Translation lattice

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-... 

For the reconstruction of the occupation number density (k) in the repeated zone scheme one uses the reciprocal form factor at lattice translation vectors R, as (k) can be written as [9]... 

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. 

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... 

Calculate the volume of a unit cell from the lattice translation vectors. 

A cell direction is designated by the vector r, which is a combination of the lattice translation vectors a, b, and c ... 

Negative directions are indicated by an overbar [1-1] and are called the one negative one one direction. All directions are relative to the origin where the three lattice translation vectors originate (see Figure 1.19). 

The cell volume, V, can be calculated using the lattice translation vectors ... 

Mathematically, this is a triple scalar product and can be used to calculate the volume of any cell, with only a knowledge of the lattice translation vectors. If the lattice parameters and interaxial angles are known, the following expression for V can be derived from the vector expression ... 

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. 

Not all incommensurate structures are composite. It is possible to have incommensurate modulations in a structure composed of a single infinite building block, particularly if a weak cation fits rather loosely into a hole in a flexible framework. The polyhedra that compose the framework tend to twist to give the cation a distorted environment. These twists can often be described by a wave with a wavelength that may or may not be commensurate with the lattice translation of the crystal. If it is commensurate, the twisting is described as... 

Reaction of the triacetylide [ 1,3,5( = ) 3- 6 3] and the corresponding monodentate ligand (L) (L = isocyanide, phosphine, phosphite) gives [ 1,3,5 (C = CAuL) 3-C(lH3]. In the crystal structures of compounds with L — CN Bu and P(OMe)3 molecules related by lattice translations are connected through aurophilic interactions to form a polymer (Figure 2.73) [363]. 

Figure 11.9. A diagram showing how an entire set of objects is generated from an initial one (No. 1) at a general position (jc, y) by the combined action of glide lines and the lattice translations.

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. 

P/. This group is so trivial, having no symmetry apart from the three independent lattice translations, that we shall not present the diagrams or coordinate tables. 

Pi. The action of the lattice translations (i.e., the symmetry of the lattice itself) upon any one inversion center (1) that we introduce is to generate others (cf. the 2D group / 2). It is conventional to place one inversion center at the origin of the unit cell. The translational symmetry of the lattice then generates another one at the center of the cell (i,U), three more at face centers (e.g., 0,, i), and three at the midpoints of the edges (e.g., 2,0,0), for a total of eight inversion centers, none of which are equivalent. 

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