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_data

from easydiffraction import ExperimentFactory from easydiffraction import Project from easydiffraction import SampleModelFactory from easydiffraction import download_data

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(
    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(
    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(
    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(
    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( 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( 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( 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( 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(
    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( 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]:

data_path = download_data(id=8, destination='data')

data_path = download_data(id=8, destination='data')

Getting data...

Data #8: La0.5Ba0.5CoO3 + Si, McStas simulation

✅ Data #8 downloaded to 'data/ed-8.xye'

Create Experiment¶

In [12]:

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

experiment = ExperimentFactory.create( name='mcstas', data_path=data_path, 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(id='1', x=45000, y=0.2)
experiment.background.add(id='2', x=50000, y=0.2)
experiment.background.add(id='3', x=55000, y=0.2)
experiment.background.add(id='4', x=65000, y=0.2)
experiment.background.add(id='5', x=70000, y=0.2)
experiment.background.add(id='6', x=75000, y=0.2)
experiment.background.add(id='7', x=80000, y=0.2)
experiment.background.add(id='8', x=85000, y=0.2)
experiment.background.add(id='9', x=90000, y=0.2)
experiment.background.add(id='10', x=95000, y=0.2)
experiment.background.add(id='11', x=100000, y=0.2)
experiment.background.add(id='12', x=105000, y=0.2)
experiment.background.add(id='13', x=110000, y=0.2)

experiment.background.add(id='1', x=45000, y=0.2) experiment.background.add(id='2', x=50000, y=0.2) experiment.background.add(id='3', x=55000, y=0.2) experiment.background.add(id='4', x=65000, y=0.2) experiment.background.add(id='5', x=70000, y=0.2) experiment.background.add(id='6', x=75000, y=0.2) experiment.background.add(id='7', x=80000, y=0.2) experiment.background.add(id='8', x=85000, y=0.2) experiment.background.add(id='9', x=90000, y=0.2) experiment.background.add(id='10', x=95000, y=0.2) experiment.background.add(id='11', x=100000, y=0.2) experiment.background.add(id='12', x=105000, y=0.2) experiment.background.add(id='13', x=110000, y=0.2)

Set Linked Phases¶

In [17]:

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

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

Add Sample Models¶

In [19]:

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

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

Show Sample Models¶

In [20]:

project.sample_models.show_names()

project.sample_models.show_names()

Defined sample models 🧩

['lbco', 'si']

Add Experiments¶

In [21]:

project.experiments.add(experiment=experiment)

project.experiments.add(experiment=experiment)

Set Excluded Regions¶

Show measured data as loaded from the file.

In [22]:

project.plot_meas(expt_name='mcstas')

project.plot_meas(expt_name='mcstas')

Add excluded regions.

In [23]:

experiment.excluded_regions.add(id='1', start=0, end=40000)
experiment.excluded_regions.add(id='2', start=108000, end=200000)

experiment.excluded_regions.add(id='1', start=0, end=40000) experiment.excluded_regions.add(id='2', start=108000, end=200000)

Show excluded regions.

In [24]:

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 [25]:

project.plot_meas(expt_name='mcstas')

project.plot_meas(expt_name='mcstas')

Show experiment as CIF.

In [26]:

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 45137.0000
9	_peak.gauss_sigma_1 -52394.0000
10	_peak.gauss_sigma_2 22998.0000
11	_peak.lorentz_gamma_0 0.0000
12	_peak.lorentz_gamma_1 0.0000
13	_peak.lorentz_gamma_2 0.0000
14	_peak.mix_beta_0 0.0055
15	_peak.mix_beta_1 0.0041
16	_peak.asym_alpha_0 0.0000
17	_peak.asym_alpha_1 0.0097
18
19	_instr.2theta_bank 94.9093
20	_instr.d_to_tof_offset 0.0000
21	_instr.d_to_tof_linear 58724.7687
22	_instr.d_to_tof_quad -0.0000
23	_instr.d_to_tof_recip 0.0000
24
25	loop_
26	_pd_phase_block.id
27	_pd_phase_block.scale
28	lbco 4.0000
29	si 0.2000
30
31	loop_
32	_excluded_region.id
33	_excluded_region.start
34	_excluded_region.end
35	1 0.0000 40000.0000
36	2 108000.0000 200000.0000
37
38	loop_
39	_pd_meas.time_of_flight
40	_pd_data.point_id
41	_pd_proc.d_spacing
42	_pd_meas.intensity_total
43	_pd_meas.intensity_total_su
44	_pd_calc.intensity_total
45	_pd_calc.intensity_bkg
46	_pd_data.refinement_status
47	41168.1286 1 0.7010 0.2154 0.0249 0.2544 0.2000 incl
48	41273.8536 2 0.7028 0.2609 0.0333 0.3380 0.2000 incl
49	41379.5785 3 0.7046 0.3043 0.0355 0.4664 0.2000 incl
50	41485.3035 4 0.7064 0.4737 0.0421 0.5860 0.2000 incl
51	41591.0285 5 0.7082 0.6003 0.0420 0.6236 0.2000 incl
52	41696.7534 6 0.7100 0.6017 0.0397 0.5577 0.2000 incl
53	41802.4784 7 0.7118 0.5012 0.0381 0.4329 0.2000 incl
54	41908.2034 8 0.7136 0.3770 0.0346 0.3153 0.2000 incl
55	42013.9284 9 0.7154 0.2753 0.0319 0.2457 0.2000 incl
56	42119.6533 10 0.7172 0.2618 0.0283 0.2273 0.2000 incl
57	...
58	145835.8489 991 2.4834 0.2241 0.0113 0.2000 0.2000 excl
59	145941.5739 992 2.4852 0.2138 0.0107 0.2000 0.2000 excl
60	146047.2988 993 2.4870 0.2093 0.0109 0.2000 0.2000 excl
61	146153.0238 994 2.4888 0.2184 0.0109 0.2000 0.2000 excl
62	146258.7488 995 2.4906 0.2161 0.0110 0.2000 0.2000 excl
63	146364.4737 996 2.4924 0.2057 0.0111 0.2000 0.2000 excl
64	146470.1987 997 2.4942 0.2262 0.0114 0.2000 0.2000 excl
65	146575.9237 998 2.4960 0.2211 0.0115 0.2000 0.2000 excl
66	146681.6486 999 2.4978 0.2086 0.0110 0.2000 0.2000 excl
67	146787.3736 1000 2.4996 0.2088 0.0109 0.2000 0.2000 excl
68
69	loop_
70	_pd_background.id
71	_pd_background.line_segment_X
72	_pd_background.line_segment_intensity
73	1 45000.0000 0.2000
74	2 50000.0000 0.2000
75	3 55000.0000 0.2000
76	4 65000.0000 0.2000
77	5 70000.0000 0.2000
78	6 75000.0000 0.2000
79	7 80000.0000 0.2000
80	8 85000.0000 0.2000
81	9 90000.0000 0.2000
82	10 95000.0000 0.2000
83	11 100000.0000 0.2000
84	12 105000.0000 0.2000
85	13 110000.0000 0.2000

Perform Analysis¶

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

Set Calculator¶

In [27]:

project.analysis.current_calculator = 'cryspy'

project.analysis.current_calculator = 'cryspy'

Current calculator changed to

cryspy

Set Minimizer¶

In [28]:

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 [29]:

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 [30]:

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 [31]:

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	301.38
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: 5.34 seconds

📏 Goodness-of-fit (reduced χ²): 3.24

📏 R-factor (Rf): 4.74%

📏 R-factor squared (Rf²): 5.12%

📏 Weighted R-factor (wR): 5.18%

📈 Fitted parameters:

	datablock	category	entry	parameter	start	fitted	uncertainty	units	change
1	lbco	cell		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		length_a	5.4315	5.4327	0.0006	Å	0.02 % ↑
5	mcstas	linked_phases	lbco	scale	4.0000	4.5577	0.0318		13.94 % ↑
6	mcstas	linked_phases	si	scale	0.2000	0.0314	0.0008		84.28 % ↓
7	mcstas	peak		gauss_sigma_0	45137.0000	45210.8969	2868.4651	µs²	0.16 % ↑
8	mcstas	peak		gauss_sigma_1	-52394.0000	-52476.8991	3797.9256	µs/Å	0.16 % ↑
9	mcstas	peak		gauss_sigma_2	22998.0000	23014.3392	1193.4306	µs²/Ų	0.07 % ↑
10	mcstas	peak		mix_beta_0	0.0055	0.0055	0.0001	deg	0.64 % ↑
11	mcstas	peak		mix_beta_1	0.0041	0.0041	0.0003	deg	0.32 % ↓
12	mcstas	peak		asym_alpha_1	0.0097	0.0097	0.0002		0.40 % ↓
13	mcstas	background	1	y	0.2000	0.2502	0.0082		25.09 % ↑
14	mcstas	background	2	y	0.2000	0.2498	0.0080		24.91 % ↑
15	mcstas	background	3	y	0.2000	0.2674	0.0047		33.68 % ↑
16	mcstas	background	4	y	0.2000	0.2551	0.0036		27.53 % ↑
17	mcstas	background	5	y	0.2000	0.2463	0.0035		23.17 % ↑
18	mcstas	background	6	y	0.2000	0.2417	0.0027		20.86 % ↑
19	mcstas	background	7	y	0.2000	0.2421	0.0033		21.07 % ↑
20	mcstas	background	8	y	0.2000	0.2374	0.0021		18.71 % ↑
21	mcstas	background	9	y	0.2000	0.2381	0.0020		19.07 % ↑
22	mcstas	background	10	y	0.2000	0.2423	0.0025		21.14 % ↑
23	mcstas	background	11	y	0.2000	0.2258	0.0022		12.88 % ↑
24	mcstas	background	12	y	0.2000	0.2228	0.0026		11.40 % ↑
25	mcstas	background	13	y	0.2000	0.2225	0.0078		11.23 % ↑

Plot Measured vs Calculated¶

In [32]:

project.plot_meas_vs_calc(expt_name='mcstas')

project.plot_meas_vs_calc(expt_name='mcstas')

In [ ]: