Lower Marginal Dimensionality. X-Ray Scattering from the Smectic-A Phase of Iquid Crystals

Abstract

In the previous chapter we have seen that the critical fluctuations associating a second order phase transition, expressed in terms of a response function X(q) directly measurable by scattering spectroscopy, may be treated in a self-consistent and correct way by simple mean field theory if the dimensionality exceeds a certain marginal dimensionality d*. In particular we stressed that d* depends on the volume in reciprocal space available to the critical fluctuations, and systems with d ≤ d* can indeed be studied experimentally. In this chapter we shall discuss the opposite limit: That the fluctuations are so strong that they prevent the onset of true long range order at all. Let us look upon an example. We postulate an ordered crystalline state in d dimensions and we want to calculate the mean squared fluctuations <u²>. If <u²> diverges in an infinite sample the postulated long range order cannot take place. First, consider a sinusoidal fluctuation with wavevector $\vec{q}$ and amplitude ${{\vec{u}}_{q}}$. For simplicity let us assume that the medium is elasrically isotropic.