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.coord_system_code = '1'

structure_1.space_group.name_h_m = 'P m -3 m' structure_1.space_group.coord_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(
    id='La',
    type_symbol='La',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    adp_iso=0.2,
    occupancy=0.5,
)
structure_1.atom_sites.create(
    id='Ba',
    type_symbol='Ba',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    adp_iso=0.2,
    occupancy=0.5,
)
structure_1.atom_sites.create(
    id='Co',
    type_symbol='Co',
    fract_x=0.5,
    fract_y=0.5,
    fract_z=0.5,
    adp_iso=0.2567,
)
structure_1.atom_sites.create(
    id='O',
    type_symbol='O',
    fract_x=0,
    fract_y=0.5,
    fract_z=0.5,
    adp_iso=1.4041,
)

structure_1.atom_sites.create( id='La', type_symbol='La', fract_x=0, fract_y=0, fract_z=0, adp_iso=0.2, occupancy=0.5, ) structure_1.atom_sites.create( id='Ba', type_symbol='Ba', fract_x=0, fract_y=0, fract_z=0, adp_iso=0.2, occupancy=0.5, ) structure_1.atom_sites.create( id='Co', type_symbol='Co', fract_x=0.5, fract_y=0.5, fract_z=0.5, adp_iso=0.2567, ) structure_1.atom_sites.create( id='O', type_symbol='O', fract_x=0, fract_y=0.5, fract_z=0.5, adp_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.coord_system_code = '1'

structure_2.space_group.name_h_m = 'F d -3 m' structure_2.space_group.coord_system_code = '1'

Set Unit Cell¶

In [9]:

structure_2.cell.length_a = 5.43

structure_2.cell.length_a = 5.43

Set Atom Sites¶

In [10]:

structure_2.atom_sites.create(
    id='Si',
    type_symbol='Si',
    fract_x=0.0,
    fract_y=0.0,
    fract_z=0.0,
    adp_type='Biso',
    adp_iso=0.89,
)

structure_2.atom_sites.create( id='Si', type_symbol='Si', fract_x=0.0, fract_y=0.0, fract_z=0.0, adp_type='Biso', adp_iso=0.89, )

🔬 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('meas-lbco-si-mcstas', destination='data')

data_path = download_data('meas-lbco-si-mcstas', destination='data')

Getting data...

Data 'meas-lbco-si-mcstas': La0.5Ba0.5CoO3 + Si, McStas simulation

✅ Data 'meas-lbco-si-mcstas' downloaded to '../../../data/meas-lbco-si-mcstas.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', )

Set Instrument¶

In [13]:

experiment.instrument.setup_twotheta_bank = 94.90931761529106
experiment.instrument.calib_d_to_tof_linear = 58724.76869981215

experiment.instrument.setup_twotheta_bank = 94.90931761529106 experiment.instrument.calib_d_to_tof_linear = 58724.76869981215

Set Peak Profile¶

In [14]:

experiment.peak.broad_gauss_sigma_0 = 45137
experiment.peak.broad_gauss_sigma_1 = -52394
experiment.peak.broad_gauss_sigma_2 = 22998
experiment.peak.decay_beta_0 = 0.0055
experiment.peak.decay_beta_1 = 0.0041
experiment.peak.rise_alpha_0 = 0
experiment.peak.rise_alpha_1 = 0.0097

experiment.peak.broad_gauss_sigma_0 = 45137 experiment.peak.broad_gauss_sigma_1 = -52394 experiment.peak.broad_gauss_sigma_2 = 22998 experiment.peak.decay_beta_0 = 0.0055 experiment.peak.decay_beta_1 = 0.0041 experiment.peak.rise_alpha_0 = 0 experiment.peak.rise_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', position=45000, intensity=0.2)
experiment.background.create(id='2', position=50000, intensity=0.2)
experiment.background.create(id='3', position=55000, intensity=0.2)
experiment.background.create(id='4', position=65000, intensity=0.2)
experiment.background.create(id='5', position=70000, intensity=0.2)
experiment.background.create(id='6', position=75000, intensity=0.2)
experiment.background.create(id='7', position=80000, intensity=0.2)
experiment.background.create(id='8', position=85000, intensity=0.2)
experiment.background.create(id='9', position=90000, intensity=0.2)
experiment.background.create(id='10', position=95000, intensity=0.2)
experiment.background.create(id='11', position=100000, intensity=0.2)
experiment.background.create(id='12', position=105000, intensity=0.2)
experiment.background.create(id='13', position=110000, intensity=0.2)

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

Set Linked Structures¶

In [17]:

experiment.linked_structures.create(structure_id='lbco', scale=4.0)
experiment.linked_structures.create(structure_id='si', scale=0.2)

experiment.linked_structures.create(structure_id='lbco', scale=4.0) experiment.linked_structures.create(structure_id='si', scale=0.2)

📦 Define Project¶

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

Create Project¶

In [18]:

project = Project(name='lbco_si_mcstas')

project = Project(name='lbco_si_mcstas')

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)

Display Structure¶

In [22]:

project.display.structure(struct_name='lbco')
project.display.structure(struct_name='si')

project.display.structure(struct_name='lbco') project.display.structure(struct_name='si')

Structure 🧩 'lbco' (Atom view type: 'covalent')

Loading plot…

drag = rotate
wheel = zoom
right-drag = pan

Structure 🧩 'si' (Atom view type: 'covalent')

Loading plot…

drag = rotate
wheel = zoom
right-drag = pan

Set Excluded Regions¶

Show measured data as loaded from the file.

In [23]:

project.display.pattern(expt_name='mcstas')

project.display.pattern(expt_name='mcstas')

Loading plot…

Add excluded regions.

In [24]:

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 [25]:

experiment.excluded_regions.show()

experiment.excluded_regions.show()

Excluded regions

	start	end
1	0.00000	40000.00000
2	108000.00000	200000.00000

Show measured data after adding excluded regions.

In [26]:

project.display.pattern(expt_name='mcstas')

project.display.pattern(expt_name='mcstas')

Loading plot…

Show experiment as text.

In [27]:

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

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

Experiment 🔬 'mcstas' as text

	Edi
1	data_mcstas
2
3	_experiment_type.sample_form powder
4	_experiment_type.beam_mode time-of-flight
5	_experiment_type.radiation_probe neutron
6	_experiment_type.scattering_type bragg
7
8	_diffrn.ambient_temperature ?
9	_diffrn.ambient_pressure ?
10	_diffrn.ambient_magnetic_field ?
11	_diffrn.ambient_electric_field ?
12
13	_calculator.type cryspy
14
15	_peak.rise_alpha_0 0.
16	_peak.rise_alpha_1 0.0097
17	_peak.decay_beta_0 0.0055
18	_peak.decay_beta_1 0.0041
19	_peak.broad_gauss_sigma_0 45137
20	_peak.broad_gauss_sigma_1 -52394
21	_peak.broad_gauss_sigma_2 22998
22	_peak.broad_gauss_size 0.
23	_peak.broad_gauss_strain 0.
24	_peak.cutoff_fwhm 0.
25	_peak.type tof-jorgensen
26
27	_instrument.setup_twotheta_bank 94.90931762
28	_instrument.calib_d_to_tof_offset 0.
29	_instrument.calib_d_to_tof_linear 58724.76869981
30	_instrument.calib_d_to_tof_quadratic 0.
31	_instrument.calib_d_to_tof_reciprocal 0.
32
33	_absorption.type none
34
35	loop_
36	_linked_structure.structure_id
37	_linked_structure.scale
38	lbco 4.
39	si 0.2
40
41	loop_
42	_excluded_region.id
43	_excluded_region.start
44	_excluded_region.end
45	1 0. 40000
46	2 108000 200000
47
48	_background.type line-segment
49
50	loop_
51	_background.id
52	_background.position
53	_background.intensity
54	1 45000 0.2
55	2 50000 0.2
56	3 55000 0.2
57	4 65000 0.2
58	5 70000 0.2
59	6 75000 0.2
60	7 80000 0.2
61	8 85000 0.2
62	9 90000 0.2
63	10 95000 0.2
64	11 100000 0.2
65	12 105000 0.2
66	13 110000 0.2
67
68	loop_
69	_data.time_of_flight
70	_data.id
71	_data.d_spacing
72	_data.intensity_meas
73	_data.intensity_meas_su
74	_data.intensity_calc
75	_data.intensity_bkg
76	_data.calc_status
77	41168.1286 1 0.70103518 0.21537107 0.02485114 0.20309954 0.2 incl
78	41273.8536 2 0.70283552 0.26087313 0.03329888 0.20793262 0.2 incl
79	41379.5785 3 0.70463587 0.30433686 0.03547088 0.21589262 0.2 incl
80	41485.3035 4 0.70643622 0.47366708 0.04206284 0.22494001 0.2 incl
81	41591.0285 5 0.70823657 0.6002652 0.04196648 0.23091504 0.2 incl
82	41696.7534 6 0.71003691 0.60174483 0.03965186 0.22996176 0.2 incl
83	41802.4784 7 0.71183726 0.50118258 0.03810349 0.22194769 0.2 incl
84	41908.2034 8 0.71363761 0.37702264 0.03458906 0.21175012 0.2 incl
85	42013.9284 9 0.71543795 0.27533183 0.03190951 0.20472412 0.2 incl
86	42119.6533 10 0.7172383 0.26182311 0.02830043 0.20226809 0.2 incl
87	...
88	145835.8489 991 2.48337886 0.2240686 0.01131008 0.2 0.2 excl
89	145941.5739 992 2.4851792 0.21377274 0.01070627 0.2 0.2 excl
90	146047.2988 993 2.48697955 0.20927726 0.0108552 0.2 0.2 excl
91	146153.0238 994 2.4887799 0.21839779 0.01091455 0.2 0.2 excl
92	146258.7488 995 2.49058024 0.21613293 0.01104795 0.2 0.2 excl
93	146364.4737 996 2.49238059 0.20571926 0.01109275 0.2 0.2 excl
94	146470.1987 997 2.49418094 0.22615941 0.01142946 0.2 0.2 excl
95	146575.9237 998 2.49598129 0.22113311 0.01151463 0.2 0.2 excl
96	146681.6486 999 2.49778163 0.2086239 0.01097638 0.2 0.2 excl
97	146787.3736 1000 2.49958198 0.20883912 0.01092893 0.2 0.2 excl
98
99	loop_
100	_refln.id
101	_refln.structure_id
102	_refln.d_spacing
103	_refln.sin_theta_over_lambda
104	_refln.index_h
105	_refln.index_k
106	_refln.index_l
107	_refln.f_calc
108	_refln.f_squared_calc
109	_refln.time_of_flight
110	1 lbco 1.74006338 0.28734586 2 1 0 0.10595474 0.01122641 102184.81943898
111	2 lbco 1.58845327 0.31477161 2 1 1 0.39025607 0.1522998 93281.55106887
112	3 lbco 1.37564089 0.36346695 2 2 0 2.33500019 5.45222587 80784.19293006
113	4 lbco 1.29696667 0.38551492 2 2 1 0.06467317 0.00418262 76164.06751137
114	5 lbco 1.29696667 0.38551492 3 0 0 0.06467317 0.00418262 76164.06751137
115	6 lbco 1.23041061 0.40636841 3 1 0 0.42233682 0.17836839 72255.57875963
116	7 lbco 1.17315049 0.42620278 3 1 1 1.7530811 3.07329336 68892.99121159
117	8 lbco 1.12320608 0.44515429 2 2 2 2.19435299 4.81518503 65960.0173204
118	9 lbco 1.0791415 0.46333127 3 2 0 0.02740221 0.00075088 63372.3347908
119	10 lbco 1.03988677 0.48082158 3 2 1 0.45063705 0.20307375 61067.10981657
120	...
121	42 si 0.85855838 0.58237158 6 2 0 2.45443961 6.0242738 50418.64255889
122	43 si 0.82806724 0.6038157 5 3 3 1.69670167 2.87879657 48628.0569552
123	44 si 0.8186033 0.61079646 6 2 2 0. 0. 48072.28948045
124	45 si 0.78375299 0.6379561 4 4 4 2.31064934 5.33910039 46025.71308049
125	46 si 0.76035209 0.65759009 5 5 1 1.59730253 2.55137538 44651.50035888
126	47 si 0.76035209 0.65759009 7 1 1 1.59730253 2.55137538 44651.50035888
127	48 si 0.75300552 0.66400576 6 4 0 0. 0. 44220.07477885
128	49 si 0.72561427 0.68907134 6 4 2 2.17528285 4.73185548 42611.53027628
129	50 si 0.70692579 0.70728782 7 3 1 1.50372657 2.26119359 41514.05330754
130	51 si 0.70692579 0.70728782 5 5 3 1.50372657 2.26119359 41514.05330754

🚀 Perform Analysis¶

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

Set Free Parameters¶

Set structure parameters to be optimized.

In [28]:

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

structure_2.cell.length_a.free = True

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

Set experiment parameters to be optimized.

In [29]:

experiment.linked_structures['lbco'].scale.free = True
experiment.linked_structures['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.rise_alpha_1.free = True
experiment.peak.decay_beta_0.free = True
experiment.peak.decay_beta_1.free = True

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

experiment.linked_structures['lbco'].scale.free = True experiment.linked_structures['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.rise_alpha_1.free = True experiment.peak.decay_beta_0.free = True experiment.peak.decay_beta_1.free = True for point in experiment.background: point.intensity.free = True

Run Fitting¶

In [30]:

project.analysis.minimizer.chi_square_change_tolerance = 1e-2

project.analysis.minimizer.chi_square_change_tolerance = 1e-2

In [31]:

project.analysis.fit()
project.display.fit.results()
project.display.fit.correlations()

project.analysis.fit() project.display.fit.results() project.display.fit.correlations()

Standard fitting

📋 Using experiment 🔬 'mcstas' for 'single' fitting

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

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.23	462.31
2	31	1.33	392.51	15.1% ↓
3	57	2.26	290.28	26.0% ↓
4	83	3.43	170.56	41.2% ↓
5	109	4.35	55.23	67.6% ↓
6	135	5.55	7.56	86.3% ↓
7	161	6.47	3.55	53.0% ↓
8	187	7.63	3.45	2.9% ↓
9	214	8.73	3.45

🏆 Best goodness-of-fit (reduced χ²) is 3.45 at iteration 213

✅ Fitting complete.

⚙️ Settings used:

	Name	Value	Description
1	max_iterations	1000	Maximum solver iterations.
2	chi_square_change_tolerance	0.01	Relative change in the objective (chi-square) used to stop fitting.
3	parameter_change_tolerance	1e-08	Relative change in fitted parameters used to stop fitting.
4	gradient_tolerance	0.0	Gradient orthogonality used to stop fitting; zero disables it.

📋 Least-squares fit results:

	Metric	Value
1	🧪 Minimizer	lmfit (leastsq)
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	8.73
4	🔁 Iterations	211
5	📏 Goodness-of-fit (reduced χ²)	3.45
6	📏 R-factor (Rf, %)	4.92
7	📏 R-factor squared (Rf², %)	5.23
8	📏 Weighted R-factor (wR, %)	4.59

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	lbco	cell		length_a	Å	3.8909	3.8905	0.0001	0.01 % ↓
2	lbco	atom_site	Co	adp_iso	Ų	0.2567	0.2974	0.0870	15.85 % ↑
3	lbco	atom_site	O	adp_iso	Ų	1.4041	2.1685	0.0290	54.44 % ↑
4	si	cell		length_a	Å	5.4300	5.4328	0.0006	0.05 % ↑
5	mcstas	linked_structure	lbco	scale		4.0000	39.6993	0.2845	892.48 % ↑
6	mcstas	linked_structure	si	scale		0.2000	0.3311	0.0088	65.56 % ↑
7	mcstas	peak		rise_alpha_1	μs/Å	0.0097	0.0096	0.0002	1.09 % ↓
8	mcstas	peak		decay_beta_0	μs	0.0055	0.0055	0.0001	0.43 % ↑
9	mcstas	peak		decay_beta_1	μs/Å	0.0041	0.0041	0.0003	0.30 % ↓
10	mcstas	peak		broad_gauss_sigma_0	μs²	45137.0000	45270.7878	2953.1291	0.30 % ↑
11	mcstas	peak		broad_gauss_sigma_1	μs/Å	-52394.0000	-52782.9993	3912.0169	0.74 % ↑
12	mcstas	peak		broad_gauss_sigma_2	μs²/Ų	22998.0000	23122.2426	1230.0476	0.54 % ↑
13	mcstas	background	1	intensity		0.2000	0.2543	0.0084	27.14 % ↑
14	mcstas	background	2	intensity		0.2000	0.2531	0.0083	26.54 % ↑
15	mcstas	background	3	intensity		0.2000	0.2686	0.0049	34.31 % ↑
16	mcstas	background	4	intensity		0.2000	0.2572	0.0037	28.59 % ↑
17	mcstas	background	5	intensity		0.2000	0.2464	0.0037	23.18 % ↑
18	mcstas	background	6	intensity		0.2000	0.2424	0.0028	21.20 % ↑
19	mcstas	background	7	intensity		0.2000	0.2422	0.0034	21.10 % ↑
20	mcstas	background	8	intensity		0.2000	0.2374	0.0022	18.68 % ↑
21	mcstas	background	9	intensity		0.2000	0.2383	0.0021	19.13 % ↑
22	mcstas	background	10	intensity		0.2000	0.2412	0.0026	20.62 % ↑
23	mcstas	background	11	intensity		0.2000	0.2257	0.0022	12.87 % ↑
24	mcstas	background	12	intensity		0.2000	0.2227	0.0027	11.36 % ↑
25	mcstas	background	13	intensity		0.2000	0.2226	0.0080	11.29 % ↑

• start = parameter value before refinement
• value = refined value from least-squares minimization
• s.u. = standard uncertainty (one sigma), from the covariance matrix
• change = relative change from start, in %; ↑ = increase, ↓ = decrease

Loading plot…

Display Pattern¶

In [32]:

project.display.pattern(expt_name='mcstas')

project.display.pattern(expt_name='mcstas')

Loading plot…

💾 Save Project¶

In [33]:

project.save_as(dir_path='projects/refine-lbco-si-mcstas')

project.save_as(dir_path='projects/refine-lbco-si-mcstas')

Saving project 📦 'lbco_si_mcstas' to '../../../projects/refine-lbco-si-mcstas'

├── 📄 project.edi
├── 📁 structures/
│   └── 📄 lbco.edi
│   └── 📄 si.edi
├── 📁 experiments/
│   └── 📄 mcstas.edi
├── 📁 analysis/
│   └── 📄 analysis.edi
└── 📁 reports/
    └── 📄 lbco_si_mcstas.html