Chapter 1: Principles of Powder Diffraction

Published:10 Mar 2008

Product Type: Textbooks
R. E. Dinnebier and S. J. L. Billinge, in Powder Diffraction: Theory and Practice, ed. R. E. Dinnebier and S. J. L. Billinge, The Royal Society of Chemistry, 2008, ch. 1, pp. 119.
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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
Xrays are electromagnetic (em) waves with a much shorter wavelength than visible light, typically on the order of 1 Å (= 1 × 10^{−10} m). The physics of emwaves 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 emwaves can be described by a sine wave that repeats periodically every 2π radians. The spatial length of each period is the wavelength λ. 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, Δ on a length scale, in the units of the wavelength, or equivalently as a phase shift, δϕ on an angular scale, such that:
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 δφ. The two extremes occur when δφ = 0 (constructive interference), where I = (A_{1} + A_{2})_{2}, and δφ = π (destructive interference), where I = (A_{1} − A_{2})_{2}. In general, I = [A_{1} +A_{2} exp (iδφ)]_{2}. When more than two waves are present, this equation becomes:
where the sum is over all the sinewaves present and the phases, ϕ_{j} are measured with respect to some origin.
Xray diffraction involves the measurement of the intensity of Xrays scattered 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).
In the case of Xray diffraction, the Fraunhofer approximation is used to calculate the detected intensities. This is a farfield approximation, where the distance, L_{1}, from the source to the place where scattering occurs (the sample), and then on to the detector, L_{2}, is much larger than the separation, D, of the scatterers. This is an excellent approximation, since in this case D/L_{1} ≈ D/L_{2} ≈ 10^{−10}. The Fraunhofer approximation greatly simplifies the mathematics. The incident Xrays form a wave such that the constant phase wave front is a plane wave. Xrays scattered by single electrons are outgoing spherical waves that again appear as plane waves in the farfield. This allows us to express the intensity of diffracted Xrays using Equation (2).
The phases φ_{j} 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 wavevectors 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 Xray diffraction experiment using single crystals of copper sulfate and zinc sulfite, proving the hypothesis that Xrays are emwaves 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 wellknown Bragg equation (W. L. Bragg, 1912), which describes the principle of Xray diffraction in terms of a reflection of Xrays by sets of lattice planes. Lattice planes are crystallographic planes, characterized by the index triplet hkl, the socalled Miller indices. Parallel planes have the same indices and are equally spaced, separated by the distance d_{hkl}. Bragg analysis treats Xrays like visible light being reflected by the surface of a mirror, with the Xrays being specularly reflected at the lattice planes. In contrast to the lower energy visible light, the Xrays penetrate deep inside the material where additional reflections occur at thousands of consecutive parallel planes. Since all Xrays 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 Δ = PN + NQ is a multiple n = 0, ±1, ±2, … of the wavelength λ:
In all other cases, destructive interference results since it is always possible to find a deeper plane, p, for which the relation pΔ = nλ with n = ±1/2, ± 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:
where d is the interplanar spacing of parallel lattice planes and 2θ is the diffraction angle, the angle between the incoming and outgoing Xray beams. Combining Equations (3) and (4) we get:
the famous Bragg equation.
This simplified derivation of the Bragg equation, although leading to the correct solution, has a serious drawback. In reality the Xrays 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:
From any textbook on trigonometry we know that:
Therefore Equation (7) becomes:
with:
from which the already known Bragg equation follows:
Another equivalent, and highly useful, expression of the Bragg equation is:
with the energy E of the Xrays in keV.
To derive the Bragg equation, we used an assumption of specular reflection, which is borne out by experiment. For crystalline materials, destructive interference 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 socalled 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, α, β, γ, and the volume V of the unit cell, a second lattice with lattice parameters of a*, b*, c*, α*, β*, γ*, and the volume V*, and with the same origin, exists such that:
This is known as the reciprocal lattice,^{†} which exists in socalled 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 consider h, k and l to be integers that index a point in the reciprocal lattice. A reciprocal lattice vector h_{hkl} is the vector from the origin of reciprocal space to the (hkl) reciprocal lattice point:^{‡}
The length of the reciprocal base vectors is defined according to:
where the scale factor x can easily be deduced using Equation (12) as:
leading to:
and vice versa:
The relationship between the reciprocal and the real lattice parameters is:
Equation (18) simplifies considerably for higher symmetry crystal systems.
We now rederive Bragg's law using vector notation. The wave vectors of the incoming and outgoing beams are given by s_{0} and s, respectively (Figure 1.6). They point in the direction of propagation of the wave and their length depends on λ. For elastic scattering (no change in wavelength on scattering), s_{0} and s have the same length.
We define the scattering vector as:
which for a specular reflection is always perpendicular to the scattering plane. The length of h is given by:
Comparison with the formula for the Bragg equation [Equation (5)]:
we get:
Setting the magnitude of s to 1/λ, we get the Bragg equation in terms of the magnitude of the scattering vector h:
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* perpendicular to the lattice planes with length 1/d. Since h is perpendicular the scattering plane, this leads to:
Diffraction can occur at different scattering angles 2θ for the same crystallographic plane, giving the different orders n of diffraction. For simplicity, the number n will be incorporated in the indexing of the lattice planes, where:
e.g., d$222*$ = 2d$111*$ and we get an alternative expression for Bragg's equation:
The vector d$hkl*$ 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 d_{hkl} in terms of real space basis vectors a, b, c. Referring to Figure 1.7, we can define that:
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 d_{hkl} originates on one plane and terminates on the next parallel plane. Therefore, OA·d = (OA)dcosα. From Figure 1.7 we see that, geometrically, (OA)cosα = d. Substituting, we get OA·d = d^{2}. Combining with Equation (27) leads to:
and consequently:
By definition, h, k, and l are divided by their largest common integer to be Miller indices. The vector d$hkl*$, from Bragg's Equation (26) points in the plane normal direction parallel to d but with length 1/d. We can now write d$hkl*$ in terms of the vector d:
which gives:
or written in terms of the reciprocal basis:
which was obtained using:
Comparing Equation (32) with Equation (13) proves the identity of d$hkl*$ and the reciprocal lattice vectorh h_{hkl}. Bragg's equation, Equation (26), can be restated as:
In other words, diffraction occurs whenever the scattering vector h equals a reciprocal lattice vector h_{hkl}. This powerful result is visualized in the useful Ewald construction that is described below.
Useful equivalent variations of the Bragg equation are:
and:
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 2π.
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 geometry of the experiment whereas the reciprocal lattice is determined by the orientation and the lattice parameters of the crystalline sample. Ewald's construction combines these two concepts in an intuitive way. A sphere of radius 1/λ 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 sphere^{§} is as follows (Figure 1.8):
Draw the incident wave vector s_{0}. This points in the direction of the incident beam with length 1/λ.
Draw a sphere centered on the tail of this vector with radius 1/λ. The incident wave vector s_{0} defines the radius of the sphere. The scattered wave vector s, also of length 1/λ, 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 = s – s_{0} completes the triangle from the tip of s to the tip of s_{0}, both lying on the surface of the sphere.
Draw the reciprocal lattice with the origin lying at the tip of s_{0}.
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 orientations 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 d$hkl*$ vector is lost and the threedimensional vector space is reduced to one dimension of the modulus of the vector d$hkl*$.
These spherical shells intersect the surface of the Ewald sphere in circles. Figure 1.10 shows a twodimensional projection. Diffracted beams emanate 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 representative reflected beams are indicated by the broken lines.
The lowest dspacing reflections accessible in the experiment are determined by the diameter of the Ewald sphere 2/λ. To increase the number of detectable reflections one must decrease the incident wavelength. In the case of an energy dispersive experiment such as timeofflight neutron powder diffraction, which makes use of a continuous distribution of wavelengths from λ_{min} to λ_{max} 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 Xrays of the reflection hkl forming coaxial cones, the socalled 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 dspacing. 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 crystalline samples, accidental overlaps can be reduced by making measurements 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 twodimensional 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. Conventional powder diffraction measurements, e.g., Bragg–Brentano geometry, take onedimensional 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 twodimensional detectors the counting statistics can be improved by integrating around the rings. If the powder is nonideal, the ring intensity is no longer uniform, as illustrated in Figure 1.12b, giving arbitrary intensities for the reflections in a onedimensional scan. To improve powder statistics, powder samples are generally rotated during measurement. However, the intensity variation around the rings can give important 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 dspacing the subject of the equation:
The uncertainty of the measured lattice spacing is given by the total derivative dd, which can be written according to the chain rule as:
leading to:
and finally:
This equation allows us to discuss several physically important phenomena.
When a crystal is strained, the dspacings vary. A macroscopic strain changes the interplanar spacing by Δd_{hkl}, giving rise to a shift in the average position of the diffraction peak of Δθ, while microscopic strains give a distribution of dspacings Δd_{hkl} which broaden the peak by δθ. 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 d_{hkl}, disproportionately affecting low angle reflections (Figure 1.13). Similarly, our ability to resolve two partially overlapping reflections separated by Δd_{hkl} is limited by the finite angular resolution Δθ 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 contribution comes from finite wavelength spread of the incident beam Δλ. From Equation (40) we get the angular dispersion to be:
This is plotted in Figure 1.14, which shows that the resolution due to a finite spread in λ is decreasing at higher angles. In a real experiment the angle dependence of the resolution function can be complicated. In traditional modeling programs the Braggpeak line shapes are modeled using empirical lineshape functions. More recently, approaches have been developed that explicitly account for the different physical processes that result in the line 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 deltafunction 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 Θ is different by an amount ε from the Bragg condition, it is always possible to find a lattice plane inside the crystal where the extra path is Δ = λ/2 producing destructive interference. For a thick crystal this is true for arbitrarily small ε, which explains the sharp Bragg reflections. For a crystal with finite dimensions, for small ε the plane for which 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 d_{hkl} (Figure 1.15) is:
The additional beam path between consecutive lattice planes at the angle θ + ε is:
The corresponding phase difference is then:
The phase difference between the top and the bottom layer, p is then:
Rearranging Equation (45) leads to:
which gives an expression for the misalignment angle in terms of the crystallite size L_{hkl} and the phase difference δφ between the reflections between the top and the bottom plane. Clearly, the scattered intensity is at a maximum for δφ = 0 (ε = 0). With increasing ε the intensity decreases giving rise to a peak of finite width. Perfect cancellation of the top and bottom waves occurs a phase difference of δφ = ±π at which point ε = ± λ/(4L_{hkl} cosθ). On a measured 2θscale the measured angular width between these points is:
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:
with a scale factor of K = 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 crystallites^{¶} 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 nanoscale crystallites, Bragg's law fails and needs to be replaced by the Debye equation (see Chapter 16).
Vectors are in bold.
The reciprocal lattice is a commonly used construct in solid state physics, but with a different normalization: a·a*=2π.
For practical reasons, plots of the Ewald “sphere” are circular cuts through the sphere and the corresponding slice of reciprocal space.
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.