Pretransition effects in the nematic phase

In the smectic phase (T T ), the amplitude of the density wave may be taken to be constant, so that only p varies. The gradient terms of F therefore become 

Comparing this with (5.3.3) it is at once clear that p is related to the layer displacement u p = ql u and B = y/ ql/My. The terms hfp/dx and hq /dy represent the tilt of the layers with respect to the director. If the director orientation is not fixed, it is the relative tilt between the layers and the director that should be considered and therefore (5.5.6) takes the generalized form 

The analogy may be extended further. By including the Frank elastic free energy terms in (5.5.6), we may define (as already shown in 5.3.3) a characteristic length A = (k/B). Making use of the condition SF /Sv o = 0 and ignoring the difference between A/j, and My, 

Above r, we may ignore the term involving the fourth power in [p and from the equipartition theorem obtain in the usual manner 

To discuss the critical behaviour of the twist and bend elastic constants in the nematic phase, we observe that the Frank free energy expression should include the contribution due to smectic short-range order  

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