Of a space group

The absence of a space group makes the spectral problem a difficult one. Our work in this area1- 3 is far from complete,... 

The identification of a space group that matches the multiplicity and symmetry is only the first step in finding a space group that can accommodate the bond graph. There are then three further conditions that must then be satisfied ... 

When studying the properties of a particular zeolitic catalyst it is not sufficient to consider the overall crystal structure in terms of a space group and pore dimensions. It is also essential to know factors such as the degree of... 

In Sections 16.4-16.6 the problem of finding the representations of a space group G at any particular k point was solved by reducing the size of the group of the wave vector... 

Since the physical system (crystal) is indistinguishable from what it was before the application of a space-group operator, and a translational symmetry operator only changes the phase of the Bloch function without affecting the corresponding energy E(k),... 

In applying the projection operator method for the calculation of symmetrized linear combinations of eigenfunctions, we shall need the effect of a space-group function operator (R ) on the FE eigenfunction mk(r), which is... 

The second part of a space group symbol consists of one, two or three entries after the initial letter symbol described above. At each position, the entry consists of one or two characters, describing a symmetry element, either an axis or a plane. 

Each choice of a space group and a set of cell dimensions attempted requires an effort equivalent to that necessary to solve the structure. If the structure determination effort is unsuccessful, it is not known whether the combination of space group and cell dimensions is wrong or whether the... 

We suppose that jR is a symmetry operator that corresponds to some proper or improper rotation, and that r is a vector in the real lattice. The vector Rr is also a vector in the real lattice since K is a symmetry operator. There are as many points in reciprocal lattice space as in the direct lattice, and each direct lattice vector corresponds to a definite vector in the reciprocal lattice. It follows that Rr corresponds to a reciprocal lattice point if r is a reciprocal lattice vector. Thus the operators R, S,. . . , that form the rotational parts of a space group are also the rotational parts of the reciprocal lattice space group. It now follows that the direct and reciprocal lattices must belong to the same crystal class, although not necessarily to the same type of translational lattice (see Eqs. 10.28-10.31). 

The purpose of this note is to introduce briefly the subject and to illustrate practically the method with the help of a few examples. In the first part, we shall introduce the concept of subgroup of a space group along with some definitions. In the second part, the use of symmetry relationships derived from group theoretical considerations will be presented with a study of some well known families of compounds. 

MAXIMAL SUBGROUPS AND MINIMAL SUPERGROUPS OF A SPACE GROUP... 

In the previous section the 230 crystallographic space groups, with the symbols listed in Table 2.16, have been introduced. Yet, one should be aware that in the symbol of a space group are included the minimum symmetries necessary to deduce the other ones and does not reflect all the symmetries that a space group involves. Here we analytically unfold such reality the present discussion follows (Chiriac-Putz-Chiriac, 2005). 

To deduce all the symmetries of a space group actually it means to determine all the equivalent points, i.e., those points that can be mutually transformed one in other through the internal symmetry operations, the translations included. 

The different Wyckoff positions are labeled by smaU Roman letters. The maximum number of different Wyckoff positions of a space group is 27 (in the group >2/, — Pmmm). The various possible sets of Wyckoff positions for all the space... 

The difference between oriented site-symmetry groups of different Wyckoff positions is due to different orientations of the elements of the site-symmetry group G, with respect to the lattice. The difference arises when similar symmetry elements (reflections in planes and rotations about twofold axes of symmetry) occur in more than one class of elements of the point group F. Only eleven site groups [C2(2), Cs m), C2h 2./m), C 2 (2mm), CsyiZmm), 2(222), Ds(322), D2d(42m), D3d 32m), D hijnrnm), and >3 (62m)] can have different orientations with respect to the Bra-vais lattice. Oriented site- symmetry symbols show how the symmetry elements at a site are related to the symmetry elements of a space group. The site-symmetry... 

The dimension of a space-group full representation (degeneracy of energy levels in a crystal) for a given k is equal to the product of the number of rays in the star k and the dimension of the point group irreducible representation (ordinary or projective). In particular, for the space group under consideration at the X point the dimensions of full representations are 6 and at the W point - 12. As to each of the... 

Any subgroup rep can generate some induced rep of a group (see Sect. 3.2.1). In the particular case of a space group G the small irreps of the little group Gk C G induce its full irreps. 

In [39] the concept of a band rep of a space group, which may be an induced rep, was introduced. Band reps were used to define the symmetry of an electron energy band as a whole entity. 

Prom the group>-theoretical point of view a band rep of a space group is a direct sum of its irreps that have the following properties ... 

In a cyclic model of a crystal with N primitive cells in the main region a band rep is an Np dimensional reducible rep of a space group. An induced rep is a particular case of a band rep as it satisfies both properties 1 and 2 with p = nqU/ n is the dimension of the site-symmetry group irrep for a point q belonging to the set of n, points in the unit cell). 

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