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.

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¶

In [2]:

import easydiffraction as ed

import easydiffraction as ed

Step 1: Define Project¶

In [3]:

project = ed.Project()

project = ed.Project()

Step 2: Define Structure¶

In [4]:

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

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

In [5]:

structure = project.structures['lbco']

structure = project.structures['lbco']

In [6]:

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'

In [7]:

structure.cell.length_a = 3.88

structure.cell.length_a = 3.88

In [8]:

structure.atom_sites.create(
    label='La',
    type_symbol='La',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    wyckoff_letter='a',
    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,
    wyckoff_letter='a',
    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,
    wyckoff_letter='b',
    adp_iso=0.5,
)
structure.atom_sites.create(
    label='O',
    type_symbol='O',
    fract_x=0,
    fract_y=0.5,
    fract_z=0.5,
    wyckoff_letter='c',
    adp_iso=0.5,
)

structure.atom_sites.create( label='La', type_symbol='La', fract_x=0, fract_y=0, fract_z=0, wyckoff_letter='a', 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, wyckoff_letter='a', 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, wyckoff_letter='b', adp_iso=0.5, ) structure.atom_sites.create( label='O', type_symbol='O', fract_x=0, fract_y=0.5, fract_z=0.5, wyckoff_letter='c', adp_iso=0.5, )

Step 3: Define Experiment¶

In [9]:

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 already present at 'data/ed-3.xye'. Keeping existing file.

In [10]:

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.

In [11]:

experiment = project.experiments['hrpt']

experiment = project.experiments['hrpt']

In [12]:

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

In [13]:

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

In [14]:

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)

In [15]:

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)

In [16]:

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

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

Step 4: Perform Analysis (no constraints)¶

In [17]:

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

In [18]:

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

In [19]:

project.analysis.fit()

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	165.02
2	28	33.58	79.7% ↓
3	45	10.82	67.8% ↓
4	63	6.49	40.0% ↓
5	81	3.35	48.4% ↓
6	98	2.24	33.1% ↓
7	116	1.91	14.7% ↓
8	133	1.50	21.3% ↓
9	150	1.45	3.6% ↓
10	167	1.34	7.8% ↓
11	185	1.29	3.4% ↓
12	276	1.29

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

✅ Fitting complete.

In [20]:

project.analysis.display.fit_results()

project.analysis.display.fit_results()

Fit results

✅ Success: True

⏱️ Fitting time: 13.87 seconds

📏 Goodness-of-fit (reduced χ²): 1.29

📏 R-factor (Rf): 5.63%

📏 R-factor squared (Rf²): 5.27%

📏 Weighted R-factor (wR): 4.41%

📈 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	0.5052	3231.7297	Ų	1.04 % ↑
3	lbco	atom_site	Ba	adp_iso	0.5000	0.5049	5252.5594	Ų	0.97 % ↑
4	lbco	atom_site	Co	adp_iso	0.5000	0.2370	0.0611	Ų	52.60 % ↓
5	lbco	atom_site	O	adp_iso	0.5000	1.3935	0.0167	Ų	178.71 % ↑
6	hrpt	linked_phases	lbco	scale	10.0000	9.1351	0.0642		8.65 % ↓
7	hrpt	peak		broad_gauss_u	0.1000	0.0816	0.0031	deg²	18.43 % ↓
8	hrpt	peak		broad_gauss_v	-0.1000	-0.1159	0.0067	deg²	15.93 % ↑
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	15.55 % ↓
11	hrpt	instrument		twotheta_offset	0.6000	0.6226	0.0010	deg	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 % ↑

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

In [21]:

project.plotter.plot_param_correlations()

project.plotter.plot_param_correlations()

In [22]:

project.plotter.plot_meas_vs_calc(expt_name='hrpt', show_residual=True)

project.plotter.plot_meas_vs_calc(expt_name='hrpt', show_residual=True)

Step 5: Perform Analysis (with constraints)¶

In [23]:

# 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 [24]:

project.analysis.show_available_minimizers()
project.analysis.show_current_minimizer()
project.analysis.current_minimizer = 'lmfit'

project.analysis.show_available_minimizers() project.analysis.show_current_minimizer() project.analysis.current_minimizer = 'lmfit'

Supported 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 (lm)	Bumps library with Levenberg-Marquardt method
5	dfols	DFO-LS library for derivative-free least-squares optimization
6	lmfit	LMFIT library using the default Levenberg-Marquardt least squares method
7	lmfit (least_squares)	LMFIT library with SciPy's trust region reflective algorithm
8	lmfit (leastsq)	LMFIT library with Levenberg-Marquardt least squares method

Current minimizer

lmfit (leastsq)

Current minimizer changed to

lmfit

In [25]:

project.analysis.fit()

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	36	1.29

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

✅ Fitting complete.

In [26]:

project.analysis.display.fit_results()

project.analysis.display.fit_results()

Fit results

✅ Success: True

⏱️ Fitting time: 1.66 seconds

📏 Goodness-of-fit (reduced χ²): 1.29

📏 R-factor (Rf): 5.63%

📏 R-factor squared (Rf²): 5.27%

📏 Weighted R-factor (wR): 4.41%

📈 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	0.5052	0.5051	0.0278	Ų	0.03 % ↓
3	lbco	atom_site	Co	adp_iso	0.2370	0.2370	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	9.1351	9.1351	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	0.00 % ↑
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	168.5585	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 % ↓

In [27]:

project.plotter.plot_param_correlations()

project.plotter.plot_param_correlations()

In [28]:

project.plotter.plot_meas_vs_calc(expt_name='hrpt', show_residual=True)

project.plotter.plot_meas_vs_calc(expt_name='hrpt', show_residual=True)