Solid State Physics. Philip Hofmann

We drop the first factor that does not contain and will thus not play a role for the interference of X‐rays emitted from different positions in the sample. The total wave field at the detector can finally be calculated by integrating over the entire volume of the crystal. As the detector is far away from the sample, the wave vector is essentially the same for all points in the sample. The result is therefore

      (1.10)

      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

(1.11)

      where we have introduced the so‐called scattering vector , which is just the difference of the outgoing and incoming wave vectors. Note that although the direction of the wave vector for the scattered waves is different from that of the incoming wave , 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 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)

      we define the reciprocal lattice as the set of vectors for which

(1.13)

      where is an integer. Equivalently, we could require that

(1.14)

Note that this equation must hold for any choice of the lattice vector and reciprocal lattice vector . We can write any as the linear combination of three vectors

(1.15)

where , and are integers. The reciprocal lattice is also a Bravais lattice. The vectors , , and spanning the reciprocal lattice can be constructed explicitly from the lattice vectors ¹

(1.16)