tation of the data, thus considerably displacing the position of the line in question. 

 If the probable constitution of the mixture can be surmised, direct comparison 

 with standard patterns will immediately disclose such situations, and errors 

 and time-consuming labor are avoided. 



Occasionally unit-cell data are available in the literature when powder data 

 are lacking (Donnay, 1954). Unit-cell data for a known material can be used 

 to establish the identity of an unknown material from which a powder-diffraction 

 pattern has been obtained. This method is practicable only if some clue suggests 

 the identity of the unknown, and the number of known materials to be compared 

 with the unknown is small. The comparison of the unit-cell data with the 

 powder-diffraction data is accomplished by application of the reciprocal-lattice 

 concept. A complete explanation of this concept is, of course, beyond the scope 

 of this chapter and the reader is referred to other sources (Clark, 1955; Davey 

 1934; Bunn, 1946). However, it can be shown that the relationship between 

 the true lattice (real space) and the reciprocal lattice (reciprocal space) can be 

 expressed by the equation 



R\ 



a = 



where d is the interplanar distance in the true lattice, d'" the interplanar distance 

 in the reciprocal lattice, A the wave length of the radiation used, and R a con- 

 stant called the "magnification factor" applied to convert the dimensions in re- 

 ciprocal space to such a magnitude that the reciprocal lattice or net can be 

 plotted easily in cm-units. If the unit-cell dimensions are not much over 10 A, 

 the value R — 10 will produce a reciprocal net of convenient dimensions. If the 

 unit cell has dimensions between 10 and 30 A, a value of R = 20 should be 

 chosen. Briefly, the procedure is the following: the a, b, and c dimensions of 

 the unit cell are converted into reciprocal-cell dimensions by means of the 

 equation above and the resulting three-dimensional net plotted in one plane by 

 folding the vertical planes down into the horizontal plane (fig. 9-11). There- 

 upon, the experimentally determined powder-diffraction data are also converted 

 into reciprocal dimensions by the same equation and the results (rings repre- 

 senting the ends of reciprocal space vectors free to turn about the origin) are 

 superimposed on the reciprocal net of the unit cell. If the unit cell fits the experi- 

 mentally determined powder-diffraction data, there will be a net intersection at 

 the end of each vector; i.e., the rings derived from the powder data will all pass 

 through one or more intersections of the three-dimensional reciprocal unit-cell 

 net. 



A mixture of minerals which are frequently difficult to differentiate by 

 optical examination, especially when examined in the form of a rather fine 

 powder, has been chosen to illustrate this method. Owing to the nature of these 



170 



