homework2 - ( ~a 2 ~a 3 ) , 2 ~a 1 ~a 2 ~a 1 ( ~a 2 ~a 3 )...

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Physics 481: Condensed Matter Physics - Homework 2 due date: Friday, Jan 28, 2011 Problem 1: Packing fractions in two and three dimensions (20 points) To find the degree of space filling for a given lattice, first find the distance d min between nearest neighbors. Then put circles (in two dimensions) or spheres (in three dimensions) of diameter d min on each lattice site, and ask what portion of space is occupied by the circles or spheres. This number is the packing fraction . a) Calculate the packing fractions for the square and hexagonal lattices in two dimensions. b) Calculate the packing fractions for the following three-dimensional lattices: simple cubic, body- centered cubic, face-centered cubic, hexagonal close-packed, and diamond. Problem 2: Reciprocal lattice in three dimensions (8 points) Consider a Bravais lattice with primitive vectors ~a 1 ,~a 2 ,~a 3 . The primitive vectors ~ b 1 , ~ b 2 , ~ b 3 of the reciprocal lattice are defined via the condition ~a i · ~ b j = 2 πδ ij . a) Verify that the vectors 2 π ~a 2 × ~a 3 ~a 1 · ( ~a 2 × ~a 3 ) , 2 π ~a 3 × ~a 1 ~a 1
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Unformatted text preview: ( ~a 2 ~a 3 ) , 2 ~a 1 ~a 2 ~a 1 ( ~a 2 ~a 3 ) are the primitive vectors of the reciprocal lattice! b) If V is the volume of the primitive cell of the direct lattice, show that the volume of the primitive cell of the reciprocal lattice is 8 3 /V . Problem 3: Face-centered and body-centered cubic lattices (12 points) The face-centered cubic (fcc) lattice can be dened by the primitive vectors a 2 (1 , 1 , 0) , a 2 (1 , , 1) , a 2 (0 , 1 , 1) where a is the lattice spacing of the conventional cubic unit cell. For the body-centered cubic (bcc) lattice, a set of primitive vectors is a 2 (1 , 1 ,-1) , a 2 (-1 , 1 , 1) , a 2 (1 ,-1 , 1) . a) Calculate the volumes of the primitive cells and compare them with the volume of the con-ventional cubic unit cell. What does this tell you about the number of particles in the cubic cell? b) Show that the reciprocal lattice of a fcc lattice is a bcc lattice and vice versa....
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This note was uploaded on 10/04/2011 for the course PHYSICS 481 taught by Professor Thomasvojta during the Spring '11 term at Missouri S&T.

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