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 Experiment
from easydiffraction import Project
from easydiffraction import SampleModel
from easydiffraction import download_from_repository

from easydiffraction import Experiment from easydiffraction import Project from easydiffraction import SampleModel from easydiffraction import download_from_repository

⚠️ 'pycrysfml' module not found. This calculation engine will not be available.
✅ 'cryspy' calculation engine is successfully imported.
✅ 'pdffit' calculation engine is successfully imported.

Define Sample Model¶

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

Create Sample Model¶

In [3]:

model = SampleModel('cosio')

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

model.atom_sites.add('Co1', 'Co', 0, 0, 0, wyckoff_letter='a', b_iso=0.5) model.atom_sites.add('Co2', 'Co', 0.279, 0.25, 0.985, wyckoff_letter='c', b_iso=0.5) model.atom_sites.add('Si', 'Si', 0.094, 0.25, 0.429, wyckoff_letter='c', b_iso=0.5) model.atom_sites.add('O1', 'O', 0.091, 0.25, 0.771, wyckoff_letter='c', b_iso=0.5) model.atom_sites.add('O2', 'O', 0.448, 0.25, 0.217, wyckoff_letter='c', b_iso=0.5) model.atom_sites.add('O3', 'O', 0.164, 0.032, 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

data_cosio

_space_group.IT_coordinate_system_code abc

_space_group.name_H-M_alt "P n m a"

_cell.angle_alpha 90.0

_cell.angle_beta 90.0

_cell.angle_gamma 90.0

_cell.length_a 10.3

_cell.length_b 6.0

_cell.length_c 4.8

loop_

_atom_site.ADP_type

_atom_site.B_iso_or_equiv

_atom_site.fract_x

_atom_site.fract_y

_atom_site.fract_z

_atom_site.label

_atom_site.occupancy

_atom_site.type_symbol

_atom_site.Wyckoff_letter

Biso 0.5 0 0 0 Co1 1.0 Co a

Biso 0.5 0.279 0.25 0.985 Co2 1.0 Co c

Biso 0.5 0.094 0.25 0.429 Si 1.0 Si c

Biso 0.5 0.091 0.25 0.771 O1 1.0 O c

Biso 0.5 0.448 0.25 0.217 O2 1.0 O c

Biso 0.5 0.164 0.032 0.28 O3 1.0 O d

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

data_cosio

_space_group.IT_coordinate_system_code abc

_space_group.name_H-M_alt "P n m a"

_cell.angle_alpha 90

_cell.angle_beta 90

_cell.angle_gamma 90

_cell.length_a 10.3

_cell.length_b 6.0

_cell.length_c 4.8

loop_

_atom_site.ADP_type

_atom_site.B_iso_or_equiv

_atom_site.fract_x

_atom_site.fract_y

_atom_site.fract_z

_atom_site.label

_atom_site.occupancy

_atom_site.type_symbol

_atom_site.Wyckoff_letter

Biso 0.5 0.0 0.0 0.0 Co1 1.0 Co a

Biso 0.5 0.279 0.25 0.985 Co2 1.0 Co c

Biso 0.5 0.094 0.25 0.429 Si 1.0 Si c

Biso 0.5 0.091 0.25 0.771 O1 1.0 O c

Biso 0.5 0.448 0.25 0.217 O2 1.0 O c

Biso 0.5 0.164 0.032 0.28 O3 1.0 O d

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')

⚠️ Warning
File 'data/co2sio4_d20.xye' already exists and will not be overwritten.

Create Experiment¶

In [11]:

expt = Experiment('d20', data_path='data/co2sio4_d20.xye')

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

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

Set Linked Phases¶

In [15]:

expt.linked_phases.add('cosio', scale=1.0)

expt.linked_phases.add('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'

Current plotter changed to
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(
    label='biso_Co1',
    param_uid=project.sample_models['cosio'].atom_sites['Co1'].b_iso.uid,
)
project.analysis.aliases.add(
    label='biso_Co2',
    param_uid=project.sample_models['cosio'].atom_sites['Co2'].b_iso.uid,
)

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

Set constraints.

In [27]:

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

project.analysis.constraints.add( 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	423.20
43	71.71	83.1% ↓
83	40.43	43.6% ↓
123	17.05	57.8% ↓
163	10.78	36.8% ↓
203	9.26	14.1% ↓
243	8.08	12.7% ↓
283	5.66	30.0% ↓
323	4.67	17.4% ↓
363	4.57	2.3% ↓
564	4.56

🏆 Best goodness-of-fit (reduced χ²) is 4.56 at iteration 562
✅ Fitting complete.

Fit results
✅ Success: True
⏱️ Fitting time: 10.80 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	atom_sites	Co1	b_iso	0.5000	0.2770	0.0802	Ų	44.59 % ↓
2	cosio	atom_sites	Co2	fract_x	0.2790	0.2794	0.0007		0.14 % ↑
3	cosio	atom_sites	Co2	fract_z	0.9850	0.9847	0.0015		0.03 % ↓
4	cosio	atom_sites	Si	b_iso	0.5000	0.3519	0.0625	Ų	29.62 % ↓
5	cosio	atom_sites	Si	fract_x	0.0940	0.0937	0.0004		0.31 % ↓
6	cosio	atom_sites	Si	fract_z	0.4290	0.4290	0.0008		0.01 % ↑
7	cosio	atom_sites	O1	b_iso	0.5000	0.6457	0.0581	Ų	29.14 % ↑
8	cosio	atom_sites	O1	fract_x	0.0910	0.0911	0.0003		0.08 % ↑
9	cosio	atom_sites	O1	fract_z	0.7710	0.7714	0.0006		0.05 % ↑
10	cosio	atom_sites	O2	b_iso	0.5000	0.5778	0.0582	Ų	15.57 % ↑
11	cosio	atom_sites	O2	fract_x	0.4480	0.4482	0.0003		0.06 % ↑
12	cosio	atom_sites	O2	fract_z	0.2170	0.2167	0.0007		0.14 % ↓
13	cosio	atom_sites	O3	b_iso	0.5000	0.8389	0.0485	Ų	67.79 % ↑
14	cosio	atom_sites	O3	fract_x	0.1640	0.1636	0.0002		0.26 % ↓
15	cosio	atom_sites	O3	fract_y	0.0320	0.0317	0.0003		1.08 % ↓
16	cosio	atom_sites	O3	fract_z	0.2800	0.2800	0.0005		0.02 % ↑
17	cosio	cell		length_a	10.3000	10.3090	0.0003	Å	0.09 % ↑
18	cosio	cell		length_b	6.0000	6.0039	0.0002	Å	0.07 % ↑
19	cosio	cell		length_c	4.8000	4.7868	0.0001	Å	0.28 % ↓
20	d20	background	8	y	500.0000	608.6305	14.5414		21.73 % ↑
21	d20	background	9	y	500.0000	580.6214	9.7278		16.12 % ↑
22	d20	background	10	y	500.0000	562.8991	9.2422		12.58 % ↑
23	d20	background	11	y	500.0000	540.2882	8.7695		8.06 % ↑
24	d20	background	12	y	500.0000	519.8428	6.0496		3.97 % ↑
25	d20	background	15	y	500.0000	507.3838	3.4846		1.48 % ↑
26	d20	background	25	y	500.0000	463.0794	3.1898		7.38 % ↓
27	d20	background	30	y	500.0000	434.2455	2.3904		13.15 % ↓
28	d20	background	50	y	500.0000	450.8166	2.2505		9.84 % ↓
29	d20	background	70	y	500.0000	430.3828	2.0250		13.92 % ↓
30	d20	background	90	y	500.0000	413.7132	2.2594		17.26 % ↓
31	d20	background	110	y	500.0000	361.2599	2.0915		27.75 % ↓
32	d20	background	130	y	500.0000	291.0612	1.9671		41.79 % ↓
33	d20	background	150	y	500.0000	238.7749	2.9764		52.25 % ↓
34	d20	instrument		twotheta_offset	0.1000	0.2884	0.0020	deg	188.44 % ↑
35	d20	linked_phases	cosio	scale	1.0000	1.1957	0.0106		19.57 % ↑
36	d20	peak		broad_gauss_u	0.3000	0.2423	0.0068	deg²	19.22 % ↓
37	d20	peak		broad_gauss_v	-0.5000	-0.5288	0.0145	deg²	5.76 % ↑
38	d20	peak		broad_gauss_w	0.4000	0.3840	0.0090	deg²	4.01 % ↓
39	d20	peak		broad_lorentz_y	0.0000	0.0159	0.0045	deg	N/A

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

alpha	90.00000
beta	90.00000
gamma	90.00000
a	10.30896
b	6.00391
c	4.78675

Atom sites

Label	Type	fract_x	fract_y	fract_z	Occupancy	B_iso
Co1	Co	0.00000	0.00000	0.00000	1.00000	0.27703
Co2	Co	0.27938	0.25000	0.98470	1.00000	0.27703
Si	Si	0.09371	0.25000	0.42904	1.00000	0.35192
O1	O	0.09107	0.25000	0.77140	1.00000	0.64570
O2	O	0.44825	0.25000	0.21671	1.00000	0.57784
O3	O	0.16358	0.03165	0.28005	1.00000	0.83894

*** EXPERIMENTS ***

Experiment datablock
🔬 d20

Experiment type
powder, neutron, constant wavelength

Wavelength
1.87000

2θ offset
0.28844

Profile type
pseudo-voigt

Peak broadening (Gaussian)

U	0.24233
V	-0.52880
W	0.38395

Peak broadening (Lorentzian)

X	0.00000
Y	0.01591

*** FITTING ***

Calculation engine
cryspy

Minimization engine
lmfit (leastsq)

Fit quality

Goodness-of-fit (reduced χ²)

4.56