Peak integration

Quantitative analysis of single crystal diffraction data relies on accurate peak integration and correct application of appropriate corrections. This is distinguished from Rietveld analysis used for powders.

Note

In contrast to a single-crystal refinement, where structural parameters are fitted to observed structure factors, in a Rietveld fit one must describe all the items that affect the complete diffraction pattern [1]. - B. H. Toby, “Rietveld versus single-crystal refinements,” International Tables for Crystallography (2019). Vol. H. ch. 4.7, p. 466.

Instead, the objective is to calculate the observed structure factors [2]. The observed integrated intensity \(I_{hkl}\) is related to its corresponding \(|F_{hkl}|^2\) through several corrections and a scale constant \(c\).

\[I_{hkl} = c M \Phi(\lambda) \varepsilon(\lambda) A(\lambda) y(\lambda) L(\lambda,\theta,\phi) |F_{hkl}|^2\]

Careful calculation of the correction factors that are often wavelength-dependent are crucial in order to obtain meaningful observed structure factors.

Integration correction factors.

\(M\)	Normalization (time, monitor count) factor
\(\Phi(\lambda)\)	Incident flux
\(\varepsilon(\lambda)\)	Detector efficiency
\(A(\lambda)\)	Absorption correction
\(y(\lambda)\)	Extinction correction
\(L(\lambda,\theta,\phi)\)	Lorentz correction

Therefore, the data reduction rather than analysis software is generally responsible for applying corrections to the data.

Integrated intensity

The integrated intensity is essentially the “area” under the intensity “curve”.

Reciprocal space

The integrated peak intesity in reciprocal space corresponds to the integral under the three-dimensional peak less the background. Often, a three-dimensional envelope is used to approximate the integral by first determining its location and shape and summing the counts inside.

Detector space

Conversely, for detector-space integration, a region-of-interest (ROI) around the peak is used to extract the intensity. A one-dimensional profile (in time-of-flight for white beam or motor angle for monochromatic beam) could be used to integrate. For quasi-Laue image frame data, a two-dimensional profile or summation about the ROI in detector space might also be used.

Normalization factor

The counting statistics used to measure the peak must be accounted for.

Monochromatic beam

The beam monitor counts accumulated during the scan could be used as the normalization factor but relies on a reliable beam monitor. Another option assuming continuous and constant source is the counting time. This is reasonable for an experiment performed on the order of minutes to a single day but may be less-reliable over the course of several days or a week if the flux of the source changes throughout a fuel cycle.

White beam

For an accelerator-based source, it is best to used to use the accumulated proton charge during the measurement. This accounts for periods when the accelerator may be intermittent.

Incident flux

The incident flux is wavelength-dependent. For a monochromatic experiment, it is generally not required to correct for the incident flux since it is essentially a scale constant. However, for a Laue measurment, the spectrum must be well-understood. It is common to use a wavelength-resolved measurement of an incoherent scatter using a material like vanadium [3] to directly characterize the spectrum [4].

Detector efficiency

The efficiency of a neutron detector (with wavelength) may vary depending on several factors. An incoherent scattering measurement can be used to estimate the efficieny.

Lorentz correction

The Lorentz correction is related to the diffraction geometry and accounts for the time a given reflection remains in the diffraction correction. The factor (and its unit) depends on how a reflection is brought into the Bragg condition by considering the Ewald sphere.

Rotating crystal method

Given a monochromatic beam where a peak is brought into the diffraction position by rotating the crystal about the vertical axis (two-axis scan), the Lorentz correction for area detectors depends on the Bragg and azimuthal angle.

\[L(\lambda,\theta,\phi) = \frac{\lambda^3}{\sin2\theta\cos\phi}\]

Laue method

For Laue diffraction with a white beam where the Bragg condition is met within the incident wavelength band, the factor depends on a fourth power of wavelength.

\[L(\lambda,\theta,\phi) = \frac{\lambda^4}{2\sin^2\theta}\]

References

[1]

Toby, B. H. Rietveld versus Single-Crystal Refinements. International Tables for Crystallography (2019). Vol. H. ch. 4.7, p. 466. https://doi.org/10.1107/97809553602060000962.

[2]

Schultz, A. J.; Teller, R. G.; Peterson, S. W.; Williams, J. M. Collection and Analysis of Single Crystal Time‐of‐flight Neutron. AIP Conference Proceedings 1982, 89 (1), 35–42. https://doi.org/10.1063/1.33654.

[3]

Mayers, J. The Use of Vanadium as a Scattering Standard for Pulsed Source Neutron Spectrometers. Nuclear Instruments and Methods in Physics Research 1984, 221 (3), 609–618. https://doi.org/10.1016/0167-5087(84)90073-5.

[4]

Ikeda, S.; Carpenter, J. M. Wide-Energy-Range, High-Resolution Measurements of Neutron Pulse Shapes of Polyethylene Moderators. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 1985, 239 (3), 536–544. https://doi.org/10.1016/0168-9002(85)90033-6.