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Unformatted text preview: In class 4 PHZ 5156 Due Thursday Nov. 20 1. The Ising model in the mean-field approximation can also be studied using a Landau free energy density L , L = atη 2 + 1 2 bη 4- Hη where H is the applied external field, and we can make the connection t = T- T c T c . In this way, we will see that t < 0 corresponds to T < T c , and likewise t > 0 corresponds to T > T c . The η is called the order parameter and, in this case, can be thought of as the average magnetization per site. The a and b are simply phenomenological parameters. a) Make a plot on the computer of L in the case where H = 0 for t < 0, t = 0, and t > 0. Also, explain what happens when H < 0 or H > 0. Take for simplicity a = b = 1 in your plots. Explain how your pictures agree with the general picture of the phase transition. b) Assume that the value of η is found by taking the partial derivative of L and setting the result equal to zero. Consider the case where H = 0. Show that η = 0 for t > 0, and η = ± q- at b for t ≤ 0. Hence we see the critical exponent β = 1 / 2 just as in the Weiss mean-field theory as discussed in lecture. c) Show that the specific heat per spin C V = 0 for t > 0 and C V = a 2 T bT 2 c for t < 0. Again this is the case for H = 0. This shows there is a discontinuity, but no divergence, in the heat capacity at T = T c . d) Show that the magnetic equation of state, found by differentiating L with respect to η , is given by atη + bη 3 = 1 2 H . This gives the critical exponent....
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This note was uploaded on 08/08/2011 for the course PHZ 5156 taught by Professor Johnson,m during the Fall '08 term at University of Central Florida.
- Fall '08