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 Experiment
from easydiffraction import Project
from easydiffraction import SampleModel
from easydiffraction import download_from_repository

from easydiffraction import Experiment from easydiffraction import Project from easydiffraction import SampleModel from easydiffraction import download_from_repository

⚠️ 'pycrysfml' module not found. This calculation engine will not be available.
✅ 'cryspy' calculation engine is successfully imported.
✅ 'pdffit' calculation engine is successfully imported.

Define Sample Models¶

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

Create Sample Model 1: LBCO¶

In [3]:

model_1 = SampleModel('lbco')

model_1 = SampleModel('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(
    'La',
    'La',
    0,
    0,
    0,
    wyckoff_letter='a',
    b_iso=0.2,
    occupancy=0.5,
)
model_1.atom_sites.add(
    'Ba',
    'Ba',
    0,
    0,
    0,
    wyckoff_letter='a',
    b_iso=0.2,
    occupancy=0.5,
)
model_1.atom_sites.add(
    'Co',
    'Co',
    0.5,
    0.5,
    0.5,
    wyckoff_letter='b',
    b_iso=0.2567,
)
model_1.atom_sites.add(
    'O',
    'O',
    0,
    0.5,
    0.5,
    wyckoff_letter='c',
    b_iso=1.4041,
)

model_1.atom_sites.add( 'La', 'La', 0, 0, 0, wyckoff_letter='a', b_iso=0.2, occupancy=0.5, ) model_1.atom_sites.add( 'Ba', 'Ba', 0, 0, 0, wyckoff_letter='a', b_iso=0.2, occupancy=0.5, ) model_1.atom_sites.add( 'Co', 'Co', 0.5, 0.5, 0.5, wyckoff_letter='b', b_iso=0.2567, ) model_1.atom_sites.add( 'O', 'O', 0, 0.5, 0.5, wyckoff_letter='c', b_iso=1.4041, )

Create Sample Model 2: Si¶

In [7]:

model_2 = SampleModel('si')

model_2 = SampleModel('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(
    'Si',
    'Si',
    0.0,
    0.0,
    0.0,
    wyckoff_letter='a',
    b_iso=0.0,
)

model_2.atom_sites.add( 'Si', 'Si', 0.0, 0.0, 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')

⚠️ Warning
File 'data/mcstas_lbco-si.xye' already exists and will not be overwritten.

Create Experiment¶

In [12]:

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

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

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

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

Set Linked Phases¶

In [17]:

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

experiment.linked_phases.add('lbco', scale=4.0) experiment.linked_phases.add('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'

Current plotter changed to
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(minimum=0, maximum=40000)
experiment.excluded_regions.add(minimum=108000, maximum=200000)

experiment.excluded_regions.add(minimum=0, maximum=40000) experiment.excluded_regions.add(minimum=108000, maximum=200000)

Show excluded regions.

In [25]:

experiment.excluded_regions.show()

experiment.excluded_regions.show()

Excluded regions

minimum	maximum
0	40000
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

data_mcstas
_expt_type.beam_mode time-of-flight
_expt_type.radiation_probe neutron
_expt_type.sample_form powder
_expt_type.scattering_type bragg
_instr.d_to_tof_linear 58724.76869981215
_instr.d_to_tof_offset 0.0
_instr.d_to_tof_quad -1e-05
_instr.d_to_tof_recip 0.0
_instr.2theta_bank 94.90931761529106
_peak.asym_alpha_0 0
_peak.asym_alpha_1 0.0097
_peak.gauss_sigma_0 45137
_peak.gauss_sigma_1 -52394
_peak.gauss_sigma_2 22998
_peak.lorentz_gamma_0 0.0
_peak.lorentz_gamma_1 0.0
_peak.lorentz_gamma_2 0.0
_peak.mix_beta_0 0.0055
_peak.mix_beta_1 0.0041
loop_
_pd_phase_block.id
_pd_phase_block.scale
lbco 4.0
si 0.2
loop_
_pd_background.line_segment_X
_pd_background.line_segment_intensity
45000 0.2
50000 0.2
55000 0.2
65000 0.2
70000 0.2
75000 0.2
80000 0.2
85000 0.2
90000 0.2
95000 0.2
100000 0.2
105000 0.2
110000 0.2
loop_
_excluded_region.maximum
_excluded_region.minimum
40000 0
200000 108000
loop_
_pd_meas.2theta_scan
_pd_meas.intensity_total
_pd_meas.intensity_total_su
41168.1286 0.21537107469310052 0.024851140698641586
41273.8536 0.2608731253560513 0.03329888041963864
41379.5785 0.3043368615904641 0.0354708759178845
41485.3035 0.473667080481105 0.04206283617542718
41591.0285 0.6002651986502522 0.04196647994561225
...
107563.4097 0.2230171962356684 0.007062820875646159
107669.1347 0.2196314236755727 0.007156715455880738
107774.8597 0.22676234246085125 0.00713080922521723
107880.5847 0.21245260466359916 0.006812625277766074
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	301.41
29	3.36	98.9% ↓
55	3.24	3.7% ↓
134	3.24

🏆 Best goodness-of-fit (reduced χ²) is 3.24 at iteration 133
✅ Fitting complete.

Fit results
✅ Success: True
⏱️ Fitting time: 2.14 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	atom_sites	Co	b_iso	0.2567	0.3233	0.0853	Ų	25.94 % ↑
2	lbco	atom_sites	O	b_iso	1.4041	2.1686	0.0282	Ų	54.45 % ↑
3	lbco	cell		length_a	3.8909	3.8904	0.0001	Å	0.01 % ↓
4	si	cell		length_a	5.4315	5.4327	0.0006	Å	0.02 % ↑
5	mcstas	background	45000	y	0.2000	0.2502	0.0082		25.09 % ↑
6	mcstas	background	50000	y	0.2000	0.2498	0.0080		24.91 % ↑
7	mcstas	background	55000	y	0.2000	0.2674	0.0047		33.68 % ↑
8	mcstas	background	65000	y	0.2000	0.2551	0.0036		27.53 % ↑
9	mcstas	background	70000	y	0.2000	0.2463	0.0035		23.17 % ↑
10	mcstas	background	75000	y	0.2000	0.2417	0.0027		20.86 % ↑
11	mcstas	background	80000	y	0.2000	0.2421	0.0033		21.07 % ↑
12	mcstas	background	85000	y	0.2000	0.2374	0.0021		18.71 % ↑
13	mcstas	background	90000	y	0.2000	0.2381	0.0020		19.07 % ↑
14	mcstas	background	95000	y	0.2000	0.2423	0.0025		21.14 % ↑
15	mcstas	background	100000	y	0.2000	0.2258	0.0022		12.88 % ↑
16	mcstas	background	105000	y	0.2000	0.2228	0.0026		11.40 % ↑
17	mcstas	background	110000	y	0.2000	0.2225	0.0078		11.23 % ↑
18	mcstas	linked_phases	lbco	scale	4.0000	4.5577	0.0318		13.94 % ↑
19	mcstas	linked_phases	si	scale	0.2000	0.0314	0.0008		84.28 % ↓
20	mcstas	peak		asym_alpha_1	0.0097	0.0097	0.0002		0.40 % ↓
21	mcstas	peak		gauss_sigma_0	45137.0000	45210.8969	2868.4651	µs²	0.16 % ↑
22	mcstas	peak		gauss_sigma_1	-52394.0000	-52476.8991	3797.9256	µs/Å	0.16 % ↑
23	mcstas	peak		gauss_sigma_2	22998.0000	23014.3392	1193.4306	µs²/Ų	0.07 % ↑
24	mcstas	peak		mix_beta_0	0.0055	0.0055	0.0001	deg	0.64 % ↑
25	mcstas	peak		mix_beta_1	0.0041	0.0041	0.0003	deg	0.32 % ↓

Plot Measured vs Calculated¶

In [33]:

project.plot_meas_vs_calc(expt_name='mcstas')

project.plot_meas_vs_calc(expt_name='mcstas')