Body-centered cell

Figure 11.13. Alternate ways of choosing a unit cell for the centered monoclinic lattice, a, fi, c, define the body-centered (/) cell a, b c define the A-centered cell a, fi, c define the F-centered cell.

A crystal lattice is an array of points arranged according to the symmetry of the crystal system. Connecting the points produces the lattice that can be divided into identical parallelepipeds. This parallelepiped is the unit cell. The space lattice can be reproduced by repeating the unit cells in three dimensions. The seven basic primitive space lattices (P) correspond to the seven systems. There are variations of the primitive cells produced by lattice points in the center of cells (body-centered cells, I) or in the center of faces (face-centered cells, F). Base-centered orthorhombic and monoclinic lattices are designated by C. Primitive cells contain one lattice point (8 x 1/8). Body-centered cells... 

We have noted that Pu has six allotropic forms including ccp (8-Pu), body-centered tetragonal (S -Pu) and bcc (e-Pu) structures. The tetragonal body-centered cell of 8 -Pu becomes the bcc cell of s-Pu when the axial ratio is unity. There are many metals having ccp and bcc structures, but Pu is the only one of these metals that also has the intermediate body-centered tetragonal structure. 

Figure 7.3. A projection of the tetragonal body-centered cell of B4CI4.

Figure 7.4. A projection along c0 of the atoms in the tetragonal cell of B5H9. Boron atoms are the smaller circles. The centers of the four B atoms in the square base of the pyramidal molecules are at 0 (and 100) and 50 giving a body-centered cell.

XeF4 is square planar as expected for a molecule with four electron bonding pairs and two unshared electron pairs on Xe. Crystals of XeF4 are monoclinic (P2i/n) with two molecules per cells (a = 5.05, b = 5.92, c = 5.77 A, and (3 = 99.6°). It is a body-centered cell and the notation is 3 2PTOT(t). 

The crystals of p-dinitrobenzene (O2NC6H4NO2) are monoclinic, with two molecules per cell, C, P2ib, a0 = 11.05, b0 = 5.42, c0 = 5.65 A, and (3 = 92°18. Figure 11.19 shows that the benzene rings are centered at the comers and the center of the cell giving a body-centered cell, 3 2PTOT(m). The long direction of the molecule in the center is inclined in the opposite direction compared to those at the corners. 

PROBLEM 7.4.4. For a monoatomic cubic crystal consisting of spherical atoms packed as close as possible, given the choices of a simple cubic crystal (SCC atom at cell edges only this structure is rarely used in nature, but is found in a-Po), a body-centered cubic crystal (BCC, atom at comers and at center of body), and a face-centered cubic crystal (FCC body at face comers and at face centers), show that the density is largest (or the void volume is smallest) for the FCC structure (see Fig. 7.12). In particular, show that the packing density of spheres is (a) 52% in a simple cubic cell (b) 68% for a body-centered cell (c) 71% for a face-centered cubic cell. 

One has to take into account, however, that the unit cell which is relevant for spectroscopy is the primitive (or Wigner-Seitz) unit cell. It is a parallelepiped from which the entire lattice may be generated by applying multiples of elementary translations. Face- and body-centered cells are multiple unit cells. The content of such a cell has to be divided by a factor m to obtain the content of a primitive unit cell. This factor m is implicitly given by the international symbol for a space group P and R denote primitive cells (m = 1), face-centered cells are denoted A, B, C (m = 2), and F m = 4), and body-centered cells are represented by I m = 2). Examples are described by Turrell (1972). 

FIGURE 3.15 The types of unit cells that form the basis for the allowable lattices of all crystals (known as the Bravais lattices). There are 15 unique lattices (see International Tables, Volume I, for further descriptions). All primitive (/ ) cells may be considered to contain a single lattice point (one-eighth of a point contributed by each of those at the corners of the cell), face-centered (C) and body-centered (/) cells contain two full points, and face-centered (F) cells contain four complete lattice points. 

We had previously concluded from geometrical considerations that the base-centered cell would produce a 001 reflection but that the body-centered cell would not. This result is in agreement with the structure-factor equations for these two cells. A detailed examination of the geometry of all possible reflections, however, would be a very laborious process compared to the straightforward calculation of the structure factor, a calculation that yields a set of rules governing the value of F for all possible values of plane indices. 

The reader may have noticed in the previous examples that some of the information given was not used in the calculations. In (a), for example, the cell was said to contain only one atom, but the shape of the cell was not specified in (b) and (c), the cells were described as orthorhombic and in (d) as cubic, but this information did not enter into the structure-factor calculations. This illustrates the important point that the structure factor is independent of the shape and size of the unit cell. For example, any body-centered cell will have missing reflections for those planes which have ft + k + 1) equal to an odd number, whether the cell is cubic, tetragonal, or orthorhombic. The rules we have derived in the above examples are therefore of wider applicability than would at first appear and demonstrate the close connection between the Bravais lattice of a substance and its diffraction pattern. They are summarized in Table 4-1. These rules are subject to... 

Figure 4.1 The 14 Bravais unit cells. The letters after the name of the crystal system denote P is the primitive unit cell B and C, the cells are centered on two faces I is the body-centered cell F is the face-centered cell.

Whenever the lattice type is something other than primitive, certain sets of planes will yield destructive interference in the centered cell where there normally would have been constructive interference in the primitive cell. The example shown In Figure I l.l I illustrates how this can happen in the case of a body-centered cell. The (10 0) reflection is missing because the body center yields a set of equivalent... 

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