HW1Solutions - MSE 102 Fall 2011 Problem Set 1 Solutions...

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Unformatted text preview: MSE 102 Fall 2011 Problem Set 1 Solutions James Mastandrea Problem Set 1 Due Date: September 1, 2011 1 Question 1 First set of primitive lattice vectors, with the corresponding [2 3] direction, and 1 3 point. Second set of primitive lattice vectors, with the corresponding [2 3] direction, and 1 3 point. Primitive unit cell. One should note that the primitive unit cell is independent of the selection of primitive lattice vectors, and there is more than one possible primitive unit cell. The criteria for a primitive unit cell is that it needs to fill all space, without overlapping, when translated by all of the R uvw lattice vectors, and contain exactly one lattice point. 2 Wigner-Seitz Cell Question 2 Along which line do the planes (1 1 2) and (3 2 1 ) interest? Using the zonal equation, 0 = uh + vk + wl (1) the (1 1 2) plane and the (3 2 1) plane can be written in the following manner. 0 = u + v + 2 w (2) 0 = 3 u + 2 v + w (3) Multiply eqn. 2 by 2. 0 = 2 u + 2 v + 4 w (4) Subtract eqn. 3 from eqn. 4, and solve for u. u = 3 w (5) Plug the expression for u into eqn. 2, and solve for v . v =- 5 w (6) [ u v w ] = [3 w- 5 w w ] = w [3- 5 1] = w [3 ¯ 5 1] (7) The w in eqn. 7 is a scalar value that changes the length of the vector, but it does not change the direction of the vector. By convention, when ignoring the signs of the components, we want to represent the directional components of the vector using the smallest possible positive integers. Consequently, since w is an integer, we will set w = 1. The vector can go in either direction and consequently [3= 1....
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This note was uploaded on 09/15/2011 for the course MSE 102 taught by Professor Staff during the Fall '08 term at University of California, Berkeley.

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HW1Solutions - MSE 102 Fall 2011 Problem Set 1 Solutions...

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