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hw5 - Homework 5 PHZ 5156 Due Thursday December 4 1...

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Homework 5 PHZ 5156 Due Thursday, December 4 1. Consider a periodic charge density ρ ( ~ r ) defined such that ρ ( ~ r + ~ R n 1 n 2 n 3 ) = ρ ( ~ r ) . (1) We define the vector ~ R n 1 n 2 n 3 to be ~ R n 1 n 2 n 3 = n 1 ~ a 1 + n 2 ~ a 2 + n 3 ~ a 3 (2) where n 1 , n 2 , and n 3 are integers and the primitive lattice vectors are defined in terms of a Cartesian system by ~ a 1 = L ˆ x , ~ a 2 = L ˆ y , and ~ a 3 = L ˆ z . a) Determine the appropriate reciprocal lattice vectors ~ G such that we can write the periodic charge density ρ ( ~ r ) as a summation ρ ( ~ r ) = X ~ G ρ ~ G exp( i ~ G · ~ r ) (3) where the summation is over all possible vectors ~ G . b) Show that the Coulomb potential due to this charge density is also periodic and given by (up to a constant), V ( ~ r ) = X ~ G V ~ G exp( i ~ G · ~ r ) (4) which can be written in terms of the ρ ~ G as, V ( ~ r ) = X ~ G 4 πρ ~ G G 2 exp( i ~ G · ~ r ) (5) with G 2 = ~ G · ~ G . Hint: Consider the Poisson equation. c) Show that for a point charge at ~ r 0 and its periodic images, that ρ ~ G = 1 Ω exp( - i ~ G~ r 0 ) with Ω = L 3 .
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