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

Step 1: Define Project¶

In [3]:

# Create minimal project without name and description
project = ed.Project()

# Create minimal project without name and description project = ed.Project()

Step 2: Define Crystal 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)

Step 3: Define Experiment¶

In [6]:

# 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 [7]:

# 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)

Step 4: Perform Analysis (no constraints)¶

In [8]:

# 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 (reduced χ²) change:

	iteration	χ²	improvement [%]
1	1	165.50
2	28	33.67	79.7% ↓
3	45	10.85	67.8% ↓
4	63	6.43	40.7% ↓
5	81	3.33	48.2% ↓
6	98	2.23	33.2% ↓
7	116	1.91	14.5% ↓
8	133	1.50	21.1% ↓
9	150	1.45	3.6% ↓
10	167	1.34	7.7% ↓
11	185	1.29	3.4% ↓
12	276	1.29

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

✅ Fitting complete.

In [9]:

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

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

Fit results

✅ Success: True

⏱️ Fitting time: 16.34 seconds

📏 Goodness-of-fit (reduced χ²): 1.29

📏 R-factor (Rf): 5.62%

📏 R-factor squared (Rf²): 5.25%

📏 Weighted R-factor (wR): 4.40%

📈 Fitted parameters:

	datablock	category	entry	parameter	start	fitted	uncertainty	units	change
1	lbco	cell		length_a	3.8800	3.8909	0.0000	Å	0.28 % ↑
2	lbco	atom_site	La	adp_iso	0.5000	0.5051	6011.8674	Ų	1.02 % ↑
3	lbco	atom_site	Ba	adp_iso	0.5000	0.5052	9770.9689	Ų	1.04 % ↑
4	lbco	atom_site	Co	adp_iso	0.5000	0.2371	0.0611	Ų	52.59 % ↓
5	lbco	atom_site	O	adp_iso	0.5000	1.3935	0.0168	Ų	178.70 % ↑
6	hrpt	linked_phases	lbco	scale	10.0000	9.1349	0.0643		8.65 % ↓
7	hrpt	peak		broad_gauss_u	0.1000	0.0816	0.0031	deg²	18.44 % ↓
8	hrpt	peak		broad_gauss_v	-0.1000	-0.1159	0.0067	deg²	15.90 % ↑
9	hrpt	peak		broad_gauss_w	0.1000	0.1204	0.0033	deg²	20.45 % ↑
10	hrpt	peak		broad_lorentz_y	0.1000	0.0844	0.0021	deg	15.57 % ↓
11	hrpt	instrument		twotheta_offset	0.6000	0.6226	0.0010	deg	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 % ↑

⚠️ Red uncertainty: exceeds the fitted value (consider adding constraints)                                                        

In [10]:

# Show parameter correlations
project.plotter.plot_param_correlations()

# Show parameter correlations project.plotter.plot_param_correlations()

Step 5: Perform Analysis (with constraints)¶

In [11]:

# 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 [12]:

# 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 (reduced χ²) change:

	iteration	χ²	improvement [%]
1	1	1.29
2	20	1.29

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

✅ Fitting complete.

In [13]:

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

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

Fit results

✅ Success: True

⏱️ Fitting time: 1.12 seconds

📏 Goodness-of-fit (reduced χ²): 1.29

📏 R-factor (Rf): 5.62%

📏 R-factor squared (Rf²): 5.25%

📏 Weighted R-factor (wR): 4.40%

📈 Fitted parameters:

	datablock	category	entry	parameter	start	fitted	uncertainty	units	change
1	lbco	cell		length_a	3.8909	3.8909	0.0000	Å	0.00 % ↑
2	lbco	atom_site	La	adp_iso	0.5051	0.5051	0.0278	Ų	0.00 % ↑
3	lbco	atom_site	Co	adp_iso	0.2371	0.2371	0.0564	Ų	0.00 % ↓
4	lbco	atom_site	O	adp_iso	1.3935	1.3935	0.0160	Ų	0.00 % ↑
5	hrpt	linked_phases	lbco	scale	9.1349	9.1349	0.0538		0.00 % ↓
6	hrpt	peak		broad_gauss_u	0.0816	0.0816	0.0031	deg²	0.00 % ↑
7	hrpt	peak		broad_gauss_v	-0.1159	-0.1159	0.0066	deg²	0.01 % ↑
8	hrpt	peak		broad_gauss_w	0.1204	0.1204	0.0032	deg²	0.00 % ↑
9	hrpt	peak		broad_lorentz_y	0.0844	0.0844	0.0021	deg	0.00 % ↓
10	hrpt	instrument		twotheta_offset	0.6226	0.6226	0.0010	deg	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 % ↓

In [14]:

# Show parameter correlations
project.plotter.plot_param_correlations()

# Show parameter correlations project.plotter.plot_param_correlations()

In [15]:

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

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

Defined experiments 🔬

['hrpt']

In [16]:

# Plot measured vs. calculated diffraction patterns
project.plotter.plot_meas_vs_calc(expt_name='hrpt', show_residual=True)

# Plot measured vs. calculated diffraction patterns project.plotter.plot_meas_vs_calc(expt_name='hrpt', show_residual=True)