Structure Refinement: LBCO, HRPT¶

This basic example is designed to show how Rietveld refinement can be performed when both the crystal structure and experiment parameters are defined using CIF files.

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

The example is intended for users who are already familiar with the EasyDiffraction library and want to quickly get started with a basic refinement.

It is also useful for those who want to see how constraints can be applied to highly correlated parameters. For a more detailed explanation of the code, please refer to the other tutorials.

🛠️ Import Library¶

In [2]:

import easydiffraction as ed

import easydiffraction as ed

📦 Define Project¶

In [3]:

# Create a minimal project with a short name
project = ed.Project(name='lbco_hrpt')

# Create a minimal project with a short name project = ed.Project(name='lbco_hrpt')

🧩 Define Structure¶

In [4]:

# Download CIF file from repository
structure_path = ed.download_data(id=1, destination='data')

# Download CIF file from repository structure_path = ed.download_data(id=1, destination='data')

Getting data...

Data #1: La0.5Ba0.5CoO3 (crystal structure)

✅ Data #1 downloaded to '../../../data/ed-1.cif'

In [5]:

# Add structure from downloaded CIF
project.structures.add_from_cif_path(structure_path)

# Add structure from downloaded CIF project.structures.add_from_cif_path(structure_path)

In [6]:

# Plot the crystal structure
project.display.structure(struct_name='lbco')

# Plot the crystal structure project.display.structure(struct_name='lbco')

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

Loading plot…

drag = rotate
wheel = zoom
right-drag = pan

🔬 Define Experiment¶

In [7]:

# Download CIF file from repository
expt_path = ed.download_data(id=2, destination='data')

# Download CIF file from repository expt_path = ed.download_data(id=2, destination='data')

Getting data...

Data #2: La0.5Ba0.5CoO3, HRPT (PSI), 300 K

✅ Data #2 downloaded to '../../../data/ed-2.cif'

In [8]:

# Add experiment from downloaded CIF
project.experiments.add_from_cif_path(expt_path)

# Add experiment from downloaded CIF project.experiments.add_from_cif_path(expt_path)

🚀 Perform Analysis¶

Without Constraints¶

In [9]:

# Start refinement. All parameters, which have standard uncertainties
# in the input CIF files, are refined by default.
project.analysis.fit()

# Start refinement. All parameters, which have standard uncertainties # in the input CIF files, are refined by default. 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	0.35	165.50
2	28	1.80	33.67	79.7% ↓
3	45	2.71	10.85	67.8% ↓
4	63	3.68	6.43	40.7% ↓
5	81	4.66	3.33	48.2% ↓
6	98	5.58	2.23	33.2% ↓
7	116	6.84	1.91	14.5% ↓
8	133	7.75	1.50	21.1% ↓
9	150	8.66	1.45	3.6% ↓
10	167	9.59	1.34	7.7% ↓
11	185	10.56	1.29	3.4% ↓
12	276	14.95	1.29

🏆 Best goodness-of-fit (reduced χ²) is 1.29 at iteration 261

✅ Fitting complete.

In [10]:

# Show fit results summary
project.display.fit.results()

# Show fit results summary 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)	14.95
4	🔁 Iterations	273
5	📏 Goodness-of-fit (reduced χ²)	1.29
6	📏 R-factor (Rf, %)	5.62
7	📏 R-factor squared (Rf², %)	5.25
8	📏 Weighted R-factor (wR, %)	4.40

📈 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	Ų	0.5000	0.5051	6011.8674	1.02 % ↑
3	lbco	atom_site	Ba	adp_iso	Ų	0.5000	0.5052	9770.9689	1.04 % ↑
4	lbco	atom_site	Co	adp_iso	Ų	0.5000	0.2371	0.0611	52.59 % ↓
5	lbco	atom_site	O	adp_iso	Ų	0.5000	1.3935	0.0168	178.70 % ↑
6	hrpt	linked_phases	lbco	scale		10.0000	9.1349	0.0643	8.65 % ↓
7	hrpt	peak		broad_gauss_u	deg²	0.1000	0.0816	0.0031	18.44 % ↓
8	hrpt	peak		broad_gauss_v	deg²	-0.1000	-0.1159	0.0067	15.90 % ↑
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	15.57 % ↓
11	hrpt	instrument		twotheta_offset	deg	0.6000	0.6226	0.0010	3.76 % ↑
12	hrpt	background	1	y		170.0000	168.4242	1.3977	0.93 % ↓
13	hrpt	background	2	y		170.0000	164.3715	1.0025	3.31 % ↓
14	hrpt	background	3	y		170.0000	166.8870	0.7391	1.83 % ↓
15	hrpt	background	4	y		170.0000	175.3977	0.6586	3.18 % ↑
16	hrpt	background	5	y		170.0000	174.3027	0.9113	2.53 % ↑

• 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)

In [11]:

# Show parameter correlations
project.display.fit.correlations()

# Show parameter correlations project.display.fit.correlations()

Loading plot…

With Constraints¶

In [12]:

# 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')

In [13]:

# Start refinement. All parameters, which have standard uncertainties
# in the input CIF files, are refined by default.
project.analysis.fit()

# Start refinement. All parameters, which have standard uncertainties # in the input CIF files, are refined by default. 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	20	1.28	1.29

🏆 Best goodness-of-fit (reduced χ²) is 1.29 at iteration 19

✅ Fitting complete.

In [14]:

# Show fit results summary
project.display.fit.results()

# Show fit results summary 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)	1.28
4	🔁 Iterations	17
5	📏 Goodness-of-fit (reduced χ²)	1.29
6	📏 R-factor (Rf, %)	5.62
7	📏 R-factor squared (Rf², %)	5.25
8	📏 Weighted R-factor (wR, %)	4.40

📈 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	Ų	0.5051	0.5051	0.0278	0.00 % ↑
3	lbco	atom_site	Co	adp_iso	Ų	0.2371	0.2371	0.0564	0.00 % ↓
4	lbco	atom_site	O	adp_iso	Ų	1.3935	1.3935	0.0160	0.00 % ↑
5	hrpt	linked_phases	lbco	scale		9.1349	9.1349	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	0.01 % ↑
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		168.4242	168.4242	1.3974	0.00 % ↑
12	hrpt	background	2	y		164.3715	164.3715	1.0023	0.00 % ↑
13	hrpt	background	3	y		166.8870	166.8871	0.7388	0.00 % ↑
14	hrpt	background	4	y		175.3977	175.3978	0.6490	0.00 % ↑
15	hrpt	background	5	y		174.3027	174.3026	0.8956	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

In [15]:

# Show parameter correlations
project.display.fit.correlations()

# Show parameter correlations project.display.fit.correlations()

Loading plot…

In [16]:

# Show defined experiment names
project.experiments.show_names()

# Show defined experiment names project.experiments.show_names()

Defined experiments 🔬

['hrpt']

In [17]:

# Plot measured vs. calculated diffraction patterns
project.display.pattern(expt_name='hrpt')

# Plot measured vs. calculated diffraction patterns project.display.pattern(expt_name='hrpt')

Loading plot…

💾 Save Project¶

In [18]:

project.save_as(dir_path='projects/ed_1_lbco_hrpt')

project.save_as(dir_path='projects/ed_1_lbco_hrpt')

Saving project 📦 'lbco_hrpt' to '../../../projects/ed_1_lbco_hrpt'

├── 📄 project.cif

├── 📁 structures/

│   └── 📄 lbco.cif

├── 📁 experiments/

│   └── 📄 hrpt.cif

├── 📁 analysis/

│   └── 📄 analysis.cif

└── 📁 reports/

    └── 📄 lbco_hrpt.html