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A general introduction to groups
International Tables for Crystallography (2016). Vol. A, ch. 1.1, pp. 2-11 [ doi:10.1107/97809553602060000919 ]
... two operations (called their composition) is again a symmetry operation. (b) Every symmetry operation can be reversed by simply moving every ... that a mapping from a set A to a set B associates to each an element , denoted by and called the ... i.e. is equivalent to itself: this is easily seen since ; (b) it is symmetric, i.e. if is equivalent to , then ...
Synoptic table of the plane groups
International Tables for Crystallography (2016). Vol. A, Section 1.5.4.2, p. 95 [ doi:10.1107/97809553602060000923 ]
Synoptic table of the plane groups 1.5.4.2. Synoptic table of the plane groups The possible plane-group symbols are listed in Table 1.5.4.3. Two cases of multiple cells are included in addition to the standard cells, namely the c centring in the square system and the h centring in the hexagonal ...
[more results from section 1.5.4 in volume A]
Transformations of coordinate systems
International Tables for Crystallography (2016). Vol. A, ch. 1.5, pp. 75-106 [ doi:10.1107/97809553602060000923 ]
... change of the basis1 H. Wondratschek a andM. I. Aroyo b 1.5.1.1. Origin shift | | Let a coordinate system be given with ... cf. the space-group tables for in Chapter 2.3 ). (b) Krstanovic (1958), ICSD No. 45520, space group , No. 141, cell ... and 2.1.3.15) Cell choice 2 cell choice 3: Unique axis b invariant Cell choice 3 cell choice 1: (Fig. 1.5.1.2a) ...
Projections
International Tables for Crystallography (2016). Vol. A, Section 1.4.5.3, pp. 71-73 [ doi:10.1107/97809553602060000922 ]
... of type c1m1 (plane group No. 5). References Vainshtein, B. K. (1994). Fundamentals of Crystals. Symmetry, and Methods of ...
[more results from section 1.4.5 in volume A]
Eigensymmetry groups and non-characteristic orbits
International Tables for Crystallography (2016). Vol. A, Section 1.4.4.4, pp. 66-67 [ doi:10.1107/97809553602060000922 ]
... orthorhombic family, but the unit cell has equal parameters a = b, then the eigensymmetry group of an orbit can belong to ...
[more results from section 1.4.4 in volume A]
Space-group diagrams
International Tables for Crystallography (2016). Vol. A, Section 1.4.2.5, pp. 56-59 [ doi:10.1107/97809553602060000922 ]
... In Fig. 1.4.2.6 (left) the symmetry elements of (unique axis b, cell choice 1) are represented graphically in a projection of the unit cell along the monoclinic axis b. The directions of the basis vectors c and a can ... symbols with tails represent the twofold screw axes parallel to b. The c-glide plane at height along b is ...
[more results from section 1.4.2 in volume A]
Space groups and their descriptions
International Tables for Crystallography (2016). Vol. A, ch. 1.4, pp. 42-74 [ doi:10.1107/97809553602060000922 ]
... their symmetry properties. 1.4.1. Symbols of space groups H. Wondratschek b 1.4.1.1. Introduction | | Space groups describe the symmetries of crystal patterns ... implemented in crystallographic software, such as Hall symbols (Hall, 1981a,b) or explicit symbols (Shmueli, 1984), the reader is referred to Chapter 1.4 of International Tables for Crystallography, Volume B (2008). For the definition of space groups and ...
Crystal families
International Tables for Crystallography (2016). Vol. A, Section 1.3.4.4.4, pp. 40-41 [ doi:10.1107/97809553602060000921 ]
Crystal families 1.3.4.4.4. Crystal families The classification into crystal systems has many important applications, but it has the disadvantage that it is not compatible with the classification into lattice systems. Space groups that belong to the hexagonal lattice system are distributed over the trigonal and the hexagonal crystal system. Conversely, space ...
[more results from section 1.3.4 in volume A]
Symmorphic and non-symmorphic space groups
International Tables for Crystallography (2016). Vol. A, Section 1.3.3.3, p. 31 [ doi:10.1107/97809553602060000921 ]
... corresponding rotation symbol N and every glide reflection symbol a, b, c, d, e, n by the symbol m for a ...
[more results from section 1.3.3 in volume A]
Reciprocal lattice
International Tables for Crystallography (2016). Vol. A, Section 1.3.2.5, pp. 27-28 [ doi:10.1107/97809553602060000921 ]
... reciprocal basis is given by Inserting the columns for a, b, c, one obtains For the direct computation, the matrix with ...
[more results from section 1.3.2 in volume A]
International Tables for Crystallography (2016). Vol. A, ch. 1.1, pp. 2-11 [ doi:10.1107/97809553602060000919 ]
... two operations (called their composition) is again a symmetry operation. (b) Every symmetry operation can be reversed by simply moving every ... that a mapping from a set A to a set B associates to each an element , denoted by and called the ... i.e. is equivalent to itself: this is easily seen since ; (b) it is symmetric, i.e. if is equivalent to , then ...
Synoptic table of the plane groups
International Tables for Crystallography (2016). Vol. A, Section 1.5.4.2, p. 95 [ doi:10.1107/97809553602060000923 ]
Synoptic table of the plane groups 1.5.4.2. Synoptic table of the plane groups The possible plane-group symbols are listed in Table 1.5.4.3. Two cases of multiple cells are included in addition to the standard cells, namely the c centring in the square system and the h centring in the hexagonal ...
[more results from section 1.5.4 in volume A]
Transformations of coordinate systems
International Tables for Crystallography (2016). Vol. A, ch. 1.5, pp. 75-106 [ doi:10.1107/97809553602060000923 ]
... change of the basis1 H. Wondratschek a andM. I. Aroyo b 1.5.1.1. Origin shift | | Let a coordinate system be given with ... cf. the space-group tables for in Chapter 2.3 ). (b) Krstanovic (1958), ICSD No. 45520, space group , No. 141, cell ... and 2.1.3.15) Cell choice 2 cell choice 3: Unique axis b invariant Cell choice 3 cell choice 1: (Fig. 1.5.1.2a) ...
Projections
International Tables for Crystallography (2016). Vol. A, Section 1.4.5.3, pp. 71-73 [ doi:10.1107/97809553602060000922 ]
... of type c1m1 (plane group No. 5). References Vainshtein, B. K. (1994). Fundamentals of Crystals. Symmetry, and Methods of ...
[more results from section 1.4.5 in volume A]
Eigensymmetry groups and non-characteristic orbits
International Tables for Crystallography (2016). Vol. A, Section 1.4.4.4, pp. 66-67 [ doi:10.1107/97809553602060000922 ]
... orthorhombic family, but the unit cell has equal parameters a = b, then the eigensymmetry group of an orbit can belong to ...
[more results from section 1.4.4 in volume A]
Space-group diagrams
International Tables for Crystallography (2016). Vol. A, Section 1.4.2.5, pp. 56-59 [ doi:10.1107/97809553602060000922 ]
... In Fig. 1.4.2.6 (left) the symmetry elements of (unique axis b, cell choice 1) are represented graphically in a projection of the unit cell along the monoclinic axis b. The directions of the basis vectors c and a can ... symbols with tails represent the twofold screw axes parallel to b. The c-glide plane at height along b is ...
[more results from section 1.4.2 in volume A]
Space groups and their descriptions
International Tables for Crystallography (2016). Vol. A, ch. 1.4, pp. 42-74 [ doi:10.1107/97809553602060000922 ]
... their symmetry properties. 1.4.1. Symbols of space groups H. Wondratschek b 1.4.1.1. Introduction | | Space groups describe the symmetries of crystal patterns ... implemented in crystallographic software, such as Hall symbols (Hall, 1981a,b) or explicit symbols (Shmueli, 1984), the reader is referred to Chapter 1.4 of International Tables for Crystallography, Volume B (2008). For the definition of space groups and ...
Crystal families
International Tables for Crystallography (2016). Vol. A, Section 1.3.4.4.4, pp. 40-41 [ doi:10.1107/97809553602060000921 ]
Crystal families 1.3.4.4.4. Crystal families The classification into crystal systems has many important applications, but it has the disadvantage that it is not compatible with the classification into lattice systems. Space groups that belong to the hexagonal lattice system are distributed over the trigonal and the hexagonal crystal system. Conversely, space ...
[more results from section 1.3.4 in volume A]
Symmorphic and non-symmorphic space groups
International Tables for Crystallography (2016). Vol. A, Section 1.3.3.3, p. 31 [ doi:10.1107/97809553602060000921 ]
... corresponding rotation symbol N and every glide reflection symbol a, b, c, d, e, n by the symbol m for a ...
[more results from section 1.3.3 in volume A]
Reciprocal lattice
International Tables for Crystallography (2016). Vol. A, Section 1.3.2.5, pp. 27-28 [ doi:10.1107/97809553602060000921 ]
... reciprocal basis is given by Inserting the columns for a, b, c, one obtains For the direct computation, the matrix with ...
[more results from section 1.3.2 in volume A]
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