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Unformatted text preview: Solid State Physics Chapter 5 The Reciprocal Lattice James Glasbrenner University of Nebraska May 22, 2007 Here we will investigate the reciprocal lattice from a general point of view. It will not be tied to any particular application. 1 Definition of Reciprocal Lattice Consider a set of points ~ R that constitute a Bravais lattice, and also a standard plane wave given by e i ~ k · ~ r . As we know, for a general ~ k , the plane wave will not be periodic in the sense of a Bravais lattice. However, we should be able to choose some wave vectors that will yield such a result. Thus, we say that the set of all wave vectors ~ K that yield plane waves with the periodicity of a given Bravais lattice is known as its reciprocal lattice . Mathematically, we say that a wave vector ~ K belongs to a Bravais lattice of points ~ R if e i ~ K · ( ~ r + ~ R ) = e i ~ K · ~ r (1) So, we gather that we can say the reciprocal lattice is the set of wave vectors ~ K that satisfy e i ~ K · ~ R = 1 (2) for all ~ R in the Bravais lattice. We must note that a reciprocal lattice defined with respect to a given Bravais lattice. Such a defining Bravais lattice is called the direct lattice . Furthermore, a set of ~ K is only a reciprocal lattice if the set of corresponding vectors ~ R is a Bravais lattice. 1.1 Proof that the Reciprocal Lattice is a Bravais Lattice Let ~a 1 , ~a 2 , and ~a 3 be a set of primitive vectors for a direct lattice. Now, let us define another set of vectors ~ b 1 = 2 π ~a 2 × ~a 3 ~a 1 · ( ~a 2 × ~a 3 ) 1 ~ b 2 = 2 π ~a 3 × ~a 1 ~a 1 · ( ~a 2 × ~a 3 ) (3) ~ b 3 = 2 π ~a 1 × ~a 2 ~a 1 · ( ~a 2 × ~a 3 ) Here, it is easy to note that ~ b i · ~a j = 2 πδ ij (4) Now, let us construct a wave vector ~ k as a linear combination of ~ b i : ~ k = k 1 ~ b 1 + k 2 ~ b 2 + k 3 ~ b 3 (5) We note that a direct lattice vector ~ R is ~ R = n 1 ~a 1 + n 2 ~a 2 + n 3 ~a 3 (6) Using the orthogonality condition from Eq. ( 4 ), it follows that ~ k · ~ R = 2 π ( k 1 n 1 + k 2 n 2 + k 3 n 3 ) (7) By referencing Eq. ( 2 ), we see that it is fulfilled if all k i are integers. Thus, ~ k ≡ ~ K when Eq. ( 5 ) has integral coefficients. Thus, comparing the form for ~ K to the definition of a Bravais lattice, we see that they are equivalent, and that the reciprocal lattice is a Bravais lattice with ~ b i as the primitive vectors. 1.2 The Reciprocal of the Reciprocal Lattice We have proven that the reciprocal lattice is itself a Bravais lattice. So, what is the reciprocal of the reciprocal lattice? As one might guess, it is simply the original direct lattice used to generate the first reciprocal lattice. It is possible to follow the previous proof through a second time, this time defining vectors ~ c i from the vectors ~ b i . If it is carried out, we would see that ~ c i = ~a i ....
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This note was uploaded on 04/21/2010 for the course MS&E 2060 taught by Professor Robinson during the Spring '10 term at Cornell University (Engineering School).
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