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,
    adp_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,
    adp_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,
    adp_iso=0.2567,
)
structure_1.atom_sites.create(
    label='O',
    type_symbol='O',
    fract_x=0,
    fract_y=0.5,
    fract_z=0.5,
    adp_iso=1.4041,
)

structure_1.atom_sites.create( label='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( label='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( label='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( label='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.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,
    adp_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, adp_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', )

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.exp_decay_beta_0 = 0.0055
experiment.peak.exp_decay_beta_1 = 0.0041
experiment.peak.exp_rise_alpha_0 = 0
experiment.peak.exp_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.exp_decay_beta_0 = 0.0055 experiment.peak.exp_decay_beta_1 = 0.0041 experiment.peak.exp_rise_alpha_0 = 0 experiment.peak.exp_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', 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(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 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	_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.gauss_sigma_0 45137
20	_peak.gauss_sigma_1 -52394
21	_peak.gauss_sigma_2 22998
22	_peak.type tof-jorgensen
23
24	_instr.2theta_bank 94.90931762
25	_instr.d_to_tof_offset 0.
26	_instr.d_to_tof_linear 58724.76869981
27	_instr.d_to_tof_quad 0.
28	_instr.d_to_tof_recip 0.
29
30	loop_
31	_pd_phase_block.id
32	_pd_phase_block.scale
33	lbco 4.
34	si 0.2
35
36	loop_
37	_excluded_region.id
38	_excluded_region.start
39	_excluded_region.end
40	1 0. 40000
41	2 108000 200000
42
43	_background.type line-segment
44
45	loop_
46	_pd_background.id
47	_pd_background.line_segment_X
48	_pd_background.line_segment_intensity
49	1 45000 0.2
50	2 50000 0.2
51	3 55000 0.2
52	4 65000 0.2
53	5 70000 0.2
54	6 75000 0.2
55	7 80000 0.2
56	8 85000 0.2
57	9 90000 0.2
58	10 95000 0.2
59	11 100000 0.2
60	12 105000 0.2
61	13 110000 0.2
62
63	loop_
64	_pd_meas.time_of_flight
65	_pd_data.point_id
66	_pd_proc.d_spacing
67	_pd_meas.intensity_total
68	_pd_meas.intensity_total_su
69	_pd_calc.intensity_total
70	_pd_calc.intensity_bkg
71	_pd_data.refinement_status
72	41168.1286 1 0.70103518 0.21537107 0.02485114 0.3052076 0.2 incl
73	41273.8536 2 0.70283552 0.26087313 0.03329888 0.46242221 0.2 incl
74	41379.5785 3 0.70463587 0.30433686 0.03547088 0.69080511 0.2 incl
75	41485.3035 4 0.70643622 0.47366708 0.04206284 0.87296344 0.2 incl
76	41591.0285 5 0.70823657 0.6002652 0.04196648 0.87808518 0.2 incl
77	41696.7534 6 0.71003691 0.60174483 0.03965186 0.71269725 0.2 incl
78	41802.4784 7 0.71183726 0.50118258 0.03810349 0.49758472 0.2 incl
79	41908.2034 8 0.71363761 0.37702264 0.03458906 0.33330217 0.2 incl
80	42013.9284 9 0.71543795 0.27533183 0.03190951 0.24597095 0.2 incl
81	42119.6533 10 0.7172383 0.26182311 0.02830043 0.21381642 0.2 incl
82	...
83	145835.8489 991 2.48337886 0.2240686 0.01131008 0.2 0.2 excl
84	145941.5739 992 2.4851792 0.21377274 0.01070627 0.2 0.2 excl
85	146047.2988 993 2.48697955 0.20927726 0.0108552 0.2 0.2 excl
86	146153.0238 994 2.4887799 0.21839779 0.01091455 0.2 0.2 excl
87	146258.7488 995 2.49058024 0.21613293 0.01104795 0.2 0.2 excl
88	146364.4737 996 2.49238059 0.20571926 0.01109275 0.2 0.2 excl
89	146470.1987 997 2.49418094 0.22615941 0.01142946 0.2 0.2 excl
90	146575.9237 998 2.49598129 0.22113311 0.01151463 0.2 0.2 excl
91	146681.6486 999 2.49778163 0.2086239 0.01097638 0.2 0.2 excl
92	146787.3736 1000 2.49958198 0.20883912 0.01092893 0.2 0.2 excl
93
94	loop_
95	_refln.id
96	_refln.phase_id
97	_refln.d_spacing
98	_refln.sin_theta_over_lambda
99	_refln.index_h
100	_refln.index_k
101	_refln.index_l
102	_refln.f_calc
103	_refln.f_squared_calc
104	_refln.time_of_flight
105	1 lbco 1.74006338 0.28734586 2 1 0 0.10595474 0.01122641 102184.81943898
106	2 lbco 1.58845327 0.31477161 2 1 1 0.39025607 0.1522998 93281.55106887
107	3 lbco 1.37564089 0.36346695 2 2 0 2.33500019 5.45222587 80784.19293006
108	4 lbco 1.29696667 0.38551492 2 2 1 0.06467317 0.00418262 76164.06751137
109	5 lbco 1.29696667 0.38551492 3 0 0 0.06467317 0.00418262 76164.06751137
110	6 lbco 1.23041061 0.40636841 3 1 0 0.42233682 0.17836839 72255.57875963
111	7 lbco 1.17315049 0.42620278 3 1 1 1.7530811 3.07329336 68892.99121159
112	8 lbco 1.12320608 0.44515429 2 2 2 2.19435299 4.81518503 65960.0173204
113	9 lbco 1.0791415 0.46333127 3 2 0 0.02740221 0.00075088 63372.3347908
114	10 lbco 1.03988677 0.48082158 3 2 1 0.45063705 0.20307375 61067.10981657
115	...
116	42 si 0.85878923 0.58221503 6 2 0 0. 0. 50432.19895265
117	43 si 0.82828988 0.60365339 5 3 3 3.31928 11.01761972 48641.13190237
118	44 si 0.8188234 0.61063228 6 2 2 6.63856 44.07047887 48085.21499475
119	45 si 0.78396372 0.63778461 4 4 4 6.63856 44.07047887 46038.08831826
120	46 si 0.76055653 0.65741333 5 5 1 3.31928 11.01761972 44663.50610299
121	47 si 0.76055653 0.65741333 7 1 1 3.31928 11.01761972 44663.50610299
122	48 si 0.75320798 0.66382727 6 4 0 0. 0. 44231.96452271
123	49 si 0.72580937 0.68888612 6 4 2 0. 0. 42622.98752015
124	50 si 0.70711586 0.7070977 7 3 1 3.31928 11.01761972 41525.21546552
125	51 si 0.70711586 0.7070977 5 5 3 3.31928 11.01761972 41525.21546552

🚀 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_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.exp_rise_alpha_1.free = True
experiment.peak.exp_decay_beta_0.free = True
experiment.peak.exp_decay_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.exp_rise_alpha_1.free = True experiment.peak.exp_decay_beta_0.free = True experiment.peak.exp_decay_beta_1.free = True for point in experiment.background: point.y.free = True

Run Fitting¶

In [30]:

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	0.06	4760.34
2	29	1.54	9.79	99.8% ↓
3	55	2.76	9.54	2.6% ↓
4	154	7.77	9.53
5	186	9.67	9.53

🏆 Best goodness-of-fit (reduced χ²) is 9.53 at iteration 159

✅ Fitting complete.

⚙️ Settings used:

	Name	Value	Description
1	max_iterations	1000	Maximum solver iterations.

📋 Least-squares fit results:

	Metric	Value
1	🧪 Minimizer	lmfit (leastsq)
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	9.67
4	🔁 Iterations	183
5	📏 Goodness-of-fit (reduced χ²)	9.53
6	📏 R-factor (Rf, %)	5.81
7	📏 R-factor squared (Rf², %)	5.35
8	📏 Weighted R-factor (wR, %)	4.92

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	lbco	cell		length_a	Å	3.8909	3.8905	0.0002	0.01 % ↓
2	lbco	atom_site	Co	adp_iso	Ų	0.2567	0.2309	0.1435	10.06 % ↓
3	lbco	atom_site	O	adp_iso	Ų	1.4041	2.1179	0.0478	50.84 % ↑
4	si	cell		length_a	Å	5.4315	5.4358	0.0019	0.08 % ↑
5	mcstas	linked_phases	lbco	scale		4.0000	4.4909	0.0536	12.27 % ↑
6	mcstas	linked_phases	si	scale		0.2000	0.0047	0.0004	97.64 % ↓
7	mcstas	peak		rise_alpha_1	μs/Å	0.0097	0.0098	0.0004	0.59 % ↑
8	mcstas	peak		decay_beta_0	μs	0.0055	0.0055	0.0002	0.85 % ↑
9	mcstas	peak		decay_beta_1	μs/Å	0.0041	0.0043	0.0005	4.31 % ↑
10	mcstas	peak		gauss_sigma_0	μs²	45137.0000	39764.0543	4803.0848	11.90 % ↓
11	mcstas	peak		gauss_sigma_1	μs/Å	-52394.0000	-43394.9893	6407.9873	17.18 % ↓
12	mcstas	peak		gauss_sigma_2	μs²/Ų	22998.0000	19741.6178	2007.9434	14.16 % ↓
13	mcstas	background	1	y		0.2000	0.2568	0.0140	28.38 % ↑
14	mcstas	background	2	y		0.2000	0.2598	0.0137	29.91 % ↑
15	mcstas	background	3	y		0.2000	0.2674	0.0081	33.69 % ↑
16	mcstas	background	4	y		0.2000	0.2691	0.0062	34.57 % ↑
17	mcstas	background	5	y		0.2000	0.2498	0.0061	24.88 % ↑
18	mcstas	background	6	y		0.2000	0.2468	0.0047	23.40 % ↑
19	mcstas	background	7	y		0.2000	0.2451	0.0057	22.57 % ↑
20	mcstas	background	8	y		0.2000	0.2412	0.0036	20.60 % ↑
21	mcstas	background	9	y		0.2000	0.2229	0.0035	11.47 % ↑
22	mcstas	background	10	y		0.2000	0.2599	0.0043	29.93 % ↑
23	mcstas	background	11	y		0.2000	0.2276	0.0037	13.78 % ↑
24	mcstas	background	12	y		0.2000	0.2215	0.0045	10.75 % ↑
25	mcstas	background	13	y		0.2000	0.2244	0.0134	12.21 % ↑

• 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 [31]:

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

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

Loading plot…

💾 Save Project¶

In [32]:

project.save_as(dir_path='projects/ed_9_lbco_si_mcstas')

project.save_as(dir_path='projects/ed_9_lbco_si_mcstas')

Saving project 📦 'lbco_si_mcstas' to '../../../projects/ed_9_lbco_si_mcstas'

├── 📄 project.cif

├── 📁 structures/

│   └── 📄 lbco.cif

│   └── 📄 si.cif

├── 📁 experiments/

│   └── 📄 mcstas.cif

├── 📁 analysis/

│   └── 📄 analysis.cif

└── 📁 reports/

    └── 📄 lbco_si_mcstas.html