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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).
1.3.1.5 The Meaning of the Reciprocal Lattice
We have now defined the reciprocal lattice in a proper way, and we will give some simple examples of its usefulness. The most important feature of the reciprocal lattice is that it facilitates the description of functions with the same periodicity as that of the lattice. To see this, consider a one‐dimensional lattice, a chain of points with a lattice constant (Fig. 1.11). We are interested in a function with the periodicity of the lattice, such as the electron concentration along the chain, . We can write this as a Fourier series of the form
(1.19)
with real coefficients and . The sum starts at , the constant is therefore outside the summation. Using complex coefficients , we can also write this in the more compact form
(1.20)
To ensure that is still a real function, we require that
(1.21)
that is, that the coefficient must be the complex conjugate of the coefficient . This description is more elegant than the one with the sine and cosine functions. How is it related to the reciprocal lattice? In one dimension, the reciprocal lattice of a chain of points with lattice constant is also a chain of points, now with spacing [see Eq. (1.17)]. This means that we can write a general reciprocal lattice “vector” as
(1.22)
where is an integer. Exactly these reciprocal lattice “vectors” appear in Eq. (1.20). In fact, Eq. (1.20) is a sum of functions with a periodicity corresponding to the lattice vector, weighted by the coefficients . Figure 1.11 illustrates these ideas by showing the lattice and reciprocal lattice for such a chain as well as two lattice‐periodic functions, both in real space and as Fourier coefficients on the reciprocal lattice points. The advantage of describing these functions by the coefficients is immediately obvious: Instead of giving for every point in a range of , the Fourier description consists of just three numbers for the upper function and five numbers for the lower function. Actually, these even reduce to two and three numbers, respectively, because of Eq. (1.21).
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