Structural analysis

The crystallographic average structure describes the unique arrangement of atoms or molecules in a crystalline material where atomic positions repeat periodically infinitely within unit cells in three dimensions. This approximation which characterize majority of crystalline materials is established based on the Rietveld refinement.

The Rietveld refinements in this thesis were carried out first for perovskites Bi2Mn4/3Ni2/3O6 and BiTi3/8Fe2/8Mg3/8O3 to obtain an average crystallographic

structure and then compare to a local structure. The same technique was also applied for BiFe0.6Mn0.4O3 to derive the incommensurate crystallographic

structure.

2.1.11.1 Profile matching

The structural analysis usually starts with profile matching which provides information about unit cell parameters, peak shapes and background function, without knowing the structural model. Profile matching is usually done by Le Bail or Pawley fit. In this thesis the Le Bail method [17] was applied for fitting background coefficients, peak shapes and lattice parameters of unit cell of collected diffraction patterns. The program used to Le Bail fit of 3 dimensional non modulated structure was GSAS (General Structure Analysis System) [18], while Jana 2006 [19] was applied to determine profile of incommensurate structure BiFe0.6Mn0.4O3.

2.1.11.2 Rietveld refinement

When a profile matching is determined (Le Bail or Pawley fit), a starting crystallographic model is needed to start a structural Rietveld refinement. The Rietveld refinement uses a least squares minimization approach, where the starting model is refined until the theoretical pattern becomes a good match to the measured profile of the experimental diffraction pattern. The calculated pattern providing crystallographic structure of atoms within unit cell, is based on the simultaneous refinement of atomic and thermal positions, lattice parameters and profile parameters of the experimental diffraction pattern. This minimization

technique must be carried out carefully, by checking the atomic positions or thermal parameters of atoms, to keep a model physically sensible.

The diffraction pattern is a function represented by intensity in terms of scattering angle (2θ), energy parameter (time of flight) or wavelength, depending on which diffractometer is applied (e.g. X-ray or neutron). In every case, the Rietveld technique utilizes the same principle where minimization technique of least- squares refinement is expressed by:

𝐿𝑆 = 𝑤_𝑖(𝐼_𝑖𝑜𝑏𝑠 ₋ 𝑖 𝐼𝑖𝑐𝑎𝑙𝑐)2 Equation 2.20 where: wi - statistical weight 𝑤_𝑖 = _𝐼1 𝑖𝑜𝑏𝑠 Equation 2.21

𝐼_𝑖𝑜𝑏𝑠 – observed, experimental intensity of the ith step

𝐼_𝑖𝑐𝑎𝑙𝑐 _{- calculated intensity of the i}th step

The diffraction pattern is represented by Bragg intensity where each individual intensity corresponds to the reflection from a specific plane, described by Miller indices hkl. These Bragg intensities in their position, height and width for every scattered material is proportional to the structure factor as described earlier in the section 2.1.3. Thus, the calculated intensity of reflections is represented by the structure factor and the background:

𝐼_𝑖𝑐𝑎𝑙𝑐 = 𝑆𝑗 𝑁𝑝𝑒𝑎𝑘𝑠𝑘=1 𝐿𝑘|𝐹𝑘,𝑗|2 𝑆𝑗 2𝜃𝑖 − 2𝜃𝑘,𝑗 𝑃𝑘,𝑗𝐴𝑗 + 𝑏𝑘𝑔𝑖 Equation 2.22

where:

Sj – phase scale factor is written as:

𝑆𝑗 = 𝑆𝐹 _𝑉𝑓𝑗

𝑗2

𝑁𝑝𝑕𝑎𝑠𝑒𝑠

𝑗 =1 Equation 2.23

SF – beam intensity

Fj – phase volume fraction

Vj – phase cell volume

Lk – Lorentz- Polarization factor, depending on the instrument geometry, e.g. for Bragg-Brentano

Fk, j – the structure factor for k, j reflection

Aj – absorption factor defined as:

𝐴_𝑗 = _2𝜇1 Equation 2.25

µ - linear absorption coefficient

Pk, j – The texture (preferred orientation function)

𝑆_𝑗 2𝜃_𝑖 − 2𝜃_𝑘,𝑗 – profile shape function, where different types are available:

G – Gaussian

V, PV – Voigt or Pseudo-Voigt

C – Cauchy

bkgi – background intensity of the ith step.

In a short simplification, the refined parameters in the calculated intensity pattern are divided into two groups: structural and profile parameters. The profile parameters are calculated during Le Bail profile match and include lattice parameters, peak shape and the background coefficients. The background is usually calculated by linear interpolation of previously selected points. It is often necessary in a case of complex background to increase the number of points in the diffraction pattern in order to improve the intensity of background.

On the other hand, the structural parameters which determine the intensity of reflections are calculated by the structure factor which include atomic positions, fractional occupancies and thermal parameters.

The quality of Rietveld refinement is usually represented by goodness of fit or several R- factors, depending if factors are weighted Rwp or not Rp [20, 21]. These

functions define the agreement between calculated and observed diffraction pattern, and their updating with every cycle are very useful, to control refinement of crystallographic structure.

1. Profile R-factor

𝑅_𝑝 = 𝑦𝑖 𝑖(𝑜𝑏𝑠 ) _𝑦 −𝑦𝑖(𝑐𝑎𝑙𝑐 ) 𝑖(𝑐𝑎𝑙𝑐 )

𝑖 Equation 2.26

2. Weighted profile R-factor

𝑅𝑤𝑝 = 𝑤𝑖|𝑦𝑖(𝑜𝑏𝑠 )−𝑦𝑖(𝑐𝑎𝑙𝑐 )|

2 𝑖

where:

wi – statistical weight defined in equation 1.19 3. Goodness of fit χ2 = _𝑅𝑅𝑤𝑝 𝑒𝑥𝑝 ² Equation 2.28 where: 𝑅𝑒𝑥𝑝2 - expected R-factor 𝑅𝑒𝑥𝑝2 = _𝑤 𝑁 𝑖(𝑦𝑖(𝑐𝑎𝑙𝑐 ))2 𝑖 Equation 2.29

The most trustworthy factor is the weighted profile R-factor where the square root of the minimized quantity is scaled by the statistical weight. Another meaningful factor is goodness of fit, which during least-squares refinement should approach to 1, which is the ideal situation when weighted R-factor is equal to the expected R-factor, while the value below 1 indicates over-fitting data.