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.coord_system_code = 'abc'
structure.space_group.name_h_m = 'P n m a'
structure.space_group.coord_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(
    id='Co1',
    type_symbol='Co',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='Co2',
    type_symbol='Co',
    fract_x=0.279,
    fract_y=0.25,
    fract_z=0.985,
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='Si',
    type_symbol='Si',
    fract_x=0.094,
    fract_y=0.25,
    fract_z=0.429,
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='O1',
    type_symbol='O',
    fract_x=0.091,
    fract_y=0.25,
    fract_z=0.771,
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='O2',
    type_symbol='O',
    fract_x=0.448,
    fract_y=0.25,
    fract_z=0.217,
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='O3',
    type_symbol='O',
    fract_x=0.164,
    fract_y=0.032,
    fract_z=0.28,
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='Co1',
    type_symbol='Co',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='Co2',
    type_symbol='Co',
    fract_x=0.279,
    fract_y=0.25,
    fract_z=0.985,
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='Si',
    type_symbol='Si',
    fract_x=0.094,
    fract_y=0.25,
    fract_z=0.429,
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='O1',
    type_symbol='O',
    fract_x=0.091,
    fract_y=0.25,
    fract_z=0.771,
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='O2',
    type_symbol='O',
    fract_x=0.448,
    fract_y=0.25,
    fract_z=0.217,
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='O3',
    type_symbol='O',
    fract_x=0.164,
    fract_y=0.032,
    fract_z=0.28,
    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 Data¶

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data_path = download_data('meas-cosio-d20', destination='data')
data_path = download_data('meas-cosio-d20', destination='data')

Getting data...

Data 'meas-cosio-d20': Co2SiO4, D20 (ILL)

✅ Data 'meas-cosio-d20' downloaded to '../../../data/meas-cosio-d20.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.peak.show_supported()
expt.peak.show_supported()

Peak types

		Type	Description
1	*	pseudo-voigt	CWL pseudo-Voigt profile
2		pseudo-voigt + berar-baldinozzi asymmetry	CWL pseudo-Voigt profile with Berar-Baldinozzi asymmetry correction.

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expt.peak.type = 'pseudo-voigt + berar-baldinozzi asymmetry'
expt.peak.type = 'pseudo-voigt + berar-baldinozzi asymmetry'

⚠️ Switching peak profile type adds these settings with defaults:                                                                 
   • asym_beba_a0=0.0                                                                                                             
   • asym_beba_a1=0.0                                                                                                             
   • asym_beba_b0=0.0                                                                                                             
   • asym_beba_b1=0.0

Peak profile type for experiment 'd20' changed to

pseudo-voigt + berar-baldinozzi 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

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expt.peak.cutoff_fwhm = 8
expt.peak.cutoff_fwhm = 8

Set Background¶

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expt.background.show_supported()
expt.background.show_supported()

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

Set Linked Structures¶

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expt.linked_structures.create(structure_id='cosio', scale=1.0)
expt.linked_structures.create(structure_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(name='cosio_d20')
project = Project(name='cosio_d20')

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project.save_as(dir_path='projects/refine-cosio-d20')
project.save_as(dir_path='projects/refine-cosio-d20')

Saving project 📦 'cosio_d20' to '../../../projects/refine-cosio-d20'

├── 📄 project.edi

├── 📁 structures/

├── 📁 experiments/

├── 📁 analysis/

│   └── 📄 analysis.edi

└── 📁 reports/

    └── 📄 cosio_d20.html

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.

Display Structure¶

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project.display.structure(struct_name='cosio')
project.display.structure(struct_name='cosio')

Structure 🧩 'cosio' (Atom view type: 'covalent')

Loading plot…

drag = rotate
wheel = zoom
right-drag = pan

Display Pattern¶

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project.display.pattern(expt_name='d20')
project.display.pattern(expt_name='d20')

Loading plot…

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

Loading plot…

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 constrained by symmetry. Ignoring free=True.

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

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

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

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

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

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

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expt.linked_structures['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_beba_b0.free = True

for point in expt.background:
    point.intensity.free = True
expt.linked_structures['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_beba_b0.free = True

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

Show free parameters after selection.

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project.display.parameters.free()
project.display.parameters.free()

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_structure	cosio	scale	1.00000	-inf	inf
25	d20	peak		asym_beba_b0	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	intensity	500.00000	-inf	inf
32	d20	background	2	intensity	500.00000	-inf	inf
33	d20	background	3	intensity	500.00000	-inf	inf
34	d20	background	4	intensity	500.00000	-inf	inf
35	d20	background	5	intensity	500.00000	-inf	inf
36	d20	background	6	intensity	500.00000	-inf	inf
37	d20	background	7	intensity	500.00000	-inf	inf
38	d20	background	8	intensity	500.00000	-inf	inf
39	d20	background	9	intensity	500.00000	-inf	inf
40	d20	background	10	intensity	500.00000	-inf	inf
41	d20	background	11	intensity	500.00000	-inf	inf
42	d20	background	12	intensity	500.00000	-inf	inf
43	d20	background	13	intensity	500.00000	-inf	inf
44	d20	background	14	intensity	500.00000	-inf	inf

Set Constraints¶

Set aliases for parameters.

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

	iteration	time (s)	χ²	change / status
1	1	0.06	424.43
2	47	4.07	73.81	82.6% ↓
3	91	7.98	39.81	46.1% ↓
4	136	12.16	19.19	51.8% ↓
5	180	15.92	12.76	33.5% ↓
6	224	19.59	7.64	40.1% ↓
7	268	23.15	4.61	39.6% ↓
8	312	26.79	4.38	5.0% ↓
9	369	31.81	4.38
10	429	37.01	4.38
11	490	42.03	4.38
12	533	47.08	4.38

🏆 Best goodness-of-fit (reduced χ²) is 4.38 at iteration 516

✅ Fitting complete.

Saving project 📦 'cosio_d20' to '../../../projects/refine-cosio-d20'

├── 📄 project.edi

├── 📁 structures/

│   └── 📄 cosio.edi

├── 📁 experiments/

│   └── 📄 d20.edi

├── 📁 analysis/

│   └── 📄 analysis.edi

└── 📁 reports/

    └── 📄 cosio_d20.html

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project.display.fit.results()
project.display.fit.results()

⚙️ Settings used:

	Name	Value	Description
1	max_iterations	1000	Maximum solver iterations.

📋 Least-squares fit results:

	Metric	Value
1	🧪 Minimizer	lmfit (leastsq)
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	47.08
4	🔁 Iterations	530
5	📏 Goodness-of-fit (reduced χ²)	4.38
6	📏 R-factor (Rf, %)	2.99
7	📏 R-factor squared (Rf², %)	4.32
8	📏 Weighted R-factor (wR, %)	4.54

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	cosio	cell		length_a	Å	10.3000	10.3084	0.0003	0.08 % ↑
2	cosio	cell		length_b	Å	6.0000	6.0036	0.0002	0.06 % ↑
3	cosio	cell		length_c	Å	4.8000	4.7864	0.0001	0.28 % ↓
4	cosio	atom_site	Co1	adp_iso	Å²	0.5000	0.6716	0.1079	34.33 % ↑
5	cosio	atom_site	Co2	fract_x		0.2790	0.2792	0.0007	0.07 % ↑
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	0.26 % ↓
8	cosio	atom_site	Si	fract_z		0.4290	0.4294	0.0008	0.08 % ↑
9	cosio	atom_site	Si	adp_iso	Å²	0.5000	0.6864	0.0884	37.28 % ↑
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	0.9148	0.0136	8.52 % ↓
13	cosio	atom_site	O1	adp_iso	Å²	0.5000	0.4662	0.0775	6.77 % ↓
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	0.9588	0.0135	4.12 % ↓
17	cosio	atom_site	O2	adp_iso	Å²	0.5000	0.6713	0.0795	34.26 % ↑
18	cosio	atom_site	O3	fract_x		0.1640	0.1636	0.0002	0.23 % ↓
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	0.9548	0.0118	4.52 % ↓
22	cosio	atom_site	O3	adp_iso	Å²	0.5000	0.8499	0.0614	69.99 % ↑
23	d20	linked_structure	cosio	scale		1.0000	1.3307	0.0289	33.07 % ↑
24	d20	peak		asym_beba_b0		0.0000	-0.0087	0.0019	N/A
25	d20	peak		broad_gauss_u	deg²	0.3000	0.2422	0.0066	19.26 % ↓
26	d20	peak		broad_gauss_v	deg²	-0.5000	-0.5309	0.0143	6.17 % ↑
27	d20	peak		broad_gauss_w	deg²	0.4000	0.3884	0.0089	2.91 % ↓
28	d20	peak		broad_lorentz_y	deg	0.0000	0.0134	0.0043	N/A
29	d20	instrument		twotheta_offset	deg	0.1000	0.2762	0.0033	176.23 % ↑
30	d20	background	1	intensity		500.0000	609.3228	14.2442	21.86 % ↑
31	d20	background	2	intensity		500.0000	581.1915	9.5285	16.24 % ↑
32	d20	background	3	intensity		500.0000	563.3757	9.0531	12.68 % ↑
33	d20	background	4	intensity		500.0000	540.7175	8.5904	8.14 % ↑
34	d20	background	5	intensity		500.0000	520.1578	5.9256	4.03 % ↑
35	d20	background	6	intensity		500.0000	507.7804	3.4171	1.56 % ↑
36	d20	background	7	intensity		500.0000	463.0315	3.1418	7.39 % ↓
37	d20	background	8	intensity		500.0000	430.9893	2.4269	13.80 % ↓
38	d20	background	9	intensity		500.0000	451.8045	2.1831	9.64 % ↓
39	d20	background	10	intensity		500.0000	429.4275	1.9786	14.11 % ↓
40	d20	background	11	intensity		500.0000	414.1454	2.1448	17.17 % ↓
41	d20	background	12	intensity		500.0000	362.4173	2.0131	27.52 % ↓
42	d20	background	13	intensity		500.0000	293.1495	1.9075	41.37 % ↓
43	d20	background	14	intensity		500.0000	241.3362	2.8958	51.73 % ↓

  • start      = parameter value before refinement
  • value      = refined value from least-squares minimization
  • s.u.       = standard uncertainty (one sigma), from the covariance matrix
  • change     = relative change from start, in %; ↑ = increase, ↓ = decrease

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project.display.fit.correlations()
project.display.fit.correlations()

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Display Pattern¶

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project.display.pattern(expt_name='d20')
project.display.pattern(expt_name='d20')

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In [33]:

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

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📊 Report¶

The HTML report is written automatically when the project is saved; enable project.report.pdf as well for a PDF version.