Solid State Physics. Philip Hofmann

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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)

      The wave vector points in the direction of the wave propagation with a length of , where 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)

Consider now the situation depicted in Figure 1.10. The source of the X‐rays is far away from the sample at the position so that the X‐ray wave at the sample can be described as a plane wave. The electric field at a point in the crystal at time can be written as

      (1.6)

      Before we proceed, we can drop the absolute amplitude from this expression because we are only interested in relative phase changes. The field at point is then

(1.7)

Figure 1.10 Illustration of X‐ray scattering from a sample. The source and detector for the X‐rays are placed at and , respectively. Both are very far away from the sample.

A small volume element located at will give rise to scattered waves in all directions. The direction we are interested in is that towards the detector, which we assume to be placed at position , in the direction of a second wave vector . We assume that the amplitude of the wave scattered in this direction will be proportional to the incoming field from Eq. (1.7) and to a factor describing the scattering probability and scattering phase. We already know that the scattering of X‐rays proceeds via the electrons in the material, and for our purpose, we can view as the electron concentration in the solid. For the field at the detector, we obtain

      (1.8)

      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

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