Structure Refinement: Co2SiO4, D20¶

This example demonstrates a Rietveld refinement of Co2SiO4 crystal structure using constant wavelength neutron powder diffraction data from D20 at ILL.

It also shows different ways to set free parameters: standard one-by-one and batch setting.

Import Library¶

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from easydiffraction import ExperimentFactory
from easydiffraction import Project
from easydiffraction import StructureFactory
from easydiffraction import download_data
from easydiffraction import ExperimentFactory
from easydiffraction import Project
from easydiffraction import StructureFactory
from easydiffraction import download_data

Define Structure¶

This section shows how to add structures and modify their parameters.

Create Structure¶

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structure = StructureFactory.from_scratch(name='cosio')
structure = StructureFactory.from_scratch(name='cosio')

Set Space Group¶

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structure.space_group.name_h_m = 'P n m a'
structure.space_group.it_coordinate_system_code = 'abc'
structure.space_group.name_h_m = 'P n m a'
structure.space_group.it_coordinate_system_code = 'abc'

Set Unit Cell¶

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structure.cell.length_a = 10.3
structure.cell.length_b = 6.0
structure.cell.length_c = 4.8
structure.cell.length_a = 10.3
structure.cell.length_b = 6.0
structure.cell.length_c = 4.8

Set Atom Sites¶

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structure.atom_sites.create(
    label='Co1',
    type_symbol='Co',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    wyckoff_letter='a',
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='Co2',
    type_symbol='Co',
    fract_x=0.279,
    fract_y=0.25,
    fract_z=0.985,
    wyckoff_letter='c',
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='Si',
    type_symbol='Si',
    fract_x=0.094,
    fract_y=0.25,
    fract_z=0.429,
    wyckoff_letter='c',
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='O1',
    type_symbol='O',
    fract_x=0.091,
    fract_y=0.25,
    fract_z=0.771,
    wyckoff_letter='c',
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='O2',
    type_symbol='O',
    fract_x=0.448,
    fract_y=0.25,
    fract_z=0.217,
    wyckoff_letter='c',
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='O3',
    type_symbol='O',
    fract_x=0.164,
    fract_y=0.032,
    fract_z=0.28,
    wyckoff_letter='d',
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='Co1',
    type_symbol='Co',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    wyckoff_letter='a',
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='Co2',
    type_symbol='Co',
    fract_x=0.279,
    fract_y=0.25,
    fract_z=0.985,
    wyckoff_letter='c',
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='Si',
    type_symbol='Si',
    fract_x=0.094,
    fract_y=0.25,
    fract_z=0.429,
    wyckoff_letter='c',
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='O1',
    type_symbol='O',
    fract_x=0.091,
    fract_y=0.25,
    fract_z=0.771,
    wyckoff_letter='c',
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='O2',
    type_symbol='O',
    fract_x=0.448,
    fract_y=0.25,
    fract_z=0.217,
    wyckoff_letter='c',
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='O3',
    type_symbol='O',
    fract_x=0.164,
    fract_y=0.032,
    fract_z=0.28,
    wyckoff_letter='d',
    adp_iso=0.5,
)

Define Experiment¶

This section shows how to add experiments, configure their parameters, and link the structures defined in the previous step.

Download Measured Data¶

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data_path = download_data(id=12, destination='data')
data_path = download_data(id=12, destination='data')

Getting data...

Data #12: Co2SiO4, D20 (ILL)

✅ Data #12 downloaded to 'data/ed-12.xye'

Create Experiment¶

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expt = ExperimentFactory.from_data_path(name='d20', data_path=data_path)
expt = ExperimentFactory.from_data_path(name='d20', data_path=data_path)

Set Instrument¶

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expt.instrument.setup_wavelength = 1.87
expt.instrument.calib_twotheta_offset = 0.1
expt.instrument.setup_wavelength = 1.87
expt.instrument.calib_twotheta_offset = 0.1

Set Peak Profile¶

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expt.show_peak_profile_types()
expt.show_peak_profile_types()

Peak profile types

		Type	Description
1	*	pseudo-voigt	CWL pseudo-Voigt profile
2		pseudo-voigt + empirical asymmetry	CWL pseudo-Voigt profile with empirical asymmetry correction.

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expt.peak_profile_type = 'pseudo-voigt + empirical asymmetry'
expt.peak_profile_type = 'pseudo-voigt + empirical asymmetry'

⚠️ Switching peak profile type discards existing peak parameters.

Peak profile type for experiment 'd20' changed to

pseudo-voigt + empirical asymmetry

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expt.peak.broad_gauss_u = 0.3
expt.peak.broad_gauss_v = -0.5
expt.peak.broad_gauss_w = 0.4
expt.peak.broad_gauss_u = 0.3
expt.peak.broad_gauss_v = -0.5
expt.peak.broad_gauss_w = 0.4

Set Background¶

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expt.show_background_types()
expt.show_background_types()

Background types

		Type	Description
1		chebyshev	Chebyshev polynomial background
2	*	line-segment	Linear interpolation between points

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expt.background.create(id='1', x=8, y=500)
expt.background.create(id='2', x=9, y=500)
expt.background.create(id='3', x=10, y=500)
expt.background.create(id='4', x=11, y=500)
expt.background.create(id='5', x=12, y=500)
expt.background.create(id='6', x=15, y=500)
expt.background.create(id='7', x=25, y=500)
expt.background.create(id='8', x=30, y=500)
expt.background.create(id='9', x=50, y=500)
expt.background.create(id='10', x=70, y=500)
expt.background.create(id='11', x=90, y=500)
expt.background.create(id='12', x=110, y=500)
expt.background.create(id='13', x=130, y=500)
expt.background.create(id='14', x=150, y=500)
expt.background.create(id='1', x=8, y=500)
expt.background.create(id='2', x=9, y=500)
expt.background.create(id='3', x=10, y=500)
expt.background.create(id='4', x=11, y=500)
expt.background.create(id='5', x=12, y=500)
expt.background.create(id='6', x=15, y=500)
expt.background.create(id='7', x=25, y=500)
expt.background.create(id='8', x=30, y=500)
expt.background.create(id='9', x=50, y=500)
expt.background.create(id='10', x=70, y=500)
expt.background.create(id='11', x=90, y=500)
expt.background.create(id='12', x=110, y=500)
expt.background.create(id='13', x=130, y=500)
expt.background.create(id='14', x=150, y=500)

Set Linked Phases¶

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expt.linked_phases.create(id='cosio', scale=1.0)
expt.linked_phases.create(id='cosio', scale=1.0)

Define Project¶

The project object is used to manage the structure, experiment, and analysis.

Create Project¶

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project = Project()
project = Project()

Add Structure¶

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project.structures.add(structure)
project.structures.add(structure)

Add Experiment¶

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project.experiments.add(expt)
project.experiments.add(expt)

Perform Analysis¶

This section shows the analysis process, including how to set up calculation and fitting engines.

Plot Measured vs Calculated¶

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project.display.plotter.plot_meas_vs_calc(expt_name='d20')
project.display.plotter.plot_meas_vs_calc(expt_name='d20')

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project.display.plotter.plot_meas_vs_calc(expt_name='d20', x_min=41, x_max=54)
project.display.plotter.plot_meas_vs_calc(expt_name='d20', x_min=41, x_max=54)

Set Free Parameters¶

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structure.cell.length_a.free = True
structure.cell.length_b.free = True
structure.cell.length_c.free = True

for atom_site in structure.atom_sites:
    for parameter in ('fract_x', 'fract_y', 'fract_z'):
        getattr(atom_site, parameter).free = True

for atom_site in structure.atom_sites:
    atom_site.adp_iso.free = True

for label in ('O1', 'O2', 'O3'):
    atom_site = structure.atom_sites[label]
    atom_site.occupancy.free = True
structure.cell.length_a.free = True
structure.cell.length_b.free = True
structure.cell.length_c.free = True

for atom_site in structure.atom_sites:
    for parameter in ('fract_x', 'fract_y', 'fract_z'):
        getattr(atom_site, parameter).free = True

for atom_site in structure.atom_sites:
    atom_site.adp_iso.free = True

for label in ('O1', 'O2', 'O3'):
    atom_site = structure.atom_sites[label]
    atom_site.occupancy.free = True

⚠️ Parameter 'cosio.atom_site.Co1.fract_x' is fixed by symmetry. Ignoring free=True.

⚠️ Parameter 'cosio.atom_site.Co1.fract_y' is fixed by symmetry. Ignoring free=True.

⚠️ Parameter 'cosio.atom_site.Co1.fract_z' is fixed by symmetry. Ignoring free=True.

⚠️ Parameter 'cosio.atom_site.Co2.fract_y' is fixed by symmetry. Ignoring free=True.

⚠️ Parameter 'cosio.atom_site.Si.fract_y' is fixed by symmetry. Ignoring free=True.

⚠️ Parameter 'cosio.atom_site.O1.fract_y' is fixed by symmetry. Ignoring free=True.

⚠️ Parameter 'cosio.atom_site.O2.fract_y' is fixed by symmetry. Ignoring free=True.

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expt.linked_phases['cosio'].scale.free = True

expt.instrument.calib_twotheta_offset.free = True

expt.peak.broad_gauss_u.free = True
expt.peak.broad_gauss_v.free = True
expt.peak.broad_gauss_w.free = True
expt.peak.broad_lorentz_y.free = True

expt.peak.asym_empir_2.free = True

for point in expt.background:
    point.y.free = True
expt.linked_phases['cosio'].scale.free = True

expt.instrument.calib_twotheta_offset.free = True

expt.peak.broad_gauss_u.free = True
expt.peak.broad_gauss_v.free = True
expt.peak.broad_gauss_w.free = True
expt.peak.broad_lorentz_y.free = True

expt.peak.asym_empir_2.free = True

for point in expt.background:
    point.y.free = True

Show free parameters after selection.

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project.analysis.display.free_params()
project.analysis.display.free_params()

Free parameters for both structures (🧩 data blocks) and experiments (🔬 data blocks)

	datablock	category	entry	parameter	value	min	max	units
1	cosio	cell		length_a	10.30000	-inf	inf	Å
2	cosio	cell		length_b	6.00000	-inf	inf	Å
3	cosio	cell		length_c	4.80000	-inf	inf	Å
4	cosio	atom_site	Co1	adp_iso	0.50000	-inf	inf	Å²
5	cosio	atom_site	Co2	fract_x	0.27900	-inf	inf
6	cosio	atom_site	Co2	fract_z	0.98500	-inf	inf
7	cosio	atom_site	Co2	adp_iso	0.50000	-inf	inf	Å²
8	cosio	atom_site	Si	fract_x	0.09400	-inf	inf
9	cosio	atom_site	Si	fract_z	0.42900	-inf	inf
10	cosio	atom_site	Si	adp_iso	0.50000	-inf	inf	Å²
11	cosio	atom_site	O1	fract_x	0.09100	-inf	inf
12	cosio	atom_site	O1	fract_z	0.77100	-inf	inf
13	cosio	atom_site	O1	occupancy	1.00000	-inf	inf
14	cosio	atom_site	O1	adp_iso	0.50000	-inf	inf	Å²
15	cosio	atom_site	O2	fract_x	0.44800	-inf	inf
16	cosio	atom_site	O2	fract_z	0.21700	-inf	inf
17	cosio	atom_site	O2	occupancy	1.00000	-inf	inf
18	cosio	atom_site	O2	adp_iso	0.50000	-inf	inf	Å²
19	cosio	atom_site	O3	fract_x	0.16400	-inf	inf
20	cosio	atom_site	O3	fract_y	0.03200	-inf	inf
21	cosio	atom_site	O3	fract_z	0.28000	-inf	inf
22	cosio	atom_site	O3	occupancy	1.00000	-inf	inf
23	cosio	atom_site	O3	adp_iso	0.50000	-inf	inf	Å²
24	d20	linked_phases	cosio	scale	1.00000	-inf	inf
25	d20	peak		asym_empir_2	0.00000	-inf	inf
26	d20	peak		broad_gauss_u	0.30000	-inf	inf	deg²
27	d20	peak		broad_gauss_v	-0.50000	-inf	inf	deg²
28	d20	peak		broad_gauss_w	0.40000	-inf	inf	deg²
29	d20	peak		broad_lorentz_y	0.00000	-inf	inf	deg
30	d20	instrument		twotheta_offset	0.10000	-inf	inf	deg
31	d20	background	1	y	500.00000	-inf	inf
32	d20	background	2	y	500.00000	-inf	inf
33	d20	background	3	y	500.00000	-inf	inf
34	d20	background	4	y	500.00000	-inf	inf
35	d20	background	5	y	500.00000	-inf	inf
36	d20	background	6	y	500.00000	-inf	inf
37	d20	background	7	y	500.00000	-inf	inf
38	d20	background	8	y	500.00000	-inf	inf
39	d20	background	9	y	500.00000	-inf	inf
40	d20	background	10	y	500.00000	-inf	inf
41	d20	background	11	y	500.00000	-inf	inf
42	d20	background	12	y	500.00000	-inf	inf
43	d20	background	13	y	500.00000	-inf	inf
44	d20	background	14	y	500.00000	-inf	inf

Set Constraints¶

Set aliases for parameters.

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project.analysis.aliases.create(
    label='biso_Co1',
    param=project.structures['cosio'].atom_sites['Co1'].adp_iso,
)
project.analysis.aliases.create(
    label='biso_Co2',
    param=project.structures['cosio'].atom_sites['Co2'].adp_iso,
)
project.analysis.aliases.create(
    label='biso_Co1',
    param=project.structures['cosio'].atom_sites['Co1'].adp_iso,
)
project.analysis.aliases.create(
    label='biso_Co2',
    param=project.structures['cosio'].atom_sites['Co2'].adp_iso,
)

Set constraints.

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project.analysis.constraints.create(expression='biso_Co2 = biso_Co1')
project.analysis.constraints.create(expression='biso_Co2 = biso_Co1')

Run Fitting¶

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project.analysis.fit()
project.analysis.fit()

Standard fitting

📋 Using experiment 🔬 'd20' for 'single' fitting

🚀 Starting fit process with 'lmfit (leastsq)'...

📈 Goodness-of-fit (reduced χ²) change:

	iteration	χ²	improvement [%]
1	1	424.43
2	47	73.78	82.6% ↓
3	91	39.41	46.6% ↓
4	136	19.47	50.6% ↓
5	180	17.19	11.7% ↓
6	224	11.03	35.8% ↓
7	268	7.94	28.0% ↓
8	312	4.67	41.2% ↓
9	356	4.38	6.2% ↓
10	577	4.38

🏆 Best goodness-of-fit (reduced χ²) is 4.38 at iteration 560

✅ Fitting complete.

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project.analysis.display.fit_results()
project.analysis.display.fit_results()

Fit results

✅ Success: True

⏱️ Fitting time: 56.36 seconds

📏 Goodness-of-fit (reduced χ²): 4.38

📏 R-factor (Rf): 2.99%

📏 R-factor squared (Rf²): 4.32%

📏 Weighted R-factor (wR): 4.54%

📈 Fitted parameters:

	datablock	category	entry	parameter	start	fitted	uncertainty	units	change
1	cosio	cell		length_a	10.3000	10.3084	0.0003	Å	0.08 % ↑
2	cosio	cell		length_b	6.0000	6.0036	0.0002	Å	0.06 % ↑
3	cosio	cell		length_c	4.8000	4.7864	0.0001	Å	0.28 % ↓
4	cosio	atom_site	Co1	adp_iso	0.5000	0.6716	0.1079	Å²	34.32 % ↑
5	cosio	atom_site	Co2	fract_x	0.2790	0.2792	0.0007		0.06 % ↑
6	cosio	atom_site	Co2	fract_z	0.9850	0.9850	0.0014		0.00 % ↓
7	cosio	atom_site	Si	fract_x	0.0940	0.0938	0.0004		0.25 % ↓
8	cosio	atom_site	Si	fract_z	0.4290	0.4293	0.0008		0.08 % ↑
9	cosio	atom_site	Si	adp_iso	0.5000	0.6856	0.0885	Å²	37.11 % ↑
10	cosio	atom_site	O1	fract_x	0.0910	0.0911	0.0003		0.14 % ↑
11	cosio	atom_site	O1	fract_z	0.7710	0.7715	0.0006		0.07 % ↑
12	cosio	atom_site	O1	occupancy	1.0000	0.9149	0.0136		8.51 % ↓
13	cosio	atom_site	O1	adp_iso	0.5000	0.4655	0.0776	Å²	6.90 % ↓
14	cosio	atom_site	O2	fract_x	0.4480	0.4482	0.0003		0.04 % ↑
15	cosio	atom_site	O2	fract_z	0.2170	0.2171	0.0007		0.04 % ↑
16	cosio	atom_site	O2	occupancy	1.0000	0.9589	0.0135		4.11 % ↓
17	cosio	atom_site	O2	adp_iso	0.5000	0.6701	0.0795	Å²	34.03 % ↑
18	cosio	atom_site	O3	fract_x	0.1640	0.1636	0.0002		0.24 % ↓
19	cosio	atom_site	O3	fract_y	0.0320	0.0315	0.0003		1.47 % ↓
20	cosio	atom_site	O3	fract_z	0.2800	0.2802	0.0005		0.07 % ↑
21	cosio	atom_site	O3	occupancy	1.0000	0.9549	0.0118		4.51 % ↓
22	cosio	atom_site	O3	adp_iso	0.5000	0.8491	0.0614	Å²	69.83 % ↑
23	d20	linked_phases	cosio	scale	1.0000	1.3308	0.0289		33.08 % ↑
24	d20	peak		asym_empir_2	0.0000	-0.0087	0.0019		N/A
25	d20	peak		broad_gauss_u	0.3000	0.2420	0.0066	deg²	19.32 % ↓
26	d20	peak		broad_gauss_v	-0.5000	-0.5306	0.0143	deg²	6.11 % ↑
27	d20	peak		broad_gauss_w	0.4000	0.3881	0.0089	deg²	2.98 % ↓
28	d20	peak		broad_lorentz_y	0.0000	0.0137	0.0044	deg	N/A
29	d20	instrument		twotheta_offset	0.1000	0.2762	0.0033	deg	176.23 % ↑
30	d20	background	1	y	500.0000	608.7247	14.2455		21.74 % ↑
31	d20	background	2	y	500.0000	580.6999	9.5299		16.14 % ↑
32	d20	background	3	y	500.0000	562.9608	9.0541		12.59 % ↑
33	d20	background	4	y	500.0000	540.3572	8.5911		8.07 % ↑
34	d20	background	5	y	500.0000	519.8448	5.9267		3.97 % ↑
35	d20	background	6	y	500.0000	507.5406	3.4171		1.51 % ↑
36	d20	background	7	y	500.0000	462.8842	3.1447		7.42 % ↓
37	d20	background	8	y	500.0000	430.7947	2.4268		13.84 % ↓
38	d20	background	9	y	500.0000	451.5176	2.2116		9.70 % ↓
39	d20	background	10	y	500.0000	429.1040	1.9992		14.18 % ↓
40	d20	background	11	y	500.0000	413.5987	2.2203		17.28 % ↓
41	d20	background	12	y	500.0000	362.0222	2.0545		27.60 % ↓
42	d20	background	13	y	500.0000	292.8711	1.9349		41.43 % ↓
43	d20	background	14	y	500.0000	241.1480	2.9203		51.77 % ↓

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project.display.plotter.plot_param_correlations()
project.display.plotter.plot_param_correlations()

Plot Measured vs Calculated¶

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project.display.plotter.plot_meas_vs_calc(expt_name='d20')
project.display.plotter.plot_meas_vs_calc(expt_name='d20')

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project.display.plotter.plot_meas_vs_calc(expt_name='d20', x_min=42, x_max=52)
project.display.plotter.plot_meas_vs_calc(expt_name='d20', x_min=42, x_max=52)

Summary¶

This final section shows how to review the results of the analysis.

Show Project Summary¶

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project.summary.show_report()
project.summary.show_report()

————————————
PROJECT INFO
————————————

Title

Untitled Project

—————————————————————
CRYSTALLOGRAPHIC DATA
—————————————————————

Phase datablock

🧩 cosio

Space group

P n m a

	Parameter	Value	Uncertainty
1	a	10.30836736	0.00027647
2	b	10.30836736	0.00027647
3	c	10.30836736	0.00027647
4	α	90.00000000
5	β	90.00000000
6	γ	90.00000000

Atom sites

	label	type	x	y	z	occ	Biso
1	Co1	Co	0.00000000	0.00000000	0.00000000	1.00000000	0.67158838
2	Co2	Co	0.27917936	0.25000000	0.98498721	1.00000000	0.67158838
3	Si	Si	0.09376329	0.25000000	0.42933149	1.00000000	0.68556173
4	O1	O	0.09112409	0.25000000	0.77152556	0.91489691	0.46550121
5	O2	O	0.44815717	0.25000000	0.21709214	0.95889236	0.67012818
6	O3	O	0.16361394	0.03153092	0.28020344	0.95492679	0.84913226

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EXPERIMENTS
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Experiment datablock

🔬 d20

Experiment type

powder, neutron, constant wavelength bragg

Calculation engine

cryspy

Wavelength

1.87000

2θ offset

0.27623

Profile type

pseudo-voigt + empirical asymmetry

Peak broadening (Gaussian)

	Parameter	Value	Uncertainty
1	U	0.24204807	0.00662677
2	V	-0.53056938	0.01427471
3	W	0.38809537	0.00893623

Peak broadening (Lorentzian)

	Parameter	Value	Uncertainty
1	X	0.00000000
2	Y	0.01367915	0.00438704

Asymmetry (Empirical)

	Parameter	Value	Uncertainty
1	p1	0.00000000
2	p2	-0.00873349	0.00189254
3	p3	0.00000000
4	p4	0.00000000

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FITTING
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Minimization engine

lmfit (leastsq)

Fit quality

	metric	value
1	Goodness-of-fit (reduced χ²)	4.38