be fine enough to diffract the much shorter X rays. But it occurred to Dr. Max Laue, of Zurich, that the reticular structure of crystals would supply the necessary grating, since the distances between the atoms in the space-lattices are of the order icr 8 cm. When, in 1912, this idea was put to the test a very surprising result was obtained. Plates cut from crystals parallel to certain faces were placed perpendicularly in the path of a thin pencil of X rays, and beyond a photographic plate was exposed. The resulting photograph (known as a Laue photograph or radiogram, Rontgenogram or Rontgen pattern, or spot photograph) shows a larger central spot representing the direct rays, whilst surrounding it is a symmetrical pattern of smaller spots. The spots may also be shown directly by projection on a screen of fluorescent mate- rial. This pattern shows the same degree of symmetry as that on the crystal face. Thus a plate from a hexagonal crystal of beryl cut parallel to the basal plane (i.e. perpendicular to the principal axis) shows a six-fold arrangement of spots symmetrical about six radial lines at 30; whilst when the plate is cut parallel to a prism face of the same crystal the spots are symmetrical about two lines at right angles. Fig. 5 is a reproduction of an actual photograph obtained by passing a pencil of X rays through a basal cleavage plate, 0-81 mm. in thickness, of the pseudo- rhombohedral chlorite, penninite. This photograph (after H. Haga and F. M. Jaeger, 1915) is selected on account of its comparative simplicity and the obvious three-fold arrangement of the spots.
The results obtained with these Laue photographs were at first explained as due to diffraction, but the problem is much more complex than diffraction by a single system of parallel lines in one plane, since we are here dealing with a lattice in three dimensions in which there are many series of lines in many planes. As explained by Sir William Bragg and his son Prof. W. L. Bragg in their book (X-rays and Crystal Structure, London, 1915; 3rd ed., 1918) it is due to the amplification of waves reflected from successive layers of atoms within the crystal. In fig. 6 a beam of X rays AB, A'B', A"B", all of the same wave-length X, strikes at a glancing angle the planes of particles, the distances between which are d (cf. fig. 3). They are reflected by successive planes as a single ray BC. Produce A'B' to D (then BD is perpendicular to the planes) and draw B N perpendicular to A'D. Then, since B'B=B'D and AB=A'N, the length of path of the ray A'B'C is greater than that of the ray ABC by the distance ND=2d sin 6. Similarly, A "B"C is longer than A'B'C by the same amount. If, now, this distance is equal to the wave-length of the rays, namely, if \ = 2d sin d, the rays reflected by successive layers of particles will be vibrating in the same phase and their amplitudes will be added together. If the glancing angle 9 be varied but slightly the reflections from the millions of layers will vary in phase and they will mutually interfere. But at certain other glancing angles 2 , 3 when 2\ = 2d sin 2 or 3\= 2d sin 03, there will again be an accumulative effect, giving reflections of the second and third orders. (See fig. 6.)
In the Bragg apparatus, called an X-ray spectrometer, homogene- ous (" monochromatic ") rays from an X-ray tube emerge through a narrow slit in a leaden screen and strike at a glancing angle the crystal plate mounted on a goniometer. The reflected beam enters an ionization chamber containing sulphur dioxide or methyl bromide and connected with an electroscope. The crystal is slowly turned on the goniometer until a maximum effect is noted in the electroscope, when the angle is read. Plotting the readings of the electroscope against those of the goniometer, a curve (X-ray " spectrum ) is obtained which shows a series of sharply defined maxima or peaks corresponding to reflections of the first, second, and other orders. Knowing the wave-length of the rays, the distance between the planes of particles can then be calculated from the above funda- mental equation; or alternatively, knowing the spacing of the planes, the wave-length of the rays can be determined. As an ex- ample, rays from a palladium anticathode (" palladium rays ") were strongly reflected from the cube face of rock-salt when the angle was 5-9, 11-85, and 18-15. Taking the spacing d between the cube planes of rock-salt as 2-81 Xio- 8 cm., the wave-length X is found to be 2X2-81 Xio- 8 sin 5-9 = 0-578X10-* cm., or 2\ = 2X2-81 Xio' 8 sin i-i85 = i-i54Xio- 8 cm.
To return now to an explanation of the spots shown by the Laue photographs. Here, instead of homogeneous rays, the rays employed are of mixed wave-lengths (as in white light). For such a bundle of rays reflected by a certain set of parallel planes (as ex- plained in fig. 6) there will be some of wave-length that will satisfy the equation X = 2d sin 0, or at jeast n\ = zd sin n . There will then be a reinforcement in the reflection of these rays from the particular set of planes. Let fig. 7 represent a plate of beryl cut perpendicular to the principal axis of the crystal, the upper and lower boundaries in the figure being then parallel to the basal plane. The rows of particles lie in the traces of two sets of planes respectively parallel to
two possible pyramidal faces of the crystal. Reflection from these will yield two spots on the photographic plate. Now, according to the hexagonal degree of symmetry possessed by beryl, there will be 6 (or 12) similar sets of planes equally inclined to the vertical axis, and corresponding to a hexagonal (or dihexagonal) pyramid; consequently 6 (or 12) similar spots will appear on the photograph equally distant from the centre. For other sets of 6 (or 12) planes inclined at other angles to the vertical axis of the crystal, and parallel to possible faces of hexagonal or dihexagonal pyramids, intensified reflections will take place for rays of other wave-lengths. The result will be a large number of spots on the photographic plate, but all of them in sets of 6 (or 12) symmetrically grouped around the centre.
Some of these Laue photographs are highly complex in appearance, but by analysis they can be reduced to simple crystallographic re- lations. Since each spot represents a structural plane in the crystal and also a possible external crystal-face, the series of spots lie in zones and their Millerian indices can be deduced. Further, it will be seen from fig. 7 that the distances of the spots from the centre are in direct relation to the inclinations of the various planes. Fig. 8 (after H. Haga and F. M.Jaeger, 1915) shows plotted on a stereo- graphic projection the series of spots of a Laue photograph on the face (oio) of anhydrite. The spots are here symmetrical with respect to two lines at right angles, corresponding with the ortho- rhombic symmetry of the crystal. The zone-circles are drawn in one-half of the diagram and the indices of the planes are given in one-quarter. It is thus possible to deduce from the Laue photo- graphs not only the zonal relations and indices of possible faces (many of which have not been observed as actual faces), but also the angles between these faces and the fundamental elements of the crystal. This information can even be obtained from an irregular fragment showing no external faces and of unknown orientation. Such a fragment is mounted on a two-circle goniometer and a series of Laue photographs taken in various positions; and a special instrument is provided for the analysis of the series of photographs.
A further point to be noticed in the Laue photographs (figs. 5 and 8) is that the spots are of different sizes and intensities (though spots repeated by the symmetry are, of course, identical). The stronger reflections are from planes of greater reticular density and indicate at once the important structural planes and the prominent faces of the crystal.
A third method of investigation has been devised by P. Debye and P. Scherrer in Germany in 1916, and independently by A. W. Hull in the United States in 1917. Here a beam of homogeneous (" monochromatic ") X rays of known wave-length is transmitted through the finely powdered crystalline material, and the reflections received on a photographic film. The tiny crystal fragments are in all manner of orientations; and to further ensure all possible orien- tations in the aggregate, the tube containing the small amount of powder is rotated during the exposure. For structural planes with the spacing d there are bound to be some of the particles in the posi- tion shown in fig. 6 in which the equation \=2d sin is satisfied: but these will be lying in all azimuths, i.e. sloping away in all direc- tions at the angle from the axis of the rays. The reflected rays will consequently lie on the surface of a cone, the angle of which is 40; and, instead of a single spot, a continuous series of spots forming a circle will appear in the photograph. Similarly, in other fragments the same set of planes with spacing d may be inclined at angle 0j giving a second order reflection as required by the equation 2\ = 2d sin 02, and producing a wider-angled cone concentric with the first. Further, other structural planes with spacing di and inclined at other values of will be provided by other fragments, giving still other conical reflections. Since, however, the experiment is per- formed with rays of one wave-length, it is only certain values of d that will satisfy the equation, so that the number of reflections is really limited. Even with this limited number, there would appear to be some difficulty in sorting out the -reflections of the different orders and those from different structural planes. Since it is only the angles of divergence of the concentric conical sheaths that are to be measured, all that need be photographed is a narrow strip through the centre. This strip is made semicircular, in order to embrace a wide field of reflected cones. Knowing and X, the equation gives, as in the Bragg method, the spacing d between the structural planes of the crystal.
Although the Debye-Scherrer method may be regarded as a modification of the Laue method, yet the results it gives are the same as those given by the Bragg method, namely the spacing be- tween the structural planes of the crystal. The Laue method gives other supplementary information, but it is mainly on the spacing between the planes of particles that ideas of structure arc built up. A large amount of experimental work on crystals of different sub- stances has been done in this direction, and deductions have been drawn as to their probable atomic arrangement. In this place only one or two examples can be briefly considered.
Rock-salt (sodium chloride) crystallizes in cubes and possesses a perfect cleavage parallel to the faces of the cube. Plates cut parallel to the faces of the cube (loo), the rhombic-dodecahedron (no), and the octahedron (in) respectively give by the Bragg method values for the spacing between the planes of particles in the ratio of I : I/ V3 : 1/V 3. These ratios are the same as those ment ioned above for the simple cubic space-lattice (figs. 1-3), and the conclusion
