scholarly journals Generation of Basis Vectors for Magnetic Structures and Displacement Modes

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Z. L. Davies ◽  
A. S. Wills
Keyword(s):  
Basis Functions ◽  
Irreducible Representations ◽  
Basis Sets ◽  
Space Groups ◽  
Magnetic Structures ◽  
Structural Distortions ◽  
Symmetry Adapted Functions ◽  
Crystallographic Space Groups ◽  
Basis Vectors ◽  
Linear Vector Space

Increasing attention is being focused on the use of symmetry-adapted functions to describe magnetic structures, structural distortions, and incommensurate crystallography. Though the calculation of such functions is well developed, significant difficulties can arise such as the generation of too many or too few basis functions to minimally span the linear vector space. We present an elegant solution to these difficulties using the concept of basis sets and discuss previous work in this area using this concept. Further, we highlight the significance of unitary irreducible representations in this method and provide the first validation that the irreducible representations of the crystallographic space groups tabulated by Kovalev are unitary.

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.

A general algorithm for generating isotropy subgroups in superspace

2017 ◽  
Vol 73 (1) ◽  
pp. 4-13 ◽  
Author(s):  
Harold T. Stokes ◽  
Branton J. Campbell
Keyword(s):  
Functional Materials ◽  
Order Parameters ◽  
General Algorithm ◽  
Group Symmetry ◽  
Space Groups ◽  
Structural Distortions ◽  
Modulated Structures ◽  
Incommensurate Modulation ◽  
Incommensurate Structures ◽  
Crystallographic Space Groups

This paper presents a general algorithm for generating the isotropy subgroups of superspace extensions of crystallographic space groups involving arbitrary superpositions of multi-korder parameters from incommensurate and commensuratekvectors. Several examples are presented in detail in order to illuminate each step of the algorithm. The practical outcome is that one can now start with any commensurate parent crystal structure and generate a structure model for any conceivable incommensurate modulation of that parent, fully parameterized in terms of order parameters of irreducible representations at the relevant wavevectors. The resulting modulated structures have (3 +d)-dimensional superspace-group symmetry. Because incommensurate structures are now commonly encountered in the context of many scientifically and technologically important functional materials, the opportunity to apply the powerful methods of group representation theory to this broader class of structural distortions is very timely.

ISOSUBGROUP: an internet tool for generating isotropy subgroups of crystallographic space groups

2016 ◽  
Vol 49 (5) ◽  
pp. 1849-1853 ◽  
Author(s):  
Harold T. Stokes ◽  
Seth van Orden ◽  
Branton J. Campbell
Keyword(s):  
Phase Transition ◽  
Order Parameters ◽  
Irreducible Representations ◽  
Software Suite ◽  
Space Groups ◽  
Program Output ◽  
User Friendly ◽  
Crystallographic Space Groups

ISOSUBGROUP, the newest member of the ISOTROPY Software Suite (http://iso.byu.edu), generates isotropy subgroups of crystallographic space groups based on superpositions of multiple irreducible representations, along with a wealth of information about each one. Like the original ISOTROPY program, its scope is general rather than being restricted to common types of order parameters of a user-specified parent structure. But like the newer ISODISTORT program, its user-friendly interface has menu-driven selections. This combination of features has been oft requested but unavailable until now. Program output includes information about the subgroup symmetry, ferroic species, phase-transition continuity, active k vectors, domains and secondary order parameters.

The symmetric group and the method of diatomics in molecules: an application to small lithium clusters

1973 ◽  
Vol 333 (1592) ◽  
pp. 69-87 ◽  
Keyword(s):  
Alkali Metal ◽  
Symmetric Group ◽  
Metal Clusters ◽  
Basis Functions ◽  
Basis Set ◽  
Irreducible Representations ◽  
Secular Equations ◽  
Alkali Metal Clusters ◽  
Lithium Clusters

A new ‘most economical’ algorithm for the construction of diatomics in molecules secular equations is described. The method does not require the basis functions to be written down explicitly, since overlap may be factored out of the equations entirely. The theory is presented in detail for the particular case of homogeneous alkali metal clusters. A knowledge of the irreducible representations of the symmetric group for the Jahn-Serber basis set is necessary. The irreducible representations are derived by a genealogical procedure. Some preliminary calculations are presented for the molecules Li 3 through Li 6 , Li + 3 and Li + 4 . The lithium clusters are found to be stable with respect to all possible dissociations, and the i.ps of Li 3 and Li 4 are in agreement with the trends for the species Na 3 , Na 4 , K 3 , K 4 , etc., whose i.ps have been measured experimentally.

scholarly journals Double crystallographic groups and their representations on the Bilbao Crystallographic Server

2017 ◽  
Vol 50 (5) ◽  
pp. 1457-1477 ◽  
Author(s):  
Luis Elcoro ◽  
Barry Bradlyn ◽  
Zhijun Wang ◽  
Maia G. Vergniory ◽  
Jennifer Cano ◽  
...  
Keyword(s):  
Irreducible Representations ◽  
Site Symmetry ◽  
Crystallographic Groups ◽  
Space Groups ◽  
Symmetry Approach ◽  
Wyckoff Position ◽  
Double Space ◽  
Elementary Band ◽  
Symmetry Relations ◽  
Symmetry Operations

A new section of databases and programs devoted to double crystallographic groups (point and space groups) has been implemented in the Bilbao Crystallographic Server (http://www.cryst.ehu.es). The double crystallographic groups are required in the study of physical systems whose Hamiltonian includes spin-dependent terms. In the symmetry analysis of such systems, instead of the irreducible representations of the space groups, it is necessary to consider the single- and double-valued irreducible representations of the double space groups. The new section includes databases of symmetry operations (DGENPOS) and of irreducible representations of the double (point and space) groups (REPRESENTATIONS DPGandREPRESENTATIONS DSG). The toolDCOMPRELprovides compatibility relations between the irreducible representations of double space groups at differentkvectors of the Brillouin zone when there is a group–subgroup relation between the corresponding little groups. The programDSITESYMimplements the so-called site-symmetry approach, which establishes symmetry relations between localized and extended crystal states, using representations of the double groups. As an application of this approach, the programBANDREPcalculates the band representations and the elementary band representations induced from any Wyckoff position of any of the 230 double space groups, giving information about the properties of these bands. Recently, the results ofBANDREPhave been extensively applied in the description of and the search for topological insulators.

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