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.

It also shows how to switch calculation engine and peak profile type.

🛠️ Import Library¶

In [2]:

from easydiffraction import ExperimentFactory
from easydiffraction import Project
from easydiffraction import StructureFactory
from easydiffraction import download_data

from easydiffraction import ExperimentFactory from easydiffraction import Project from easydiffraction import StructureFactory from easydiffraction import download_data

🧩 Define Structure¶

This section shows how to add structures and modify their parameters.

Create Structure¶

In [3]:

structure = StructureFactory.from_scratch(name='si')

structure = StructureFactory.from_scratch(name='si')

Set Space Group¶

In [4]:

structure.space_group.name_h_m = 'F d -3 m'
structure.space_group.coord_system_code = '2'

structure.space_group.name_h_m = 'F d -3 m' structure.space_group.coord_system_code = '2'

Set Unit Cell¶

In [5]:

structure.cell.length_a = 5.431

structure.cell.length_a = 5.431

Set Atom Sites¶

In [6]:

structure.atom_sites.create(
    id='Si',
    type_symbol='Si',
    fract_x=0.125,
    fract_y=0.125,
    fract_z=0.125,
    adp_iso=0.5,
)

structure.atom_sites.create( id='Si', type_symbol='Si', fract_x=0.125, fract_y=0.125, fract_z=0.125, adp_iso=0.5, )

🔬 Define Experiment¶

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

Download Data¶

In [7]:

data_path = download_data('meas-si-sepd', destination='data')

data_path = download_data('meas-si-sepd', destination='data')

Getting data...

Data 'meas-si-sepd': Si, SEPD (Argonne)

✅ Data 'meas-si-sepd' already present at '../../../data/meas-si-sepd.xye'. Keeping existing.

Create Experiment¶

In [8]:

expt = ExperimentFactory.from_data_path(
    name='sepd',
    data_path=data_path,
    beam_mode='time-of-flight',
)

expt = ExperimentFactory.from_data_path( name='sepd', data_path=data_path, beam_mode='time-of-flight', )

Set Instrument¶

In [9]:

expt.instrument.setup_twotheta_bank = 144.845
expt.instrument.calib_d_to_tof_offset = -10.0
expt.instrument.calib_d_to_tof_linear = 7476.91
expt.instrument.calib_d_to_tof_quadratic = -1.54

expt.instrument.setup_twotheta_bank = 144.845 expt.instrument.calib_d_to_tof_offset = -10.0 expt.instrument.calib_d_to_tof_linear = 7476.91 expt.instrument.calib_d_to_tof_quadratic = -1.54

Set Peak Profile¶

In [10]:

expt.peak.show_supported()

expt.peak.show_supported()

Peak types

		Type	Description
1		pseudo-voigt	TOF non-convoluted pseudo-Voigt profile
2	*	jorgensen	TOF Jorgensen profile: back-to-back exponentials ⊗ Gaussian
3		jorgensen-von-dreele	TOF Jorgensen-Von Dreele profile: back-to-back exponentials ⊗ pseudo-Voigt
4		double-jorgensen-von-dreele	TOF Double-Jorgensen-Von Dreele profile: double back-to-back exponentials ⊗ pseudo-Voigt (Z-Rietveld type0m)

In [11]:

expt.peak.type = 'jorgensen-von-dreele'

expt.peak.type = 'jorgensen-von-dreele'

⚠️ Switching peak profile type adds these settings with defaults:                                                                 
   • broad_lorentz_gamma_0=0.0                                                                                                    
   • broad_lorentz_gamma_1=0.0                                                                                                    
   • broad_lorentz_gamma_2=0.0                                                                                                    
   • broad_lorentz_size_l=0.0                                                                                                     
   • broad_lorentz_strain_l=0.0                                                                                                   

Peak profile type for experiment 'sepd' changed to

jorgensen-von-dreele

In [12]:

expt.peak.broad_gauss_sigma_0 = 3.0148
expt.peak.broad_gauss_sigma_1 = 33.3451
expt.peak.broad_gauss_sigma_2 = 0.0
expt.peak.broad_lorentz_gamma_0 = 0.0
expt.peak.broad_lorentz_gamma_1 = 2.5489
expt.peak.broad_lorentz_gamma_2 = 0.0
expt.peak.rise_alpha_0 = 0.0
expt.peak.rise_alpha_1 = 0.5971
expt.peak.decay_beta_0 = 0.0408
expt.peak.decay_beta_1 = 0.0123

expt.peak.broad_gauss_sigma_0 = 3.0148 expt.peak.broad_gauss_sigma_1 = 33.3451 expt.peak.broad_gauss_sigma_2 = 0.0 expt.peak.broad_lorentz_gamma_0 = 0.0 expt.peak.broad_lorentz_gamma_1 = 2.5489 expt.peak.broad_lorentz_gamma_2 = 0.0 expt.peak.rise_alpha_0 = 0.0 expt.peak.rise_alpha_1 = 0.5971 expt.peak.decay_beta_0 = 0.0408 expt.peak.decay_beta_1 = 0.0123

In [13]:

expt.peak.cutoff_fwhm = 8.2

expt.peak.cutoff_fwhm = 8.2

Set Background¶

In [14]:

expt.background.auto_estimate()

expt.background.auto_estimate()

Set Linked Structures¶

In [15]:

expt.linked_structures.create(structure_id='si', scale=600.0)

expt.linked_structures.create(structure_id='si', scale=600.0)

📦 Define Project¶

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

Create Project¶

In [16]:

project = Project(name='si_sepd')

project = Project(name='si_sepd')

Add Structure¶

In [17]:

project.structures.add(structure)

project.structures.add(structure)

Add Experiment¶

In [18]:

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.

Display Structure¶

In [19]:

project.display.structure(struct_name='si')

project.display.structure(struct_name='si')

Structure 🧩 'si' (Atom view type: 'covalent')

Loading plot…

drag = rotate
wheel = zoom
right-drag = pan

Display Pattern¶

In [20]:

project.display.pattern(expt_name='sepd')
project.display.pattern(expt_name='sepd', x_min=23200, x_max=23700)

project.display.pattern(expt_name='sepd') project.display.pattern(expt_name='sepd', x_min=23200, x_max=23700)

Loading plot…

Loading plot…

Perform Fit 1/4¶

Set parameters to be refined.

In [21]:

structure.cell.length_a.free = True

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

structure.cell.length_a.free = True expt.linked_structures['si'].scale.free = True expt.instrument.calib_d_to_tof_offset.free = True

Show free parameters after selection.

In [22]:

project.display.parameters.free()

project.display.parameters.free()

Free parameters for both structures (🧩 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_structure	si	scale	600.00000		-inf	inf
3	sepd	instrument		d_to_tof_offset	-10.00000		-inf	inf	μs

Run Fitting¶

In [23]:

project.analysis.minimizer.type = 'bumps (lm)'

project.analysis.minimizer.type = 'bumps (lm)'

Current minimizer changed to

bumps (lm)

In [24]:

project.analysis.fit()
project.display.fit.results()

project.analysis.fit() project.display.fit.results()

Standard fitting

📋 Using experiment 🔬 'sepd' for 'single' fitting

🚀 Starting fit process with 'bumps (lm)'...

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.30	9.20
2	5	1.76	6.92	24.7% ↓
3	22	7.21	6.92
4	28	10.98	6.92

🏆 Best goodness-of-fit (reduced χ²) is 6.92 at iteration 22

✅ Fitting complete.

⚙️ Settings used:

	Name	Value	Description
1	max_iterations	1000	Maximum solver iterations.

📋 Least-squares fit results:

	Metric	Value
1	🧪 Minimizer	bumps (lm)
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	10.98
4	📏 Goodness-of-fit (reduced χ²)	6.92
5	📏 R-factor (Rf, %)	11.59
6	📏 R-factor squared (Rf², %)	6.24
7	📏 Weighted R-factor (wR, %)	5.16

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	si	cell		length_a	Å	5.4310	5.4308	0.0001	0.00 % ↓
2	sepd	linked_structure	si	scale		600.0000	609.6045	1.5939	1.60 % ↑
3	sepd	instrument		d_to_tof_offset	μs	-10.0000	-8.2700	0.0760	17.30 % ↓

• start = parameter value before refinement
• value = refined value from least-squares minimization
• s.u. = standard uncertainty (one sigma), from the covariance matrix
• change = relative change from start, in %; ↑ = increase, ↓ = decrease

Display Pattern¶

In [25]:

project.display.pattern(expt_name='sepd')

project.display.pattern(expt_name='sepd')

Loading plot…

In [26]:

project.display.pattern(expt_name='sepd', x_min=23200, x_max=23700)

project.display.pattern(expt_name='sepd', x_min=23200, x_max=23700)

Loading plot…

Perform Fit 2/4¶

Set more parameters to be refined.

In [27]:

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

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

Show free parameters after selection.

In [28]:

project.display.parameters.free()

project.display.parameters.free()

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

	datablock	category	entry	parameter	value	uncertainty	min	max	units
1	si	cell		length_a	5.43079	0.00006	-inf	inf	Å
2	sepd	linked_structure	si	scale	609.60449	1.59390	-inf	inf
3	sepd	instrument		d_to_tof_offset	-8.26996	0.07596	-inf	inf	μs
4	sepd	background	1	intensity	213.55062		-inf	inf
5	sepd	background	2	intensity	117.61669		-inf	inf
6	sepd	background	3	intensity	147.70005		-inf	inf
7	sepd	background	4	intensity	122.26237		-inf	inf
8	sepd	background	5	intensity	163.04903		-inf	inf
9	sepd	background	6	intensity	124.58762		-inf	inf
10	sepd	background	7	intensity	120.30738		-inf	inf
11	sepd	background	8	intensity	186.02117		-inf	inf
12	sepd	background	9	intensity	133.38534		-inf	inf
13	sepd	background	10	intensity	108.60000		-inf	inf
14	sepd	background	11	intensity	139.80000		-inf	inf
15	sepd	background	12	intensity	272.05501		-inf	inf
16	sepd	background	13	intensity	114.09603		-inf	inf
17	sepd	background	14	intensity	177.06320		-inf	inf

Run Fitting¶

In [29]:

project.analysis.fit()
project.display.fit.results()

project.analysis.fit() project.display.fit.results()

Standard fitting

📋 Using experiment 🔬 'sepd' for 'single' fitting

🚀 Starting fit process with 'bumps (lm)'...

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.30	6.93
2	19	6.50	3.71	46.5% ↓
3	37	12.75	3.71
4	58	26.43	3.71

🏆 Best goodness-of-fit (reduced χ²) is 3.71 at iteration 37

✅ Fitting complete.

⚙️ Settings used:

	Name	Value	Description
1	max_iterations	1000	Maximum solver iterations.

📋 Least-squares fit results:

	Metric	Value
1	🧪 Minimizer	bumps (lm)
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	26.43
4	📏 Goodness-of-fit (reduced χ²)	3.71
5	📏 R-factor (Rf, %)	8.29
6	📏 R-factor squared (Rf², %)	4.19
7	📏 Weighted R-factor (wR, %)	3.12

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	si	cell		length_a	Å	5.4308	5.4309	0.0000	0.00 % ↑
2	sepd	linked_structure	si	scale		609.6045	627.6584	1.2045	2.96 % ↑
3	sepd	instrument		d_to_tof_offset	μs	-8.2700	-8.4435	0.0541	2.10 % ↑
4	sepd	background	1	intensity		213.5506	203.7850	0.4110	4.57 % ↓
5	sepd	background	2	intensity		117.6167	103.7439	0.4466	11.79 % ↓
6	sepd	background	3	intensity		147.7001	125.1165	0.8726	15.29 % ↓
7	sepd	background	4	intensity		122.2624	119.8161	0.9651	2.00 % ↓
8	sepd	background	5	intensity		163.0490	127.4777	2.7426	21.82 % ↓
9	sepd	background	6	intensity		124.5876	123.1558	1.6805	1.15 % ↓
10	sepd	background	7	intensity		120.3074	121.7257	1.7646	1.18 % ↑
11	sepd	background	8	intensity		186.0212	140.3609	3.2718	24.55 % ↓
12	sepd	background	9	intensity		133.3853	135.1065	2.4405	1.29 % ↑
13	sepd	background	10	intensity		108.6000	129.9435	1.3425	19.65 % ↑
14	sepd	background	11	intensity		139.8000	143.2305	2.1413	2.45 % ↑
15	sepd	background	12	intensity		272.0550	175.4444	4.7257	35.51 % ↓
16	sepd	background	13	intensity		114.0960	166.4957	5.1274	45.93 % ↑
17	sepd	background	14	intensity		177.0632	203.9661	10.8187	15.19 % ↑

• start = parameter value before refinement
• value = refined value from least-squares minimization
• s.u. = standard uncertainty (one sigma), from the covariance matrix
• change = relative change from start, in %; ↑ = increase, ↓ = decrease

Display Pattern¶

In [30]:

project.display.pattern(expt_name='sepd')

project.display.pattern(expt_name='sepd')

Loading plot…

In [31]:

project.display.pattern(expt_name='sepd', x_min=23200, x_max=23700)

project.display.pattern(expt_name='sepd', x_min=23200, x_max=23700)

Loading plot…

Perform Fit 3/4¶

Fix background points.

In [32]:

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

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

Set more parameters to be refined.

In [33]:

expt.peak.broad_gauss_sigma_0.free = True
expt.peak.broad_gauss_sigma_1.free = True
expt.peak.broad_lorentz_gamma_1.free = True

expt.peak.broad_gauss_sigma_0.free = True expt.peak.broad_gauss_sigma_1.free = True expt.peak.broad_lorentz_gamma_1.free = True

Show free parameters after selection.

In [34]:

project.display.parameters.free()

project.display.parameters.free()

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

	datablock	category	entry	parameter	value	uncertainty	min	max	units
1	si	cell		length_a	5.43093	0.00004	-inf	inf	Å
2	sepd	linked_structure	si	scale	627.65841	1.20447	-inf	inf
3	sepd	peak		broad_lorentz_gamma_1	2.54890		-inf	inf	μs/Å
4	sepd	peak		broad_gauss_sigma_0	3.01480		-inf	inf	μs²
5	sepd	peak		broad_gauss_sigma_1	33.34510		-inf	inf	μs/Å
6	sepd	instrument		d_to_tof_offset	-8.44347	0.05410	-inf	inf	μs

Run Fitting¶

In [35]:

project.analysis.fit()
project.display.fit.results()

project.analysis.fit() project.display.fit.results()

Standard fitting

📋 Using experiment 🔬 'sepd' for 'single' fitting

🚀 Starting fit process with 'bumps (lm)'...

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.31	3.70
2	8	2.81	3.63	2.1% ↓
3	31	13.57	3.62

🏆 Best goodness-of-fit (reduced χ²) is 3.62 at iteration 31

✅ Fitting complete.

⚙️ Settings used:

	Name	Value	Description
1	max_iterations	1000	Maximum solver iterations.

📋 Least-squares fit results:

	Metric	Value
1	🧪 Minimizer	bumps (lm)
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	13.57
4	📏 Goodness-of-fit (reduced χ²)	3.62
5	📏 R-factor (Rf, %)	8.32
6	📏 R-factor squared (Rf², %)	4.23
7	📏 Weighted R-factor (wR, %)	3.08

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	si	cell		length_a	Å	5.4309	5.4309	0.0000	0.00 % ↓
2	sepd	linked_structure	si	scale		627.6584	627.0144	1.2525	0.10 % ↓
3	sepd	peak		broad_lorentz_gamma_1	μs/Å	2.5489	2.2430	0.0734	12.00 % ↓
4	sepd	peak		broad_gauss_sigma_0	μs²	3.0148	6.3137	0.3700	109.42 % ↑
5	sepd	peak		broad_gauss_sigma_1	μs/Å	33.3451	32.7072	0.6820	1.91 % ↓
6	sepd	instrument		d_to_tof_offset	μs	-8.4435	-8.4025	0.0568	0.49 % ↓

• start = parameter value before refinement
• value = refined value from least-squares minimization
• s.u. = standard uncertainty (one sigma), from the covariance matrix
• change = relative change from start, in %; ↑ = increase, ↓ = decrease

Display Pattern¶

In [36]:

project.display.pattern(expt_name='sepd')

project.display.pattern(expt_name='sepd')

Loading plot…

In [37]:

project.display.pattern(expt_name='sepd', x_min=23200, x_max=23700)

project.display.pattern(expt_name='sepd', x_min=23200, x_max=23700)

Loading plot…

Perform Fit 4/4¶

Set more parameters to be refined.

In [38]:

structure.atom_sites['Si'].adp_iso.free = True

expt.peak.decay_beta_0.free = True
expt.peak.decay_beta_1.free = True

structure.atom_sites['Si'].adp_iso.free = True expt.peak.decay_beta_0.free = True expt.peak.decay_beta_1.free = True

Show free parameters after selection.

In [39]:

project.display.parameters.free()

project.display.parameters.free()

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

	datablock	category	entry	parameter	value	uncertainty	min	max	units
1	si	cell		length_a	5.43090	0.00004	-inf	inf	Å
2	si	atom_site	Si	adp_iso	0.50000		-inf	inf	Ų
3	sepd	linked_structure	si	scale	627.01444	1.25251	-inf	inf
4	sepd	peak		decay_beta_0	0.04080		-inf	inf	μs
5	sepd	peak		decay_beta_1	0.01230		-inf	inf	μs/Å
6	sepd	peak		broad_lorentz_gamma_1	2.24297	0.07343	-inf	inf	μs/Å
7	sepd	peak		broad_gauss_sigma_0	6.31373	0.37001	-inf	inf	μs²
8	sepd	peak		broad_gauss_sigma_1	32.70719	0.68203	-inf	inf	μs/Å
9	sepd	instrument		d_to_tof_offset	-8.40249	0.05683	-inf	inf	μs

Run Fitting¶

In [40]:

project.analysis.fit()
project.display.fit.results()

project.analysis.fit() project.display.fit.results()

Standard fitting

📋 Using experiment 🔬 'sepd' for 'single' fitting

🚀 Starting fit process with 'bumps (lm)'...

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.30	3.63
2	21	7.26	3.60
3	38	16.94	3.60

🏆 Best goodness-of-fit (reduced χ²) is 3.60 at iteration 38

✅ Fitting complete.

⚙️ Settings used:

	Name	Value	Description
1	max_iterations	1000	Maximum solver iterations.

📋 Least-squares fit results:

	Metric	Value
1	🧪 Minimizer	bumps (lm)
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	16.94
4	📏 Goodness-of-fit (reduced χ²)	3.60
5	📏 R-factor (Rf, %)	8.20
6	📏 R-factor squared (Rf², %)	4.08
7	📏 Weighted R-factor (wR, %)	2.89

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	si	cell		length_a	Å	5.4309	5.4309	0.0001	0.00 % ↓
2	si	atom_site	Si	adp_iso	Ų	0.5000	0.5223	0.0036	4.45 % ↑
3	sepd	linked_structure	si	scale		627.0144	634.1959	1.7129	1.15 % ↑
4	sepd	peak		decay_beta_0	μs	0.0408	0.0406	0.0002	0.43 % ↓
5	sepd	peak		decay_beta_1	μs/Å	0.0123	0.0124	0.0002	0.87 % ↑
6	sepd	peak		broad_lorentz_gamma_1	μs/Å	2.2430	2.2124	0.0799	1.36 % ↓
7	sepd	peak		broad_gauss_sigma_0	μs²	6.3137	5.5649	0.4291	11.86 % ↓
8	sepd	peak		broad_gauss_sigma_1	μs/Å	32.7072	33.2039	0.7002	1.52 % ↑
9	sepd	instrument		d_to_tof_offset	μs	-8.4025	-8.4056	0.0810	0.04 % ↑

• start = parameter value before refinement
• value = refined value from least-squares minimization
• s.u. = standard uncertainty (one sigma), from the covariance matrix
• change = relative change from start, in %; ↑ = increase, ↓ = decrease

Display Correlations¶

In [41]:

project.display.fit.correlations()

project.display.fit.correlations()

Loading plot…

Display Pattern¶

In [42]:

project.display.pattern(expt_name='sepd')

project.display.pattern(expt_name='sepd')

Loading plot…

In [43]:

project.display.pattern(expt_name='sepd', x_min=23200, x_max=23700)

project.display.pattern(expt_name='sepd', x_min=23200, x_max=23700)

Loading plot…

In [44]:

project.display.pattern(expt_name='sepd', x='d_spacing')

project.display.pattern(expt_name='sepd', x='d_spacing')

Loading plot…

💾 Save Project¶

In [45]:

project.save_as(dir_path='projects/refine-si-sepd')

project.save_as(dir_path='projects/refine-si-sepd')

Saving project 📦 'si_sepd' to '../../../projects/refine-si-sepd'

├── 📄 project.edi

├── 📁 structures/

│   └── 📄 si.edi

├── 📁 experiments/

│   └── 📄 sepd.edi

├── 📁 analysis/

│   └── 📄 analysis.edi

└── 📁 reports/

    └── 📄 si_sepd.html