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for example, for the leftmost plane in Figure 1.9.

      2 We take the “reciprocal value” of these three numbers. For our example, this gives .

      3 We multiply the numbers obtained in this manner with some factor so that we arrive at the smallest set of integers having the same ratio. In the example given, this is not necessary as all number are already integers.

      Such a set of three integers can then be used to denote any given lattice plane. Later, we will encounter a different and more elegant definition of the Miller indices.

      In practice, the X‐ray diffraction peaks are so sharp that it is difficult to align and move the sample so that the incoming and reflected X‐rays lie in a plane normal to a certain crystal plane. An elegant way to circumvent this problem is to use a powder consisting of very small crystals instead of a large single crystal. This will not only ensure that some of the many crystals are oriented correctly to get constructive interference from a certain set of crystal planes, it will also automatically yield the interference pattern for all possible crystal planes.

      1.3.1.3 General Diffraction Theory

      The Bragg theory for X‐ray diffraction is useful for extracting the distances between lattice planes in a crystal, but it has its limitations. Most importantly, it does not provide any information on what the lattice actually consists of, that is, the basis. Also, the fact that the X‐rays are described as being reflected by planes is physically somewhat obscure. In the following, we will therefore discuss a more general description of X‐ray diffraction that goes back to M. von Laue.

      The physical process leading to X‐ray scattering is that the electromagnetic field of the X‐rays forces the electrons in the material to oscillate with the same frequency as that of the field. The oscillating electrons then emit new X‐rays that give rise to an interference pattern. For the following discussion, however, it is merely important that something scatters the X‐rays, not what it is.

      It is highly beneficial to use the complex notation for describing the electromagnetic X‐ray waves. For the electric field, a general plane wave can be written as

      (1.4)bold-script upper E left-parenthesis bold r comma t right-parenthesis equals bold-script upper E 0 normal e Superscript normal i bold k dot bold r minus normal i omega t Baseline period

      The wave vector bold k points in the direction of the wave propagation with a length of 2 normal pi slash lamda, where lamda is the wavelength. The convention is that the physical electric field is obtained as the real part of the complex field and the intensity of the wave is obtained as

      (1.5)upper I left-parenthesis bold r right-parenthesis equals StartAbsoluteValue bold-script upper E 0 normal e Superscript normal i bold k dot bold r minus normal i omega t Baseline EndAbsoluteValue squared equals StartAbsoluteValue bold-script upper E 0 EndAbsoluteValue squared period

      (1.6)bold-script upper E left-parenthesis bold r comma t right-parenthesis equals bold-script upper E 0 normal e Superscript normal i bold k dot left-parenthesis bold r minus bold upper R right-parenthesis minus normal i omega t Baseline period

      Before we proceed, we can drop the absolute amplitude bold-script upper E 0 from this expression because we are only interested in relative phase changes. The field at point bold r is then

Schematic illustration of illustration of X-ray scattering from a sample. The source and detector for the X-rays are placed at R and R′, respectively. Both are very far away from the sample.

      (1.8)bold-script upper E left-parenthesis bold upper R prime comma t right-parenthesis proportional-to bold-script upper E left-parenthesis bold r comma t right-parenthesis 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 period

      Again, we have assumed that the detector is very far away from the sample so that the scattered wave at the detector can be written as a plane wave. Inserting Eq. (1.7) gives the field at the detector as

      (1.9)bold-script upper E left-parenthesis bold upper R prime comma t right-parenthesis 
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