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Rh CROZIER, WILLIAM (1855- ), American soldier (see 7.520), was detailed in 1912 as president of the Army War College and the following year was reappointed chief of ordnance with the rank of brigadier-general. He was made major-general, chief of ordnance, U.S.A., in 1017, and the provision of munitions in the World War was under his charge until Dec. 1917. He was then made a member of the War Council, and in the discharge of this office was in France and Italy for the first half of 1918. For the remainder of the year he was commandant of the N.E. Department, U.S.A., retiring from active service in December. CRYSTALLOGRAPHY (sec 7.569). The geometry of the ex- ternal forms of crystals may be said to have been completely worked out. The 32 crystal-classes differing from one another in their type and degree of symmetry and the six crystal-systems into which these classes can be grouped are now well established. The same is also true of the geometrical conceptions of the in- ternal structure of crystals (though a good general account is still wanting). It is known that there are 230 possible types of homogeneous point-systems and that these are referable to 14 kinds of space-lattices. Recent work has been in the direction of attempting to trace a connexion between the internal structure of crystals and their chemical constitution. Here there is ample scope for speculation; but since 1912, when X rays provided a new method of investigation, some real advance has been made. By this method it is possible not only to determine the internal structure of crystals, but also actually to measure the distance between the atoms.

Crystals consist of a homogeneous assemblage of particles, and these particles are marshalled in certain definite ways. The grouping around any one particle (except those on the bounda- ries of the crystal) is the same as that around every other par- ticle of the same kind. Further, the particles are arranged at regular intervals along straight lines. Throughout the structure there are several parallel sets of such lines, and these lie in several parallel sets of planes also at regular intervals apart.

An example of such a structure is the simple cubic space-lattice represented in fig. I. Here the particles (all of the same kind) are placed at equal distances, say a, along parallel lines in three sets at right angles; the distance between the parallel lines in each plane and between the parallel planes of lines being also a. That is, the particles are situated at the points of intersection of a system of lines that form a square network or lattice in three dimensions. Or the structure may be regarded as a stack of small cubes each with a quarter of a particle at every corner; the four adjoining cubes at each corner then providing the whole particle. In this grouping, any one particle is surrounded by a set of 6 similar particles at distance a; further, it is surrounded by 12 particles at distance V2 (i.e. the diagonal of the square); and by 8 other particles at distance V3 (i.e. the diagonal of the cube).

It is clear from fig. I that the three sets of lines are parallel to the edges of the cube, and that they lie in planes parallel to the faces of the cube. But it is to be noticed that the particles also lie in other sets of parallel lines, and that these lines fall in other sets of parallel planes. Certain of these additional lines and planes of particles are represented more prominently in fig. 2, which is drawn on a smaller scale with a larger number of particles (but to avoid confusion only those on the surface of the solid are marked). In this figure the three front edges of a portion of the main cube are truncated by planes of the rhombic-dodecahedron, and one corner has been cut off symmetrically by a face of the octahedron. (Since the octahedron face intersects both the cube and the rhombic-dodecahedron faces, its outline is hexagonal.) It will be seen that the several layers of particles parallel to any one of these faces are continuous over the other faces, although the particles themselves are ranged along lines of different directions. Hundreds of different planes of particles can, in fact, be traced out in such a structure; and it is important to remember that these structure planes are parallel to possible external faces on the crystal. A close relation exists between the Millerian indices of these faces and the number of particles along certain lines in the corresponding planes. The dotted lines on the front cube face in fig. 2 represent the intersections or traces of such planes with the indices: (ill), (2ii),_(3ii), etc.; (221), (321), etc.; (331), (431), etc. ; respectively for the lines from right to left. The seven planes of which the indices have just been given necessitate by symmetrical repetition the presence of 93 other structure planes, or, in all, 200 external crystal faces.

It will be further seen from a study of fig. 2 that the spacing between the particles is not the same on each of the faces (allowance being made for foreshortening in the drawing: only on the front

cube face are the particles represented at their true distance apart). On the cube faces the distances each way are, of course, a. On the faces of the rhombic-dodecahedron they are spaced at distance a in one direction, but along the second direction at right angles at distance V2a. On the octahedral face there is, instead of a rec- tangular grouping, a triangular and hexagonal pattern with the particles spaced at distances V2a in three directions. It follows therefore that the number of particles on each of the faces is not the same for equal areas. The network of particles is closer on the cube face than on the rhombic-dodecahedron, and more open on the octahedron. This " reticular density " of the different faces is a question of importance and is closely related to the cleavage of crystals. Minerals with cubic cleavage (e.g. rock-salt and galena) would be expected to be of this structure.

In addition to the spacing of the particles in the planes, there is also to be considered the distances between the planes themselves. This is represented in fig. 3 by mea-ns of vertical sections through the structure (fig. 2) perpendicular to the respective planes. In fig. 33 the spaces between the cube planes is, of course, a, and the particles are also spaced at distance a; the pattern being, in fact, that on a cube face perpendicular to the first. In fig. 3b the distance be- tween the rhombic-dodecahedron planes is given by half the diagonal of the cube face, namely a / V2, and the particles are at distances a apart. Here, however, the section-plane intersects lines of particles only in alternate rhombic-dodecahedron planes. In fig. 3c the distance between octahedron planes is given by one-third the diagonal of the cube, namely a / V3 ; and the particles are at distance V6a apart along the traces of the octahedron planes, though only at distances a or V2a across these planes. (In figs. 3b and 3c the sec- tion-plane is the same, since it is perpendicular to both the rhombic- dodecahedron and the octahedron, and the particles intersected are also the same; but to avoid confusion in the drawing the two sets of planes are separated in the two figures.) Other section-planes could, of course, be drawn perpendicular to the planes in question, but, whilst the distances between the planes would be the same, the spacing of the particles would be different.

In addition to the simplest type of cubic lattice discussed in some detail above, there are two other types. The three are represented together for comparison in fig. 4. In fig. 4b there is an additional point at the centre of each cube this may be called the centred cubic lattice; and in fig. 4c there are additional points at the centre of each face, giving the face-centred cubic lattice. The different relations afforded by these types need not be discussed here. But it may be pointed out that in the centred cubic lattice the greatest reticular density is in the rhombic-dodecahedron planes, whilst in the face-centred cubic lattice the particles are most closely packed in the octahedron planes. These would be expected to correspond to cubic crystals showing rhombic-dodecahedral and octahedral cleavage (e.g. zinc-blende and fluor-spar) respectively.

Types of lattices other than the cubic are deduced by varying the distances of the particles along the different axes and by varying the angles between these axes, in a manner similar to that in which the six crystal-systems are deduced. In fact the elements of the ele- mentary cells of the lattice, namely the lengths and inclination of their edges, are identical (except in certain cases) with the para- meters a:b:c and the axial angles a, /3 and 7 deduced from the ex- ternal crystal faces.

The " particles " referred to above may be crystal molecules, chemical molecules, or even atoms. They are represented in the diagrams as spots without committing ourselves as to their shape or size (in relation to their distance apart). Some authors represent them as spheres in contact with one another, regarding these as the spheres of influence of each atom. If the spheres are of equal size, the number of points of contact and the closeness of the packing will vary with the type of lattice. Or again, we may regard the par- ticles (all of the same size) as completely filling space. In this case the particles in the simple cubic lattice will be cubes, each in contact with six other cubes; in the centred cubic lattice they are cubp-octahedra with 14 surfaces of contact ; and in the face-centred cubic lattice they are rhombic-dodecahedra with 12 surfaces of contact.

The above outline of the geometrical structure of crystals has been necessary for the purpose of introducing the new X-ray methods of investigating the internal structure of crystals.

X rays, or Rb'ntgen rays, are propagated as waves in the same manner as rays of ordinary light, but they are of much smaller wave-length. The wave-length of yellow (sodium) light is 0-0000589 cm. (i.e. of the order io b cm.), whilst the wave- lengths of X rays are of the order to' 8 or io- 9 cm., or one thou- sand to ten thousand times smaller. The very fine rulings of parallel lines (about 7,000 to a cm.) of diffraction gratings being of a magnitude (io- 4 cm.) comparable with the wave-lengths of light, they produce well-known diffraction effects. It would be impossible to produce mechanically a grating which would