Powder Diffraction: Theory and Practice / Edition 1

by R E Dinnebier, S J L Billinge

Overview

Powder diffraction is a widely used scientific technique in the characterization of materials with broad application in materials science, chemistry, physics, geology, pharmacology and archaeology. Powder Diffraction: Theory and Practice provides an advanced introductory text about modern methods and applications of powder diffraction in research and industry. The authors begin with a brief overview of the basic theory of diffraction from crystals and powders. Data collection strategies are described including x-ray, neutron and electron diffraction setups using modern day apparatus including synchrotron sources. Data corrections, essential for quantitative analysis are covered before the authors conclude with a discussion of the analysis methods themselves. The information is presented in a way that facilitates understanding the information content of the data, as well as best practices for collecting and analyzing data for quantitative analysis. This long awaited book condenses the knowledge of renowned experts in the field into a single, authoritative, overview of the application of powder diffraction in modern materials research. The book contains essential theory and introductory material for students and researchers wishing to learn how to apply the frontier methods of powder diffraction

Product Details

ISBN-13:	9780854042319
Publisher:	RSC
Publication date:	03/10/2008
Pages:	604
Product dimensions:	6.30(w) x 9.30(h) x 1.50(d)

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Powder Diffraction

Theory and Practice

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By Robert E. Dinnebier, Simon J. L. Billinge

The Royal Society of Chemistry

Copyright © 2008 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-0-85404-231-9

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CHAPTER 1

Principles of Powder Diffraction

ROBERT E. DINNEBIER AND SIMON J. L. BILLINGE

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 [Angstrom] (= 1 × 10-10 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 2 π 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, δ[??]on an angular scale, such that:

Δ/λ = δφ/2π [??] δφ = 2π/Δ (1)

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 + A2)2, and δφ= π (destructive interference), where I = (A1 – A2)2. In general, I = [A1 +A2 exp (iδφ)]2. When more than two waves are present, this equation becomes:

I = [[summation over j] Aj exp (iφj)]2, (2)

where the sum is over all the sine-waves present and the phases, [??]j are measured with respect to some origin.

X-ray diffraction involves the measurement of the intensity of X-rays 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 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 ≈ D/L2 ≈ 10-10. The Fraunhofer approximation greatly simplifies the mathematics. 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 φj 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 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 Δ = PN + NQ is a multiple n = 0, [+ or -] 1, [+ or -] 2, ... of the wavelength λ:

Δ = nλ (3)

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 = [+ or -] 1/2, [+ or -] 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:

Δ = 2d sin θ (4)

where d is the interplanar spacing of parallel lattice planes and 2θ is the diffraction angle, the angle between the incoming and outgoing X-ray beams. Combining Equations (3) and (4) we get:

n λ = 2d sin θ (5)

the famous Bragg equation.

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:

nλ = MN cos (180° - (α + θ)) + MN cos (α - θ) = MN-cos (α + θ) + cos (α - θ)] (6)

From any textbook on trigonometry we know that:

cos(α + θ) = cos α cos θ - sin α sin θ cos(α - θ) = cos α cos θ + sin α sin θ (7)

Therefore Equation (7) becomes:

nλ = MN [2sin α sin θ] (8)

with:

d = MN sin α (9)

from which the already known Bragg equation follows:

nlambda] = 2d sin θ (10)

Another equivalent, and highly useful, expression of the Bragg equation is:

Ed = 6.199/sin θ with λ [[Angstrom]] = 12.398/E[keV] (11)

with the energy E of the X-rays 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 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, α, β, y, and the volume V of the unit cell, a second lattice with lattice parameters of a*, b*, c*, α*, β *, y*, and the volume V*, and with the same origin, exists such that:

a • b* = a • c* = b • c* = a* • b = a* • c = b* • c = 0+ (12) a • a* = b • b* = c • c* = 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 consider 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:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

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

a* = x (b × c) (14)

where the scale factor x can easily be deduced using Equation (12) as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

leading to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

and vice versa:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

The relationship between the reciprocal and the real lattice parameters is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

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 s0 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), s0 and s have the same length.

We define the scattering vector as:

h = (s - s0) (19)

which for a specular reflection is always perpendicular to the scattering plane. The length of h is given by:

h/s = 2 sin θ (20)

Comparison with the formula for the Bragg equation [Equation (5)]:

nλ/d = 2 sin θ (21)

we get:

n θ/d = h/s (22)

Setting the magnitude of s to 1/λ, we get the Bragg equation in terms of the magnitude of the scattering vector h:

h = n/d (23)

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:

h = nd* (24)

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:

d*nh,nk,nl = nd*hkl (25)

e.g., d*222 = 2d*111 and we get an alternative expression for Bragg's equation:

h = d*hkl (26)

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 dhkl in terms of real space basis vectors a, b, c. Referring to Figure 1.7, we can define that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

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 = (OA)dcosα. From Figure 1.7 we see that, geometrically, (OA)cosα = d. Substituting, we get OA • d = d2. Combining with Equation (27) leads to:

1/h a • d = d2 (28)

and consequently:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

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:

d*hkl = d/d2 (30)

which gives:

d*hkl = ha + kb + lc (31)

or written in terms of the reciprocal basis:

d*hkl = ha* + kb* + lc*(32)

which was obtained using:

d*hkl • a* = ha • a* + kb • a* + lc • a* = h (33)

d*hkl • b* = ha • b* + kb • b* + lc • b* = k

d*hkl • c* = ha • c* + kb • c* + lc • c* = l

Comparing Equation (32) with Equation (13) proves the identity of d*hkl and the reciprocal lattice vector hhkl Bragg's equation, Equation (26), can be restated as:

h = hhkl (34)

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:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

and:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

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):

1. Draw the incident wave vector s0. This points in the direction of the incident beam with length 1/λ.

2. Draw a sphere centered on the tail of this vector with radius 1/λ. The incident wave vector s 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 – 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 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 three-dimensional 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 two-dimensional 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 d-spacing 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 time-of-flight neutron powder diffraction, which makes use of a continuous distribution of wavelengths from λmin to λmax at fixed angle, all smeared out cones between the two limiting Ewald spheres can be detected.

(Continues...)

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Excerpted from Powder Diffraction by Robert E. Dinnebier, Simon J. L. Billinge. Copyright © 2008 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Chapter 1: Principles of Powder Diffraction;
Chapter 2: Experimental Setups;
Chapter 3: The Intensity of a Bragg Reflection;
Chapter 4: General Data Reduction;
Chapter 5: The Profile of a Bragg Reflection for Extracting Intensities;
Chapter 6: Instrumental Contributions to the Line Profile in X-ray Powder Diffraction. Example of the Diffractometer with Bragg-Brentano Geometry;
Chapter 7: Indexing and Space Group Determination;
Chapter 8: Crystal Structure Determination;
Chapter 9: Rietveld Refinement;
Chapter 10: The Derivative Difference Minimization Method;
Chapter 11: Quantitative Phase Analysis;
Chapter 12: Microstructural Properties: Texture and Macrostress Effects;
Chapter 13: Microstructural Properties: Lattice Defects and Domain Size Effects;
Chapter 14: Two-dimensional Powder Diffraction using Area Detectors: The Experimental Setup;
Chapter 15: Powder Diffraction at Non-ambient Conditions;
Chapter 16: Local Structure from Total Scattering and Atomic Pair Distribution Function (PDF) Analysis;
Chapter 17: Computer Software for Powder Diffraction

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"This book provides an excellent overview and much detail of the state-of the-art in powder diffraction methods." "Powder diffraction theory and practice will be an invaluable resource for experienced practitioners and new research students alike. It provides an approachable, comprehensive description of modern methods and their computational implementations,..."

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"The book has some excellent contributions that have perhaps seven or eight real highlights."...an excellent starting point for many researchers and will be a useful addition to the shelves of those people involved with, or actively practising, powder diffraction methods."

1476 - 3508 online Crystallography Reviews

"The authors of chapters have done a very good job of summarizing current ideas in the field of powder diffraction, and the editors and publisher have likewise maintained the coherence, consistency and quality of the finished product. It provides an excellent bridge between elementary texts and current research literature. I believe that there is room for this book on the shelf (and more importantly, in the hands) of nearly everyone who is using powder diffraction as a research technique. "

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"This book provides an excellent overview and much detail of the state-of the-art in powder diffraction methods."Powder diffraction theory and practice will be an invaluable resource for experienced practitioners and new research students alike. It provides an approachable, comprehensive description of modern methods and their computational implementations,..."

Chemistry World

"This book provides an excellent overview and much detail of the state-of the-art in powder diffraction methods." "Powder diffraction theory and practice will be an invaluable resource for experienced practitioners and new research students alike. It provides an approachable, comprehensive description of modern methods and their computational implementations,..."

From the Publisher