Order true long-range

In this book we wish to discuss partially disordered systems and cannot thus always expect true long range order to exist. A proper treatment of the statistical mechanics of a partially ordered system is extremely difficult, though some success has been achieved by numerical modelling... 

In this region the periodic potential stabilizes true long-range order of the CDW the phase is everywhere close to one of the q classical ground states yy . The depinning transition from the lattice for K f Kw is again of KT type. The correlation length w in the low-iT ordered phase is defined by w(ln 1... 

After these remarks on situations where fluctuations are strong enough to destroy true long range order, we now assume that de < d for the system of interest, and discuss the critical behavior. At Tc the power law decay of the correlation function, G(x) = G x (d 2+n> [eq. (43)] implies... 

The smectic A phase is a liquid in two dimensions, i.e. in the layer planes, but behaves elastically as a solid in the remaining direction. However, true long-range order in this one-dimensional solid is suppressed by logarithmic growth of thermal layer fluctuations, an effect known as the Landau-Peierls instability [11,12 and 13]... 

These results imply that smectic A does not possess true long-range translational order. Hence the conventional picture of the smectic A structure with the molecules forming well defined layers (fig. 1.1.5(a)), though useful conceptually, is far from accurate. Fig. 5.3.2 gives a more realistic representation of the actual situation. X-ray experiments confirm... 

Fig. 5.3.3. X-ray scattering intensity profile from the smectic A phase of 80CB at two reduced temperatures, ( an V an = 9 x 10" (filled circles) and 10 (open circles). The dashed line is the experimental resolution function as would be seen if smectic A had true long-range order. The full lines are the best fits of the theoretical line shape 1 —folded with the resolution function. The values of tj so obtained agree with those calculated from (5.3.12) using experimentally determi ned values of B and k. (Als-Nielson et al. )...

As in the S phase, Sg lacks true long-range translational order because of the Peierls-Landau instability. Theoretically, the existence of molecular tilt implies that there must be a certain degree of bond-orientational order in the Sg phase. This has been verified experimentally by a high resolution synchrotron X-ray study of the transition from Sg to S, in a monodomain freely suspended film. It will be recalled that S, is a tilted hexatic phase. The bond-orientational order parameter Cg (as defined in (5.7.1), but appropriately modified to allow for the fact that the molecules are tilted) is plotted as a function of temperature in fig. 5.8.2. The weak bond-orientational order in evolves continuously into S, showing... 

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