Brillouin-zone database on theBilbao Crystallographic Server

2014 ◽  
Vol 70 (2) ◽  
pp. 126-137 ◽  
Author(s):  
Mois I. Aroyo ◽  
Danel Orobengoa ◽  
Gemma de la Flor ◽  
Emre S. Tasci ◽  
J. Manuel Perez-Mato ◽  
...  
Keyword(s):  
New York ◽  
Brillouin Zone ◽  
Irreducible Representations ◽  
Group Representations ◽  
General Introduction ◽  
Space Groups ◽  
Symmetry Properties ◽  
Parameter Ranges ◽  
Representations Of Space

The Brillouin-zone database of theBilbao Crystallographic Server(http://www.cryst.ehu.es) offersk-vector tables and figures which form the background of a classification of the irreducible representations of all 230 space groups. The symmetry properties of the wavevectors are described by the so-called reciprocal-space groups and this classification scheme is compared with the classification of Cracknellet al.[Kronecker Product Tables, Vol. 1,General Introduction and Tables of Irreducible Representations of Space Groups(1979). New York: IFI/Plenum]. The compilation provides a solution to the problems of uniqueness and completeness of space-group representations by specifying the independent parameter ranges of general and specialkvectors. Guides to thek-vector tables and figures explain the content and arrangement of the data. Recent improvements and modifications of the Brillouin-zone database, including new tables and figures for the trigonal, hexagonal and monoclinic space groups, are discussed in detail and illustrated by several examples.

scholarly journals Layer groups: Brillouin-zone and crystallographic databases on the Bilbao Crystallographic Server

2021 ◽  
Vol 77 (6) ◽  
Author(s):  
Gemma de la Flor ◽  
Bernd Souvignier ◽  
Gotzon Madariaga ◽  
Mois I. Aroyo
Keyword(s):  
New York ◽  
Brillouin Zone ◽  
Maximal Subgroups ◽  
Group Representations ◽  
Crystallographic Databases ◽  
Character Tables ◽  
Symmetry Properties ◽  
Parameter Ranges ◽  
Layer Groups

The section of the Bilbao Crystallographic Server (https://www.cryst.ehu.es/) dedicated to subperiodic groups contains crystallographic and Brillouin-zone databases for the layer groups. The crystallographic databases include the generators/general positions (GENPOS), Wyckoff positions (WYCKPOS) and maximal subgroups (MAXSUB). The Brillouin-zone database (LKVEC) offers k-vector tables and Brillouin-zone figures of all 80 layer groups which form the background of the classification of their irreducible representations. The symmetry properties of the wavevectors are described applying the so-called reciprocal-space-group approach and this classification scheme is compared with that of Litvin & Wike [(1991), Character Tables and Compatibility Relations of the Eighty Layer Groups and Seventeen Plane Groups. New York: Plenum Press]. The specification of independent parameter ranges of k vectors in the representation domains of the Brillouin zones provides a solution to the problems of uniqueness and completeness of layer-group representations. The Brillouin-zone figures and k-vector tables are described in detail and illustrated by several examples.

SIMPLIFICATION IN FINDING THE ALLOWABLE REPRESENTATIONS OF SPACE GROUPS

1962 ◽  
Vol 40 (10) ◽  
pp. 1490-1495 ◽  
Author(s):  
I. V. V. Raghavacharyulu ◽  
I. Bhavanacharyulu
Keyword(s):  
Brillouin Zone ◽  
Space Group ◽  
Space Groups ◽  
Representations Of Space

A simplification in finding the allowable representations of the groups of typical wave vectors of a space group is considered, which makes use of the symmetry of the first Brillouin zone of the space group. By this method, one also obtains the compatibility relations of the allowable representations of the space group as a by product.

Irreducible representations of space groups and superspace symmetry

1984 ◽  
Vol 53 (1) ◽  
pp. 293-296 ◽  
Author(s):  
J. M. Perez Mato ◽  
G. Madariaga ◽  
M. J. Tello
Keyword(s):  
Irreducible Representations ◽  
Space Groups ◽  
Representations Of Space

Theory of spin-space groups

1966 ◽  
Vol 294 (1438) ◽  
pp. 343-358 ◽  
Keyword(s):  
Rare Earth Metals ◽  
Spin Waves ◽  
Irreducible Representations ◽  
Spin Space ◽  
Excitation Spectra ◽  
Spin Structures ◽  
Space Group ◽  
Space Groups ◽  
Conduction Electrons ◽  
Symmetry Properties

The symmetry properties of a magnetically ordered crystal are normally described in terms of the magnetic space group. But the dominant interactions, Heisenberg exchange and anisotropy fields, have more symmetry than this in that the spins may be rotated independently of the lattice. The ‘spin-space groups’ appropriate to this symmetry are defined and described, and methods are given for finding the irreducible representations of their ‘groups of k'. The theory is applicable to discussions of the excitation spectra in these systems, especially spin waves and conduction electrons. Compatibility relations between the spin-space group and the magnetic space group are considered—these allow discussion of the modification of the spectra brought about by the smaller interactions which only have the lower symmetry. The groups for antiferromagnetic rutile structures, spinels and garnets are examined in detail, and applied to spin waves. A group for helical spin structures, as found in rare earth metals, is discussed in relation to the energy band structure.

Decomposition of Plane Waves into Irreducible Representations of Space Groups

1995 ◽  
Vol 50 (6) ◽  
pp. 577-583
Author(s):  
H. Teuscher ◽  
P. Kramer
Keyword(s):  
Dirichlet Problem ◽  
Representation Theory ◽  
Boundary Problem ◽  
Plane Waves ◽  
Neumann Boundary ◽  
Basis Functions ◽  
Irreducible Representations ◽  
Space Groups ◽  
Crystallographic Space Groups ◽  
Representations Of Space

Abstract Using a relation between representation theory of crystallographic space groups and a Dirichlet type of boundary problem for the Laplacian, we derive the solutions for the Dirichlet problem, as well as for a similar Neumann boundary problem, by a complete decomposition of plane waves into irreducible representations of a particular space group. This decomposition corresponds to a basis transformation in L2(Ω) and yields a new set of basis functions adapted to the symmetry of the lattice considered.

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