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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 ]
... as five-, eight- or twelvefold axes. For a review, see Janssen & Janner (1987). We denote a basis for the lattice ... previous section. Conceptually, it corresponds (de Wolff, 1974, 1977; Janner & Janssen, 1977) to enlarging the class of symmetry transformations admitted, from ... be obtained directly by considering as additional coordinate the phase t of the modulation. The positions in three dimensions are ...

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 ]
Incommensurate and commensurate modulated structures This chapter discusses incommensurate and commensurate modulated structures. After an introduction to the concepts used to describe modulated structures, including the basic ideas of higher-dimensional crystallography, four-dimensional space groups and displacive and occupational modulation, superspace-group determination is outlined. Tables of Bravais lattices, geometric ...

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 ]
... to the a, b and c axes, and internal coordinate t, the generators of the (3 + 1)-dimensional superspace group Pcmn ... group are, respectively: for m, n, r, s integers and t real. These conditions are never satisfied simultaneously. It depends on ... 1 it also depends on the value of the `phase' t, or more precisely on the product [tau] = 4st. The ...
     [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 ]
... j being of species A when the internal position is t. In particular, for a given atomic species, without occupational modulation ...
     [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 ]
... edited by Th. Hahn, fifth ed. Heidelberg: Springer. Yamamoto, A., Janssen, T., Janner, A. & de Wolff, P. M. (1985). A note ...
     [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 and satellites. The main reflections span a reciprocal lattice generated by , , . Considerations are here restricted for simplicity to the one-dimensional modulated case ...

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 ]
... satellites (m = 1). One may introduce an additional coordinate t and generalize (9.8.1.27) to which has (3 + 1)-dimensional space ...
     [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 ]
Incommensurate and commensurate modulated structures This chapter discusses incommensurate and commensurate modulated structures. After an introduction to the concepts used to describe modulated structures, including the basic ideas of higher-dimensional crystallography, four-dimensional space groups and displacive and occupational modulation, superspace-group determination is outlined. Tables of Bravais lattices, geometric ...

Representations of crystallographic groups
Janssen, T.  International Tables for Crystallography (2013). Vol. D, ch. 1.2, pp. 34-71 [ doi:10.1107/97809553602060000901 ]
... i.e. a discrete subgroup of the group of all translations T(n) in n dimensions, generated by n linearly independent translations. ... more involved. A derivation can, for example, be found in Janssen (1973). The group O(3) is again the union ... the same basis. Then there is a nonsingular linear operator T such that (). In this case, the representations and ...

Elasticity tensor for an icosahedral quasicrystal
Janssen, T.  International Tables for Crystallography (2013). Vol. D, Section 1.10.4.6.5, p. 259 [ doi:10.1107/97809553602060000909 ]
Elasticity tensor for an icosahedral quasicrystal 1.10.4.6.5. Elasticity tensor for an icosahedral quasicrystal The point group of an icosahedral quasicrystal is 532(5232) with generatorsin physical space, with , , and in internal space (c.f. Table 1.10.5.2). The phonon and phason strain tensors form a 6D, respectively 9D, vector space. The generators ...
     [more results from section 1.10.4 in volume D]

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