There are many different ways in which stacking disordered ice (ice Isd) can be prepared in the lab. Depending on how ice Isd is made and its thermal history the stacking disorder will be different. This manifests mainly in different fractions of cubic and hexagonal stacking. The presence of memory effects in the stacking sequences can introduce additonal complexity.
The picture below shows a possible stacking sequence in ice Isd. Hydrogen atoms have been omitted here for clarity, and "H" and "C" denote hexagonal and cubic stacking, respectively.
In order to calculate a diffraction pattern DIFFaX requires information about the atomic structure of the layers, the probabilities of the various types of stacking and the symmetry relationships between stacked layers. In addition, the calculated pattern is convolved with peak profile parameters.
DIFFaX itself is not capable of refining the various parameters. This is why we have "embedded" DIFFaX in a least-square environment within our MCDIFFaX software. MCDIFFaX uses a simple Monte Carlo based approach to refine the various unknown parameters including stacking probabilities, lattice constants, zero shift and peak profile parameters (u, v, w and GL ratio). The programme randomly suggests a change in one of the parameters, DIFFaX is then asked to calculate the diffraction pattern and if this leads to an improvement of the fit, the change in the parameter is accepted. The idea is that this process goes on until the best possible fit to the experimental data is obtained. The user is required to specify which parameters should be refined and also the step size of a change. In addition, it is possible to allow a certain percentage of unfavourable changes in order to avoid ending up in local minima. Ideally, the step size is chosen quite large initially and then decreased successively as the fit converges. Similar to Rietveld refinements, MCDIFFaX requires the user to develop a bit of a "feeling" for the various parameters. For example, it is advisable to first refine the lattice constants and to then keep them constant while other parameters are refined.
The following picture shows the results from an MCDIFFaX run where all parameters apart from the stacking probabilities were already known. In our opinion, it is good practise to initially assume that there are no memory effects present in the sample. Once the best fit has been obtained, 1st or even 2nd-order memory effects can be switched on in MCDIFFaX to investigate if this improves the fit to the pattern further.
To illustrate this kind of analysis we first investigate ideal gas mixtures which have randomly placed molecules. For these mixtures, there is no defined first coordination shell and therefore an arbitrary distance was chosen to define the local environment. In fact, the distance over which the local environments are analysed does not influence the outcome of our analysis as long as this distance is the same for both species and all environments. The figure below shows how the local environments change as the mole fraction of A molecules, xA, increases. XXX As expected, only BB environments are found at xA = 0. Upon increasing xA to 0.1 the percentage of BB environments decreases by about 10% since the mole fraction of B decreases. Owing to the small amounts of A at xA = 0.1 very few BA environments exist. The local environments of A are initially almost all AB. The chances of finding AA environments are very small. Upon increasing xA to 0.5 the percentage of BB environments decreases substantially, and increasing numbers of AB, BA and AA environments are found. At xA = 0.5 equal percentages of all four environments, i.e. 25%, are found as expected. Upon further increasing xA, the AA environments begin to dominate and the percentages of all other local environments tend to zero as xA approaches one. In the figure below, the changes in environments are shown from the perspective of A molecules only.