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Introduction
International Tables for Crystallography (2006). Vol. C, Section 9.8.1, pp. 907-913 [ doi:10.1107/97809553602060000624 ]
Introduction 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 ...
Four-dimensional space groups
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4.5, pp. 912-913 [ doi:10.1107/97809553602060000624 ]
... on the basis of algorithms developed by Zassenhaus (1948), Janssen, Janner & Ascher (1969a,b), Brown (1969), and Fast & Janssen (1971). ... by Brown, Bülow, Neubüser, Wondratschek & Zassenhaus, quoted above, a mathematical characterization of the basic crystallographic concepts is given together ... one, two, three, and four. One finds there, in particular, a full list of four-dimensional space groups. The list ...
Generalized nomenclature
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4.4, p. 912 [ doi:10.1107/97809553602060000624 ]
Generalized nomenclature 9.8.1.4.4. Generalized nomenclature In Section 9.8.4, the theory is extended to structures containing d modulations, with . In this case, each point-group transformation in internal space is given by and the associated internal translation by the (d-dimensional) vector . Thus, The transformations R and are represented by ...
Four-dimensional crystallography
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4.3, pp. 911-912 [ doi:10.1107/97809553602060000624 ]
... obtained in the previous paragraph. The matrices [Gamma](R) form a faithful integral representation of the three-dimensional point group K with a four-dimensional carrier space . It is a reducible representation having as invariant subspaces the physical three- ...
Description in four dimensions
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4.2, p. 911 [ doi:10.1107/97809553602060000624 ]
... 9.8.1.4.2. Description in four dimensions The matrices [Gamma]*(R) form a faithful integral representation of the three-dimensional point group K. ... possible to consider them as four-dimensional orthogonal transformations leaving a lattice with basis vectors (9.8.1.14) invariant. Indeed, one can consider ... 9.8.4, these vectors span the four-dimensional reciprocal lattice for a periodic structure having as three-dimensional intersection (say defined ...
Bravais classes of vector modules
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4.1, pp. 910-911 [ doi:10.1107/97809553602060000624 ]
... of vector modules 9.8.1.4.1. Bravais classes of vector modules For a modulated crystal structure with a one-dimensional modulation, the positions of the diffraction spots are given by vectors This set of vectors is a vector module M*. The vectors form a basis ...
Basic symmetry considerations
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4, pp. 910-913 [ doi:10.1107/97809553602060000624 ]
... Basic symmetry considerations 9.8.1.4.1. Bravais classes of vector modules | | For a modulated crystal structure with a one-dimensional modulation, the positions of the diffraction spots are given by vectors This set of vectors is a vector module M*. The vectors form a basis ...
The symmetry
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.3.2, pp. 909-910 [ doi:10.1107/97809553602060000624 ]
... the measuring process limits the precision in the determination of a modulation wavevector. Accordingly, one can try an approximation of the modulation wavevector q by a commensurate one: an irrational number can be approximated arbitrarily well by a rational one. There are two main disadvantages in this ...
The diffraction pattern
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.3.1, p. 909 [ doi:10.1107/97809553602060000624 ]
... diffraction pattern To introduce what follows, the simple case of a displacively modulated crystal structure is considered. The point-atom approximation is adopted and the modulation is supposed to be a sinusoidal plane wave. This means that the structure can be described in terms of atomic positions of a basic structure with three-dimensional space-group symmetry, periodically ...
The simple case of a displacively modulated crystal
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.3, pp. 909-910 [ doi:10.1107/97809553602060000624 ]
The simple case of a displacively modulated crystal 9.8.1.3. The simple case of a displacively modulated crystal 9.8.1.3.1. The diffraction pattern | | To introduce what follows, the simple case of a displacively modulated crystal structure is considered. The point-atom ...
International Tables for Crystallography (2006). Vol. C, Section 9.8.1, pp. 907-913 [ doi:10.1107/97809553602060000624 ]
Introduction 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 ...
Four-dimensional space groups
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4.5, pp. 912-913 [ doi:10.1107/97809553602060000624 ]
... on the basis of algorithms developed by Zassenhaus (1948), Janssen, Janner & Ascher (1969a,b), Brown (1969), and Fast & Janssen (1971). ... by Brown, Bülow, Neubüser, Wondratschek & Zassenhaus, quoted above, a mathematical characterization of the basic crystallographic concepts is given together ... one, two, three, and four. One finds there, in particular, a full list of four-dimensional space groups. The list ...
Generalized nomenclature
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4.4, p. 912 [ doi:10.1107/97809553602060000624 ]
Generalized nomenclature 9.8.1.4.4. Generalized nomenclature In Section 9.8.4, the theory is extended to structures containing d modulations, with . In this case, each point-group transformation in internal space is given by and the associated internal translation by the (d-dimensional) vector . Thus, The transformations R and are represented by ...
Four-dimensional crystallography
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4.3, pp. 911-912 [ doi:10.1107/97809553602060000624 ]
... obtained in the previous paragraph. The matrices [Gamma](R) form a faithful integral representation of the three-dimensional point group K with a four-dimensional carrier space . It is a reducible representation having as invariant subspaces the physical three- ...
Description in four dimensions
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4.2, p. 911 [ doi:10.1107/97809553602060000624 ]
... 9.8.1.4.2. Description in four dimensions The matrices [Gamma]*(R) form a faithful integral representation of the three-dimensional point group K. ... possible to consider them as four-dimensional orthogonal transformations leaving a lattice with basis vectors (9.8.1.14) invariant. Indeed, one can consider ... 9.8.4, these vectors span the four-dimensional reciprocal lattice for a periodic structure having as three-dimensional intersection (say defined ...
Bravais classes of vector modules
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4.1, pp. 910-911 [ doi:10.1107/97809553602060000624 ]
... of vector modules 9.8.1.4.1. Bravais classes of vector modules For a modulated crystal structure with a one-dimensional modulation, the positions of the diffraction spots are given by vectors This set of vectors is a vector module M*. The vectors form a basis ...
Basic symmetry considerations
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.4, pp. 910-913 [ doi:10.1107/97809553602060000624 ]
... Basic symmetry considerations 9.8.1.4.1. Bravais classes of vector modules | | For a modulated crystal structure with a one-dimensional modulation, the positions of the diffraction spots are given by vectors This set of vectors is a vector module M*. The vectors form a basis ...
The symmetry
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.3.2, pp. 909-910 [ doi:10.1107/97809553602060000624 ]
... the measuring process limits the precision in the determination of a modulation wavevector. Accordingly, one can try an approximation of the modulation wavevector q by a commensurate one: an irrational number can be approximated arbitrarily well by a rational one. There are two main disadvantages in this ...
The diffraction pattern
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.3.1, p. 909 [ doi:10.1107/97809553602060000624 ]
... diffraction pattern To introduce what follows, the simple case of a displacively modulated crystal structure is considered. The point-atom approximation is adopted and the modulation is supposed to be a sinusoidal plane wave. This means that the structure can be described in terms of atomic positions of a basic structure with three-dimensional space-group symmetry, periodically ...
The simple case of a displacively modulated crystal
International Tables for Crystallography (2006). Vol. C, Section 9.8.1.3, pp. 909-910 [ doi:10.1107/97809553602060000624 ]
The simple case of a displacively modulated crystal 9.8.1.3. The simple case of a displacively modulated crystal 9.8.1.3.1. The diffraction pattern | | To introduce what follows, the simple case of a displacively modulated crystal structure is considered. The point-atom ...
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