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Next: Landau Theory Up: Mean Field Approach Previous: Flory Theory

Subsections


Self-Consistent Field Theory

Main Idea

Energy of Ising system is

\begin{displaymath}
E = -H\sum_i \sigma_i -\frac{J}{2}\sum_{(i,j)}\sigma_i\sigma_j\end{displaymath}

It can be written as

\begin{displaymath}
E = -\sum_i \sigma_i H_{\text{eff}}(i)\end{displaymath}

with molecular field

\begin{displaymath}
H_{\text{eff}}(i) = H + \frac{J}{2}\sum_{\text{neighbors of $i$}}
 \sigma_j \end{displaymath}

We substitute $H_{\text{eff}}(i)$ by its average value

\begin{displaymath}
H_{\text{eff}}=\left\langle H_{\text{eff}}(i)\right\rangle= H + Jz\left\langle \sigma\right\rangle\end{displaymath}

and choose $\left\langle \sigma\right\rangle$ from consistency condition!

One-Particle Hamiltonian

Let us write

\begin{displaymath}
\sigma_i = M + \delta_i\end{displaymath}

The product:

\begin{displaymath}
\sigma_i\sigma_j = (M+\delta_i)(M+\delta_j) = M^2 + M\delta_i +
 M\delta_j + \delta_i\delta_j\end{displaymath}

Approximation: neglect $\delta_i\delta_j$. Result:

\begin{displaymath}
E=\sum_i E_i\end{displaymath}

with

 
 \begin{displaymath}
 \begin{split}
 E_i(\sigma_i) &= -MH-\delta_i(H + JzM) - \fr...
 ...^2 \  &= -\sigma_i(H + JzM) + \frac{1}{2}JzM^2 \  \end{split}\end{displaymath} (4)
We separated energy into independent contributions from each spin!

Free energy

Partition function:

\begin{displaymath}
Q = \prod_i Q_i = Q_1^N\end{displaymath}

Partition function for one spin

\begin{displaymath}
\begin{split}
 Q_i &= \exp\bigl(-\beta E_i(-1)\bigr) + \exp\...
 ...= 2\exp(-JzM^2/2kT)\cosh\bigl(\beta(H + JzM)\bigr)
 \end{split}\end{displaymath}

Free energy:  
 \begin{displaymath}
 \begin{split}
 A &= -NkT\ln Q_i \  &= \frac{JzM^2N}{2} - NkT\ln\Bigl(2\cosh\bigl(\beta(H + JzM)\bigr)\Bigr)
 \end{split}\end{displaymath} (5)
This depends on the magnetization M!

Calculation of M

There are two ways to calculate M:

Self-Consistency Equation

By definition, $M=\left\langle \sigma_i\right\rangle$. Since only Ei depends on $\sigma_i$, we have:

\begin{displaymath}
M = \frac{1}{Q_i}\left[\exp\bigl(-\beta E_i(1)\bigr) - \exp\bigl(-\beta
 E_i(-1)\bigr) \right]\end{displaymath}

or, from (4)  
 \begin{displaymath}
 M = \tanh\bigl(\beta(H+JzM)\bigr)\end{displaymath} (6)
This is a transcendent equation for M.

Minimization of Free Energy

Free energy (5) depends on M. It should be minimal as function of M:

\begin{displaymath}
\frac{\partial A}{\partial M} = 0\end{displaymath}

or

\begin{displaymath}
JzMN - NJz\frac{\sinh\bigl(\beta(H + JzM)\bigr)}{\cosh\bigl(\beta(H
 + JzM)\bigr)} \end{displaymath}

This gives  
 \begin{displaymath}
 M = \tanh\bigl(\beta(H+JzM)\bigr)\end{displaymath} (7)
The results (6) and (7) coincide! Our theory is self-consistent.

Critical Point and Phase Diagram

We can rewrite equation (7) as  
 \begin{displaymath}
 H = kT\tanh^{-1} M - JzM\end{displaymath} (8)

1.
At large T, this is monotonic function:
\begin{figure}
 \psfrag{H}{$H$}
 \psfrag{M}{$M$}
 \psfrag{kT\gt Jz}{$T\gt Jz$}
 \includegraphics{MH_highT}
 \end{figure}
This means one-phase regime

2.
At low T, this is non-monotonic function:
\begin{figure}
 \psfrag{M}{$M$}
 \psfrag{H}{$H$}
 \psfrag{kT<Jz}{$kT<Jz$}
 \includegraphics{MH_lowT}
 \end{figure}
From stability condition $(\partial M/\partial H)\gt$ we see, that this means phase separation

3.
At critical temperature Tc, we have only one point, where $(\partial H/\partial M)=0$:
\begin{figure}
 \psfrag{M}{$M$}
 \psfrag{H}{$H$}
 \psfrag{kT=Jz}{$kT=Jz$}
 \includegraphics{MH_critT}
 \end{figure}
At this temperature

\begin{displaymath}
\left.\frac{\partial H}{\partial M}\right\rvert_{M=0} = 0
 \end{displaymath}

or, from (8)

kT=Jz

Phase diagram:
\begin{figure}
 \psfrag{T}{$T$}
 \psfrag{Tc}{$T_c$}
 \psfrag{phi}{$\phi$}
 \psfr...
 ...nodal}
 \psfrag{Spinodal}{Spinodal}
 \includegraphics{phase_diagram}\end{figure}

Flory Theory and Self-Consistent Field Theory

Advantages of SCF:

1.
We know how to extend it--just make a better approximation for $\delta_i\delta_j$.
2.
Works better farther from critical point

next up previous
Next: Landau Theory Up: Mean Field Approach Previous: Flory Theory

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