Crystals primitive unit cell

The rocksalt stmcture is illustrated in figure Al.3.5. This stmcture represents one of the simplest compound stmctures. Numerous ionic crystals fonn in the rocksalt stmcture, such as sodium chloride (NaCl). The conventional unit cell of the rocksalt stmcture is cubic. There are eight atoms in the conventional cell. For the primitive unit cell, the lattice vectors are the same as FCC. The basis consists of two atoms one at the origin and one displaced by one-half the body diagonal of the conventional cell. 

Translationengleiche subgroups have an unaltered translation lattice, i.e. the translation vectors and therefore the size of the primitive unit cells of group and subgroup coincide. The symmetry reduction in this case is accomplished by the loss of other symmetry operations, for example by the reduction of the multiplicity of symmetry axes. This implies a transition to a different crystal class. The example on the right in Fig. 18.1 shows how a fourfold rotation axis is converted to a twofold rotation axis when four symmetry-equivalent atoms are replaced by two pairs of different atoms the translation vectors are not affected. 

An infinite three-dimensional crystal lattice is described by a primitive unit cell which generates the lattice by simple translations. The primitive cell can be represented by three basic lattice vectors such as and h defined above. They may or may not be mutually perpendicular, depending on the crystal... 

The unit cell is defined by the lengths (a, b, and c) of the crystal axes, and by the angles (a, f>, and y) between these. The usual convention is that a defines the angle between the b- and c-axes, p the angle between the a- and c-axes, and y the angle between the a- and 6-axes. There are seven fundamental types of primitive unit cell (whose characteristics are provided in Table 7.1), and these unit cell characteristics define the seven crystal classes. If the size of the unit cell is known (i.e., a, (i, y, a, b, and c have been determined), then the unit cell volume (V) may be used... 

The extension of the quantum-mechanic interpretation of the vibrational motion of atoms to a crystal lattice is obtained by extrapolating the properties of the diatomic molecule. In this case there are 3 ( independent harmonic oscillators (9l is here the number of atoms in the primitive unit cell—e.g., fayalite has four... 

Certain factors are likely to influence future analyses of more complex viruses. Crystal stability is governed by packing interactions and, as can be seen from Fig. 16.4, is, to a first approximation, inversely proportional to the square of the virus radius, presumably underl)dng the problems with crystal stability for analyses such as that of PRDl. Even assuming that well-ordered, stable crystals can be formed, technical considerations will place an upper limit on the unit cell size from which useful data can be collected. Nevertheless, with some improvements in beam and detector technology, we expect that data collection from cells up to 2000 A should be feasible for even a primitive unit cell. 

Thus, the reciprocal lattice of a simple cubic lattice is also simple cubic. It is shown in Fig. 5.7 in the xy plane, where it is clear that the bisectors of the first nearest-neighbour (100) reciprocal lattice vectors from a closed volume about the origin which is not cut by the second or any further near-neighbour bisectors. Hence, the Brillouin zone is a cube of volume (2n/a)2 that from eqn (2.38) contains as many allowed points as there are primitive unit cells in the crystal. The second, third, and fourth zones can... 

Calcium carbonate crystallizes in several different forms. In aragonite4 there are four formula units in an orthorhombic primitive unit cell with dimensions a = 4.94 x 10-10 m, b — 7.94 x 10 10 m and c — 5.72 x 10-10 m. 

The pattern points associated with a particular lattice are referred to as the basis so that the description of a crystal pattern requires the specification of the space lattice by ai a2 a3 and the specification of the basis by giving the location of the pattern points in one unit cell by K, i= 1,2,. .., (Figure 16.1(b), (c)). The choice of the fundamental translations is a matter of convenience. For example, in a face-centred cubic fee) lattice we could choose orthogonal fundamental translation vectors along OX, OY, OZ, in which case the unit cell contains (Vg)8 + (l/2)6 = 4 lattice points (Figure 16.2(a)). Alternatively, we might choose a primitive unit cell with the fundamental translations... 

Figure 16.2. Conventional (non-primitive) unit cells of (a) the face-centered cubic and (b) the body-centered cubic lattices, showing the fundamental vectors a1 a2, and a3 of the primitive unit cells. (A conventional unit cell is one that displays the macroscopic symmetry of the crystal.)...

On the (111) crystal face of an fee metal the (2x2) unit cell is a rhombohedron while the c(4x2) primitive unit cell is a rectangle. 

The rocksalt crystal structure belongs to the cubic system with space group 0 (Pm3m). It consists of two face-centered cubic (fee) sublattices, which are occupied by one atom species each. The two sublattices are shifted along one half of the diagonal of the primitive unit cell against each other. The rocksalt lattice is sixfold coordinated. 

As previously mentioned, the primitive unit cell is the smallest unit of a crystal that reproduces itself by translations. Figure 1-37 illustrates the difference between a primitive and a centered or nonprimitive cell. The primitive cell can be defined by the lines a and c. Alternatively, we could have defined it by the lines a and c. Choosing the cell defined by the lines a" and c" gives us a nonprimitive cell or centered cell, which has twice the volume and two repeat units. Table 1-11 illustrates the symbolism used for the various types of lattices and records the number of repeat units in the cell for a primitive and a nonprimitive lattice. The spectroscopist is concerned with the primitive (Bravais) unit cell in dealing with lattice vibrations. For factor group selection rules, it is necessary to convert the number of molecules per crystallographic unit cell Z to Z, discussed later, which is the number of molecules per primitive cell. For example,... 

In order to discuss the selection rules for crystalline lattices it is necessary to consider elementary theory of solid vibrations. The treatment essentially follows that of Mitra (47). A crystal can be regarded as a mechanical system of nN particles, where n is the number of particles (atoms) per unit cell and N is the number of primitive cells contained in the crystal. Since N is very large, a crystal has a huge number of vibrations. However, the observed spectrum is relatively simple because, as shown later, only where equivalent atoms in primitive unit cells are moving in phase as they are observed in the IR or Raman spectrum. In order to describe the vibrational spectrum of such a solid, a frequency distribution or a distribution relationship is necessary. The development that follows is for a simple one-dimensional crystalline diatomic linear lattice. See also Turrell (48). 

To derive factor group (space group) selection rules, it is necessary to utilize X-ray data for a molecule from a literature source or from Wyckoffs (54) Crystal Structures. The factor group and site symmetries of the ion, molecule, or atoms must be available, as well as the number of molecules per unit cell reduced to a primitive unit cell. 

In direct analogy with two dimensions, we can define a primitive unit cell that when repeated by translations in space, generates a 3D space lattice. There are only 14 unique ways of connecting lattice points in three dimensions, which define unit cells (Bravais, 1850). These are the 14 three-dimensional Bravais lattices. The unit cells of the Bravais lattices may be described by six parameters three translation vectors (a, b, c) and three interaxial angle (a, (3, y). These six parameters differentiate the seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

Plastic crystals are almost crystalline solids, except that the molecular constituents in the primitive unit cell rotate freely in place this confers to them a certain degree of plasticity. Examples are certain cage-shaped boranes (e.g., B10H14), carboranes (e.g., Bi0C2Hi2), and buckminsterfullerene (Ceo) at room temperature. These plastic crystals usually order into normal crystalline solids at low temperatures. 

It is frequently formd that it is not possible to find a primitive rrrrit cell with edges parallel to crystal axes chosen on the basis of symmetry. In such a case the crystal axes, chosen on the basis of symmetry, are proportiorral to the edges of a unit of stracture that is larger than a primitive unit cell. Such a rrrrit is called a nonprimitive unit cell, and there is more than one lattice point per nonprimitive unit cell. If the nonprimitive unit cell is chosen as small as possible consistent with the symmetry desired, it is found that the extra lattice points (those other than the comer points) lie in the center of the unit cell or at the centers of some or all of the faces of the imit cell. The coordinates of the lattice points in such a case are therefore either integers or half-integers. 

The crystal structure of l2(i). The primitive unit cell, outlined in heavy lines, contains two molecules, identified hy dots at the atomic centers (one-half molecule each at the upper left and lower right corners, and one molecule in the body center). The light lines outline an orthorhomhic nonprimitive unit cell of dimensions Of) = 0.727 nm, feo = 0.479 nm, Co = 0.979 nm. All molecules are in planes parallel to the b and c axes. (Not all molecules in the orthorhombic cell are shown.)... 

The number of discrete k values is N/2, the number of primitive unit cells in the crystal. Each of these is assumed to yield the same set of 12 branch frequencies Vj. Thus we can simplify Eq. (30) to... 

These are denoted as F, I, and C, respectively, while primitive cells are denoted as P, and rhombohedral as R. Several symmetry-related copies of the asymmetric unit may be contained in the nonprimitive unit cell, which can generate the entire crystal structure by means of translation in three dimensions. Although primitive unit cells are smaller than nonprimitive unit cells, the nonprimitive unit cell may be preferred if it possesses higher symmetry. In general, the unit cell used is the smallest one with the highest symmetry. 

Figure 2.6-2 Variation of the frequencies by the incorporation of a tetraatomic molecule with two degenerate vibrational states ( ) in a crystal lattice, a spectrum of the free molecule, R = rotations, T = translations b static influence of the crystal lattice. The degenerate states split, the free rotations change into librations L c dynamic coupling of the vibrations of molecules within a primitive unit cell with z = 2 molecules. Each vibrational level of a molecule splits into z components and 3 z - 3 translational vibrations TS and 3 z librations L appear d dependence of the vibrational frequencies on the wave vector k of the coupled vibrations of all unit cells in the lattice. The three acoustic branches arise from the three free translations with = 0 (for k 0) of the unit cell all vibrations of the unit cells with / 0 (for k 0) give optical branches .

Every vibrational mode is due to motions of the entire chemical species (the molecule or the primitive unit cell) as a whole. In principle, molecular and crystal dynamics calculations should define rigorously the motions of every atom of the chemical species upon a vibrational mode. This approach gives rise to a very complex picture (at least for large and complex chemical species), so that the results are sometimes not easily interpreted and comparison between the vibrational behavior of similar species is frequently difficult. 

For all possible crystals there are seven basic or primitive unit cells, which are shown in Fig. 1.2. We will represent the lengths of the sides as a, b and c and the angles as... 

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