Fluctuations Peierls-Landau instability

The smectic A phase is a liquid in two dimensions, i.e. in tire layer planes, but behaves elastically as a solid in the remaining direction. However, tme long-range order in tliis one-dimensional solid is suppressed by logaritlimic growth of tliennal layer fluctuations, an effect known as tire Landau-Peierls instability [H, 12 and 13]... 

The weak logarithmic divergence with the sample size L is known as the Landau-Peierls instability. As a result, for sufficiently large L the fluctuations become of the order of the layer spacing, which means that the layer structure would be wiped out. However, for samples in the miUimolar range and typical values of the elastic moduli K 10 N and B 10 N/m, the layer displacement amplitude a = (ifi) does not exceed 0.5-0.7 run. For a typical smectic period d ra 3 nm this gives relative displacements ajd k, 0.2 the smectic layers are still well defined. Nevertheless, the displacements are large compared to those of a typical 3D crystal for which ... 

It is seen that for smectics the fluctuation logarithmically diverges with L.+ This was first shown by Peierls and Landau and is called Landau-Peierls instability. This is behavior is very different from the fluctuations in crystal and columnar liquid crystals (3D and 2D elasticity, respectively) where the amplitude of the fluctuations remain finite even for infinite samples, as u )cryM = (1/Z-1/L), and (u ) columnar I Inserting typical... 

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