La₀.₅Ba₀.₅CoO₃ on HRPT¶

This tutorial fits the cubic perovskite La₀.₅Ba₀.₅CoO₃ to constant-wavelength neutron powder data measured on HRPT at PSI. It builds the structure by hand, loads the measured pattern from EasyDiffraction's collection, describes the experiment, links the two and fits them, showing the pattern and the parameters after each step. The other tutorials load their structures from a CIF.

Fitting Single pd neut cwl Bragg Crystal structure

Import the library¶

import easydiffraction as edi

Define the project¶

A new project is empty: it holds no structure and no experiment until you add them.

project = edi.Project(name='lbco_hrpt')

Define the structure¶

La and Ba share the corner site of the cubic cell, Co sits at its centre and O on the face centres, in space group P m -3 m. Every atom starts with an isotropic displacement of 0.5 Å².

project.structures.create(name='lbco')
structure = project.structures['lbco']
structure.space_group.name_h_m = 'P m -3 m'
structure.cell.length_a = 3.88

structure.atom_sites.create(
    id='La',
    type_symbol='La',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    wyckoff_letter='a',
    occupancy=0.5,
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='Ba',
    type_symbol='Ba',
    fract_x=0,
    fract_y=0,
    fract_z=0,
    wyckoff_letter='a',
    occupancy=0.5,
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='Co',
    type_symbol='Co',
    fract_x=0.5,
    fract_y=0.5,
    fract_z=0.5,
    wyckoff_letter='b',
    adp_iso=0.5,
)
structure.atom_sites.create(
    id='O',
    type_symbol='O',
    fract_x=0,
    fract_y=0.5,
    fract_z=0.5,
    wyckoff_letter='c',
    adp_iso=0.5,
)

project.display.structure('lbco')

Define the experiment¶

The measured pattern comes from EasyDiffraction's collection: 2θ, the counts and their standard uncertainties.

data_path = edi.download_data('pd-neut-cwl_lbco-hrpt-hrpt-data', destination='data')
project.experiments.add_from_data_path(
    name='hrpt',
    data_path=data_path,
    sample_form='powder',
    beam_mode='constant wavelength',
    radiation_probe='neutron',
    scattering_type='bragg',
)
experiment = project.experiments['hrpt']

Data 'pd-neut-cwl_lbco-hrpt-hrpt-data' is at data/pd-neut-cwl_lbco-hrpt-hrpt.xye

HRPT's wavelength is 1.494 Å. The 2θ zero starts at 0.3°: from 0 the low-angle peaks sit more than a peak width away and the fit settles in a wrong minimum. The profile is the Thompson-Cox-Hastings pseudo-Voigt, its terms to one significant figure.

experiment.instrument.setup_wavelength = 1.494
experiment.instrument.calib_twotheta_offset = 0.3
experiment.peak.broad_gauss_u = 0.08
experiment.peak.broad_gauss_v = -0.1
experiment.peak.broad_gauss_w = 0.1
experiment.peak.broad_lorentz_y = 0.08

A flat background of five points, and the two ends of the pattern left out of the fit.

for position in (10, 30, 50, 110, 165):
    experiment.background.create(position=position, intensity=160)
experiment.excluded_regions.create(start=0, end=5)
experiment.excluded_regions.create(start=165, end=180)

The experiment shows the structure it links, here lbco with a scale of 9.

experiment.linked_structures.create(structure_id='lbco', scale=9)

project.display.pattern('hrpt')

Perform the analysis¶

The fit goes in stages, freeing one group of parameters at a time, so each stage starts close to its answer.

Scale, 2θ zero and cell¶

The scale sets the overall height of the pattern, and the zero shift and the cell length put the peaks where they are measured, so they come first.

experiment.linked_structures['lbco'].scale.free = True
experiment.instrument.calib_twotheta_offset.free = True
structure.cell.length_a.free = True

project.display.parameters.free()

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

edi fit: project lbco_hrpt, 3098 pts fitted / 3098 loaded, 3 free
   iter   time (s)   Rwp (%)  reduced χ²   change
                       53.67        71.21
      1      0.001     26.78        17.72   75.1% ↓
      2      0.002     14.11         4.92   72.2% ↓
      3      0.003     12.56         3.90   20.7% ↓
      4      0.005     12.56         3.90   
      5      0.006     12.56         3.90   
  done · converged · 5 iters · reduced χ² 3.90 · Rwp 12.56 % · 0.01 s

project.display.pattern('hrpt')

Background¶

With the peaks in place, the five background points follow the level between them.

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

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

edi fit: project lbco_hrpt, 3098 pts fitted / 3098 loaded, 8 free
   iter   time (s)   Rwp (%)  reduced χ²   change
                       12.56         3.91
      1      0.002     10.92         2.95   24.4% ↓
      2      0.004     10.92         2.95   
  done · converged · 2 iters · reduced χ² 2.95 · Rwp 10.92 % · 0.01 s

project.display.pattern('hrpt')

Peak widths¶

The profile terms now fit the widths and shapes of the peaks.

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

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

edi fit: project lbco_hrpt, 3098 pts fitted / 3098 loaded, 12 free
   iter   time (s)   Rwp (%)  reduced χ²   change
                       10.92         2.96
      1      0.002     10.75         2.86   3.2% ↓
      2      0.003     10.75         2.86   
      3      0.005     10.75         2.86   
  done · converged · 3 iters · reduced χ² 2.86 · Rwp 10.75 % · 0.01 s

project.display.pattern('hrpt')

Displacements¶

Every atom sits on a special position of P m -3 m, so there are no coordinates to fit. The displacements change the peak intensities with angle and come last.

structure.atom_sites['La'].adp_iso.free = True
structure.atom_sites['Ba'].adp_iso.free = True
structure.atom_sites['Co'].adp_iso.free = True
structure.atom_sites['O'].adp_iso.free = True

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

edi fit: project lbco_hrpt, 3098 pts fitted / 3098 loaded, 16 free
   iter   time (s)   Rwp (%)  reduced χ²   change
                       10.75         2.87
      1      0.002      7.76         1.50   47.8% ↓
      2      0.004      7.24         1.30   13.0% ↓
      3      0.007      7.23         1.30   0.2% ↓
      4      0.008      7.23         1.30   
      5      0.011      7.23         1.30   
  done · converged · 5 iters · reduced χ² 1.30 · Rwp 7.23 % · 0.01 s

project.display.pattern('hrpt')

project.display.fit.correlations()

Constrain La and Ba¶

The correlations show the displacements of La and Ba moving together almost exactly, with opposite signs: they share one site and scatter neutrons similarly, so the pattern cannot tell them apart, and their s.u. are large. Give each an alias and constrain Ba's to follow La's, then fit again. Ba's displacement is no longer fitted on its own.

structure.atom_sites['Ba'].adp_iso.free = False
project.analysis.aliases.create(id='biso_La', param=structure.atom_sites['La'].adp_iso)
project.analysis.aliases.create(id='biso_Ba', param=structure.atom_sites['Ba'].adp_iso)
project.analysis.constraints.create(id='biso_Ba', expression='biso_Ba = biso_La')
project.display.constraints()

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

edi fit: project lbco_hrpt, 3098 pts fitted / 3098 loaded, 15 free
   iter   time (s)   Rwp (%)  reduced χ²   change
                        7.23         1.30
      1      0.003      7.23         1.30   
      2      0.005      7.23         1.30   
  done · converged · 2 iters · reduced χ² 1.30 · Rwp 7.23 % · 0.01 s

project.display.pattern('hrpt')

project.display.pattern('hrpt', x_min=38, x_max=41)

The pair of La and Ba has gone from the correlations, and La's displacement now has a small s.u.

project.display.fit.correlations()

Save the project¶

Save the fitted project, and its report as one HTML file that opens without a network connection.

project.save_as('lbco_hrpt')
project.report.save_html('lbco_hrpt-report.html', offline=True)

Saving project to 'lbco_hrpt'
Saving report to 'lbco_hrpt-report.html'