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Irreducible representations of three-dimensional space groups
International Tables for Crystallography (2013). Vol. D, Section 1.10.3.3, pp. 252-253 [ doi:10.1107/97809553602060000909 ]
Irreducible representations of three-dimensional space groups 1.10.3.3. Irreducible representations of three-dimensional space groups A third way to describe the symmetry of a quasiperiodic function is by means of irreducible representations of a space group. For the theory of these representations we refer to Chapter 1.2 on representations of crystallographic ...
[more results from section 1.10.3 in volume D]
Superspace groups
International Tables for Crystallography (2013). Vol. D, Section 1.10.2.3, p. 250 [ doi:10.1107/97809553602060000909 ]
... . Superspace groups for quasicrystals of rank are given in Janssen (1988). Table 1.10.2.1| | Allowable three-dimensional point groups for ... 12 16 20 24 8 16 24 32 40 48 T 12 23 O 24 432, I 60 532 24 48 ... higher-dimensional symmetry groups is discussed in two IUCr reports (Janssen et al., 1999, 2002). References International Tables for ...
[more results from section 1.10.2 in volume D]
Embedding in superspace
International Tables for Crystallography (2013). Vol. D, Section 1.10.1.3, pp. 247-248 [ doi:10.1107/97809553602060000909 ]
... is periodic, the array is left invariant if one replaces t by , and for every lattice vector of the basic structure the array is left invariant if one replaces simultaneously t by . This means that the array is left invariant ...
[more results from section 1.10.1 in volume D]
Tensors in quasiperiodic structures
International Tables for Crystallography (2013). Vol. D, ch. 1.10, pp. 246-270 [ doi:10.1107/97809553602060000909 ]
... also called incommensurate systems if they are not lattice periodic (Janssen & Janner, 1987).] It is not a strict classification, because ... is periodic, the array is left invariant if one replaces t by , and for every lattice vector of the basic structure the array is left invariant if one replaces simultaneously t by . This means that the array is left ...
Glossary
International Tables for Crystallography (2013). Vol. D, Section 1.2.8, pp. 70-71 [ doi:10.1107/97809553602060000901 ]
... dimension of irreducible representation order of class class multiplication constants T tetrahedral group O octahedral group I icosahedral group projective representation ...
Invariant tensors
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4.6, pp. 68-70 [ doi:10.1107/97809553602060000901 ]
... This can be written with the tensor as This tensor T is invariant under the group. Example (3). Dimension 3 ...
[more results from section 1.2.7 in volume D]
Tables
International Tables for Crystallography (2013). Vol. D, Section 1.2.6, pp. 56-62 [ doi:10.1107/97809553602060000901 ]
... of order n, for the dihedral group of order 2n, T, O and I the tetrahedral, octahedral and icosahedral groups, respectively ... n odd) (n odd) (n even) (n even, ) () (n odd, ) T 23 O 432 I 532 Table 1.2.6.2. Among the infinite ... 6 6 32 6 8 422 , 8 8 622 , 12 T 23 12 12 16 O 432 24 24 24 ...
Co-representations
International Tables for Crystallography (2013). Vol. D, Section 1.2.5.5, pp. 55-56 [ doi:10.1107/97809553602060000901 ]
Co-representations 1.2.5.5. Co-representations Suppose the magnetic point group G has an orthochronous subgroup H and an antichronous coset for some antichronous element a. The elements of H are represented by unitary operators, those of by anti-unitary operators. These operators correspond to matrices in the following way. Suppose are ...
[more results from section 1.2.5 in volume D]
Clebsch-Gordan coefficients
International Tables for Crystallography (2013). Vol. D, Section 1.2.4.3, pp. 52-53 [ doi:10.1107/97809553602060000901 ]
Clebsch-Gordan coefficients 1.2.4.3. Clebsch-Gordan coefficients The tensor product of two irreducible representations of a group K is, in general, reducible. If is a basis for the irreducible representation () and one for ( ), a basis for the tensor product space is given byOn this basis, the matrix representation is, in general ...
[more results from section 1.2.4 in volume D]
Double space groups and their representations
International Tables for Crystallography (2013). Vol. D, Section 1.2.3.5, pp. 50-51 [ doi:10.1107/97809553602060000901 ]
Double space groups and their representations 1.2.3.5. Double space groups and their representations In Section 1.2.2.9, it was mentioned that the transformation properties of spin- particles under rotations are not given by the orthogonal group O(3), but by the covering group SU(2). Hence, the transformation of a spinor ...
[more results from section 1.2.3 in volume D]
International Tables for Crystallography (2013). Vol. D, Section 1.10.3.3, pp. 252-253 [ doi:10.1107/97809553602060000909 ]
Irreducible representations of three-dimensional space groups 1.10.3.3. Irreducible representations of three-dimensional space groups A third way to describe the symmetry of a quasiperiodic function is by means of irreducible representations of a space group. For the theory of these representations we refer to Chapter 1.2 on representations of crystallographic ...
[more results from section 1.10.3 in volume D]
Superspace groups
International Tables for Crystallography (2013). Vol. D, Section 1.10.2.3, p. 250 [ doi:10.1107/97809553602060000909 ]
... . Superspace groups for quasicrystals of rank are given in Janssen (1988). Table 1.10.2.1| | Allowable three-dimensional point groups for ... 12 16 20 24 8 16 24 32 40 48 T 12 23 O 24 432, I 60 532 24 48 ... higher-dimensional symmetry groups is discussed in two IUCr reports (Janssen et al., 1999, 2002). References International Tables for ...
[more results from section 1.10.2 in volume D]
Embedding in superspace
International Tables for Crystallography (2013). Vol. D, Section 1.10.1.3, pp. 247-248 [ doi:10.1107/97809553602060000909 ]
... is periodic, the array is left invariant if one replaces t by , and for every lattice vector of the basic structure the array is left invariant if one replaces simultaneously t by . This means that the array is left invariant ...
[more results from section 1.10.1 in volume D]
Tensors in quasiperiodic structures
International Tables for Crystallography (2013). Vol. D, ch. 1.10, pp. 246-270 [ doi:10.1107/97809553602060000909 ]
... also called incommensurate systems if they are not lattice periodic (Janssen & Janner, 1987).] It is not a strict classification, because ... is periodic, the array is left invariant if one replaces t by , and for every lattice vector of the basic structure the array is left invariant if one replaces simultaneously t by . This means that the array is left ...
Glossary
International Tables for Crystallography (2013). Vol. D, Section 1.2.8, pp. 70-71 [ doi:10.1107/97809553602060000901 ]
... dimension of irreducible representation order of class class multiplication constants T tetrahedral group O octahedral group I icosahedral group projective representation ...
Invariant tensors
International Tables for Crystallography (2013). Vol. D, Section 1.2.7.4.6, pp. 68-70 [ doi:10.1107/97809553602060000901 ]
... This can be written with the tensor as This tensor T is invariant under the group. Example (3). Dimension 3 ...
[more results from section 1.2.7 in volume D]
Tables
International Tables for Crystallography (2013). Vol. D, Section 1.2.6, pp. 56-62 [ doi:10.1107/97809553602060000901 ]
... of order n, for the dihedral group of order 2n, T, O and I the tetrahedral, octahedral and icosahedral groups, respectively ... n odd) (n odd) (n even) (n even, ) () (n odd, ) T 23 O 432 I 532 Table 1.2.6.2. Among the infinite ... 6 6 32 6 8 422 , 8 8 622 , 12 T 23 12 12 16 O 432 24 24 24 ...
Co-representations
International Tables for Crystallography (2013). Vol. D, Section 1.2.5.5, pp. 55-56 [ doi:10.1107/97809553602060000901 ]
Co-representations 1.2.5.5. Co-representations Suppose the magnetic point group G has an orthochronous subgroup H and an antichronous coset for some antichronous element a. The elements of H are represented by unitary operators, those of by anti-unitary operators. These operators correspond to matrices in the following way. Suppose are ...
[more results from section 1.2.5 in volume D]
Clebsch-Gordan coefficients
International Tables for Crystallography (2013). Vol. D, Section 1.2.4.3, pp. 52-53 [ doi:10.1107/97809553602060000901 ]
Clebsch-Gordan coefficients 1.2.4.3. Clebsch-Gordan coefficients The tensor product of two irreducible representations of a group K is, in general, reducible. If is a basis for the irreducible representation () and one for ( ), a basis for the tensor product space is given byOn this basis, the matrix representation is, in general ...
[more results from section 1.2.4 in volume D]
Double space groups and their representations
International Tables for Crystallography (2013). Vol. D, Section 1.2.3.5, pp. 50-51 [ doi:10.1107/97809553602060000901 ]
Double space groups and their representations 1.2.3.5. Double space groups and their representations In Section 1.2.2.9, it was mentioned that the transformation properties of spin- particles under rotations are not given by the orthogonal group O(3), but by the covering group SU(2). Hence, the transformation of a spinor ...
[more results from section 1.2.3 in volume D]
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