2.3
Structural analysis
The task of solving the structure of an unknown compound is often non-trivial and requires single crystal diffraction data, although in some cases it is possible with high quality SXRD or NPD data. This was not necessary in the current work, as all possible perovskite structures resulting from distortions are well established, as discussed in chapter 1. These structures can in some cases be differentiated by visual inspection of the diffraction pattern.
Firstly, peak splitting indicates that the symmetry is lower than cubic, and further inspection of the splitting can indicate the type of crystal system. If the splitting is significant and characteristic of a particular crystal system, the possible structures can be identified with the aid of the work of Howard et al. [12, 20].
Another feature of a diffraction pattern that can be used to identify the struc- ture is the presence of superlattice peaks. For both single and double perovskites, octahedral tilting produces additional, sometimes weaker reflections, labelled by the point on the Brillouin zone as described by Howard et al. [12], R-, M -, or X-point. These correspond to the presence of out-of-phase, in-phase and both in- and out-of-phase tilting, respectively, in ABX3 perovskites.
The combination of peak splitting and the presence or absence of these peaks is a powerful indicator of the structure. However, using these can often be misleading for a number of reasons. When a high degree of pseudo-symmetry is present, as is sometimes the case with perovskites, the peak splitting indicative of a distorted structure may be absent such that the system appears to adopt a higher symmetry cell than it actually does. Additionally, the presence of rock-salt ordering in double perovskites also contributes to the R-point reflections, and so the presence of intensity of these can indicate either ordering or out-of-phase tilting, or both. The final complication is that the super-lattice reflections caused by octahedral tilting are relatively very weak (in the presence of heavy cations contained in the samples of interest) in X-ray patterns.
These challenges may be overcome by the use of a combination of neutron and X-ray diffraction techniques. Where possible, combined refinements, that is
2.3. Structural analysis 25
using both X-ray and neutron data sets, were performed in order to exploit the advantages of both techniques.
2.3.1
Rietveld method
Rietveld refinement of a structural model is a way of obtaining information about the crystal structure of a material, from powder diffraction data. The Rietveld method fits a calculated model to an entire data set using a least squares min- imisation process. By fitting the entire pattern, the obstacle of peak overlap is overcome such that the splitting of overlapped peaks does not need to be observed. The general structure including space group, approximate lattice parameters, and atomic positions, must be known prior to the refinement process. An unreason- able or incorrect initial model can give an unstable refinement or result in a false minimum.
The Rietveld method determines a calculated intensity, given by Equation2.2, at each point of the pattern, yic. The variables are the background intensity, yib,
the normalised peak profile function, Gik, the intensity of the kth Bragg reflection
at point i is Ik, and p is the number of phases present.
yic= yib+ X p kp2 X k=kp1 GpikIk (2.2)
Individual parameters are refined, starting with global parameters which are dependent upon the instrument and sample environment. These may include the background, zero-offset, peak shapes, absorption, and instrumental parame- ters. The structural parameters, which depend upon the chemical structure of the sample, may be refined once the global parameters are stable. These include cell parameters, atomic displacement parameters (ADP), atomic positions, and occupancies.
2.3. Structural analysis 26
The refinement is theoretically complete when all parameters can be refined simultaneously and the refinement remains stable. This should produce the model of best fit which can be assessed using various R-factors and the goodness of fit term, χ2, where the deviation from 1 reflects the difference between the calculated
model and obtained data (Equation 2.5). However, problems such as reaching a local minimum instead of a global one, or correlations among the parameters can give an incorrect result. To avoid this, parameters must be constantly monitored to make sure none become unreasonable. That is the refined structural model must be chemically sensible and the various parameters physically possible. Refinements in this project were performed using the Rietveld method, implemented in the programmes Rietica [51], GSAS [52, 53] or Fullprof [54], as appropriate.
For most diffraction patterns, isotropic displacement parameters were refined and the background estimated using a fifth-order polynomial in two theta. Each diffractometer will give a unique profile dependent upon the instrumental set- up, and so a combination of Gaussian and Laurentian functions must be used to describe the peak shapes. A pseudo-Voigt function coupled with a Finger-Cox- Jephcoat asymmetry [52] correction was appropriate for most refinements.
The high resolution patterns obtained on the Echidna diffractometer [55] al- lowed anisotropic displacement parameters to be accurately refined, significantly improving the fit to some data. Where the background had considerable structure, such as from the capillary, the background was estimated using a linear interpo- lation between a set of manually selected points, rather than a mathematical function.
In some cases, both SXRD and NPD data were collected and the model was refined against both data sets simultaneously, in order to improve the accuracy and precision of the refinements. For each refinement, the lattice parameter was initially refined using the value obtained from the SXRD refinement. In some cases, the wavelength of the less precise measurement was relaxed in order to assist the combined refinement.
The quality of the refinements is judged by firstly evaluating how reasonable each parameter is, and secondly by examining the statistical measures of the
2.3. Structural analysis 27
quality of the fit calculated by the program. These include the profile factor Rp
(Equation 2.3), the weighted profile factor Rwp (Equation2.4), and the goodness
of fit term χ2 (Equation 2.5).
Rp = P |yiobs− yicalc| P yiobs (2.3) Rwp= P wi(yiobs− yicalc)2 P wiyiobs2 1/2 (2.4) χ2 = P wi(yiobs− yicalc) 2 N − P (2.5)
Where wi is the weight assigned to the individual step intensity and is given
by wi = σ12 i
, with σ2
i being the variance at the ith step. The yiobs and yicalc are
the set of observed and calculated intensities, respectively, at each step, N is the number of observed steps, and P are the number of refined parameters.
While a lower value indicates a better fit for all of these, it is important to note that the χ2 term depends strongly on the signal-to-noise. A larger χ2 value for a
fit of a SXRD pattern compared to the fit of a NPD pattern does not necessarily reflect a better model, as the much improved signal-to-noise will impact the χ2
value. Therefore, comparison of these values between different diffraction patterns is not always useful, although these values can often distinguish between different structural models. Ultimately, the residuals can’t be solely relied on and the model must be thoroughly examined to ensure that it is chemically sensible.