Structure Refinement: LBCO+Si, McStas¶

This example demonstrates a Rietveld refinement of La0.5Ba0.5CoO3 crystal structure with a small amount of Si phase using time-of-flight neutron powder diffraction data simulated with McStas.

Import Library¶

In [2]:

from easydiffraction import ExperimentFactory
from easydiffraction import Project
from easydiffraction import SampleModelFactory
from easydiffraction import download_from_repository

from easydiffraction import ExperimentFactory from easydiffraction import Project from easydiffraction import SampleModelFactory from easydiffraction import download_from_repository

Define Sample Models¶

This section shows how to add sample models and modify their parameters.

Create Sample Model 1: LBCO¶

In [3]:

model_1 = SampleModelFactory.create(name='lbco')

model_1 = SampleModelFactory.create(name='lbco')

Set Space Group¶

In [4]:

model_1.space_group.name_h_m = 'P m -3 m'
model_1.space_group.it_coordinate_system_code = '1'

model_1.space_group.name_h_m = 'P m -3 m' model_1.space_group.it_coordinate_system_code = '1'

Set Unit Cell¶

In [5]:

model_1.cell.length_a = 3.8909

model_1.cell.length_a = 3.8909

Set Atom Sites¶

In [6]:

model_1.atom_sites.add_from_args(
    label='La',
    type_symbol='La',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    wyckoff_letter='a',
    b_iso=0.2,
    occupancy=0.5,
)
model_1.atom_sites.add_from_args(
    label='Ba',
    type_symbol='Ba',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    wyckoff_letter='a',
    b_iso=0.2,
    occupancy=0.5,
)
model_1.atom_sites.add_from_args(
    label='Co',
    type_symbol='Co',
    fract_x=0.5,
    fract_y=0.5,
    fract_z=0.5,
    wyckoff_letter='b',
    b_iso=0.2567,
)
model_1.atom_sites.add_from_args(
    label='O',
    type_symbol='O',
    fract_x=0,
    fract_y=0.5,
    fract_z=0.5,
    wyckoff_letter='c',
    b_iso=1.4041,
)

model_1.atom_sites.add_from_args( label='La', type_symbol='La', fract_x=0, fract_y=0, fract_z=0, wyckoff_letter='a', b_iso=0.2, occupancy=0.5, ) model_1.atom_sites.add_from_args( label='Ba', type_symbol='Ba', fract_x=0, fract_y=0, fract_z=0, wyckoff_letter='a', b_iso=0.2, occupancy=0.5, ) model_1.atom_sites.add_from_args( label='Co', type_symbol='Co', fract_x=0.5, fract_y=0.5, fract_z=0.5, wyckoff_letter='b', b_iso=0.2567, ) model_1.atom_sites.add_from_args( label='O', type_symbol='O', fract_x=0, fract_y=0.5, fract_z=0.5, wyckoff_letter='c', b_iso=1.4041, )

Create Sample Model 2: Si¶

In [7]:

model_2 = SampleModelFactory.create(name='si')

model_2 = SampleModelFactory.create(name='si')

Set Space Group¶

In [8]:

model_2.space_group.name_h_m = 'F d -3 m'
model_2.space_group.it_coordinate_system_code = '2'

model_2.space_group.name_h_m = 'F d -3 m' model_2.space_group.it_coordinate_system_code = '2'

Set Unit Cell¶

In [9]:

model_2.cell.length_a = 5.43146

model_2.cell.length_a = 5.43146

Set Atom Sites¶

In [10]:

model_2.atom_sites.add_from_args(
    label='Si',
    type_symbol='Si',
    fract_x=0.0,
    fract_y=0.0,
    fract_z=0.0,
    wyckoff_letter='a',
    b_iso=0.0,
)

model_2.atom_sites.add_from_args( label='Si', type_symbol='Si', fract_x=0.0, fract_y=0.0, fract_z=0.0, wyckoff_letter='a', b_iso=0.0, )

Define Experiment¶

This section shows how to add experiments, configure their parameters, and link the sample models defined in the previous step.

Download Data¶

In [11]:

download_from_repository('mcstas_lbco-si.xye', destination='data')

download_from_repository('mcstas_lbco-si.xye', destination='data')

Downloading...

File 'mcstas_lbco-si.xye' from 'easyscience/diffraction-lib'

Create Experiment¶

In [12]:

experiment = ExperimentFactory.create(
    name='mcstas',
    data_path='data/mcstas_lbco-si.xye',
    sample_form='powder',
    beam_mode='time-of-flight',
    radiation_probe='neutron',
    scattering_type='bragg',
)

experiment = ExperimentFactory.create( name='mcstas', data_path='data/mcstas_lbco-si.xye', sample_form='powder', beam_mode='time-of-flight', radiation_probe='neutron', scattering_type='bragg', )

Data loaded successfully

Experiment 🔬 'mcstas'. Number of data points: 1000

Set Instrument¶

In [13]:

experiment.instrument.setup_twotheta_bank = 94.90931761529106
experiment.instrument.calib_d_to_tof_offset = 0.0
experiment.instrument.calib_d_to_tof_linear = 58724.76869981215
experiment.instrument.calib_d_to_tof_quad = -0.00001

experiment.instrument.setup_twotheta_bank = 94.90931761529106 experiment.instrument.calib_d_to_tof_offset = 0.0 experiment.instrument.calib_d_to_tof_linear = 58724.76869981215 experiment.instrument.calib_d_to_tof_quad = -0.00001

Set Peak Profile¶

In [14]:

# experiment.peak_profile_type = 'pseudo-voigt * ikeda-carpenter'
experiment.peak.broad_gauss_sigma_0 = 45137
experiment.peak.broad_gauss_sigma_1 = -52394
experiment.peak.broad_gauss_sigma_2 = 22998
experiment.peak.broad_mix_beta_0 = 0.0055
experiment.peak.broad_mix_beta_1 = 0.0041
experiment.peak.asym_alpha_0 = 0
experiment.peak.asym_alpha_1 = 0.0097

# experiment.peak_profile_type = 'pseudo-voigt * ikeda-carpenter' experiment.peak.broad_gauss_sigma_0 = 45137 experiment.peak.broad_gauss_sigma_1 = -52394 experiment.peak.broad_gauss_sigma_2 = 22998 experiment.peak.broad_mix_beta_0 = 0.0055 experiment.peak.broad_mix_beta_1 = 0.0041 experiment.peak.asym_alpha_0 = 0 experiment.peak.asym_alpha_1 = 0.0097

Set Background¶

Select the background type.

In [15]:

experiment.background_type = 'line-segment'

experiment.background_type = 'line-segment'

Background type for experiment 'mcstas' changed to

line-segment

Add background points.

In [16]:

experiment.background.add_from_args(x=45000, y=0.2)
experiment.background.add_from_args(x=50000, y=0.2)
experiment.background.add_from_args(x=55000, y=0.2)
experiment.background.add_from_args(x=65000, y=0.2)
experiment.background.add_from_args(x=70000, y=0.2)
experiment.background.add_from_args(x=75000, y=0.2)
experiment.background.add_from_args(x=80000, y=0.2)
experiment.background.add_from_args(x=85000, y=0.2)
experiment.background.add_from_args(x=90000, y=0.2)
experiment.background.add_from_args(x=95000, y=0.2)
experiment.background.add_from_args(x=100000, y=0.2)
experiment.background.add_from_args(x=105000, y=0.2)
experiment.background.add_from_args(x=110000, y=0.2)

experiment.background.add_from_args(x=45000, y=0.2) experiment.background.add_from_args(x=50000, y=0.2) experiment.background.add_from_args(x=55000, y=0.2) experiment.background.add_from_args(x=65000, y=0.2) experiment.background.add_from_args(x=70000, y=0.2) experiment.background.add_from_args(x=75000, y=0.2) experiment.background.add_from_args(x=80000, y=0.2) experiment.background.add_from_args(x=85000, y=0.2) experiment.background.add_from_args(x=90000, y=0.2) experiment.background.add_from_args(x=95000, y=0.2) experiment.background.add_from_args(x=100000, y=0.2) experiment.background.add_from_args(x=105000, y=0.2) experiment.background.add_from_args(x=110000, y=0.2)

Set Linked Phases¶

In [17]:

experiment.linked_phases.add_from_args(id='lbco', scale=4.0)
experiment.linked_phases.add_from_args(id='si', scale=0.2)

experiment.linked_phases.add_from_args(id='lbco', scale=4.0) experiment.linked_phases.add_from_args(id='si', scale=0.2)

Define Project¶

The project object is used to manage sample models, experiments, and analysis.

Create Project¶

In [18]:

project = Project()

project = Project()

Set Plotting Engine¶

In [19]:

project.plotter.engine = 'plotly'

project.plotter.engine = 'plotly'

Add Sample Models¶

In [20]:

project.sample_models.add(model_1)
project.sample_models.add(model_2)

project.sample_models.add(model_1) project.sample_models.add(model_2)

Show Sample Models¶

In [21]:

project.sample_models.show_names()

project.sample_models.show_names()

Defined sample models 🧩

['lbco', 'si']

Add Experiments¶

In [22]:

project.experiments.add(experiment)

project.experiments.add(experiment)

Set Excluded Regions¶

Show measured data as loaded from the file.

In [23]:

project.plot_meas(expt_name='mcstas')

project.plot_meas(expt_name='mcstas')

Add excluded regions.

In [24]:

experiment.excluded_regions.add_from_args(start=0, end=40000)
experiment.excluded_regions.add_from_args(start=108000, end=200000)

experiment.excluded_regions.add_from_args(start=0, end=40000) experiment.excluded_regions.add_from_args(start=108000, end=200000)

Show excluded regions.

In [25]:

experiment.excluded_regions.show()

experiment.excluded_regions.show()

Excluded regions

	start	end
1	0	40000
2	108000	200000

Show measured data after adding excluded regions.

In [26]:

project.plot_meas(expt_name='mcstas')

project.plot_meas(expt_name='mcstas')

Show experiment as CIF.

In [27]:

project.experiments['mcstas'].show_as_cif()

project.experiments['mcstas'].show_as_cif()

Experiment 🔬 'mcstas' as cif

	CIF
1	data_mcstas
2
3	_expt_type.sample_form powder
4	_expt_type.beam_mode time-of-flight
5	_expt_type.radiation_probe neutron
6	_expt_type.scattering_type bragg
7
8	_peak.gauss_sigma_0 45137
9	_peak.gauss_sigma_1 -52394
10	_peak.gauss_sigma_2 22998
11	_peak.lorentz_gamma_0 0.0
12	_peak.lorentz_gamma_1 0.0
13	_peak.lorentz_gamma_2 0.0
14	_peak.mix_beta_0 0.0055
15	_peak.mix_beta_1 0.0041
16	_peak.asym_alpha_0 0
17	_peak.asym_alpha_1 0.0097
18
19	_instr.2theta_bank 94.90931761529106
20	_instr.d_to_tof_offset 0.0
21	_instr.d_to_tof_linear 58724.76869981215
22	_instr.d_to_tof_quad -1e-05
23	_instr.d_to_tof_recip 0.0
24
25	loop_
26	_pd_phase_block.id
27	_pd_phase_block.scale
28	lbco 4.0
29	si 0.2
30
31	loop_
32	_excluded_region.start
33	_excluded_region.end
34	0 40000
35	108000 200000
36
37	loop_
38	_pd_background.line_segment_X
39	_pd_background.line_segment_intensity
40	45000 0.2
41	50000 0.2
42	55000 0.2
43	65000 0.2
44	70000 0.2
45	75000 0.2
46	80000 0.2
47	85000 0.2
48	90000 0.2
49	95000 0.2
50	100000 0.2
51	105000 0.2
52	110000 0.2
53
54	loop_
55	_pd_meas.time_of_flight
56	_pd_meas.intensity_total
57	_pd_meas.intensity_total_su
58	41168.1286 0.21537107469310052 0.024851140698641586
59	41273.8536 0.2608731253560513 0.03329888041963864
60	41379.5785 0.3043368615904641 0.0354708759178845
61	41485.3035 0.473667080481105 0.04206283617542718
62	41591.0285 0.6002651986502522 0.04196647994561225
63	...
64	107563.4097 0.2230171962356684 0.007062820875646159
65	107669.1347 0.2196314236755727 0.007156715455880738
66	107774.8597 0.22676234246085125 0.00713080922521723
67	107880.5847 0.21245260466359916 0.006812625277766074
68	107986.3096 0.21452950815000962 0.006898412477966333

Perform Analysis¶

This section outlines the analysis process, including how to configure calculation and fitting engines.

Set Calculator¶

In [28]:

project.analysis.current_calculator = 'cryspy'

project.analysis.current_calculator = 'cryspy'

Current calculator changed to

cryspy

Set Minimizer¶

In [29]:

project.analysis.current_minimizer = 'lmfit (leastsq)'

project.analysis.current_minimizer = 'lmfit (leastsq)'

Current minimizer changed to

lmfit (leastsq)

Set Fitting Parameters¶

Set sample model parameters to be optimized.

In [30]:

model_1.cell.length_a.free = True
model_1.atom_sites['Co'].b_iso.free = True
model_1.atom_sites['O'].b_iso.free = True

model_2.cell.length_a.free = True

model_1.cell.length_a.free = True model_1.atom_sites['Co'].b_iso.free = True model_1.atom_sites['O'].b_iso.free = True model_2.cell.length_a.free = True

Set experiment parameters to be optimized.

In [31]:

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

experiment.peak.broad_gauss_sigma_0.free = True
experiment.peak.broad_gauss_sigma_1.free = True
experiment.peak.broad_gauss_sigma_2.free = True

experiment.peak.asym_alpha_1.free = True
experiment.peak.broad_mix_beta_0.free = True
experiment.peak.broad_mix_beta_1.free = True

for point in experiment.background:
    point.y.free = True

experiment.linked_phases['lbco'].scale.free = True experiment.linked_phases['si'].scale.free = True experiment.peak.broad_gauss_sigma_0.free = True experiment.peak.broad_gauss_sigma_1.free = True experiment.peak.broad_gauss_sigma_2.free = True experiment.peak.asym_alpha_1.free = True experiment.peak.broad_mix_beta_0.free = True experiment.peak.broad_mix_beta_1.free = True for point in experiment.background: point.y.free = True

Perform Fit¶

In [32]:

project.analysis.fit()

project.analysis.fit()

Using experiment 🔬 'mcstas' for 'single' fitting

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

📈 Goodness-of-fit (reduced χ²) change:

	iteration	χ²	improvement [%]
1	1	301.41
2	29	3.36	98.9% ↓
3	55	3.24	3.7% ↓
4	134	3.24

🏆 Best goodness-of-fit (reduced χ²) is 3.24 at iteration 133

✅ Fitting complete.

Fit results

✅ Success: True

⏱️ Fitting time: 5.04 seconds

📏 Goodness-of-fit (reduced χ²): 3.24

📏 R-factor (Rf): 4.76%

📏 R-factor squared (Rf²): 5.16%

📏 Weighted R-factor (wR): 5.22%

📈 Fitted parameters:

	datablock	category	entry	parameter	start	fitted	uncertainty	units	change
1	lbco	cell	None	length_a	3.8909	3.8904	0.0001	Å	0.01 % ↓
2	lbco	atom_site	Co	b_iso	0.2567	0.3233	0.0853	Ų	25.94 % ↑
3	lbco	atom_site	O	b_iso	1.4041	2.1686	0.0282	Ų	54.45 % ↑
4	si	cell	None	length_a	5.4315	5.4327	0.0006	Å	0.02 % ↑
5	mcstas	peak	None	gauss_sigma_0	45137.0000	45210.8969	2868.4651	µs²	0.16 % ↑
6	mcstas	peak	None	gauss_sigma_1	-52394.0000	-52476.8991	3797.9256	µs/Å	0.16 % ↑
7	mcstas	peak	None	gauss_sigma_2	22998.0000	23014.3392	1193.4306	µs²/Ų	0.07 % ↑
8	mcstas	peak	None	mix_beta_0	0.0055	0.0055	0.0001	deg	0.64 % ↑
9	mcstas	peak	None	mix_beta_1	0.0041	0.0041	0.0003	deg	0.32 % ↓
10	mcstas	peak	None	asym_alpha_1	0.0097	0.0097	0.0002		0.40 % ↓
11	mcstas	linked_phases	lbco	scale	4.0000	4.5577	0.0318		13.94 % ↑
12	mcstas	linked_phases	si	scale	0.2000	0.0314	0.0008		84.28 % ↓
13	mcstas	background	45000	y	0.2000	0.2502	0.0082		25.09 % ↑
14	mcstas	background	50000	y	0.2000	0.2498	0.0080		24.91 % ↑
15	mcstas	background	55000	y	0.2000	0.2674	0.0047		33.68 % ↑
16	mcstas	background	65000	y	0.2000	0.2551	0.0036		27.53 % ↑
17	mcstas	background	70000	y	0.2000	0.2463	0.0035		23.17 % ↑
18	mcstas	background	75000	y	0.2000	0.2417	0.0027		20.86 % ↑
19	mcstas	background	80000	y	0.2000	0.2421	0.0033		21.07 % ↑
20	mcstas	background	85000	y	0.2000	0.2374	0.0021		18.71 % ↑
21	mcstas	background	90000	y	0.2000	0.2381	0.0020		19.07 % ↑
22	mcstas	background	95000	y	0.2000	0.2423	0.0025		21.14 % ↑
23	mcstas	background	100000	y	0.2000	0.2258	0.0022		12.88 % ↑
24	mcstas	background	105000	y	0.2000	0.2228	0.0026		11.40 % ↑
25	mcstas	background	110000	y	0.2000	0.2225	0.0078		11.23 % ↑

Plot Measured vs Calculated¶

In [33]:

project.plot_meas_vs_calc(expt_name='mcstas')

project.plot_meas_vs_calc(expt_name='mcstas')