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Subsections


Landau Theory

Lifshits Term

In homogeneous phase we had

\begin{displaymath}
A = A_0+V\left[aM^2+dM^4-HM\right]\end{displaymath}

Now we divide system into layers, and

\begin{displaymath}
A = \int_{-\infty}^{\infty} A(x)\,dx\end{displaymath}

with A(x) being the free energy of one layer.

We want to add the dependence on $\partial^2M/\partial
x^2$. Lowest order: linear dependence on $\partial^2M/\partial
x^2$. But this is not symmetric with respect to $M\to-M$. Solution: use

\begin{displaymath}
M\frac{\partial^2 M(x)}{\partial x^2}\end{displaymath}

Integrating by parts:  
 \begin{displaymath}
 \begin{split}
 &\int_{-\infty}^{\infty}M(x)\frac{\partial^2...
 ...left(\frac{\partial M(x)}{\partial
 x}\right)^2\,dx \end{split}\end{displaymath} (2)
If there is bulk phase at $x\to\pm\infty$, first term is zero. Otherwise we can include it in the boundary conditions. Result:

\begin{displaymath}
\begin{split}
 &A = A_0+ \\  & W\int\left[aM^2+dM^4-HM+g\left(\frac{\partial
 M}{\partial x}\right)^2\right]\,dx
 \end{split}\end{displaymath}

where W is the cross-section normal to x

General case: substitute $\partial/\partial x$ by $\nabla$: 
 \begin{displaymath}
 A= A_0 + \int\left[aM^2+dM^4-HM+g\left(\nabla M\right)^2\right]\,d\mathbf{r}\end{displaymath} (3)
The term $g(\nabla M)^2$ is called Lifshits term, and free energy (3)--Landau Hamiltonian (or Landau-Ginsburg Hamiltonian).

Estimates for Lifshits Term

1.
Sign of g: if g<0, ``wavy'' structures are favored:
\begin{figure}
 \psfrag{x}{$x$}
 \psfrag{M}{$M$}
 \includegraphics{waves}\end{figure}
Therefore g>0.

2.
Value of g: the dimensionality of g/d is $\mathrm{cm}^2$ (Why?). From previous section it is clear, that this of the order of the square of the molecular size a0:

\begin{displaymath}
g/d\sim a_0^2
 \end{displaymath}

Since $a\to0$ at $T\to T_c$, we obtain $g/\left\lvert a\right\rvert\to\infty$. We will see that this is square of the correlation length:

\begin{displaymath}
g/\left\lvert a\right\rvert\sim \xi^2
 \end{displaymath}


next up previous
Next: Mathematical Digression: Euler-Lagrange Equation Up: Spatial Inhomogeneity. Interfaces Previous: Self-Consistent Field Theory

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