Pair distribution function coordination number

We turn now to the orientational correlations which are of particular relevance for liquid crystals that is involving the orientations of the molecules with each other, with the vector joining them and with the director [17, 28]. In principal they can be characterised by a pair distribution function but in view of the large number of orientational coordinates the representation of the multi-dimensional distribution can be rather difficult. An alternative is to use distance dependent orientational correlation coefficients which are related to the coefficients in an expansion of the distribution function in an appropriate basis set [17, 28]. 

A simple property which has been the subject of many investigations is the coordination number (CN). We recall that the average coordination number can be obtained from the pair distribution function (section 2.5). Here, we are interested in more detailed information on the distribution of CN. 

Examples of structural data relevant to the oxides presented later in this chapter include X-ray diffraction, the static structure function, the radial distribution function (RDF), the pair distribution function (PDF), coordination numbers, bond lengths and bond angles, ring structures, Q" species (from NMR nomenclature). 

Fig. 7.4 Pair distribution function (a) and accumulated coordination numbers (b) for P, Si and Al in cerium aluminophosphosilicate glass [20]...

The local structure of the modified network can be characterized further by computing partial pair distribution functions gap r). We now can count all P-atoms as nearest neighbors of an o-atom which are closer in distance from a selected a-atom than the location of the first minimum of gop(r). In this way, one can characterize the corresponding coordination number z for... 

If all the particles in the system are identical then r i and r2 can be the coordinates of any two particles in the system. It is sometimes convenient to use a normalization that will express the fact that, if we look at the corresponding neighborhoods of ri and T2, the probability to find these neighborhoods occupied by any two particles increases in a statistically determined way with the number of particles in the system. This is achievedbymultiplyingthejoint distributionfunctionP / (ri, r2) by the number, N(N — 1), of distinct pairs in the system. This yields the pair distribution function... 

If one takes g n gir) as the pair correlation function defined in Eq. (5.2), the radial distribution function represents the number of particles of 5 in a shell up to rj around A. If r, is then (see Fig. 5.7), one can regard Eq. (5.2) as giving the coordination number of A in the liquid. In the example chosen for simplicity, species A is the same as species B but this of course is only true for radial distributions of monatomics, e.g., sodium. It is found in practice that in a liquid, g B r) settles to unity by the third or fourth atom away from the reference atom A. 

---