Space Lattices and Unit Cells

At an earlier point (p. 185) the unit cell of a crystalline structure was described as one of a large number of identical prisms, which, when oriented in the same way and stacked together in three dimensions, form a perfect crystal. The corners of an array of unit cells put together in this way are said to be the points of a space lattice the surroundings about each point of the lattice must be identical to the surroundings about every other point. Additional lattice points may sometimes be put at the face centers or at the body centers of the unit cells in crystalline sodium chloride (Fig. 12-2), for example, chloride ions are located both at the corners and the face centers of the unit cell, and an observer at a corner would have the same surroundings as one at a face center. The description of the structure of a crystalline solid is then a description of the size and shape of the unit cell and of the locations of the atoms within it. 

Each unit cell is a parallelepiped whose three axes may or may not be equal in length and whose interaxial angles may or may not be 90 degrees. Relationships between the values of the angles and between the lengths of the axes form the basis for classification into seven types of crystal 

Hexagonal Two equal Two right angles (120° between equal axes) 

Unit cells are further subclassified as simple cubic/ face-centered cubic, body-centered cubic, base-centered rhombic, etc. but in order to avoid duplication in classification (Exercise 3), certain of the possibilities are left out (for example, face-centered tetragonal, side-centered rhombic), Actually 14 distinct types of space lattice are recognized. A number of cubic unit ceils and one body-centered tetragonal cell 

To specify the position of a point within or on the surface of a unit cell, a system of coordinates similar to ordinary Cartesian coordinates is used. The corner of a cell is selected as the origin, and distances to the point are measured parallel to each axis these distances are expressed in terms of the identity distances along the respective axes. This is best shown in a two dimensional lattice (Fig. 20-3). Here there are four two-dimensional 

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