Lattice translation operation

Thus far we have addressed the symmetry of crystalline arrays only in terms of the proper rotations and the rotation-inversion operations (the latter including simple inversion, as 1, and reflection, as 2) that occur in point symmetries, along with the lattice translation operations. However, for a complete discussion of symmetry in crystalline solids, we require two more types of operation in which translation is combined with either reflection or rotation. These are, respectively, glide-reflections (or, as commonly called, glides) and screw-rotations. 

Till now, we have only considered a mathematical set of points. However, a material, in reality, is not merely an array of points, but the group of points is a lattice. A real crystalline material is constituted of atoms periodically arranged in the structure, where the condition of periodicity implies a translational invariance with respect to a translation operation, and where a lattice translation operation, T, is defined as a vector connecting two lattice points, given by Equation 1.1 as... 

Any linear combination of functions labeled by the same any lattice translation operator and corresponds to the same k. translation operator. 

These include rotation axes of orders two, tliree, four and six and mirror planes. They also include screM/ axes, in which a rotation operation is combined witii a translation parallel to the rotation axis in such a way that repeated application becomes a translation of the lattice, and glide planes, where a mirror reflection is combined with a translation parallel to the plane of half of a lattice translation. Each space group has a general position in which the tln-ee position coordinates, x, y and z, are independent, and most also have special positions, in which one or more coordinates are either fixed or constrained to be linear fimctions of other coordinates. The properties of the space groups are tabulated in the International Tables for Crystallography vol A [21]. 

For the nanotubes, then, the appropriate symmetries for an allowed band crossing are only present for the serpentine ([ , ]) and the sawtooth ([ ,0]) conformations, which will both have C point group symmetries that will allow band crossings, and with rotation groups generated by the operations equivalent by conformal mapping to the lattice translations Rj -t- R2 and Ri, respectively. However, examination of the graphene model shows that only the serpentine nanotubes will have states of the correct symmetry (i.e., different parities under the reflection operation) at the K point where the bands can cross. Consider the K point at (K — K2)/3. The serpentine case always sat-... 

The special positions include the following points and those obtained from them by the translational operations of the face-centered lattice ... 

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. 

Several types of symmetry operations can be distinguished in a crystalline substance. Purely translational operations, such as the translations defining the crystal lattice, are represented by I 1, n3, with nu n2, n3 being integers. 

In conclusion, the invariance under lattice translations means that if 14 ) is a wave function of the Hamilton operator, is a solution as well. It can be... 

The symmetry operations, G, of the space group acting on an atom placed at an arbitrary point in space will generate a set of mo equivalent atoms in the unit cell. Operation of the lattice translations, R, acting on this set generates an infinite array of such atoms, with the finite set of ma atoms being repeated at each point on the lattice. This is illustrated in Fig. 10.1 in which nia = 4 and each of the rectangles defined by the horizontal and vertical lines represents a unit cell that is identical with the one outlined with heavy lines. 

Upon adding each one of these we create, respectively, the symmetries / 2, p3, p4, and p6. In each case it can be seen in Figure 11.7 that the combined effect of the explicitly introduced rotation axis and the translational operations generates further symmetry axes. Thus, inp2, in addition to the set of twofold axes explicitly introduced, two other sets, lying at the midpoints of the translations connecting them, arise automatically because of the repetitive nature of the lattice. In the case of p3 a second, independent set of threefold axes arises. The reader can see by inspection of Figure 11.7 the additional axes that arise in p4 and p6. 

In each of the equations (l)-(4) the crystal pattern appears the same after carrying out the operation signified. It follows from eq. (2) that the pattern, and therefore the subset of lattice translations... 

Figure 6. The simple transformation for switching between fee and hep lattices. The diagram shows six close-packed u v) layers. (The additional bracketed layer at the bottom is the periodic image of the layer at the top.) The circles show the boundaries of particles located at the sites of the two close-packed structures. In the lattice switch operation the top pair of planes are left unaltered, while the other pairs of planes are relocated by translations, specified by the black and white arrows. The switch operation is discrete The relocations occur through the wormholes. (Taken from Fig. 4 of Ref. 48.)...

Bravais lattice — used to describe atomic structure of crystalline -> solid materials [i,ii], is an infinite array of points generated by a set of discrete translation operations, providing the same arrangement and orientation when viewed from any lattice point. A three-dimensional Bravais lattice consists of all points with position vectors R ... 

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