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normal e Superscript normal i bold k dot left-parenthesis bold r minus bold upper R right-parenthesis Baseline rho left-parenthesis bold r right-parenthesis normal e Superscript normal i bold k Super Superscript prime Superscript dot left-parenthesis bold upper R bold prime minus bold r right-parenthesis Baseline normal e Superscript minus normal i omega t Baseline equals normal e Superscript normal i left-parenthesis bold k prime dot bold upper R Super Superscript bold prime Superscript minus bold k dot bold upper R right-parenthesis Baseline rho left-parenthesis bold r right-parenthesis normal e Superscript normal i left-parenthesis bold k minus bold k Super Superscript prime Superscript right-parenthesis dot bold r Baseline normal e Superscript minus normal i omega t Baseline period"/>

      (1.10)bold-script upper E left-parenthesis bold upper R prime comma t right-parenthesis proportional-to normal e Superscript minus normal i omega t Baseline integral Underscript upper V Endscripts rho left-parenthesis bold r right-parenthesis normal e Superscript normal i left-parenthesis bold k minus bold k Super Superscript prime Superscript right-parenthesis dot bold r Baseline normal d upper V period

      In most cases, it will only be possible to measure the intensity of the X‐rays and not the field amplitude. This intensity is given by

      where we have introduced the so‐called scattering vector bold upper K equals bold k prime minus bold k, which is just the difference of the outgoing and incoming wave vectors. Note that although the direction of the wave vector bold k prime for the scattered waves is different from that of the incoming wave bold k, their lengths are the same because we consider elastic scattering only.

      Equation 1.11 is our final result. It relates the measured intensity to the electron concentration in the sample. Except for very light elements, most of the electrons are located close to the ion cores and the electron concentration that scatters the X‐rays is essentially identical to the geometrical arrangement of the atomic cores. Hence, Eq. (1.11) can be used for the desired structure determination. To this end, one could try to measure the intensity as a function of scattering vector bold upper K and to infer the structure from the result. This is a formidable task, however. It is greatly simplified by the fact that the specimen under investigation is a crystal with a periodic lattice. In the following, we introduce the mathematical tools and concepts that are needed to exploit the crystalline structure in the analysis. The most important of these is the so‐called reciprocal lattice.

      1.3.1.4 The Reciprocal Lattice

      The concept of the reciprocal lattice is fundamental to solid state physics because it permits us to exploit crystal symmetry in the analysis of many problems. Here, we will use it to describe X‐ray diffraction from periodic structures and we will continue to meet it again in the following chapters. Unfortunately, the meaning of the reciprocal lattice turns out to be difficult to grasp. We will start out with a formal definition and provide some of its mathematical properties. We then go on to discuss the meaning of the reciprocal lattice before we come back to X‐ray diffraction. The full importance of the concept will become apparent in the course of this book.

      For a given Bravais lattice

      (1.12)bold upper R equals m bold a bold 1 plus n bold a bold 2 plus o bold a bold 3 comma

      we define the reciprocal lattice as the set of vectors bold upper G for which

      where l is an integer. Equivalently, we could require that