Subsections

# Landau Theory

## Main Idea

From previous section we learned that:

1.
Free energy A is an analytical function of magnetization M
2.
At T>Tc it has one minimum, at T<Tc--two

Landau idea: let us use only these facts!

## Expansion

Let us expand free energy A in powers of M.

First, a symmetric system: H=0. Then M=0 is either a minimum (T>Tc) or a maximum (T<Tc) of the free energy. Due to symmetry only even powers are present: Assumption:
We neglect all powers higher than 4.
At large free energy should be large--so d>0. We will see below that at a>0 there is one phase, at a<0--two. So a should change sign at T=Tc. We will expand it: With the same accuracy: Result: (9)

## Phase Diagram

Differentiate (9): Solutions: (10)
At T>Tc only the first solution is realized--one phase. At T<Tc we have two solutions--two phases: Equation of binodal is (10): (11)
This is parabola.

Equation of spinodal: or (12)
This is another parabola, inside the binodal

Free energy: (13)

## Entropy and Specific Heat

Entropy is Differentiating (13): Specific heat: There is a jump in specific heat near Tc: ## Susceptibility

Now we add a small external field H: (14)
and want to calculate Minimizing (14): (15)

Two cases:

1.
If T>Tc, then M(H=0)=0. For small H we neglect M3, and Susceptibility This diverges at .

2.
If T<Tc, then with . Let Expanding (15), we obtain and This also diverges

We obtained a -like curve:    Next: Quiz Up: Mean Field Approach Previous: Self-Consistent Field Theory

© 1997 Boris Veytsman and Michael Kotelyanskii
Tue Oct 14 22:58:59 EDT 1997