Plane lattices symmetry operations

A chirality classification of crystal structures that distinguishes between homochiral (type A), heterochiral (type B), and achiral (type C) lattice types has been provided by Zorkii, Razumaeva, and Belsky [11] and expounded by Mason [12], In the type A structure, the molecules occupy a homochiral system, or a system of equivalent lattice positions. Secondary symmetry elements (e.g., inversion centers, mirror or glide planes, or higher-order inversion axes) are precluded in type A lattices. In the racemic type B lattice, the molecules occupy heterochiral systems of equivalent positions, and opposite enantiomers are related by secondary lattice symmetry operations. In type C structures, the molecules occupy achiral systems of equivalent positions, and each molecule is located on an inversion center, on a mirror plane, or on a special position of a higher-order inversion axis. If there are two or more independent sets of equivalent positions in a crystal lattice, the type D lattice becomes feasible. This structure consists of one set of type B and another of type C, but it is rare. Of the 5,000 crystal structures studied, 28.4% belong to type A, 55.6% are of type B, 15.7% belong to type C, and only 0.3% are considered as type D. 

Knowledge of basic crystallography starts from the conception of symmetry, symmetry planes and symmetry operations necessary to identify the parameters, like lattice, lattice planes, crystal lattices (i.e. Bravis lattice) describing different crystal symmetries. Miller indices h, k, 1) to identify crystal planes etc. are explained here. 

Thus, the planes of the lattice are found to be important and can be defined by moving along one or more of the lattice directions of the unitcell to define them. Also important are the symmetry operations that can be performed within the unit-cell, as we have illustrated in the preceding diagram. These give rise to a total of 14 different lattices as we will show below. But first, let us confine our discussion to just the simple cubic lattice. 

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

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 ... 

An example is the (110) plane of III-V semiconductors, such as GaAs(llO). The only nontrivial symmetry operation is a mirror reflection through a line connecting two Ga (or As) nuclei in the COOl] direction, which we labeled as the X axis. The Bravais lattice is orthorhombic primitive (op). In terms of real Fourier components, the possible corrugation functions are... 

Only certain symmetry operations are possible in crystals composed of identical unit cells. In three dimensions these are one-, two-, three-, four- and six-fold rotations and each of these axes combined with inversion through a centre to give I, 2 ( = m, mirror plane), 3, 4, and 6 operations. Five-fold rotations and rotations of order 7 and higher, while possible in a finite molecule, are not compatible with a three-dimensional lattice. 

Let s now put some p orbitals on the square lattice, with the direction perpendicular to the lattice taken as z. The pz orbitals will be separated from py and px by their symmetry. Reflection in the plane of the lattice remains a good symmetry operation at all k. The pz(z) orbitals will give a band structure similar to that of the s orbital, since the topology of the interaction of these orbitals is similar. This is why in the one-dimensional case we could talk at one and the same time about chains of H atoms and polyenes. 

BCC real-space lattices are completely determined by the condition that each inner vector, k, go over into another by all the symmetry operations. This is not the case for the tmncated octahedron. The surface of the Wigner-Seitz cell is only fixed at the truncating planes, not the octahedral planes. Nonetheless, the volume enclosed by the truncated octahedron is taken to be the first BZ for the FCC real-space lattice (Bouckaert et ak, 1936). The special high-symmetry points are shown in Table 4.5. 

Let us first consider the ID lattice (1) with one s orbital per site. Shown in (27) are three different types of symmetry planes. According to equations (23) and (26), a general 4>s k) BO can be represented as in (28), where on top of each lattice point the exp(ik R) coefficient is given and the dotted fines denote the reference cell. When k = 0, that is, x = 0, all coefficients are one (29) and the BO is symmetric with respect to the symmetry planes a, a", and a. When k = n/a, that is, X = 1 /2, the coefficients are alternately one and minus one (30), so that the BO is symmetric with respect to the a and a" planes but antisymmetric with respect to the a plane. Thus the BOs s(0) and 4>s nla) are either symmetric or antisymmetric with respect to the a, a, and a" planes. This is not the case for an arbitrary value of the wave vector, as shown in (31) for k = tt/Su, that is, x = 1/10. This BO is symmetric with respect to a but is neither symmetric nor antisymmetric with respect to the a and a" planes. The reason for this observation is very simple. A BO is an eigenfunction of a symmetry operator if under application of this operator the same BO multiplied by a constant is generated. Because... 

Symmetry operation A symmetry operation or a series of symmetry operations converts an object into an exact replica of itself. In crystal structures, the possible symmetry operations are axes of rotation and rotatory inversion, screw axes, and glide planes, as well as lattice translations. Proper operations, which convert an object into a replica of itself, are translation and rotation. Improper operations, which convert an object into the mirror image of its replica, are reflection and inversion. 

The entire content of the unit cell can be established from its asymmetric unit using the combination of symmetry operations present in the unit cell. Here, this operation is a rotation by 180° around the line perpendicular to the plane of the projection at the center of the unit cell. It is worth noting that the rotation axis shown in the upper left comer of Figure 1.6 is not the only axis present in this crystal lattice - identical axes are found at the beginning and in the middle of every unit cell edge as shown in one of the neighboring cells. ... 

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