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¶

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='pbso4')

model = SampleModelFactory.create(name='pbso4')

Set Space Group¶

In [4]:

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

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

Set Unit Cell¶

In [5]:

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¶

In [6]:

model.atom_sites.add_from_args(
    label='Pb',
    type_symbol='Pb',
    fract_x=0.1876,
    fract_y=0.25,
    fract_z=0.167,
    wyckoff_letter='c',
    b_iso=1.37,
)
model.atom_sites.add_from_args(
    label='S',
    type_symbol='S',
    fract_x=0.0654,
    fract_y=0.25,
    fract_z=0.684,
    wyckoff_letter='c',
    b_iso=0.3777,
)
model.atom_sites.add_from_args(
    label='O1',
    type_symbol='O',
    fract_x=0.9082,
    fract_y=0.25,
    fract_z=0.5954,
    wyckoff_letter='c',
    b_iso=1.9764,
)
model.atom_sites.add_from_args(
    label='O2',
    type_symbol='O',
    fract_x=0.1935,
    fract_y=0.25,
    fract_z=0.5432,
    wyckoff_letter='c',
    b_iso=1.4456,
)
model.atom_sites.add_from_args(
    label='O3',
    type_symbol='O',
    fract_x=0.0811,
    fract_y=0.0272,
    fract_z=0.8086,
    wyckoff_letter='d',
    b_iso=1.2822,
)

model.atom_sites.add_from_args( label='Pb', type_symbol='Pb', fract_x=0.1876, fract_y=0.25, fract_z=0.167, wyckoff_letter='c', b_iso=1.37, ) model.atom_sites.add_from_args( label='S', type_symbol='S', fract_x=0.0654, fract_y=0.25, fract_z=0.684, wyckoff_letter='c', b_iso=0.3777, ) model.atom_sites.add_from_args( label='O1', type_symbol='O', fract_x=0.9082, fract_y=0.25, fract_z=0.5954, wyckoff_letter='c', b_iso=1.9764, ) model.atom_sites.add_from_args( label='O2', type_symbol='O', fract_x=0.1935, fract_y=0.25, fract_z=0.5432, wyckoff_letter='c', b_iso=1.4456, ) model.atom_sites.add_from_args( label='O3', type_symbol='O', fract_x=0.0811, fract_y=0.0272, fract_z=0.8086, wyckoff_letter='d', 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¶

In [7]:

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

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

Downloading...

File 'd1a_pbso4.dat' from 'easyscience/diffraction-lib'

Create Experiment¶

In [8]:

expt1 = ExperimentFactory.create(
    name='npd',
    data_path='data/d1a_pbso4.dat',
    radiation_probe='neutron',
)

expt1 = ExperimentFactory.create( name='npd', data_path='data/d1a_pbso4.dat', radiation_probe='neutron', )

Data loaded successfully

Experiment 🔬 'npd'. Number of data points: 1801

Set Instrument¶

In [9]:

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¶

In [10]:

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.

In [11]:

expt1.background_type = 'line-segment'

expt1.background_type = 'line-segment'

Background type for experiment 'npd' changed to

line-segment

Add background points.

In [12]:

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_from_args(x=x, y=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_from_args(x=x, y=y)

Set Linked Phases¶

In [13]:

expt1.linked_phases.add_from_args(id='pbso4', scale=1.5)

expt1.linked_phases.add_from_args(id='pbso4', scale=1.5)

Experiment 2: xrd¶

Download Data¶

In [14]:

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

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

Downloading...

File 'lab_pbso4.dat' from 'easyscience/diffraction-lib'

Create Experiment¶

In [15]:

expt2 = ExperimentFactory.create(
    name='xrd',
    data_path='data/lab_pbso4.dat',
    radiation_probe='xray',
)

expt2 = ExperimentFactory.create( name='xrd', data_path='data/lab_pbso4.dat', radiation_probe='xray', )

Data loaded successfully

Experiment 🔬 'xrd'. Number of data points: 3601

Set Instrument¶

In [16]:

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¶

In [17]:

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.

In [18]:

expt2.background_type = 'chebyshev polynomial'

expt2.background_type = 'chebyshev polynomial'

Background type for experiment 'xrd' changed to

chebyshev polynomial

Add background points.

In [19]:

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_from_args(order=x, coef=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_from_args(order=x, coef=y)

Set Linked Phases¶

In [20]:

expt2.linked_phases.add_from_args(id='pbso4', scale=0.001)

expt2.linked_phases.add_from_args(id='pbso4', scale=0.001)

Define Project¶

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

Create Project¶

In [21]:

project = Project()

project = Project()

Add Sample Model¶

In [22]:

project.sample_models.add(model)

project.sample_models.add(model)

Add Experiments¶

In [23]:

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¶

In [24]:

project.analysis.current_calculator = 'cryspy'

project.analysis.current_calculator = 'cryspy'

Current calculator changed to

cryspy

Set Fit Mode¶

In [25]:

project.analysis.fit_mode = 'joint'

project.analysis.fit_mode = 'joint'

Current fit mode changed to

joint

Set Minimizer¶

In [26]:

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.

In [27]:

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.

In [28]:

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

In [29]:

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¶

In [30]:

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	1	37.01
2	25	16.30	56.0% ↓
3	136	16.21

🏆 Best goodness-of-fit (reduced χ²) is 16.21 at iteration 135

✅ Fitting complete.

Fit results

✅ Success: True

⏱️ Fitting time: 12.36 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	None	length_a	8.4700	8.4693	0.0002	Å	0.01 % ↓
2	pbso4	cell	None	length_b	5.3900	5.3912	0.0001	Å	0.02 % ↑
3	pbso4	cell	None	length_c	6.9500	6.9504	0.0002	Å	0.01 % ↑
4	npd	peak	None	broad_gauss_u	0.1390	0.2901	0.0350	deg²	108.67 % ↑
5	npd	peak	None	broad_gauss_v	-0.4120	-0.6390	0.0531	deg²	55.10 % ↑
6	npd	peak	None	broad_gauss_w	0.3860	0.4552	0.0186	deg²	17.92 % ↑
7	npd	peak	None	broad_lorentz_y	0.0880	0.0931	0.0034	deg	5.76 % ↑
8	npd	linked_phases	pbso4	scale	1.5000	1.4622	0.0049		2.52 % ↓
9	npd	instrument	None	twotheta_offset	-0.1406	-0.1385	0.0019	deg	1.47 % ↓
10	xrd	peak	None	broad_gauss_u	0.3041	0.5724	0.0196	deg²	88.21 % ↑
11	xrd	peak	None	broad_gauss_v	-0.1126	-0.2799	0.0142	deg²	148.57 % ↑
12	xrd	peak	None	broad_gauss_w	0.0213	0.0452	0.0024	deg²	112.27 % ↑
13	xrd	peak	None	broad_lorentz_y	0.0577	0.0587	0.0014	deg	1.75 % ↑
14	xrd	linked_phases	pbso4	scale	0.0010	0.0010	0.0000		1.52 % ↓
15	xrd	instrument	None	twotheta_offset	-0.0518	-0.0637	0.0009	deg	22.93 % ↑
16	xrd	background	0	coef	119.1950	85.8587	1.3293		27.97 % ↓
17	xrd	background	1	coef	6.2210	-19.3112	1.8805		410.42 % ↓
18	xrd	background	2	coef	-45.7250	5.3658	1.7778		111.74 % ↓
19	xrd	background	3	coef	8.1190	11.8842	1.7472		46.37 % ↑
20	xrd	background	4	coef	54.5520	18.5560	1.5670		65.98 % ↓
21	xrd	background	5	coef	-20.6610	-6.4093	1.5359		68.98 % ↓

Plot Measured vs Calculated¶

In [31]:

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)

In [32]:

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)