Co₂SiO₄ on D20¶
This tutorial fits the olivine Co₂SiO₄ to constant-wavelength neutron powder data measured on D20 at the ILL. Its structure and its measured pattern come from EasyDiffraction's collection; the tutorial describes the experiment, links the two and fits the coordinates, the displacements and the oxygen occupancies.
Fitting 
Single pd neut cwl Bragg 
Crystal structure
Import the library¶
import easydiffraction as edi
Define the project¶
project = edi.Project(name='cosio_d20')
Define the structure¶
Load the structure from the collection. The CIF gives the space group P n m a, the cell and the six sites, and marks the cell, coordinates, displacements and oxygen occupancies to fit. The fit below frees them in stages, so start with all of them fixed.
structure_path = edi.download_data('pd-neut-cwl_cosio-d20-cosio', destination='data')
project.structures.add_from_cif_path(structure_path)
structure = project.structures['cosio']
for parameter in project.free_parameters:
parameter.free = False
Data 'pd-neut-cwl_cosio-d20-cosio' is at data/pd-neut-cwl_cosio-d20-cosio.edi
project.display.structure('cosio')
Define the experiment¶
The measured pattern also comes from the collection.
data_path = edi.download_data('pd-neut-cwl_cosio-d20-d20-data', destination='data')
project.experiments.add_from_data_path(
name='d20',
data_path=data_path,
sample_form='powder',
beam_mode='constant wavelength',
radiation_probe='neutron',
scattering_type='bragg',
)
experiment = project.experiments['d20']
Data 'pd-neut-cwl_cosio-d20-d20-data' is at data/pd-neut-cwl_cosio-d20-d20.xye
D20's wavelength is 1.87 Å. The profile is the Thompson-Cox-Hastings pseudo-Voigt, and a flat background runs through fourteen points.
experiment.instrument.setup_wavelength = 1.87
experiment.instrument.calib_twotheta_offset = 0
experiment.peak.broad_gauss_u = 0.2
experiment.peak.broad_gauss_v = -0.5
experiment.peak.broad_gauss_w = 0.4
experiment.peak.broad_lorentz_y = 0.01
for position in (8, 9, 10, 11, 12, 15, 25, 30, 50, 70, 90, 110, 130, 150):
experiment.background.create(position=position, intensity=340)
experiment.linked_structures.create(structure_id='cosio', scale=1)
project.display.pattern('d20')
Perform the analysis¶
The fit goes in stages, freeing one group of parameters at a time.
Scale, 2θ zero and cell¶
The scale sets the overall height of the pattern, and the zero shift and the three cell lengths put the peaks where they are measured.
experiment.linked_structures['cosio'].scale.free = True
experiment.instrument.calib_twotheta_offset.free = True
structure.cell.length_a.free = True
structure.cell.length_b.free = True
structure.cell.length_c.free = True
project.display.parameters.free()
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_d20, 1418 pts fitted / 1418 loaded, 5 free
iter time (s) Rwp (%) reduced χ² change
40.51 452.01
1 0.003 32.67 293.90 35.0% ↓
2 0.006 28.52 224.01 23.8% ↓
3 0.010 24.07 159.57 28.8% ↓
4 0.013 20.61 117.03 26.7% ↓
5 0.016 20.10 111.27 4.9% ↓
6 0.018 20.09 111.14 0.1% ↓
7 0.021 20.09 111.14
8 0.025 20.09 111.14
done · converged · 8 iters · reduced χ² 111.14 · Rwp 20.09 % · 0.03 s
project.display.pattern('d20')
Background¶
With the peaks in place, the fourteen background points follow the level between them.
for point in experiment.background:
point.intensity.free = True
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_d20, 1418 pts fitted / 1418 loaded, 19 free
iter time (s) Rwp (%) reduced χ² change
20.09 112.25
1 0.005 5.51 8.45 92.5% ↓
2 0.008 5.51 8.45
3 0.013 5.51 8.45
done · converged · 3 iters · reduced χ² 8.45 · Rwp 5.51 % · 0.02 s
project.display.pattern('d20')
Peak widths¶
The profile terms now fit the widths and shapes of the peaks. diffraction-lib's tutorial also fits a peak asymmetry; this case's profile has none.
experiment.peak.broad_gauss_u.free = True
experiment.peak.broad_gauss_v.free = True
experiment.peak.broad_gauss_w.free = True
experiment.peak.broad_lorentz_y.free = True
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_d20, 1418 pts fitted / 1418 loaded, 23 free
iter time (s) Rwp (%) reduced χ² change
5.51 8.47
1 0.005 4.84 6.54 22.8% ↓
2 0.009 4.80 6.43 1.7% ↓
3 0.012 4.80 6.42 0.1% ↓
4 0.016 4.80 6.42
5 0.021 4.80 6.42
done · converged · 5 iters · reduced χ² 6.42 · Rwp 4.80 % · 0.03 s
project.display.pattern('d20')
Atom positions¶
The peak intensities now depend mostly on where the atoms are. Co1 sits on a centre of symmetry and the other sites on the mirror plane, so only their free coordinates are fitted.
for site in structure.atom_sites:
for coordinate in (site.fract_x, site.fract_y, site.fract_z):
coordinate.free = not coordinate.symmetry_constrained
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_d20, 1418 pts fitted / 1418 loaded, 34 free
iter time (s) Rwp (%) reduced χ² change
4.80 6.47
1 0.006 4.15 4.84 25.2% ↓
2 0.011 4.14 4.82 0.4% ↓
3 0.016 4.14 4.82
4 0.022 4.14 4.82
done · converged · 4 iters · reduced χ² 4.82 · Rwp 4.14 % · 0.03 s
project.display.pattern('d20')
Displacements¶
The displacements change the intensities more at high angle.
for site in structure.atom_sites:
site.adp_iso.free = True
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_d20, 1418 pts fitted / 1418 loaded, 40 free
iter time (s) Rwp (%) reduced χ² change
4.14 4.84
1 0.007 3.99 4.50 7.0% ↓
2 0.012 3.99 4.49 0.1% ↓
3 0.020 3.99 4.49
done · converged · 3 iters · reduced χ² 4.49 · Rwp 3.99 % · 0.03 s
project.display.pattern('d20')
project.display.fit.correlations()
Constrain the two Co sites¶
The correlations show no strong pair for the two Co sites, yet the fit gives them very different displacements, Co1's even negative, which no atom can have: both are Co and scatter alike, so the pattern cannot tell their thermal motions apart. Give each an alias and constrain Co2's to follow Co1's, starting again from 0.5 Ų, then fit again; the shared displacement is positive and its s.u. smaller than either had alone.
structure.atom_sites['Co2'].adp_iso.free = False
structure.atom_sites['Co1'].adp_iso = 0.5
project.analysis.aliases.create(id='biso_Co1', param=structure.atom_sites['Co1'].adp_iso)
project.analysis.aliases.create(id='biso_Co2', param=structure.atom_sites['Co2'].adp_iso)
project.analysis.constraints.create(id='biso_Co2', expression='biso_Co2 = biso_Co1')
project.display.constraints()
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_d20, 1418 pts fitted / 1418 loaded, 39 free
iter time (s) Rwp (%) reduced χ² change
4.05 4.64
1 0.006 4.02 4.57 1.6% ↓
2 0.012 4.02 4.57
3 0.019 4.02 4.57
done · converged · 3 iters · reduced χ² 4.57 · Rwp 4.02 % · 0.03 s
project.display.pattern('d20')
Oxygen occupancies¶
Last, the oxygen occupancies, which change the intensities as the displacements do.
structure.atom_sites['O1'].occupancy.free = True
structure.atom_sites['O2'].occupancy.free = True
structure.atom_sites['O3'].occupancy.free = True
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_d20, 1418 pts fitted / 1418 loaded, 42 free
iter time (s) Rwp (%) reduced χ² change
4.02 4.58
1 0.006 3.98 4.48 2.2% ↓
2 0.010 3.97 4.45 0.5% ↓
3 0.017 3.97 4.45
done · converged · 3 iters · reduced χ² 4.45 · Rwp 3.97 % · 0.02 s
project.display.pattern('d20')
project.display.pattern('d20', x_min=40, x_max=60)
project.display.fit.correlations()
project.display.structure('cosio')
Save the project¶
project.save_as('cosio_d20')
Saving project to 'cosio_d20'