Structure Refinement: Si, SEPD¶

This example demonstrates a Rietveld refinement of Si crystal structure using time-of-flight neutron powder diffraction data from SEPD at Argonne.

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='si')

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

Set Space Group¶

In [4]:

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

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

Set Unit Cell¶

In [5]:

model.cell.length_a = 5.431

model.cell.length_a = 5.431

Set Atom Sites¶

In [6]:

model.atom_sites.add_from_args(
    label='Si',
    type_symbol='Si',
    fract_x=0.125,
    fract_y=0.125,
    fract_z=0.125,
    b_iso=0.5,
)

model.atom_sites.add_from_args( label='Si', type_symbol='Si', fract_x=0.125, fract_y=0.125, fract_z=0.125, b_iso=0.5, )

Define Experiment¶

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

Download Measured Data¶

In [7]:

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

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

Downloading...

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

Create Experiment¶

In [8]:

expt = ExperimentFactory.create(
    name='sepd', data_path='data/sepd_si.xye', beam_mode='time-of-flight'
)

expt = ExperimentFactory.create( name='sepd', data_path='data/sepd_si.xye', beam_mode='time-of-flight' )

Data loaded successfully

Experiment 🔬 'sepd'. Number of data points: 5600

Set Instrument¶

In [9]:

expt.instrument.setup_twotheta_bank = 144.845
expt.instrument.calib_d_to_tof_offset = 0.0
expt.instrument.calib_d_to_tof_linear = 7476.91
expt.instrument.calib_d_to_tof_quad = -1.54

expt.instrument.setup_twotheta_bank = 144.845 expt.instrument.calib_d_to_tof_offset = 0.0 expt.instrument.calib_d_to_tof_linear = 7476.91 expt.instrument.calib_d_to_tof_quad = -1.54

Set Peak Profile¶

In [10]:

expt.peak_profile_type = 'pseudo-voigt * ikeda-carpenter'
expt.peak.broad_gauss_sigma_0 = 3.0
expt.peak.broad_gauss_sigma_1 = 40.0
expt.peak.broad_gauss_sigma_2 = 2.0
expt.peak.broad_mix_beta_0 = 0.04221
expt.peak.broad_mix_beta_1 = 0.00946

expt.peak_profile_type = 'pseudo-voigt * ikeda-carpenter' expt.peak.broad_gauss_sigma_0 = 3.0 expt.peak.broad_gauss_sigma_1 = 40.0 expt.peak.broad_gauss_sigma_2 = 2.0 expt.peak.broad_mix_beta_0 = 0.04221 expt.peak.broad_mix_beta_1 = 0.00946

Peak profile type for experiment 'sepd' changed to

pseudo-voigt * ikeda-carpenter

Set Peak Asymmetry¶

In [11]:

expt.peak.asym_alpha_0 = 0.0
expt.peak.asym_alpha_1 = 0.5971

expt.peak.asym_alpha_0 = 0.0 expt.peak.asym_alpha_1 = 0.5971

Set Background¶

In [12]:

expt.background_type = 'line-segment'
for x in range(0, 35000, 5000):
    expt.background.add_from_args(x=x, y=200)

expt.background_type = 'line-segment' for x in range(0, 35000, 5000): expt.background.add_from_args(x=x, y=200)

Background type for experiment 'sepd' changed to

line-segment

Set Linked Phases¶

In [13]:

expt.linked_phases.add_from_args(id='si', scale=10.0)

expt.linked_phases.add_from_args(id='si', scale=10.0)

Define Project¶

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

Create Project¶

In [14]:

project = Project()

project = Project()

Set Plotting Engine¶

In [15]:

project.plotter.engine = 'plotly'

project.plotter.engine = 'plotly'

Add Sample Model¶

In [16]:

project.sample_models.add(model)

project.sample_models.add(model)

Add Experiment¶

In [17]:

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 [18]:

project.analysis.current_calculator = 'cryspy'

project.analysis.current_calculator = 'cryspy'

Current calculator changed to

cryspy

Set Minimizer¶

In [19]:

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

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

Current minimizer changed to

lmfit (leastsq)

Plot Measured vs Calculated¶

In [20]:

project.plot_meas_vs_calc(expt_name='sepd', show_residual=True)
project.plot_meas_vs_calc(expt_name='sepd', x_min=23200, x_max=23700, show_residual=True)

project.plot_meas_vs_calc(expt_name='sepd', show_residual=True) project.plot_meas_vs_calc(expt_name='sepd', x_min=23200, x_max=23700, show_residual=True)

Perform Fit 1/5¶

Set parameters to be refined.

In [21]:

model.cell.length_a.free = True

expt.linked_phases['si'].scale.free = True
expt.instrument.calib_d_to_tof_offset.free = True

model.cell.length_a.free = True expt.linked_phases['si'].scale.free = True expt.instrument.calib_d_to_tof_offset.free = True

Show free parameters after selection.

In [22]:

project.analysis.show_free_params()

project.analysis.show_free_params()

Free parameters for both sample models (🧩 data blocks) and experiments (🔬 data blocks)

	datablock	category	entry	parameter	value	uncertainty	min	max	units
1	si	cell		length_a	5.43100		-inf	inf	Å
2	sepd	linked_phases	si	scale	10.00000		-inf	inf
3	sepd	instrument		d_to_tof_offset	0.00000		-inf	inf	µs

Run Fitting¶

In [23]:

project.analysis.fit()

project.analysis.fit()

Using experiment 🔬 'sepd' for 'single' fitting

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

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

	iteration	χ²	improvement [%]
1	1	113.06
2	7	72.20	36.1% ↓
3	11	66.76	7.5% ↓
4	30	66.72

🏆 Best goodness-of-fit (reduced χ²) is 66.72 at iteration 26

✅ Fitting complete.

Fit results

✅ Success: True

⏱️ Fitting time: 8.53 seconds

📏 Goodness-of-fit (reduced χ²): 66.72

📏 R-factor (Rf): 23.08%

📏 R-factor squared (Rf²): 12.55%

📏 Weighted R-factor (wR): 12.51%

📈 Fitted parameters:

	datablock	category	entry	parameter	start	fitted	uncertainty	units	change
1	si	cell	None	length_a	5.4310	5.4314	0.0002	Å	0.01 % ↑
2	sepd	linked_phases	si	scale	10.0000	13.3619	0.1153		33.62 % ↑
3	sepd	instrument	None	d_to_tof_offset	0.0000	-9.2543	0.2503	µs	N/A

Plot Measured vs Calculated¶

In [24]:

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

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

In [25]:

project.plot_meas_vs_calc(expt_name='sepd', x_min=23200, x_max=23700, show_residual=True)

project.plot_meas_vs_calc(expt_name='sepd', x_min=23200, x_max=23700, show_residual=True)

Perform Fit 2/5¶

Set more parameters to be refined.

In [26]:

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

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

Show free parameters after selection.

In [27]:

project.analysis.show_free_params()

project.analysis.show_free_params()

Free parameters for both sample models (🧩 data blocks) and experiments (🔬 data blocks)

	datablock	category	entry	parameter	value	uncertainty	min	max	units
1	si	cell		length_a	5.43135	0.00018	-inf	inf	Å
2	sepd	linked_phases	si	scale	13.36187	0.11531	-inf	inf
3	sepd	instrument		d_to_tof_offset	-9.25432	0.25033	-inf	inf	µs
4	sepd	background	0	y	200.00000		-inf	inf
5	sepd	background	5000	y	200.00000		-inf	inf
6	sepd	background	10000	y	200.00000		-inf	inf
7	sepd	background	15000	y	200.00000		-inf	inf
8	sepd	background	20000	y	200.00000		-inf	inf
9	sepd	background	25000	y	200.00000		-inf	inf
10	sepd	background	30000	y	200.00000		-inf	inf

Run Fitting¶

In [28]:

project.analysis.fit()

project.analysis.fit()

Using experiment 🔬 'sepd' for 'single' fitting

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

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

	iteration	χ²	improvement [%]
1	1	66.80
2	14	3.38	94.9% ↓
3	48	3.38

🏆 Best goodness-of-fit (reduced χ²) is 3.38 at iteration 47

✅ Fitting complete.

Fit results

✅ Success: True

⏱️ Fitting time: 11.87 seconds

📏 Goodness-of-fit (reduced χ²): 3.38

📏 R-factor (Rf): 9.29%

📏 R-factor squared (Rf²): 6.33%

📏 Weighted R-factor (wR): 5.95%

📈 Fitted parameters:

	datablock	category	entry	parameter	start	fitted	uncertainty	units	change
1	si	cell	None	length_a	5.4314	5.4314	0.0000	Å	0.00 % ↑
2	sepd	linked_phases	si	scale	13.3619	14.6317	0.0265		9.50 % ↑
3	sepd	instrument	None	d_to_tof_offset	-9.2543	-9.2534	0.0515	µs	0.01 % ↓
4	sepd	background	0	y	200.0000	268.6002	0.9745		34.30 % ↑
5	sepd	background	5000	y	200.0000	144.7589	0.4071		27.62 % ↓
6	sepd	background	10000	y	200.0000	120.0247	0.4282		39.99 % ↓
7	sepd	background	15000	y	200.0000	135.8494	0.8169		32.08 % ↓
8	sepd	background	20000	y	200.0000	132.6887	1.4317		33.66 % ↓
9	sepd	background	25000	y	200.0000	175.1775	2.8755		12.41 % ↓
10	sepd	background	30000	y	200.0000	180.4556	5.8525		9.77 % ↓

Plot Measured vs Calculated¶

In [29]:

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

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

In [30]:

project.plot_meas_vs_calc(expt_name='sepd', x_min=23200, x_max=23700, show_residual=True)

project.plot_meas_vs_calc(expt_name='sepd', x_min=23200, x_max=23700, show_residual=True)

Perform Fit 3/5¶

Fix background points.

In [31]:

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

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

Set more parameters to be refined.

In [32]:

expt.peak.broad_gauss_sigma_0.free = True
expt.peak.broad_gauss_sigma_1.free = True
expt.peak.broad_gauss_sigma_2.free = True

expt.peak.broad_gauss_sigma_0.free = True expt.peak.broad_gauss_sigma_1.free = True expt.peak.broad_gauss_sigma_2.free = True

Show free parameters after selection.

In [33]:

project.analysis.show_free_params()

project.analysis.show_free_params()

Free parameters for both sample models (🧩 data blocks) and experiments (🔬 data blocks)

	datablock	category	entry	parameter	value	uncertainty	min	max	units
1	si	cell		length_a	5.43137	0.00004	-inf	inf	Å
2	sepd	peak		gauss_sigma_0	3.00000		-inf	inf	µs²
3	sepd	peak		gauss_sigma_1	40.00000		-inf	inf	µs/Å
4	sepd	peak		gauss_sigma_2	2.00000		-inf	inf	µs²/Ų
5	sepd	linked_phases	si	scale	14.63167	0.02651	-inf	inf
6	sepd	instrument		d_to_tof_offset	-9.25336	0.05153	-inf	inf	µs

Run Fitting¶

In [34]:

project.analysis.fit()

project.analysis.fit()

Using experiment 🔬 'sepd' for 'single' fitting

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

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

	iteration	χ²	improvement [%]
1	1	3.38
2	10	3.21	5.0% ↓
3	39	3.21

🏆 Best goodness-of-fit (reduced χ²) is 3.21 at iteration 38

✅ Fitting complete.

Fit results

✅ Success: True

⏱️ Fitting time: 8.44 seconds

📏 Goodness-of-fit (reduced χ²): 3.21

📏 R-factor (Rf): 8.99%

📏 R-factor squared (Rf²): 5.52%

📏 Weighted R-factor (wR): 4.88%

📈 Fitted parameters:

	datablock	category	entry	parameter	start	fitted	uncertainty	units	change
1	si	cell	None	length_a	5.4314	5.4314	0.0000	Å	0.00 % ↑
2	sepd	peak	None	gauss_sigma_0	3.0000	5.7727	0.4206	µs²	92.42 % ↑
3	sepd	peak	None	gauss_sigma_1	40.0000	44.2827	0.7966	µs/Å	10.71 % ↑
4	sepd	peak	None	gauss_sigma_2	2.0000	1.2962	0.1680	µs²/Ų	35.19 % ↓
5	sepd	linked_phases	si	scale	14.6317	14.7057	0.0257		0.51 % ↑
6	sepd	instrument	None	d_to_tof_offset	-9.2534	-9.2506	0.0546	µs	0.03 % ↓

Plot Measured vs Calculated¶

In [35]:

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

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

In [36]:

project.plot_meas_vs_calc(expt_name='sepd', x_min=23200, x_max=23700, show_residual=True)

project.plot_meas_vs_calc(expt_name='sepd', x_min=23200, x_max=23700, show_residual=True)

Perform Fit 4/5¶

Set more parameters to be refined.

In [37]:

model.atom_sites['Si'].b_iso.free = True

model.atom_sites['Si'].b_iso.free = True

Show free parameters after selection.

In [38]:

project.analysis.show_free_params()

project.analysis.show_free_params()

Free parameters for both sample models (🧩 data blocks) and experiments (🔬 data blocks)

	datablock	category	entry	parameter	value	uncertainty	min	max	units
1	si	cell		length_a	5.43143	0.00004	-inf	inf	Å
2	si	atom_site	Si	b_iso	0.50000		-inf	inf	Ų
3	sepd	peak		gauss_sigma_0	5.77274	0.42055	-inf	inf	µs²
4	sepd	peak		gauss_sigma_1	44.28265	0.79664	-inf	inf	µs/Å
5	sepd	peak		gauss_sigma_2	1.29621	0.16795	-inf	inf	µs²/Ų
6	sepd	linked_phases	si	scale	14.70568	0.02567	-inf	inf
7	sepd	instrument		d_to_tof_offset	-9.25065	0.05458	-inf	inf	µs

Run Fitting¶

In [39]:

project.analysis.fit()

project.analysis.fit()

Using experiment 🔬 'sepd' for 'single' fitting

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

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

	iteration	χ²	improvement [%]
1	1	3.21
2	44	3.19

🏆 Best goodness-of-fit (reduced χ²) is 3.19 at iteration 43

✅ Fitting complete.

Fit results

✅ Success: True

⏱️ Fitting time: 10.04 seconds

📏 Goodness-of-fit (reduced χ²): 3.19

📏 R-factor (Rf): 9.01%

📏 R-factor squared (Rf²): 5.35%

📏 Weighted R-factor (wR): 4.47%

📈 Fitted parameters:

	datablock	category	entry	parameter	start	fitted	uncertainty	units	change
1	si	cell	None	length_a	5.4314	5.4314	0.0000	Å	0.00 % ↑
2	si	atom_site	Si	b_iso	0.5000	0.5201	0.0034	Ų	4.02 % ↑
3	sepd	peak	None	gauss_sigma_0	5.7727	4.8233	0.4394	µs²	16.45 % ↓
4	sepd	peak	None	gauss_sigma_1	44.2827	45.2122	0.8016	µs/Å	2.10 % ↑
5	sepd	peak	None	gauss_sigma_2	1.2962	1.1931	0.1667	µs²/Ų	7.95 % ↓
6	sepd	linked_phases	si	scale	14.7057	14.8594	0.0366		1.05 % ↑
7	sepd	instrument	None	d_to_tof_offset	-9.2506	-9.2823	0.0548	µs	0.34 % ↑

Plot Measured vs Calculated¶

In [40]:

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

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

In [41]:

project.plot_meas_vs_calc(expt_name='sepd', x_min=23200, x_max=23700, show_residual=True)

project.plot_meas_vs_calc(expt_name='sepd', x_min=23200, x_max=23700, show_residual=True)