X-Rays and Crystal Structure

The Rotation Method.—For the rotation method originally due to de Broglie, an apparatus like fig. 8 is used The X-rays from a tube (not shown) pass through the fine barrel aperture and fall on the crystal, which is rotated uniformly on the spindle, driven by the motor. As each crystal plane comes into a reflecting position, the reflected ray registers on the photographic plate, producing a pattern such as shown in fig. 9. Each plane re flects four times per revolution, giving the photograph a sym metrical appearance. From the position of the spot on the plate, the spacings or the indices of the planes can be calculated. An important property of the rotation method is that if the crystal is rotated about a translation a of the lattice, all the reflected rays lie on a set of cones of angles where sincp= these cones intersect the plate in hyperbolae called layer lines (see fig. 9) and their distance apart is the best measure of the true axes of the crystal. It can be seen from the photograph that the spots are of very different intensities. These are usually estimated by eye, as strong, medium, weak, etc., so that the method is definitely in ferior in this respect to the ionisa tion spectrometer.

The Powder Method.—In the powder method, due to Debye and to Hull and Scherrer, a crys talline powder is used instead of a single crystal, depending on the fact that among so many grains some will be in the exact position to reflect. Consequently from the central beam diverge a set of cones each corresponding to the reflection from a single plane and forming circular rings on a plate (see fig. roA) or so-called Debye curves on a film bent round a cylinder (see fig. roB). Unfor tunately, many planes often have the same or nearly the same spacings, which causes them to be confused and limits the method to simple crystals. However, as it is the only method for sub stances which cannot be obtained as single crystals, it is very use ful, particularly for metals.

The Laue Method.—This de pends on a different principle from the first three. Here the crystal is kept fixed, and a beam of white X-rays, that is with wave lengths ranging from between .25 to .5 A. (Angstrom units = cm.), is passed through it, usually parallel to a crystallographic axis. The apparatus used is the same as before (see fig. 8). For a great number of planes there will always be some wave length which is right for reflection and a spot is formed, the position of which depends only on the angular position of the reflecting plane.

Thus the Laue method gives no information as to spacing. But, as can be seen from the photographs (see figs. i ra and r lb) it gives an excellent picture of the symmetry of the crystal and also of the relative intensities of a great number of reflecting planes.

Plainly, though any one of these methods but the last could be and has been used alone for crystal analysis, it is much better to use all to amplify and check each other's results. In this way we arrive at the experimental data for crystal analysis; the spacings and intensities of the X-ray re flections of a number of planes of known indices.

Stages in Structure Analy sis.—With these data the actual analysis divides into two parts. It is carried out schematically as follows: I. Determination of Cell Size.

The lengths of the three axes, a, b, c, are found from rotation photographs or from the spacing of planes given by the formula for orthogonal crystals. For monoclinic and triclinic crystals it is more complicated and here the angles between the axes are usually obtained from the external crystal form. It is easiest to use the pinacoid spacings (ioo) (cio) (ooi), which give the axes directly, but these may be halved, so for the true cell, one general plane (hid) at least must be taken.

2. Determination of the number of molecules per cell.

If D is the density of the crystal, and V its volume in cu.A. (V =abc for an orthogonal cell), then the cell will contain Z molecules of molecular weight M, where ZxMX 1.66 =DXV. (1.66X gm. being the weight of an atom of hydrogen.) 3. Determination of the Lat tice Type. This is done by means of the halvings of planes of the type (hkl). In simple lattices there are no inner regularities and all kinds of planes appear, whereas in centred lattices, cer tain planes are inter-spaced iden tically so that they do not reflect in odd orders and are said to be halved. A halving when HA is odd will, for instance, indicate an a c face-centred lattice r,„' (see fig. 4 [xiii.]).

4. Determination of Space Group.

It is first necessary to know the crystal class. This must be done by the methods of ordinary crystallography. Laue photographs are useful in detecting axes, but unfortunately, X -ray methods cannot distinguish directly be tween crystals with or without a centre of symmetry. Next, screw axes and glide planes can be detected, for the former cause all orders of the plane normal to it to disappear except that corre• sponding to a multiple of the screw translation—e.g., in quartz, owing to the trigonal screw axis, perpendicular to the c plane, only the third, sixth, ninth, etc., orders of this plane reflect, Glide planes, on the other hand, halve whole sets of planes of the type (hko), where h-l-k is odd. Space groups are usually found by the use of tables, of which those of Astbury and Yardley and Wyckoff are most used. They are denoted by a complex symbol, such as where D6n stands for the symmetry class and 4 is the ordinal number in a space group table, usually that of Schoenflies.