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Results for DC.creator="A." AND DC.creator="Looijenga-Vos" in section 9.8.4 of volume C page 1 of 2 pages. |
Theoretical foundation
International Tables for Crystallography (2006). Vol. C, Section 9.8.4, pp. 937-945 [ doi:10.1107/97809553602060000624 ]
Theoretical foundation 9.8.4. Theoretical foundation 9.8.4.1. Lattices and metric | | A periodic crystal structure is defined in a three-dimensional Euclidean space V and is invariant with respect ... three fundamental ones : These translations are linearly independent and span a lattice [Lambda]. The dimension of [Lambda] is the ...
Equivalent positions and modulation relations
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.4.2, pp. 940-941 [ doi:10.1107/97809553602060000624 ]
... positions and modulation relations 9.8.4.4.2. Equivalent positions and modulation relations A (3 + d)-dimensional space group that leaves a function invariant maps points in (3 + d)-space to points ... the function has the same value. The atomic positions of a modulated crystal represent such a pattern, and the superspace ...
Symmetry elements
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.4.1, p. 940 [ doi:10.1107/97809553602060000624 ]
Symmetry elements 9.8.4.4.1. Symmetry elements The elements of a (3 + d)-dimensional superspace group are pairs of Euclidean transformations ... direct product of the corresponding Euclidean groups. The elements form a three-dimensional space group, but the same does not hold ... translation v in V [see (9.8.4.32)]. In other words, a basis of the lattice does not simply split into ...
Superspace groups
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.4, pp. 940-941 [ doi:10.1107/97809553602060000624 ]
... groups 9.8.4.4. Superspace groups 9.8.4.4.1. Symmetry elements | | The elements of a (3 + d)-dimensional superspace group are pairs of Euclidean transformations ... direct product of the corresponding Euclidean groups. The elements form a three-dimensional space group, but the same does not hold ... translation v in V [see (9.8.4.32)]. In other words, a basis of the lattice does not simply split into ...
Bravais classes
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.3.3, p. 940 [ doi:10.1107/97809553602060000624 ]
... are arithmetically equivalent. This means that each of them admits a lattice basis of standard form such that their holohedral point ...
Crystallographic systems
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.3.2, p. 940 [ doi:10.1107/97809553602060000624 ]
Crystallographic systems 9.8.4.3.2. Crystallographic systems Definition 5.A crystallographic system is a set of lattices having geometrically equivalent holohedral point groups. In this way, a given holohedral point group (and even each crystallographic point group ...
Holohedry
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.3.1, pp. 939-940 [ doi:10.1107/97809553602060000624 ]
... 9.8.4.3.1. Holohedry The Laue group of the diffraction pattern is a three-dimensional point group that leaves the positions (and the intensities)3 of the diffraction spots as a set invariant, thus the vector module M* also. As discussed ... vector module M* is the projection. Conversely, if one has a point group that leaves the (3 + d)-dimensional lattice ...
Systems and Bravais classes
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.3, pp. 939-940 [ doi:10.1107/97809553602060000624 ]
... 9.8.4.3.1. Holohedry | | The Laue group of the diffraction pattern is a three-dimensional point group that leaves the positions (and the intensities)3 of the diffraction spots as a set invariant, thus the vector module M* also. As discussed ... vector module M* is the projection. Conversely, if one has a point group that leaves the (3 + d)-dimensional lattice ...
Geometric and arithmetic crystal classes
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.2.2, p. 939 [ doi:10.1107/97809553602060000624 ]
... to the previous section, in the case of modulated structures a standard basis can be chosen (for M* and correspondingly for ... operation R that leaves the diffraction pattern invariant, there is a point-group transformation in the external space (the physical one, so that ) and a point-group transformation in the internal space, such that ...
Laue class
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.2.1, pp. 938-939 [ doi:10.1107/97809553602060000624 ]
... group in three dimensions that transforms every diffraction peak into a peak of the same intensity.2 Because all diffraction vectors are ... is given by The (3 + d) × (3 + d) matrices form a finite group of integral matrices for K equal to or to one of its subgroups. A well known theorem in algebra states that then there ...
International Tables for Crystallography (2006). Vol. C, Section 9.8.4, pp. 937-945 [ doi:10.1107/97809553602060000624 ]
Theoretical foundation 9.8.4. Theoretical foundation 9.8.4.1. Lattices and metric | | A periodic crystal structure is defined in a three-dimensional Euclidean space V and is invariant with respect ... three fundamental ones : These translations are linearly independent and span a lattice [Lambda]. The dimension of [Lambda] is the ...
Equivalent positions and modulation relations
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.4.2, pp. 940-941 [ doi:10.1107/97809553602060000624 ]
... positions and modulation relations 9.8.4.4.2. Equivalent positions and modulation relations A (3 + d)-dimensional space group that leaves a function invariant maps points in (3 + d)-space to points ... the function has the same value. The atomic positions of a modulated crystal represent such a pattern, and the superspace ...
Symmetry elements
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.4.1, p. 940 [ doi:10.1107/97809553602060000624 ]
Symmetry elements 9.8.4.4.1. Symmetry elements The elements of a (3 + d)-dimensional superspace group are pairs of Euclidean transformations ... direct product of the corresponding Euclidean groups. The elements form a three-dimensional space group, but the same does not hold ... translation v in V [see (9.8.4.32)]. In other words, a basis of the lattice does not simply split into ...
Superspace groups
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.4, pp. 940-941 [ doi:10.1107/97809553602060000624 ]
... groups 9.8.4.4. Superspace groups 9.8.4.4.1. Symmetry elements | | The elements of a (3 + d)-dimensional superspace group are pairs of Euclidean transformations ... direct product of the corresponding Euclidean groups. The elements form a three-dimensional space group, but the same does not hold ... translation v in V [see (9.8.4.32)]. In other words, a basis of the lattice does not simply split into ...
Bravais classes
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.3.3, p. 940 [ doi:10.1107/97809553602060000624 ]
... are arithmetically equivalent. This means that each of them admits a lattice basis of standard form such that their holohedral point ...
Crystallographic systems
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.3.2, p. 940 [ doi:10.1107/97809553602060000624 ]
Crystallographic systems 9.8.4.3.2. Crystallographic systems Definition 5.A crystallographic system is a set of lattices having geometrically equivalent holohedral point groups. In this way, a given holohedral point group (and even each crystallographic point group ...
Holohedry
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.3.1, pp. 939-940 [ doi:10.1107/97809553602060000624 ]
... 9.8.4.3.1. Holohedry The Laue group of the diffraction pattern is a three-dimensional point group that leaves the positions (and the intensities)3 of the diffraction spots as a set invariant, thus the vector module M* also. As discussed ... vector module M* is the projection. Conversely, if one has a point group that leaves the (3 + d)-dimensional lattice ...
Systems and Bravais classes
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.3, pp. 939-940 [ doi:10.1107/97809553602060000624 ]
... 9.8.4.3.1. Holohedry | | The Laue group of the diffraction pattern is a three-dimensional point group that leaves the positions (and the intensities)3 of the diffraction spots as a set invariant, thus the vector module M* also. As discussed ... vector module M* is the projection. Conversely, if one has a point group that leaves the (3 + d)-dimensional lattice ...
Geometric and arithmetic crystal classes
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.2.2, p. 939 [ doi:10.1107/97809553602060000624 ]
... to the previous section, in the case of modulated structures a standard basis can be chosen (for M* and correspondingly for ... operation R that leaves the diffraction pattern invariant, there is a point-group transformation in the external space (the physical one, so that ) and a point-group transformation in the internal space, such that ...
Laue class
International Tables for Crystallography (2006). Vol. C, Section 9.8.4.2.1, pp. 938-939 [ doi:10.1107/97809553602060000624 ]
... group in three dimensions that transforms every diffraction peak into a peak of the same intensity.2 Because all diffraction vectors are ... is given by The (3 + d) × (3 + d) matrices form a finite group of integral matrices for K equal to or to one of its subgroups. A well known theorem in algebra states that then there ...
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