INDUSTRIAL A N D ENGINEERING CHEMISTRY
July, 1924
689
Crystal Analysis by the Diffraction of X-Rays’ By R. A. Patterson RENSSELAER POLYTECHNIC INSTITUTE, TROY,N. Y.
EDITOR s NOTE.-Because of the growillg importance of crystal analysis by the diffraction of Xrays-a method that has already found practical application in the industries--we are presenting this discussion and that by c E. Bain for the information of our readers and in the hope that others may be stimulated to engage upon similar research in the application of its results.
Sir William Bragg and his son, w. L- Bragg, were the first to determine the exact arrangement Of the atoms in rock salt, as illustrated in Fig. 1. The distance between the atoms was computed from the known density and atomic masses of sodium and chlorine. It was but a step further to establish the fact that X-rays had a wave length about one-ten thousandth that of ordinary light. To these two investigators we owe the following simple explanation of the diffraction of X-rays by a crystal.
Following the discooery by oon Laue that in crystals we haw a natural diffraction grating for X-rays, Bragg and others haoe eoolued simple methods of computing the distances between atoms in crystalline substance. Hull’s method of applying X-ray diffraction to the analysis of crystals, especially those of metals, is described. .By this method it is possible to measure the distances between the different layers of atoms in the crystals, and, by carrying it further, to determine the arrangement of the atoms in the crystal. An illustration of this method applied to copper and copper-manganese alloy is gioen’
-ILAYS have revealed the number and motion of electrons within atoms and the configuration of atoms in crystals. The purpose of this article is to explain in a simple and nonmathematical manner how the distances between atoms in crystals may be determined.
X
X-RAYS-WHAT ARE THEY X-rays are identical in nature with ordinary light except for wave length. Whereas light waves have a length of 0.00006 cm., X-rays are ten thousand times shorter. They are produced when swiftly moving electrons are suddenly stopped by collision with a heavy metal. In the Coolidge X-ray tube the electrons are supplied by a heated filament similar to that used in radio tubes. These electrons are driven against a tungsten “target” by applying to them a very high electromotive force. Under 100,000 volts they strike the target with a maximum speed of over 100,000 miles per second. At the collisions electromagnetic disturbances are radiated in all directions from the target. These are X-rays. The higher the voltage, the harder the collision and the more penetrating the rays, To their shortness in wave length is due the great penetrating power of X-rays.
X-RAYs The great advantages of diffraction gratings in the spectral analysis of ultraviolet and visible light are well known. Such gratings consist of a large number (20,000 per inch) of equally spaced lines ruled on some polished surface by means of a diamond. A diffraction grating pic. l-ARRANGEMGNTOF for X-rays would require a diamond ATOMSIN ROCKSALT point so fine as to rule lines an The white circles represent atom’s width apart. This is of chlorine atoms; the black ones, impossible, sodium atoms; or vice versa The distance dloo = 2,814 I n 1912, however, 11.von Laue called attention to the fact that ( l O ) - S cm. nature herself had provided such a grating in crystalsa For here we have regularly spaced atoms (or molecules) the proper distance apart. To be sure, these scattering centers are distributed in space in three directions, instead of one, as are the lines of a simple diffraction grating. But this merely complicates the general mathematical analysis of the diffraction patterns obtained. The action that takes place is essentially the same as that with which we are familiar in the diffraction of light. 1
Received May 16, 1924.
A-
FIG 2-REFLECTION
OF
X-RAYSBY CRYSTALS
THEORYOF X-RAYDIFFRACTION Let us consider what happens to a beam of X-rays which falls upon a crystal as in Fig. 2 . When the incident beam, I , reaches the first layer of atoms, AA’, these atoms may be regarded as stirred into vibration and, therefore, as themselves sending out little wave disturbances in all directions. The energy thus scattered in any one direction by a single atom is very small. There is, however, a direction, R, in which all the wavelets scattered by these atoms will meet in like phase and reinforce each other. This direction is such that the glancing angle, 8 , between the incident beam and AA’ will equal the angle between the direction of R and AA’, also 8. This is, of course, merely the mechanism of the reflection of light from a plane, smooth surface. The wavelets sent out in other directions by the atoms in AA’ meet in such phase as to neutralize each other. We may therefore regard the incident beam, I , as partially reflected by this layer of atoms according to the ordinary law of reflection. X-rays, however, are very penetrating, and consequently, little energy is reflected by the layer AA’, although this reflected energy, as in the case of light, may be of any wave length (or color). Most of the incident energy passes on through the crystal to the layers BB’, CC’, DD’, EE’, etc. Each of these successive layers will also reflect a portion of the incident energy in the direction of R. The waves reflected in this direction from the successive layers will reinforce each other only if they meet in like phase. The necessary and sufficient condition that they meet in like phase is given by the equation
690
INDUSTRIAL AND ENGINEERING CHEMI$TR Y
nX = 2d sin 8 where X = wave length of incident rays n = 1, 2, 3, . . . . . . order of interference d = distance between successive layers
Only one particular wave length, therefore, will be reflected in the direction of R with sufficient intensity to be detected, the other wave lengths being neutralized by interference between waves reflected from different layers. Consequently, there are two respects in which this picture of the reflection of X-rays differs from that of ordinary light: first, the reflection is not a surface layer phenomenon; second, only one wave length (or color) is reflected at each angle of incidence. If, then, X-rays are incident upon a crystal for which d is known, the wave lengths of the incident rays may be determined by measuring the angles a t which the reflections take place. Or, if the incident rays are of known wave length, X, then the spacing of the atomic layers within the crystal may be determined by again measuring 0. The former constitutes the analysis of X-ray spectra; the latter, the analysis of crystal structure. The firbt delves into the inner secrets of atoms; the second reveals the configuratim of atoms in crystals, HULL’S EXPERIMENTAL METHOD
It is beyond the scope of this article to describe or compare the several methods of applying X-ray diffraction t o the analysis of crystals. One only has been selected, largely because it is in wide use in this country and is particularly adapted to the crystal analysis of metals.2 The method W. Hull. chosen is that developed by -4.
Vol. 16, N Q . 3
distance between these particular layers in the crystal’s structure. Some of the incident X-ray energy will then be reflected and intensified by interference a t an angle 8 to AA’ and will register a photographic impression a t P. This impression will be nearly an image of the specimen X. Since the equation may be written in the form d_ = - =x x n
2 sin 8
2 sin
(&)
the distance d between the successive layers of atoms that produce the image may be computed by measuring the distance ~t: from the shadow cast by the specimen a t S’ to the line P , and substituting it in the foregoing equation. We thus determine the spacing between one set of layers of atoms in the specimen.
(a)
’\
(b)
(C 1
ARRANGEMENT OF ATOMSIN CUBICAL CRYSTALS ( a ) Simple cubic; ( b ) body-centered cubic: ( 6 ) face-centered cubic. In ( a ) and (c) t h e dotted lines show the 111 planes; in ( b ) they show the 110 These three sets of planes are the principal planes in each type planes of lattice. F r o . 4-POSSIBLE
Each of the minute crystals, however, has many sets of parallel layers of atoms, each set having a different spacing, or value of d. For instance, in Fig. 4 the three possible arrangements of atoms giving rise t o cubical symmetry in crystals are shown. The planes of atoms parallel to the faces of the cubes are called the 100 planes, using Miller’s indices. Their separation in each case is represented by dloa. I n the simple cubic and face-centered cubic arrangement, the dotted I n the bodylines show the 111 planes separated by &I. centered cubic arr‘angement, the 110 layers are indicated b y dotted lines and their spacing, dllo. I n each arrangement: d1o0, dllO,and dill differ from each other. Consequently, the fundamental equation requires that reflections from these different spacings take place at different angles. If, then, the specimen consists of a very large number of minute crystals, cubical in their symmetry and oriented in all possible manners, some of them will be properly oriented to reflect rays from their 100 planes, others to reflect rays from their 110 planes, and so on for all the possible sets of planes. But the reflections from different sets of planes will take place a t different angles and give rise to different iriterference images along the film. There will be several such images on the film, each one due to a certain order of interference from a definite set of layers of atoms in the crystal. The spacing for each set of planes can be determined as described above for the particular image, P. As a matter of convenience, a scale is provided which gives directly the spacing (divided b y the order of interference) due to any line on the film when the shadow cast by the specimen is placed opposite the zero point of the scale. We can thus measure the distances between the different layers of atoms in crystals.
‘ i
FIG.3-ARRANGEMENT O F APPARATUS I N HULL’SMETHOD T h e X-ray tube is a t the left. W conceals the hot filament which emits t h e electrons. T i s t h e target against which t h e electrons collide, generating X-rays.
The arrangement of the apparatus is schematically shown in Fig. 3. A narrow beam of X-rays, limited by the slits, LI and L,, irradiates the specimen, X, to be analyzed. This specimen may be a small block of metal, a thin ribbon or wire, or a cylinder or slab of finely powdered material (not necessarily metallic). I n all these cases it consists, as is well known, of a multitude of minute crystals heterogeneously oriented with respect to one another. A photographic film, FF’, is bent in the arc of a circle at a distance D from the specimen S . By proper voltage control and filtering screens it is possible to render the rays that finally reach the film practically monochromatic, or of a single wave length. Suppose, then, that in this specimen a large number of discrete crystals are so oriented that they present a series of equally spaced layers of atoms parallel to AA’, and that the angle 8 between the incident beam and AA’ is such that the fundamental condition nX = 2d sin 8 is satisfied for the particular wave lengths used. The symbol d, of course, represents the 2 E C Bain describes results already obtained for metals in a n article on page 692 of this issue.
ARRANGEMENT OF THE ATOMS One might‘assume the analysis complete a t this point. It is very desirable, however, to have a clear picture of just how the atoms are situated with respect to one another. It is easy to obtain this picture in the case of cubical crystals in which the atoms are all alike, but considerable difficulty is experienced in crystals of lower degree of symmetry and in
INDUSTRIAL AND E,VGl 'NEERI.VG CHEMISTRY
July, 1924
the case of compounds in which there arc two or more different kinds of atoms. A. \$'. Hull and W. P. Davey have devised charts by which a solution of this prohlem in the case of crystals belonging to the cubic, hexagonal, and tctragonal systems may be quickly and easily obtained. In the case of compounds, the heavier atoms are more effect.ire in scattering (or reflecting) X-ray energy than the lighter ones. Consequently, differences in intensity in the interference patterns on tho films become significant. Consideratioti of these intensity effects is usually necessary in arriving at the arrangcmcnt of atoms in crystalline compounds. But it is again beyond the seope of ihis art.icle to deal with I~irialyses involving these i~ii,nplii~i~ti~j,i~. Returning tu the specimen S , which TW have dread iinicd to be coinposed of crystals with cuhical sgmmet.ry, kt us forthor assume that its atoms are alike, or iiearlp so. Wlint is their confignration?
691
tically equal in reflecting power. As the X-rays nscd consisted of two slightly different wave lengths (0.708 and 0.712 (lo)-* em.), each image on the film is a double instead of a single linc. I t is at once apparent that both the alloy and copper pniduce the same pattern and tlicrcfore possess the same crystal lattice. A comparison with the patterns in Fig. 5 shows tlie lattice to be of the face-centered t,ype. Tlie displacemeot of cadi of tlie lines in tlie lower half bo tlic left of the corresponding line due to copper slrows that the spacing o l tlie layers of atoms in tlie alloy is greater than in the pure nret,al. In fact, mcasorement gives 3.00 em. fur t,he side of tlie clemcntary cube in copper, slid 3.70 (10) 8 cm. in the alloy. Evirleiitly, wlicn manganese atom.: are added t o copper tlicy rcplace copper atoms ill the spnw Iatbioe of the 1a.tter element a i d a t th nrc t,irne extend it- --tireixipper lattice acting BY a solvent.
fiO1.UTIiJX 01: CVBlC.41, CRYSTAIS
Tlir atoms of cubical cryitnls can have only three possiblo corrfiauratiuns. They arc slioirii in Fig. 4, tlie simple cubic, body-rcntored cubic, and face-centered cubic. I ' h m the of these confignratians it is easily slimsrr that. t.he following~ratiushukl true: Simple cubic:
d*o:d~,.:d,,, = i
Body-centered cubic:
d , o o : d , l o : d , ~= t 1:
Face-centered cubic:
a
I
1
: -
4 4
1 v'3 1 2 diou:diiu:diil = 1 :---: : -
I I I / lIlll\
42:
~
531 a i o 21 I (2) Omnrnlin P-HOM CoPPSn A N D C " P P L U . M * N O * X E S I
811 I l l CY1
1111 PIC. 6-PX"T"
