Structure Refinement: Tb2TiO7, HEiDi¶

Crystal structure refinement of Tb2TiO7 using single crystal neutron diffraction data from HEiDi at FRM II.

🛠️ Import Library¶

In [2]:

import easydiffraction as ed

import easydiffraction as ed

📦 Define Project¶

In [3]:

# Create a minimal project with a short name
project = ed.Project(name='tbti_heidi')
project.info.title = 'Tb2Ti2O7 at HEiDi@FRMII'
project.info.description = """This project demonstrates a standard
refinement of the crystal structure of Tb2Ti2O7 using single crystal 
neutron diffraction data from HEiDi at FRM II."""

project.save_as(dir_path='projects/ed_14_tbti_heidi')

# Create a minimal project with a short name project = ed.Project(name='tbti_heidi') project.info.title = 'Tb2Ti2O7 at HEiDi@FRMII' project.info.description = """This project demonstrates a standard refinement of the crystal structure of Tb2Ti2O7 using single crystal neutron diffraction data from HEiDi at FRM II.""" project.save_as(dir_path='projects/ed_14_tbti_heidi')

Saving project 📦 'tbti_heidi' to '../../../projects/ed_14_tbti_heidi'

├── 📄 project.cif

├── 📁 structures/

├── 📁 experiments/

├── 📁 analysis/

│   └── 📄 analysis.cif

└── 📁 reports/

    └── 📄 tbti_heidi.html

🧩 Define Structure¶

In [4]:

# Download CIF file from repository
structure_path = ed.download_data(id=20, destination='data')

# Download CIF file from repository structure_path = ed.download_data(id=20, destination='data')

Getting data...

Data #20: Tb2Ti2O7 (crystal structure)

✅ Data #20 downloaded to '../../../data/ed-20.cif'

In [5]:

project.structures.add_from_cif_path(structure_path)

project.structures.add_from_cif_path(structure_path)

In [6]:

project.structures.show_names()

project.structures.show_names()

Defined structures 🧩

['tbti']

In [7]:

structure = project.structures['tbti']

structure = project.structures['tbti']

In [8]:

structure.show_as_cif()

structure.show_as_cif()

Structure 🧩 'tbti' as cif

	CIF
1	data_tbti
2
3	_cell.length_a 10.13
4	_cell.length_b 10.13
5	_cell.length_c 10.13
6	_cell.angle_alpha 90.
7	_cell.angle_beta 90.
8	_cell.angle_gamma 90.
9
10	_space_group.name_H-M_alt "F d -3 m"
11	_space_group.IT_coordinate_system_code 2
12
13	_geom.min_bond_distance_cutoff 0.
14	_geom.bond_distance_incr 0.25
15
16	loop_
17	_atom_site.label
18	_atom_site.type_symbol
19	_atom_site.fract_x
20	_atom_site.fract_y
21	_atom_site.fract_z
22	_atom_site.Wyckoff_symbol
23	_atom_site.site_symmetry_multiplicity
24	_atom_site.occupancy
25	_atom_site.B_iso_or_equiv
26	_atom_site.ADP_type
27	Tb Tb 0.5 0.5 0.5 d 16 1. 0.530(20) Biso
28	Ti Ti 0. 0. 0. c 16 1. 0.480(30) Biso
29	O1 O 0.328040(90) 0.125 0.125 f 48 1.000(10) 0.450(20) Biso
30	O2 O 0.375 0.375 0.375 b 8 0.970(20) 0.230(40) Biso

In [9]:

structure.atom_sites['Tb'].adp_type = 'Uiso'
structure.atom_sites['Ti'].adp_type = 'Uiso'
structure.atom_sites['O1'].adp_type = 'Uiso'
structure.atom_sites['O2'].adp_type = 'Uiso'

structure.atom_sites['Tb'].adp_type = 'Uiso' structure.atom_sites['Ti'].adp_type = 'Uiso' structure.atom_sites['O1'].adp_type = 'Uiso' structure.atom_sites['O2'].adp_type = 'Uiso'

In [10]:

print(structure.atom_sites.as_cif)

print(structure.atom_sites.as_cif)

loop_
_atom_site.label
_atom_site.type_symbol
_atom_site.fract_x
_atom_site.fract_y
_atom_site.fract_z
_atom_site.Wyckoff_symbol
_atom_site.site_symmetry_multiplicity
_atom_site.occupancy
_atom_site.U_iso_or_equiv
_atom_site.ADP_type
Tb Tb 0.5 0.5 0.5 d 16 1. 0.007(20) Uiso
Ti Ti 0. 0. 0. c 16 1. 0.006(30) Uiso
O1 O 0.328040(90) 0.125 0.125 f 48 1.000(10) 0.006(20) Uiso
O2 O 0.375 0.375 0.375 b 8 0.970(20) 0.003(40) Uiso

In [11]:

structure.atom_sites['Tb'].adp_iso = 0.0
structure.atom_sites['Ti'].adp_iso = 0.0
structure.atom_sites['O1'].adp_iso = 0.0
structure.atom_sites['O2'].adp_iso = 0.0

structure.atom_sites['Tb'].adp_iso = 0.0 structure.atom_sites['Ti'].adp_iso = 0.0 structure.atom_sites['O1'].adp_iso = 0.0 structure.atom_sites['O2'].adp_iso = 0.0

Display Structure (ADP)¶

Select the ADP atom view. With the starting isotropic displacements every atom is drawn as a sphere; after the anisotropic refinement below, the same view will draw probability ellipsoids for the anisotropic sites.

In [12]:

project.structure_style.atom_view = 'adp'

project.structure_style.atom_view = 'adp'

In [13]:

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

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

Structure 🧩 'tbti' (Atom view type: 'adp')

Loading plot…

drag = rotate
wheel = zoom
right-drag = pan

🔬 Define Experiment¶

In [14]:

# Download data file from repository
data_path = ed.download_data(id=19, destination='data')

# Download data file from repository data_path = ed.download_data(id=19, destination='data')

Getting data...

Data #19: Tb2Ti2O7, HEiDi (MLZ)

✅ Data #19 downloaded to '../../../data/ed-19.xye'

In [15]:

project.experiments.add_from_data_path(
    name='heidi',
    data_path=data_path,
    sample_form='single crystal',
    beam_mode='constant wavelength',
    radiation_probe='neutron',
)

project.experiments.add_from_data_path( name='heidi', data_path=data_path, sample_form='single crystal', beam_mode='constant wavelength', radiation_probe='neutron', )

Data loaded successfully

Experiment 🔬 'heidi'. Number of data points: 220.

In [16]:

experiment = project.experiments['heidi']

experiment = project.experiments['heidi']

In [17]:

experiment.linked_crystal.id = 'tbti'
experiment.linked_crystal.scale = 1.0

experiment.linked_crystal.id = 'tbti' experiment.linked_crystal.scale = 1.0

In [18]:

experiment.instrument.setup_wavelength = 0.793

experiment.instrument.setup_wavelength = 0.793

In [19]:

experiment.extinction.mosaicity = 35000
experiment.extinction.radius = 10

experiment.extinction.mosaicity = 35000 experiment.extinction.radius = 10

🚀 Perform Analysis¶

ADP iso¶

In [20]:

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

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

Loading plot…

In [21]:

structure.atom_sites['O1'].fract_x.free = True

structure.atom_sites['Ti'].occupancy.free = True
structure.atom_sites['O1'].occupancy.free = True
structure.atom_sites['O2'].occupancy.free = True

structure.atom_sites['Tb'].adp_iso.free = True
structure.atom_sites['Ti'].adp_iso.free = True
structure.atom_sites['O1'].adp_iso.free = True
structure.atom_sites['O2'].adp_iso.free = True

structure.atom_sites['O1'].fract_x.free = True structure.atom_sites['Ti'].occupancy.free = True structure.atom_sites['O1'].occupancy.free = True structure.atom_sites['O2'].occupancy.free = True structure.atom_sites['Tb'].adp_iso.free = True structure.atom_sites['Ti'].adp_iso.free = True structure.atom_sites['O1'].adp_iso.free = True structure.atom_sites['O2'].adp_iso.free = True

In [22]:

experiment.linked_crystal.scale.free = True
experiment.extinction.radius.free = True

experiment.linked_crystal.scale.free = True experiment.extinction.radius.free = True

In [23]:

# Start refinement. All parameters, which have standard uncertainties
# in the input CIF files, are refined by default.
project.analysis.fit()

# Start refinement. All parameters, which have standard uncertainties # in the input CIF files, are refined by default. project.analysis.fit()

Standard fitting

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

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

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.05	355.79
2	14	0.65	186.41	47.6% ↓
3	25	1.14	42.28	77.3% ↓
4	36	1.65	12.97	69.3% ↓
5	47	2.14	12.75	1.6% ↓
6	81	3.70	12.75

🏆 Best goodness-of-fit (reduced χ²) is 12.75 at iteration 80

✅ Fitting complete.

Saving project 📦 'tbti_heidi' to '../../../projects/ed_14_tbti_heidi'

├── 📄 project.cif

├── 📁 structures/

│   └── 📄 tbti.cif

├── 📁 experiments/

│   └── 📄 heidi.cif

├── 📁 analysis/

│   └── 📄 analysis.cif

└── 📁 reports/

    └── 📄 tbti_heidi.html

In [24]:

# Show fit results summary
project.display.fit.results()

# Show fit results summary 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)	3.70
4	🔁 Iterations	78
5	📏 Goodness-of-fit (reduced χ²)	12.75
6	📏 R-factor (Rf, %)	7.65
7	📏 R-factor squared (Rf², %)	8.10
8	📏 Weighted R-factor (wR, %)	8.59

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	tbti	atom_site	Tb	adp_iso	Ų	0.0000	0.0068	0.0003	N/A
2	tbti	atom_site	Ti	occupancy		1.0000	0.9730	0.0178	2.70 % ↓
3	tbti	atom_site	Ti	adp_iso	Ų	0.0000	0.0054	0.0006	N/A
4	tbti	atom_site	O1	fract_x		0.3280	0.3280	0.0001	0.00 % ↑
5	tbti	atom_site	O1	occupancy		1.0000	0.9989	0.0104	0.11 % ↓
6	tbti	atom_site	O1	adp_iso	Ų	0.0000	0.0057	0.0002	N/A
7	tbti	atom_site	O2	occupancy		0.9700	0.9688	0.0183	0.12 % ↓
8	tbti	atom_site	O2	adp_iso	Ų	0.0000	0.0030	0.0005	N/A
9	heidi	extinction		radius	μm	10.0000	26.4348	1.0122	164.35 % ↑
10	heidi	linked_crystal		scale		1.0000	2.9329	0.0586	193.29 % ↑

• 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 [25]:

structure.show_as_cif()

structure.show_as_cif()

Structure 🧩 'tbti' as cif

	CIF
1	data_tbti
2
3	_cell.length_a 10.13
4	_cell.length_b 10.13
5	_cell.length_c 10.13
6	_cell.angle_alpha 90.
7	_cell.angle_beta 90.
8	_cell.angle_gamma 90.
9
10	_space_group.name_H-M_alt "F d -3 m"
11	_space_group.IT_coordinate_system_code 2
12
13	_geom.min_bond_distance_cutoff 0.
14	_geom.bond_distance_incr 0.25
15
16	loop_
17	_atom_site.label
18	_atom_site.type_symbol
19	_atom_site.fract_x
20	_atom_site.fract_y
21	_atom_site.fract_z
22	_atom_site.Wyckoff_symbol
23	_atom_site.site_symmetry_multiplicity
24	_atom_site.occupancy
25	_atom_site.U_iso_or_equiv
26	_atom_site.ADP_type
27	Tb Tb 0.5 0.5 0.5 d 16 1. 0.00682(29) Uiso
28	Ti Ti 0. 0. 0. c 16 0.973(18) 0.00540(60) Uiso
29	O1 O 0.328044(94) 0.125 0.125 f 48 0.999(10) 0.00573(25) Uiso
30	O2 O 0.375 0.375 0.375 b 8 0.969(18) 0.00303(48) Uiso

In [26]:

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

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

Structure 🧩 'tbti' (Atom view type: 'adp')

Loading plot…

drag = rotate
wheel = zoom
right-drag = pan

In [27]:

project.experiments.show_names()

project.experiments.show_names()

Defined experiments 🔬

['heidi']

In [28]:

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

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

Loading plot…

ADP aniso¶

In [29]:

structure.atom_sites['Tb'].adp_type = 'Uani'
structure.atom_sites['Ti'].adp_type = 'Uani'
structure.atom_sites['O1'].adp_type = 'Uani'
# structure.atom_sites['O2'].adp_type = 'Uani'

structure.atom_sites['Tb'].adp_type = 'Uani' structure.atom_sites['Ti'].adp_type = 'Uani' structure.atom_sites['O1'].adp_type = 'Uani' # structure.atom_sites['O2'].adp_type = 'Uani'

In [30]:

print(structure.atom_site_aniso.as_cif)

print(structure.atom_site_aniso.as_cif)

loop_
_atom_site_aniso.label
_atom_site_aniso.U_11
_atom_site_aniso.U_22
_atom_site_aniso.U_33
_atom_site_aniso.U_12
_atom_site_aniso.U_13
_atom_site_aniso.U_23
Tb 0.00681683 0.00681683 0.00681683 0. 0. 0.
Ti 0.00539742 0.00539742 0.00539742 0. 0. 0.
O1 0.00572628 0.00572628 0.00572628 0. 0. 0.

In [31]:

structure.atom_site_aniso['Tb'].adp_11.free = True
structure.atom_site_aniso['Tb'].adp_12.free = True
structure.atom_site_aniso['Ti'].adp_11.free = True
structure.atom_site_aniso['Ti'].adp_12.free = True
structure.atom_site_aniso['O1'].adp_11.free = True
structure.atom_site_aniso['O1'].adp_22.free = True
structure.atom_site_aniso['O1'].adp_23.free = True
# structure.atom_site_aniso['O2'].adp_11.free = True

structure.atom_site_aniso['Tb'].adp_11.free = True structure.atom_site_aniso['Tb'].adp_12.free = True structure.atom_site_aniso['Ti'].adp_11.free = True structure.atom_site_aniso['Ti'].adp_12.free = True structure.atom_site_aniso['O1'].adp_11.free = True structure.atom_site_aniso['O1'].adp_22.free = True structure.atom_site_aniso['O1'].adp_23.free = True # structure.atom_site_aniso['O2'].adp_11.free = True

In [32]:

project.analysis.fit()

project.analysis.fit()

Standard fitting

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

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

📈 Goodness-of-fit progress:

	iteration	time (s)	χ²	change / status
1	1	0.34	13.00
2	18	1.42	3.02	76.8% ↓
3	33	2.41	2.94	2.5% ↓
4	79	5.40	2.94

🏆 Best goodness-of-fit (reduced χ²) is 2.94 at iteration 78

✅ Fitting complete.

Saving project 📦 'tbti_heidi' to '../../../projects/ed_14_tbti_heidi'

├── 📄 project.cif

├── 📁 structures/

│   └── 📄 tbti.cif

├── 📁 experiments/

│   └── 📄 heidi.cif

├── 📁 analysis/

│   └── 📄 analysis.cif

└── 📁 reports/

    └── 📄 tbti_heidi.html

In [33]:

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)	5.40
4	🔁 Iterations	76
5	📏 Goodness-of-fit (reduced χ²)	2.94
6	📏 R-factor (Rf, %)	4.21
7	📏 R-factor squared (Rf², %)	6.61
8	📏 Weighted R-factor (wR, %)	13.29

📈 Refined parameters:

	datablock	category	entry	parameter	units	start	value	s.u.	change
1	tbti	atom_site	Ti	occupancy		0.9730	0.9661	0.0085	0.71 % ↓
2	tbti	atom_site	O1	fract_x		0.3280	0.3280	0.0000	0.01 % ↓
3	tbti	atom_site	O1	occupancy		0.9989	0.9940	0.0050	0.50 % ↓
4	tbti	atom_site	O2	occupancy		0.9688	0.9718	0.0088	0.31 % ↑
5	tbti	atom_site	O2	adp_iso	Ų	0.0030	0.0033	0.0002	7.87 % ↑
6	tbti	atom_site_aniso	Tb	adp_11	Ų	0.0068	0.0066	0.0001	3.45 % ↓
7	tbti	atom_site_aniso	Tb	adp_12	Ų	0.0000	-0.0023	0.0001	N/A
8	tbti	atom_site_aniso	Ti	adp_11	Ų	0.0054	0.0049	0.0003	9.56 % ↓
9	tbti	atom_site_aniso	Ti	adp_12	Ų	0.0000	-0.0005	0.0002	N/A
10	tbti	atom_site_aniso	O1	adp_11	Ų	0.0057	0.0067	0.0002	17.67 % ↑
11	tbti	atom_site_aniso	O1	adp_22	Ų	0.0057	0.0049	0.0001	14.61 % ↓
12	tbti	atom_site_aniso	O1	adp_23	Ų	0.0000	0.0019	0.0001	N/A
13	heidi	extinction		radius	μm	26.4348	24.8317	0.4931	6.06 % ↓
14	heidi	linked_crystal		scale		2.9329	2.8744	0.0274	1.99 % ↓

• 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

In [34]:

project.display.fit.correlations()

project.display.fit.correlations()

Loading plot…

In [35]:

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

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

Loading plot…

In [36]:

structure.show_as_cif()

structure.show_as_cif()

Structure 🧩 'tbti' as cif

	CIF
1	data_tbti
2
3	_cell.length_a 10.13
4	_cell.length_b 10.13
5	_cell.length_c 10.13
6	_cell.angle_alpha 90.
7	_cell.angle_beta 90.
8	_cell.angle_gamma 90.
9
10	_space_group.name_H-M_alt "F d -3 m"
11	_space_group.IT_coordinate_system_code 2
12
13	_geom.min_bond_distance_cutoff 0.
14	_geom.bond_distance_incr 0.25
15
16	loop_
17	_atom_site.label
18	_atom_site.type_symbol
19	_atom_site.fract_x
20	_atom_site.fract_y
21	_atom_site.fract_z
22	_atom_site.Wyckoff_symbol
23	_atom_site.site_symmetry_multiplicity
24	_atom_site.occupancy
25	_atom_site.U_iso_or_equiv
26	_atom_site.ADP_type
27	Tb Tb 0.5 0.5 0.5 d 16 1. 0.00658189 Uani
28	Ti Ti 0. 0. 0. c 16 0.9661(85) 0.00488144 Uani
29	O1 O 0.327996(47) 0.125 0.125 f 48 0.9940(50) 0.00550586 Uani
30	O2 O 0.375 0.375 0.375 b 8 0.9718(88) 0.00327(24) Uiso
31
32	loop_
33	_atom_site_aniso.label
34	_atom_site_aniso.U_11
35	_atom_site_aniso.U_22
36	_atom_site_aniso.U_33
37	_atom_site_aniso.U_12
38	_atom_site_aniso.U_13
39	_atom_site_aniso.U_23
40	Tb 0.00658(14) 0.00658189 0.00658189 -0.00228(11) -0.00228459 -0.00228459
41	Ti 0.00488(29) 0.00488144 0.00488144 -0.00048(25) -0.00048176 -0.00048176
42	O1 0.00674(18) 0.00489(13) 0.0048898 0. 0. 0.00189(13)

Display Structure (final)¶

Tb, Ti and O1 were refined with anisotropic displacements (Uani), so the ADP view now draws them as probability ellipsoids, while O2 stayed isotropic (Uiso) and remains a sphere — highlighting how the view changed from the initial spheres to ellipsoids.

In [37]:

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

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

Structure 🧩 'tbti' (Atom view type: 'adp')

Loading plot…

drag = rotate
wheel = zoom
right-drag = pan

📊 Report¶

In [38]:

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

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

Saving project 📦 'tbti_heidi' to '../../../projects/ed_14_tbti_heidi'

├── 📄 project.cif

├── 📁 structures/

│   └── 📄 tbti.cif

├── 📁 experiments/

│   └── 📄 heidi.cif

├── 📁 analysis/

│   └── 📄 analysis.cif

└── 📁 reports/

    ├── 📄 tbti_heidi.html

    └── 📄 tbti_heidi.pdf