How to quote lattice parameters with their esds#

Read each lattice parameter’s row with RefinementResult.parameter, the volume from RefinementResult.cell_volumes, and write each as value(esd) with rx.format_su.

The esd measures how precisely the pattern fixes a parameter within the model you fitted. How close the cell is to the true one depends on the wavelength and the position corrections you declared, which the esd does not include.

Read the cell and its volume#

This runs mccusker_structural on the fluorapatite pattern that ships with rietx, a hexagonal cell:


import rietx as rx
from rietx.examples import examples_dir

data = rx.read_pattern(examples_dir() / "FAP.XRA")


def refine(instrument):
    structure = rx.Structure.from_cif(examples_dir() / "fluorapatite.cif")
    instrument.background = rx.BackgroundChebyshev.with_terms(6)
    return rx.Refinement(structure, instrument).fit(
        data, plan="mccusker_structural", two_theta_limits=(15, 130))


result = refine(rx.Instrument.bragg_brentano(radiation="CuKa"))
for path in ("phases.0.cell.a", "phases.0.cell.b", "phases.0.cell.c"):
    row = result.parameter(path)
    print(f"{path:<16} {rx.format_su(row.value, row.stderr)}  free {row.vary}")
volume = result.cell_volumes[0]
print(f"V = {rx.format_su(volume.volume, volume.stderr)} ų")

phases.0.cell.a  9.370962(83)  free True
phases.0.cell.b  9.370962(83)  free False
phases.0.cell.c  6.885324(80)  free True
V = 523.629(12) ų

In a hexagonal cell b is tied to a, so its row has vary set to False and carries a’s esd. The angles are fixed by symmetry, so they have no rows in the result and no esds to quote. rx.format_su gives the esd two significant figures and rounds the value to match.

The volume’s esd is propagated through the full covariance of the cell parameters, so the correlation between a and c is included. CellVolume.stderr_diagonal is the same esd with that correlation left out, 0.011 ų here. Quote stderr.

Quote the esd as it is#

result.statistics.esd_inflation is the factor already applied to every esd:

print(f"Bérar-Lelann factor {result.statistics.esd_inflation:.2f}")

Bérar-Lelann factor 2.17

Neighbouring points of a step scan have correlated residuals, so they carry less independent information than their number suggests. The Bérar-Lelann factor widens the esds to account for it. The esds above already include it, so do not multiply them by it, or by the goodness of fit, again. GSAS’s esds do not include it, so compare them with stderr_uncorrected, the same esd without the factor. Estimation gives the formula.

Check what the esd leaves out#

A lattice parameter scales with the wavelength you declare. rietx’s Cu Kα₁ is 1.5405929 Å, and the GSAS tutorial this pattern comes from declares 1.5405 Å, 60 ppm shorter. Refining again with only the tutorial’s wavelengths:

gsas = rx.read_gsas_exp(examples_dir() / "FAP.EXP")
tutorial = rx.Instrument.bragg_brentano(radiation="CuKa")
for line, wavelength in zip(tutorial.source.lines, gsas.histogram().wavelengths):
    line.wavelength.value = wavelength
other = refine(tutorial)

a, b = result.parameter("phases.0.cell.a"), other.parameter("phases.0.cell.a")
print(f"a with rietx's Cu Ka      {rx.format_su(a.value, a.stderr)}")
print(f"a with the tutorial's     {rx.format_su(b.value, b.stderr)}")
print(f"difference {(a.value / b.value - 1) * 1e6:.0f} ppm, {(a.value - b.value) / a.stderr:.1f} esd")

a with rietx's Cu Ka      9.370962(83)
a with the tutorial's     9.370341(84)
difference 66 ppm, 7.5 esd

A 60 ppm change in the declared wavelength moves a by 7.5 esd, and the esd itself does not change. GSAS’s own refinement of this pattern, at the tutorial’s wavelengths and with its own peak-position model, gives a = 9.371724(36) Å, 148 ppm above the second fit.

To quote an accurate cell, state the wavelength you used. On a laboratory instrument, measure a certified standard such as NIST SRM 660c LaB₆ on the same setup, and check that its refined cell matches the certificate before you trust a sample’s. At a synchrotron, use the wavelength the beamline calibrated for your run.

See also#