Phys560Notes-4 - Reciprocal Lattice Direct lattice given by...

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Reciprocal Lattice Direct lattice given by ii i n Ra Definition: reciprocal lattice is a Bravais lattice given by K = reciprocal lattice vectors i b , where i = integers, and 23 1 12 3 2  aa b aa a ; 31 2 2 b ; 12 3 2 b Useful for 3d Fourier analysis involving crystal axes. Theorem : 2 ij i j   ab Examples: 11 21 2, 0 , etc. Theorem : reciprocal of reciprocal = original lattice Or, 1 2 bb a bb b ; 2 2 a ; 3 2 a Theorem :  exp 1 i KR for any pair of and . Pf: 22 i n t e g e r i i i i i nn      b a  Theorem : Any vector K satisfying   exp 1 i for all R is a reciprocal lattice vector. Pf: resolve in terms of Kb K b i x  ; 1 i n   exp 1 i integer xn for arbitrary i n i x must be integers. Example : bcc aaa 1, 1, 1 ; 1, 1, 1 ; 1, 1, 1 222  a 123 41 0, 1, 1 ; 1, 0, 1 ; 1, 1, 0 a2  bbb Theorem : Reciprocal of bcc (lattice constant = a ) is fcc (lattice constant 4 a ) Reciprocal of fcc (lattice constant = a ) is bcc (lattice constant 4 a ) Fourier analysis based on K and R , using   exp 1 i Consider 1D first
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Direct lattice Rn a Reciprocal lattice 2 Kb a   Consider a function that is periodic wrt to the direct lattice:    , Vx Vx a  e.g., potential energy of an e in a crystal. Fourier series:  2 exp Vx V i x a      Easy to verify  , Fourier coefficient: o 12 exp a VV xi x d x aa  With 2 K a , o exp 1 exp K K a K V i K x x i K x d x a Generalize to 3D: Assume r V is periodic; i.e. rr R (for example, crystal potential) a b 3 exp 1 exp v i i d r v  K K K rK r r 12 3 v  aa a volume of primitive cell. Proof for the 3d case: First, show i ev  Kr KR rR , known as the completeness relation.
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Phys560Notes-4 - Reciprocal Lattice Direct lattice given by...

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