Powder Diffraction

Theory and Practice

R E Dinnebier, S J L Billinge, Peter G Bruce, R E Dinnebier, S J L Billinge

• Publisher: Royal Society of Chemistry
• Year: 2015
• ISBN:
  9781782625995
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physical & Theoretical Chemistry

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

CHAPTER 1

Principles of Powder Diffraction

ROBERT E. DINNEBIER^a AND SIMON J. L. BILLINGE^b

^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 Å (= 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 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:

(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 = (A₁ + 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:

(2)

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

Figure 1.1 Graphical illustration of the phase shift between two sine waves of equal amplitude.

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).

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 ≈ 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 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 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, Knippi...