Physics 212
homework 11
Due Friday, April 24
April 17, 2010
1. Consider a system with an effective free energy given by
f (h, t, m) = −hm + q(t) + r(t)m2 + s(t)m4 + u(t)m6
where t = (T −Tc )/T c. To investigate the spontaneous magnetization, one minimizes
f with respect to m. This gives m0 as a function of the parameters r and s. Determine the phase structure by showing the following:
(a) For r > 0 and s > −(3ur)1/2 , show m0 = 0 is the only real solution.
(b) For r > √
0 and −(4ur)1/2 < s ≤ −(3ur)1/2 we can have m0 = 0 or m0 = ±m1
with m21 = ( s2 − 3ur − s)/(3u). However, the minimum of f at m0 = 0 is lower
than the minima at m0 = ±m1 so the true equilibrium value of m0 is 0.
(c) For r > 0 and s < −(4ur)1/2 , m0 = ±(r/u)1/4 or 0 and the minimum of f
at m0 = 0 is at the same value of f as the ones at m0 = ±m1 .
(d) For r > 0 and s < −(4ur)1/2 , show m0 = ±m1 . This is a first order phase
transition because the two possible states have ms that differ by a finite amount.
(e) For r = 0 and s < 0, m0 = ±(2|s|/3u)1/2 .
(f) For r < 0, m0 = ±m1 for all s. As r → 0, m1 → 0 if s is positive.
(g) For r = 0 and s > 0, m0 = 0 is the only solution. Combining this result
with (f) we conclude that the line r = 0 , with s > 0, is a line of second order phase
transitions since the two states differ by a vanishing amount of m.
1
The lines of first-order and second-order phase transitions meet at the point r = 0,
s = 0, known as the tricritical point.
2. In problem 1, set s = 0 and approach the tricritical point along the r-axis, setting
r ≈ r1 t. Show that the critical exponents near the tricritical point are
1
α= ,
2
1
β= ,
4
2
γ = 1,
δ = 5.