distance to physical space form a three-dimensional space that intersects the fourth coordinate axis in the point xs = (0, 0, 0, t). Physical space is recovered for t = 0. 3)] x¯si (j) = x¯i (j) = li + x0i (µ) x¯s4 (j) = x¯4 (j) = t + q· x ¯(j) . 24), the number j stands for atom µ in unit cell L of the basic structure. 5). 9) and Fig. 21) where x ¯s4 has been substituted for x ¯4 in the argument of the modulation functions. 21) for 0 t < 1. 22) where the property has been used that ai is the projection of asi onto physical space with xi = xsi (i = 1, 2, 3).

For example, optical activity and other non-linear properties are absent in crystals with inversion symmetry. Extensive experimental studies have been made towards establishing the presence of optical activity in incommensurately modulated crystals that have but hidden inversion symmetry (Kobayashi, 1990). , 1993).

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