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47The packing ratio is defined as the fraction of the total volume of the cell that is ' filled by atoms. Determine the maximum values of this ratio for equal spheres located at the points of simple-cubic, body-centered-cubic, and face-centered-cubic crystals. Crystal Shuctures and Interatomic Forces 5. Repeat Problem 4 for simple hexagonal, and rhombohedral lattices. 6. Repeat Problem 4 for an hcp structure. 7. Consider a face-centered-cubic cell. Construct a primitive cell within this larger cell, and compare the two.

3 BRAGG'S LAW 4h /,{J When a monochYomatic x-ray beam is incident on the surface of a crystal, it is reflected. Ho-wever, the reflection takes place only when the angle of incidence has certain values. These values depend on the wavelength and the lattice constants of the crystal, and consequently it seems reasonable to attempt to explain the selective reflectivity in terms of interference effects, as in physical optics. 2(a),where the crystal is represented by a set of parallel planes, corresponding to the atomic planes.

B) Reflected intensity from a KBr crystal. The reflecting planes for the various peaks are indicated. difference between the paths of any two consecutive rays is an integral multiple of the wavelength. 3) where ,t is the wavelength and rays I ,? a positive integer. The path difference A between and 2 in the figure is L:TB+Ee -Ae,:2TB-Te,. , that the angles of incidence equal the angles of reflection. when the interplanar distance is denoted by d, it follows from the figure that -AB: disinl and -AC, :T-Ccosl: (2dltanO) x cosg, where 0 is the glancing angle between the incident beam and the reflecting planes.