Local Order Hidden Inside the Average Order in PZN
This is a lengthier version of the post found at Crystallography365.
Short Range Order
Modelling short-range order (SRO) is tricky. Since in a system with SRO you cannot assume that all unit cells are the same, conventional crystallographic ideas — like Rietveld refinement or conventional single crystal refinement of diffraction data — just don’t work., They are useful for giving you the gloal average structure, but they don’t tell you about the local configurations that make up that average. It is a bit like looking at the average of a room full of people. The average person might be 45% male and 55% female, 1.7m tall, 68kg in weight… but there is a lot more information to be had. What is the average weight of the males? Average height of the females? And, of course, the average person does not really exist, so knowing this average might not be very useful.
Similarly with materials, if we want to know how the useful properties arise from the structure and we only have the average structure, then if the system shows disorder, then this average might not ever actually occur, so doing crystal chemical calculations based on it will lead us astray.
A good example of this is the family of relaxor ferroelectrics, one of the best known being PZN, PbZn1/3Nb2/3O3. It is a perovskite, which we’ve seen plenty of this year. In this case, the PZN unit cell can be modelled by considering the Pb sites to really consist of 12 Pb sites, one along each of the  directions.
Evidence for SRO shows up in the diffuse scattering, the scattering between the Bragg reflections. Some diffuse scattering for PZN looks like this:
Now, the diffuse scattering in PZN can be modelled quite well by assuming that the Pb displace along  and these displacements are correlated in certain ways, and that the other atoms then relax around this Pb configuration. The problem is that these models are largely descriptive and artificial. They are based on human interpretation of the data, and do not necessarily relate directly to the underlying chemistry.
Bond Valence Sums
Bond valence sums are empirical measures of how ‘happy’ an atom is in its environment, and these can be related to the chemistry of the constituents. Hence, maybe instead of using an artificial potential to force the system into the posited configuration, the Zn/Nb and O can be made to relax around the Pb by making their bond valence sums as ‘happy’ as possible.
One measure of this is the global instability index. Now, the valence of a bond is like how many electrons have been donated to it. An atom with, for example, a 3+ change should be donating a total of 3 electrons to all the bonds around it. Empirical parameters have been determined which allow a notional calculation of how much electron is on a bond as this depends on the atoms involved and their charge states. So the valence of a single bond from i to j is called vij and looks like this:
where dij is the distance from atom to atom and Rij and b are parameters. Then, if these are all summed for an atom, i, the total is Vi:
Now, if this total is equal to the formal charge (oxidation state) of the atom, the atom is ‘happy’ in its bonding environment. A measure of this can be obtained by looking at the mean squared deviation of the Vi away from the ideal valences, Vi0 for all the atoms:
This is the global instability index (squared). If this is minimised the diffuse scattering should be well modelled. Since it is not a vector calculation, it can not model the displacement behaviour of the Pb atoms, which relies on the polarity of the Pb2+ and its lone pair. So this had to be taken as a given.
Modelling the Diffuse Scattering
In the simulation, atomic positions were swapped around and configurations kept or rejected based on whether they reduced the global instability index or not. This was after initial random distributions had been established based on the average structure. This means that the histograms of atomic separations always give the correct average and overall atomic displacement parameters, so all this SRO can be ‘hidden’ inside a conventional average structure.
And this approach really can give diffuse scattering which looks a lot like the observed and which reveals what is ‘inside’ the average and see how the Nb behaves compared to the Zn, and so on…so below we can see that the O-Nb distances change compared to the O-Zn, even though there is no long-range order in the distribution of Nb and Zn.
The highly structured diffuse scattering, and the local ordering that gives rise to it, can exist in the system without influencing the outcomes of conventional structural studies. This indicates that short-range order may be present, and crucial, and unsuspected, in many systems whose structures are thought to be thoroughly determined.