280 H. K. SCHACHMAN AND R. C. WILLIAMS 



scattering center, and a phenomenon quite analogous to light scattering will 

 be originated. But since there is an ordered array of such centers the effects 

 of interference among the scattered wavelets will be predominant, with the 

 result that there will be constructive interference only in certain, highly 

 restricted directions, and destructive interference in all others. 



The existence of repeating, ordered arrangements of atoms within a 

 crystal makes it convenient to think in terms of intracrystalline planes. 

 Suppose we have a crystal made of a multitude of just two kinds of atoms, 

 A and B. . . . Planes can be drawn through the A atoms in numerous ways; 

 the most significant for our purposes are those that contain the highest 

 density of atomic population. Suppose one of these planes is designated with 

 respect to its orientation by three direction indices (Miller indices) h, k, and 

 I. Numerous parallel planes can be drawn through the A atoms, all having 

 the same {h k I) values. But each B atom in a crystal is spatially related in a 

 particular way to each A atom. Hence, for every {h k I) plane drawn tln-ough 

 the A atoms an equivalent plane can be drawn tlirough the B atoms. In a 

 crystal whose molecules have a complex character it is clear that for each 

 planar orientation of a given {h k I) designation there will be sets of parallel 

 planes tlirough atoms A, B, C, . . . 



Since a crystal being analyzed in an X-ray apparatus is always small m 

 comparison with the distance from the X-ray source to the receiver, all 

 parallel planes are geometrically equivalent. If the conditions (discussed 

 below) are correct for constructive interference to exist along a certain 

 direction for the wavelets scattered by atoms in a given {h k I) plane, they 

 will be correct for all planes of this MiUer index whether the atoms contained 

 therein are of species A, or B, or . . , The X-ray beam is usually small in 

 cross section, and well collimated, so that the trace of a constructively 

 interfered beam on the receiving plane (such as a photographic film) is usually 

 a spot. From the measured coordinates of a "spot" one can ascertain the 

 value of {h k I) for the parallel planes that gave rise to it. 



If a single crystal is held fixed in front of a beam of monochromatic 

 X-rays, very few spots will be found on the receiving plane despite the 

 infinite number of {h k I) values of the planes within the crystal. This cir- 

 cumstance is due to the very stringent conditions that are imposed upon 

 constructive interference from a three-dimensional lattice. W. L. Bragg 

 showed m 1915 that the conditions can be very simply expressed. Suppose 

 a given set of planes make the grazing angle 6 with respect to the incident 

 X-ray beam, and suppose further that the distance between the parallel 

 planes is d. The simultaneous conditions for constructive interference are 

 then: 



(1) ^ = 6', where 6' is the angle between the planes and the constructively 

 interfered beam of scattered wavelets; 



