text: However, fluctuations that cause the correlations between pairs of atoms to decrease as their separation increases, causes the Bragg  peaks in the structure factor of a crystal to broaden. To see how this works, we consider a one-dimensional toy model: a stack of plates with mean spacing . The derivation follows that in chapter 9 of Guinier's textbook.  This model has been pioneered by and applied to a number of materials by Hosemann and collaborators over a number of years. Guinier and they termed this disorder of the second kind, and Hosemann in particular referred to this imperfect crystalline ordering as paracrystalline ordering.  Disorder of the first kind is the source of the Debye–Waller factor.
  To derive the model we start with the definition (in one dimension) of the
  To start with we will consider, for simplicity an infinite crystal, i.e., . We will consider a finite crystal with disorder of the second-type below.
  For our infinite crystal, we want to consider pairs of lattice sites. For large each plane of an infinite crystal, there are two neighbours   planes away, so the above double sum becomes a single sum over pairs of neighbours either side of an atom, at positions  and  lattice spacings away, times . So, then
  where  is the probability density function for the separation  of a pair of planes,  lattice spacings apart. For the separation of neighbouring planes we assume for simplicity that the fluctuations around the mean neighbour spacing of a are Gaussian, i.e., that
  and we also assume that the fluctuations between a plane and its neighbour, and between this neighbour and the next plane, are independent. Then  is just the convolution of two s, etc. As the convolution of two Gaussians is just another Gaussian, we have that
  The sum in  is then just a sum of Fourier transforms of Gaussians, and so
  for . The sum is just the real part of the sum  and so the structure factor of the infinite but disordered crystal is
  This has peaks at maxima , where . These peaks have heights
  i.e., the height of successive peaks drop off as the order of the peak (and so ) squared. Unlike finite-size effects that broaden peaks but do not decrease their height, disorder lowers peak heights. Note that here we assuming that the disorder is relatively weak, so that we still have relatively well defined peaks. This is the limit , where . In this limit, near a peak we can approximate , with and obtain
  which is a Lorentzian or Cauchy function, of FWHM , i.e., the FWHM increases as the square of the order of peak, and so as the square of the wave vector  at the peak.
  Finally, the product of the peak height and the FWHM is constant and equals , in the  limit. For the first few peaks where  is not large, this is just the  limit.