Y
The Chemistry Student SOME ELEMENTARY PRINCIPLES OF X-RAY CRYSTAL ANALYSIS. I Olro R E I ~ HAssocrilrs , EDITOR
The student of general chemistry hears frequent mention of the X-ray analysis of crystals. Perhaps he receives rather vague and generalized accounts of the principles involved in the making of such analyses. But he has great difficulty in finding any clear and detailed explanation of how the thing is actually accomplished. His instrnctor will tell him, quite truthfully, that the time allotted to a general chemistry course does not permit any rambling into interesting bypaths. If he refers to treatises and textbooks he finds too often that they have been written for advanced students or specialists. He finds writers assuming that he knows things which he does not know and he soon becomes swamped in a maze of diagrams and mathematical formulas. He is likely to conclude that the matter is entirely too involved for him and give it up in despair. Now the subject of crystal analysis is not altogether an easy and simple one. We venture to assert that no one can present its fundamentals in snch fashion that the reader can assimilate them without any effort whatever. The present writer is undkr no illusion that he is about to produce a piece of light and interesting readinfwhich all comers can peruse with unalloyed pleasure. On the contrary, he addresses himself only to those who would really like to understand the elements of the subject and who are willing to employ a certain degree of serious effort to that end if they can feel reasonably assured that it will not go for naught. To snch as these we offer, not a ride in a limousine, but companionship, guidance and some slight assistancein a hike through interestingly variegated, though at times difficult,territory.
Crystal Analysis a By-Product Idea The experiment which first suggested the possibility of investigating the structure of crystals by means of X-rays was designed, not to study a y s tals, but to test the nature of X-rays themselves. There had been a great deal of discussion as to whether X-rays were fundamentally like visible light or whether they constituted a new and altogether different type of radiation. It had been suggested that if X-rays were really a form of light, they must be light of very short wave-length and that, if only it were possible to secure fine enough diffraction gratings, they should display dif138
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fraction effects., It had also been argued that the regular and symmetrical external forms of crystals must indicate a regular and symmetrical internal structure also. In 1912 Laue pointed out that if both these contentions were correct a crystal should furnish the very fine diffraction grating which was needed, and that if X-rays were allowed to shine through a crystal upon a photographic plate, some sort of diffractionpattern should result. Subsequent experiment proved that he was right. The Diraction of Waves Before we proceed farther let us review a little of our physics and remind ourselves just how certain diffraction effects arise. Although X-ray diffraction can be explained upon the basis of the quantum theory,' we shall avoid confusion by adhering to the more familiar and, in some respects, simpler wave-front theory. We sometimes say that a beam of X-rays is "reflected" by a plane, or by planes, of atoms in a crystal. A little consideration couviuces us, however, that this sort of "reflection" must be different from the reflection of ordinary light by a mirror, or from the rebound of a billiard ball directed against a cushion. In comparison with the wave-length of ordinary light, a mirror is to all practical intents and purposes just what it appears to the eyea plane surface. Just so a billiard cushion presents what is, with relation to the ball, a perfectly smooth and regular edge. But when we direct a beam of X-rays against a plane of atoms in a crystal i t is more as though we were shooting a charge of buckshot ?gainst the side of a pyramid of cannon halls, to borrow Dr. Germer's simile, and in this case we must imagine that there are considerable spaces between tBe cannon balls also. How can anything resembling regular reflection result from such an operation? Some Familiar Analogies In order to make the problem easier let us begin with a familiar analogy in two dimensions. When that has been made clear we can amplify our ideas a little to make them applicable to three-dimensional space. Let us suppose that there is a series of straight ripples or wave-fronts traveling across the surface of an otherwise smooth pond. When they encounter a post set upright in the pond, what happens? The physicists tell us, and we can observe for ourselves, that a new, or secondary, series of ripples is set up. These secondary ripples take the form of concentric rings spreading outward from the'post. (See Figure 1.) Just so when an X-ray wavetrain encounters an atom in space a new series of waves is set up. In that case, however, we are dealing with three dimensions rather than two and we must think of the new waves as spreading, concentric spheres rather than as concentric rings. 1 Davey, Gen. Elec. Reu, 27, 742-8 (Nov., 1924).
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The reader may object that an atom isno more a dense, "solid" body than is the side of a pyramid of cannon balls a true, "smooth" plane. He may say that we must think of the X-ray waves as encountering electrons rather than atoms. And in so saying he will be right. The atom acts like a dense .-- body in the way i t spreads light waves only because its electrons together operate to produce a resultant of effects which is very similar to the single effect which would be produced by a dense body. To illustrate and clarify this statement let us return to our pond and post analogy. Only let us now suppose that, instead of .a single large post, we -- have a -aroup. of small posts set comparaF I G ~1.-STRAIGHT E WAVEFRONTS tively close together. When a series of TRAVELING ACROSS THE SURFACE OF A POND SETUP A SER~ES OF SECONDARY straight ripples encounters this obstrucCIRCULARRIPPLESWHEN THEYEN- tion each small post becomes the origin COUNTER A POST of a series of outward-spreading, circular, concentric ripples. A glance a t Figure 2 shows us how these series of small circular ripples combine to form a resultant series of large, nearly circular ripples emanating from the group of posts as a whole. We can readily conceive that the group of electrons making up the outer shell of an atom would, by a somewhat similar process, scatter X-ray waves in the f o m of spreading, concentric sphere's which would merge to form a series of larger, nearly spherical wave-fronts ap.......... parently emanating from the atom as a .. whole.
*
'LReflection" from Regularly Spaced Points Since we have been able to justify the simplification in this manner, and since we realize clearly that we are OF SMALL F~crmr(2.-A GROUP dealing with a resultant, we can conEFFECT Posts HASMUCHTEE SAME tinue, for the sake of convenience, to AS ONE LARGE poST consider the atom as a unit in the scattering of X-rays. Let us now inquire what would be the composite effect of a number of these pseudo-units, regularly spaced. Again we shall approach the answer to our question by way of the pond. This time, instead of observing a number of closely grouped small posts, or a single large post, we shall suppose that we have a straight row of regularly spaced, large posts. Recording our observations diagrammatically in Figure 3, we
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note that a series of straight wave-fronts approaching the line of posts a t an angle is partially converted into a number of series of circular wave-fronts spreading outward from the individual posts. Again we find that the circular wave-fronts combine to produce resultant waves. But now the resultant wave-fronts are (nearly) straight, like the original, or "incident" wave-fronts. We see also that the lines of motion of the resultant waves and of the incident waves make equal angles with the line of , ... the posts. Applying our discoveries concerning the general nature of refraction to X-rays and crystals, we perceive that, when a beam of Xrays is directed a t an angle against a single layer or plane of atoms, a diffracted ray should leave the plane a t an equal angle. I n this FIGURE3.-A STRAIGHI. LINE OR RBGUZARLY one respect the plane of SPACED POSTS SETS UP A TRAIN OR' "RER'LECTED" WAVE-FRONTS atoms can be said to act like a mirror and the beam of X-rays can be said to act as though i t were reflected. It is therefore possible t6 simplify Figure 3 into a diagram like Pigure 4. E We do this merely as a convenience and with certain mental reservations. We are not deluding ourselves as to the real nature of diffraction phe&> nomena. We merely take &" . advantage of the fact that r ,.,d ,. a simplified diagram will ,.d work just as well for certain ~a .. purposes that we have in "'%n,e mind as a more elaborate + a,' \* '%,,,, picture, and that it will be 'U,#, easier to handle. So long OR' FIGURE 3 FIG& 4.-A SIMPLIPICATION as we realize that this diagram is only useful where the analogy between reflection and these diffraction effectsholds good, and so long as we do not try to use our diagram where it is not applicable,this expedient is entirely legitimate and highly commendable. "Reflection" from Parallel Planes of Points DiEers from Mirror Reflection There is one important difference between the reflection of ordinary light
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from the surface of a mirror and the diffraction of X-rays by crystals which we must keep in mind. The angle that a reflected heam of light makes with a reflecting surface is always equal to the angle a t which the incident beam strikes the surface. This much holds true for X-rays and crystals, if we interpret "surface" to mean a plane of atoms within the crystal. But in the case of ordinary light directed against a mirror the reflected beam will always have the same intensity, no matter what the angle of the incident beam may be. This is not true of monochromaticZX-rays directed against a crystal. This peculiarity arises out of the fact that even an exceedingly thin crystal is made up of a great many parallel planes of atoms. A single atomic plane, if such a thing could be obtained, would "reflect" a beam of constant intensity, regardless of the angle of the incident beam. Let us consider why a group of parallel planes cannot do so. It is evident from Figure 3 that only a minute fraction of the lieht of " an incident beam is "reflected" by a single plane of atoms. A great deal of light passes through, unaltered in direction, to be partially "reflected by successive deeper layers or planes of atoms. We see, then, from Figure 5 that the "reflections" from a great many parallel, incident rays may follow a common track in emereing. FIGURE ~.-"RRPLE~~ON" paox SuccHsswE PLANES from the c"Ystal. This being true, i t becomes a matter of great importance whether ornot these "reflected" wave-trains are all in phases with each other.
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The Part Played by Interference We learned in elementary optics that, when one train of waves is super% X-rav . light. . . like ordinanr visible light, includes arange of wave-lenahths. When we separate rays of n single wave-lcnnh from risible white light we call them o n e - a h r
or "monochromatic" rays. By analogy, mixed X-ray light is sometimes called "white" X-raylight, and X-rays of a single wave-length are called monochromatic rays. What n to ~ h i t e a n dto we have just said about themzrror r ~ f l ~ ~ l ofi oziriblelinhtapplic~equnlly . -monochromatic light. a The use of the wave analogy t o explain the action of light implies the idea of some sort of crest and trough formation. When the crests and troughs of one waveh i n exactly concide with the crests and troughs of another wave-train we say that the two trains are exactly in phase. When the crests of one train coincide with the troughs of the other, we say that the trains are opposite in phase. (See Figure 6.) ~
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imposed upon another with which it is exactly in phase, the two trains reenforce each other and combine to form a more intense ray of light. If the two trains are exactly opposite in phase they nullify each other and no resultant ray is produced. The latter phenomenon is known as interference. In order to study this effect in a little more detail as i t applies to X-ray crystal analysis, let us take a portion of Figure 5 and amplify it somewhat until W m v r m a / ~ s /N Pndsc it looks like Figure 7. The heavy lines designated p, p', and .. .. . pa represent atomic planes. Z1C and :+xi -x.-A+: . . . . A'B are lncldent beams of monochro- =':::::=::::.=:::::=:;::=::::: matic X-rays CF and B'CF are "reflected" beams.
Putting a Little of Our Plane Geometry WAVETRAINS OPPOSITE/N
%SL
to Practical Use It is obvious that the total path A'B'CF is longer than the path ACF. -------------.In order to discover how much longer .....-. ....- .... ..-. let us extend A'BL to E and drop per( A 1 WAVELENGTH) pendiculars to the paths of the incident FIGURE 6.-Two CONVENTIONS ROE rays from A to A', from B' to B, and from R~PRBSENTING W~VEIUINS C to C'. Let us also drop a perpendicular to the atomic planes from C to E. ' The paths to be compared are AB + BC CF and A'B' B'C CF. Since AB = A'B' and CF is
-
+
+
+
A
L
R C ~?.-A E DETAILED ANALYSISoII "REPLBM'ION" OM SuCCESSIVB PLANES
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common to both paths, we need compare only BC and B'C. The reader can easily satisfy himself that B'CE is an isosceles triangle4 and that B'C = B%. But B'C' = BC. Hence the difference in length of the two paths ABCF and A'B'CF must be equal to C'E. Since the light-beams come from the same source, they are perfectly in phase when they pass through the points A and A', respectively; it depends upon the difference in length of the paths traveled whether the "reflected" rays will still be in phase when they arrive at F. If the distance C'E, is equal to one wave-length, or any whole number of wave-lengths, the "reflected" rays will be in phase and maximum "reflection" will be observed; if it is not, "reflection" will be minimal or will disappear entirely. Translating the Report of O w Findings into a More Elegant and Exact Language than English The fact which we have been trying to bring out can be summed up as follows: When X-rays of a given wave-length are directed against a given crystal a t a given angle we will observe reflection, if the line C% on a diagram corresponding to F i p r e 7 is equal in length to one, or any whole number, of wave-lengths. You see for yourself what a vague, clumsy expressioa that is. So don't blame the physicist when he tries to express what we have found out more neatly and exactly by resorting to matbematics-give him credit. After all there is nothing very terrible about his formula when one sees how he gets it. He says: let X = the wave-length 6f the X-rays, and let n = any whole number; then we will observe reflectionwhen:
Trigonometry Helps Us Make the Translation That is a great improvement over our k s t attempt to tell what we have learned, but i t is not altogether satisfactory yet. The person who sees this formula may not have our diagram a t hand, and if he draws one of his own he may employ a different geometrical construction or he may letter i t differently. We ought to find some way of expressing C% so that no mistake could possibly arise. If we study Figure 7 we find that the lengtb represented by C'E depends upon just two t h i n g s t h e magnitude of the angle of incidence which the light rays make with the crystal planes, and the distance between the 4 It is an accident of construction that B'CE is also an equilateral triangle in this drawing. That is not a general case.
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planes. The angle we shall call 8, and the distance between the planes, d. If now we can express the length represented by C% in terms of 8 and d we shall have gotten down to fundamental causes and we shall have eliminated the necessity for reference to any specific diagram. Fortunately there is in trigonometry a very useful series of relationships between the angles and sides of a right triangle. Since the triangle CEC' is a right triangle we can properly select one of these relationships to express the length of C% as a function6of 0 and d. (See Figure 8.) The particular function which we shall select is called the "sine," and we write it sin 0. Expressing what we have just said mathematically: sin
--.. -.
CE
=
sin 8
(2)
Again referring to Figure 6, we see that CE is equal to 2d, so we may write:
C'E = 2d sin 0 %.
Substituting this value in equation (I), we obtain the usual statement of the physicist that "reflection" will be observed when:
nX = 2d sin 83‘
. - - - _ _ _ _,-'*
/
,
FIGURE 8,-ILLUSTRATING mE FuNcrroNs AN ANGLB
(4P
So we see that, after all, there is nothing mysterious or difficult to understand about this temhle-looking formula. "Elephant" is just as easy to spell as "cat" when one knows how it's done.
Figures 9A and 9B are intended t o sum up and emphasize the essential points which we have just considered. In Figure 9A we note that 2d sin 0 is equal t o X; hence n is 1, a whole number, and "reflection" is observed. This is what is known as a "first-order reflection;" if n had been 2, we should have had a "second-order reflection." In short, the "order" of any "reflection" is determined by the value of n. (It is an accident of construction that the diagonal distance between planes as measured along the paths of the incident X-rays, happens to be a whole-number multiple of X When we say that one quantity is a "function" of another we mean that the magnitnde of the one in some way depends upon the magnitude of the other. This is known as Bragg's Law.
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JANUARY, 1930
in this diagram. That is not an essential to "reflection.") In Figure 9B Zd sin 0 is equal to 3/2 A; hence n is 3/2, which is not a whole number. Consequently the refracted rays nullify each other and no "reflection" is observed. Calculating a Crystal Dimension We now have a formula which describes X-ray "reflection" from crystal planes briefly and exactly, but, as a practical working tool, it is still lacking. It contains too many unknowns. We can measure and control the angle 0 and we might manage to select satisfactory values for n, but X and d remain unknown. If we are to use this for. + .. .. .1 mula to calculate values .' "\.* \ \a *\ ::'{ " C for. either from experi0 mental data, we must find n \ 4../' some independent means .. .. of evaluating the other. %. %:\ i/ \ \,; ...' There is more than one
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.
d
'-.
\*
, \..\
9
' , ?-...\ , /-'-
possible line of approach to the solution of this FIGURE~A.-"REFLECTED"WAVETRAINS IN PHASE. problem, but the one THISISAN EXAMPLE OF "FIRST-ORDER REFLECTION" occurs to us most readily is one which the early investigators actually used. It is entirely possible as * 4 we shall see later, to ded \, , ' termine the structure of a ;. ; 4 . . , , crystal without knowing
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