Reference no: EM132780238
Rietveld Practical
The Task
Your task is to implement a Rietveld refinement for the data set that you have been given. This refinement will allow you to extract information on the crystal structure of the material you are studying. You will report the procedure you undertook to complete the refinement, provide evidence for the quality of your fit, and discuss the limitation of the model and data. You should report the structural parameters that you extract as part of this refinement, providing figures to illustrate your results.
Before you get started, you should spend some time looking at the Rietveld refinement of the silicon standard data in the Excel spreadsheet. Play around with changing a few parameters and then try using the Excel Solver to perform a refinement. Try and get a feeling for how far each parameter can be from a "good" value for the refinement to still converge. Once you have gained some feeling for this, proceed to the second tab in Excel and start your task.
A detailed procedure you could use for setting up your model is given below. Setting up the initial model
Try and work systemically through filling in the blank columns to complete the spreadsheet that you can use to calculate and initial model for your Rietveld refinement. You should refer regularly back to the Si spreadsheet to check that you are implementing various equations correctly at each step. As an overview, you will need to go through the following steps to simulate your initial powder diffraction pattern:
• Make a plot of the observed experimental intensity yobs versus 2θ. It's always good practice to look at your data first!
• Identify the list of hkls that are allowed for the space group of the sample you are studying for the range of data you wish to fit. Rank these from low to high 2θ and enter then in the relevant columns.
• Input the multiplicity (m) of each reflection. This is the number of symmetry equivalent reflections for each hkl.
• Using an appropriate formula, calculate the d-spacing, 2θ and sin(θ)/λ values associated with each hkl.
• From the tabulated form factor parameters, calculate the scattering factor at the given sin(θ)/λ values for each type of atom in your structure.
• Check that the Lorentz-Polarisation factor (LP) has been calculated for each reflection and associated sin(θ)/λ.
• Calculate the structure factor part (An) for each symmetry unique atom (n) by summing over all symmetry equivalents (i) in the unit cell. A list of starting atom coordinates is given to help you out. Remember for a centrosymmetric structure (with a centre of symmetry at the origin). The individual An parts for any one symmetry unique atom can be given as:
Σ cos(hxi + kyi + lzi)
where we sum over all of the symmetry equivalent (i) atoms. In Excel this may be more conveniently written as a sum of several dot products. The dot product command in Excel may be invoked as SUMPRODUCT((h,k,l),(xi,yi,zi)), where the cells containing (h,k,l) and (xi,yi,zi) are referenced as appropriate.
• Calculate your structure factor F(hkl) by summing each of the prior An value together having first pre-multiplying them by their associated form factor values and expression for their isotropic displacement parameters. In principle each symmetry unique atom can have its own thermal displacement parameter (BISO) but it may be necessary to constrain these to be equal for different symmetry unique atoms. Remember, BISO describes the mean squared atomic displacement of an atom in Å2 and must therefore be a positive value.
• The total intensity observed in your diffraction pattern (Ihkl) for each hkl is given by multiplying F(hkl)2 by the multiplicity of the reflection, the LP factor and the overall Rietveld scale factor.
• The intensity of each of these reflections needs to be convoluted with a resolution function describing the instrument/beam characteristics and crystallinity (size and strain) of the sample. The effect of this convolution is to spread the intensity associated with Ihkl at 2θhkl across a range of 2θ values. This process can be modelled by multiplying Ihkl by a unit peakshape function for a Lorentzian whose FWHM varies according to FWHM = kytan(2θhkl) + kx/cos(2θhkl). For a given 2θhkl this will return a FWHM for a Lorentzian function. This FWHM value will be used in the next step to distribute the calculated total intensity of each Ihkl across the whole powder diffraction pattern.
• For each Ihkl the intensity it contributes to each experimentally measured 2θ value must now be calculated on the right hand side of the spreadsheet using the unit Lorentzian peak shape function described above and associated FWHM. Note that the theoretical 2θ values are corrected by the 2θ zero point error of the diffractometer, so that they are in closer agreement with the experimental data.
• The total powder diffraction pattern intensity (ycalc) at each 2θ is calculated by summing over all of these columns for each Ihkl and the background function (which uses a simple polynomial to describe its variation as a function of 2θ).
• Add ycalc versus experimental 2θ value to your initial plot.
• If you have done all of the above steps correctly, after manually adjusting the overall scale factor, you should have a simulation (ycal) that approximately matches up with your data (yobs). Large discrepancies (such as peaks very far away from overlap or very large discrepancies between predicted and observed intensities) are likely to originate from errors in your model. If this is the case, you will need to troubleshoot your model to find out where you went wrong. There will be a session on trouble shooting on 5th February at 14:30 on MS Teams.
• Next we want to quantify how well the experimental intensity (yobs) is fit by the model. This is given by yobs- ycalc. You could add a plot showing this function to your original graph.
• Finally, we need our minimisation function that we weight (w) according to 1/yobs:
w(yobs-ycalc)2. It is the sum of this final column that you will minimise against using the Excel Solver. Note, it is also customary to monitor and report the weighted profile R-factor for the refinement which may be written in a simplified form as:
Rwp = 100 . J∑ w(yobc-ycalc)2/∑yobc provided that w has been set equal to 1/yobs
Once your model is working, you will need to adjust certain parameters to get a good starting point for your refinement. Generally speaking you will only get convergence if there is some overlap between yobs and ycalc at points in your dataset where reflection are clearly resolvable. To achieve this starting point you may need to manually adjust the overall scale parameter, perturb lattice parameters by a small mount and adjust the background function. Once you have a good starting point, you can start using the Solver routine in Excel to perform your Rietveld refinement. If the Solver doesn't appear automatically at the far right of the Data ribbon, follow these instructions to install it. You should set your "Objective" cell according to the function you wish to minimise which should be your correctly weighed Sum(yobs-ycalc)2. Next you should carefully think about which parameters to optimise first. Generally speaking these should be ones that you deem to be furthest from their optimised value.
You should subsequently add in parameters to your refinements that have less sensitivity to the data such as Uiso and the xyzs. Once added in, you should not take out parameters arbitrarily during your refinement since you are trying to find the global minimum and certain parameters will be highly correlated with each other. Your final solution should be a refinement of all of the parameters that you feel it is reasonable to fit. Throughout the process you should monitor your refinement by looking both at the Rwp and visually at the fit shown in your plot. You may need to zoom in or create extra plots to show the very weak diffraction peaks. Although these are weak they still contain very important information describing distortions in the structure you are studying.
Finally, you should pick one parameter and test the sensitivity of your data to this. You can do this by varying the parameter through various fixed values, performing a refinement at each point and recording the Rwp. You should plot the results in a graph.
You should submit your final annotated spreadsheet with your report. The spreadsheet should include clear and correctly labelled figures showing the quality of you fit.
Assessment critical
This assessment is worth 33% of your overall mark for this module. You will be graded on the following.
• Implementation of refinement (Excel spreadsheet)
• Qualify of fit and results (Excel spreadsheet)
• Presentation of fits and parameter sensitivity (Excel spreadsheet and write-up)
• Experimental description of how you performed the Rietveld refinement (Write-up)
• Figures showing refined structure (Write-up)
• Discussion of limitations and sensitivity of the data to parameters refined in your model (Write-up).
Your write-up should be no longer than two A4 pages (submitted as a PDF) including figures. Your write-up and spreadsheet need to be submitted by 9am on the 19th of February.
Attachment:- Rietveld Practical.rar