Structure Refinement: LBCO, HRPT¶

This minimalistic example is designed to show how Rietveld refinement can be performed when both the crystal structure and experiment are defined directly in code. Only the experimentally measured data is loaded from an external file. It also shows how to switch calculation engine.

For this example, constant-wavelength neutron powder diffraction data for La0.5Ba0.5CoO3 from HRPT at PSI is used.

It does not contain any advanced features or options, and includes no comments or explanations — these can be found in the other tutorials. Default values are used for all parameters if not specified. Only essential and self-explanatory code is provided.

The example is intended for users who are already familiar with the EasyDiffraction library and want to quickly get started with a simple refinement. It is also useful for those who want to see what a refinement might look like in code. For a more detailed explanation of the code, please refer to the other tutorials.

🛠️ Import Library¶

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import easydiffraction as ed
import easydiffraction as ed

📦 Define Project¶

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project = ed.Project(name='lbco_hrpt')
project = ed.Project(name='lbco_hrpt')

🧩 Define Structure¶

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project.structures.create(name='lbco')
project.structures.create(name='lbco')

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structure = project.structures['lbco']
structure = project.structures['lbco']

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structure.space_group.name_h_m = 'P m -3 m'
structure.space_group.it_coordinate_system_code = '1'
structure.space_group.name_h_m = 'P m -3 m'
structure.space_group.it_coordinate_system_code = '1'

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structure.cell.length_a = 3.88
structure.cell.length_a = 3.88

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structure.atom_sites.create(
    label='La',
    type_symbol='La',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    adp_iso=0.5,
    occupancy=0.5,
)
structure.atom_sites.create(
    label='Ba',
    type_symbol='Ba',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    adp_iso=0.5,
    occupancy=0.5,
)
structure.atom_sites.create(
    label='Co',
    type_symbol='Co',
    fract_x=0.5,
    fract_y=0.5,
    fract_z=0.5,
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='O',
    type_symbol='O',
    fract_x=0,
    fract_y=0.5,
    fract_z=0.5,
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='La',
    type_symbol='La',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    adp_iso=0.5,
    occupancy=0.5,
)
structure.atom_sites.create(
    label='Ba',
    type_symbol='Ba',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    adp_iso=0.5,
    occupancy=0.5,
)
structure.atom_sites.create(
    label='Co',
    type_symbol='Co',
    fract_x=0.5,
    fract_y=0.5,
    fract_z=0.5,
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='O',
    type_symbol='O',
    fract_x=0,
    fract_y=0.5,
    fract_z=0.5,
    adp_iso=0.5,
)

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

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

Loading plot…

drag = rotate
wheel = zoom
right-drag = pan

🔬 Define Experiment¶

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data_path = ed.download_data(id=3, destination='data')
data_path = ed.download_data(id=3, destination='data')

Getting data...

Data #3: La0.5Ba0.5CoO3, HRPT (PSI), 300 K

✅ Data #3 downloaded to '../../../data/ed-3.xye'

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project.experiments.add_from_data_path(
    name='hrpt',
    data_path=data_path,
    sample_form='powder',
    beam_mode='constant wavelength',
    radiation_probe='neutron',
)
project.experiments.add_from_data_path(
    name='hrpt',
    data_path=data_path,
    sample_form='powder',
    beam_mode='constant wavelength',
    radiation_probe='neutron',
)

Data loaded successfully

Experiment 🔬 'hrpt'. Number of data points: 3098.

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experiment = project.experiments['hrpt']
experiment = project.experiments['hrpt']

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experiment.instrument.setup_wavelength = 1.494
experiment.instrument.calib_twotheta_offset = 0.6
experiment.instrument.setup_wavelength = 1.494
experiment.instrument.calib_twotheta_offset = 0.6

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experiment.peak.broad_gauss_u = 0.1
experiment.peak.broad_gauss_v = -0.1
experiment.peak.broad_gauss_w = 0.1
experiment.peak.broad_lorentz_y = 0.1
experiment.peak.broad_gauss_u = 0.1
experiment.peak.broad_gauss_v = -0.1
experiment.peak.broad_gauss_w = 0.1
experiment.peak.broad_lorentz_y = 0.1

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experiment.background.create(id='1', x=10, y=170)
experiment.background.create(id='2', x=30, y=170)
experiment.background.create(id='3', x=50, y=170)
experiment.background.create(id='4', x=110, y=170)
experiment.background.create(id='5', x=165, y=170)
experiment.background.create(id='1', x=10, y=170)
experiment.background.create(id='2', x=30, y=170)
experiment.background.create(id='3', x=50, y=170)
experiment.background.create(id='4', x=110, y=170)
experiment.background.create(id='5', x=165, y=170)

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experiment.excluded_regions.create(id='1', start=0, end=5)
experiment.excluded_regions.create(id='2', start=165, end=180)
experiment.excluded_regions.create(id='1', start=0, end=5)
experiment.excluded_regions.create(id='2', start=165, end=180)

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experiment.linked_phases.create(id='lbco', scale=10.0)
experiment.linked_phases.create(id='lbco', scale=10.0)

🚀 Perform Analysis¶

Without Constraints¶

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structure.cell.length_a.free = True

structure.atom_sites['La'].adp_iso.free = True
structure.atom_sites['Ba'].adp_iso.free = True
structure.atom_sites['Co'].adp_iso.free = True
structure.atom_sites['O'].adp_iso.free = True
structure.cell.length_a.free = True

structure.atom_sites['La'].adp_iso.free = True
structure.atom_sites['Ba'].adp_iso.free = True
structure.atom_sites['Co'].adp_iso.free = True
structure.atom_sites['O'].adp_iso.free = True

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experiment.instrument.calib_twotheta_offset.free = True

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

experiment.background['1'].y.free = True
experiment.background['2'].y.free = True
experiment.background['3'].y.free = True
experiment.background['4'].y.free = True
experiment.background['5'].y.free = True

experiment.linked_phases['lbco'].scale.free = True
experiment.instrument.calib_twotheta_offset.free = True

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

experiment.background['1'].y.free = True
experiment.background['2'].y.free = True
experiment.background['3'].y.free = True
experiment.background['4'].y.free = True
experiment.background['5'].y.free = True

experiment.linked_phases['lbco'].scale.free = True

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

Standard fitting

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

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

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.29	165.02
2	28	1.52	33.58	79.7% ↓
3	45	2.28	10.82	67.8% ↓
4	63	3.10	6.49	40.0% ↓
5	81	3.95	3.35	48.4% ↓
6	98	4.72	2.24	33.1% ↓
7	116	5.77	1.91	14.7% ↓
8	133	6.55	1.50	21.3% ↓
9	150	7.32	1.45	3.6% ↓
10	167	8.09	1.34	7.8% ↓
11	185	8.92	1.29	3.4% ↓
12	276	13.17	1.29

🏆 Best goodness-of-fit (reduced χ²) is 1.29 at iteration 275

✅ Fitting complete.

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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)	13.17
4	🔁 Iterations	273
5	📏 Goodness-of-fit (reduced χ²)	1.29
6	📏 R-factor (Rf, %)	5.63
7	📏 R-factor squared (Rf², %)	5.27
8	📏 Weighted R-factor (wR, %)	4.41

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	lbco	cell		length_a	Å	3.8800	3.8909	0.0000	0.28 % ↑
2	lbco	atom_site	La	adp_iso	Å²	0.5000	0.5052	3231.7297	1.04 % ↑
3	lbco	atom_site	Ba	adp_iso	Å²	0.5000	0.5049	5252.5594	0.97 % ↑
4	lbco	atom_site	Co	adp_iso	Å²	0.5000	0.2370	0.0611	52.60 % ↓
5	lbco	atom_site	O	adp_iso	Å²	0.5000	1.3935	0.0167	178.71 % ↑
6	hrpt	linked_phases	lbco	scale		10.0000	9.1351	0.0642	8.65 % ↓
7	hrpt	peak		broad_gauss_u	deg²	0.1000	0.0816	0.0031	18.43 % ↓
8	hrpt	peak		broad_gauss_v	deg²	-0.1000	-0.1159	0.0067	15.93 % ↑
9	hrpt	peak		broad_gauss_w	deg²	0.1000	0.1204	0.0033	20.45 % ↑
10	hrpt	peak		broad_lorentz_y	deg	0.1000	0.0844	0.0021	15.55 % ↓
11	hrpt	instrument		twotheta_offset	deg	0.6000	0.6226	0.0010	3.76 % ↑
12	hrpt	background	1	y		170.0000	168.5585	1.3671	0.85 % ↓
13	hrpt	background	2	y		170.0000	164.3357	0.9992	3.33 % ↓
14	hrpt	background	3	y		170.0000	166.8881	0.7388	1.83 % ↓
15	hrpt	background	4	y		170.0000	175.4006	0.6578	3.18 % ↑
16	hrpt	background	5	y		170.0000	174.2813	0.9105	2.52 % ↑

  • 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

⚠️ Red s.u.: exceeds the refined value (consider adding constraints)

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

Loading plot…

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

Loading plot…

With Constraints¶

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# As can be seen from the parameter-correlation plot, the isotropic
# displacement parameters of La and Ba are highly correlated. Because
# La and Ba share the same mixed-occupancy site, their contributions to
# the neutron diffraction pattern are difficult to separate, especially
# since their coherent scattering lengths are not very different.
# Therefore, it is necessary to constrain them to be equal. First we
# define aliases and then use them to create a constraint.
project.analysis.aliases.create(
    label='biso_La',
    param=project.structures['lbco'].atom_sites['La'].adp_iso,
)
project.analysis.aliases.create(
    label='biso_Ba',
    param=project.structures['lbco'].atom_sites['Ba'].adp_iso,
)
project.analysis.constraints.create(expression='biso_Ba = biso_La')
# As can be seen from the parameter-correlation plot, the isotropic
# displacement parameters of La and Ba are highly correlated. Because
# La and Ba share the same mixed-occupancy site, their contributions to
# the neutron diffraction pattern are difficult to separate, especially
# since their coherent scattering lengths are not very different.
# Therefore, it is necessary to constrain them to be equal. First we
# define aliases and then use them to create a constraint.
project.analysis.aliases.create(
    label='biso_La',
    param=project.structures['lbco'].atom_sites['La'].adp_iso,
)
project.analysis.aliases.create(
    label='biso_Ba',
    param=project.structures['lbco'].atom_sites['Ba'].adp_iso,
)
project.analysis.constraints.create(expression='biso_Ba = biso_La')

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project.analysis.minimizer.show_supported()
project.analysis.minimizer.type = 'lmfit'
project.analysis.minimizer.show_supported()
project.analysis.minimizer.type = 'lmfit'

Minimizer types

		Type	Description
1		bumps	BUMPS library using the default Levenberg-Marquardt method
2		bumps (amoeba)	BUMPS library with Nelder-Mead simplex method
3		bumps (de)	BUMPS library with differential evolution method
4		bumps (dream)	BUMPS library with DREAM Bayesian sampling
5		bumps (lm)	BUMPS library with Levenberg-Marquardt method
6		dfols	DFO-LS library for derivative-free least-squares optimization
7		emcee	emcee affine-invariant ensemble Bayesian sampling
8		lmfit	LMFIT library using the default Levenberg-Marquardt method
9		lmfit (least_squares)	LMFIT library with SciPy's trust region reflective algorithm
10	*	lmfit (leastsq)	LMFIT library with Levenberg-Marquardt least squares method

Current minimizer changed to

lmfit

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

Standard fitting

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

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

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.05	1.29
2	36	2.00	1.29

🏆 Best goodness-of-fit (reduced χ²) is 1.29 at iteration 35

✅ Fitting complete.

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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
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	2.00
4	🔁 Iterations	33
5	📏 Goodness-of-fit (reduced χ²)	1.29
6	📏 R-factor (Rf, %)	5.63
7	📏 R-factor squared (Rf², %)	5.27
8	📏 Weighted R-factor (wR, %)	4.41

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	lbco	cell		length_a	Å	3.8909	3.8909	0.0000	0.00 % ↑
2	lbco	atom_site	La	adp_iso	Å²	0.5052	0.5051	0.0278	0.03 % ↓
3	lbco	atom_site	Co	adp_iso	Å²	0.2370	0.2370	0.0564	0.00 % ↑
4	lbco	atom_site	O	adp_iso	Å²	1.3935	1.3935	0.0160	0.00 % ↓
5	hrpt	linked_phases	lbco	scale		9.1351	9.1351	0.0538	0.00 % ↓
6	hrpt	peak		broad_gauss_u	deg²	0.0816	0.0816	0.0031	0.00 % ↑
7	hrpt	peak		broad_gauss_v	deg²	-0.1159	-0.1159	0.0066	0.00 % ↑
8	hrpt	peak		broad_gauss_w	deg²	0.1204	0.1204	0.0032	0.00 % ↑
9	hrpt	peak		broad_lorentz_y	deg	0.0844	0.0844	0.0021	0.00 % ↓
10	hrpt	instrument		twotheta_offset	deg	0.6226	0.6226	0.0010	0.00 % ↑
11	hrpt	background	1	y		168.5585	168.5585	1.3669	0.00 % ↓
12	hrpt	background	2	y		164.3357	164.3357	0.9990	0.00 % ↑
13	hrpt	background	3	y		166.8881	166.8881	0.7386	0.00 % ↑
14	hrpt	background	4	y		175.4006	175.4006	0.6488	0.00 % ↑
15	hrpt	background	5	y		174.2813	174.2813	0.8944	0.00 % ↓

  • 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()

Loading plot…

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

Loading plot…

Switch Calculator¶

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experiment.calculator.show_supported()
experiment.calculator.show_supported()

Calculator types

		Type	Description
1		crysfml	CrysFML library for crystallographic calculations
2	*	cryspy	CrysPy library for crystallographic calculations

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experiment.calculator.type = 'crysfml'
experiment.calculator.type = 'crysfml'

Calculator for experiment 'hrpt' changed to

crysfml

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

Standard fitting

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

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

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²
1	1	0.05	1.29
2	117	5.07	1.29
3	122	5.36	1.29

🏆 Best goodness-of-fit (reduced χ²) is 1.29 at iteration 112

✅ Fitting complete.

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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
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	5.36
4	🔁 Iterations	119
5	📏 Goodness-of-fit (reduced χ²)	1.29
6	📏 R-factor (Rf, %)	5.62
7	📏 R-factor squared (Rf², %)	5.27
8	📏 Weighted R-factor (wR, %)	4.44

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	lbco	cell		length_a	Å	3.8909	3.8908	0.0000	0.00 % ↓
2	lbco	atom_site	La	adp_iso	Å²	0.5051	0.5192	0.0255	2.79 % ↑
3	lbco	atom_site	Co	adp_iso	Å²	0.2370	0.2163	0.0550	8.74 % ↓
4	lbco	atom_site	O	adp_iso	Å²	1.3935	1.3955	0.0155	0.14 % ↑
5	hrpt	linked_phases	lbco	scale		9.1351	9.1561	0.0516	0.23 % ↑
6	hrpt	peak		broad_gauss_u	deg²	0.0816	0.0815	0.0029	0.10 % ↓
7	hrpt	peak		broad_gauss_v	deg²	-0.1159	-0.1159	0.0063	0.06 % ↓
8	hrpt	peak		broad_gauss_w	deg²	0.1204	0.1204	0.0030	0.05 % ↓
9	hrpt	peak		broad_lorentz_y	deg	0.0844	0.0846	0.0021	0.13 % ↑
10	hrpt	instrument		twotheta_offset	deg	0.6226	0.6200	0.0007	0.41 % ↓
11	hrpt	background	1	y		168.5585	169.6910	1.3655	0.67 % ↑
12	hrpt	background	2	y		164.3357	164.7858	0.9957	0.27 % ↑
13	hrpt	background	3	y		166.8881	167.0204	0.7365	0.08 % ↑
14	hrpt	background	4	y		175.4006	175.2816	0.6506	0.07 % ↓
15	hrpt	background	5	y		174.2813	175.1232	0.8851	0.48 % ↑

  • 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()

Loading plot…

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

Loading plot…

💾 Save Project¶

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project.save_as(dir_path='projects/ed_2_lbco_hrpt')
project.save_as(dir_path='projects/ed_2_lbco_hrpt')

Saving project 📦 'lbco_hrpt' to '../../../projects/ed_2_lbco_hrpt'

├── 📄 project.cif

├── 📁 structures/

│   └── 📄 lbco.cif

├── 📁 experiments/

│   └── 📄 hrpt.cif

├── 📁 analysis/

│   └── 📄 analysis.cif

└── 📁 reports/

    └── 📄 lbco_hrpt.html