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Incommensurate and commensurate modulated structures
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de  International Tables for Crystallography (2006). Vol. C, ch. 9.8, pp. 907-955 [ doi:10.1107/97809553602060000624 ]
... examples. 9.8.1. Introduction 9.8.1.1. Modulated crystal structures | | Lattice periodicity is a fundamental concept in crystallography. This property is widely considered as essential for the characterization of the concept of a crystal. In recent decades, however, more and more long-range ... physical properties. It is convenient to extend the concept of a crystal in such a way that it includes these ...

Incommensurate and commensurate modulated structures
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de  International Tables for Crystallography (2006). Vol. C, ch. 9.8, pp. 907-955 [ doi:10.1107/97809553602060000624 ]
... dimension of M* is m, its rank n. Basis of a vector module M* of rank n; if n = 4 and ... to modulated crystals), the basis of M* is chosen as a*, b*, c*, q, with a*, b*, c* a basis of the lattice of main ...

The incommensurate versus the commensurate case
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de  International Tables for Crystallography (2006). Vol. C, Section 9.8.5.2, pp. 942-943 [ doi:10.1107/97809553602060000624 ]
... use the description as developed for incommensurate crystal phases for a (commensurate) superstructure. In fact, very often the modulation wavevector also ... incommensurate) values, several rational values at various phase transitions of a given crystal or in different compounds of a given structural family. In these cases, there is three- ...
     [more results from section 9.8.5 in volume C]

Structure factor
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de  International Tables for Crystallography (2006). Vol. C, Section 9.8.4.4.3, p. 941 [ doi:10.1107/97809553602060000624 ]
Structure factor 9.8.4.4.3. Structure factor The scattering from a set of atoms at positions is described in the kinematic ... projection of on . This structure factor is expressed by a sum of the products of atomic scattering factors and phase ... For an incommensurate phase, the number of particles in such a unit cell is infinite: for a given atom in ...
     [more results from section 9.8.4 in volume C]

Ambiguities in the notation
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de  International Tables for Crystallography (2006). Vol. C, Section 9.8.3.6, pp. 936-937 [ doi:10.1107/97809553602060000624 ]
... the notation The invariant part of the translation part of a (3 + 1)-dimensional superspace-group element is uniquely determined by ... that for each element of the point group there is a translation for which the invariant part is unique up to lattice vectors. The reason is that, for a given element R of the point group and given ...
     [more results from section 9.8.3 in volume C]

Outline for a superspace-group determination
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de  International Tables for Crystallography (2006). Vol. C, Section 9.8.2, pp. 913-915 [ doi:10.1107/97809553602060000624 ]
Outline for a superspace-group determination 9.8.2. Outline for a superspace-group determination In the case of a modulated structure, the diffraction pattern consists of main reflections ...

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. 913 [ doi:10.1107/97809553602060000624 ]
... type of modulation, the occupation modulation, can be treated in a way similar to the displacive modulation. As an example consider ... the position r is occupied by an atom of type A with probability and by one of type B with probability ... and that p is periodic. The probability of finding an A atom at site is with . In this case, ...
     [more results from section 9.8.1 in volume C]

Incommensurate and commensurate modulated structures
Janssen, T., Janner, A., Looijenga-Vos, A. and Wolff, P. M. de  International Tables for Crystallography (2006). Vol. C, ch. 9.8, pp. 907-955 [ doi:10.1107/97809553602060000624 ]
... is illustrated with different examples. In the following, we give a short definition of the most important notions appearing in the ... to the case of modulated crystal phases. [i] Vector module. A set of all integral linear combinations of a finite number of vectors. The dimension of the vector ...

Guide to the use of the space-group tables
Hahn, Th., Looijenga-Vos, A., Aroyo, M. I., Flack, H. D., Momma, K. and Konstantinov, P.  International Tables for Crystallography (2016). Vol. A, ch. 2.1, pp. 142-174 [ doi:10.1107/97809553602060000926 ]
... the use of the space-group tables This chapter forms a guide to understanding and using the plane-group and space ... group tables in Chapters 2.2 and 2.3. It explains in a systematic fashion, with many examples and figures, all entries and ... closely follow those in the fifth (2002) edition of Volume A, hereafter IT A (2002). This chapter is a ...

Monoclinic space groups
Hahn, Th. and Looijenga-Vos, A.  International Tables for Crystallography (2016). Vol. A, Section 2.1.3.15, pp. 169-172 [ doi:10.1107/97809553602060000926 ]
... cases, they are given in six versions. The description of a monoclinic crystal structure in this volume, including its Hermann-Mauguin ... two shorter vectors span the `reduced mesh' (where mesh means a two-dimensional unit cell), here e and f; for this ... vectors in the monoclinic plane are possible, provided they span a primitive mesh. It turns out, however, that the space- ...
     [more results from section 2.1.3 in volume A]

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