STRUCTURE OF CRYSTALS — WYCKOFF. 205 



the reflections from the most important faces of a number of crystals, 

 such as sodium and potassium chlorides, iron pyrites, the diamond, 

 carborundum, and various members of the calcite group of minerals. 13 

 Probably in no case can the structures which are thus obtained be said 

 with absolute surety to correspond with the arrangements of the 

 atoms within these crystals. 



The general method of studying the structure of crystals. — It con- 

 sequently becomes necessary, if we are to make any definite and 

 certain progress toward unraveling the structures of cystalline com- 

 pounds, that a method be developed which will make it possible to 

 determine uniquely such structures, or at least that will indicate the 

 degree of probability with which a particular structure has been 

 determined. Fortunately the basis for such a method was already 

 prepared, for the geometrical theory of space groups developed many 

 years ago by Federov, Schonflies, and Barlow can be made to give 

 all of the possible ways of arranging points in space so that the group- 

 ing which results will exhibit crystallographic symmetry. 14 Since a 

 crystal is an orderly arrangement of atoms in space, it must corre- 

 spond with one of these space groups. Consequently, when the space 

 groups are given a suitable analytical representation, a means quite 

 independent of any X-ray experiments is provided for writing down 

 all of the possible positions which the atoms of any compound can 

 occupy. After this has been done- the particular data which will 

 serve to distinguish between these various possible arrangements 

 can be selected and those methods of experimentation employed which 

 will yield most readily the needed facts. Such a method has the 

 advantage of being equally applicable to complicated and to very 

 simple structures. The point of view involving such a use of the 

 theory of space groups was used first by Nishikawa in studying 

 spinel. 15 The geometry of the method arising from it, which has 

 been in the course of development for several years, 18 may now be said 

 to be nearly completely developed. 



THE METHODS OF OBTAINING DIFFRACTION EFFECTS FROM CRYSTALS. 



The spectrometer method. — Three ways of obtaining definite X-ray 

 diffractions have been mentioned: (1) By reflecting X rays from 



13 W. H. and W. L. Bragg, op. cit., chaps, vii and viii. C. L. Burdick and E. A. Owen, 

 J. Am. Chem. Soc., 40, 1749, 1918 ; C. L. Burdick and J. H. Ellis, ibid., 39, 2518, 1917 ; 

 R. G. Dickinson, ibid., 42, 85 (1920), and others. 



"E. Federov, Z. Kryst, 24, 209, 1895. A Schoenflies, " Krystallsysteme u. Krystall- 

 struktur " (1891). W. Barlow, Z. Kryst., 23, 1, 1894. 



16 S. Nishikawa, Proc. Tokyo Math. Phys. Soc, 8, 199, 1915. 



."This development has been going on independently in Germany and in tbis country. 

 For instance, see P. Niggli ; Geometrische Krystallographie des Discontinuums (1919); 

 Ralph W. G. Wyckoff, J. Am. Chem. Soc, 42, 1100, 1920; Am. J. Sci., 50, 317, 1920; ibid., 

 1, 127, 1921. 



