The Lattice Translations

FIGURE 3.6 In (a), the disposition of asymmetric units related by a 4j screw axis is illustrated. As with the 2i screw axis, continued application of the symmetry operator in a crystal simply generates asymmetric units in adjacent unit cells which were already present due to unit cell translations. In (b), a 61 screw axis produces six identical asymmetric units whose equivalent positions are specified according to a hexagonal coordinate system. It follows that such a symmetry axis could only be compatible with a unit cell having a hexagonal face (i.e., a hexagonal prism). 

The final stage in the creation of a crystal is to sequentially translate the unit cell and its contents along a, b, c, by distances a, b, c, respectively, many times, to generate a contiguous three-dimensional array of periodically repeated unit cells. The continuous solid so formed constitutes a crystal and exhibits all of its properties. In a sense, this final step of translation of the unit cell contents along a, b, and c is conceptually redundant with the previous definition of the unit cell parameters, which defines the directions and magnitudes of the translations. It is nonetheless necessary to actually carry out the operations in order to generate the physical crystal. 

The idea of a lattice, which expresses the translational periodicity within a crystal as the systematic repetition of the molecular contents of a unit cell, is a salient concept in X-ray diffraction analysis. A lattice, mathematically, is a discrete, discontinuous function. A lattice is absolutely zero everywhere except at very specific, predictable, periodically distributed points where it takes on a value of one. We can begin to see, from the discussion 

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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-... 

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. 

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 ... 

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... 

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. 

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. 

Figure 16.11. (a) bm is a vector from the origin O to a lattice point P in the reciprocal lattice representation, and plane 1 is normal to bm. The lattice translation a is a vector from O to another lattice point P2 on plane 1. Plane 0 is parallel to plane 1 through O. (b) a intersects plane 1 at one of the other lattice points in plane 1. If a lies along ai, n2 and n3 are zero and a = ttiHi. Similarly for a2, a3. 

Here x a refers to the ath atomic orbital in the unit cell of the crystal described by the lattice translation vector, t. The CRYSTAL code is also capable of calculating charge density in a solid using the density functional theory (DFT) at local density approximation (LDA) or at generalized gradient approximation (GGA). 

Let us examine now the limitations on the screw axes. In a lattice the screw axes must be parallel to a translation direction. After n rotations by an angle cp and n translations by the distance T, that is, after n translations along the screw axis, the total amount of translation distance in the direction of this axis must be equal to some multiple of the lattice translation mt,... 

Figure 2(a) shows Raman spectra in a wide frequency range in proton disordered Ih (thin spectra) and ordered XI phase (thick spectra). In spite of the low resolution (Av 15cm ), changes of spectra by the transition can be seen clearly in the lattice (translational, librational) modes and also in the stretching bands of water molecules. Particularly, peaks... 

The periodic structure consists of a motif which is repeated by the lattice translations. 

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