Ionic Conductors: Characterisation of Defect Structure Lecture 15 Total scattering analysis Dr. I. Abrahams Queen Mary University of London Lectures co-financed by the European Union in scope of the European Social Fund

What is total scattering analysis ? A diffraction pattern of a fully ordered solid will contain peaks corresponding to reflections from particular sets of planes in the crystal lattice. In this type of solid, conventional analysis of the diffraction pattern gives an accurate picture of both long range and short range structure. We have already seen that in disordered solids conventional diffraction analysis gives an average picture of the structure. From which carefull analysis of the average picture can yield information on the defect structure. The long range order in a solid manifests itself in the sharp Bragg peaks observed in the diffraction pattern. However in addition to these Bragg peaks where is diffuse scattering that comes from short range correlations. Usually this information is not used. In the total scattering approach both diffuse scattering and Bragg scattering are used to give a more complete picture of the structure.

Lectures co-financed by the European Union in scope of the European Social Fund Neutron diffraction patterns for Bi 3 Nb 1-x Y x O 7-x

Lectures co-financed by the European Union in scope of the European Social Fund Diffuse Neutron Scattering Analysis of diffuse scattering gives information on local structure Analysis of Bragg scattering gives long range structure. Together the analysis gives a more complete picture of the defect structure

Lectures co-financed by the European Union in scope of the European Social Fund The basic theory behind total scattering analysis is essentially the same as for analysis of diffuse scattering for glasses, except that the Bragg data are additionally analysed. As in analysis of glass diffraction data we transform the data to Q space where: Theory of total scattering analysis  = Bragg angle, = wavelength and d = d-spacing.

Lectures co-financed by the European Union in scope of the European Social Fund In a diffraction experiment the Intensity I(Q) measured at the detector of angle d  is given by: Where  is the scattering cross section,  is the flux and The differential total scattering cross section has components from Bragg and diffuse scattering. For a system containing N atoms: is the differential scattering cross section which is defined as: Differential scattering cross section

Lectures co-financed by the European Union in scope of the European Social Fund For a system with N atoms of n chemical species: F(Q) is the total interference function, c  is the fraction of chemical species  and b  is the scattering length of species . As we are mostly interested in the distribution of one species (  ) around another (  ) we can define F(Q) as: Where S  is known as the partial structure factor. Structure factors and correlation functions

Lectures co-financed by the European Union in scope of the European Social Fund The partial structure factor is given by: Where r ij is the radial distance between scatterers i and j and the symbol   denotes thermal average. The partial pair distribution functions g  are obtained by Fourier transformation of S  Where  0 is the total number density of atoms = N/V (V = volume)

Lectures co-financed by the European Union in scope of the European Social Fund The number of  atoms around  atoms in a spherical shell i.e. the partial coordination number is given by integration of the partial radial distribution function. The total pair correlation function G(r) is derived by Fourier transform of the total interference function F(Q). The total correlation function T(r) is given by: where

Lectures co-financed by the European Union in scope of the European Social Fund Reverse Monte Carlo modelling RMC is a general simulation method based on experimental data, therefore the models can be simulated without bias. The procedure is a variation of the standard Metropolis Monte Carlo simulation, and is based on the random sampling of atom positions to drive structural models to be as consistent with the experimental data as possible. The process involves an arrangement of N atoms (the configuration) that are generated within certain ranges in a three-dimensional box. Some of the atoms are selected randomly and moved a random amount, under periodic boundary conditions. Each time, the difference (usually of the structure factors) between the new model and the data are recalculated, and only the difference minimizing movements are accepted otherwise the movements are rejected.

Lectures co-financed by the European Union in scope of the European Social Fund RMC methodology 1. Generation of the configuration of the system. N atoms are arranged in a three-dimensional box. The box is based on the unit cell obtained from the Rietveld refinements and made by generating a 10  10  10 supercell of the crystallographic cell in P -1 symmetry. 2. Calculation of the correlation functions from the atom positions in the configuration usually: S(Q), G(r), I Bragg (Q) and G O-O (r) where, I Bragg (Q) is the intensity of Bragg scattering profile and G O-O (r) is the pair distribution function for O-O pairs).

Lectures co-financed by the European Union in scope of the European Social Fund 3. Calculation of the difference (  2 ) between the measured correlation functions and the functions calculated from the configuration. e.g. for S(Q) Where the summation is over all n experimental data points, each with error σ (Q i ). The total  2 is summed over all correlation functions. 4. Atom movements. One atom is selected at random. This atom is moved randomly in both direction and distance, which generates a new configuration. From the new configuration, a new set of correlation functions are calculated. If the value of  2 is lower than its previous value and the model satisfies the constraints, e.g. the minimum distance to other atoms in the system, the movement is accepted and saved. Otherwise, the atom is returned to its previous position.

Lectures co-financed by the European Union in scope of the European Social Fund 5. Cycling. The calculations and movements are repeated by returning to step 3. This procedure continues until the  2 value reaches equilibrium. At this point, the model can be said to have converged. In the case of multiple data sets (e.g. X-ray and neutron diffraction data), the overall agreement parameter  2 all includes a summation over all data types.

Lectures co-financed by the European Union in scope of the European Social Fund Worked example - Bi 3 YO 6 Conventional analysis of Bi 3 YO 6 Fit to neutron diffraction data for Bi 3 YO 6 Detail of Y coordination in Bi 3 YO 6

Lectures co-financed by the European Union in scope of the European Social Fund Bi 3 YO 6 thermal variation of oxide ion distribution There is a small change in the oxide ion distribution with temperature Oxide ions per cell Site 25 o C 800 o C % change 8c f i Vac

Lectures co-financed by the European Union in scope of the European Social Fund Problems with Conventional Approach to Structure Elucidation The conventional approach to analysis of defect structure relies on detailed analysis of the average crystal structure and assumes ions/atoms behave according to their known crystal chemistry to derive models of local structure. There are several problems associated with this approach. Atoms/ions do not always behave according to their established crystal chemistry. In “fully disordered” systems where, the conventional approach ignores the diffuse scattering which contains information on local ordering. All crystallographic approaches effectively use a static model to model a dynamic system.

Lectures co-financed by the European Union in scope of the European Social Fund RT 800  C S(Q)S(Q)G(r)G(r) Total scattering for Bi 3 YO 6

Lectures co-financed by the European Union in scope of the European Social Fund Pair Correlations – g(r) 25 o C 800 o C

Lectures co-financed by the European Union in scope of the European Social Fund Coordination in Bi 3 YO 6 25 o C: Bi CN av = o C Bi CN av = o C: Y CN av = o C Y CN av = 4.9

Lectures co-financed by the European Union in scope of the European Social Fund Vacancy ordering in Bi 3 YO 6 ordering O Angle ratio 6:6:3 ordering Angle ratio 6:7:2 ordering Angle ratio 7:6:2 There are three characteristic O-Bi-O angles for the ideal fluorite structure: 70 , 109 , 180 . The ratio of the number of these angles per Bi atom gives information on the vacancy ordering. Ideal Angle ratio 12:12:4

Lectures co-financed by the European Union in scope of the European Social Fund Angular Distribution Function A O-M-O (  ) in Bi 3 YO  180  70  109  Angles 71.4 o o 180 o Ratio 6 : 8 : 2 Angles 72.8 o o 180 o Ratio 5 : 9 : o C 25 o C

Lectures co-financed by the European Union in scope of the European Social Fund Vacancy ordering Angle ratio 4:3:2 Vacancy ordering Angle ratio 6:12:2 Vacancy ordering Angle ratio 10:8:2 If one considers the ions on the 48i site as a Frenkel defect, leaving a vacancy on the fluorite site, then in Bi 3 YO 6 there are 3 vacancies per metal atom