inclass4 - In class 4 PHZ 5156 Due Thursday Nov 20 1 The...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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.

Page1 / 3

inclass4 - In class 4 PHZ 5156 Due Thursday Nov 20 1 The...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online