Subsurface Laboratory Methods 213 



coincide: that is, they are in phase and diflfraction effects from this par- 

 ticular family of planes are observed. A sketch set into the figure shows 

 how this phenomenon is related to conditions in the X-ray camera. 



This simple two-dimensional analogy can be applied to the three- 

 dimensional diffraction of X-rays by crystalline matter if the posts are 

 replaced by a regular assemblage of points (atoms or ions) distributed 

 in space at a distance that is of the same order of magnitude as the wave 

 lengths of X-rays. Spherical waves are created when X-rays, which are 

 electromagnetic waves, cause forced oscillations of the planetary elec- 

 trons of the atoms which they traverse, the electrons absorbing energy 

 from the X-rays when moving away from the nucleus and radiating energy 

 in all directions when moving toward the nucleus. Inspection reveals that 

 this three-dimensional point system will produce very narrow pencils of 

 rays only in those directions in which these spherical waves are in phase. 

 These reinforced waves are the rays that produce the individual spots in 

 X-ray patterns (Laue, rotation, Weissenberg, etc.) obtained from single 

 crystals. If the single crystal is replaced by a large number of smaller 

 crystals, that is, a powder, the 29 angle with the undiffracted beam must 

 remain constant since, in Bragg's equation, d for the particular set of 

 planes and the wave length. A, of the X-rays from a particular target 

 material are fixed. The crystals of the powder with their statistical 

 orientation, unless preferred orientation effects result owing to peculiar 

 crystal shapes, then must produce a whole series of such discrete pencils, 

 so that as a result a continuous diffraction cone with an apex angle of 4^ 

 is obtained. If this cone is now recorded on a photographic film placed 

 perpendicular to the cone axis, the diffraction effect is obtained as a line 

 which is in the form of a ring. A pattern on which the diffraction rings 

 from all families of planes have been recorded is usually referred to as a 

 powder pattern and consists of a series of concentric rings on a flat film, 

 or arcs of rings on a cylindrical strip of film. 



Figure 92 reveals that a fixed space arrangement of atoms with 

 definite fixed distances between them must always produce precisely the 

 same X-ray pattern. Furthermore, if the same space arrangement is re- 

 tained but the distances between atom centers are changed,^ the X-ray 

 pattern will retain its same general appearance but will either expand or 

 contract. On the other hand, if the space arrangement is altered, the 

 pattern is changed. Consequently, X-ray-diffraction patterns are a sort 

 of fingerprint of crystalline materials. Each individual substance present 

 in a mixture will produce its unique diffraction effects, so that the pattern 

 derived from the mixture is a composite of the patterns of all the materials 

 or compounds in the mixture. Furthermore, the intensities of the lines of 

 the individual patterns are a function of the relative amount of the mate- 

 rial present in the mixture, so that the method also has quantitative aspects. 



'Atomic diameters vary from one element to the next; that is, the silicon atom as an ion has a dia- 

 meter of 0.8 angstroms (lA ^ lO"* cm.), calcium 2.0 A., potassium 2.66 A., etc. 



