Translation and Point Symmetry of Crystals

The point qj of the orbit has a site-symmetry group Fj=RjFRj isomorphic to Fi. Thus, an orbit may be characterized by a site group Fi, (or any other from the set of groups Fj). The number of points in an orbit is equal to the index t=np/np. of the group Fj in F. 

If the elements Rj in (2.1) form a group P then the group F may be factorized in the form F = PFj. The group P is called the permutation symmetry group of an orbit with a site-symmetry group Fj (or orbital group). 

The number of atoms in an orbit is given in brackets. For example, in a molecule XY4Z (see Fig. 2.1) the atoms are distributed over three orbits atoms X and Z occupy 

As we can see, atoms of the same chemical element may occupy different orbits, i.e. may be nonequivalent with regard to symmetry. 

In crystals, systems of equivalent points (orbits) are called Wyckoff positions. As we shall see, the total number of possible splittings of space of a crystal on systems of equivalent points is finite and for the three-dimensional periodicity case equals 230 (number of space groups of crystals).The various ways of filhng of equivalent points by atoms generate a huge (hundreds of thousands) number of real crystalline structures. 

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Translation and Point Symmetry of Crystals 9 Example. The list of orbits in the group F = Ciy is... 

Translation and Point Symmetry of Crystals 13 Table 2 1- Distribution of crystal classes F and Bravais lattices on singonies F°... 

The plane waves exp(ik a ) seem to be the most convenient as interpolation functions for the integrand in the BZ integration, where a = rejUj are direct lattice translation vectors and a are primitive translations. It is easy with plane waves to take into account the translational and point symmetry of the crystal. 

Thus the combination of 3-D translational and point symmetry operations leads to an infinite number of sets of symmetry operations. Mathematically, each of these sets forms a group, and they are called space groups. It can be shown that all possible periodic crystals can be described by only 230 space groups. These 230 space groups are described in tables, for example the International Tables for Crystallography [3.8]. 

Here, as in the discussion of the Smoluchowski effect, we have made expHdt reference to the discrete, periodic stracture of the lattice. The symmetry ofa periodic lattice is substantially lower than the continuous translational and rotational symmetry of the uniform jeUium model. Only translations involving a Bravais lattice vector and a discrete number of point symmetry operations remain and the constant potential has to be replaced by a crystal potential, which reflects the lowered symmetry. This has profound consequences for the electronic structure. 

A crystal is similar to three-dimensional wallpaper, in that it is an endless repetition of some motif (a group of atoms or molecules). The motif is created by point group operations, while the wallpaper, which we caU the space lattice, is generated by translation of the motif, either with or without rotation or reflection. The symmetry of the motif is the true point group symmetry of the crystal and the S5mmetry of the crystal s external (morphological) form can be no higher than the point symmetry of the lattice. The... 

The local translational and orientational order of atoms or molecules in a sample may be destroyed by singular points, lines or walls. The discontinuities associated with the translational order are the dislocations while the defects associated with the orientational order are the disclinations. Another kind of defect, dispirations, are related to the singularities of the chiral symmetry of a medium. The dislocations were observed long after the research on them began. The dislocations in crystals have been extensively studied because of the requirement in industry for high strength materials. On the contrary, the first disclination in liquid crystals was observed as early as when the liquid crystal was discovered in 1888, but the theoretical treatment on disclinations was quite a recent endeavor. 

The symmetry elements and point groups of molecules and ions in the free state have been discussed in Sec. 1.5. For molecules and ions in crystals, however, it is necessary to consider some additional symmetry operations that characterize translational symmetries in the lattice. Addition of these translational operations results in the formation of the space groups that can be used to classify the symmetry of molecules and ions in crystals. 

In a further development of detail, one can take into account how the atoms of the solid are distributed spatially. The issue of symmetry in context with a fixed point in the crystal, and the symmetry of Bravais lattices, has been addressed, but in order to describe the entire crystal the effects of two new types of symmetry operation must be included. A space group determined in this way describes the spatial symmetry of the crystal. By definition, a crystallographic space group is the set of geometrical symmetry operations that take a three-dimensional periodic crystal into itself The set of operations that make up the space group must form a group in the mathematical sense and must include the primitive lattice translations as well as other symmetry operations. 

Crystalline lattices are commonly described in terms of their symmetry properties. Crystals can have two fundamentally different kinds of symmetry translational symmetry and point symmetry. Translational symmetry is a repeating type of symmetry that describes the method of propagation between lattice points lying on a screw axis or a glide plane, both of which are illustrated in Figure 11.3. Point symmetry, on the other hand, is the type where at least one point within the object remains unchanged with respect to every kind of symmetry operation. Point symmetry operations include proper rotations, reflections, improper rotations (rotation-reflection), and inversion these were already discussed in Chapter 8. 

The set of operations of symmetry of a crystal forms its group of symmetry G called a space group of symmetry. Group G includes both translations, operations from point groups of symmetry, and also the combined operations. The structure of space groups of symmetry of crystals and their irreducible representations is much... 

For crystaUine solids the translation symmetry of the Hamiltonian is taken into account in any electronic-structure calculations as it allows calculations to be made for the basis connected only with the primitive unit cell. In the translation-symmetry-adapted basis the matrix JI has a qnasidiagonal structure with identical blocks related to an irrep k of the translation symmetry group T. As the latter is Abelian its irreps are one-dimensional. The translation s3Tnmetry adapted functions are known as Bloch functions and numbered by wavevector k. Use of the point symmetry of a crystal allows the number of Bloch functions calculated to be decreased and further block-diagonaUzation of Hamiltonian of a crystal to be made. 

The functions < (k) = (5r,r (k)) are periodic in the reciprocal space with periods determined by the basic translation vectors b of the reciprocal lattice and having the fuU point symmetry of the crystal F is a point group of order np of the crystal). 

Crystal lattices can be depicted not only by the lattice translation defined in Eq. (7.2), but also by the performance of various point symmetry operations. A symmetry operation is defined as an operation that moves the system into a new configuration that is equivalent to and indistinguishable from the original one. A symmetry element is a point, line, or plane with respect to which a symmetry operation is performed. The complete ensemble of symmetry operations that define the spatial properties of a molecule or its crystal are referred to as its group. In addition to the fundamental symmetry operations associated with molecular species that define the point group of the molecule, there are additional symmetry operations necessary to define the space group of its crystal. These will only be briefly outlined here, but additional information on molecular symmetry [10] and solid-state symmetry [11] is available. 

Notice that the symmetry operations of each point group by continued repetition always bring us back to the point from which we started. Considering, however, a space crystalline pattern, additional symmetry operations can be observed. These involve translation and therefore do not occur in point groups (or crystal classes). These additional operations are glide planes which correspond to a simultaneous reflection and translation and screw axis involving simultaneous rotation and translation. With subsequent application of these operations we do not obtain the point from which we started but another, equivalent, point of the lattice. The symbols used for such operations are exemplified as follows ... 

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