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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)

From this, one can derive the simple but useful property²

(1.17)

which can easily be verified. Equation (1.17) can then be used to verify that the reciprocal lattice vectors defined by Eqs. (1.15) and (1.16) do indeed fulfill the fundamental property of Eq. (1.13) defining the reciprocal lattice (see also Problem 6).

Another way to view the vectors of the reciprocal lattice is as wave vectors that yield plane waves with the periodicity of the Bravais lattice, because

(1.18)

Using the reciprocal lattice, we can finally define the Miller indices in a much simpler way: The Miller indices define a plane that is perpendicular to the reciprocal lattice vector (see Problem 9).