Stat mech problem on order parameter

[quasar_4]

Homework Statement

A system has a two component order parameter: [phi1, phi2]. The mean-field Landau free energy is $$E(\phi_1,\phi_2) = E0+at(\phi_1^2+\phi_2^2)+b(\phi_1^2+\phi_2^2)^2+c(\phi_1^4+\phi_2^4)$$

where t, a, and b are positive constants.
Represent the order paramter as a vector on a plane. Use polar coordinates to write
$$\phi_1=\phi \cos{\theta}, \phi_2=\phi \sin{\theta}$$.

Minimize the free energy with respect to phi, and show that its minimum occurs at the minimum of

$$b_1 = b + c(\cos^4{\theta} + \sin^4{\theta})$$

Homework Equations

The Attempt at a Solution

I don't understand the language at the end of the problem. It's easy to substitute in the polar coordinates, then I get

$$E(\phi) = E0 + at \phi^2 + \phi^4 (b+c (\cos^4{\theta}+\sin^4{\theta})$$

This is fine. To minimize, I look for critical points:

$$\frac{d E(\phi)}{d\phi} = 0 \Rightarrow \phi = \pm \frac{\sqrt{-4 (b+c(\cos^4{\theta}+\sin^4{\theta}) at}}{2 (b+c(\cos^4{\theta}+\sin^4{\theta})}$$

Ok, so now what? What does it mean to have the "minimum at the minimum of"? Am I supposed to solve for b1 in terms of phi by inverting the change of coordinates to polar? That seems to be more than they really want. Also, b1 isn't the value of E(phi) at the critical point. So, I am completely lost as to what this problem wants. Can anyone please help?

 

[quasar_4]

The tex is all screwed up and it appears ok in the edited version, but not in the actual version. Instead of showing the critical point twice, it's supposed to say I get from polar coords:

$$E(\phi) = E0 + at \phi^2 + \phi^4 (b+c (\cos^4{\theta}+\sin^4{\theta})$$