unit prism will contain the following equivalent positions. The unit of groups having this lattice, with the primitive translations 2Tx; Ty, Tz; Ty, Tz, is best considered as a prism, two of whose sides are centered (figure 4a). There are then two crystal molecules in the unit. The coordinates of the equivalent positions are those of the equivalent points about a corner of the lattice (the origin) and those of the crystal molecule about a lattice point which is at the center of a side. Thus since the coordinates of this second lattice point are 0, Ty, Tz, the equivalent positions within the unit prism are:

xyz;

X, Ty+Y, T2+2; X, Ty−Y, T2+2; X,τy — Y, τz — Z; x,Ty+Y,T2−2.

Ꮓ

The positions of the equivalent points in the unit of structure can be obtained in a similar manner for any space group.

The Significance of the Space Groups in the Study of the Structure of Crystals. If an atom occurs at a general position x, y, z within the unit of structure, then in order that the conditions of symmetry may be fulfilled there must be as many more atoms of the same kind in the unit prism as there are equivalent points. For instance in the case of the holohedry of the cubic system the unit cube of a space group having the simple cubic lattice (T) as a basis has 48 equivalent positions contained within it; the unit cell of a space group having the face-centered lattice (F) with four points of the lattice (crystal molecules) associated with it, has 192 equivalent positions. In the first of these two cases, if an atom, say an oxygen atom, has a general position (xy) within the cube, then there must be 47 other oxygen atoms of the same sort within the unit; or in the latter case there would have to be 191 other similar oxygen atoms.

In actual practice, for the present, we are dealing with simple compounds having relatively few atoms in the chemical molecule; X-ray spectrum measurements seem to indicate at the same time that only a small number of chemical molecules are associated with the unit cell. Consequently but few atoms of the same kind (less than the number of general equivalent positions) occur within the unit and, as a result, these atoms must take up special

positions such that two or more of the general equivalent points have the same position. If, for instance, the coordinates of a point in the unit are such that it lies in a plane of symmetry or on a trigonal axis, two or, in the second case, three sets of coordinates three sets of coordinates of the most generally placed equivalent points will coincide. In the case of the space group C2 (fig. 3) if ≈ is made equal to T. that is, to one half of the height of the unit prism, then the four equivalent points of the unit occupy two equivalent positions (M coincides with M" and M' with

IZ

FIG. 3. -A portion of the monoclinic space group Cah'. The point group C ̧1 is placed at the points of the monoclinic lattice T, as at points A, A, B, C, etc. OADBGCFE is the unit of structure. M (x y z), M' (x = BD

--

− x = x; y = AD y = 2ty — y = y; z), M'' (x y z), and M the equivalent points within the unit.

x = 2Tx (xyz) are

M'''). Again sodium chloride crystallizes in the holohedry of the cubic system. Since there are as many as 192 equivalent positions within the unit cube, there would in the most general case be as many as 192 sodium atoms and an equal number of chlorine atoms in the unit, if all the sodium atoms are alike and if all the chlorine atoms are also alike. X-ray spectrum measurements, however, seem to indicate that but four chemical molecules are associated in this case with the unit cell. If this is true, then, the sodium atoms and the chlorine atoms must

occupy such special positions that the 192 equivalent points reduce to four.

It is thus seen that with compounds which crystallize in the systems of higher symmetry, these special positions become of the utmost importance. A knowledge of all of these special cases is highly desirable as an aid in determining the structure of crystals. Niggli1 records the simpler cases."

17

双下

0

C

丸

FIG. 4 (A). The side-centered monoclinic unit of structure, Tm'. O (000) and P (0, Ty, T2) are taken as the two points associated with the unit.

FIG. 4 (B). The simple cubic unit of structure, Te. O (000) may be taken as the single point of the lattice associated with the unit cube.

FIG. 4 (C).The body-centered unit of structure, T.". O (000) and P,,, (Tx,

Ty, T) are the lattice points associated with the cube.

FIG. 4 (D).-The face-centered unit of structure, T. O (000), P (0, Ty, Tz), P, (7x, 7, 0) and P,, (7x, 0, 7y) are the lattice points associated with the unit.

16 P. Niggli, op. cit.

17 In the course of the development of a generally useful method for studying crystals, the writer has been engaged for some time in working out all of these special cases and expects to be able to present them in the near future. Some of the results to be given in the following paper are based upon this work.

The positions of the atoms in the crystal molecule are defined by having as many groups of equivalent points associated with each point of the lattice as there are kinds of crystallographically different atoms in the unit. This number will be as great as, and may be greater than, the number of different kinds of atoms in the chemical molecule. For instance, in the case of calcite's the positions of the carbon atoms must be assigned to at least one set of equivalent points, the positions of the calcium. atoms to another set and the oxygen atoms to still another. With two chemical molecules associated with the unit cell, it might be conceivable for the two carbon atoms and the two calcium atoms to be alike and for all six of the oxygen atoms to be crystallographically alike, for four of them to be alike and two different,19 or that there should be three sets of two like atoms or two sets of three that are alike. There might thus be as many as seven different groups of points associated with each unit of calcite.

The manner of obtaining, with the aid of the theory of space groups, all of the crystallographically possible ways of arranging the atoms of a compound in the fundamental unit has been illustrated in detail in dealing with calcite. So detailed an application of the theory to the case of magnesium oxide, which follows, is not possible here because of the large number of space groups that must be considered.

Summary.

Such details of the theory of space groups as are of importance in the application of this theory to the determination of the structure of crystals are briefly considered. Point groups, space lattices and space groups are illustrated by simple examples. The relations between space groups and crystals is discussed and those modifications in the results of the theory of space groups that are required in order that it may serve as the basis for a general method for the study of the structure of crystals, are indicated.

Geophysical Laboratory,

18

Carnegie Institution of Washington,

Washington, D. C.

October, 1920.

Ralph W. G. Wyckoff, this Journal, (4) 50, 317, 1920.

Some of these possibilities are actually ruled out by considerations of symmetry.

ART. IX.-The Crystal Structure of Magnesium Oxide; by RALPH W. G. WYCKOFF.

Previous measurements on powdered magnesium oxide1 have led to the conclusion that this compound has the same crystal structure as sodium chloride. Such a

FIG. 1. The Laue photograph obtained by passing the X-rays in a direction roughly normal to the (100) face of magnesium oxide.

study did not, however, furnish a unique solution. The following determination is based upon the general method for studying the structures of crystals which has been developed from the theory of space groups and which is given in a very general form in the preceding article."

In the case of a cubic crystal there is no uncertainty as to the choice of those coordinate axes which will give the simplest unit of structure. The number of chemical

2

W. P. Davey and E. O. Hoffman, Phys. Rev., (2), 15, 333, 1920.

Ralph W. G. Wyckoff, see the preceding article in this number.