R. W. G. Wychoff— Theory of Space Groups, 3 2T 



Art. YIII.— ^^ Outline of the Application of the 

 Theory of Space Groups to the Study of the Struc- 

 ture of Crystals; by Ralph W. G. Wyckoff. 



The theory of space groups^ defines all of the ways of 

 symmetrically arranging points in space. It thns 

 supplies the basis npon which an entirely general method 

 for the study of the structures of crystals can be built. 

 Such a method has been in the course of development for 

 several years ; in its earlier stages it was used by Nishi- 

 kawa in studying spineP and other crystals, and has 

 been employed by the writer^ for the last three years. 

 It has now reached a degree of completeness such as to 

 be generally applicable to the problem of determining 

 complicated structures of crystals.* A large amount of 

 material, which is for the present purposes quite extran- ^ 

 eous, is involved in the development of the theory of space 

 groups. The present paper is an attempt to present only 

 those details which are required in order that the results 

 of this theory may be immediately applicable to the deter- 

 mination of the structure of crystals. It has been 

 written with special reference to the accompanying deter- 

 mination of the structure of magnesium oxide.^ 



Entirely independently of this development, various 

 authors^ have connnented upon the connection between 

 the space groups and the structures of crystals as deduced 

 from their effects upon X-rays. Niggli has recently 

 made quite an extended application to the determination 

 of crystal structures.'^ 



The great advantage in using space groups lies in the 



^L. Sohiicke, Entwickehmg einer Theorie der Krystallstruktur (1879) ; E. 

 Federov, Z. Kryst., 24, 209, 1895; A. Schonflies, Krystallsysteme imd Krys- 

 tallstruktur, 1891; W. Barlow, Z. Kryst., 23, 1, 1894. Federov's work ap- 

 peared in Russian before any of the other contributions. The last three 

 studies are compared by H. Hilton, Mathematical Crystallography, 1903. 



' S. Mshikawa, Proc. Tokyo Math. Phys. Soc, 8, 199, 1915. 



^Ealph W. G. Wyckoff, J. Am. Chem. Soc, 42, 1100, 1920; Phys. Eev., 

 (2), 16, 149, 1920; this Journal, 50, 317, 1920. 



*A more orderly discussion of the entire method will probably appear 

 shortly in book form. 



'Ealph W. G. Wyckoff. See the following article. 



®Such as A. Schonflies, Z. Kryst., 54, 545, 1915; A. Johnsen, Physik. Z., 

 16, 269, 1915. 



■^ P. Niggli, Geometrische Krystallographie des Discontinuums, 1919. His 

 development is not, however, entirely complete, and is carried out from a 

 point of view which does not seem to be the simplest and best for the deter- 

 mination of the structure of crystals. 



