HW-13 - 1 Theory of Solids HW-13 Solution By Qifeng Shan...

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Unformatted text preview: 1 Theory of Solids HW-13 Solution By Qifeng Shan Ashcroft and Mermin’s book 17-3 Properties of the Coulomb and Screened Coulomb Potentials (a) From the integral representation of the delta function, , 2 3 r k k r i e d (17.70) and the fact that the Coulomb potential φ ( r ) = e / r satisfies Poisson’s equation, , 4 2 r r e (17.71) argue that the electron-electron pair potential, V ( r ) = e φ ( r ) = e 2 / r can be written in the form: , 2 3 k k r r k V e d V i (17.72) Where the Fourier transform V ( k ) is given by , e 4 2 2 k V k (17.73) (b) Show that the Fourier transform of the screened Coulomb interaction V s ( r ) = (e 2 / r ) e k0r is , e 4 2 2 2 k k V s k (17.74) By substituting (17.74) into the Fourier integral , 2 3 k k r r k s i s V e d V (17.75) and evaluating that integral in spherical coordinates. (The radial integral is best done as a and evaluating that integral in spherical coordinates....
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This note was uploaded on 10/04/2011 for the course PHYS 4720 taught by Professor Shan during the Fall '09 term at Rensselaer Polytechnic Institute.

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HW-13 - 1 Theory of Solids HW-13 Solution By Qifeng Shan...

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