Co₂SiO₄ on HEiDi¶
Fit one Co₂SiO₄ crystal measured on HEiDi at MLZ at 300 K, 55 K and 2.5 K. Each temperature has its own structure and its own reflections, and the three are fitted independently: each fit sees only its own data and gives its own goodness of fit and uncertainties. At 2.5 K the cobalt moments order, and the magnetic structure is fitted too.
The data are those of Sazonov et al., Acta Cryst. B 64 (2008) 661-668. The 2.5 K file has no usable uncertainties, so they are estimated from the 55 K data of the same crystal (see the case's provenance); the fitted 2.5 K uncertainties depend on that estimate.
Fitting 
Independent sc neut cwl Bragg 
Crystal structure 
Magnetic structure
Import the library¶
import easydiffraction as edi
Define the project¶
The independent fitting mode fits each experiment on its own, against the structure it links.
project = edi.Project(name='cosio_heidi')
project.analysis.fitting_mode = 'independent'
Define the structures¶
Load the three structures from EasyDiffraction's collection, one per temperature. Each starts from the published positions with a small displacement tensor on every site. The 2.5 K structure also carries the magnetic space group Pnma.1 and the moment directions it allows, Co1 in any direction and Co2 along b, with every moment at zero. The files mark what the fit frees; the fit below frees it in stages, so start with all of it fixed.
path = edi.download_data('sc-neut-cwl_cosio-heidi-cosio_300k', destination='data')
project.structures.add_from_cif_path(path)
path = edi.download_data('sc-neut-cwl_cosio-heidi-cosio_55k', destination='data')
project.structures.add_from_cif_path(path)
path = edi.download_data('sc-neut-cwl_cosio-heidi-cosio_2k5', destination='data')
project.structures.add_from_cif_path(path)
for parameter in project.free_parameters:
parameter.free = False
Data 'sc-neut-cwl_cosio-heidi-cosio_300k' is at data/sc-neut-cwl_cosio-heidi-cosio_300k.edi
Data 'sc-neut-cwl_cosio-heidi-cosio_55k' is at data/sc-neut-cwl_cosio-heidi-cosio_55k.edi
Data 'sc-neut-cwl_cosio-heidi-cosio_2k5' is at data/sc-neut-cwl_cosio-heidi-cosio_2k5.edi
project.display.structure('cosio_2k5')
Define the experiments¶
Load the reflections of each temperature and link each dataset to its structure. HEiDi's wavelength is 0.552 Å; the extinction is FullProf's anisotropic Larson model, with its six coefficients starting at zero.
data_paths = {
'300k': edi.download_data('sc-neut-cwl_cosio-heidi-heidi_300k-data', destination='data'),
'55k': edi.download_data('sc-neut-cwl_cosio-heidi-heidi_55k-data', destination='data'),
'2k5': edi.download_data('sc-neut-cwl_cosio-heidi-heidi_2k5-data', destination='data'),
}
for temperature, path in data_paths.items():
project.experiments.add_from_data_path(
name=f'heidi_{temperature}',
data_path=path,
sample_form='single crystal',
beam_mode='constant wavelength',
radiation_probe='neutron',
scattering_type='bragg',
)
experiment = project.experiments[f'heidi_{temperature}']
experiment.instrument.setup_wavelength = 0.552
experiment.extinction.type = 'larson-anisotropic'
experiment.linked_structures.create(structure_id=f'cosio_{temperature}', scale=10)
Data 'sc-neut-cwl_cosio-heidi-heidi_300k-data' is at data/sc-neut-cwl_cosio-heidi-heidi_300k.hkl
Data 'sc-neut-cwl_cosio-heidi-heidi_55k-data' is at data/sc-neut-cwl_cosio-heidi-heidi_55k.hkl
Data 'sc-neut-cwl_cosio-heidi-heidi_2k5-data' is at data/sc-neut-cwl_cosio-heidi-heidi_2k5.hkl
Names for the three structures and experiments.
structure_300k = project.structures['cosio_300k']
structure_55k = project.structures['cosio_55k']
structure_2k5 = project.structures['cosio_2k5']
experiment_300k = project.experiments['heidi_300k']
experiment_55k = project.experiments['heidi_55k']
experiment_2k5 = project.experiments['heidi_2k5']
Perform the analysis¶
An experiment with nothing free is left out of a fit, so the first stages fit 300 K alone. Each stage frees one new group: the structure's coordinates and displacement components are the ones its site symmetry leaves free.
The 300 K reflections against the starting model:
project.display.pattern('heidi_300k')
300 K: scale¶
The scale brings the calculated intensities to the measured ones.
experiment_300k.linked_structures['cosio_300k'].scale.free = True
project.display.parameters.free()
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_heidi, 1624 pts fitted / 1624 loaded, 1 free
iter time (s) wR(F²) (%) reduced χ² change
33.37 14.23
1 0.001 15.51 3.07 78.4% ↓
2 0.003 15.51 3.07
done · converged · 2 iters · reduced χ² 3.07 · wR(F²) 15.51 % · 0.00 s
project.display.pattern('heidi_300k')
300 K: coordinates¶
for site in structure_300k.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_heidi, 1624 pts fitted / 1624 loaded, 12 free
iter time (s) wR(F²) (%) reduced χ² change
15.51 3.09
1 0.002 15.34 3.03 2.2% ↓
2 0.004 15.34 3.03
3 0.006 15.34 3.03
done · converged · 3 iters · reduced χ² 3.03 · wR(F²) 15.34 % · 0.01 s
project.display.pattern('heidi_300k')
300 K: displacements¶
for tensor in structure_300k.atom_site_aniso:
for component in (
tensor.adp_11,
tensor.adp_22,
tensor.adp_33,
tensor.adp_12,
tensor.adp_13,
tensor.adp_23,
):
component.free = not component.symmetry_constrained
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_heidi, 1624 pts fitted / 1624 loaded, 40 free
iter time (s) wR(F²) (%) reduced χ² change
15.34 3.08
1 0.004 7.73 0.78 74.6% ↓
2 0.010 7.67 0.77 1.7% ↓
3 0.016 7.67 0.77
done · converged · 3 iters · reduced χ² 0.77 · wR(F²) 7.67 % · 0.02 s
project.display.pattern('heidi_300k')
300 K: extinction¶
Larson's anisotropic extinction, its six coefficients starting at zero.
experiment_300k.extinction.coefficient_11.free = True
experiment_300k.extinction.coefficient_22.free = True
experiment_300k.extinction.coefficient_33.free = True
experiment_300k.extinction.coefficient_12.free = True
experiment_300k.extinction.coefficient_13.free = True
experiment_300k.extinction.coefficient_23.free = True
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_heidi, 1624 pts fitted / 1624 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
7.67 0.77
1 0.006 6.25 0.51 33.6% ↓
2 0.013 6.16 0.50 2.9% ↓
3 0.019 6.15 0.50
4 0.025 6.15 0.50
done · converged · 4 iters · reduced χ² 0.50 · wR(F²) 6.15 % · 0.03 s
project.display.pattern('heidi_300k')
55 K: scale¶
The 300 K parameters stay free, so the 300 K experiment is fitted again from where it ended.
experiment_55k.linked_structures['cosio_55k'].scale.free = True
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_heidi, 1624 pts fitted / 1624 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
6.15 0.50
1 0.005 6.15 0.50
edi fit: project cosio_heidi, 1465 pts fitted / 1465 loaded, 1 free
iter time (s) wR(F²) (%) reduced χ² change
36.92 69.94
1 0.001 8.90 4.06 94.2% ↓
2 0.003 8.90 4.06
3 0.005 8.90 4.06
done · converged · 4 iters · reduced χ² 2.21 · wR(F²) 8.38 % · 0.02 s
project.display.pattern('heidi_55k')
55 K: coordinates¶
for site in structure_55k.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_heidi, 1624 pts fitted / 1624 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
6.15 0.50
1 0.007 6.15 0.50
edi fit: project cosio_heidi, 1465 pts fitted / 1465 loaded, 12 free
iter time (s) wR(F²) (%) reduced χ² change
8.90 4.09
1 0.002 8.78 3.98 2.7% ↓
2 0.005 8.78 3.98
3 0.007 8.78 3.98
done · converged · 4 iters · reduced χ² 2.17 · wR(F²) 8.28 % · 0.02 s
project.display.pattern('heidi_55k')
55 K: extinction¶
At 55 K the displacements are small. Freed before the extinction, they would take its place and Co1's tensor would stop being positive definite, so the extinction comes before them here.
experiment_55k.extinction.coefficient_11.free = True
experiment_55k.extinction.coefficient_22.free = True
experiment_55k.extinction.coefficient_33.free = True
experiment_55k.extinction.coefficient_12.free = True
experiment_55k.extinction.coefficient_13.free = True
experiment_55k.extinction.coefficient_23.free = True
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_heidi, 1624 pts fitted / 1624 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
6.15 0.50
1 0.005 6.15 0.50
edi fit: project cosio_heidi, 1465 pts fitted / 1465 loaded, 18 free
iter time (s) wR(F²) (%) reduced χ² change
8.78 4.00
1 0.002 6.80 2.40 40.1% ↓
2 0.004 6.58 2.25 6.2% ↓
3 0.006 6.58 2.25 0.1% ↓
4 0.008 6.58 2.25
done · converged · 5 iters · reduced χ² 1.33 · wR(F²) 6.49 % · 0.02 s
project.display.pattern('heidi_55k')
55 K: displacements¶
for tensor in structure_55k.atom_site_aniso:
for component in (
tensor.adp_11,
tensor.adp_22,
tensor.adp_33,
tensor.adp_12,
tensor.adp_13,
tensor.adp_23,
):
component.free = not component.symmetry_constrained
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_heidi, 1624 pts fitted / 1624 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
6.15 0.50
1 0.006 6.15 0.50
edi fit: project cosio_heidi, 1465 pts fitted / 1465 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
6.58 2.29
1 0.004 3.80 0.76 66.6% ↓
2 0.009 3.79 0.76 0.4% ↓
3 0.013 3.79 0.76
done · converged · 4 iters · reduced χ² 0.62 · wR(F²) 4.41 % · 0.03 s
project.display.pattern('heidi_55k')
2.5 K: scale¶
The nuclear structure first, with every moment still at zero, in the order 55 K needed.
experiment_2k5.linked_structures['cosio_2k5'].scale.free = True
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_heidi, 1624 pts fitted / 1624 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
6.15 0.50
1 0.008 6.15 0.50
edi fit: project cosio_heidi, 1465 pts fitted / 1465 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
3.79 0.76
1 0.006 3.79 0.76
edi fit: project cosio_heidi, 1424 pts fitted / 1424 loaded, 1 free
iter time (s) wR(F²) (%) reduced χ² change
39.10 63.48
1 0.001 13.12 7.15 88.7% ↓
2 0.003 13.12 7.15
3 0.004 13.12 7.15
done · converged · 5 iters · reduced χ² 2.72 · wR(F²) 8.82 % · 0.03 s
project.display.pattern('heidi_2k5')
2.5 K: coordinates¶
for site in structure_2k5.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_heidi, 1624 pts fitted / 1624 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
6.15 0.50
1 0.006 6.15 0.50
edi fit: project cosio_heidi, 1465 pts fitted / 1465 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
3.79 0.76
1 0.004 3.79 0.76
edi fit: project cosio_heidi, 1424 pts fitted / 1424 loaded, 12 free
iter time (s) wR(F²) (%) reduced χ² change
13.12 7.21
1 0.002 13.01 7.08 1.7% ↓
2 0.004 13.01 7.08
3 0.006 13.01 7.08
done · converged · 5 iters · reduced χ² 2.69 · wR(F²) 8.75 % · 0.03 s
project.display.pattern('heidi_2k5')
2.5 K: extinction¶
experiment_2k5.extinction.coefficient_11.free = True
experiment_2k5.extinction.coefficient_22.free = True
experiment_2k5.extinction.coefficient_33.free = True
experiment_2k5.extinction.coefficient_12.free = True
experiment_2k5.extinction.coefficient_13.free = True
experiment_2k5.extinction.coefficient_23.free = True
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_heidi, 1624 pts fitted / 1624 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
6.15 0.50
1 0.007 6.15 0.50
edi fit: project cosio_heidi, 1465 pts fitted / 1465 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
3.79 0.76
1 0.005 3.79 0.76
edi fit: project cosio_heidi, 1424 pts fitted / 1424 loaded, 18 free
iter time (s) wR(F²) (%) reduced χ² change
13.01 7.11
1 0.002 11.97 6.02 15.4% ↓
2 0.005 11.88 5.93 1.4% ↓
3 0.008 11.88 5.93
4 0.011 11.88 5.93
done · converged · 6 iters · reduced χ² 2.32 · wR(F²) 8.12 % · 0.04 s
project.display.pattern('heidi_2k5')
2.5 K: displacements¶
for tensor in structure_2k5.atom_site_aniso:
for component in (
tensor.adp_11,
tensor.adp_22,
tensor.adp_33,
tensor.adp_12,
tensor.adp_13,
tensor.adp_23,
):
component.free = not component.symmetry_constrained
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_heidi, 1624 pts fitted / 1624 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
6.15 0.50
1 0.013 6.15 0.50
edi fit: project cosio_heidi, 1465 pts fitted / 1465 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
3.79 0.76
1 0.007 3.79 0.76
edi fit: project cosio_heidi, 1424 pts fitted / 1424 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
11.88 6.05
1 0.007 10.52 4.74 21.6% ↓
2 0.013 10.51 4.74 0.1% ↓
3 0.020 10.51 4.74
done · converged · 5 iters · reduced χ² 1.92 · wR(F²) 7.36 % · 0.07 s
Some reflections stay far from their measured intensity: (1 0 0), (0 0 1) and (1 1 0) among them. Pnma forbids them for nuclear scattering, so with the moments at zero their calculated intensity is zero; at 2.5 K they are magnetic.
project.display.pattern('heidi_2k5')
2.5 K: the magnetic moments¶
Free the moment components the magnetic space group allows. They start at zero, where the magnetic intensity has no slope, so the fit starts them at 1 μB and says so.
structure_2k5.atom_site_moment['Co1'].crystalaxis_x.free = True
structure_2k5.atom_site_moment['Co1'].crystalaxis_y.free = True
structure_2k5.atom_site_moment['Co1'].crystalaxis_z.free = True
structure_2k5.atom_site_moment['Co2'].crystalaxis_y.free = True
project.analysis.fit()
project.display.fit.results()
edi fit: project cosio_heidi, 1624 pts fitted / 1624 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
6.15 0.50
1 0.013 6.15 0.50
edi fit: project cosio_heidi, 1465 pts fitted / 1465 loaded, 46 free
iter time (s) wR(F²) (%) reduced χ² change
3.79 0.76
1 0.008 3.79 0.76
edi fit: project cosio_heidi, 1424 pts fitted / 1424 loaded, 50 free
iter time (s) wR(F²) (%) reduced χ² change
10.51 4.76
1 0.010 9.16 3.61 24.1% ↓
2 0.020 3.94 0.67 81.5% ↓
3 0.029 3.77 0.61 8.2% ↓
4 0.039 3.77 0.61
5 0.049 3.77 0.61
done · converged · 7 iters · reduced χ² 0.62 · wR(F²) 4.18 % · 0.10 s
project.display.pattern('heidi_2k5')
Sazonov et al., Acta Cryst. B 65 (2009) 664-675, Table 4, give Co1 (1.20(3), 3.64(2), 0.57(9)) μB and Co2 3.35(2) μB along b. The fitted moments' uncertainties rest on the estimated 2.5 K uncertainties.
project.display.structure('cosio_2k5')
Save the project¶
project.save_as('sc-neut-cwl_cosio-heidi')
project.report.save_html('sc-neut-cwl_cosio-heidi.html')
Saving project to 'sc-neut-cwl_cosio-heidi'
Saving report to 'sc-neut-cwl_cosio-heidi.html'