Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) Structures
Symmetry operations of layers periodic in two dimensions restrict the geometry the lattice according to the five two-dimensional Bravais types of lattices. In order-disorder (OD) structures, the operations relating equivalent layers generally leave invariant only a sublattice of the layers. The thus...
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| Format: | Article |
| Language: | English |
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MDPI AG
2014-07-01
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| Series: | Symmetry |
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| Online Access: | http://www.mdpi.com/2073-8994/6/3/589 |
| _version_ | 1811304205624803328 |
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| author | Berthold Stöger |
| author_facet | Berthold Stöger |
| author_sort | Berthold Stöger |
| collection | DOAJ |
| description | Symmetry operations of layers periodic in two dimensions restrict the geometry the lattice according to the five two-dimensional Bravais types of lattices. In order-disorder (OD) structures, the operations relating equivalent layers generally leave invariant only a sublattice of the layers. The thus resulting restrictions can be expressed in terms of linear relations of the a2, b2 and a · b scalar products of the lattice basis vectors with rational coefficients. To characterize OD families and to check their validity, these lattice restrictions are expressed in the bases of different layers and combined. For a more familiar notation, they can be expressed in terms of the lattice parameters a, b and . Alternatively, the description of the lattice restrictions may be simplified by using centered lattices. The representation of the lattice restrictions in terms of scalar products is dependent on the chosen basis. A basis-independent classification of the lattice restrictions is outlined. |
| first_indexed | 2024-04-13T08:02:59Z |
| format | Article |
| id | doaj.art-e933231054cb4e219d1d24ba5536b541 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-13T08:02:59Z |
| publishDate | 2014-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-e933231054cb4e219d1d24ba5536b5412022-12-22T02:55:15ZengMDPI AGSymmetry2073-89942014-07-016358962110.3390/sym6030589sym6030589Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) StructuresBerthold Stöger0Institute of Chemical Technologies and Analytics, Vienna University of Technology, Getreidemarkt 9/194-SC, Vienna 1060, AustriaSymmetry operations of layers periodic in two dimensions restrict the geometry the lattice according to the five two-dimensional Bravais types of lattices. In order-disorder (OD) structures, the operations relating equivalent layers generally leave invariant only a sublattice of the layers. The thus resulting restrictions can be expressed in terms of linear relations of the a2, b2 and a · b scalar products of the lattice basis vectors with rational coefficients. To characterize OD families and to check their validity, these lattice restrictions are expressed in the bases of different layers and combined. For a more familiar notation, they can be expressed in terms of the lattice parameters a, b and . Alternatively, the description of the lattice restrictions may be simplified by using centered lattices. The representation of the lattice restrictions in terms of scalar products is dependent on the chosen basis. A basis-independent classification of the lattice restrictions is outlined.http://www.mdpi.com/2073-8994/6/3/589order-disorder theorylocal symmetrylattice |
| spellingShingle | Berthold Stöger Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) Structures Symmetry order-disorder theory local symmetry lattice |
| title | Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) Structures |
| title_full | Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) Structures |
| title_fullStr | Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) Structures |
| title_full_unstemmed | Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) Structures |
| title_short | Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) Structures |
| title_sort | non crystallographic layer lattice restrictions in order disorder od structures |
| topic | order-disorder theory local symmetry lattice |
| url | http://www.mdpi.com/2073-8994/6/3/589 |
| work_keys_str_mv | AT bertholdstoger noncrystallographiclayerlatticerestrictionsinorderdisorderodstructures |