2.3 Fitting of Combinatorial X-Ray Diffraction Patterns
25
fitting the top of peaks only [10]. However, this is not feasible when there are
multiple phases present for which overlapping peaks result in shifts in apparent peak
positions and only using peaks that have no overlap would result in poor precision.
To overcome this, an in-house software was written in Yorick to fit the entire scans
using a nonlinear least squares fitting algorithm [59]. Fitting the entire patterns had
the added benefit of extracting peak width information that was used to calculate
crystallite sizes and strains.
Using a degree six polynomial to describe the background was found to converge
very slowly. This was resolved by using a cubic function and two broad asymmetric
Gaussians centered near 20 and 45
◦ to describe the background. This procedure was
found to work well for all samples, including spinel samples where manganese fluorescence gives rise to a complicated background when the three frames are stitched
together as shown in Fig. 2.4 (b).
In the program, each experimental peak was described by the convolution of the
sample scattering with the machine broadening as outlined by Warren [60]:
I k (2θ ) =
F k (2θ − z)M(z)dz
(2.1)
where 2θ is the scattering angle, F k is the sample scattering due to the k-th peak, and
M is the machine broadening normalized to have an area of unity. The integration
over z, the scattering angle of the machine broadening, was done numerically. The
sample scattering was described with a pseudo-Voigt function:
F k (2θ ) = I
◦
k [ηC(2θ , 2θ k , H k ) + (1 − η)G(2θ , 2θ k , H k )]
(2.2)
where I
◦
k is the integrated peak intensity of the k-th peak, H k is the full width at
half maximum (FWHM), 2θ k is the position of the center of the peak, η is the
Lorentzian component and is kept constant for all scattering angles, and C and G are
the Lorentzian and Gaussian functions, respectively, each normalized to have an area
of unity. The peak intensities were fitting parameters, thereby avoiding problems with
distortions due to the stitching of frames. The position of each peak was calculated
from the h, k and l values given in the Joint Committee on Powder Diffraction
Standards (JCPDS) database and from the adjustable lattice parameters. The Kα 1
and Kα 2 peaks were included in a 2:1 ratio in the calculation in order to fit the high
angle peaks accurately.
The machine broadening function, M, was determined by measuring the scattering
from corundum (NIST standard 1976a), the expected scattering from which is given
in JCPDS #46-1212. The scan was fit using pseudo-Voigt functions with η = 0.5 and
letting the FWHM of each peak, H (θ), be a fitting parameter. Although corundum is
not commonly used as a standard for machine broadening, it is sufficient to describe
the extensive broadening obtained with the 0.5 mm wide beam. This was determined
by measuring the scattering from a silicon wafer which was ground. The results
gave peak widths comparable to those obtained with the corundum, however the five
peaks showed far more scatter such that corundum proved to be the better standard.
Figure 2.5 shows that the resulting peak widths can be described by the Williamson–
Hall expression : H (θ ) · cosθ = A + Bsinθ , where A and B are constants. The
25
fitting the top of peaks only [10]. However, this is not feasible when there are
multiple phases present for which overlapping peaks result in shifts in apparent peak
positions and only using peaks that have no overlap would result in poor precision.
To overcome this, an in-house software was written in Yorick to fit the entire scans
using a nonlinear least squares fitting algorithm [59]. Fitting the entire patterns had
the added benefit of extracting peak width information that was used to calculate
crystallite sizes and strains.
Using a degree six polynomial to describe the background was found to converge
very slowly. This was resolved by using a cubic function and two broad asymmetric
Gaussians centered near 20 and 45
◦ to describe the background. This procedure was
found to work well for all samples, including spinel samples where manganese fluorescence gives rise to a complicated background when the three frames are stitched
together as shown in Fig. 2.4 (b).
In the program, each experimental peak was described by the convolution of the
sample scattering with the machine broadening as outlined by Warren [60]:
I k (2θ ) =
F k (2θ − z)M(z)dz
(2.1)
where 2θ is the scattering angle, F k is the sample scattering due to the k-th peak, and
M is the machine broadening normalized to have an area of unity. The integration
over z, the scattering angle of the machine broadening, was done numerically. The
sample scattering was described with a pseudo-Voigt function:
F k (2θ ) = I
◦
k [ηC(2θ , 2θ k , H k ) + (1 − η)G(2θ , 2θ k , H k )]
(2.2)
where I
◦
k is the integrated peak intensity of the k-th peak, H k is the full width at
half maximum (FWHM), 2θ k is the position of the center of the peak, η is the
Lorentzian component and is kept constant for all scattering angles, and C and G are
the Lorentzian and Gaussian functions, respectively, each normalized to have an area
of unity. The peak intensities were fitting parameters, thereby avoiding problems with
distortions due to the stitching of frames. The position of each peak was calculated
from the h, k and l values given in the Joint Committee on Powder Diffraction
Standards (JCPDS) database and from the adjustable lattice parameters. The Kα 1
and Kα 2 peaks were included in a 2:1 ratio in the calculation in order to fit the high
angle peaks accurately.
The machine broadening function, M, was determined by measuring the scattering
from corundum (NIST standard 1976a), the expected scattering from which is given
in JCPDS #46-1212. The scan was fit using pseudo-Voigt functions with η = 0.5 and
letting the FWHM of each peak, H (θ), be a fitting parameter. Although corundum is
not commonly used as a standard for machine broadening, it is sufficient to describe
the extensive broadening obtained with the 0.5 mm wide beam. This was determined
by measuring the scattering from a silicon wafer which was ground. The results
gave peak widths comparable to those obtained with the corundum, however the five
peaks showed far more scatter such that corundum proved to be the better standard.
Figure 2.5 shows that the resulting peak widths can be described by the Williamson–
Hall expression : H (θ ) · cosθ = A + Bsinθ , where A and B are constants. The