Although paracrystalline theory (PT) is a reasonable first attempt to
deal with disorder in multilamellar arrays, one might prefer a scattering
theory that allows for bending of the bilayers in addition to fluctuations
in the mean spacings between bilayers as well as one that is based on energetics
of the fluctuations rather than just an arbitrary stochastic assumption.
Such a theory was originally presented by Caillé (1972) who built upon
the thermodynamic theory of DeGennes (1974) for smectic liquid crystals.
This theory arrives at some quite different conclusions than PT. For example,
the mean square fluctuations 
in the multilayered sample diverge logarithmically with n instead of linearly
as in Eq. 2. This has strong consequences
for scattering. In particular, for powder samples (Roux and Safinya, 1988;
Zhang et al, 1994), the tails of the scattering peaks decay according
to the power law behavior
The parameter 
involves the bending modulus K of lipid bilayers and the bulk modulus
B for compression
Because 
varies as 
and because each peak is well separated from other peaks, it is appropriate
to report just 
,
which is defined to be the value of 
at 
for the
h=1 first order peak, recognizing that 
near the h-th order peak is given by
Detailed fitting using classical Caille and domain size theory has given quite good visual fits to scattering peaks from a variety of smectic liquid crystalline systems (Roux and Safinya, 1988).
We have modified the Caillé theory in a recent theoretical paper (Zhang et al, 1994). Our modifications did not affect any of the qualitative results in the preceding paragraph, but they were necessary for obtaining better quantitative fits to data, and particularly for extracting the correct form factors to be used for obtaining electron density profiles. The present paper will use Eqs. (80) and (82) in Zhang et al. (1994): this will be described as modified Caillé theory (MCT).
Despite the much richer and more realistic model for multilamellar arrays,
MCT has effectively the same number of parameters for fitting scattering
peaks as paracrystalline theory. The parameter 
in MCT is basically a disorder parameter much like 
in PT. Numerical values of 
also translate to mean square nearest neighbor distance fluctuations 
given by Eq. 1. The derivation requires use
of the pair correlation functions (Zhang et al, 1994) and yields
The other two basic parameters in both MCT and in PT are the mean size
of domains 
and the root mean square distribution 
of domain size in Eq. 6.
