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Ginzburg-Landau functional for crystallization

One can attempt to study crystallization by means of GL functional of density $\rho(r)$. This is not hopeless if the transition is only weakly first order, that is the order parameter below the transition temperature is not very large. The formation of crystal is signified by the appearance of one or multiple non-zero Fourier components of density, $\rho_Q\ne 0$. In a weak first order phase transition, we expect $\rho_Q\ll \rho_0$. In contrast, the limiting case of periodically arranged delta-function atoms corresponds to all harmonics being of equal value, $\rho_{nQ} = \rho_0$. Thermodynamically, weakly first order phase transition has small latent heat. Real-life crystallization occurs most commonly through a first order phase transition, which satisfies the Lindemann criterion, according to which atoms fluctuate by 15 to 30% of interatomic distance at the transition. This translates into the need to keep from 3 to 7 Fourier harmonics, far beyond what is typically handled by GL theory. Still, even there it is believed that GL can provide guidance as to what kinds of crystal structures can be stabilized or favored.

Generic GL functional for translationally invariant system is:

$F = r(q-Q)|\rho_q|^2 + \lambda (\{q_i\}) \rho_{q_1}\rho_{q_2}\rho_{q_3}\delta(\sum{q_i}) + \mu (\{q_i\}) \rho_{q_1}\rho_{q_2}\rho_{q_3}\rho_{q_4}\delta(\sum{q_i}) + \frac{u_0}{2} |\rho_{q}|^4$.

The cubic term tends to stabilize hexagonal crystals in 2D and BCC in 3D. However, here we are mainly after the properties of the quatric term; it turns out to have strong temperature dependence and can be responsible for formation of quasicrystals and other non-hexagonal states.

For simplicity we will assume that the momentum dependence of the 2nd order term is very strong, favoring momenta on a shell of radius $Q$. Then, in 2D, $\mu$ is only a function of an angle between $q_1$ and $q_2$ ($|q_i| = Q$), and in 3D a function of two angles only. The self-interaction, as we will show can be related to the mutual interaction. Hence, the GL in 2D is

$F = r(q-Q)\sum_i|\rho_{q_i}|^2 + \sum_{i,j,k}\lambda (\{q_i\}) \rho_{q_i}\rho_{q_j}\rho_{q_k}\delta(\sum{q_i}) +\sum_{i\ne j} u (\alpha_{ij}) |\rho_{q_i}|^2|\rho_{q_j}|^2+ \frac{u_0}{2} \sum_i|\rho_{q_i}|^4$.

In 3D, there is also a possibility of having non-coplanar terms, i.e. ones that not only contain $\pm q_1$ and $\pm q_2$.

$$ \begin{eqnarray} F &=& \sum\nolimits_\mathbf{q_i} \tilde r(q) |\rho_\mathbf{q}|^2 + \tilde \lambda_3(q_0) \sum\nolimits_{\triangle} \rho_{q_1}\rho_{q_2}\rho_{q_3}\delta(\sum{q_i}) \nonumber\\ &&+ \frac 12 \sum\nolimits_{\mathbf{q}_i\ne \mathbf{q}_j }[\lambda_4 + u(\alpha_{ij})] |\rho_{q_i}|^2 |\rho_{q_j}|^2 + \frac 14\sum\nolimits_{\mathbf{q}_i}[\lambda_4 + u(0)] |\rho_{q_i}|^4 + \sum\nolimits_\Box [\lambda_4 + w(\{\mathbf q_i\})]\rho_{q_1}\rho_{q_2}\rho_{q_3}\rho_{q_4}\label{eq:F} \end{eqnarray} $$

The symbols $\sum\nolimits_{\triangle}$ and $\sum\nolimits_\Box$ indicate summation over unique triangles and non-planar quadrilaterals of $\mathbf{q_i}$.

This functional can be used, in particular, to determine the relative stability of different variational states.