5. F. Pavkovic
Loyola University of Chicago Chicago, lllinois 60626
x - R ~Diffraction ~ by Crystal Planes in Real Space and Reciprocal Space
Interaction of X-rays .with crystalline matter is accounted for by Bragg's law, n X = 2 d sin 0, in which sets of parallel planes in the crystal separated by a distance d , reflect X-rays of wavelength X when their angle to the beam (theta) satisfies the expression. Accordingly, the specification and identification of reflecting crystal planes is of fundamental importance in interpreting the X-ray diffraction process. There are several means of describing crystal planes which use the unit cell as a reference coordinate system. Since all planes meet each unit cell axis at some point in the crystal, intercepts along a, b, and c can be used to designate any particular plane, for no two planes have identical intercepts. Alternately, the inverse of a reduced intercept designation may be taken to generate hkl indices which also specify planes (I). Indices are more useful for diffraction purposes because each hkl set specifies an entire stack of parallel planes all having the same interplanar separation d (i.e., a family of planes). I n the powder method, different hkl families with like normals reflect a t the same theta angle and diffraction "lines" are often compound in origin (2). However, with moving-film single-crystal devices such as precession and Weissenberg cameras, separate reflections from individual hkl families are detected. Both of these instruments generally record reflections in which one of the hkl indices remains constant while the other two vary. For example, all measurable hkO to 720 reflections are recorded on one film, hkl to iiz1 on a second film, hk2 to ZK2 on still another film, etc., with each 1 "level" separately isolated by means of mechanical adjustments on the cameras (3); in this manner all available hkl reflections can be individually registered. Indexing such films, i.e., assigning hkl indices to diffraction spots, proceeds quite readily when planes are considered in a so-called reciprocal-space lattice (4). Reciprocal Space
This concept is developed by means of the following construction (1) For use as a real space planar cell model, select a parallelagram. A model of dimensions 3 om (side a) by 4 cm (side c) with an angle of 75' (beta) between sides is convenient for illustrative purposes. (2) Within this cell, draw and label the plane closest t o the origin for several h01 families (it is clearer t o use a separate cell motif for each lane diaerammed). The followine families are sufficientlyrepr&ntativeUfor the method:
(3) Now construct a. normal from the origin t o each of the dis, grammed planes. (4) A. Measure each normal t o three significant figures. This length is dhrl,the real space interplanar normal. B. Calculate the inverse magnitude of each dntr. This is the reciprocal space interplanar distance. C. Prepare a table listing hkl, dm, and d*ntr. ( 5 ) On a clear sheet of paper, select and mark a central point. This will serve as the origin in "reciprocal space" t o which the hkl planes will be referred. (6) Then for each plane diagrammed in real space, locate and mark a. point on the reciprocal space sheet a t a distance d'nxz from the origin, and in a direction specified by its hkl normal constructed in step 3. Finally, label each reciprocal space point aocording t o the hklfamily of planes it represents.
Figure 1. A red-space or planor cell with its 100, 001, 102 plonss and normals.
Figure 1 shows the ac plane of the unit cell model suggested in step 1, along with its 100, 001, and 102 planes and normals. The normals have lengths of dl00 = 2.90, &I = 3.86, and dIoP= 1.84 cm; their inverse magnitudes are d*lm = 0.345, d*oa~= 0.259, and d*loz = 0.543. Points in reciprocal space for these and several other planes appear in Figure 2. Note that joining points in directions parallel to Plw and defines a grid network whose lattice pattern is clearly related to the original model cell motif. In reciprocal space the axis containing hOO points is labeled a* and similarly c* contains 001 points; the angle between them is P* = 180" - 8 . If remaining model cell parameters were known, then construction could be continued for hll, h21, h31 and higher k levels. This results in a three-dimensional reciprocal space Volume 49, Number 4, April 1972
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Figure 3. Ewald constwstion illustrating the Bmgg refleeing condition in reciprocal-space.
A = 2 sin R/d*.w = 2dm sin e
Figure 2. Points in reciprocal-space for s w a r d planes belonging to the red-spaso cell of Figure 1 .
lattice, in which each hkl point represents a unique family of reflecting planes. Throughout reciprocal space the distance from origin to any point is always inverse to the corresponding real space interplanar = l / d l l h lwithout exception. separation, so that Diffraction in Reciprocal Space
The concept of a reciprocal space lattice leads to complete rationalization of the X-ray diffraction process (6). Illustration of this significant feature is readily acccmplished in two dimensions and easily extended to three. For as shown in Figure 3, if the origin of reciprocal space (000) is anchored to a circle of radius l / h at a point containing the diameter (direct X-ray beam), then as the crystal located at A rotates or precesses about an axis perpendicular to both circle and beam, its coupled reciprocal lattice duplicates this motion about D. And for that instant when any reciprocal lattice point lies on the circle, the following conditions apply: BC is parallel to a reflecting hkl crystal plane at A ; L CBD = 8, the angle between hkl plane and direct beam; L CAD = 28, the angle between reflected ray and direct beam; CD = d*,,,, distance between origin and hkl lattice point in reciprocal space; and ABCD is a right triangle. Thus sin 0
=
CDIBAD = d*hrz.A/2
This situation completely fulfills the first-order Bragg reflecting condition, and it follows that whenever a reciprocal-lattice point meets the circle of reflection as it is called (the Ewald sphere in three-dimensions), Bragg's law is satisfied and a reflected ray darts from crystal to film or detector and records itself (barring extinctions). Conversely,those planes with d*nrlgreater than 2 / h lie outside the reflection circle or sphere and cannot be detected. Then taken from this viewpoint, diffraction films are simply photographic representations of the reciprocal lattice, and what Weissenberg and precession cameras really record is the reciprocal space lattice itself, one level at a time (6). The precession technique produces an enlarged but otherwise true replica of the reciprocal space pattern, so that precession films are particularly eagy to index (7, 8). Then given crystal-to-film distance and X-ray wavelength, it is a relatively straightforward matter to determine lattice spacings and angles in reciprocal space from such film records. And with this information realspace unit-cell parameters are obtained more or less directly, depending on of what crystal.system the specimen is a member. Thus in actuality, one works backwards from a reciprocal-space diffraction record to real-space cell parameters by reversing the steps in the first construction, and using units and constants appropriate to the type of camera and X-radiation employed. Literature Cited (1) RemmTR, 0. J., J. C n ~ u . E o u o .7,860 . (18301.. (2) CVLLITT. B. D., "Elements of X-ray Diffraction." Addison-Weslsy
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(5) (6) (7)
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Co..Reading, Mass., 1956.p.301. STOUT. G. H.. A N D J E N ~ ~ L. N . H.. "X-IBVStruotur. Determination." ~he'~aomiilan~ New o . , ~ o r k1968, , ~haiter5. A.*ROPF, L. v.. "Elements of X-ray Cryataiiography," MoGra,v-Hill Book Co., N e r v York, 1968, p. 136. B u ~ n o r n .M. J., "X-ray Cryataiiography," John Wilev & Sons, Ino.. New York. 1942,Chspter 7. Bnenosn, M. J.. "Contemporaw Crystailography." MoGraw-Hill Book Co.. New York. 1970, Chapters 8andY. BaEnoBn, M. J., "The Precession Method in X-ray Crystallography," John Wiiey & Sons, Ino., New York. 1964. Wnsm, J., J. C n ~ u . E n u o .45,446 , (1968). Figur. 12.
