The generalized unit cell

In Fig. 4.2, each face of an orthorhombic unit cell is labeled with the indices of the set of planes that includes that face. (The crossed arrows lie upon the labeled face.) 

The set of planes including and parallel to the be face, and hence normal to the x-axis, is designated (100) because there is one such plane per lattice point in the x direction. In like manner, the planes parallel to and including the ac face are called (010) planes (one plane per lattice point along y). Finally, the ab faces of the cell determine the (001) planes. In the Bragg model of diffraction as reflection from parallel sets of planes, any of these sets of planes can be the source of one diffracted X-ray beam. (Remember that an entire set of parallel planes, not just one plane, acts as a single diffractor and produces one 

In Fig. 4.3, an additional set of planes, and thus an additional source of diffraction, is indicated. The lattice (dark lines) is shown in section parallel to the ab faces or the xy plane. The dashed lines represent the intersection of a set of equivalent, parallel planes that are perpendicular to the xy plane of the paper. Note that the planes cut each a edge into two parts and each b edge into one part, so these planes have indices 210. Because all (210) planes are parallel to the z axis (which is perpendicular to the plane of the paper), the / index is zero. [Or equivalently, because the planes are infinite in extent, and are coincident with c edges, and thus do not cut edges parallel to the z axis, there are zero (210) planes per unit cell in the z direction.] As another example, for any plane in the set shown in Fig. 4.4, the first plane encountered from any lattice point cuts that unit cell at a/2 and b 3, so the indices are 230. 

All planes perpendicular to the xy plane have indices hkO. Planes perpendicular to the xz plane have indices hOk, and so forth. Many additional sets of planes are not perpendicular to x,y, or z. For example, the (234) planes cut the unit cell edges a into two parts, b into three parts and c into four parts. (See Fig. 4.5.) 

Finally indices can be negative as well as positive. The (210) planes are the same as (-2 -1 0), whereas the (2-10) or (—2 1 0) planes tilt in the direction opposite to the (210) planes (Fig. 4.6). (The negative signs are often printed on top of the indices, but for clarity I will present them as shown here.) 

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The general unit cell in a triclinic (lowest symmetry) crystal. The unit cell has sides a, b, c, angles a, ft, and y and volume V. 

The 2 1 layer type has two tetrahedral sheets sandwiching an octahedral sheet. The three clay groups with this structure are (illitic) mica, vermicu-lite, and smectite (montmorillonite), each with the general unit cell chemical formula ... 

Figure 21.28 shows typical unit cells for these 1 1 ionic crystals. The names of the unit cells are taken from common compounds that typify the general unit-cell structure. Zincblende is a common name for zinc sulfide, ZnS, which typifies the unit-cell structure. 

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