Structure Refinement: YAlO3+Al2O3, SPODI¶

This example demonstrates a staged two-phase Rietveld refinement of yttrium aluminium perovskite YAlO3 (or YAP) with a small Al2O3 impurity using constant wavelength neutron powder diffraction data measured at 3 K on SPODI at MLZ.

The workflow defines both structures, configures the experiment, and refines the cell, scale, profile, background, and atom parameters of both phases in stages.

🛠️ Import Library¶

In [2]:

import easydiffraction as edi

import easydiffraction as edi

📦 Define Project¶

Create Project¶

In [3]:

project = edi.Project(
    name='yap_3k',
    description='Two-phase YAlO3 and Al2O3 refinement using 3 K data from SPODI at MLZ.',
)

project = edi.Project( name='yap_3k', description='Two-phase YAlO3 and Al2O3 refinement using 3 K data from SPODI at MLZ.', )

Save Initial Project¶

In [4]:

project.save_as(dir_path='projects/refine-yap-3k')

project.save_as(dir_path='projects/refine-yap-3k')

Saving project 📦 'yap_3k' to '../../../projects/refine-yap-3k'

├── 📄 project.edi
├── 📁 structures/
├── 📁 experiments/
├── 📁 analysis/
│   └── 📄 analysis.edi
└── 📁 reports/
    └── 📄 yap_3k.html

🧩 Define Structures¶

Create Structure 1: YAlO3¶

Preserve the orthorhombic Pbnm setting used in FullProf. In EasyDiffraction this is represented by the standard space-group symbol P n m a with coordinate-system code cab. The cell axes and atom coordinates below therefore stay in the original Pbnm setting.

FullProf's PCR occupancies include the site multiplicity divided by the general-position multiplicity. Here each atom site is fully occupied: the PCR values 0.5 for Y, Al, and O1, and 1.0 for O2, all become an occupancy of 1.0. Displacement parameters are entered as Biso, matching the PCR file.

In [5]:

yap_cif = """
data_yap

_cell.length_a 5.18
_cell.length_b 5.33
_cell.length_c 7.37
_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 cab

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
Y  Y   0.0100  0.5500 0.2500 1.0 0.12 Biso
Al Al  0.0000  0.0000 0.0000 1.0 0.13 Biso
O1 O  -0.0800 -0.0200 0.2500 1.0 0.06 Biso
O2 O   0.2000  0.2900 0.0400 1.0 0.14 Biso
"""

yap_cif = """ data_yap _cell.length_a 5.18 _cell.length_b 5.33 _cell.length_c 7.37 _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 cab 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 Y Y 0.0100 0.5500 0.2500 1.0 0.12 Biso Al Al 0.0000 0.0000 0.0000 1.0 0.13 Biso O1 O -0.0800 -0.0200 0.2500 1.0 0.06 Biso O2 O 0.2000 0.2900 0.0400 1.0 0.14 Biso """

In [6]:

project.structures.add_from_cif_str(yap_cif)

project.structures.add_from_cif_str(yap_cif)

In [7]:

yap = project.structures['yap']

yap = project.structures['yap']

Display Structure 1: YAlO3¶

In [8]:

yap.show_as_text()

yap.show_as_text()

Structure 🧩 'yap' as text

	Edi
1	data_yap
2
3	_cell.length_a 5.18
4	_cell.length_b 5.33
5	_cell.length_c 7.37
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 cab
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	Y Y 0.01 0.55 0.25 c 4 1. 0.12 Biso
28	Al Al 0. 0. 0. a 4 1. 0.13 Biso
29	O1 O -0.08 -0.02 0.25 c 4 1. 0.06 Biso
30	O2 O 0.2 0.29 0.04 d 8 1. 0.14 Biso

In [9]:

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

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

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

Loading plot…

drag = rotate
wheel = zoom
right-drag = pan

Create Structure 2: Al2O3¶

Define the corundum impurity in the hexagonal setting of R-3c. The FullProf PCR occupancies of 2/3 for Al and 1 for O also describe fully occupied sites. Refine its two independent cell lengths, Al z and O x coordinates, and both Biso values, as specified by the PCR codewords. The PCR contains a negative Al Biso. EasyDiffraction requires a nonnegative input value, so start this parameter at 0.1 Ų and refine it alongside O Biso.

In [10]:

alumina = edi.StructureFactory.from_scratch(name='alumina')

alumina = edi.StructureFactory.from_scratch(name='alumina')

Set Space Group¶

In [11]:

alumina.space_group.name_h_m = 'R -3 c'
alumina.space_group.coord_system_code = 'h'

alumina.space_group.name_h_m = 'R -3 c' alumina.space_group.coord_system_code = 'h'

Set Unit Cell¶

In [12]:

alumina.cell.length_a = 4.75
alumina.cell.length_c = 12.95

alumina.cell.length_a = 4.75 alumina.cell.length_c = 12.95

Set Atom Sites¶

In [13]:

alumina.atom_sites.create(
    id='Al1',
    type_symbol='Al',
    fract_x=0.0,
    fract_y=0.0,
    fract_z=0.33351,
    occupancy=1.0,
    adp_type='Biso',
    adp_iso=0.1,
)
alumina.atom_sites.create(
    id='O1',
    type_symbol='O',
    fract_x=0.3503,
    fract_y=0.0,
    fract_z=0.25,
    occupancy=1.0,
    adp_type='Biso',
    adp_iso=1.22884,
)

alumina.atom_sites.create( id='Al1', type_symbol='Al', fract_x=0.0, fract_y=0.0, fract_z=0.33351, occupancy=1.0, adp_type='Biso', adp_iso=0.1, ) alumina.atom_sites.create( id='O1', type_symbol='O', fract_x=0.3503, fract_y=0.0, fract_z=0.25, occupancy=1.0, adp_type='Biso', adp_iso=1.22884, )

In [14]:

project.structures.add(alumina)

project.structures.add(alumina)

Display Structure 2: Al2O3¶

In [15]:

alumina.show_as_text()

alumina.show_as_text()

Structure 🧩 'alumina' as text

	Edi
1	data_alumina
2
3	_cell.length_a 4.75
4	_cell.length_b 4.75
5	_cell.length_c 12.95
6	_cell.angle_alpha 90.
7	_cell.angle_beta 90.
8	_cell.angle_gamma 120.
9
10	_space_group.name_h_m "R -3 c"
11	_space_group.coord_system_code h
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	Al1 Al 0. 0. 0.33351 c 12 1. 0.1 Biso
28	O1 O 0.3503 0. 0.25 e 18 1. 1.22884 Biso

In [16]:

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

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

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

Loading plot…

drag = rotate
wheel = zoom
right-drag = pan

🔬 Define Experiment¶

Download Measured Data¶

Download the YAlO3 + Al2O3 pattern from the EasyDiffraction online data repository. The three columns contain 2-theta in degrees, intensity, and its standard uncertainty. They are copied from the original SPODI dataset without changing the measured values.

In [17]:

data_path = edi.download_data('meas-yap-spodi', destination='data')

data_path = edi.download_data('meas-yap-spodi', destination='data')

Getting data...

Data 'meas-yap-spodi': YAlO3 + Al2O3, SPODI (MLZ), 3 K

✅ Data 'meas-yap-spodi' downloaded to '../../../data/meas-yap-spodi.xye'

In [18]:

project.experiments.add_from_data_path(
    name='yap_3k',
    data_path=data_path,
    sample_form='powder',
    beam_mode='constant wavelength',
    radiation_probe='neutron',
)

project.experiments.add_from_data_path( name='yap_3k', data_path=data_path, sample_form='powder', beam_mode='constant wavelength', radiation_probe='neutron', )

Data loaded successfully

Experiment 🔬 'yap_3k'. Number of data points: 3000.

In [19]:

expt = project.experiments['yap_3k']

expt = project.experiments['yap_3k']

Set Instrument¶

Use the neutron wavelength reported for the SPODI dataset and start with zero 2-theta offset.

In [20]:

expt.instrument.setup_wavelength = 1.54816
expt.instrument.calib_twotheta_offset = 0.0

expt.instrument.setup_wavelength = 1.54816 expt.instrument.calib_twotheta_offset = 0.0

Set Peak Profile¶

Use a pseudo-Voigt profile with Bérar-Baldinozzi asymmetry.

In [21]:

expt.peak.show_supported()

expt.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 [22]:

expt.peak.type = 'pseudo-voigt + berar-baldinozzi asymmetry'

expt.peak.type = 'pseudo-voigt + berar-baldinozzi asymmetry'

⚠️ Switching peak profile type adds these settings with defaults:
• asym_beba_a0=0.0
• asym_beba_a1=0.0
• asym_beba_b0=0.0
• asym_beba_b1=0.0

Peak profile type for experiment 'yap_3k' changed to

pseudo-voigt + berar-baldinozzi asymmetry

In [23]:

expt.peak.broad_gauss_u = 0.04
expt.peak.broad_gauss_v = -0.05
expt.peak.broad_gauss_w = 0.10
expt.peak.broad_lorentz_x = 0.0
expt.peak.broad_lorentz_y = 0.01

expt.peak.broad_gauss_u = 0.04 expt.peak.broad_gauss_v = -0.05 expt.peak.broad_gauss_w = 0.10 expt.peak.broad_lorentz_x = 0.0 expt.peak.broad_lorentz_y = 0.01

In [24]:

expt.peak.asym_beba_a0 = 0.0
expt.peak.asym_beba_b0 = 0.0
expt.peak.asym_beba_a1 = 0.0
expt.peak.asym_beba_b1 = 0.0

expt.peak.asym_beba_a0 = 0.0 expt.peak.asym_beba_b0 = 0.0 expt.peak.asym_beba_a1 = 0.0 expt.peak.asym_beba_b1 = 0.0

In [25]:

expt.peak.cutoff_fwhm = 8.0

expt.peak.cutoff_fwhm = 8.0

Set Absorption¶

Apply the cylindrical-sample Hewat correction with the absorption radius product from FullProf.

In [26]:

expt.absorption.type = 'cylinder-hewat'
expt.absorption.mu_r = 0.0221

expt.absorption.type = 'cylinder-hewat' expt.absorption.mu_r = 0.0221

Absorption type changed to

cylinder-hewat

Set Excluded Regions¶

In [27]:

expt.excluded_regions.create(id='1', start=0.0, end=4.0)
expt.excluded_regions.create(id='2', start=153.95, end=180.0)

expt.excluded_regions.create(id='1', start=0.0, end=4.0) expt.excluded_regions.create(id='2', start=153.95, end=180.0)

Set Background¶

Estimate the initial line-segment background from the measured pattern. This first estimate does not use a calculated structural model.

In [28]:

expt.background.auto_estimate(use_model=False)

expt.background.auto_estimate(use_model=False)

In [29]:

expt.background.show()

expt.background.show()

Line-segment background points

	Position	Intensity
1	4.05000	1303.00000
2	5.95000	2193.97204
3	18.20000	1997.00000
4	37.35000	2147.12282
5	47.70000	2092.00000
6	59.25000	1977.00000
7	151.95000	3468.48635

Set Linked Structures¶

Give each phase an independent scale factor. Scale factors are fitted intensity multipliers, rather than phase weight fractions.

In [30]:

expt.linked_structures.create(structure_id='yap', scale=30)
expt.linked_structures.create(structure_id='alumina', scale=0.1)

expt.linked_structures.create(structure_id='yap', scale=30) expt.linked_structures.create(structure_id='alumina', scale=0.1)

In [31]:

expt.show_as_text()

expt.show_as_text()

Experiment 🔬 'yap_3k' as text

	Edi
1	data_yap_3k
2
3	_experiment_type.sample_form powder
4	_experiment_type.beam_mode "constant wavelength"
5	_experiment_type.radiation_probe neutron
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.asym_beba_a0 0.
16	_peak.asym_beba_b0 0.
17	_peak.asym_beba_a1 0.
18	_peak.asym_beba_b1 0.
19	_peak.broad_gauss_u 0.04
20	_peak.broad_gauss_v -0.05
21	_peak.broad_gauss_w 0.1
22	_peak.broad_lorentz_x 0.
23	_peak.broad_lorentz_y 0.01
24	_peak.cutoff_fwhm 8.
25	_peak.type cwl-pseudo-voigt-berar-baldinozzi-asymmetry
26
27	_instrument.setup_wavelength 1.54816
28	_instrument.setup_wavelength_2 0.
29	_instrument.setup_wavelength_2_to_1_ratio 0.
30	_instrument.calib_twotheta_offset 0.
31	_instrument.calib_sample_displacement 0.
32	_instrument.calib_sample_transparency 0.
33
34	_absorption.type cylinder-hewat
35	_absorption.mu_r 0.0221
36
37	loop_
38	_linked_structure.structure_id
39	_linked_structure.scale
40	yap 30
41	alumina 0.1
42
43	loop_
44	_excluded_region.id
45	_excluded_region.start
46	_excluded_region.end
47	1 0. 4.
48	2 153.95 180.
49
50	_background.type line-segment
51
52	loop_
53	_background.id
54	_background.position
55	_background.intensity
56	1 4.05 1303.
57	2 5.95 2193.97203967
58	3 18.2 1997.
59	4 37.35 2147.12281802
60	5 47.7 2092.
61	6 59.25 1977.
62	7 151.95 3468.48635134
63
64	loop_
65	_data.two_theta
66	_data.id
67	_data.d_spacing
68	_data.intensity_meas
69	_data.intensity_meas_su
70	_data.intensity_calc
71	_data.intensity_bkg
72	_data.calc_status
73	2. 1 0. 0. 1. 0. 0. excl
74	2.05 2 0. 0. 1. 0. 0. excl
75	2.1 3 0. 0. 1. 0. 0. excl
76	2.15 4 0. 0. 1. 0. 0. excl
77	2.2 5 0. 0. 1. 0. 0. excl
78	2.25 6 0. 0. 1. 0. 0. excl
79	2.3 7 0. 0. 1. 0. 0. excl
80	2.35 8 0. 0. 1. 0. 0. excl
81	2.4 9 0. 0. 1. 0. 0. excl
82	2.45 10 0. 0. 1. 0. 0. excl
83	...
84	151.5 2991 0.79865385 3354. 57. 4902.79239884 3461.24612633 incl
85	151.55 2992 0.79856544 3438. 58. 4938.57755495 3462.05059578 incl
86	151.6 2993 0.79847719 3441. 58. 4948.05195479 3462.85506522 incl
87	151.65 2994 0.79838912 3396. 58. 4927.349773 3463.65953467 incl
88	151.7 2995 0.79830122 3429. 58. 4876.02736329 3464.46400411 incl
89	151.75 2996 0.79821349 3448. 58. 4796.74526503 3465.26847356 incl
90	151.8 2997 0.79812593 3532. 59. 4694.51730196 3466.072943 incl
91	151.85 2998 0.79803854 3693. 60. 4575.78062248 3466.87741245 incl
92	151.9 2999 0.79795132 3780. 61. 4447.50441615 3467.68188189 incl
93	151.95 3000 0.79786427 3784. 61. 4316.44341414 3468.48635134 incl
94
95	loop_
96	_refln.id
97	_refln.structure_id
98	_refln.d_spacing
99	_refln.sin_theta_over_lambda
100	_refln.index_h
101	_refln.index_k
102	_refln.index_l
103	_refln.f_calc
104	_refln.f_squared_calc
105	_refln.two_theta
106	1 yap 7.37 0.06784261 0 0 1 0. 0. 12.05792913
107	2 yap 5.33 0.09380863 0 1 0 0. 0. 16.7012855
108	3 yap 5.18 0.0965251 1 0 0 0. 0. 17.18852072
109	4 yap 4.31891055 0.11576994 0 1 1 0. 0. 20.64987025
110	5 yap 4.23794054 0.11798183 1 0 1 0.17288679 0.02988984 21.0488605
111	6 yap 3.71471093 0.13459998 1 1 0 0.10027655 0.01005539 24.05513521
112	7 yap 3.685 0.13568521 0 0 2 0.02219992 0.00049284 24.25201992
113	8 yap 3.31717219 0.15073079 1 1 1 0.86359044 0.74578846 26.98939657
114	9 yap 3.03110835 0.16495616 0 1 2 0. 0. 29.5920322
115	10 yap 3.00271487 0.16651598 1 0 2 0. 0. 29.878338
116	...
117	365 alumina 0.81930006 0.61027702 4 -1 11 1.19108015 1.41867193 141.75014806
118	366 alumina 0.81616218 0.61262334 5 0 2 4.22226892 17.82755481 143.04165903
119	367 alumina 0.81529214 0.6132771 5 -2 -8 0.52947251 0.28034113 143.40934214
120	368 alumina 0.81138823 0.61622782 2 -1 15 5.6072072 31.44077264 145.11538603
121	369 alumina 0.80534833 0.62084936 4 0 10 0.67018386 0.44914641 147.96362502
122	370 alumina 0.79864545 0.62606004 4 -2 12 0.91581608 0.8387191 151.50474768
123	371 alumina 0.79738004 0.62705357 5 0 -4 0.18323927 0.03357663 152.22995668
124	372 alumina 0.79495324 0.62896781 3 -1 -14 2.05217352 4.21141616 153.68237943
125	373 alumina 0.79414921 0.6296046 1 0 16 2.43194259 5.91434477 154.18330783
126	374 alumina 0.79166667 0.63157895 6 -3 0 9.73709759 94.8110695 155.80110779

🚀 Perform Analysis¶

Display Initial Pattern¶

In [32]:

project.display.pattern(expt_name='yap_3k')

project.display.pattern(expt_name='yap_3k')

Loading plot…

In [33]:

project.display.pattern(expt_name='yap_3k', x_min=134, x_max=146)

project.display.pattern(expt_name='yap_3k', x_min=134, x_max=146)

Loading plot…

Select Calculator¶

In [34]:

expt.calculator.show_supported()

expt.calculator.show_supported()

Calculator types

		Type	Description
1		crysfml	CrysFML library for crystallographic calculations
2	*	cryspy	CrysPy library for crystallographic calculations

In [35]:

expt.calculator.type = 'cryspy'

expt.calculator.type = 'cryspy'

Calculator for experiment 'yap_3k' already set to

cryspy

Select Minimizer¶

In [36]:

project.analysis.minimizer.show_supported()

project.analysis.minimizer.show_supported()

Minimizer types

		Type	Description
1		bumps	BUMPS library using the default Levenberg-Marquardt method
2		bumps (amoeba)	BUMPS library with Nelder-Mead simplex method
3		bumps (de)	BUMPS library with differential evolution method
4		bumps (dream)	BUMPS library with DREAM Bayesian sampling
5		bumps (lm)	BUMPS library with Levenberg-Marquardt method
6		dfols	DFO-LS library for derivative-free least-squares optimization
7		emcee	emcee affine-invariant ensemble Bayesian sampling
8		lmfit	LMFIT library using the default Levenberg-Marquardt method
9		lmfit (least_squares)	LMFIT library with SciPy's trust region reflective algorithm
10	*	lmfit (leastsq)	LMFIT library with Levenberg-Marquardt least squares method

In [37]:

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

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

⚠️ Switching minimizer type removes these settings:
• gradient_tolerance

Current minimizer changed to

bumps (lm)

In [38]:

project.analysis.minimizer.max_iterations = 500
project.analysis.minimizer.chi_square_change_tolerance = 1e-2

project.analysis.minimizer.max_iterations = 500 project.analysis.minimizer.chi_square_change_tolerance = 1e-2

Perform Fit 1/3: Cell, Scale, and Background¶

First refine the independent cell lengths of both phases, both phase scales, the instrument zero offset, and the automatically estimated background intensities. Hexagonal symmetry couples the Al2O3 b length to a, leaving only a and c independent.

In [39]:

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

alumina.cell.length_a.free = True
alumina.cell.length_c.free = True

expt.linked_structures['yap'].scale.free = True
expt.linked_structures['alumina'].scale.free = True

expt.instrument.calib_twotheta_offset.free = True

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

yap.cell.length_a.free = True yap.cell.length_b.free = True yap.cell.length_c.free = True alumina.cell.length_a.free = True alumina.cell.length_c.free = True expt.linked_structures['yap'].scale.free = True expt.linked_structures['alumina'].scale.free = True expt.instrument.calib_twotheta_offset.free = True for point in expt.background: point.intensity.free = True

In [40]:

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	yap	cell		length_a	5.18000		-inf	inf	Å
2	yap	cell		length_b	5.33000		-inf	inf	Å
3	yap	cell		length_c	7.37000		-inf	inf	Å
4	alumina	cell		length_a	4.75000		-inf	inf	Å
5	alumina	cell		length_c	12.95000		-inf	inf	Å
6	yap_3k	linked_structure	yap	scale	30.00000		-inf	inf
7	yap_3k	linked_structure	alumina	scale	0.10000		-inf	inf
8	yap_3k	instrument		twotheta_offset	0.00000		-inf	inf	deg
9	yap_3k	background	1	intensity	1303.00000		-inf	inf
10	yap_3k	background	2	intensity	2193.97204		-inf	inf
11	yap_3k	background	3	intensity	1997.00000		-inf	inf
12	yap_3k	background	4	intensity	2147.12282		-inf	inf
13	yap_3k	background	5	intensity	2092.00000		-inf	inf
14	yap_3k	background	6	intensity	1977.00000		-inf	inf
15	yap_3k	background	7	intensity	3468.48635		-inf	inf

In [41]:

project.analysis.fit()

project.analysis.fit()

Standard fitting

📋 Using experiment 🔬 'yap_3k' for 'single' fitting

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

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.37	1170.67
2	17	2.61	266.00	77.3% ↓
3	33	4.80	74.80	71.9% ↓
4	49	7.37	66.39	11.3% ↓
5	67	17.89	66.03

🏆 Best goodness-of-fit (reduced χ²) is 66.03 at iteration 67

✅ Fitting complete.

In [42]:

project.display.fit.results()

project.display.fit.results()

⚙️ Settings used:

	Name	Value	Description
1	max_iterations	500	Maximum solver iterations.
2	chi_square_change_tolerance	0.01	Relative change in the objective (chi-square) used to stop fitting.
3	parameter_change_tolerance	1e-08	Relative change in fitted parameters used to stop fitting.

📋 Least-squares fit results:

	Metric	Value
1	🧪 Minimizer	bumps (lm)
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	17.89
4	📏 Goodness-of-fit (reduced χ²)	66.03
5	📏 R-factor (Rf, %)	8.19
6	📏 R-factor squared (Rf², %)	10.54
7	📏 Weighted R-factor (wR, %)	10.63

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	yap	cell		length_a	Å	5.1800	5.1736	0.0001	0.12 % ↓
2	yap	cell		length_b	Å	5.3300	5.3278	0.0001	0.04 % ↓
3	yap	cell		length_c	Å	7.3700	7.3624	0.0001	0.10 % ↓
4	alumina	cell		length_a	Å	4.7500	4.7556	0.0012	0.12 % ↑
5	alumina	cell		length_c	Å	12.9500	12.9893	0.0063	0.30 % ↑
6	yap_3k	linked_structure	yap	scale		30.0000	27.0708	0.0851	9.76 % ↓
7	yap_3k	linked_structure	alumina	scale		0.1000	0.1940	0.0166	93.98 % ↑
8	yap_3k	instrument		twotheta_offset	deg	0.0000	0.0044	0.0010	N/A
9	yap_3k	background	1	intensity		1303.0000	1364.5791	89.8261	4.73 % ↑
10	yap_3k	background	2	intensity		2193.9720	2366.3256	42.1391	7.86 % ↑
11	yap_3k	background	3	intensity		1997.0000	2014.4755	28.6320	0.88 % ↑
12	yap_3k	background	4	intensity		2147.1228	2231.3583	33.2468	3.92 % ↑
13	yap_3k	background	5	intensity		2092.0000	2238.6534	39.8940	7.01 % ↑
14	yap_3k	background	6	intensity		1977.0000	2069.8340	21.0995	4.70 % ↑
15	yap_3k	background	7	intensity		3468.4864	2414.2626	24.4757	30.39 % ↓

• 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

Perform Fit 2/3: Peak Profile¶

Add the Gaussian and Lorentzian broadening and asymmetry parameters to the refinement. The background intensities remain free.

In [43]:

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

expt.peak.asym_beba_a0.free = True
# expt.peak.asym_beba_b0.free = True
# expt.peak.asym_beba_a1.free = True
expt.peak.asym_beba_b1.free = True

expt.peak.broad_gauss_u.free = True expt.peak.broad_gauss_v.free = True expt.peak.broad_gauss_w.free = True expt.peak.broad_lorentz_y.free = True expt.peak.asym_beba_a0.free = True # expt.peak.asym_beba_b0.free = True # expt.peak.asym_beba_a1.free = True expt.peak.asym_beba_b1.free = True

In [44]:

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	yap	cell		length_a	5.17364	0.00007	-inf	inf	Å
2	yap	cell		length_b	5.32778	0.00007	-inf	inf	Å
3	yap	cell		length_c	7.36239	0.00010	-inf	inf	Å
4	alumina	cell		length_a	4.75560	0.00119	-inf	inf	Å
5	alumina	cell		length_c	12.98925	0.00626	-inf	inf	Å
6	yap_3k	linked_structure	yap	scale	27.07084	0.08509	-inf	inf
7	yap_3k	linked_structure	alumina	scale	0.19398	0.01660	-inf	inf
8	yap_3k	peak		asym_beba_a0	0.00000		-inf	inf
9	yap_3k	peak		asym_beba_b1	0.00000		-inf	inf
10	yap_3k	peak		broad_gauss_u	0.04000		-inf	inf	deg²
11	yap_3k	peak		broad_gauss_v	-0.05000		-inf	inf	deg²
12	yap_3k	peak		broad_gauss_w	0.10000		-inf	inf	deg²
13	yap_3k	peak		broad_lorentz_y	0.01000		-inf	inf	deg
14	yap_3k	instrument		twotheta_offset	0.00445	0.00096	-inf	inf	deg
15	yap_3k	background	1	intensity	1364.57909	89.82607	-inf	inf
16	yap_3k	background	2	intensity	2366.32560	42.13908	-inf	inf
17	yap_3k	background	3	intensity	2014.47546	28.63205	-inf	inf
18	yap_3k	background	4	intensity	2231.35827	33.24678	-inf	inf
19	yap_3k	background	5	intensity	2238.65341	39.89401	-inf	inf
20	yap_3k	background	6	intensity	2069.83400	21.09954	-inf	inf
21	yap_3k	background	7	intensity	2414.26259	24.47574	-inf	inf

In [45]:

project.analysis.fit()

project.analysis.fit()

Standard fitting

📋 Using experiment 🔬 'yap_3k' for 'single' fitting

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

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.15	66.17
2	23	4.77	59.24	10.5% ↓
3	45	9.89	58.60	1.1% ↓
4	69	22.28	58.50

🏆 Best goodness-of-fit (reduced χ²) is 58.50 at iteration 69

✅ Fitting complete.

In [46]:

project.display.fit.results()

project.display.fit.results()

⚙️ Settings used:

	Name	Value	Description
1	max_iterations	500	Maximum solver iterations.
2	chi_square_change_tolerance	0.01	Relative change in the objective (chi-square) used to stop fitting.
3	parameter_change_tolerance	1e-08	Relative change in fitted parameters used to stop fitting.

📋 Least-squares fit results:

	Metric	Value
1	🧪 Minimizer	bumps (lm)
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	22.28
4	📏 Goodness-of-fit (reduced χ²)	58.50
5	📏 R-factor (Rf, %)	7.66
6	📏 R-factor squared (Rf², %)	10.03
7	📏 Weighted R-factor (wR, %)	9.99

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	yap	cell		length_a	Å	5.1736	5.1728	0.0001	0.02 % ↓
2	yap	cell		length_b	Å	5.3278	5.3270	0.0001	0.01 % ↓
3	yap	cell		length_c	Å	7.3624	7.3610	0.0002	0.02 % ↓
4	alumina	cell		length_a	Å	4.7556	4.7577	0.0011	0.04 % ↑
5	alumina	cell		length_c	Å	12.9893	12.9761	0.0059	0.10 % ↓
6	yap_3k	linked_structure	yap	scale		27.0708	28.0693	0.1137	3.69 % ↑
7	yap_3k	linked_structure	alumina	scale		0.1940	0.1941	0.0161	0.07 % ↑
8	yap_3k	peak		asym_beba_a0		0.0000	-0.1818	0.0152	N/A
9	yap_3k	peak		asym_beba_b1		0.0000	-0.0612	0.0050	N/A
10	yap_3k	peak		broad_gauss_u	deg²	0.0400	0.0364	0.0020	8.95 % ↓
11	yap_3k	peak		broad_gauss_v	deg²	-0.0500	-0.0510	0.0041	2.07 % ↑
12	yap_3k	peak		broad_gauss_w	deg²	0.1000	0.0985	0.0022	1.55 % ↓
13	yap_3k	peak		broad_lorentz_y	deg	0.0100	0.0298	0.0019	198.09 % ↑
14	yap_3k	instrument		twotheta_offset	deg	0.0044	0.0059	0.0027	33.39 % ↑
15	yap_3k	background	1	intensity		1364.5791	1368.9347	84.5448	0.32 % ↑
16	yap_3k	background	2	intensity		2366.3256	2355.8542	39.6696	0.44 % ↓
17	yap_3k	background	3	intensity		2014.4755	2037.0481	27.0053	1.12 % ↑
18	yap_3k	background	4	intensity		2231.3583	2140.9844	32.0606	4.05 % ↓
19	yap_3k	background	5	intensity		2238.6534	2025.1406	41.3151	9.54 % ↓
20	yap_3k	background	6	intensity		2069.8340	1954.5600	21.6874	5.57 % ↓
21	yap_3k	background	7	intensity		2414.2626	2293.7219	26.6836	4.99 % ↓

• 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

Perform Fit 3/3: Model-Guided Background and Atom Parameters¶

Replace the initial background with a new estimate based on the fitted peak model. Automatically generated points are fixed by default, so mark their intensities free before fitting them with the atom parameters.

In [47]:

expt.background.auto_estimate(use_model=True)

expt.background.auto_estimate(use_model=True)

In [48]:

expt.background.show()

expt.background.show()

Line-segment background points

	Position	Intensity
1	4.05000	1303.00000
2	5.85000	2173.00000
3	7.30000	2414.80830
4	17.85000	2010.28100
5	24.75000	2041.00000
6	35.00000	1968.46433
7	39.35000	1606.73948
8	41.45000	0.00000
9	43.95000	1695.97793
10	44.95000	2367.00000
11	47.65000	1691.86967
12	48.15000	1321.52032
13	51.60000	1934.23588
14	74.60000	1881.37866
15	101.10000	1996.38773
16	122.20000	1525.63663
17	129.65000	3108.73377
18	142.50000	2493.86153
19	144.75000	2295.53422
20	151.95000	2451.55313

In [49]:

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

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

Refine the independent Y and O coordinates in Pbnm and the isotropic displacement parameters of both phases. Symmetry keeps Y and O1 in YAlO3 at z = 1/4 and Al at the origin. For Al2O3, refine Al1 z and O1 x. Occupancies remain fixed at 1.0, and all coordinates fixed by symmetry remain fixed.

In [50]:

yap.atom_sites['Y'].fract_x.free = True
yap.atom_sites['Y'].fract_y.free = True
yap.atom_sites['O1'].fract_x.free = True
yap.atom_sites['O1'].fract_y.free = True
yap.atom_sites['O2'].fract_x.free = True
yap.atom_sites['O2'].fract_y.free = True
yap.atom_sites['O2'].fract_z.free = True

alumina.atom_sites['Al1'].fract_z.free = True
alumina.atom_sites['O1'].fract_x.free = True

for structure in (yap, alumina):
    for atom in structure.atom_sites:
        atom.adp_iso.free = True

yap.atom_sites['Y'].fract_x.free = True yap.atom_sites['Y'].fract_y.free = True yap.atom_sites['O1'].fract_x.free = True yap.atom_sites['O1'].fract_y.free = True yap.atom_sites['O2'].fract_x.free = True yap.atom_sites['O2'].fract_y.free = True yap.atom_sites['O2'].fract_z.free = True alumina.atom_sites['Al1'].fract_z.free = True alumina.atom_sites['O1'].fract_x.free = True for structure in (yap, alumina): for atom in structure.atom_sites: atom.adp_iso.free = True

In [51]:

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	yap	cell		length_a	5.17278	0.00012	-inf	inf	Å
2	yap	cell		length_b	5.32698	0.00012	-inf	inf	Å
3	yap	cell		length_c	7.36105	0.00017	-inf	inf	Å
4	yap	atom_site	Y	fract_x	0.01000		-inf	inf
5	yap	atom_site	Y	fract_y	0.55000		-inf	inf
6	yap	atom_site	Y	adp_iso	0.12000		-inf	inf	Ų
7	yap	atom_site	Al	adp_iso	0.13000		-inf	inf	Ų
8	yap	atom_site	O1	fract_x	-0.08000		-inf	inf
9	yap	atom_site	O1	fract_y	-0.02000		-inf	inf
10	yap	atom_site	O1	adp_iso	0.06000		-inf	inf	Ų
11	yap	atom_site	O2	fract_x	0.20000		-inf	inf
12	yap	atom_site	O2	fract_y	0.29000		-inf	inf
13	yap	atom_site	O2	fract_z	0.04000		-inf	inf
14	yap	atom_site	O2	adp_iso	0.14000		-inf	inf	Ų
15	alumina	cell		length_a	4.75771	0.00112	-inf	inf	Å
16	alumina	cell		length_c	12.97614	0.00589	-inf	inf	Å
17	alumina	atom_site	Al1	fract_z	0.33351		-inf	inf
18	alumina	atom_site	Al1	adp_iso	0.10000		-inf	inf	Ų
19	alumina	atom_site	O1	fract_x	0.35030		-inf	inf
20	alumina	atom_site	O1	adp_iso	1.22884		-inf	inf	Ų
21	yap_3k	linked_structure	yap	scale	28.06935	0.11372	-inf	inf
22	yap_3k	linked_structure	alumina	scale	0.19413	0.01609	-inf	inf
23	yap_3k	peak		asym_beba_a0	-0.18183	0.01524	-inf	inf
24	yap_3k	peak		asym_beba_b1	-0.06117	0.00501	-inf	inf
25	yap_3k	peak		broad_gauss_u	0.03642	0.00197	-inf	inf	deg²
26	yap_3k	peak		broad_gauss_v	-0.05103	0.00409	-inf	inf	deg²
27	yap_3k	peak		broad_gauss_w	0.09845	0.00218	-inf	inf	deg²
28	yap_3k	peak		broad_lorentz_y	0.02981	0.00192	-inf	inf	deg
29	yap_3k	instrument		twotheta_offset	0.00593	0.00268	-inf	inf	deg
30	yap_3k	background	1	intensity	1303.00000		-inf	inf
31	yap_3k	background	2	intensity	2173.00000		-inf	inf
32	yap_3k	background	3	intensity	2414.80830		-inf	inf
33	yap_3k	background	4	intensity	2010.28100		-inf	inf
34	yap_3k	background	5	intensity	2041.00000		-inf	inf
35	yap_3k	background	6	intensity	1968.46433		-inf	inf
36	yap_3k	background	7	intensity	1606.73948		-inf	inf
37	yap_3k	background	8	intensity	0.00000		-inf	inf
38	yap_3k	background	9	intensity	1695.97793		-inf	inf
39	yap_3k	background	10	intensity	2367.00000		-inf	inf
40	yap_3k	background	11	intensity	1691.86967		-inf	inf
41	yap_3k	background	12	intensity	1321.52032		-inf	inf
42	yap_3k	background	13	intensity	1934.23588		-inf	inf
43	yap_3k	background	14	intensity	1881.37866		-inf	inf
44	yap_3k	background	15	intensity	1996.38773		-inf	inf
45	yap_3k	background	16	intensity	1525.63663		-inf	inf
46	yap_3k	background	17	intensity	3108.73377		-inf	inf
47	yap_3k	background	18	intensity	2493.86153		-inf	inf
48	yap_3k	background	19	intensity	2295.53422		-inf	inf
49	yap_3k	background	20	intensity	2451.55313		-inf	inf

In [52]:

project.analysis.fit()

project.analysis.fit()

Standard fitting

📋 Using experiment 🔬 'yap_3k' for 'single' fitting

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

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.14	78.60
2	51	9.00	9.93	87.4% ↓
3	101	17.52	9.28	6.6% ↓
4	153	43.73	9.25

🏆 Best goodness-of-fit (reduced χ²) is 9.25 at iteration 153

✅ Fitting complete.

Inspect Results¶

Review the fit statistics, refined parameters, and correlations. Inspect the full pattern and a closer view with impurity reflections.

In [53]:

project.display.fit.results()

project.display.fit.results()

⚙️ Settings used:

	Name	Value	Description
1	max_iterations	500	Maximum solver iterations.
2	chi_square_change_tolerance	0.01	Relative change in the objective (chi-square) used to stop fitting.
3	parameter_change_tolerance	1e-08	Relative change in fitted parameters used to stop fitting.

📋 Least-squares fit results:

	Metric	Value
1	🧪 Minimizer	bumps (lm)
2	✅ Overall status	success
3	⏱️ Fitting time (seconds)	43.73
4	📏 Goodness-of-fit (reduced χ²)	9.25
5	📏 R-factor (Rf, %)	3.13
6	📏 R-factor squared (Rf², %)	3.71
7	📏 Weighted R-factor (wR, %)	3.96

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	yap	cell		length_a	Å	5.1728	5.1728	0.0000	0.00 % ↑
2	yap	cell		length_b	Å	5.3270	5.3271	0.0000	0.00 % ↑
3	yap	cell		length_c	Å	7.3610	7.3613	0.0001	0.00 % ↑
4	yap	atom_site	Y	fract_x		0.0100	0.0123	0.0001	22.93 % ↑
5	yap	atom_site	Y	fract_y		0.5500	0.5541	0.0001	0.74 % ↑
6	yap	atom_site	Y	adp_iso	Ų	0.1200	0.0711	0.0086	40.77 % ↓
7	yap	atom_site	Al	adp_iso	Ų	0.1300	0.0938	0.0149	27.81 % ↓
8	yap	atom_site	O1	fract_x		-0.0800	-0.0839	0.0001	4.91 % ↑
9	yap	atom_site	O1	fract_y		-0.0200	-0.0219	0.0001	9.38 % ↑
10	yap	atom_site	O1	adp_iso	Ų	0.0600	0.0280	0.0096	53.41 % ↓
11	yap	atom_site	O2	fract_x		0.2000	0.2047	0.0001	2.33 % ↑
12	yap	atom_site	O2	fract_y		0.2900	0.2948	0.0001	1.64 % ↑
13	yap	atom_site	O2	fract_z		0.0400	0.0442	0.0001	10.53 % ↑
14	yap	atom_site	O2	adp_iso	Ų	0.1400	0.0894	0.0075	36.17 % ↓
15	alumina	cell		length_a	Å	4.7577	4.7580	0.0004	0.01 % ↑
16	alumina	cell		length_c	Å	12.9761	12.9748	0.0024	0.01 % ↓
17	alumina	atom_site	Al1	fract_z		0.3335	0.3409	0.0036	2.22 % ↑
18	alumina	atom_site	Al1	adp_iso	Ų	0.1000	0.0789	0.3262	21.13 % ↓
19	alumina	atom_site	O1	fract_x		0.3503	0.3402	0.0077	2.88 % ↓
20	alumina	atom_site	O1	adp_iso	Ų	1.2288	1.0775	0.2565	12.32 % ↓
21	yap_3k	linked_structure	yap	scale		28.0693	28.0057	0.0755	0.23 % ↓
22	yap_3k	linked_structure	alumina	scale		0.1941	0.1847	0.0123	4.87 % ↓
23	yap_3k	peak		asym_beba_a0		-0.1818	-0.1825	0.0064	0.35 % ↑
24	yap_3k	peak		asym_beba_b1		-0.0612	-0.0596	0.0021	2.51 % ↓
25	yap_3k	peak		broad_gauss_u	deg²	0.0364	0.0303	0.0007	16.83 % ↓
26	yap_3k	peak		broad_gauss_v	deg²	-0.0510	-0.0458	0.0015	10.26 % ↓
27	yap_3k	peak		broad_gauss_w	deg²	0.0985	0.0964	0.0009	2.03 % ↓
28	yap_3k	peak		broad_lorentz_y	deg	0.0298	0.0326	0.0009	9.53 % ↑
29	yap_3k	instrument		twotheta_offset	deg	0.0059	0.0084	0.0011	42.18 % ↑
30	yap_3k	background	1	intensity		1303.0000	1389.3685	36.3054	6.63 % ↑
31	yap_3k	background	2	intensity		2173.0000	2260.6485	32.2917	4.03 % ↑
32	yap_3k	background	3	intensity		2414.8083	2363.8752	17.4837	2.11 % ↓
33	yap_3k	background	4	intensity		2010.2810	2000.3479	14.0653	0.49 % ↓
34	yap_3k	background	5	intensity		2041.0000	2093.0766	14.7535	2.55 % ↑
35	yap_3k	background	6	intensity		1968.4643	2089.9668	18.0042	6.17 % ↑
36	yap_3k	background	7	intensity		1606.7395	2267.8070	28.6865	41.14 % ↑
37	yap_3k	background	8	intensity		0.0000	1651.2100	51.4432	N/A
38	yap_3k	background	9	intensity		1695.9779	2069.5408	49.6652	22.03 % ↑
39	yap_3k	background	10	intensity		2367.0000	2093.7316	38.8953	11.54 % ↓
40	yap_3k	background	11	intensity		1691.8697	2062.8503	34.5231	21.93 % ↑
41	yap_3k	background	12	intensity		1321.5203	1738.0206	37.4917	31.52 % ↑
42	yap_3k	background	13	intensity		1934.2359	2008.8316	14.0618	3.86 % ↑
43	yap_3k	background	14	intensity		1881.3787	2028.2740	12.7842	7.81 % ↑
44	yap_3k	background	15	intensity		1996.3877	2072.5901	13.5330	3.82 % ↑
45	yap_3k	background	16	intensity		1525.6366	1925.4302	15.7703	26.21 % ↑
46	yap_3k	background	17	intensity		3108.7338	2266.1180	24.4631	27.10 % ↓
47	yap_3k	background	18	intensity		2493.8615	2561.2009	23.8398	2.70 % ↑
48	yap_3k	background	19	intensity		2295.5342	2546.2284	28.5615	10.92 % ↑
49	yap_3k	background	20	intensity		2451.5531	2754.9245	37.8898	12.37 % ↑

• 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

⚠️ Red s.u.: exceeds the refined value (consider adding constraints)

In [54]:

project.display.fit.correlations()

project.display.fit.correlations()

Loading plot…

In [55]:

project.display.pattern(expt_name='yap_3k')

project.display.pattern(expt_name='yap_3k')

Loading plot…

In [56]:

project.display.pattern(expt_name='yap_3k', x_min=134, x_max=146)

project.display.pattern(expt_name='yap_3k', x_min=134, x_max=146)

Loading plot…

💾 Save Project¶

Save the refined model and analysis results in the project directory.

In [57]:

project.save()

project.save()

Saving project 📦 'yap_3k' to '../../../projects/refine-yap-3k'

├── 📄 project.edi
├── 📁 structures/
│   └── 📄 yap.edi
│   └── 📄 alumina.edi
├── 📁 experiments/
│   └── 📄 yap_3k.edi
├── 📁 analysis/
│   └── 📄 analysis.edi
└── 📁 reports/
    └── 📄 yap_3k.html