CHAPTER 1 Principles of Powder Diffraction ROBERT E. DINNEBIERa AND SIMON J. L. BILLINGEb a Max-Planck-Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany; b Department of Physics and Astronomy, 4268 Biomed. Phys. Sci. Building, Michigan State University, East Lansing, MI 48824, USA 1.1 INTRODUCTION This chapter presents some very basic results about the geometry of diffraction from crystals. This is developed in much greater detail in many textbooks but a concise statement of the basic concepts greatly facilitates the understanding of the advanced later chapters so we reproduce it here for the convenience of the reader. Since the results are so basic, we do not make any attempt to reference the original sources. The bibliography at the end of the chapter lists a selection of some of our favorite introductory books on powder diffraction. 1.2 FUNDAMENTALS X-rays are electromagnetic (em) waves with a much shorter wavelength than visible light, typically on the order of 1 A˚ (¼ 1  10À10 m). The physics of em-waves is well understood and excellent introductions to the subject are found in every textbook on optics. Here we briefly review the results most important for understanding the geometry of diffraction from crystals. Classical em-waves can be described by a sine wave that repeats periodically every 2p radians. The spatial length of each period is the wavelength l. If two identical waves are not coincident, they are said to have a ‘‘phase shift’’ with respect to each other (Figure 1.1). This is either measured as a linear shift, D on a length scale, in the units of the wavelength, or equivalently as a phase shift, df on an angular scale, such that: D dj 2p ¼ ) dj ¼ D ð1Þ l 2p l 1 2 Chapter 1 Figure 1.1 Graphical illustration of the phase shift between two sine waves of equal amplitude. The detected intensity, I, is the square of the amplitude, A, of the sine wave. With two waves present, the resulting amplitude is not just the sum of the individual amplitudes but depends on the phase shift dj. The two extremes 2 occur when dj ¼ 0 (constructive interference), where I ¼ (A1 + A2) , and 2 dj ¼ p (destructive interference), where I ¼ (A1 À A2) . In general, 2 I ¼ [A1 +A2 exp (idj)] . When more than two waves are present, this equation becomes: "# X 2 I ¼ Aj exp ðijjÞ ; ð2Þ j where the sum is over all the sine-waves present and the phases, fj are measured with respect to some origin. X-ray diffraction involves the measurement of the intensity of X-rays scat- tered from electrons bound to atoms. Waves scattered at atoms at different positions arrive at the detector with a relative phase shift. Therefore, the measured intensities yield information about the relative atomic positions (Figure 1.2). Principles of Powder Diffraction 3 Figure 1.2 Scattering of a plane wave by a one-dimensional chain of atoms. Wave front and wave vectors of different orders are given. Dashed lines indicate directions of incident and scattered wave propagation. The labeled orders of diffraction refer to the directions where intensity maxima occur due to constructive interference of the scattered waves. In the case of X-ray diffraction, the Fraunhofer approximation is used to calculate the detected intensities. This is a far-field approximation, where the distance, L1, from the source to the place where scattering occurs (the sample), and then on to the detector, L2, is much larger than the separation, D, of the scatterers. This is an excellent approximation, since in this case D/L1 E À10 D/L2 E 10 . The Fraunhofer approximation greatly simplifies the mathemat- ics. The incident X-rays form a wave such that the constant phase wave front is a plane wave. X-rays scattered by single electrons are outgoing spherical waves that again appear as plane waves in the far-field. This allows us to express the intensity of diffracted X-rays using Equation (2). The phases jj introduced in Equation (2), and therefore the measured intensity I, depend on the position of the atoms, j, and the directions of the incoming and the scattered plane waves (Figure 1.2). Since the wave-vectors of the incident and scattered waves are known, we can infer the relative atomic positions from the detected intensities. From optics we know that diffraction only occurs if the wavelength is comparable to the separation of the scatterers. In 1912, Friedrich, Knipping and Max von Laue performed the first X-ray diffraction experiment using single crystals of copper sulfate and zinc sulfite, proving the hypothesis that X-rays are em-waves of very short wavelength, on the order of the separation of the atoms in a crystalline lattice. Four years later (1916), Debye and Scherrer reported the first powder diffraction pattern with a procedure that is named after them. 1.3 DERIVATION OF THE BRAGG EQUATION The easiest access to the structural information in powder diffraction is via the well-known Bragg equation (W. L. Bragg, 1912), which describes the principle 4 Chapter 1 of X-ray diffraction in terms of a reflection of X-rays by sets of lattice planes. Lattice planes are crystallographic planes, characterized by the index triplet hkl, the so-called Miller indices. Parallel planes have the same indices and are equally spaced, separated by the distance dhkl. Bragg analysis treats X-rays like visible light being reflected by the surface of a mirror, with the X-rays being specularly reflected at the lattice planes. In contrast to the lower energy visible light, the X-rays penetrate deep inside the material where additional reflections occur at thousands of consecutive parallel planes. Since all X-rays are reflected in the same direction, superposition of the scattered rays occurs. From Figure 1.3 it follows that the second wave travels a longer distance PN before and NQ after reflection occurs. Constructive interference occurs only if D ¼ PN + NQ is a multiple n ¼ 0, Æ1, Æ2, . of the wavelength l: D ¼ nl ð3Þ In all other cases, destructive interference results since it is always possible to find a deeper plane, p, for which the relation pD ¼ nl with n ¼1/2, Æ 3/2, . (i.e., perfect destructive interference) exactly holds. Thus, sharp intensity maxima emerge from the sample only at the special angles where Equation (3) holds, with no intensity in between. As can be easily seen from Figure 1.3, geometrically: D ¼ 2d sin y ð4Þ where d is the interplanar spacing of parallel lattice planes and 2y is the diffraction angle, the angle between the incoming and outgoing X-ray beams. Combining Equations (3) and (4) we get: nl ¼ 2d sin y ð5Þ the famous Bragg equation. Figure 1.3 Illustration of the geometry used for the simplified derivation of Bragg’s law. Principles of Powder Diffraction 5 This simplified derivation of the Bragg equation, although leading to the correct solution, has a serious drawback. In reality the X-rays are not reflected by planes but are scattered by electrons bound to the atoms. Crystal planes are not like shiny optical mirrors, but contain discrete atoms separated by regions of much lower electron intensity, and, in general, atoms in one plane will not lie exactly above atoms in the plane below. How is it then that the simplified picture shown in Figure 1.3 results in the correct result? A more general description (Bloss, 1971) shows that Equation (5) is also valid, if the atom of the lower lattice plane in Figure 1.3 is shifted by an arbitrary amount parallel to the plane (Figure 1.4). The phase shift can immediately be deduced from Figure 1.4 as: nl ¼ MN cosð180 ða þ yÞÞ þ MN cosða À yÞ ð6Þ ¼ MN½À cosða þ yÞþcosða À yÞ From any textbook on trigonometry we know that: cosða þ yÞ¼cos a cos y À sin a sin y ð7Þ cosða À yÞ¼cos a cos y þ sin a sin y Therefore Equation (7) becomes: nl ¼ MN½2 sin a sin y ð8Þ with: d ¼ MN sin a ð9Þ from which the already known Bragg equation follows: nl ¼ 2d sin y ð10Þ Another equivalent, and highly useful, expression of the Bragg equation is: 6:199 12:398 Ed ¼ with l½A˚ ¼ ð11Þ sin y E½keV with the energy E of the X-rays in keV. θ M α α−θ d θ P Q θ α N 180-(α+θ) Figure 1.4 Illustration of the geometry in the general case where scattering takes place at the position of atoms in consecutive planes. 6 Chapter 1 To derive the Bragg equation, we used an assumption of specular reflection, which is borne out by experiment. For crystalline materials, destructive inter- ference completely destroys intensity in all directions except where Equation (5) holds. This is no longer true for disordered materials where diffracted intensity can be observed in all directions away from reciprocal lattice points, known as diffuse scattering, as discussed in Chapter 16. 1.4 THE BRAGG EQUATION IN THE RECIPROCAL LATTICE As a prerequisite, the so-called reciprocal lattice needs to be introduced. Notably, it is not the intention of this book to reproduce basic crystallographic knowledge but, for completeness, some important formalism that recurs throughout the book is briefly presented. The reciprocal lattice was invented by crystallographers as a simple and convenient representation of the physics of diffraction by a crystal. It is an extremely useful tool for describing all kinds of diffraction phenomena occurring in powder diffraction (Figure 1.5).