Cell Refinement
Unit Cell Refinement
Files needed: peak_positions.raw, hardcopy of data set
Learning Outcomes: This tutorial will introduce you to the factors which influence peak position in a Rietveld refinement. You’ll use approximate cell parameters to index a material then refine the cell parameters and zero point error in excel. You’ll then perform an equivalent Pawley refinement to check your results.
Information: Data were collected using a Ka1/Ka2 Cu source and have been Ka2 stripped. Use a wavelength of 1.54055 Angstroms. The approximate cell parameters of this tetragonal material are a=4.5, c=2.96 A. For a tetragonal material the d-spacing is given by:
1/d^2 = (h^2+k^2)/a^2 + l^2/c^2
Expected Knowledge: how to fit functions in excel.
Instructions
1. Use excel to create a spreadsheet with cells in which you can enter unit-cell parameters a and c and with columns headed “h,k,l,d_calc,2th_calc,dobs,2thobs,delta_d^2,delta_2th^2”.
2. Create a list of potential hkl values and use these to calculate d-spacings and 2-theta values you would expect to observe (the cheat at the bottom might save you typing).
3. Using your calculated d-spacings, index the experimental powder pattern. Enter observed d-spacings and 2 theta values in your table.
4. Use the excel solver function to refine the cell parameters by minimising (2th_calc-2th_obs)^2.
5. If you need help getting started try using the excel sheet tio2_pks_start.xls as a template.
6. Check your indexing is correct!
7. Introduce a 2-theta zero point into your equation for 2th_calc and re-refine the cell.
8. Perform a Pawley fit on the file peak_positions.raw from 10-70 degrees and compare refined cell parameters with those in excel. Use space group P42nm.
9. In excel try minimising (d_calc-d_obs)^2 and see how your cell parameters differ. Which values do you trust more?
Extra Work
Try setting up topas to mimic the excel refinement by inserting individual peaks at 2-theta values predicted by simple equations. A little hard!
Try basing it on macros such as:
prm !lam 1.540596
macro get2th(h,k,l) { = (360/Pi) ArcSin((lam/(2*alp)) ((h)^2 + (k)^2 + (l)^2)^0.5);}
macro peakmacro(h,k,l,p1,v1,p2,v2,int) {xo_Is xo get2th(h,k,l) peak_type pv pv_lor p1 v1 pv_fwhm p2 v2 max = 0.25; I @ int }
peakmacro( 0, 1, 0,lor,1,fwhm,0.25,2.25620897)
peakmacro( 1, 1, 0,lor,1,fwhm,0.25,1e-009)
Data
To save typing, a sensible set of hkls to use and observed peak d-spacings and 2-thetas are in tio2_pks_start.xls and below:
h k l
0 0 1
0 0 2
0 1 0
0 1 1
0 1 2
0 2 0
0 2 1
0 2 2
0 3 1
0 3 2
1 1 0
1 1 1
1 1 2
1 2 0
1 2 1
1 2 2
1 3 1
1 3 2
2 1 2
2 2 0
2 2 1
2 2 2
2 3 1
2 3 2
3 1 0
d-space 2th
3.24505 27.463
2.48511 36.113
2.29395 39.241
2.18518 41.281
2.05272 44.079
1.68599 54.371
1.62283 56.674
1.47801 62.820
1.45167 64.094
1.35899 69.066
1.34562 69.840