The simplest model for disorder in multilamellar vesicles is to suppose
that the local spacing D between each neighboring pair of bilayers is a
random variable with a mean value of 
and mean square fluctuations defined by
Paracrystalline theory (PT) assumes that these nearest neighbor fluctuations are independent for each pair of neighboring bilayers in the multilamellar array. Therefore, the mean square fluctuation in the distance between bilayers separated by n-1 intervening bilayers is given by
The divergence of 
for large n contrasts strongly with truly crystalline systems for which
the mean square fluctuations remain bounded at large distances. It should
be emphasized that this is a stochastic model with no Hamiltonian dynamics.
It is also assumed that each bilayer in the array remains flat, with no
bending undulations, as shown in Fig.1, thus
treating multilamellar arrays as pure one-dimensional systems, as emphasized
by Guinier (1963). Although our results will indicate that PT is not appropriate
for multilamellar arrays of lipid bilayers, it is worth noting that such
a theory may be appropriate for one-dimensional systems, such as helices
(Worthington and Elliott, 1989) and that it is also possible that multilamellar
arrays of nerve myelin (Blaurock and Nelander, 1976) or retinal rods (Schwartz
et al., 1975) may be adequately described by disorder of the paracrystalline
kind.
It was shown by Guinier (1963) that, when the mean square fluctuations in Eq. 2 are incorporated into the phase factors for scattering between pairs of bilayers, the basic scattering formula for the structure factor S(q) for N oriented bilayers (domain size L = ND) becomes
where the average spacing 
is now written just as D, and q is 
.
To avoid terminological confusion, it should be emphasized that the structure
factor in Eq. 3 only gives the scattering
from infinitely thin bilayers. (For comparison, if the sample consisted
of large crystals, S(q) would be given by delta functions of equal amplitude
at each Bragg peak.) The actual scattering I(q) is given (Zhang et al,
1994) by
where the factor of 
is the Lorentz factor for unoriented powder samples and F(q) is the form
factor that is related to the electron density profile 
by
For smectic liquid crystals Eq. 3 allows for finite size effects by taking finite values of N. Since it is unlikely that all domains in a sample will have precisely the same number of bilayers, it is also appropriate to consider a distribution of N or L values. We will assume the distribution function
which is essentially a Gaussian except that 
for L<0. The mean values of this distribution will be designated
(which is
generally not equal to 
)
and the mean square fluctuation (distribution) in L will be designated
.
Figure 2 shows the first five scattering
peaks from paracrystalline theory (PT) plotted versus 
for particular values of the parameters, 
,
and 
that are close to those that emerge from our data. For higher orders (not
shown) the peaks continue to decrease in height and diffuse scattering
between the peaks increases until S(q) approaches a constant. (Even the
first five peaks for S(q) decrease rapidly in height with increasing h
and that is why qS(q) is plotted in Fig. 2.)
Figure 3 shows the first three PT peaks
at higher angular resolution for a smaller value of 
.
The peaks grow broader proportional to 
(Schwartz et al, 1975) and the tails of the peaks are essentially
Lorentzian with
