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Next: Quiz Up: Mean Field Approach Previous: Self-Consistent Field Theory

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:

\begin{displaymath}
A = A_0 + V(a M^2 + d M^4 + \dots)\end{displaymath}

Assumption:
We neglect all powers higher than 4.
At large $\left\lvert M\right\rvert$ 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:

\begin{displaymath}
a = \alpha(T-T_c)\end{displaymath}

With the same accuracy:

\begin{displaymath}
d= \mathit{const}\end{displaymath}

Result:  
 \begin{displaymath}
 A = A_0+V\left(\alpha(T-T_c)M^2+dM^4\right)\end{displaymath} (9)

Phase Diagram

Differentiate (9):

\begin{displaymath}
2\alpha(T-T_c)M+4dM^3=0\end{displaymath}

Solutions:  
 \begin{displaymath}
 M =0\quad
 \text{or}\quad
 M^2 = -\frac{\alpha(T-T_c)}{2d}\end{displaymath} (10)
At T>Tc only the first solution is realized--one phase. At T<Tc we have two solutions--two phases:
\begin{figure}
 \psfrag{A}{$A$}
 \psfrag{M}{$M$}
 \psfrag{T\gt Tc}{$T\gt T_c$}
 \psfrag{T<Tc}{$T<T_c$}
 \includegraphics{Alandau}\end{figure}

Equation of binodal is (10):  
 \begin{displaymath}
 2dM^2+\alpha(T-T_c)=0\end{displaymath} (11)
This is parabola.

Equation of spinodal: $\partial^2 A/\partial M^2=0$ or  
 \begin{displaymath}
 6dM^2+\alpha(T-T_c)=0\end{displaymath} (12)
This is another parabola, inside the binodal

Free energy:  
 \begin{displaymath}
 A = 
 \begin{cases}
 A_0,& T\gt T_c\  A_0 -V\alpha^2(T-T_c)^2/4d, & T<T_c
 \end{cases}\end{displaymath} (13)

Entropy and Specific Heat

Entropy is

\begin{displaymath}
S = -\frac{\partial A}{\partial T}\end{displaymath}

Differentiating (13):

\begin{displaymath}
S = 
 \begin{cases}
 S_0, T\gt T_c\  S_0+V\alpha^2(T-T_c)/2d, & T<T_c
 \end{cases}\end{displaymath}

Specific heat:

\begin{displaymath}
C = 
 \begin{cases}
 C_0,& T\gt T_c\  C_0+ V\alpha^2T/2d, & T<T_c
 \end{cases}\end{displaymath}

There is a jump in specific heat near Tc:
\begin{figure}
 \psfrag{T}{$T$}
 \psfrag{C}{$C$}
 \psfrag{Tc}{$T_c$}
 \includegraphics{Clandau}\end{figure}

Susceptibility

Now we add a small external field H:  
 \begin{displaymath}
 A = A_0+V(\alpha(T-T_c)M^2+dM^4-HM)\end{displaymath} (14)
and want to calculate

\begin{displaymath}
\chi = \frac{\partial M}{\partial H}\end{displaymath}

Minimizing (14):  
 \begin{displaymath}
 2\alpha(T-T_c)M+4dM^3=H\end{displaymath} (15)

Two cases:

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

\begin{displaymath}
M = \frac{H}{2\alpha(T-T_c)}
 \end{displaymath}

Susceptibility

\begin{displaymath}
\chi = \frac{1}{2\alpha(T-T_c)}
 \end{displaymath}

This diverges at $T\to T_c$.

2.
If T<Tc, then $M(H=0)=\pm M_0$ with $M_0 =
 \pm\sqrt{-\alpha(T-T_c)/2d}$. Let

\begin{displaymath}
M = M_0+\delta M
 \end{displaymath}

Expanding (15), we obtain

\begin{displaymath}
\delta M = - \frac{H}{4\alpha(T-T_c)}
 \end{displaymath}

and

\begin{displaymath}
\chi = -\frac{1}{4\alpha(T-T_c)}
 \end{displaymath}

This also diverges

We obtained a $\lambda$-like curve:
\begin{figure}
 \psfrag{T}{$T$}
 \psfrag{C}{$\chi$}
 \psfrag{Tc}{$T_c$}
 \includegraphics{chilandau}\end{figure}


next up previous
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