Statistical Mechanics Mean Field Model

[Chris B]

I don't think I've fully grasped the underlying ideas of this class, so at the moment I'm just sort of flailing for equations to plug stuff into...

Homework Statement

Show that in the mean field model, M is proportional to H^1/3 at T=T_c and that at H=0, M is proportional to (T_c - T)^1/2

Homework Equations

It'll take me forever to write it out here, but M equals some coefficients times tanh(x) where x depends on H, M, and T

The Attempt at a Solution

In class we approximated M at T=0 by taking the first term in the Taylor series expansion of tanh(x) which turns out to just be x. We were given the hint that we need to take the next term in the series (why?) to do this problem. The next term in the series is -x³/3 so the whole thing is
M=N/V gμ_bS(x-x³/3)
where x = gμ_bS(H+λM)/kT
I also know that T_c = λN(gμ_bS)²/VK
I plugged T_c in for T and by rearranging got something that was proportional to H^1/3 minus H/λ
For the next part I'm not sure what to do. If I plug in zero for H the whole thing is zero. I don't see how I'm going to get (T_c - T)^1/2 if I'm substituting T_c for T.

 

[RUber]

These are two different scenarios. In part A, you assume T=Tc. In part B, the only assumption is that H=0.