Structure Refinement: Co2SiO4, D20¶

This example demonstrates a Rietveld refinement of Co2SiO4 crystal structure using constant wavelength neutron powder diffraction data from D20 at ILL.

Import Library¶

In [2]:

from easydiffraction import ExperimentFactory
from easydiffraction import Project
from easydiffraction import SampleModelFactory
from easydiffraction import download_from_repository

from easydiffraction import ExperimentFactory from easydiffraction import Project from easydiffraction import SampleModelFactory from easydiffraction import download_from_repository

Define Sample Model¶

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

Create Sample Model¶

In [3]:

model = SampleModelFactory.create(name='cosio')

model = SampleModelFactory.create(name='cosio')

Set Space Group¶

In [4]:

model.space_group.name_h_m = 'P n m a'
model.space_group.it_coordinate_system_code = 'abc'

model.space_group.name_h_m = 'P n m a' model.space_group.it_coordinate_system_code = 'abc'

Set Unit Cell¶

In [5]:

model.cell.length_a = 10.3
model.cell.length_b = 6.0
model.cell.length_c = 4.8

model.cell.length_a = 10.3 model.cell.length_b = 6.0 model.cell.length_c = 4.8

Set Atom Sites¶

In [6]:

model.atom_sites.add_from_args(
    label='Co1', type_symbol='Co', fract_x=0, fract_y=0, fract_z=0, wyckoff_letter='a', b_iso=0.5
)
model.atom_sites.add_from_args(
    label='Co2',
    type_symbol='Co',
    fract_x=0.279,
    fract_y=0.25,
    fract_z=0.985,
    wyckoff_letter='c',
    b_iso=0.5,
)
model.atom_sites.add_from_args(
    label='Si',
    type_symbol='Si',
    fract_x=0.094,
    fract_y=0.25,
    fract_z=0.429,
    wyckoff_letter='c',
    b_iso=0.5,
)
model.atom_sites.add_from_args(
    label='O1',
    type_symbol='O',
    fract_x=0.091,
    fract_y=0.25,
    fract_z=0.771,
    wyckoff_letter='c',
    b_iso=0.5,
)
model.atom_sites.add_from_args(
    label='O2',
    type_symbol='O',
    fract_x=0.448,
    fract_y=0.25,
    fract_z=0.217,
    wyckoff_letter='c',
    b_iso=0.5,
)
model.atom_sites.add_from_args(
    label='O3',
    type_symbol='O',
    fract_x=0.164,
    fract_y=0.032,
    fract_z=0.28,
    wyckoff_letter='d',
    b_iso=0.5,
)

model.atom_sites.add_from_args( label='Co1', type_symbol='Co', fract_x=0, fract_y=0, fract_z=0, wyckoff_letter='a', b_iso=0.5 ) model.atom_sites.add_from_args( label='Co2', type_symbol='Co', fract_x=0.279, fract_y=0.25, fract_z=0.985, wyckoff_letter='c', b_iso=0.5, ) model.atom_sites.add_from_args( label='Si', type_symbol='Si', fract_x=0.094, fract_y=0.25, fract_z=0.429, wyckoff_letter='c', b_iso=0.5, ) model.atom_sites.add_from_args( label='O1', type_symbol='O', fract_x=0.091, fract_y=0.25, fract_z=0.771, wyckoff_letter='c', b_iso=0.5, ) model.atom_sites.add_from_args( label='O2', type_symbol='O', fract_x=0.448, fract_y=0.25, fract_z=0.217, wyckoff_letter='c', b_iso=0.5, ) model.atom_sites.add_from_args( label='O3', type_symbol='O', fract_x=0.164, fract_y=0.032, fract_z=0.28, wyckoff_letter='d', b_iso=0.5, )

Symmetry Constraints¶

Show CIF output before applying symmetry constraints.

In [7]:

model.show_as_cif()

model.show_as_cif()

Sample model 🧩 'cosio' as cif

	CIF
1	data_cosio
2
3	_cell.length_a 10.3
4	_cell.length_b 6.0
5	_cell.length_c 4.8
6	_cell.angle_alpha 90.0
7	_cell.angle_beta 90.0
8	_cell.angle_gamma 90.0
9
10	_space_group.name_H-M_alt "P n m a"
11	_space_group.IT_coordinate_system_code abc
12
13	loop_
14	_atom_site.label
15	_atom_site.type_symbol
16	_atom_site.fract_x
17	_atom_site.fract_y
18	_atom_site.fract_z
19	_atom_site.Wyckoff_letter
20	_atom_site.occupancy
21	_atom_site.B_iso_or_equiv
22	_atom_site.adp_type
23	Co1 Co 0 0 0 a 1.0 0.5 Biso
24	Co2 Co 0.279 0.25 0.985 c 1.0 0.5 Biso
25	Si Si 0.094 0.25 0.429 c 1.0 0.5 Biso
26	O1 O 0.091 0.25 0.771 c 1.0 0.5 Biso
27	O2 O 0.448 0.25 0.217 c 1.0 0.5 Biso
28	O3 O 0.164 0.032 0.28 d 1.0 0.5 Biso

Apply symmetry constraints.

In [8]:

model.apply_symmetry_constraints()

model.apply_symmetry_constraints()

Show CIF output after applying symmetry constraints.

In [9]:

model.show_as_cif()

model.show_as_cif()

Sample model 🧩 'cosio' as cif

	CIF
1	data_cosio
2
3	_cell.length_a 10.3
4	_cell.length_b 6.0
5	_cell.length_c 4.8
6	_cell.angle_alpha 90.0
7	_cell.angle_beta 90.0
8	_cell.angle_gamma 90.0
9
10	_space_group.name_H-M_alt "P n m a"
11	_space_group.IT_coordinate_system_code abc
12
13	loop_
14	_atom_site.label
15	_atom_site.type_symbol
16	_atom_site.fract_x
17	_atom_site.fract_y
18	_atom_site.fract_z
19	_atom_site.Wyckoff_letter
20	_atom_site.occupancy
21	_atom_site.B_iso_or_equiv
22	_atom_site.adp_type
23	Co1 Co 0.0 0.0 0.0 a 1.0 0.5 Biso
24	Co2 Co 0.279 0.25 0.985 c 1.0 0.5 Biso
25	Si Si 0.094 0.25 0.429 c 1.0 0.5 Biso
26	O1 O 0.091 0.25 0.771 c 1.0 0.5 Biso
27	O2 O 0.448 0.25 0.217 c 1.0 0.5 Biso
28	O3 O 0.164 0.032 0.28 d 1.0 0.5 Biso

Define Experiment¶

This section shows how to add experiments, configure their parameters, and link the sample models defined in the previous step.

Download Measured Data¶

In [10]:

download_from_repository('co2sio4_d20.xye', destination='data')

download_from_repository('co2sio4_d20.xye', destination='data')

Downloading...

File 'co2sio4_d20.xye' from 'easyscience/diffraction-lib'

Create Experiment¶

In [11]:

expt = ExperimentFactory.create(name='d20', data_path='data/co2sio4_d20.xye')

expt = ExperimentFactory.create(name='d20', data_path='data/co2sio4_d20.xye')

Data loaded successfully

Experiment 🔬 'd20'. Number of data points: 1418

Set Instrument¶

In [12]:

expt.instrument.setup_wavelength = 1.87
expt.instrument.calib_twotheta_offset = 0.1

expt.instrument.setup_wavelength = 1.87 expt.instrument.calib_twotheta_offset = 0.1

Set Peak Profile¶

In [13]:

expt.peak.broad_gauss_u = 0.3
expt.peak.broad_gauss_v = -0.5
expt.peak.broad_gauss_w = 0.4

expt.peak.broad_gauss_u = 0.3 expt.peak.broad_gauss_v = -0.5 expt.peak.broad_gauss_w = 0.4

Set Background¶

In [14]:

expt.background.add_from_args(x=8, y=500)
expt.background.add_from_args(x=9, y=500)
expt.background.add_from_args(x=10, y=500)
expt.background.add_from_args(x=11, y=500)
expt.background.add_from_args(x=12, y=500)
expt.background.add_from_args(x=15, y=500)
expt.background.add_from_args(x=25, y=500)
expt.background.add_from_args(x=30, y=500)
expt.background.add_from_args(x=50, y=500)
expt.background.add_from_args(x=70, y=500)
expt.background.add_from_args(x=90, y=500)
expt.background.add_from_args(x=110, y=500)
expt.background.add_from_args(x=130, y=500)
expt.background.add_from_args(x=150, y=500)

expt.background.add_from_args(x=8, y=500) expt.background.add_from_args(x=9, y=500) expt.background.add_from_args(x=10, y=500) expt.background.add_from_args(x=11, y=500) expt.background.add_from_args(x=12, y=500) expt.background.add_from_args(x=15, y=500) expt.background.add_from_args(x=25, y=500) expt.background.add_from_args(x=30, y=500) expt.background.add_from_args(x=50, y=500) expt.background.add_from_args(x=70, y=500) expt.background.add_from_args(x=90, y=500) expt.background.add_from_args(x=110, y=500) expt.background.add_from_args(x=130, y=500) expt.background.add_from_args(x=150, y=500)

Set Linked Phases¶

In [15]:

expt.linked_phases.add_from_args(id='cosio', scale=1.0)

expt.linked_phases.add_from_args(id='cosio', scale=1.0)

Define Project¶

The project object is used to manage the sample model, experiment, and analysis.

Create Project¶

In [16]:

project = Project()

project = Project()

Set Plotting Engine¶

In [17]:

project.plotter.engine = 'plotly'

project.plotter.engine = 'plotly'

Add Sample Model¶

In [18]:

project.sample_models.add(model)

project.sample_models.add(model)

Add Experiment¶

In [19]:

project.experiments.add(expt)

project.experiments.add(expt)

Perform Analysis¶

This section shows the analysis process, including how to set up calculation and fitting engines.

Set Calculator¶

In [20]:

project.analysis.current_calculator = 'cryspy'

project.analysis.current_calculator = 'cryspy'

Current calculator changed to

cryspy

Set Minimizer¶

In [21]:

project.analysis.current_minimizer = 'lmfit (leastsq)'

project.analysis.current_minimizer = 'lmfit (leastsq)'

Current minimizer changed to

lmfit (leastsq)

Plot Measured vs Calculated¶

In [22]:

project.plot_meas_vs_calc(expt_name='d20', show_residual=True)

project.plot_meas_vs_calc(expt_name='d20', show_residual=True)

In [23]:

project.plot_meas_vs_calc(expt_name='d20', x_min=41, x_max=54, show_residual=True)

project.plot_meas_vs_calc(expt_name='d20', x_min=41, x_max=54, show_residual=True)

Set Free Parameters¶

In [24]:

model.cell.length_a.free = True
model.cell.length_b.free = True
model.cell.length_c.free = True

model.atom_sites['Co2'].fract_x.free = True
model.atom_sites['Co2'].fract_z.free = True
model.atom_sites['Si'].fract_x.free = True
model.atom_sites['Si'].fract_z.free = True
model.atom_sites['O1'].fract_x.free = True
model.atom_sites['O1'].fract_z.free = True
model.atom_sites['O2'].fract_x.free = True
model.atom_sites['O2'].fract_z.free = True
model.atom_sites['O3'].fract_x.free = True
model.atom_sites['O3'].fract_y.free = True
model.atom_sites['O3'].fract_z.free = True

model.atom_sites['Co1'].b_iso.free = True
model.atom_sites['Co2'].b_iso.free = True
model.atom_sites['Si'].b_iso.free = True
model.atom_sites['O1'].b_iso.free = True
model.atom_sites['O2'].b_iso.free = True
model.atom_sites['O3'].b_iso.free = True

model.cell.length_a.free = True model.cell.length_b.free = True model.cell.length_c.free = True model.atom_sites['Co2'].fract_x.free = True model.atom_sites['Co2'].fract_z.free = True model.atom_sites['Si'].fract_x.free = True model.atom_sites['Si'].fract_z.free = True model.atom_sites['O1'].fract_x.free = True model.atom_sites['O1'].fract_z.free = True model.atom_sites['O2'].fract_x.free = True model.atom_sites['O2'].fract_z.free = True model.atom_sites['O3'].fract_x.free = True model.atom_sites['O3'].fract_y.free = True model.atom_sites['O3'].fract_z.free = True model.atom_sites['Co1'].b_iso.free = True model.atom_sites['Co2'].b_iso.free = True model.atom_sites['Si'].b_iso.free = True model.atom_sites['O1'].b_iso.free = True model.atom_sites['O2'].b_iso.free = True model.atom_sites['O3'].b_iso.free = True

In [25]:

expt.linked_phases['cosio'].scale.free = True

expt.instrument.calib_twotheta_offset.free = True

expt.peak.broad_gauss_u.free = True
expt.peak.broad_gauss_v.free = True
expt.peak.broad_gauss_w.free = True
expt.peak.broad_lorentz_y.free = True

for point in expt.background:
    point.y.free = True

expt.linked_phases['cosio'].scale.free = True expt.instrument.calib_twotheta_offset.free = True expt.peak.broad_gauss_u.free = True expt.peak.broad_gauss_v.free = True expt.peak.broad_gauss_w.free = True expt.peak.broad_lorentz_y.free = True for point in expt.background: point.y.free = True

Set Constraints¶

Set aliases for parameters.

In [26]:

project.analysis.aliases.add_from_args(
    label='biso_Co1',
    param_uid=project.sample_models['cosio'].atom_sites['Co1'].b_iso.uid,
)
project.analysis.aliases.add_from_args(
    label='biso_Co2',
    param_uid=project.sample_models['cosio'].atom_sites['Co2'].b_iso.uid,
)

project.analysis.aliases.add_from_args( label='biso_Co1', param_uid=project.sample_models['cosio'].atom_sites['Co1'].b_iso.uid, ) project.analysis.aliases.add_from_args( label='biso_Co2', param_uid=project.sample_models['cosio'].atom_sites['Co2'].b_iso.uid, )

Set constraints.

In [27]:

project.analysis.constraints.add_from_args(
    lhs_alias='biso_Co2',
    rhs_expr='biso_Co1',
)

project.analysis.constraints.add_from_args( lhs_alias='biso_Co2', rhs_expr='biso_Co1', )

Apply constraints.

In [28]:

project.analysis.apply_constraints()

project.analysis.apply_constraints()

Run Fitting¶

In [29]:

project.analysis.fit()

project.analysis.fit()

Using experiment 🔬 'd20' for 'single' fitting

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

📈 Goodness-of-fit (reduced χ²) change:

	iteration	χ²	improvement [%]
1	1	423.20
2	43	71.71	83.1% ↓
3	83	40.43	43.6% ↓
4	123	17.05	57.8% ↓
5	163	10.78	36.8% ↓
6	203	9.26	14.1% ↓
7	243	8.08	12.7% ↓
8	283	5.66	30.0% ↓
9	323	4.67	17.4% ↓
10	363	4.57	2.3% ↓
11	564	4.56

🏆 Best goodness-of-fit (reduced χ²) is 4.56 at iteration 546

✅ Fitting complete.

Fit results

✅ Success: True

⏱️ Fitting time: 19.55 seconds

📏 Goodness-of-fit (reduced χ²): 4.56

📏 R-factor (Rf): 3.04%

📏 R-factor squared (Rf²): 4.54%

📏 Weighted R-factor (wR): 4.87%

📈 Fitted parameters:

	datablock	category	entry	parameter	start	fitted	uncertainty	units	change
1	cosio	cell	None	length_a	10.3000	10.3090	0.0003	Å	0.09 % ↑
2	cosio	cell	None	length_b	6.0000	6.0039	0.0002	Å	0.07 % ↑
3	cosio	cell	None	length_c	4.8000	4.7868	0.0001	Å	0.28 % ↓
4	cosio	atom_site	Co1	b_iso	0.5000	0.2770	0.0802	Ų	44.59 % ↓
5	cosio	atom_site	Co2	fract_x	0.2790	0.2794	0.0007		0.14 % ↑
6	cosio	atom_site	Co2	fract_z	0.9850	0.9847	0.0015		0.03 % ↓
7	cosio	atom_site	Si	fract_x	0.0940	0.0937	0.0004		0.31 % ↓
8	cosio	atom_site	Si	fract_z	0.4290	0.4290	0.0008		0.01 % ↑
9	cosio	atom_site	Si	b_iso	0.5000	0.3519	0.0625	Ų	29.62 % ↓
10	cosio	atom_site	O1	fract_x	0.0910	0.0911	0.0003		0.08 % ↑
11	cosio	atom_site	O1	fract_z	0.7710	0.7714	0.0006		0.05 % ↑
12	cosio	atom_site	O1	b_iso	0.5000	0.6457	0.0581	Ų	29.14 % ↑
13	cosio	atom_site	O2	fract_x	0.4480	0.4482	0.0003		0.06 % ↑
14	cosio	atom_site	O2	fract_z	0.2170	0.2167	0.0007		0.14 % ↓
15	cosio	atom_site	O2	b_iso	0.5000	0.5778	0.0582	Ų	15.57 % ↑
16	cosio	atom_site	O3	fract_x	0.1640	0.1636	0.0002		0.26 % ↓
17	cosio	atom_site	O3	fract_y	0.0320	0.0317	0.0003		1.08 % ↓
18	cosio	atom_site	O3	fract_z	0.2800	0.2800	0.0005		0.02 % ↑
19	cosio	atom_site	O3	b_iso	0.5000	0.8389	0.0485	Ų	67.79 % ↑
20	d20	peak	None	broad_gauss_u	0.3000	0.2423	0.0068	deg²	19.22 % ↓
21	d20	peak	None	broad_gauss_v	-0.5000	-0.5288	0.0145	deg²	5.76 % ↑
22	d20	peak	None	broad_gauss_w	0.4000	0.3840	0.0090	deg²	4.01 % ↓
23	d20	peak	None	broad_lorentz_y	0.0000	0.0159	0.0045	deg	N/A
24	d20	linked_phases	cosio	scale	1.0000	1.1957	0.0106		19.57 % ↑
25	d20	instrument	None	twotheta_offset	0.1000	0.2884	0.0020	deg	188.44 % ↑
26	d20	background	8	y	500.0000	608.6305	14.5414		21.73 % ↑
27	d20	background	9	y	500.0000	580.6214	9.7278		16.12 % ↑
28	d20	background	10	y	500.0000	562.8991	9.2422		12.58 % ↑
29	d20	background	11	y	500.0000	540.2882	8.7695		8.06 % ↑
30	d20	background	12	y	500.0000	519.8428	6.0496		3.97 % ↑
31	d20	background	15	y	500.0000	507.3838	3.4846		1.48 % ↑
32	d20	background	25	y	500.0000	463.0794	3.1898		7.38 % ↓
33	d20	background	30	y	500.0000	434.2455	2.3904		13.15 % ↓
34	d20	background	50	y	500.0000	450.8166	2.2505		9.84 % ↓
35	d20	background	70	y	500.0000	430.3828	2.0250		13.92 % ↓
36	d20	background	90	y	500.0000	413.7132	2.2594		17.26 % ↓
37	d20	background	110	y	500.0000	361.2599	2.0915		27.75 % ↓
38	d20	background	130	y	500.0000	291.0612	1.9671		41.79 % ↓
39	d20	background	150	y	500.0000	238.7749	2.9764		52.25 % ↓

Plot Measured vs Calculated¶

In [30]:

project.plot_meas_vs_calc(expt_name='d20', show_residual=True)

project.plot_meas_vs_calc(expt_name='d20', show_residual=True)

In [31]:

project.plot_meas_vs_calc(expt_name='d20', x_min=41, x_max=54, show_residual=True)

project.plot_meas_vs_calc(expt_name='d20', x_min=41, x_max=54, show_residual=True)

Summary¶

This final section shows how to review the results of the analysis.

Show Project Summary¶

In [32]:

project.summary.show_report()

project.summary.show_report()

PROJECT INFO
━━━━━━━━━━━━

Title

Untitled Project

CRYSTALLOGRAPHIC DATA
━━━━━━━━━━━━━━━━━━━━━

Phase datablock

🧩 cosio

Space group

P n m a

Cell parameters

	Parameter	Value
1	a	10.30896
2	b	6.00391
3	c	4.78675
4	alpha	90.00000
5	beta	90.00000
6	gamma	90.00000

Atom sites

	label	type	x	y	z	occ	Biso
1	Co1	Co	0.00000	0.00000	0.00000	1.00000	0.27703
2	Co2	Co	0.27938	0.25000	0.98470	1.00000	0.27703
3	Si	Si	0.09371	0.25000	0.42904	1.00000	0.35192
4	O1	O	0.09107	0.25000	0.77140	1.00000	0.64570
5	O2	O	0.44825	0.25000	0.21671	1.00000	0.57784
6	O3	O	0.16358	0.03165	0.28005	1.00000	0.83894

EXPERIMENTS
━━━━━━━━━━━

Experiment datablock

🔬 d20

Experiment type

SampleFormEnum.POWDER, RadiationProbeEnum.NEUTRON, BeamModeEnum.CONSTANT_WAVELENGTH

Wavelength

1.87000

2θ offset

0.28844

Profile type

PeakProfileTypeEnum.PSEUDO_VOIGT

Peak broadening (Gaussian)

	Parameter	Value
1	U	0.24233
2	V	-0.52880
3	W	0.38395

Peak broadening (Lorentzian)

	Parameter	Value
1	X	0.00000
2	Y	0.01591

FITTING
━━━━━━━

Calculation engine

cryspy

Minimization engine

lmfit (leastsq)

Fit quality

	metric	value
1	Goodness-of-fit (reduced χ²)	4.56