Consideration of the structure factor (and our assumption about translational invariance) is complicated by the fact that atoms in the crystal may be displaced from their respective lattice sites. Taking the scattering potential to be proportional to the density of scattering matter, we rewrite the structure factor.
The integral from here onwards is understood to be taken over the unit cell.  is the density of scattering matter. The angle brackets indicate a temporal average of each unit cell followed by a spatial average over every unit cell. We further assume that each atom is displaced independently of the other atoms.
The number of atoms in the unit cell is  and the occupancy factor for atom  is .  represents the point in the unit cell for which we would like to know the density of scattering matter.  is the density of scattering matter from atom  at a position separated from the nuclear position  by a vector .  is the probability density function for displacement.  is the reference lattice site from which atom  may be displaced to a new position . If   is symmetrical enough (e.g. spherically symmetrical),  is simply the mean nuclear position. When considering X-ray scattering, the scattering matter density consists of electron density around the nucleus. For neutron scattering, we have -functions weighted by a scattering length  for the respective nucleus (see Fermi pseudopotential). Note that in the above discussion, we assumed the atoms were not deformable. With this in mind, (9) may be plugged into expression (8) for the structure factor.
Now we see the overall structure factor may be represented as a weighted sum of structure factors  corresponding to each atom. Set the displacement between the location in space for which we would like to know the scattering density and the reference position for the nucleus equal to a new variable . Do the same for the displacement between the displaced and reference nuclear positions . Substitute into (10).
Within the square brackets of (11), we convolve the density of scattering matter of atom  with the probability density function for some nuclear displacement. Then, in the curly brackets, we Fourier transform the resulting convolution. The final step is to multiply by a phase depending on the reference (e.g. mean) position of atom . But, according to the convolution theorem, Fourier transforming a convolution is the same as multiplying the two Fourier transformed functions. Set the displacement between the location in space for which we would like to know the scattering density and the position for the nucleus equal to a new variable .
Substitute (12) into (10).
That is:
 is the atomic form factor of the atom ; it determines how the distribution of scattering matter about the nuclear position influences scattering.  is the atomic Debye–Waller factor; it determines how the propensity for nuclear displacement from the reference lattice position influences scattering. The expression given for  in the article's opening is different because of 1) the decision to take the thermal or time average, 2) the arbitrary choice of negative sign in the exponential, and 3) the decision to square the factor (which more directly connects it to the observed intensity).