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homework4 - Physics 481 Condensed Matter Physics Homework 4...

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Physics 481: Condensed Matter Physics - Homework 4 due date: Friday, Feb 11, 2011 Problem 1: Nematic order parameter (15 points) Consider a three-dimensional nematic liquid crystal consisting of N rod-like molecules. The nematic order parameter (i.e., a quantity characterizing the degree of nematic order) can be defined as S = N X j =1 ¿ cos 2 Θ j - 1 3 where Θ j is the angle between the axis of molecule j and the director ˆ n . h . . . i indicates the thermodynamic average. a) Show that S is indeed a nematic order parameter. To this end, show that it vanishes if the rods point in random directions. Also calculate S for a state with perfect nematic order. b) In a magnet, an order parameter can be defined as M = j h cos Θ j i where Θ j is the angle between the spin j and the magnetization direction ˆ n . Is this an order parameter for the nematic, too? Explain your answer. c) Consider hypothetical molecules shaped like a + sign (symmetric under 90 rotations). Sug- gest an order parameter characterizing the degree of order in their relative orientations. Hint:
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