Rietveld Refinement

Powder (X-ray) diffraction is often used as a quick method for identifying phase purity of samples, with databases of known phases now containing a vast number of structures to be compared against, such as the Inorganic Crystal Structure Database (ICSD).171 However, with high quality data, structural information can be derived in combination with the Rietveld method, which has been used extensively in this thesis.

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The Rietveld method was first developed by H. M. Rietveld in the 1960s for use with neutron diffraction.172, 173 In this thesis, refinements were carried out using the GSAS package,174 and Topas Academic V5.175 The Rietveld method allows structural variables to be refined against observed data, in order to generate a calculated diffraction pattern. A good fit between observed and calculated data can mean that the model is correct, although one has to be careful that the structure is physically and chemically sensible, as well as statistically significant. The method needs a starting model that is reasonably accurate in modelling the crystal structure. The difference between the experimental diffraction pattern and the theoretical pattern is expressed as a residual function, Sy:

∑

Equation 2.14

Equation 2.15

Where _{and are observed and calculated intensities at the ith step, is the weight} of the squared difference in intensities. is the uncertainty estimate of ,176, 177 shown in Equation 2.15. The calculated intensity, _{, is determined by summing all contributions to} the Bragg reflections for the structure factor, Fhkl, for the proposed structural model. This can

be expressed as follows:

_∑

| |

Equation 2.16

Where is the background intensity, typically generated by a separate polynomial function,

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describes preferred orientation, is an absorption factor for the sample and is a profile function that describes peak shape.

When user defined parameters of a starting model are refined and good convergence is obtained between observed data and calculated model, the is incorporated into the residual function (see Equation 2.14), then minimised through least squares methods.

In a Rietveld refinement, firstly all non-structurally related parameters should be considered, such as zero shift, background and profile function, as well as lattice parameters. These parameters can be difficult to fit however, if the starting structural model is far from the actual structure, giving a poor fit. One way of temporarily ignoring the structural model is by fitting the data by Le Bail178 or Pawley179 procedures, with the latter used in this thesis. Peak positions can be fit by refining the lattice parameters and zero shift. The background intensity is usually fit with a polynomial function. The peak profile parameters can also be refined at this point before importing structural information for refinement by the Rietveld method, giving a better starting point. It is also one way to show the best fit that should be attainable for a particular data set prior to performing a Rietveld refinement, as it just fits intensity to peaks without considering structural parameters. When refining structural parameters that control peak intensities (fractional coordinates, thermal displacements, site occupancies) depending on the differences between starting model and observed data, parameters may need to be refined in different orders from refinement to refinement.

After each iteration, the quality of the fit should be scrutinised, with care being taken regarding the physical meaning of refined values for the structural model. The quality of the refinement can be evaluated through statistical outputs known as R-factors. The profile factor, RP, is defined as:

∑ | _|

∑

Equation 2.17

The weighted profile R-factor, Rwp, is the most straightforward statistical parameter that

55 √

∑

∑ √_∑

Equation 2.18

The best possible Rwp for a specific data set that can be obtained is called the expected

R-factor, Rexp, which is given in the following equation:

√

∑

Equation 2.19

Where is the number of observations and is the number of refined parameters. In typical powder diffraction experiments, we expect to be much greater than . If a refined model gave a perfect fit, with only difference occurring due to statistical variation, the Rexp would represent the Rwp. The parameter for goodness of fit, χ, can be defined in terms of

Rexp and Rwp:

(

) Equation 2.20

Since the should not be greater than , the should always be ≥ 1. Under certain conditions however, the can be misleadingly low, such as when too many variables are being used. When a sufficient number of data points have been collected and counting time has been carried out so that the data is not dominated by a high background intensity, (see Equation 2.16), a < 2 should be obtained for a good structural model.

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Despite the importance of the statistical fit factors, it remains vitally important to inspect the refined pattern and compare it to the observed pattern, as it can show where the misfits occur, and what parameters need to be refined. The structural model also needs to be considered, making sure that the model is chemically sensible.167