Representations of Space Groups

Vol. 32 H.F. Franzen Second-Order Phase Transitions and the Irreducible Representation of Space Groups. VI, 98 pages. 1982. (out of print)... 

Hurley, A. C. (1966) Ray representations of point groups and the irreducible representations of space groups and double space groups. Phil. Trans. Roy. Soc. (London) A260, 1-36. 

Miller, S. C. and Love, W. F. (1967) Tables of Irreducible Representations of Space Groups and Co-representations of Magnetic Space Groups. Boulder, CO Pruett. Morgan, D. J. (1969) Group theory and electronic states in perfect crystals. In Landsberg, P. T. (ed.) Solid State Theory, chaps. IV and V. London Wiley Interscience. 

In the following discussion, for simplicity we make use of the pair of symmetry diagrams that constitute a representation of space group P2 jc (No. 14) in International Tables for X-Ray Crystallography Volume /, as displayed in Fig. 9.3.4. 

J. Zak The Irreducible Representations of Space Groups (Benjamin, New York 1969)... 

Miller, S.C., and W.F. Love, 1967, Tables of Irreducible Representations of Space Groups and Co-Representations of Magnetic Space Groups (Pruett Press, Boulder, CO). 

Several good sources treat space groups and group theory of the Brillouin zone (Koster, 1957 Slater, 1965). In particular, applications of lattice dynamics have been treated by Maradudin and Vosko (1968) and by Montgomery (1969). Venkataraman and Sahni (1970) have included a discussion of symmetry in their review. In recent years, several books have appeared which contain comprehensive listings of the irreducible representations of space groups (Zak, 1969 Slater, 1965 Kovalev, 1965 Miller and Love, 1967). 

Second-Order Phase Ihtnsitions and the Irreducible Representation of Space Groups... 

The space symmetry of crystalhne orbitals is defined by irreducible representations of space groups. As will be shown in the next sections the structure of these irreps is... 

Projective representations were first introduced by Shur [32], who developed a general theory of projective representations and worked out methods for constructing projective representations of finite groups. The connection between projective representations of point groups and representations of space group was demonstrated by Lyubarskii, Kovalev, Bir [27,31,33]. To find with the factor qrstem (3.40)... 

The numbering of some small representations of space groups is different in different tables. For example, four small two- dimensional representations at the X point are ordered in different ways on the site [16] and in tables [17]. The ray of stars L,X and W chosen for the small-representation generation can also be different, as can the origin choice for space-group description. All these differences do not change the full representations of space groups. 

When the space group is realized in a crystalline structure the atomic states included in the LCAO basis define the symmetry of crystalline orbitals appearing in the electronic-structure calculations. The symmetry connection of atomic and crystalline orbitals is given by induced representations of space groups considered in the next subsection. 

Site Symmetry and Induced Representations of Space Groups 67... 

Site Symmetry and Induced Representations of Space Groups Table 3.6. Induced irreps of the point group Td... 

The theory of induced representations of space groups gives the answer to the question of whether it is possible to generate in the space of states of a given energy band the basis of localized functions The answer to this question allows the symmetry connection between delocalized Bloch-type and localized Wannier-type crystalline orbitals to be obtained. This point is discussed in Sect. 3.3. 

For the symmetry directions in the BriUouin zone A(FX), A FL), E (see Fig. 3.2) the small representations of both space groups arep-equivalent to ordinary irreducible representations of point groups C4 ,C3 and C v The notations of these representations are taken from [17]. For symmetry directions on the surface of the BriUouin zone Z XW),S smaU representations of space group are p-equivalent to ordinary irreducible representations of point group C v, for the symmetry direction - to ordinary irreducible representations of group G. For the nonsymmorphic space group... 

Table 3.15. Band representations of space groups and 0 for upper valence bands of...

The set of one-electron functions transforming according to the n j-dimensional irrep d( ) is called the shell. For molecules, these shells are connected with irreps of the point-symmetry group. For a crystal, f3 = ( fe,7) - full irreducible representation of space group G, defined by the star of wavevector k and irrep 7 of the point group of this vector. Taking into consideration the spin states a a) we have 2n/ one-electron states in the shell. The functions ( )Q ([Pg.110]

We shall illustrate the discussed procedure using the example of the SrZrOs crystal with the cubic perovskite structure. In Table 9.16 band representations of space group Oj induced by those atomic states that take part in the valence band formation are given. Each line of this table dehnes the connection of the q-basis for site-symmetry groups Oh Si), Dih 0) with the fe-basis at the symmetry points of BZ, see also Sect. 3.3. 

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