The scientist describes what is; the engineer creates what never was.
Theodore von Kármán, quoted in A. L. Mackay, Dictionary of Scientific Quotations (1994)
In this chapter, we will introduce the concept of reciprocal space. We will show that reciprocal space allows us to interpret the Miller indices h, k, and l of a plane as the components of a vector; not just any vector, but the normal to the plane (hkl). We will also show that the length of this vector is related to the spacing between consecutive (hkl) planes. This will involve the concept of the reciprocal metric tensor, a device used for computations in reciprocal space. We conclude this chapter with a series of example computations.
At first, you will probably find this whole reciprocal space business a bit abstract and difficult to understand. This is normal. It will take a while for you to really understand what is meant by reciprocal space. So, be patient; reciprocal space is probably one of the most abstract topics in this book, which means that an understanding will not come immediately. It is important, however, that you persist in trying to understand this topic, because it is of fundamental importance for everything that has to do with diffraction experiments.
The reciprocal basis vectors
In the previous chapter, we introduced a compact notation for an arbitrary plane in an arbitrary crystal system. The Miller indices (hkl) form a triplet of integer numbers and fully characterize the plane. It is tempting to interpret the Miller indices as the components of a vector, similar to the components [uvw] of a lattice vector t. This raises a few questions: if h, k, and lare indeed the components of a vector, then how does this vector relate to the plane (hkl)? Furthermore, since vector components are always taken with respect to a set of basis vectors, we must ask which are the relevant basis vectors for the components (h, k, l)?