Structure Refinement: PbSO4, NPD + XRD¶

This example demonstrates a more advanced use of the EasyDiffraction library by explicitly creating and configuring sample models and experiments before adding them to a project. It could be more suitable for users who are interested in creating custom workflows. This tutorial provides minimal explanation and is intended for users already familiar with EasyDiffraction.

The tutorial covers a Rietveld refinement of PbSO4 crystal structure based on the joint fit of both X-ray and neutron diffraction data.

Import Library¶

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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¶

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model = SampleModel('pbso4')
model = SampleModel('pbso4')

Set Space Group¶

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

Set Unit Cell¶

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model.cell.length_a = 8.47
model.cell.length_b = 5.39
model.cell.length_c = 6.95
model.cell.length_a = 8.47
model.cell.length_b = 5.39
model.cell.length_c = 6.95

Set Atom Sites¶

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model.atom_sites.add('Pb', 'Pb', 0.1876, 0.25, 0.167, b_iso=1.37)
model.atom_sites.add('S', 'S', 0.0654, 0.25, 0.684, b_iso=0.3777)
model.atom_sites.add('O1', 'O', 0.9082, 0.25, 0.5954, b_iso=1.9764)
model.atom_sites.add('O2', 'O', 0.1935, 0.25, 0.5432, b_iso=1.4456)
model.atom_sites.add('O3', 'O', 0.0811, 0.0272, 0.8086, b_iso=1.2822)
model.atom_sites.add('Pb', 'Pb', 0.1876, 0.25, 0.167, b_iso=1.37)
model.atom_sites.add('S', 'S', 0.0654, 0.25, 0.684, b_iso=0.3777)
model.atom_sites.add('O1', 'O', 0.9082, 0.25, 0.5954, b_iso=1.9764)
model.atom_sites.add('O2', 'O', 0.1935, 0.25, 0.5432, b_iso=1.4456)
model.atom_sites.add('O3', 'O', 0.0811, 0.0272, 0.8086, b_iso=1.2822)

Define Experiments¶

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

Experiment 1: npd¶

Download Data¶

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download_from_repository('d1a_pbso4.dat', destination='data')
download_from_repository('d1a_pbso4.dat', destination='data')

⚠️ Warning
File 'data/d1a_pbso4.dat' already exists and will not be overwritten.

Create Experiment¶

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expt1 = Experiment('npd', radiation_probe='neutron', data_path='data/d1a_pbso4.dat')
expt1 = Experiment('npd', radiation_probe='neutron', data_path='data/d1a_pbso4.dat')

Data loaded successfully
Experiment 🔬 'npd'. Number of data points: 1801

Set Instrument¶

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expt1.instrument.setup_wavelength = 1.91
expt1.instrument.calib_twotheta_offset = -0.1406
expt1.instrument.setup_wavelength = 1.91
expt1.instrument.calib_twotheta_offset = -0.1406

Set Peak Profile¶

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expt1.peak.broad_gauss_u = 0.139
expt1.peak.broad_gauss_v = -0.412
expt1.peak.broad_gauss_w = 0.386
expt1.peak.broad_lorentz_x = 0
expt1.peak.broad_lorentz_y = 0.088
expt1.peak.broad_gauss_u = 0.139
expt1.peak.broad_gauss_v = -0.412
expt1.peak.broad_gauss_w = 0.386
expt1.peak.broad_lorentz_x = 0
expt1.peak.broad_lorentz_y = 0.088

Set Background¶

Select the background type.

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expt1.background_type = 'line-segment'
expt1.background_type = 'line-segment'

Background type for experiment 'npd' changed to
line-segment

Add background points.

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for x, y in [
    (11.0, 206.1624),
    (15.0, 194.75),
    (20.0, 194.505),
    (30.0, 188.4375),
    (50.0, 207.7633),
    (70.0, 201.7002),
    (120.0, 244.4525),
    (153.0, 226.0595),
]:
    expt1.background.add(x, y)
for x, y in [
    (11.0, 206.1624),
    (15.0, 194.75),
    (20.0, 194.505),
    (30.0, 188.4375),
    (50.0, 207.7633),
    (70.0, 201.7002),
    (120.0, 244.4525),
    (153.0, 226.0595),
]:
    expt1.background.add(x, y)

Set Linked Phases¶

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expt1.linked_phases.add('pbso4', scale=1.5)
expt1.linked_phases.add('pbso4', scale=1.5)

Experiment 2: xrd¶

Download Data¶

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download_from_repository('lab_pbso4.dat', destination='data')
download_from_repository('lab_pbso4.dat', destination='data')

⚠️ Warning
File 'data/lab_pbso4.dat' already exists and will not be overwritten.

Create Experiment¶

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expt2 = Experiment('xrd', radiation_probe='xray', data_path='data/lab_pbso4.dat')
expt2 = Experiment('xrd', radiation_probe='xray', data_path='data/lab_pbso4.dat')

Data loaded successfully
Experiment 🔬 'xrd'. Number of data points: 3601

Set Instrument¶

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expt2.instrument.setup_wavelength = 1.540567
expt2.instrument.calib_twotheta_offset = -0.05181
expt2.instrument.setup_wavelength = 1.540567
expt2.instrument.calib_twotheta_offset = -0.05181

Set Peak Profile¶

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expt2.peak.broad_gauss_u = 0.304138
expt2.peak.broad_gauss_v = -0.112622
expt2.peak.broad_gauss_w = 0.021272
expt2.peak.broad_lorentz_x = 0
expt2.peak.broad_lorentz_y = 0.057691
expt2.peak.broad_gauss_u = 0.304138
expt2.peak.broad_gauss_v = -0.112622
expt2.peak.broad_gauss_w = 0.021272
expt2.peak.broad_lorentz_x = 0
expt2.peak.broad_lorentz_y = 0.057691

Set Background¶

Select background type.

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expt2.background_type = 'chebyshev polynomial'
expt2.background_type = 'chebyshev polynomial'

Background type for experiment 'xrd' changed to
chebyshev polynomial

Add background points.

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for x, y in [
    (0, 119.195),
    (1, 6.221),
    (2, -45.725),
    (3, 8.119),
    (4, 54.552),
    (5, -20.661),
]:
    expt2.background.add(x, y)
for x, y in [
    (0, 119.195),
    (1, 6.221),
    (2, -45.725),
    (3, 8.119),
    (4, 54.552),
    (5, -20.661),
]:
    expt2.background.add(x, y)

Set Linked Phases¶

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expt2.linked_phases.add('pbso4', scale=0.001)
expt2.linked_phases.add('pbso4', scale=0.001)

Define Project¶

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

Create Project¶

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project = Project()
project = Project()

Add Sample Model¶

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project.sample_models.add(model)
project.sample_models.add(model)

Add Experiments¶

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project.experiments.add(expt1)
project.experiments.add(expt2)
project.experiments.add(expt1)
project.experiments.add(expt2)

Perform Analysis¶

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

Set Calculator¶

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project.analysis.current_calculator = 'cryspy'
project.analysis.current_calculator = 'cryspy'

Current calculator changed to
cryspy

Set Fit Mode¶

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project.analysis.fit_mode = 'joint'
project.analysis.fit_mode = 'joint'

Current fit mode changed to
joint

Set Minimizer¶

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

Current minimizer changed to
lmfit (leastsq)

Set Fitting Parameters¶

Set sample model parameters to be optimized.

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

Set experiment parameters to be optimized.

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expt1.linked_phases['pbso4'].scale.free = True

expt1.instrument.calib_twotheta_offset.free = True

expt1.peak.broad_gauss_u.free = True
expt1.peak.broad_gauss_v.free = True
expt1.peak.broad_gauss_w.free = True
expt1.peak.broad_lorentz_y.free = True
expt1.linked_phases['pbso4'].scale.free = True

expt1.instrument.calib_twotheta_offset.free = True

expt1.peak.broad_gauss_u.free = True
expt1.peak.broad_gauss_v.free = True
expt1.peak.broad_gauss_w.free = True
expt1.peak.broad_lorentz_y.free = True

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expt2.linked_phases['pbso4'].scale.free = True

expt2.instrument.calib_twotheta_offset.free = True

expt2.peak.broad_gauss_u.free = True
expt2.peak.broad_gauss_v.free = True
expt2.peak.broad_gauss_w.free = True
expt2.peak.broad_lorentz_y.free = True

for term in expt2.background:
    term.coef.free = True
expt2.linked_phases['pbso4'].scale.free = True

expt2.instrument.calib_twotheta_offset.free = True

expt2.peak.broad_gauss_u.free = True
expt2.peak.broad_gauss_v.free = True
expt2.peak.broad_gauss_w.free = True
expt2.peak.broad_lorentz_y.free = True

for term in expt2.background:
    term.coef.free = True

Perform Fit¶

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project.analysis.fit()
project.analysis.fit()

Using all experiments 🔬 ['npd', 'xrd'] for 'joint' fitting
🚀 Starting fit process with 'lmfit (leastsq)'...
📈 Goodness-of-fit (reduced χ²) change:

iteration	χ²	improvement [%]
1	37.01
25	16.30	56.0% ↓
136	16.21

🏆 Best goodness-of-fit (reduced χ²) is 16.21 at iteration 135
✅ Fitting complete.

Fit results
✅ Success: True
⏱️ Fitting time: 7.47 seconds
📏 Goodness-of-fit (reduced χ²): 16.21
📏 R-factor (Rf): 13.18%
📏 R-factor squared (Rf²): 17.19%
📏 Weighted R-factor (wR): 17.09%
📈 Fitted parameters:

	datablock	category	entry	parameter	start	fitted	uncertainty	units	change
1	pbso4	cell		length_a	8.4700	8.4693	0.0002	Å	0.01 % ↓
2	pbso4	cell		length_b	5.3900	5.3912	0.0001	Å	0.02 % ↑
3	pbso4	cell		length_c	6.9500	6.9504	0.0002	Å	0.01 % ↑
4	npd	instrument		twotheta_offset	-0.1406	-0.1385	0.0019	deg	1.47 % ↓
5	npd	linked_phases	pbso4	scale	1.5000	1.4622	0.0049		2.52 % ↓
6	npd	peak		broad_gauss_u	0.1390	0.2901	0.0350	deg²	108.67 % ↑
7	npd	peak		broad_gauss_v	-0.4120	-0.6390	0.0531	deg²	55.10 % ↑
8	npd	peak		broad_gauss_w	0.3860	0.4552	0.0186	deg²	17.92 % ↑
9	npd	peak		broad_lorentz_y	0.0880	0.0931	0.0034	deg	5.76 % ↑
10	xrd	background	0	chebyshev_coef	119.1950	85.8587	1.3293		27.97 % ↓
11	xrd	background	1	chebyshev_coef	6.2210	-19.3112	1.8805		410.42 % ↓
12	xrd	background	2	chebyshev_coef	-45.7250	5.3658	1.7778		111.74 % ↓
13	xrd	background	3	chebyshev_coef	8.1190	11.8842	1.7472		46.37 % ↑
14	xrd	background	4	chebyshev_coef	54.5520	18.5560	1.5670		65.98 % ↓
15	xrd	background	5	chebyshev_coef	-20.6610	-6.4093	1.5359		68.98 % ↓
16	xrd	instrument		twotheta_offset	-0.0518	-0.0637	0.0009	deg	22.93 % ↑
17	xrd	linked_phases	pbso4	scale	0.0010	0.0010	0.0000		1.52 % ↓
18	xrd	peak		broad_gauss_u	0.3041	0.5724	0.0196	deg²	88.21 % ↑
19	xrd	peak		broad_gauss_v	-0.1126	-0.2799	0.0142	deg²	148.57 % ↑
20	xrd	peak		broad_gauss_w	0.0213	0.0452	0.0024	deg²	112.27 % ↑
21	xrd	peak		broad_lorentz_y	0.0577	0.0587	0.0014	deg	1.75 % ↑

Plot Measured vs Calculated¶

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project.plot_meas_vs_calc(expt_name='npd', x_min=35.5, x_max=38.3, show_residual=True)
project.plot_meas_vs_calc(expt_name='npd', x_min=35.5, x_max=38.3, show_residual=True)

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project.plot_meas_vs_calc(expt_name='xrd', x_min=29.0, x_max=30.4, show_residual=True)
project.plot_meas_vs_calc(expt_name='xrd', x_min=29.0, x_max=30.4, show_residual=True)