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Subsections


Self-Consistent Field Theory

Inhomogeneous systems

In the previous lecture we obtained for Ising lattice:  
 \begin{displaymath}
 A = \frac{JzM^2N}{2} - NkT\ln\Bigl(2\cosh\bigl(\beta(H +
 JzM)\bigr)\Bigr) \end{displaymath} (1)
This is for homogeneous systems. Now M depends on x. We introduce layers normal to x axis:
\begin{figure}
 \psfrag{z}{$x$}
 \psfrag{l}{$l$}
 \psfrag{l+1}{$l+1$}
 \psfrag{l-1}{$l-1$}
 \includegraphics{Ml}\end{figure}
and M becomes M(l).

Consider a simple cubic lattice. Each spin in the layer l has z=6 neighbors:

Changes to equation (1): substitute N by sum over layers. Then:

1.
The first term describes mean interaction of a given spin with z neighbors. Substitute zM2 for

\begin{displaymath}
M(l)\bigl[M(l+1)+M(l-1)+(z-2)M(l)\bigr]
 \end{displaymath}

2.
The second term describes molecular field acting on a given spin (in the layer l). Substitute zM by

\begin{displaymath}
\bigl[M(l+1)+M(l-1)+(z-2)M(l)\bigr]
 \end{displaymath}

Gradient expansion

Assumption:
Magnetization M(l) changes at distances much larger than the lattice size a0
This means that the difference between neighboring layers is small:

\begin{displaymath}
M(l+1)-M(l)\ll M(l)\end{displaymath}

We substitute M(l) by a smooth function M(x) and use the expansion

\begin{displaymath}
\begin{aligned}
 M(x+a_0) &= M(x) + a_0\frac{\partial M(x)}{...
 ...2}{2}\frac{\partial^2 M(x)}{\partial x^2}+\dots 
 \end{aligned}\end{displaymath}

and

\begin{displaymath}
\begin{split}
 \bigl[M(x+a_0)+M(x-a_0)+(z-2)M(x)\bigr] &= \\ zM(x) +
 a_0^2\frac{\partial^2 M(x)}{\partial x^2} 
 \end{split}\end{displaymath}

Conclusion

We see that in SCF A can be represented as

\begin{displaymath}
A = \sum_{\text{layers}}A_l\end{displaymath}

with

\begin{displaymath}
A_l = A_l\left(M(x),a_0^2\frac{\partial^2 M(x)}{\partial x^2} \right) \end{displaymath}

We will not solve the resulting non-linear equations, but rather use this in the Landau theory.


next up previous
Next: Landau Theory Up: Spatial Inhomogeneity. Interfaces Previous: Problem

© 1997 Boris Veytsman and Michael Kotelyanskii
Thu Oct 16 20:58:44 EDT 1997