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 1010 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 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 10 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, 1, 2, . . . 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, 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 yÞ ð6Þ ¼ MN½ cosða þ yÞþcosða yÞ From any textbook on trigonometry we know that: cosða þ yÞ¼cos a cos y sin a sin y ð7Þ cosða 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). Imagine that besides the ‘‘normal’’ crystal lattice with the lattice parameters a, b, c, a, b, g, and the volume V of the unit cell, a second lattice with lattice parameters of a*, b*, c*, a*, b*, g*, and the volume V*, and with the same origin, exists such that:

w a b ¼ a c ¼ b c ¼ a b ¼ a c ¼ b c ¼ 0 ð12Þ a a ¼ b b ¼ c c ¼ 1

This is known as the reciprocal lattice, which exists in so-called reciprocal space. As we will see, it turns out that the points in the reciprocal lattice are related to the vectors defining the crystallographic planes. There is one point in the reciprocal lattice for each crystallographic plane, hkl. For now, just con- sider h, k and l to be integers that index a point in the reciprocal lattice. A reciprocal lattice vector hhkl is the vector from the origin of reciprocal space to the (hkl) reciprocal lattice point:z

hhkl ¼ ha þ kb þ lc ; h; k; l 2 Z ð13Þ

The length of the reciprocal base vectors is defined according to:

a ¼ xðb cÞð14Þ where the scale factor x can easily be deduced using Equation (12) as: 1 aa ¼ xðb caÞ¼xV ) x ¼ ð15Þ V w Vectors are in bold. z The reciprocal lattice is a commonly used construct in solid state physics, but with a different normalization: a a ¼ 2p: Principles of Powder Diffraction 7

Figure 1.5 Two-dimensional monoclinic lattice and its corresponding reciprocal lattice. leading to: 1 1 1 a ¼ ðb cÞ; b ¼ ðc aÞ; c ¼ ða bÞ; ð16Þ V V V and vice versa: 1 1 1 a ¼ ðb cÞ; b ¼ ðc aÞ; c ¼ ða bÞð17Þ V V V

The relationship between the reciprocal and the real lattice parameters is: bc sin a a ¼ ; V ac sin b b ¼ ; V ab sin g c ¼ ; V cos b cos g cos a cos a ¼ ; ð18Þ sin b sin g cos b cos g cos a cos b ¼ ; sin b sin g cos b cos g cos a cos a ¼ ; psinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib sin g V ¼ abc 1 þ 2cosa cos b cos g cos2 a cos2 b cos2 g

Equation (18) simplifies considerably for higher symmetry crystal systems. 8 Chapter 1

S0 0

S-S 2θ

S θ S0 M

r d θ P θ ε Q

N ε 0

Figure 1.6 Illustration of the important wave and scattering vectors in the case of elastic Bragg scattering.

We now rederive Bragg’s law using vector notation. The wave vectors of the incoming and outgoing beams are given by s0 and s, respectively (Figure 1.6). They point in the direction of propagation of the wave and their length depends on l. For elastic scattering (no change in wavelength on scattering), s0 and s have the same length. We define the scattering vector as:

h ¼ðs s0Þð19Þ which for a specular reflection is always perpendicular to the scattering plane. The length of h is given by: h ¼ 2 sin y ð20Þ s Comparison with the formula for the Bragg equation [Equation (5)]: nl ¼ 2 sin y ð21Þ d we get: nl h ¼ ð22Þ d s Setting the magnitude of s to 1/l, we get the Bragg equation in terms of the magnitude of the scattering vector h: n h ¼ ð23Þ d This shows that diffraction occurs when the magnitude of the scattering vector h is an integral number of reciprocal lattice spacings 1/d. We define a vector d* Principles of Powder Diffraction 9 perpendicular to the lattice planes with length 1/d. Since h is perpendicular the scattering plane, this leads to: h ¼ nd ð24Þ Diffraction can occur at different scattering angles 2y for the same crystallo- graphic plane, giving the different orders n of diffraction. For simplicity, the number n will be incorporated in the indexing of the lattice planes, where: dnh;nk;nl ¼ ndhkl ð25Þ

* * e.g., d222 ¼ 2d111 and we get an alternative expression for Bragg’s equation: h ¼ dhkl ð26Þ * The vector dhkl points in a direction perpendicular to a real space lattice plane. We would like to express this vector in terms of reciprocal space basis vectors a*, b*, c*. First we define dhkl in terms of real space basis vectors a, b, c. Referring to Figure 1.7, we can define that: 1 1 1 OA ¼ a; OB ¼ b; OC ¼ c ð27Þ h k l with h, k, and l being integers as required by the periodicity of the lattice. These three integers are the Miller indices that provide a unique definition for the set of parallel planes. The plane perpendicular vector dhkl originates on one plane and terminates on the next parallel plane. Therefore, OA d ¼ (OA)dcosa. From Figure 1.7 we see that, geometrically, (OA)cosa ¼ d. Substituting, we get OA d ¼ d2. Com- bining with Equation (27) leads to: 1 a d ¼ d2 ð28Þ h

c

b

B C D d

α A a O Figure 1.7 Geometrical description of a lattice plane in terms of real space basis vectors. 10 Chapter 1 and consequently:

d d d h ¼ a ; k ¼ b ; l ¼ c ð29Þ d2 d2 d2 By definition, h, k, and l are divided by their largest common integer to be * Miller indices. The vector dhkl, from Bragg’s Equation (26) points in the plane * normal direction parallel to d but with length 1/d. We can now write dhkl in terms of the vector d: d d ¼ ð30Þ hkl d2 which gives:

dhkl ¼ ha þ kb þ lc ð31Þ or written in terms of the reciprocal basis:

dhkl ¼ ha þ kb þ lc ð32Þ which was obtained using:

dhkl a ¼ ha a þ kb a þ lc a ¼ h dhkl b ¼ ha b þ kb b þ lc b ¼ k ð33Þ dhkl c ¼ ha c þ kb c þ lc c ¼ l

* Comparing Equation (32) with Equation (13) proves the identity of dhkl and the reciprocal lattice vectorh hhkl. Bragg’s equation, Equation (26), can be restated as:

h ¼ hhkl ð34Þ

In other words, diffraction occurs whenever the scattering vector h equals a reciprocal lattice vector hhkl. This powerful result is visualized in the useful Ewald construction that is described below. Useful equivalent variations of the Bragg equation are: 2 sin y 1 jjh ¼ jjs s ¼ ¼ ð35Þ 0 l d and:

4p sin y 2p jjQ ¼ ¼ ð36Þ l d The vector Q is the physicist’s equivalent of the crystallographer’s h. The physical meaning of Q is the momentum transfer on scattering and differs from the scattering vector h by a factor of 2p. Principles of Powder Diffraction 11

1.5 THE EWALD CONSTRUCTION The Bragg equation shows that diffraction occurs when the scattering vector equals a reciprocal lattice vector. The scattering vector depends on the geo- metry of the experiment whereas the reciprocal lattice is determined by the orientation and the lattice parameters of the crystalline sample. Ewald’s con- struction combines these two concepts in an intuitive way. A sphere of radius 1/l is constructed and positioned in such a way that the Bragg equation is satisfied, and diffraction occurs, whenever a reciprocal lattice point coincides with the surface of the sphere (Figure 1.8). The recipe for constructing Ewald’s spherey is as follows (Figure 1.8):

1. Draw the incident wave vector s0. This points in the direction of the incident beam with length 1/l. 2. Draw a sphere centered on the tail of this vector with radius 1/l. The incident wave vector s0 defines the radius of the sphere. The scattered wave vector s, also of length 1/l, points in a direction from the sample to the detector. This vector is drawn also starting from the center of the sphere and also terminates at a point on the surface. The scattering vector h ¼ s – s0 completes the triangle from the tip of s to the tip of s0, both lying on the surface of the sphere. 3. Draw the reciprocal lattice with the origin lying at the tip of s0. 4. Find all the places on the surface of the sphere, where reciprocal lattice points lie.

This construction places a reciprocal lattice point at one end of h. By definition, the other end of h lies on the surface of the sphere. Thus, Bragg’s law is only satisfied, when another reciprocal lattice point coincides with the surface of the sphere. Diffraction is emanating from the sample in these directions. To detect the diffracted intensity, one simply moves the detector to the right position. Any vector between two reciprocal lattice points has the potential to produce a Bragg peak. The Ewald sphere construction additionally indicates which of these possible reflections satisfy experimental constraints and are therefore experimentally accessible. Changing the orientation of the crystal reorients the reciprocal lattice bringing different reciprocal lattice points on to the surface of the Ewald sphere. An ideal powder contains individual crystallites in all possible orientat- ions with equal probability. In the Ewald construction, every reciprocal lattice point is smeared out onto the surface of a sphere centered on the origin of * reciprocal space. This is illustrated in Figure 1.9. The orientation of the dhkl vector is lost and the three-dimensional vector space is reduced to one dimen- * sion of the modulus of the vector dhkl. These spherical shells intersect the surface of the Ewald sphere in circles. Figure 1.10 shows a two-dimensional projection. Diffracted beams emanate y For practical reasons, plots of the Ewald ‘‘sphere’’ are circular cuts through the sphere and the corresponding slice of reciprocal space. 12 Chapter 1

s Ewald phere

1/ λ

θ s θ θ

0 d*=s-s

0

s

0

Figure 1.8 Geometrical construction of the Ewald circle. The ‘‘0’’ marks the origin of reciprocal space. The vectors are defined in the text. from the sample in the directions where the thin circles from the smeared reciprocal lattice intersect the thick circle of the Ewald sphere. A few repre- sentative reflected beams are indicated by the broken lines. The lowest d-spacing reflections accessible in the experiment are determined by the diameter of the Ewald sphere 2/l. To increase the number of detectable reflections one must decrease the incident wavelength. In the case of an energy dispersive experiment such as time-of-flight neutron powder diffraction, which makes use of a continuous distribution of wavelengths from lmin to lmax at fixed angle, all smeared out cones between the two limiting Ewald spheres can be detected. In three dimensions, the circular intersection of the smeared reciprocal lattice with the Ewald sphere results in the diffracted X-rays of the reflection hkl forming coaxial cones, the so-called Debye–Scherrer cones (Figure 1.11). The smearing of reciprocal space in a powder experiment makes the measurement easier but results in a loss of information. Reflections overlap from lattice planes whose vectors lie in different directions but which have the same d-spacing. These cannot be resolved in the measurement. Some of these overlaps are dictated by symmetry (systematic overlaps) and others are accidental. Systematic overlaps are less serious for equivalent reflections [e.g., the six Bragg peaks (100), ( 100), (010), . . . from the faces of a cube] since the multiplicity is known from the symmetry. For highly crystal- line samples, accidental overlaps can be reduced by making measurements Principles of Powder Diffraction 13

0

Figure 1.9 Illustration of the reciprocal lattice associated with a single-crystal lattice (left) and a large number of randomly oriented crystallites (right). A real powder consists of so many grains that the dots of the reciprocal lattice form into continuous lines. with higher resolution, or by taking data at different temperatures in an attempt to remove the overlap by differential thermal expansion of different cell parameters. To obtain the maximum amount of information, a spherical shell detector would be desirable, though currently impractical. Often, a flat two-dimensional detector, either film, image plate, or CCD is placed perpendicular to the direct beam. In this case, the Debye–Scherrer cones appear as circles as shown in Figure 1.12a. For an ideal powder, the intensity around the rings is isotropic. Conven- tional powder diffraction measurements, e.g., Bragg–Brentano geometry, take one-dimensional cuts through the rings, either horizontally or vertically depending on the geometry of the diffractometer. If the full rings, or fractions of them, are detected with two-dimensional detectors the counting statistics can be improved by integrating around the rings. If the powder is non-ideal, the ring intensity is no longer uniform, as illustrated in Figure 1.12b, giving arbitrary intensities for the reflections in a one-dimensional scan. To improve powder statistics, powder samples are generally rotated during measure- ment. However, the intensity variation around the rings can give important 14 Chapter 1

d>=λ phere /2 g s itin m Li

eam

Primary b

Reflected beams

Figure 1.10 Illustration of the region of reciprocal space that is accessible in a powder measurement. The smaller circle represents the Ewald sphere. As shown in Figure 1.9, in a powder the reciprocal lattice is rotated to sample all orientations. An equivalent operation is to rotate the Ewald sphere in all possible orientations around the origin of reciprocal space. The volume swept out (area in the figure) is the region of reciprocal space accessible in the experiment.

hkl Reflected beam

Primary beam

hkl

Reflected beam hkl 4

Primary beam Θ

hkl Figure 1.11 Comparison between the scattered beams originating from a single crystal (top) and a powder (bottom). For the latter, some Debye– Scherrer cones are drawn. Principles of Powder Diffraction 15

Figure 1.12 Debye–Scherrer rings from an ideal fine grained (a, left) and a grainy (b, right) powder sample. information about the sample such as preferred orientation of the crystallites or texture.

1.6 TAKING DERIVATIVES OF THE BRAGG EQUATION Several important relationships in crystallography directly follow from a derivative of the Bragg equation [Equation (5)]. First we rewrite Bragg’s law making the d-spacing the subject of the equation:

nl d ¼ ð37Þ 2 sin y

The uncertainty of the measured lattice spacing is given by the total deri- vative dd, which can be written according to the chain rule as:

@d @d dd ¼ dy þ dl ð38Þ @y @l leading to: nl cosy n dd ¼ dy þ dl ð39Þ 2 sin y sin y 2 sin y and finally: dd dy dl ¼ þ ð40Þ d tan y l

This equation allows us to discuss several physically important phenomena. 16 Chapter 1

Figure 1.13 Percentage error in measured d-spacing as a function of scattering angle arising from a constant angular misalignment of DY for a well aligned (0.0011), a typically aligned, (0.011) and a poorly aligned (0.051) diffractometer.

When a crystal is strained, the d-spacings vary. A macroscopic strain changes the interplanar spacing by Ddhkl, giving rise to a shift in the average position of the diffraction peak of Dy, while microscopic strains give a distribution of d-spacings Ddhkl which broaden the peak by dy. This is discussed in detail in Chapters 12 and 13. A constant angular offset due to misalignment of the diffractometer gives rise to a nonlinear error in our determination of dhkl, disproportionately affecting low angle reflections (Figure 1.13). Similarly, our ability to resolve two partially overlapping reflections separated by Ddhkl is limited by the finite angular resolution Dy of the diffractometer. There are many geometrical contributions to the angular resolution (e.g., angular width of the receiving slit in front of the detector). Another con- tribution comes from finite wavelength spread of the incident beam Dl. From Equation (40) we get the angular dispersion to be:

dy tan y ¼ ð41Þ dl l

This is plotted in Figure 1.14, which shows that the resolution due to a finite spread in l is decreasing at higher angles. In a real experiment the angle dependence of the resolution function can be complicated. In traditional modeling programs the Bragg-peak line shapes are modeled using empirical line-shape functions. More recently, approaches have been developed that explicitly account for the different physical processes that result in the line Principles of Powder Diffraction 17

Figure 1.14 Angular dependence of the intrinsic peak width (resolution function) of the diffractometer due to the wavelength spread between Cu-Ka1 and Cu-Ka2 (about 12 eV). shapes. This is called the fundamental parameters approach and is described in Chapters 5, 6 and 13.

1.7 BRAGG’S LAW FOR FINITE SIZE CRYSTALLITES Assuming an infinite stack of lattice planes, Bragg’s equation gives the position of delta-function Bragg peaks. Finite size crystallites give rise to Bragg peaks of finite width. This size broadening is described by the Scherrer equation. We now reproduce the simple derivation following Klug and Alexander (1974; see Bibliography). Figure 1.15 shows the path length difference versus the depth of the lattice plane. When the angle between the incoming beam and the lattice plane Y is different by an amount e from the Bragg condition, it is always possible to find a lattice plane inside the crystal where the extra path is D ¼ l/2 producing destructive interference. For a thick crystal this is true for arbitrarily small e, which explains the sharp Bragg reflections. For a crystal with finite dimensions, 1 for small e the plane for which D ¼ðn þ 2Þl holds will not be reached. In this case there is not perfect cancellation of the intensity away from the Bragg condition, thus leading to an intensity distribution over some small angular range. We can use this idea to estimate the broadening of a Bragg reflection due to size effects. The thickness of a crystallite in the direction perpendicular to p (hkl) planes of separation dhkl (Figure 1.15) is:

Lhkl ¼ pdhkl ð42Þ 18 Chapter 1

+½) (λ =0 ∆ ∆ ∆ =n ∆ ∆ θ+εΘ+ε 2 3 1

2 d 3 2θ 4

p-1

p Figure 1.15 Path length difference of the scattered ray versus the depth of the lattice plane in the crystal.

The additional beam path between consecutive lattice planes at the angle y + e is:

D ¼ 2d sinðy þ eÞ ¼ 2dðsin y cos e þ cos y sin eÞ ð43Þ ¼ nl cos e þ sin e2d cos y E nl þ sin e2d cos y

The corresponding phase difference is then: D 4p 4ped cos y dj ¼ 2p ¼ 2pn þ ed cos y ¼ ð44Þ l l l

The phase difference between the top and the bottom layer, p is then: 4ped cos y 4pL e cos y dj ¼ p ¼ hkl ð45Þ l l

Rearranging Equation (45) leads to: ldj e ¼ ð46Þ 4pLhkl cos y which gives an expression for the misalignment angle in terms of the crystallite size Lhkl and the phase difference dj between the reflections between the top and the bottom plane. Clearly, the scattered intensity is at a maximum for dj ¼ 0 (e ¼ 0). With increasing e the intensity decreases giving rise to a peak of finite width. Perfect cancellation of the top and bottom waves occurs a phase difference of dj ¼p at which point e ¼l/(4Lhkl cosy). On a measured 2y-scale the Principles of Powder Diffraction 19 measured angular width between these points is: l bhkl ¼ 4e ¼ ð47Þ Lhkl cos y giving us some measure of the peak width in radians due to the finite particle size. A full treatment taking into account the correct form for the intensity distribution gives: Kl bhkl ¼ ð48Þ Lhkl cos y with a scale factor of K ¼ 0.89 for perfect spheres. In general K depends on the shape of the grains (e.g., K is 0.94 for cubic shaped grains) but is always close to unity. This equation is not valid for crystallitesz that are too large or too small. With large crystallites the peak width is governed by the coherence of the incident beam and not by particle size. For nano-scale crystallites, Bragg’s law fails and needs to be replaced by the Debye equation (see Chapter 16).

BIBLIOGRAPHY 1. V. K. Pecharsky and P. Y. Zavalij, Fundamentals of Powder Diffraction and Structural Characterization of Materials, Kluwer Academic Publishers, Boston, 2003. 2. D. L. Bish, J. E. Post (Eds.), Modern Powder Diffraction, Reviews in Mineralogy Volume 20, Mineralogical Society of America, Chantilly VA, 1989. 3. H. P. Klug and L. E. Alexander, X-ray Diffraction Procedures, 2nd edn., John Wiley & Sons, New York, 1974.

z Strictly speaking, the term crystallite size here refers to the dimension of a coherently scattering domain. Only in a perfect crystal, is this the grain size. http://www.springer.com/978-0-85404-231-9