Subsections

• Lifshits Term
• Estimates for Lifshits Term

Landau Theory

Lifshits Term

In homogeneous phase we had

$$A = A_0+V\left[aM^2+dM^4-HM\right]$$

Now we divide system into layers, and

$$A = \int_{-\infty}^{\infty} A(x)\,dx$$

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

$$M\frac{\partial^2 M(x)}{\partial x^2}$$

Integrating by parts:  

$$\begin{split} &\int_{-\infty}^{\infty}M(x)\frac{\partial^2... ...left(\frac{\partial M(x)}{\partial x}\right)^2\,dx \end{split}$$ (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:

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

where W is the cross-section normal to x

General case: substitute $\partial/\partial x$ by $\nabla$: 

$$A= A_0 + \int\left[aM^2+dM^4-HM+g\left(\nabla M\right)^2\right]\,d\mathbf{r}$$ (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:

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:

$$g/d\sim a_0^2$$

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:

$$g/\left\lvert a\right\rvert\sim \xi^2$$