Structure Refinement: LaM(7)O3, P02.1 Synchrotron XRD¶

This example refines the compositionally complex LaM(7)O3 perovskite, with La on the A site and an equimolar mixture of Ti, Cr, Mn, Fe, Co, Ni, and Cu on the B site. The low-temperature synchrotron X-ray powder diffraction pattern was collected at the P02.1 beamline at PETRA III. The workflow starts from approximate structural and profile parameters, estimates the background from the measured pattern, and improves the model in two fitting stages.

🛠️ Import Library¶

In [2]:

import easydiffraction as edi

import easydiffraction as edi

📦 Define Project¶

The project manages the structures, experiments, analysis, and saved results used throughout the tutorial.

Create Project¶

In [3]:

project = edi.Project(
    name='lam7o3_p021',
    description='LaM(7)O3 refinement using P02.1 synchrotron X-ray data.',
)

project = edi.Project( name='lam7o3_p021', description='LaM(7)O3 refinement using P02.1 synchrotron X-ray data.', )

Save Initial Project¶

Create the project directory before fitting so that analysis results can be written as they are produced.

In [4]:

project.save_as(dir_path='projects/refine-lam7o3-p021')

project.save_as(dir_path='projects/refine-lam7o3-p021')

Saving project 📦 'lam7o3_p021' to '../../../projects/refine-lam7o3-p021'

├── 📄 project.edi

├── 📁 structures/

├── 📁 experiments/

├── 📁 analysis/

│   └── 📄 analysis.edi

└── 📁 reports/

    └── 📄 lam7o3_p021.html

🧩 Define Structure¶

The Pnma structure is initialized from approximate values rather than the final refined values.

Create Structure¶

Define the structure as an inline CIF. Empty uncertainty parentheses, such as 5.5(), mark a parameter as free without assigning an initial standard uncertainty. Values without parentheses remain fixed.

In [5]:

structure_cif = """
data_lam7o3

_cell.length_a 5.5()
_cell.length_b 7.7()
_cell.length_c 5.5()
_cell.angle_alpha 90.
_cell.angle_beta  90.
_cell.angle_gamma 90.

_space_group.name_h_m "P n m a"
_space_group.coord_system_code abc

loop_
_atom_site.id
_atom_site.type_symbol
_atom_site.fract_x
_atom_site.fract_y
_atom_site.fract_z
_atom_site.occupancy
_atom_site.adp_iso
_atom_site.adp_type
La La  0.48()  0.25   0.004()   1.        0.1() Biso
Ti Ti  0.      0.     0.        0.14286   0.1() Biso
Cr Cr  0.      0.     0.        0.14286   0.1   Biso
Mn Mn  0.      0.     0.        0.14286   0.1   Biso
Fe Fe  0.      0.     0.        0.14286   0.1   Biso
Co Co  0.      0.     0.        0.14286   0.1   Biso
Ni Ni  0.      0.     0.        0.14286   0.1   Biso
Cu Cu  0.      0.     0.        0.14286   0.1   Biso
O1 O   0.51()  0.25   0.57()    1.        0.1() Biso
O2 O   0.22()  0.03() 0.27()    1.        0.1   Biso
"""

structure_cif = """ data_lam7o3 _cell.length_a 5.5() _cell.length_b 7.7() _cell.length_c 5.5() _cell.angle_alpha 90. _cell.angle_beta 90. _cell.angle_gamma 90. _space_group.name_h_m "P n m a" _space_group.coord_system_code abc loop_ _atom_site.id _atom_site.type_symbol _atom_site.fract_x _atom_site.fract_y _atom_site.fract_z _atom_site.occupancy _atom_site.adp_iso _atom_site.adp_type La La 0.48() 0.25 0.004() 1. 0.1() Biso Ti Ti 0. 0. 0. 0.14286 0.1() Biso Cr Cr 0. 0. 0. 0.14286 0.1 Biso Mn Mn 0. 0. 0. 0.14286 0.1 Biso Fe Fe 0. 0. 0. 0.14286 0.1 Biso Co Co 0. 0. 0. 0.14286 0.1 Biso Ni Ni 0. 0. 0. 0.14286 0.1 Biso Cu Cu 0. 0. 0. 0.14286 0.1 Biso O1 O 0.51() 0.25 0.57() 1. 0.1() Biso O2 O 0.22() 0.03() 0.27() 1. 0.1 Biso """

In [6]:

project.structures.add_from_cif_str(structure_cif)

project.structures.add_from_cif_str(structure_cif)

In [7]:

project.structures.show_names()

project.structures.show_names()

Defined structures 🧩

['lam7o3']

Use a short alias to access the structure parameters below.

In [8]:

structure = project.structures['lam7o3']

structure = project.structures['lam7o3']

Display Structure¶

Inspect the structure as text and as an interactive crystal model.

In [9]:

structure.show_as_text()

structure.show_as_text()

Structure 🧩 'lam7o3' as text

	Edi
1	data_lam7o3
2
3	_cell.length_a 5.5()
4	_cell.length_b 7.7()
5	_cell.length_c 5.5()
6	_cell.angle_alpha 90.
7	_cell.angle_beta 90.
8	_cell.angle_gamma 90.
9
10	_space_group.name_h_m "P n m a"
11	_space_group.coord_system_code abc
12
13	_geom.min_bond_distance_cutoff 0.
14	_geom.bond_distance_inc 0.25
15
16	loop_
17	_atom_site.id
18	_atom_site.type_symbol
19	_atom_site.fract_x
20	_atom_site.fract_y
21	_atom_site.fract_z
22	_atom_site.wyckoff_letter
23	_atom_site.multiplicity
24	_atom_site.occupancy
25	_atom_site.adp_iso
26	_atom_site.adp_type
27	La La 0.48() 0.25 0.004() c 4 1. 0.1() Biso
28	Ti Ti 0. 0. 0. a 4 0.14286 0.1() Biso
29	Cr Cr 0. 0. 0. a 4 0.14286 0.1 Biso
30	Mn Mn 0. 0. 0. a 4 0.14286 0.1 Biso
31	Fe Fe 0. 0. 0. a 4 0.14286 0.1 Biso
32	Co Co 0. 0. 0. a 4 0.14286 0.1 Biso
33	Ni Ni 0. 0. 0. a 4 0.14286 0.1 Biso
34	Cu Cu 0. 0. 0. a 4 0.14286 0.1 Biso
35	O1 O 0.51() 0.25 0.57() c 4 1. 0.1() Biso
36	O2 O 0.22() 0.03() 0.27() d 8 1. 0.1 Biso

In [10]:

project.display.structure(struct_name='lam7o3')

project.display.structure(struct_name='lam7o3')

Structure 🧩 'lam7o3' (Atom view type: 'covalent')

Loading plot…

drag = rotate
wheel = zoom
right-drag = pan

🔬 Define Experiment¶

Load the measured pattern, configure the instrument and peak profile, and link the structure to the experiment.

Download Data¶

The first two columns contain 2-theta and intensity. When a third column of standard uncertainties is absent, EasyDiffraction estimates it from the square root of the intensity.

In [11]:

data_path = edi.download_data('meas-hep7c-xray-synchrotron', destination='data')

data_path = edi.download_data('meas-hep7c-xray-synchrotron', destination='data')

Getting data...

Data 'meas-hep7c-xray-synchrotron': La high-entropy perovskite (7 B-cations), synchrotron X-ray

✅ Data 'meas-hep7c-xray-synchrotron' downloaded to '../../../data/meas-hep7c-xray-synchrotron.dat'

Create P02.1 Experiment¶

In [12]:

project.experiments.add_from_data_path(
    name='p021',
    data_path=data_path,
    sample_form='powder',
    beam_mode='constant wavelength',
    radiation_probe='xray',
)

project.experiments.add_from_data_path( name='p021', data_path=data_path, sample_form='powder', beam_mode='constant wavelength', radiation_probe='xray', )

⚠️ No uncertainty (sy) column provided. Defaulting to sqrt(y).                                                                    

Data loaded successfully

Experiment 🔬 'p021'. Number of data points: 1459.

Use a short alias to access the P02.1 experiment parameters below.

In [13]:

experiment = project.experiments['p021']

experiment = project.experiments['p021']

Set Linked Structures¶

Link the structural model to the measured pattern and use an order-of-magnitude estimate for the scale factor.

In [14]:

experiment.linked_structures.create(
    structure_id='lam7o3',
    scale=0.000005,
)

experiment.linked_structures.create( structure_id='lam7o3', scale=0.000005, )

Set P02.1 Instrument Parameters¶

Set the monochromatic X-ray wavelength reported for the P02.1 beamline measurement and initialize the unknown 2-theta zero shift at zero.

In [15]:

experiment.instrument.setup_wavelength = 0.207109
experiment.instrument.calib_twotheta_offset = 0.0

experiment.instrument.setup_wavelength = 0.207109 experiment.instrument.calib_twotheta_offset = 0.0

Set Peak Profile¶

Select a pseudo-Voigt profile and provide approximate broadening parameters. U, V, and W define the Gaussian contribution; X and Y define the Lorentzian contribution.

In [16]:

experiment.peak.show_supported()

experiment.peak.show_supported()

Peak types

		Type	Description
1	*	pseudo-voigt	CWL pseudo-Voigt profile
2		pseudo-voigt + berar-baldinozzi asymmetry	CWL pseudo-Voigt profile with Berar-Baldinozzi asymmetry correction.

In [17]:

experiment.peak.type = 'pseudo-voigt'

experiment.peak.type = 'pseudo-voigt'

Peak profile type for experiment 'p021' changed to

pseudo-voigt

In [18]:

experiment.peak.broad_gauss_u = 0.04
experiment.peak.broad_gauss_v = -0.01
experiment.peak.broad_gauss_w = 0.001
experiment.peak.broad_lorentz_x = 0.1
experiment.peak.broad_lorentz_y = 0.0

experiment.peak.broad_gauss_u = 0.04 experiment.peak.broad_gauss_v = -0.01 experiment.peak.broad_gauss_w = 0.001 experiment.peak.broad_lorentz_x = 0.1 experiment.peak.broad_lorentz_y = 0.0

Set Excluded Regions¶

Restrict the fit to the useful measured range from 2 to 15 degrees.

In [19]:

experiment.excluded_regions.create(id='1', start=0.0, end=2.0)
experiment.excluded_regions.create(id='2', start=15.0, end=20.0)

experiment.excluded_regions.create(id='1', start=0.0, end=2.0) experiment.excluded_regions.create(id='2', start=15.0, end=20.0)

Set Background¶

Estimate initial background points from the measured pattern alone.

In [20]:

experiment.background.auto_estimate(use_model=False)

experiment.background.auto_estimate(use_model=False)

In [21]:

experiment.background.show()

experiment.background.show()

Line-segment background points

	Position	Intensity
1	2.00330	584.02820
2	3.19170	616.43436
3	3.49660	608.10597
4	3.72800	594.63226
5	3.95940	574.02994
6	4.94790	457.72470
7	5.17920	443.13978
8	5.46320	433.01395
9	6.02050	432.92677
10	6.97750	421.29697
11	7.75570	444.89668
12	8.61800	453.76679
13	11.28920	346.83990
14	11.93060	357.71449
15	14.73850	302.43079
16	14.99090	278.31547

Inspect Experiment¶

Display the configured experiment as text.

In [22]:

experiment.show_as_text()

experiment.show_as_text()

Experiment 🔬 'p021' as text

	Edi
1	data_p021
2
3	_experiment_type.sample_form powder
4	_experiment_type.beam_mode "constant wavelength"
5	_experiment_type.radiation_probe xray
6	_experiment_type.scattering_type bragg
7
8	_diffrn.ambient_temperature ?
9	_diffrn.ambient_pressure ?
10	_diffrn.ambient_magnetic_field ?
11	_diffrn.ambient_electric_field ?
12
13	_calculator.type cryspy
14
15	_peak.broad_gauss_u 0.04
16	_peak.broad_gauss_v -0.01
17	_peak.broad_gauss_w 0.001
18	_peak.broad_lorentz_x 0.1
19	_peak.broad_lorentz_y 0.
20	_peak.cutoff_fwhm 0.
21	_peak.type cwl-pseudo-voigt
22
23	_instrument.setup_wavelength 0.207109
24	_instrument.setup_wavelength_2 0.
25	_instrument.setup_wavelength_2_to_1_ratio 0.
26	_instrument.calib_twotheta_offset 0.
27	_instrument.calib_sample_displacement 0.
28	_instrument.calib_sample_transparency 0.
29	_instrument.setup_polarization_coefficient 0.
30	_instrument.setup_monochromator_twotheta 0.
31
32	_absorption.type none
33
34	loop_
35	_linked_structure.structure_id
36	_linked_structure.scale
37	lam7o3 0.000005
38
39	loop_
40	_excluded_region.id
41	_excluded_region.start
42	_excluded_region.end
43	1 0. 2.
44	2 15. 20.
45
46	_background.type line-segment
47
48	loop_
49	_background.id
50	_background.position
51	_background.intensity
52	1 2.0033 584.0282
53	2 3.1917 616.4343557
54	3 3.4966 608.10596651
55	4 3.728 594.63226302
56	5 3.9594 574.02993766
57	6 4.9479 457.7247
58	7 5.1792 443.13977867
59	8 5.4632 433.01395145
60	9 6.0205 432.92676858
61	10 6.9775 421.2969741
62	11 7.7557 444.89668148
63	12 8.618 453.76678528
64	13 11.2892 346.83989649
65	14 11.9306 357.71448643
66	15 14.7385 302.43079253
67	16 14.9909 278.31546798
68
69	loop_
70	_data.two_theta
71	_data.id
72	_data.d_spacing
73	_data.intensity_meas
74	_data.intensity_meas_su
75	_data.intensity_calc
76	_data.intensity_bkg
77	_data.calc_status
78	0.0053 1 0. 0. 1. 0. 0. excl
79	0.0158 2 0. 0. 1. 0. 0. excl
80	0.0263 3 0. 0. 1. 0. 0. excl
81	0.0368 4 0. 0. 1. 0. 0. excl
82	0.0473 5 0. 0. 1. 0. 0. excl
83	0.0578 6 0. 0. 1. 0. 0. excl
84	0.0684 7 0. 0. 1. 0. 0. excl
85	0.0789 8 0. 0. 1. 0. 0. excl
86	0.0894 9 0. 0. 1. 0. 0. excl
87	0.0999 10 0. 0. 1. 0. 0. excl
88	...
89	15.2432 1450 0. 65.524826 8.09474064 0. 0. excl
90	15.2538 1451 0. 43.702377 6.61077734 0. 0. excl
91	15.2643 1452 0. 31.145777 5.58084017 0. 0. excl
92	15.2748 1453 0. 20.166813 4.49074749 0. 0. excl
93	15.2853 1454 0. 14.170188 3.76433102 0. 0. excl
94	15.2958 1455 0. 7.3013401 2.7020992 0. 0. excl
95	15.3063 1456 0. 12.101576 3.47873195 0. 0. excl
96	15.3169 1457 0. 6.531918 2.55576173 0. 0. excl
97	15.3274 1458 0. 3.1248069 1.76771234 0. 0. excl
98	15.3379 1459 0. 3.6774037 1.91765578 0. 0. excl
99
100	loop_
101	_refln.id
102	_refln.structure_id
103	_refln.d_spacing
104	_refln.sin_theta_over_lambda
105	_refln.index_h
106	_refln.index_k
107	_refln.index_l
108	_refln.f_calc
109	_refln.f_squared_calc
110	_refln.two_theta
111	1 lam7o3 7.7 0.06493506 0 1 0 0. 0. 1.54114667
112	2 lam7o3 5.5 0.09090909 0 0 1 0. 0. 2.15766778
113	3 lam7o3 5.5 0.09090909 1 0 0 0. 0. 2.15766778
114	4 lam7o3 4.47553409 0.11171851 0 1 1 3.59310475 12.91040172 2.65164569
115	5 lam7o3 4.47553409 0.11171851 1 1 0 0. 0. 2.65164569
116	6 lam7o3 3.8890873 0.12856487 1 0 1 84.40914375 7124.90354832 3.05158341
117	7 lam7o3 3.85 0.12987013 0 2 0 85.84489383 7369.34579669 3.08257218
118	8 lam7o3 3.47142809 0.14403294 1 1 1 24.66596148 608.40965573 3.4188328
119	9 lam7o3 3.15404289 0.1585267 1 2 0 0. 0. 3.76298134
120	10 lam7o3 3.15404289 0.1585267 0 2 1 0. 0. 3.76298134
121	...
122	605 lam7o3 0.63080858 0.79263348 0 10 5 0. 0. 18.89705877
123	606 lam7o3 0.63072392 0.79273987 5 2 7 107.42119834 11539.3138528 18.89961836
124	607 lam7o3 0.63072392 0.79273987 7 2 5 90.76949065 8239.10043243 18.89961836
125	608 lam7o3 0.63021671 0.79337789 2 4 8 97.82362003 9569.46063649 18.91496894
126	609 lam7o3 0.63021671 0.79337789 8 4 2 76.04580662 5782.96470497 18.91496894
127	610 lam7o3 0.62937406 0.79444012 3 6 7 108.33293447 11736.02469019 18.94052659
128	611 lam7o3 0.62937406 0.79444012 7 6 3 78.37832345 6143.16158631 18.94052659
129	612 lam7o3 0.62845104 0.79560693 6 3 6 53.64286926 2877.55742291 18.9686017
130	613 lam7o3 0.62778229 0.79645445 8 5 0 76.32628817 5825.702266 18.98899484
131	614 lam7o3 0.62778229 0.79645445 0 5 8 0. 0. 18.98899484

🚀 Perform Analysis¶

Select the refinement parameters, apply the shared-Biso constraints, and improve the model in two fitting stages.

Set Free Parameters¶

The independent cell lengths, selected fractional coordinates, and three independent Biso values were marked free by () in the inline CIF. They do not need to be selected again here.

Refine the scale factor, zero shift, U/V/W/X profile terms, and active background-point intensities.

In [23]:

experiment.linked_structures['lam7o3'].scale.free = True

experiment.instrument.calib_twotheta_offset.free = True

experiment.peak.broad_gauss_u.free = True
experiment.peak.broad_gauss_v.free = True
experiment.peak.broad_gauss_w.free = True
experiment.peak.broad_lorentz_x.free = True

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

experiment.linked_structures['lam7o3'].scale.free = True experiment.instrument.calib_twotheta_offset.free = True experiment.peak.broad_gauss_u.free = True experiment.peak.broad_gauss_v.free = True experiment.peak.broad_gauss_w.free = True experiment.peak.broad_lorentz_x.free = True for point in experiment.background: point.intensity.free = True

Display all parameters selected for refinement.

In [24]:

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	lam7o3	cell		length_a	5.50000		-inf	inf	Å
2	lam7o3	cell		length_b	7.70000		-inf	inf	Å
3	lam7o3	cell		length_c	5.50000		-inf	inf	Å
4	lam7o3	atom_site	La	fract_x	0.48000		-inf	inf
5	lam7o3	atom_site	La	fract_z	0.00400		-inf	inf
6	lam7o3	atom_site	La	adp_iso	0.10000		-inf	inf	Ų
7	lam7o3	atom_site	Ti	adp_iso	0.10000		-inf	inf	Ų
8	lam7o3	atom_site	O1	fract_x	0.51000		-inf	inf
9	lam7o3	atom_site	O1	fract_z	0.57000		-inf	inf
10	lam7o3	atom_site	O1	adp_iso	0.10000		-inf	inf	Ų
11	lam7o3	atom_site	O2	fract_x	0.22000		-inf	inf
12	lam7o3	atom_site	O2	fract_y	0.03000		-inf	inf
13	lam7o3	atom_site	O2	fract_z	0.27000		-inf	inf
14	p021	linked_structure	lam7o3	scale	0.00001		-inf	inf
15	p021	peak		broad_gauss_u	0.04000		-inf	inf	deg²
16	p021	peak		broad_gauss_v	-0.01000		-inf	inf	deg²
17	p021	peak		broad_gauss_w	0.00100		-inf	inf	deg²
18	p021	peak		broad_lorentz_x	0.10000		-inf	inf	deg
19	p021	instrument		twotheta_offset	0.00000		-inf	inf	deg
20	p021	background	1	intensity	584.02820		-inf	inf
21	p021	background	2	intensity	616.43436		-inf	inf
22	p021	background	3	intensity	608.10597		-inf	inf
23	p021	background	4	intensity	594.63226		-inf	inf
24	p021	background	5	intensity	574.02994		-inf	inf
25	p021	background	6	intensity	457.72470		-inf	inf
26	p021	background	7	intensity	443.13978		-inf	inf
27	p021	background	8	intensity	433.01395		-inf	inf
28	p021	background	9	intensity	432.92677		-inf	inf
29	p021	background	10	intensity	421.29697		-inf	inf
30	p021	background	11	intensity	444.89668		-inf	inf
31	p021	background	12	intensity	453.76679		-inf	inf
32	p021	background	13	intensity	346.83990		-inf	inf
33	p021	background	14	intensity	357.71449		-inf	inf
34	p021	background	15	intensity	302.43079		-inf	inf
35	p021	background	16	intensity	278.31547		-inf	inf

Set Constraints¶

Create aliases for the constrained Biso parameters. The seven elements share one B site and therefore one Biso value; O1 and O2 are also modeled with one shared Biso value.

In [25]:

# B sites: Ti, Cr, Mn, Fe, Co, Ni, Cu
project.analysis.aliases.create(
    id='biso_Ti',
    param=structure.atom_sites['Ti'].adp_iso,
)
project.analysis.aliases.create(
    id='biso_Cr',
    param=structure.atom_sites['Cr'].adp_iso,
)
project.analysis.aliases.create(
    id='biso_Mn',
    param=structure.atom_sites['Mn'].adp_iso,
)
project.analysis.aliases.create(
    id='biso_Fe',
    param=structure.atom_sites['Fe'].adp_iso,
)
project.analysis.aliases.create(
    id='biso_Co',
    param=structure.atom_sites['Co'].adp_iso,
)
project.analysis.aliases.create(
    id='biso_Ni',
    param=structure.atom_sites['Ni'].adp_iso,
)
project.analysis.aliases.create(
    id='biso_Cu',
    param=structure.atom_sites['Cu'].adp_iso,
)

# O sites: O1, O2
project.analysis.aliases.create(
    id='biso_O1',
    param=structure.atom_sites['O1'].adp_iso,
)
project.analysis.aliases.create(
    id='biso_O2',
    param=structure.atom_sites['O2'].adp_iso,
)

# B sites: Ti, Cr, Mn, Fe, Co, Ni, Cu project.analysis.aliases.create( id='biso_Ti', param=structure.atom_sites['Ti'].adp_iso, ) project.analysis.aliases.create( id='biso_Cr', param=structure.atom_sites['Cr'].adp_iso, ) project.analysis.aliases.create( id='biso_Mn', param=structure.atom_sites['Mn'].adp_iso, ) project.analysis.aliases.create( id='biso_Fe', param=structure.atom_sites['Fe'].adp_iso, ) project.analysis.aliases.create( id='biso_Co', param=structure.atom_sites['Co'].adp_iso, ) project.analysis.aliases.create( id='biso_Ni', param=structure.atom_sites['Ni'].adp_iso, ) project.analysis.aliases.create( id='biso_Cu', param=structure.atom_sites['Cu'].adp_iso, ) # O sites: O1, O2 project.analysis.aliases.create( id='biso_O1', param=structure.atom_sites['O1'].adp_iso, ) project.analysis.aliases.create( id='biso_O2', param=structure.atom_sites['O2'].adp_iso, )

Apply the equality constraints using the aliases.

In [26]:

project.analysis.constraints.create(id='1', expression='biso_Cr = biso_Ti')
project.analysis.constraints.create(id='2', expression='biso_Mn = biso_Ti')
project.analysis.constraints.create(id='3', expression='biso_Fe = biso_Ti')
project.analysis.constraints.create(id='4', expression='biso_Co = biso_Ti')
project.analysis.constraints.create(id='5', expression='biso_Ni = biso_Ti')
project.analysis.constraints.create(id='6', expression='biso_Cu = biso_Ti')

project.analysis.constraints.create(id='7', expression='biso_O2 = biso_O1')

project.analysis.constraints.create(id='1', expression='biso_Cr = biso_Ti') project.analysis.constraints.create(id='2', expression='biso_Mn = biso_Ti') project.analysis.constraints.create(id='3', expression='biso_Fe = biso_Ti') project.analysis.constraints.create(id='4', expression='biso_Co = biso_Ti') project.analysis.constraints.create(id='5', expression='biso_Ni = biso_Ti') project.analysis.constraints.create(id='6', expression='biso_Cu = biso_Ti') project.analysis.constraints.create(id='7', expression='biso_O2 = biso_O1')

Display the defined constraints.

In [27]:

project.analysis.display.constraints()

project.analysis.display.constraints()

User defined constraints

	id	expression
1	1	biso_Cr = biso_Ti
2	2	biso_Mn = biso_Ti
3	3	biso_Fe = biso_Ti
4	4	biso_Co = biso_Ti
5	5	biso_Ni = biso_Ti
6	6	biso_Cu = biso_Ti
7	7	biso_O2 = biso_O1

Constraints enabled: True

Fit Initial Model¶

The first fit uses a background estimated directly from the measured pattern.

Display Pattern (Before Fit)¶

Compare the measured pattern with the calculation from the starting model before optimization.

In [28]:

project.display.pattern(expt_name='p021')

project.display.pattern(expt_name='p021')

Loading plot…

In [29]:

project.display.pattern(expt_name='p021', x_min=2.2, x_max=4.0)

project.display.pattern(expt_name='p021', x_min=2.2, x_max=4.0)

Loading plot…

Run Fitting¶

In [30]:

project.analysis.fit()

project.analysis.fit()

Standard fitting

📋 Using experiment 🔬 'p021' for 'single' fitting

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

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.21	276.18
2	20	5.71	276.18
3	39	10.84	46.46	83.2% ↓
4	57	16.14	46.46
5	75	21.08	15.97	65.6% ↓
6	93	26.39	15.97
7	111	31.49	3.89	75.7% ↓
8	129	36.58	3.89
9	146	41.80	3.89
10	149	42.41	1.71	55.9% ↓
11	167	47.65	1.71
12	185	52.68	1.12	34.9% ↓
13	203	57.71	1.12
14	220	62.88	1.12
15	221	63.08	0.88	20.9% ↓
16	239	68.22	0.88
17	257	73.47	0.88
18	270	78.57	0.88
19	283	83.72	0.88
20	296	89.05	0.88
21	309	94.19	0.88
22	322	99.24	0.88
23	340	104.29	0.88
24	358	109.42	0.88
25	375	114.42	0.88
26	392	119.62	0.88
27	410	124.72	0.88
28	427	129.83	0.88
29	438	133.76	0.88

🏆 Best goodness-of-fit (reduced χ²) is 0.88 at iteration 437

✅ Fitting complete.

Saving project 📦 'lam7o3_p021' to '../../../projects/refine-lam7o3-p021'

├── 📄 project.edi

├── 📁 structures/

│   └── 📄 lam7o3.edi

├── 📁 experiments/

│   └── 📄 p021.edi

├── 📁 analysis/

│   └── 📄 analysis.edi

└── 📁 reports/

    └── 📄 lam7o3_p021.html

Display the initial fit summary and the strongest parameter correlations.

In [31]:

project.display.fit.results()

project.display.fit.results()

⚙️ Settings used:

	Name	Value	Description
1	max_iterations	1000	Maximum solver iterations.

📋 Least-squares fit results:

	Metric	Value
1	🧪 Minimizer	lmfit (leastsq)
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	133.76
4	🔁 Iterations	435
5	📏 Goodness-of-fit (reduced χ²)	0.88
6	📏 R-factor (Rf, %)	2.39
7	📏 R-factor squared (Rf², %)	2.55
8	📏 Weighted R-factor (wR, %)	1.91

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	lam7o3	cell		length_a	Å	5.5000	5.4623	0.0002	0.69 % ↓
2	lam7o3	cell		length_b	Å	7.7000	7.7332	0.0004	0.43 % ↑
3	lam7o3	cell		length_c	Å	5.5000	5.4903	0.0002	0.18 % ↓
4	lam7o3	atom_site	La	fract_x		0.4800	0.4766	0.0002	0.72 % ↓
5	lam7o3	atom_site	La	fract_z		0.0040	0.0053	0.0006	31.47 % ↑
6	lam7o3	atom_site	La	adp_iso	Ų	0.1000	0.5942	0.0216	494.21 % ↑
7	lam7o3	atom_site	Ti	adp_iso	Ų	0.1000	0.3425	0.0291	242.51 % ↑
8	lam7o3	atom_site	O1	fract_x		0.5100	0.5070	0.0021	0.58 % ↓
9	lam7o3	atom_site	O1	fract_z		0.5700	0.5589	0.0035	1.96 % ↓
10	lam7o3	atom_site	O1	adp_iso	Ų	0.1000	0.5941	0.1573	494.11 % ↑
11	lam7o3	atom_site	O2	fract_x		0.2200	0.2303	0.0033	4.68 % ↑
12	lam7o3	atom_site	O2	fract_y		0.0300	0.0378	0.0020	25.89 % ↑
13	lam7o3	atom_site	O2	fract_z		0.2700	0.2794	0.0029	3.47 % ↑
14	p021	linked_structure	lam7o3	scale		0.0000	0.0000	0.0000	27.30 % ↓
15	p021	peak		broad_gauss_u	deg²	0.0400	0.0485	0.0174	21.19 % ↑
16	p021	peak		broad_gauss_v	deg²	-0.0100	-0.0066	0.0022	33.63 % ↓
17	p021	peak		broad_gauss_w	deg²	0.0010	0.0012	0.0001	18.94 % ↑
18	p021	peak		broad_lorentz_x	deg	0.1000	0.1637	0.0042	63.65 % ↑
19	p021	instrument		twotheta_offset	deg	0.0000	-0.0004	0.0001	N/A
20	p021	background	1	intensity		584.0282	624.0578	4.4509	6.85 % ↑
21	p021	background	2	intensity		616.4344	988.2570	4.8160	60.32 % ↑
22	p021	background	3	intensity		608.1060	654.0695	6.6736	7.56 % ↑
23	p021	background	4	intensity		594.6323	626.9762	6.4339	5.44 % ↑
24	p021	background	5	intensity		574.0299	565.5692	4.4366	1.47 % ↓
25	p021	background	6	intensity		457.7247	449.1935	3.8087	1.86 % ↓
26	p021	background	7	intensity		443.1398	431.1709	5.7132	2.70 % ↓
27	p021	background	8	intensity		433.0140	423.0607	4.3495	2.30 % ↓
28	p021	background	9	intensity		432.9268	424.7706	3.6020	1.88 % ↓
29	p021	background	10	intensity		421.2970	406.9977	3.1857	3.39 % ↓
30	p021	background	11	intensity		444.8967	425.4509	3.2033	4.37 % ↓
31	p021	background	12	intensity		453.7668	462.3426	2.3750	1.89 % ↑
32	p021	background	13	intensity		346.8399	348.6887	2.3885	0.53 % ↑
33	p021	background	14	intensity		357.7145	349.4878	2.1675	2.30 % ↓
34	p021	background	15	intensity		302.4308	290.9154	2.2866	3.81 % ↓
35	p021	background	16	intensity		278.3155	258.7194	5.5684	7.04 % ↓

• 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

In [32]:

project.display.fit.correlations(max_parameters=5)

project.display.fit.correlations(max_parameters=5)

Loading plot…

Display Pattern (After Initial Fit)¶

Inspect the full fitted pattern and the nonuniform low-angle background region.

In [33]:

project.display.pattern(expt_name='p021')

project.display.pattern(expt_name='p021')

Loading plot…

In [34]:

project.display.pattern(expt_name='p021', x_min=2.2, x_max=4.0)

project.display.pattern(expt_name='p021', x_min=2.2, x_max=4.0)

Loading plot…

Improve Background Estimate¶

With a fitted peak model available, repeat automatic estimation using the calculated peak contribution. This replaces the original points with a model-guided estimate. Replacement points are fixed by default, so mark their intensities free again.

In [35]:

experiment.background.auto_estimate(use_model=True)

experiment.background.auto_estimate(use_model=True)

In [36]:

experiment.background.show()

experiment.background.show()

Line-segment background points

	Position	Intensity
1	2.00330	576.07930
2	2.17160	638.44269
3	2.23470	662.65186
4	2.29780	695.09265
5	2.40300	752.13934
6	2.75000	948.88708
7	2.79210	960.58868
8	2.83410	967.29542
9	2.87620	968.71605
10	2.91820	964.86104
11	2.97080	953.16247
12	3.14960	876.76893
13	3.32840	781.24539
14	3.49660	699.51489
15	3.64390	640.61808
16	3.72800	601.48366
17	3.81210	556.51417
18	4.03300	428.78047
19	4.08550	400.97938
20	4.14860	371.19048
21	4.21170	346.60070
22	4.26430	331.07531
23	4.43260	318.49330
24	4.50620	328.33475
25	4.77960	400.14924
26	4.94790	435.19050
27	5.02150	440.50098
28	5.46320	422.43300
29	5.82070	425.38983
30	6.08360	420.22464
31	6.43070	417.57909
32	6.78820	410.64869
33	7.40870	409.16031
34	7.75570	427.93405
35	7.86090	433.58138
36	8.17640	441.57902
37	8.33410	442.13621
38	8.97560	435.40766
39	9.97460	432.32693
40	10.38480	396.20254
41	10.85800	351.40050
42	11.69930	347.23352
43	11.85700	347.83931
44	12.53010	340.09138
45	14.24420	301.93226
46	14.99090	260.58927

In [37]:

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

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

Run Fitting¶

In [38]:

project.analysis.fit()

project.analysis.fit()

Standard fitting

📋 Using experiment 🔬 'p021' for 'single' fitting

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

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.27	2.77
2	18	5.36	2.77
3	35	10.37	2.77
4	53	15.49	2.77
5	69	20.04	0.16	94.1% ↓
6	86	25.24	0.16
7	104	30.32	0.16
8	122	35.45	0.16
9	140	40.93	0.16
10	158	45.99	0.16
11	176	51.26	0.16
12	194	56.33	0.16
13	212	61.44	0.16
14	230	66.96	0.16
15	248	72.09	0.16
16	266	77.52	0.16
17	283	82.59	0.16
18	301	87.73	0.16
19	319	92.79	0.16
20	336	98.10	0.16
21	354	103.19	0.16
22	372	108.32	0.16
23	389	113.55	0.16
24	400	122.00	0.16

🏆 Best goodness-of-fit (reduced χ²) is 0.16 at iteration 399

✅ Fitting complete.

Saving project 📦 'lam7o3_p021' to '../../../projects/refine-lam7o3-p021'

├── 📄 project.edi

├── 📁 structures/

│   └── 📄 lam7o3.edi

├── 📁 experiments/

│   └── 📄 p021.edi

├── 📁 analysis/

│   └── 📄 analysis.edi

└── 📁 reports/

    └── 📄 lam7o3_p021.html

In [39]:

project.display.fit.results()

project.display.fit.results()

⚙️ Settings used:

	Name	Value	Description
1	max_iterations	1000	Maximum solver iterations.

📋 Least-squares fit results:

	Metric	Value
1	🧪 Minimizer	lmfit (leastsq)
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	122.00
4	🔁 Iterations	397
5	📏 Goodness-of-fit (reduced χ²)	0.16
6	📏 R-factor (Rf, %)	1.01
7	📏 R-factor squared (Rf², %)	1.61
8	📏 Weighted R-factor (wR, %)	1.48

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	lam7o3	cell		length_a	Å	5.4623	5.4622	0.0001	0.00 % ↓
2	lam7o3	cell		length_b	Å	7.7332	7.7331	0.0002	0.00 % ↓
3	lam7o3	cell		length_c	Å	5.4903	5.4902	0.0001	0.00 % ↓
4	lam7o3	atom_site	La	fract_x		0.4766	0.4767	0.0001	0.03 % ↑
5	lam7o3	atom_site	La	fract_z		0.0053	0.0045	0.0003	15.23 % ↓
6	lam7o3	atom_site	La	adp_iso	Ų	0.5942	0.6260	0.0095	5.35 % ↑
7	lam7o3	atom_site	Ti	adp_iso	Ų	0.3425	0.3376	0.0126	1.44 % ↓
8	lam7o3	atom_site	O1	fract_x		0.5070	0.5083	0.0010	0.25 % ↑
9	lam7o3	atom_site	O1	fract_z		0.5589	0.5708	0.0016	2.14 % ↑
10	lam7o3	atom_site	O1	adp_iso	Ų	0.5941	0.3583	0.0654	39.69 % ↓
11	lam7o3	atom_site	O2	fract_x		0.2303	0.2218	0.0014	3.71 % ↓
12	lam7o3	atom_site	O2	fract_y		0.0378	0.0340	0.0009	9.90 % ↓
13	lam7o3	atom_site	O2	fract_z		0.2794	0.2748	0.0014	1.63 % ↓
14	p021	linked_structure	lam7o3	scale		0.0000	0.0000	0.0000	0.49 % ↑
15	p021	peak		broad_gauss_u	deg²	0.0485	0.0513	0.0075	5.81 % ↑
16	p021	peak		broad_gauss_v	deg²	-0.0066	-0.0076	0.0010	15.15 % ↑
17	p021	peak		broad_gauss_w	deg²	0.0012	0.0012	0.0000	2.99 % ↑
18	p021	peak		broad_lorentz_x	deg	0.1637	0.1669	0.0025	2.00 % ↑
19	p021	instrument		twotheta_offset	deg	-0.0004	-0.0005	0.0001	25.23 % ↑
20	p021	background	1	intensity		576.0793	583.7057	4.4597	1.32 % ↑
21	p021	background	2	intensity		638.4427	637.0138	4.1454	0.22 % ↓
22	p021	background	3	intensity		662.6519	661.4446	5.3649	0.18 % ↓
23	p021	background	4	intensity		695.0927	691.5221	4.8034	0.51 % ↓
24	p021	background	5	intensity		752.1393	751.6693	3.3433	0.06 % ↓
25	p021	background	6	intensity		948.8871	967.4276	3.8652	1.95 % ↑
26	p021	background	7	intensity		960.5887	968.7470	7.8377	0.85 % ↑
27	p021	background	8	intensity		967.2954	981.1793	8.0417	1.44 % ↑
28	p021	background	9	intensity		968.7160	991.6844	8.0566	2.37 % ↑
29	p021	background	10	intensity		964.8610	978.2161	7.5124	1.38 % ↑
30	p021	background	11	intensity		953.1625	961.7701	5.2263	0.90 % ↑
31	p021	background	12	intensity		876.7689	877.8444	4.0526	0.12 % ↑
32	p021	background	13	intensity		781.2454	775.4285	3.7077	0.74 % ↓
33	p021	background	14	intensity		699.5149	689.0055	3.7740	1.50 % ↓
34	p021	background	15	intensity		640.6181	649.7581	4.0686	1.43 % ↑
35	p021	background	16	intensity		601.4837	615.5274	4.5230	2.33 % ↑
36	p021	background	17	intensity		556.5142	597.0830	3.4872	7.29 % ↑
37	p021	background	18	intensity		428.7805	551.2716	3.5847	28.57 % ↑
38	p021	background	19	intensity		400.9794	548.1756	5.2366	36.71 % ↑
39	p021	background	20	intensity		371.1905	583.1060	5.3080	57.09 % ↑
40	p021	background	21	intensity		346.6007	514.8713	5.4911	48.55 % ↑
41	p021	background	22	intensity		331.0753	523.4231	7.0097	58.10 % ↑
42	p021	background	23	intensity		318.4933	508.0645	4.9871	59.52 % ↑
43	p021	background	24	intensity		328.3347	497.3337	3.1594	51.47 % ↑
44	p021	background	25	intensity		400.1492	467.0731	2.6516	16.72 % ↑
45	p021	background	26	intensity		435.1905	452.4149	3.2856	3.96 % ↑
46	p021	background	27	intensity		440.5010	438.9108	2.5440	0.36 % ↓
47	p021	background	28	intensity		422.4330	420.9219	2.0607	0.36 % ↓
48	p021	background	29	intensity		425.3898	426.8501	2.1298	0.34 % ↑
49	p021	background	30	intensity		420.2246	418.4449	2.8849	0.42 % ↓
50	p021	background	31	intensity		417.5791	417.3651	2.0005	0.05 % ↓
51	p021	background	32	intensity		410.6487	411.7042	1.8668	0.26 % ↑
52	p021	background	33	intensity		409.1603	410.5187	2.0424	0.33 % ↑
53	p021	background	34	intensity		427.9340	431.2367	2.5606	0.77 % ↑
54	p021	background	35	intensity		433.5814	435.4782	2.5649	0.44 % ↑
55	p021	background	36	intensity		441.5790	441.4029	2.4132	0.04 % ↓
56	p021	background	37	intensity		442.1362	440.3573	1.9173	0.40 % ↓
57	p021	background	38	intensity		435.4077	436.7917	1.4636	0.32 % ↑
58	p021	background	39	intensity		432.3269	440.6909	1.6017	1.93 % ↑
59	p021	background	40	intensity		396.2025	399.2762	1.7705	0.78 % ↑
60	p021	background	41	intensity		351.4005	345.2537	1.4677	1.75 % ↓
61	p021	background	42	intensity		347.2335	344.8956	1.8053	0.67 % ↓
62	p021	background	43	intensity		347.8393	350.5874	1.6163	0.79 % ↑
63	p021	background	44	intensity		340.0914	339.5041	1.1339	0.17 % ↓
64	p021	background	45	intensity		301.9323	302.4185	1.0953	0.16 % ↑
65	p021	background	46	intensity		260.5893	273.9354	1.5148	5.12 % ↑

• 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

In [40]:

project.display.fit.correlations(max_parameters=5)

project.display.fit.correlations(max_parameters=5)

Loading plot…

Display Pattern (After Final Fit)¶

In [41]:

project.display.pattern(expt_name='p021')

project.display.pattern(expt_name='p021')

Loading plot…

In [42]:

project.display.pattern(expt_name='p021', x_min=2.2, x_max=4.0)

project.display.pattern(expt_name='p021', x_min=2.2, x_max=4.0)

Loading plot…

📊 Report¶

The HTML report is written automatically when the project is saved. PDF generation can be enabled before the final save when required.

In [43]:

# Enable PDF report generation before the last save (time consuming)
# project.report.pdf = True

# Enable PDF report generation before the last save (time consuming) # project.report.pdf = True

💾 Save Project¶

In [44]:

project.save()

project.save()

Saving project 📦 'lam7o3_p021' to '../../../projects/refine-lam7o3-p021'

├── 📄 project.edi

├── 📁 structures/

│   └── 📄 lam7o3.edi

├── 📁 experiments/

│   └── 📄 p021.edi

├── 📁 analysis/

│   └── 📄 analysis.edi

└── 📁 reports/

    └── 📄 lam7o3_p021.html

In [ ]: