Structure factor disordered system

The trapping site structure with surrounding ions and atoms of neighbour molecules in crystals and the solvation structure in disordered systems can in many cases be elucidated by ENDOR and ESEEM measurements. The point-dipole approximation is frequently used to assign the observed hfc to specific nuclei. More elaborate models are needed when the unpaired electron is delocalized over several atoms, exemplified below, and in presence of g-factor anisotropy, see Chapter 6. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

We derive here the governing equations necessary to describe the structure factor, S (q, t), for a complex liquid mixture subject to flow. Since this observable is the Fourier transformation of the spatial correlation of concentration functions, it is first required to develop an equation of motion for 5c (r, t). The approach described here employs a modified Cahn-Hilliard equation and is described in greater detail in the book by Goldenfeld [91]. To describe the physical system, an order parameter, q/ (r, ), is introduced. In a complex mixture, this parameter would simply be /(r, 0 = c(r, r)-(c), where c(r, t) is the local concentration of one of the constituents and mean concentration. The order parameter has the property of being zero in a disordered, or on phase region, and non-zero in the ordered or two-phase region. The observed structure factor, which is the object of this calculation, is simply... 

FIG. 12 Pattern of light scattered from a single layer of colloidal particles in the disordered phase. The particles are polystyrene spheres, of diameter 2 /glass plates. Except for the contribution of the form factor P(k), which depends on the scattering angle, and normalization and geometrical factors, this picture shows directly the static structure factor of the system. 

The direct reconstruction of a disordered system by EELS is an important technique by virtue of its wide applicability. The theory for this problem requires only simple expressions that connect the inelastic cross section with the structural characteristics of the system. The second goal of this chapter will be to develop this theory. Note that the modern source of information about distribution functions is the Fourier transform of the static structure factors for a system. 

If the system is sufficiently disordered (i.e. a high Debye-Waller factor), the diffractometer resolution is sufficiently low and the Q range is sufficiently large, then the experimental structure factor can be transformed directly to gE(r) and the comparison made withgc(r) (Keen et ah, 1990b). However, this procedure is not applicable in many cases, i.e. there are still oscillations in AE(Q) at the maximum Q measured. In addition the low Q resolution leads to a loss of real space resolution in the RMC model, i.e. a broader distribution of atoms around their crystal sites. 

In the framework presented so far, the structure factor for X-ray scattering in a randomly disordered system can be written as ... 

The technique can also be applied to the examination of semi-crystalline polymer systems, for example. Fig. 2.29 shows the WAXS and broad Q neutron diffraction from the same sample of d-PVC. In the X-ray scattering data (Fig. 2.29a) the material appears very disordered, whilst the neutron scattering from the same perdeuterated sample of PVC shows a typical melt structure factor similar to polyethylene although the first peak is somewhat sharper reflecting a higher level of order which perhaps is a consequence of the more polar structure (Mitchell 2011). 

---