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X-Ray Diffraction and the Bragg Equation Christopher G. Pope Chemistry Department, University of Otago, Dunedin, New Zealand A typical derivation of the Bragg equation (1) is easy to understand but leaves a good deal unexplained. A slightly more critical approach, although a little more demanding, is more interesting and provides a better introduction to the use of X-ray diffraction in structure determination. Usually, the treatment begins with a diagram that shows a regular array of scattering points, drawn so that they lie on a set of parallel lines, distance d apart. Interference between parallel beams of X-rays, reflected as if from a mirror from these lines of points, is then considered. The condition for constructive interference, that the rays reflected from adjacent lines should differ in path length by an integral number, n, of wavelengths, then easily leads to the Bragg equation. n λ = 2d sinθ
mance of a X-ray powder diffractometer. A schematic diagram of the device is shown in Figure 3. The curved crystal focuses radiation onto the detector just as a curved mirror would be expected to focus visible light, and so increases the signal received. Because the “mirror” action of the crystal occurs only at the precisely required angle for X-rays of a single wavelength, it only reflects selected monochromatic radiation,
(1)
However, in this derivation it is not explained why the angles of incidence and reflection should be equal. For example, why should a diagram such as Figure 1 not have been used? Here, the condition for constructive interference between the parallel rays scattered from X and Y is easily seen to be n λ = d (sinθ1 + sinθ2)
(2)
It is clearly not necessary for θ 1 to equal θ2, and it is easy to see that the same reflection condition will apply to the whole set of scattering centers that lie on the line XY or its extension, and are placed d apart. However, it is important to realize that in X-ray diffraction experiments we are relying on a cooperative effect, and that we will only observe intense diffracted X-ray beams at sharply defined angles if very large numbers of scattering centers (typically more than 107) are producing radiation that is in phase. We therefore need to establish what is required to achieve this condition for all the centers in a regular three-dimensional array. The easiest way to begin this problem is to look at the path difference that exists between rays scattered from centers that lie a distance , apart in a line in the top plane, as shown in Figure 2. The path difference between rays AB and CD is path difference = , (cosθ 1 { cosθ 2)
Figure 1. Alternative diagram, showing the path difference between parallel rays.
(3)
Constructive interference between all the scattering points at any spacing in such a line is only possible if θ1 = θ2. If the argument is extended to consider scattering points that exist in different parallel lines drawn across the surface—that is, to consider all the scattering points in the plane—it can be seen that the incident and scattered beams must both lie in a plane perpendicular to the surface. Of course, vector algebra (2) can be used to give these results more succinctly and elegantly, but there is a danger that a clear physical understanding of the Bragg conditions for reflection will then be lost. A rather useful way to illustrate both the similarity and difference between X-ray reflections from a crystal and the reflection of light by a mirror is to examine the behavior of a curved crystal focusing monochromator (3), which is sometimes used to improve the perfor-
Figure 2. Reflection from points in the top Bragg plane.
Figure 3. Curved crystal monochromator. A is the X-ray source, B the sample, C the curved crystal, and D the detector. S are slits limiting the width of the X-ray beam.
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and X-rays of a different wavelength (due for example to X-ray fluorescence) do not get focused onto the detector. This can significantly improve the signal-to-noise ratio in the observed X-ray diffraction pattern. As X-ray scattering is caused by the extranuclear electrons in the atoms of the sample being examined, the diagrams used often lead to the misconception that the points shown must represent the positions of the centers of the atoms in the structure of the solid. It is most important to establish why this is not true, as unless this is done, it is not possible to appreciate how the intensities of the diffracted beams can be used to determine crystal structures. Fortunately, it is only necessary to think about the diffraction pattern we would expect to observe from a schematic two-dimensional crystal, to see what principles are involved. Extension of these ideas to the real three-dimensional case is easy once this has been done. Suppose that the center of a particular atom is arbitrarily chosen as the position of one of the lattice points. This is always legitimate, as there is no special a priori reason for using any specific location in space as the position of a given lattice point. Of course, once this initial choice has been made, the repeating pattern must be such that the center of an atom of the same kind, with an exactly similar environment of surrounding atoms, must be at all the other lattice points. For example, if the “atom” in question is a cesium ion in cesium chloride, there must be cesium ions centered at all the other lattice points, and the arrangement of chloride ions around each cesium ion must look the same from the center of any one of the cesium ions in the crystal. All this follows from the definition of a lattice point—a point chosen so that the environment of each such point is identical—and has nothing specifically to do with the fact that the regular patterns we are concerned with are made up from atoms or ions. Figure 4 shows three different two-dimensional arrangements of “atoms” having regularly repeating patterns. For most purposes, to understand what determines the relative intensities of the different possible X-ray reflections allowed by the Bragg equation, we only need to consider the scattering effect associated with each unit cell, as the complete description of one cell defines the exact geometric pattern that is repeated an enormous number of times to produce a picture of the complete crystal. Clearly, in general, all the atoms in the unit cell must be expected to contribute to the amplitude of the reflected beam. The relative size of the contribution is determined mainly by the kind of atom involved and the position of the atom in the cell that controls the perpendicular distance of the atom from the nearest Bragg plane giving rise to the reflection being considered. When the contributions from all the atoms in the unit cell are added, we get the net scattering effect which can be associated with each lattice point. This depends on the reflection being considered and on the structure of the solid, and is called the structure factor. In Figure 4(a), which might represent a solid element, the atoms are all the same, and are all located at lattice points in a very simple arrangement. In this case the contributions of each atom to the reflected beam are in phase when the Bragg condition for reflection from the layers shown a distance d apart is satisfied. Figure 4(b) represents a compound AB containing equal numbers of two kinds of atoms, shown as the larger and smaller circles. This structure has the same unit cell size and shape as structure 4(a), so the Bragg condition
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for reflection to occur is also the same for (a) and (b). However, the atoms B (large circles) are now in positions exactly halfway between the Bragg planes, so that if there is a one-wavelength difference in path length between rays reflected from the atoms A (small circles) in the top and second layers, there must be a half-wavelength difference between rays reflected from the A and B atoms. Hence the contributions from A and B will interfere destructively. As the scattering powers of atoms A and B will generally be different, we expect still to observe a reflection, as the cancellation of the waves will not be complete, but the amplitude and hence the intensity of the beam will be altered. Notice that each reflection has to be considered individually. For example, if the Bragg equation were satisfied with n = 2 (a second order reflection from the same set of planes in which the new angle θ is such that the path difference between rays reflected from A atoms in the top and second layer is now two wavelengths), then the rays scattered from A and B atoms would be one wavelength different in phase, and hence interfere constructively. Figure 4(c) represents a solid with the same chemical composition as the one shown in Figure 4(b), but with a different atom arrangement. It also has the same unit cell size and shape as structures 4(a) and 4(b), so the Bragg condition for reflection must result in reflected beams at exactly the same angles for each of them, but of course, the relative intensities of the beams from each structure will differ.
Figure 4. Three schematic two-dimensional solid structures.
Figure 5. Powder XRD patterns for KBr and KCl. Lines 1–6 appear in both patterns, but lines A, B, and C appear only in the pattern for KBr.
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Comparison of the powder XRD patterns produced by samples of potassium bromide and potassium chloride illustrates some of the points just discussed (Fig. 5). These two solids have the same shape (cubic) unit cell and the same relative positions for the ions in the cell, but the larger ionic radius of Br{ compared with Cl{ causes KBr to have the larger unit cell. The similarity in structure results in the similarity in both relative position and relative intensity in the diffraction patterns, as far as peaks 1–6 are concerned. The extra peaks A, B, and C in the KBr pattern arise because whilst K+ and Cl{ are isoelectronic ions and so scatter X-rays almost equally well, the ions K+ and Br{ do not. For the peaks corresponding to the Bragg angles A, B, and C, the scattering contributions for cations and anions are exactly half a wavelength out of phase with each other, and so cancel out for KCl but not for KBr. The contributions of cations and anions are in phase for the peaks 1–6, and so these reflections appear in the patterns for both solids and are characteristically more intense. To summarize so far, it can be said that if you only want to know the size and shape of the unit cell for a solid phase (which seems unlikely), all you need to know are the angles at which Bragg reflection occurs as a crystal is moved through all possible orientations with respect to a monochromatic X-ray beam of known wavelength. This is most easily achieved in an experiment in which a single crystal is mounted so that it can be rotated about its three symmetry axes in turn. In principle, of course, only six independent Bragg angles are needed to determine the three unit cell edge lengths, and three angles between the cell axes, but as every reflection must be consistent with these same six parameters, the extra reflections observed provide a very useful check that the unit cell geometry has been correctly established. If you wish to identify a solid phase, this can be achieved by using the angles and relative intensities of a fairly small number of the stronger reflections to “fingerprint” the sample. This is usually done using the powder technique and is analogous to using an infrared spectrum to identify an organic compound, although the Xray method is usually much more specific. In this technique, the sample consists of very many small crystals, oriented at every angle with respect to the X-ray beam throughout the experiment, so that all Bragg planes with a particular spacing give rise to a reflection at a single observed angle of deflection, and some information is lost. Despite this, used in conjunction with standard sets of tables, powder diffraction can discriminate positively between tens of thousands of known solids. If you need to know exactly what the arrangement of atoms in a solid structure is, you then need accurate information about the intensities of a large number of X-ray reflections, and the problem can usually be solved only if you have a suitable single crystal of the material you wish to examine. A considerable amount of computation is also required. However, in many cases the complete process can now be carried out almost routinely
and without an extensive knowledge of the theory of Xray crystallography. As well as explaining in outline why the amplitudes and hence intensities of reflected X-ray beams depend on how atoms are arranged within the unit cell, an introduction to the Bragg equation can also be used to explain why solids consisting of very small crystallites or imperfectly crystallized materials in which there is some disorder in the positions of the atoms in the structure produce rather broad, weak diffraction lines. This follows directly from diffraction theory, which shows that as the number of regularly spaced scattering centers increases, the angles at which diffracted beams are observed become more and more sharply defined. Thus, the angular spread of reflected X-ray beams can give a useful measure of the particle size of very finely divided solids (5). The semiempirical Scherrer equation t = Kλ (4) B cos θ describes this effect. K is a constant, which depends somewhat on particle shape, but is usually about 0.90; λ is the wavelength of the X-rays; B is the angular width of the beam at which the intensity falls to half its maximum value; and t measures the average dimension of the particles in a direction perpendicular to the Bragg planes giving rise to the reflection. Line broadening due to instrumental factors must of course be subtracted from the observed beam width before the equation is applied, and this prevents the method being useful once the regions of crystallinity exceed about 50 nm. Particles smaller than about 5 nm usually lead to reflected beams too broad and diffuse for accurate measurement, so in fact the equation can be applied successfully only over a rather limited size range. However, it is possible to distinguish between ordering in different directions in a crystal. This is apparent, for example, in the diffraction patterns obtained from samples of carbon black, which usually show only reflections that arise from approximately evenly spaced graphite layers, and no other detail. The approach in this note is aimed at making better use of a piece of theory that is introduced into a large number of courses, without demanding much extra teaching time. The excellent Symposium on Teaching Crystallography (6) shows many ways in which this material could be followed up. Literature Cited 1. Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University: Oxford, 1994; p 728. 2. Moore, W. J. Physical Chemistry, 5th ed.; Longman: London, 1972; p 841. 3. Bertin, E. P. Introduction to X-ray Spectrometric Analysis; Plenum: New York, 1978; p 152. 4. For example, Willard, H. H.; Merritt, L. L., Jr.; Dean, J. A. Instrumental Methods of Analysis, 5th ed.; van Nostrand: New York, 1974; p 291. 5. Anderson, J. R. Structure of Metallic Catalysts; Academic: London, 1975; p 365. 6. Symposium on Teaching Crystallography. J. Chem. Educ. 1988, 65, 472.
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