Nonprimitive lattices

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,... 

If nonprimitive lattice cells are used, the vector from the origin to any point in the lattice will now have components which are nonintegral multiples of the unitcell vectors a, b, c. The position of any lattice point in a cell may be given in terms of its coordinates, if the vector from the origin of the unit cell to the given point has components xa, yb, zc, where x, y, and z are fractions, then the coordinates of the point are x y z. Thus, point A in Fig. 2-7, taken as the origin, has coordinates 0 0 0 while points B, C, and D, when referred to cubic axes, have coordinates 0 0 i, and i 0, respectively. Point E has coordinates i 1 and... 

Such nonprimitive or centered cells can occur in all crystal systems higher than triclinic and introduce an extra seven lattice types to those already defined by the crystal system. The conventional types of nonprimitive lattice are C-centered (C), with an extra lattice point in the center of the (001) face at xmit cell position (V2,V2,0) body-centered (I),with an extra lattice point at the unit cell center at ( /2,V2,V2,) face-centered (F), with extra lattice points in the center of the unit cell faces at positions (V2V2O), (V2OV2), (O a A) and the rhombohedrai R setting of the trigonal lattice. There are also nonstandard settings of these, such as A-centered monoclinic cells, but these do not represent distinct lattice types. 

Primitive three-dimensional lattices have been classified into seven crystalline systems triclinic, monoclinic, orthorombic, tetragonal, cubic, trigonal, and hexagonal. They are different in the relative lengths of the basis vectors as well as in the angles they form. An additional seven nonprimitive lattices, belonging to the same crystalline systems, are added to the seven primitive lattices, which thus completes the set of all conceivable lattices in ordinary space. These 14 different types of lattices are known as Bravais lattices (Figure 3). 

Here, the subscript h indexes different helices in the unit cell. In this convention, the fractional coordinates for each unique helix in the unit cell are specified with the helix axis directed along the z-coordinate and passing through the origin. Each helix may have a distinct setting angle, o)/, and h and k may take fractional values to indicate helices at nonprimitive lattice positions. The lattice index / is replaced by the helix repeat index, n. 

A nonprimitive lattice with a pair of lattice points on opposite unit cell faces is said to be side centered and designated as A, B, or C, depending upon whether the lattice point is in the middle of the be, ac, or ab face. In practice, unit cells are always chosen in such a way that they are A or C, but not B centered. If there is a lattice point on all six faces of the unit cell, this is a face-centered lattice (such as the face-centered cubic structure of some metals) and designated F. If the unit cell has a lattice point at the body center of the unit cell, it is designated /, as in the case of the body-centered cubic structure of some metals. 

On the other hand, the orthorhombic lattice can have body-centered, base-centered, and face-centered lattices, shown in Figure 4.12d through f, which are not redundant because the three nonprimitive lattice vectors all have different lengths. 

M.L. Frankheim in 1842 was the first to classify the possible crystal lattices including the special body-centered and face-centered nonprimitive lattices. However, he had mistakenly added a 15th structure that turned out to be redimdant. Auguste Bravais was the first in 1845 to correctly characterize the 14 unique lattices that now bear his name. 

The nonprimitive reciprocal lattice vectors A, B, and C are given by Equation 6.9 using the nonprimitive lattice translation vectors a, b, c. These nonprimitive reciprocal lattice vectors are along the x, y, z axes and form a cubic unit cell with length lit/a and a basis of (000), (110), (101), (Oil) as may be seen in Figure 6.1. Note that the primitive reciprocal lattice vector A = - /2 A. ... 

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. 

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... 

C2 = nonprimitive centered lattice with a twofold axis of rotation. 

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. 

It is often convenient to choose a unit cell larger than the primitive unit cell. Nonprimitive unit cells contain extra lattice points, not at the vertices. For example, in three dimensions, nonprimitive unit cells may be of three kinds ... 

Bravais lattice Classification of fourteen three-dimensional lattices based on primitive and nonprimitive unit cells. Named after Auguste Bravais, who first used them. 

Sometimes the smallest, or primitive, unit cell does not have the full symmetry of the crystal lattice. If so, a larger nonprimitive unit cell that does have the characteristic symmetry is deliberately chosen (Fig. 21.8). Only three types of nonprimitive cells are commonly used in the description of crystals body-centered, face-centered, and side-centered. They are shown in Figure 21.9. 

FIGURE 21.18 Two ionic lattices in the fee system. A single (nonprimitive) cubic unit cell of each is shown. 

The fourteen Bravais lattices are described in Table 2-1 and illustrated in Fig. 2-3, where the symbols P, F, /, etc., have the following meanings. We must first distinguish between simple, or primitive, cells (symbol P or R) and nonprimitive cells (any other symbol) primitive cells have only one lattice point per cell while nonprimitive have more than one. A lattice point in the interior of a cell belongs to that cell, while one in a cell face is shared by two cells and one at a corner is shared by eight. The number of lattice points per cell is therefore given by... 

The lattice points in a nonprimitive unit cell can be extended through space by repeated applications of the unit-cell vectors a, b, c just like those of a primitive cell. We may regard the lattice points associated with a unit cell as being translated one by one or as a group. In either case, equivalent lattice points in adjacent unit cells are separated by one of the vectors a, b, c, wherever these points happen to be located in the cell (Fig. 2-5). 

Any of the fourteen Bravais lattices may be referred to a primitive unit cell. For example, the face-centered cubic lattice shown in Fig. 2-7 may be referred to the primitive cell indicated by dashed lines. The latter cell is rhombohedral, its axial angle a is 60°, and each of its axes is l/ /2 times the length of the axes of the cubic cell. Each cubic cell has four lattice points associated with it, each rhombohedral cell has one, and the former has, correspondingly, four times the volume of the latter. Nevertheless, it is usually more convenient to use the cubic cell rather than the rhombohedral one because the former immediately suggests the cubic symmetry which the lattice actually possesses. Similarly, the other centered nonprimitive cells listed in Table 2-1 are preferred to the primitive cells possible in their respective lattices. 

The silver halides, with the exception of one crystal modification of Agl, have cubic crystal structures. Crystal structure data, relevant lattice properties and low temperature dielectric constants, are collected in Table 3. AgF, AgCl, and AgBr all have the NaCl rocksalt structure in which there are four silver halide pairs per nonprimitive unit cell with cation-anion nearest neighbor distances equal to one-half a lattice constant. 

Rhombohedral is the primitive cell. Lattice parameters given are for the nonprimitive hexagonal cell. 

Commensurism (POP) p,q,r, and are all integers. All the overlayer lattice points lie simultaneously on two primitive substrate lattice lines and coincide with symmetry-equivalent substrate points (Fig. 7a). This can be described alternatively as POP coincidence. Each primitive overlayer lattice vector is an integral multiple of an identically oriented (primitive or nonprimitive) substrate lattice vector. This condition is generally regarded the energetically most favorable one with respect to the overlayer-substrate interface because the surface potentials of the two opposing lattices, which have periodicities that conform to the lattice dimensions, are phase coherent. This... 

Taking into account such nonprimitive unit cells, the structure of any crystalline solid can be represented in terms of one of 14 possible basic types called Bravais lattices (Figure 6.14). 

Consider a symmetry operation of the space group, designatepoint group operation executed at the chosen origin. The nonprimitive translation associated with S is designated v(S) and T is a lattice translation vector. The result of this operation on a vector R of the lattice is defined by... 

These designations of the operations define the nonprimitive translations y(S) in terms of the lattice basis vectors aj, a, aj. For example, we have... 

FIGURE 2.18 Left the paradigm of generating a periodic structure by the combination between Base and Reason is noted the possibility of identify inside the lattice of nonprimitive unit cells (NP) and those primitive (P) right a structural realization formed by the cubic lattice with the faces centered with the fullerene base (C60) after Heyes (1999). 

Rhombohedral is the primitive cell. Lattice parameters given are for the nonprimitive hexagonal cell. At 220 K (-53°C), after Darnell (1963). 

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