ART. VIII.-An Outline of the Application of the Theory of Space Groups to the Study of the Structure of Crystals; by RALPH W. G. WYCKOFF.

The theory of space groups' defines all of the ways of symmetrically arranging points in space. It thus supplies the basis upon 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 Nishikawa in studying spinel 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 extraneous, 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 determination of the structure of crystals. It has been written with special reference to the accompanying determination of the structure of magnesium oxide.5

Entirely independently of this development, various authors have commented 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.

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The great advantage in using space groups lies in the

1 L. Sohncke, Entwickelung einer Theorie der Krystallstruktur (1879); E. Federov, Z. Kryst., 24, 209, 1895; A. Schönflies, Krystallsysteme und Krystallstruktur, 1891; W. Barlow, Z. Kryst., 23, 1, 1894. Federov's work appeared in Russian before any of the other contributions. The last three studies are compared by H. Hilton, Mathematical Crystallography, 1903. * S. Nishikawa, Proc. Tokyo Math. Phys. Soc., 8, 199, 1915.

Ralph W. G. Wyckoff, J. Am. Chem. Soc., 42, 1100, 1920; Phys. Rev., (2), 16, 149, 1920; this Journal, 50, 317, 1920.

A more orderly discussion of the entire method will probably appear shortly in book form.

B

Ralph W. G. Wyckoff. See the following article.

Such as A. Schönflies, 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 determination of the structure of crystals.

fact that if in a particular case the number of molecules associated with the unit of structures and the symmetry are known, all of the possible atomic arrangements can be written down and considered in the light of further X-ray measurements. It can then be told, with a given amount of experimental data, whether the particular structure under examination can or can not be uniquely defined.

The Theory of Space Groups."

The geometrical theory of space groups can be developed in the following way. If all of the n operations of symmetry that are characteristic of some one of the thirty-two classes of crystal symmetry are made to operate upon a point in space, n points will result which are all crystallographically equivalent. The n equivalent points arising from the operations of symmetry of one of the crystal classes, or these operations themselves, can be taken to define one of the thirty-two point groups. To take a simple example: the holohedry of the monoclinic system possesses two elements of symmetry, a 180° axis which will be taken to coincide with the Z-axis and a plane of symmetry at right angles to this axis (the XY plane). If these two elements of symmetry are caused to operate upon any point in space, three other crystallographically equivalent points will result. The four symmetry operations are (using Schönflies "10 notation): 1, A (TT), Sh A (7) Sh where

1 (the identity) may be thought of as a rotation of 2 π, A (T) is a rotation of angle,

S is a mirroring against the horizontal (XY) plane and the product A (T)S is the combination of the rotation A(7) and the mirroring S. 1 and A() correspond to the operation of the 180° axis, Sh and A (T) are mirrorings of 1 and A in the XY plane. In figure 1 where Z is the 180° axis and XY the mirroring plane,

This, of course, can be told from X-ray spectroscopy.

In descriptive crystallography we are accustomed to consider the elements of symmetry-axes, planes and centers-to be of chief importance. For studying the internal structure of crystals, at least, it is much better to think of particular grouping of points (or atoms) as characterizing the different classes of symmetry. The elements of symmetry of these various arrangements then become interesting, but for most purposes unnecessary and complicating, details. Those who are well versed in the more customary crystallography are asked to read the following discussion from this other point of view, forgetting for the time being all associations with planes, axes and the like, except as they may be introduced into the argument.

10 A. Schönflies, op. cit.