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Occupation modulation
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de, International Tables for Crystallography (2006). Vol. C, Section 9.8.1.5, p. [ doi:10.1107/97809553602060000624 ]
is the sum of δ functions over the reciprocal lattice of the basic structure: Consequently, the diffraction peaks occur at positions H given by (9.8.1.7) . For a simple sinusoidal modulation [ m = ±1 in (9.8.1.29) ], there are only main ...

Basic symmetry considerations
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de, International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4, p. [ doi:10.1107/97809553602060000624 ]
*). An element R of K then transforms the basic vectors,,, q into ones of the form (9.8.1.15) . If one denotes, as in (9.8.1.2), q by, this implies with Γ*(R) a 4 × 4 matrix with integral entries. In the case ...

Four-dimensional space groups
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de, International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4.5, p. [ doi:10.1107/97809553602060000624 ]
The groups were determined on the basis of algorithms developed by Zassenhaus (1948), Janssen, Janner & Ascher (1969 a, b), Brown (1969), and Fast & Janssen (1971). In the book by Brown, Bülow, Neubüser, Wondratschek & Zassenhaus ...

The diffraction pattern
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de, International Tables for Crystallography (2006). Vol. C, Section 9.8.1.3.1, p. [ doi:10.1107/97809553602060000624 ]
in the structure factor, which is given by the expression where is the atomic scattering factor (which still, in general, depends on H). Using the Jacobi–Anger relation, one can rewrite (9.8.1.5) as where is the m th-order Bessel ...

Description in four dimensions
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de, International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4.2, p. [ doi:10.1107/97809553602060000624 ]
leaving a lattice with basis vectors (9.8.1.14) invariant. Indeed, one can consider the vectors (9.8.1.15) as projections of four-dimensional lattice vectors, which can be written as where [ cf. (9.8.1.14) ] m has now been ...

The symmetry
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de, International Tables for Crystallography (2006). Vol. C, Section 9.8.1.3.2, p. [ doi:10.1107/97809553602060000624 ]
transformations all elements ({ R | v }, {ɛ|Δ}) satisfying (9.8.1.13) . These form a space group in four dimensions. The reciprocal to the basis (9.8.1.11) is A general reciprocal-lattice vector is now The projection of this reciprocal ...

Introduction
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de, International Tables for Crystallography (2006). Vol. C, Section 9.8.1, p. [ doi:10.1107/97809553602060000624 ]
cases, cover essentially the one-dimensional modulated case [ d = 1 in equation (9.8.1.1) ]. 9.8.1.2. The basic ideas of higher-dimensional crystallography | top | pdf | Incommensurate modulated crystals are systems ...

The basic ideas of higher-dimensional crystallography
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de, International Tables for Crystallography (2006). Vol. C, Section 9.8.1.2, p. [ doi:10.1107/97809553602060000624 ]
as crystalline phases and generalize for that reason the concept of a crystal. The positions of the Bragg diffraction peaks given in (9.8.1.1) are a special case. In general, they are elements of a vector module M * and can be written as This leads ...

Generalized nomenclature
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de, International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4.4, p. [ doi:10.1107/97809553602060000624 ]
...

The simple case of a displacively modulated crystal
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de, International Tables for Crystallography (2006). Vol. C, Section 9.8.1.3, p. [ doi:10.1107/97809553602060000624 ]
there is an integer N such that N q belongs to the reciprocal lattice, one may restrict the values of m in (9.8.1.7) to the range from 0 to N − 1. 9.8.1.3.2. The symmetry | top | pdf | There is more than one way ...

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