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 StructureFactory
from easydiffraction import download_data

from easydiffraction import ExperimentFactory from easydiffraction import Project from easydiffraction import StructureFactory from easydiffraction import download_data

Define Structures¶

This section shows how to add structures and modify their parameters.

Create Structure 1: LBCO¶

In [3]:

structure_1 = StructureFactory.from_scratch(name='lbco')

structure_1 = StructureFactory.from_scratch(name='lbco')

Set Space Group¶

In [4]:

structure_1.space_group.name_h_m = 'P m -3 m'
structure_1.space_group.it_coordinate_system_code = '1'

structure_1.space_group.name_h_m = 'P m -3 m' structure_1.space_group.it_coordinate_system_code = '1'

Set Unit Cell¶

In [5]:

structure_1.cell.length_a = 3.8909

structure_1.cell.length_a = 3.8909

Set Atom Sites¶

In [6]:

structure_1.atom_sites.create(
    label='La',
    type_symbol='La',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    wyckoff_letter='a',
    b_iso=0.2,
    occupancy=0.5,
)
structure_1.atom_sites.create(
    label='Ba',
    type_symbol='Ba',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    wyckoff_letter='a',
    b_iso=0.2,
    occupancy=0.5,
)
structure_1.atom_sites.create(
    label='Co',
    type_symbol='Co',
    fract_x=0.5,
    fract_y=0.5,
    fract_z=0.5,
    wyckoff_letter='b',
    b_iso=0.2567,
)
structure_1.atom_sites.create(
    label='O',
    type_symbol='O',
    fract_x=0,
    fract_y=0.5,
    fract_z=0.5,
    wyckoff_letter='c',
    b_iso=1.4041,
)

structure_1.atom_sites.create( label='La', type_symbol='La', fract_x=0, fract_y=0, fract_z=0, wyckoff_letter='a', b_iso=0.2, occupancy=0.5, ) structure_1.atom_sites.create( label='Ba', type_symbol='Ba', fract_x=0, fract_y=0, fract_z=0, wyckoff_letter='a', b_iso=0.2, occupancy=0.5, ) structure_1.atom_sites.create( label='Co', type_symbol='Co', fract_x=0.5, fract_y=0.5, fract_z=0.5, wyckoff_letter='b', b_iso=0.2567, ) structure_1.atom_sites.create( label='O', type_symbol='O', fract_x=0, fract_y=0.5, fract_z=0.5, wyckoff_letter='c', b_iso=1.4041, )

Create Structure 2: Si¶

In [7]:

structure_2 = StructureFactory.from_scratch(name='si')

structure_2 = StructureFactory.from_scratch(name='si')

Set Space Group¶

In [8]:

structure_2.space_group.name_h_m = 'F d -3 m'
structure_2.space_group.it_coordinate_system_code = '2'

structure_2.space_group.name_h_m = 'F d -3 m' structure_2.space_group.it_coordinate_system_code = '2'

Set Unit Cell¶

In [9]:

structure_2.cell.length_a = 5.43146

structure_2.cell.length_a = 5.43146

Set Atom Sites¶

In [10]:

structure_2.atom_sites.create(
    label='Si',
    type_symbol='Si',
    fract_x=0.0,
    fract_y=0.0,
    fract_z=0.0,
    wyckoff_letter='a',
    b_iso=0.0,
)

structure_2.atom_sites.create( 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 structures 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.from_data_path(
    name='mcstas',
    data_path=data_path,
    sample_form='powder',
    beam_mode='time-of-flight',
    radiation_probe='neutron',
    scattering_type='bragg',
)

experiment = ExperimentFactory.from_data_path( 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' already set to

line-segment

Add background points.

In [16]:

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

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

Set Linked Phases¶

In [17]:

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

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

Define Project¶

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

Create Project¶

In [18]:

project = Project()

project = Project()

Add Structures¶

In [19]:

project.structures.add(structure_1)
project.structures.add(structure_2)

project.structures.add(structure_1) project.structures.add(structure_2)

Show Structures¶

In [20]:

project.structures.show_names()

project.structures.show_names()

Defined structures 🧩

['lbco', 'si']

Add Experiments¶

In [21]:

project.experiments.add(experiment)

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

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

Show excluded regions.

In [24]:

experiment.excluded_regions.show()

experiment.excluded_regions.show()

Excluded regions

	start	end
1	0.00000	40000
2	108000.00000	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	_diffrn.ambient_temperature None
9	_diffrn.ambient_pressure None
10	_diffrn.ambient_magnetic_field None
11	_diffrn.ambient_electric_field None
12
13	_peak.asym_alpha_0 0.00000000
14	_peak.asym_alpha_1 0.00970000
15	_peak.gauss_sigma_0 45137.00000000
16	_peak.gauss_sigma_1 -52394.00000000
17	_peak.gauss_sigma_2 22998.00000000
18	_peak.lorentz_gamma_0 0.00000000
19	_peak.lorentz_gamma_1 0.00000000
20	_peak.lorentz_gamma_2 0.00000000
21	_peak.mix_beta_0 0.00550000
22	_peak.mix_beta_1 0.00410000
23
24	_instr.2theta_bank 94.90931762
25	_instr.d_to_tof_offset 0.00000000
26	_instr.d_to_tof_linear 58724.76869981
27	_instr.d_to_tof_quad -0.00001000
28	_instr.d_to_tof_recip 0.00000000
29
30	loop_
31	_pd_phase_block.id
32	_pd_phase_block.scale
33	lbco 4.00000000
34	si 0.20000000
35
36	loop_
37	_excluded_region.id
38	_excluded_region.start
39	_excluded_region.end
40	1 0.00000000 40000.00000000
41	2 108000.00000000 200000.00000000
42
43	loop_
44	_pd_meas.time_of_flight
45	_pd_data.point_id
46	_pd_proc.d_spacing
47	_pd_meas.intensity_total
48	_pd_meas.intensity_total_su
49	_pd_calc.intensity_total
50	_pd_calc.intensity_bkg
51	_pd_data.refinement_status
52	41168.12860000 1 0.70103524 0.21537107 0.02485114 0.25454340 0.20000000 incl
53	41273.85360000 2 0.70283531 0.26087313 0.03329888 0.33827395 0.20000000 incl
54	41379.57850000 3 0.70463575 0.30433686 0.03547088 0.46671493 0.20000000 incl
55	41485.30350000 4 0.70643619 0.47366708 0.04206284 0.58607008 0.20000000 incl
56	41591.02850000 5 0.70823626 0.60026520 0.04196648 0.62332828 0.20000000 incl
57	41696.75340000 6 0.71003669 0.60174483 0.03965186 0.55740167 0.20000000 incl
58	41802.47840000 7 0.71183713 0.50118258 0.03810349 0.43268835 0.20000000 incl
59	41908.20340000 8 0.71363756 0.37702264 0.03458906 0.31517712 0.20000000 incl
60	42013.92840000 9 0.71543800 0.27533183 0.03190951 0.24565766 0.20000000 incl
61	42119.65330000 10 0.71723844 0.26182311 0.02830043 0.22739950 0.20000000 incl
62	...
63	145835.84890000 991 2.48337892 0.22406860 0.01131008 0.20000000 0.20000000 excl
64	145941.57390000 992 2.48517936 0.21377274 0.01070627 0.20000000 0.20000000 excl
65	146047.29880000 993 2.48697943 0.20927726 0.01085520 0.20000000 0.20000000 excl
66	146153.02380000 994 2.48877986 0.21839779 0.01091455 0.20000000 0.20000000 excl
67	146258.74880000 995 2.49058030 0.21613293 0.01104795 0.20000000 0.20000000 excl
68	146364.47370000 996 2.49238037 0.20571926 0.01109275 0.20000000 0.20000000 excl
69	146470.19870000 997 2.49418081 0.22615941 0.01142946 0.20000000 0.20000000 excl
70	146575.92370000 998 2.49598124 0.22113311 0.01151463 0.20000000 0.20000000 excl
71	146681.64860000 999 2.49778168 0.20862390 0.01097638 0.20000000 0.20000000 excl
72	146787.37360000 1000 2.49958175 0.20883912 0.01092893 0.20000000 0.20000000 excl
73
74	loop_
75	_pd_background.id
76	_pd_background.line_segment_X
77	_pd_background.line_segment_intensity
78	1 45000.00000000 0.20000000
79	2 50000.00000000 0.20000000
80	3 55000.00000000 0.20000000
81	4 65000.00000000 0.20000000
82	5 70000.00000000 0.20000000
83	6 75000.00000000 0.20000000
84	7 80000.00000000 0.20000000
85	8 85000.00000000 0.20000000
86	9 90000.00000000 0.20000000
87	10 95000.00000000 0.20000000
88	11 100000.00000000 0.20000000
89	12 105000.00000000 0.20000000
90	13 110000.00000000 0.20000000

Perform Analysis¶

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

Set Minimizer¶

In [27]:

project.analysis.current_minimizer = 'lmfit'

project.analysis.current_minimizer = 'lmfit'

Current minimizer changed to

lmfit

Set Fitting Parameters¶

Set structure parameters to be optimized.

In [28]:

structure_1.cell.length_a.free = True
structure_1.atom_sites['Co'].b_iso.free = True
structure_1.atom_sites['O'].b_iso.free = True

structure_2.cell.length_a.free = True

structure_1.cell.length_a.free = True structure_1.atom_sites['Co'].b_iso.free = True structure_1.atom_sites['O'].b_iso.free = True structure_2.cell.length_a.free = True

Set experiment parameters to be optimized.

In [29]:

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 [30]:

project.analysis.fit()
project.analysis.show_fit_results()

project.analysis.fit() project.analysis.show_fit_results()

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: 4.40 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		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		asym_alpha_1	0.0097	0.0097	0.0002		0.40 % ↓
8	mcstas	peak		gauss_sigma_0	45137.0000	45210.8969	2868.4651	µs²	0.16 % ↑
9	mcstas	peak		gauss_sigma_1	-52394.0000	-52476.8991	3797.9256	µs/Å	0.16 % ↑
10	mcstas	peak		gauss_sigma_2	22998.0000	23014.3392	1193.4306	µs²/Ų	0.07 % ↑
11	mcstas	peak		mix_beta_0	0.0055	0.0055	0.0001	deg	0.64 % ↑
12	mcstas	peak		mix_beta_1	0.0041	0.0041	0.0003	deg	0.32 % ↓
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 [31]:

project.plot_meas_vs_calc(expt_name='mcstas')

project.plot_meas_vs_calc(expt_name='mcstas')

In [ ]: