Structure Refinement: LBCO, HRPT¶

This minimalistic example is designed to be as compact as possible for a Rietveld refinement of a crystal structure using constant-wavelength neutron powder diffraction data for La0.5Ba0.5CoO3 from HRPT at PSI.

It does not contain any advanced features or options, and includes no comments or explanations—these can be found in the other tutorials. Default values are used for all parameters if not specified. Only essential and self-explanatory code is provided.

The example is intended for users who are already familiar with the EasyDiffraction library and want to quickly get started with a simple refinement. It is also useful for those who want to see what a refinement might look like in code. For a more detailed explanation of the code, please refer to the other tutorials.

Import Library¶

In [2]:

import easydiffraction as ed

import easydiffraction as ed

⚠️ 'pycrysfml' module not found. This calculation engine will not be available.
✅ 'cryspy' calculation engine is successfully imported.
✅ 'pdffit' calculation engine is successfully imported.

Step 1: Define Project¶

In [3]:

project = ed.Project()

project = ed.Project()

Step 2: Define Sample Model¶

In [4]:

project.sample_models.add(name='lbco')

project.sample_models.add(name='lbco')

In [5]:

sample_model = project.sample_models['lbco']

sample_model = project.sample_models['lbco']

In [6]:

sample_model.space_group.name_h_m = 'P m -3 m'
sample_model.space_group.it_coordinate_system_code = '1'

sample_model.space_group.name_h_m = 'P m -3 m' sample_model.space_group.it_coordinate_system_code = '1'

In [7]:

sample_model.cell.length_a = 3.88

sample_model.cell.length_a = 3.88

In [8]:

sample_model.atom_sites.add('La', 'La', 0, 0, 0, b_iso=0.5, occupancy=0.5)
sample_model.atom_sites.add('Ba', 'Ba', 0, 0, 0, b_iso=0.5, occupancy=0.5)
sample_model.atom_sites.add('Co', 'Co', 0.5, 0.5, 0.5, b_iso=0.5)
sample_model.atom_sites.add('O', 'O', 0, 0.5, 0.5, b_iso=0.5)

sample_model.atom_sites.add('La', 'La', 0, 0, 0, b_iso=0.5, occupancy=0.5) sample_model.atom_sites.add('Ba', 'Ba', 0, 0, 0, b_iso=0.5, occupancy=0.5) sample_model.atom_sites.add('Co', 'Co', 0.5, 0.5, 0.5, b_iso=0.5) sample_model.atom_sites.add('O', 'O', 0, 0.5, 0.5, b_iso=0.5)

Step 3: Define Experiment¶

In [9]:

ed.download_from_repository('hrpt_lbco.xye', destination='data')

ed.download_from_repository('hrpt_lbco.xye', destination='data')

⚠️ Warning
File 'data/hrpt_lbco.xye' already exists and will not be overwritten.

In [10]:

project.experiments.add(
    name='hrpt',
    sample_form='powder',
    beam_mode='constant wavelength',
    radiation_probe='neutron',
    data_path='data/hrpt_lbco.xye',
)

project.experiments.add( name='hrpt', sample_form='powder', beam_mode='constant wavelength', radiation_probe='neutron', data_path='data/hrpt_lbco.xye', )

Loading measured data from ASCII file
/Users/runner/work/diffraction-lib/diffraction-lib/docs/tutorials/data/hrpt_lbco.xye

Data loaded successfully
Experiment 🔬 'hrpt'. Number of data points: 3098

In [11]:

experiment = project.experiments['hrpt']

experiment = project.experiments['hrpt']

In [12]:

experiment.instrument.setup_wavelength = 1.494
experiment.instrument.calib_twotheta_offset = 0.6

experiment.instrument.setup_wavelength = 1.494 experiment.instrument.calib_twotheta_offset = 0.6

In [13]:

experiment.peak.broad_gauss_u = 0.1
experiment.peak.broad_gauss_v = -0.1
experiment.peak.broad_gauss_w = 0.1
experiment.peak.broad_lorentz_y = 0.1

experiment.peak.broad_gauss_u = 0.1 experiment.peak.broad_gauss_v = -0.1 experiment.peak.broad_gauss_w = 0.1 experiment.peak.broad_lorentz_y = 0.1

In [14]:

experiment.background.add(x=10, y=170)
experiment.background.add(x=30, y=170)
experiment.background.add(x=50, y=170)
experiment.background.add(x=110, y=170)
experiment.background.add(x=165, y=170)

experiment.background.add(x=10, y=170) experiment.background.add(x=30, y=170) experiment.background.add(x=50, y=170) experiment.background.add(x=110, y=170) experiment.background.add(x=165, y=170)

In [15]:

experiment.excluded_regions.add(minimum=0, maximum=5)
experiment.excluded_regions.add(minimum=165, maximum=180)

experiment.excluded_regions.add(minimum=0, maximum=5) experiment.excluded_regions.add(minimum=165, maximum=180)

In [16]:

experiment.linked_phases.add(id='lbco', scale=10.0)

experiment.linked_phases.add(id='lbco', scale=10.0)

Step 4: Perform Analysis¶

In [17]:

sample_model.cell.length_a.free = True

sample_model.atom_sites['La'].b_iso.free = True
sample_model.atom_sites['Ba'].b_iso.free = True
sample_model.atom_sites['Co'].b_iso.free = True
sample_model.atom_sites['O'].b_iso.free = True

sample_model.cell.length_a.free = True sample_model.atom_sites['La'].b_iso.free = True sample_model.atom_sites['Ba'].b_iso.free = True sample_model.atom_sites['Co'].b_iso.free = True sample_model.atom_sites['O'].b_iso.free = True

In [18]:

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_y.free = True

experiment.background['10'].y.free = True
experiment.background['30'].y.free = True
experiment.background['50'].y.free = True
experiment.background['110'].y.free = True
experiment.background['165'].y.free = True

experiment.linked_phases['lbco'].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_y.free = True experiment.background['10'].y.free = True experiment.background['30'].y.free = True experiment.background['50'].y.free = True experiment.background['110'].y.free = True experiment.background['165'].y.free = True experiment.linked_phases['lbco'].scale.free = True

In [19]:

project.analysis.fit()

project.analysis.fit()

Using experiment 🔬 'hrpt' for 'single' fitting
🚀 Starting fit process with 'lmfit (leastsq)'...
📈 Goodness-of-fit (reduced χ²) change:

iteration	χ²	improvement [%]
1	165.02
28	33.58	79.7% ↓
45	10.82	67.8% ↓
63	6.49	40.0% ↓
81	3.35	48.4% ↓
98	2.24	33.1% ↓
116	1.91	14.7% ↓
133	1.50	21.3% ↓
150	1.45	3.6% ↓
167	1.34	7.8% ↓
185	1.29	3.4% ↓
276	1.29

🏆 Best goodness-of-fit (reduced χ²) is 1.29 at iteration 275
✅ Fitting complete.

Fit results
✅ Success: True
⏱️ Fitting time: 2.07 seconds
📏 Goodness-of-fit (reduced χ²): 1.29
📏 R-factor (Rf): 5.63%
📏 R-factor squared (Rf²): 5.27%
📏 Weighted R-factor (wR): 4.41%
📈 Fitted parameters:

	datablock	category	entry	parameter	start	fitted	uncertainty	units	change
1	lbco	atom_sites	La	b_iso	0.5000	0.5052	3231.6242	Ų	1.04 % ↑
2	lbco	atom_sites	Ba	b_iso	0.5000	0.5049	5252.3879	Ų	0.97 % ↑
3	lbco	atom_sites	Co	b_iso	0.5000	0.2370	0.0611	Ų	52.60 % ↓
4	lbco	atom_sites	O	b_iso	0.5000	1.3935	0.0167	Ų	178.71 % ↑
5	lbco	cell		length_a	3.8800	3.8909	0.0000	Å	0.28 % ↑
6	hrpt	background	10	y	170.0000	168.5585	1.3671		0.85 % ↓
7	hrpt	background	30	y	170.0000	164.3357	0.9992		3.33 % ↓
8	hrpt	background	50	y	170.0000	166.8881	0.7388		1.83 % ↓
9	hrpt	background	110	y	170.0000	175.4006	0.6578		3.18 % ↑
10	hrpt	background	165	y	170.0000	174.2813	0.9105		2.52 % ↑
11	hrpt	instrument		twotheta_offset	0.6000	0.6226	0.0010	deg	3.76 % ↑
12	hrpt	linked_phases	lbco	scale	10.0000	9.1351	0.0642		8.65 % ↓
13	hrpt	peak		broad_gauss_u	0.1000	0.0816	0.0031	deg²	18.43 % ↓
14	hrpt	peak		broad_gauss_v	-0.1000	-0.1159	0.0067	deg²	15.93 % ↑
15	hrpt	peak		broad_gauss_w	0.1000	0.1204	0.0033	deg²	20.45 % ↑
16	hrpt	peak		broad_lorentz_y	0.1000	0.0844	0.0021	deg	15.55 % ↓

In [20]:

project.plot_meas_vs_calc(expt_name='hrpt', show_residual=True)

project.plot_meas_vs_calc(expt_name='hrpt', show_residual=True)