International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 2.3, pp. 66-69
|
Profile fitting has greatly advanced powder diffractometry by making it possible to calculate the intensities, peak positions, widths, and shapes of the reflections with a far greater precision than had been possible with manual measurements or visual inspection of the experimental data. The method has better resolution than the original data and the entire scattering distribution is used instead of only a few features such as the peak and width. Individual profiles and clusters of reflections can be fitted, or the entire pattern as in the Rietveld method (Chapter 8.6
).
The procedure is based on the least-squares fitting of theoretical profile intensities to the digitized powder pattern. The profile intensity at the ith step is calculated by where
is the background intensity,
is the integrated intensity of the jth reflection,
is the peak-maximum position,
is the profile function to represent the profile shape, and
is taken over j, in which the
has a finite value at
. Unlike the Rietveld method, a structure model is not used. In the least-squares fitting,
and
are refined together with background and profile shape parameters in
. Smoothing the experimental data is not required because it underestimates the estimated standard deviations for the least-squares parameters, which are based on the counting statistics.
The experimental profiles are a convolution of the X-ray line spectrum λ and all the combined instrumental and geometrical aberrations G with the true diffraction effects of the specimen S (Parrish, Huang & Ayers, 1976), i.e.
The profile shapes and resolution differ in the various diffractometer geometries and there is no universal profile-fitting function. In conventional X-ray tube focusing methods, the profiles are asymmetric and the shapes change continually across the scattering-angle range owing to the aberrations and the Kα doublet. To avoid problems caused by the Kα doublet, a few authors used the Kβ line, but it has only about 1/7 the intensity. The profiles obtained with synchrotron radiation are symmetrical and narrower, and the widths increase with increasing 2θ. The different shapes and rates of decay of the tails make it necessary to find an analytical function that best fits the particular experimental profile. Langford (1987) and Young & Wiles (1982
) have compiled and reviewed various profile-fitting functions and several are described below. Howard & Preston (1989
) give details of the computations in their review of the method.
Early profile analyses used Gaussian or Lorentzian (Cauchy) curves. Fig. 2.3.3.9(a)
shows that the most obvious difference is the rate of decay of the tails. X-ray synchrotron profiles lie between the two as shown in Figs. 2.3.3.9(c)–(f)
. The function must fit the tails as well as the main body and single-element functions are generally unsatisfactory. The Voigt function is a convolution of Lorentzian (L) and Gaussian (G) functions of different widths:
where x corresponds to
(Langford, 1978
). It has been used for profile fitting and also to determine certain physical properties such as crystallite size and strain broadening from the constituent L and G profiles (see, for example, Langford, 1978
; Suortti, Ahtee & Unonius, 1979
; de Keijser, Langford, Mittemeijer & Vogels, 1982
; Langford, Delhez, de Keijser & Mittemeijer, 1988
). Evaluation of a symmetrical Voigt function involves the real part of the complex error function and an algorithm for calculating this has been given by Langford (1992
).
The two most frequently used functions are at present the pseudo-Voigt (Wertheim, Butler, West & Buchanan, 1974) and the Pearson VII (Hall, Veeraraghavan, Rubin & Winchell, 1977
). They cannot be easily deconvoluted analytically and have no direct physical interpretation but the equivalent Voigt parameters can be derived. Both can be split into symmetric and asymmetric portions to adjust better to the profile asymmetries and tails. The shapes can also be varied systematically by changing the L/G ratio (de Keijser, Langford, Mittemeijer & Vogels, 1982
).
The pseudo-Voigt function is similar to the Voigt except that an addition is used in place of the convolution. It is easier to use and Wertheim et al. (1974) found there was only a small difference in the intensities and widths obtained with this approximation. It is defined as
where η is the ratio of Lorentz to Gauss and they have the same widths. The refined η and width of the full fitted profile can be related by a polynomial expansion (Hastings, Thomlinson & Cox, 1984
; David, 1986
; Cox, Toby & Eddy, 1988
) to the widths of the L and G components of the original Voigt function. It is frequently used to fit synchrotron-radiation profiles. In the particular case shown in Figs. 2.3.3.9(c)–(f)
, the pseudo-Voigt has the best fit as shown by the difference curve at half-height and the lowest Rp [= R(PF)] value.
The Pearson VII function is defined as where m is a refinable parameter based on the G/L content (for m = 1, the curve is 100% Lorentzian and for m = ∞ it is 100% Gaussian), and 1/b = 2[21/m− 1]1/2/W, where W is the FWHM.
The peak asymmetry can be incorporated into the profile function in several ways. One is to multiply (or add) the symmetrical profile function with an asymmetric function (Rietveld, 1969). Another is to dispose two or three functions asymmetrically (Parrish, Huang & Ayers, 1976
). A third is to use a split-type function, consisting of two profile functions, each of which defines one-half the total peak, i.e. the low- or high-angle sides of the peak and each has different profile widths and shapes but the same height (Toraya, Yoshimura & Somiya, 1983
; Howard & Snyder, 1983
).
Some other functions that have been used include the double Gaussian [Fig. 2.3.3.9(b)] for low-resolution synchrotron data (Will, Masciocchi, Parrish & Hart, 1987
), a Gaussian with shifted Lorentzian component to account for the asymmetry on the low-2θ side of the tail (Will, Masciocchi, Parrish & Lutz, 1990
), profile modelling of single isolated peaks with a rational function, e.g. the ratio of two polynomials (Pyrros & Hubbard, 1983
). In contrast to these analytical-type functions, some empirical functions have been developed. They are the `learned' (experimental) peak-shape function (Hepp & Baerlocher, 1988
) and the direct fitting of experimental data represented by Fourier series (Mortier & Constenoble, 1973
).
The sum of Lorentzians has been used for X-ray tube focusing profiles (Parrish & Huang, 1980; Taupin, 1973
). The instrument function
is determined (see below) by a sum of Lorentzian curves, three each for Kα1 and Kα2 and one for the weak Kα3 satellite. Three Lorentzians were used to match the asymmetry although a greater or lesser number could be used depending on the profile symmetry. Each curve has three parameters (intensity, half-width at half-height, and peak position) and the 21 parameters are adjusted by the computer program to give the best fit to the experimental data, which may contain 150 to 300 points. This is done only once for each particular instrument set-up. After
is determined, the profile fitting is easy and fast because only the specimen contribution S must be convoluted with
. If the specimen has no asymmetric broadening other than
, S can be approximated by a single symmetrical Lorentzian for each reflection; a split Lorentzian can be used if there is asymmetric broadening.
A function can be tested using isolated profiles of a standard specimen such as silicon, tungsten, quartz, and others which have 10 µm particles and no specimen broadening (Fawcett et al., 1988
). It is necessary to do this test carefully and whenever the instrument parameters are changed.
The instrument function can be measured using isolated profiles of standard specimens as stated above. The measurements should be made with small ° steps and count times long enough to accumulate about 25 000 to 50 000 counts on the Kα1 peaks for good counting statistics. By measuring a number of profiles separated by no more than about 5 to 10°, the instrument function can be established for the angular range of interest. A linear interpolation of the profile-fitting parameter between adjacent profiles gives a continuous function for use at any 2θ in the range. The intensities of the derived profile parameters are normalized and stored in the computer for later use. Note that any change in the X-ray spectrum or instrument geometry requires another set of measurements. The instrument function is also an important aid in computer graphics as described in Subsection 2.3.3.9
.
The fitting of conventional diffractometer profiles was considerably improved by the use of a convolution function, in which the Pearson VII function is convoluted with the observed instrument function (Toraya et al., 1983; Toraya, 1988
). Enzo, Fagherazzi, Benedetti & Polizzi (1988
; Benedetti, Fagherazzi, Enzo & Battagliarin, 1988
) used the convolution of a pseudo-Voigt function as the true data function and the convolution of exponential and pseudo-Voigt functions as the instrumental function for crystallite size and strain analysis. These functions have advantages in analysing the crystallite size and strain, although they require longer computation time for calculating the convolution.
Background intensity is usually included in the refinement. A first- or second-order polynomial is used to represent the background function B(x) in equation (2.3.3.12) in a small 2θ range, and the polynomial coefficients are adjusted during the least-squares refinement. In some cases, the background is subtracted from the pattern before the refinement by using the lowest intensities between the reflections. The background in the vicinity of high-intensity peaks and peak clusters is usually higher and should be avoided.
In the least-squares refinement, the following quantity is minimized: where N is the number of observations,
is the weight assigned to the ith observation, and
is the observed profile intensity. A statistical weighting factor such as
, where
, is frequently used. The quality of the fitting procedure is generally expressed by R factors such as
, the weighted R factor for profile intensity, which includes the entire scattering range and the background. The definitions of these factors are summarized by Young, Prince & Sparks (1982
). The
and
factors are given as
If the selected function is inappropriate, it will show up on the difference curve (experimental – calculated; see Fig. 2.3.3.9
), and high
and
factors.
A whole-powder-pattern fitting technique without using the structural model was proposed for analysing neutron powder data (Pawley, 1981) and then extended to X-ray data (Toraya, 1986
). The method executes the whole pattern decomposition (i.e. fitting all the profiles) in one step. In this technique, the peak position Tj in equation (2.3.3.12
) is a function of unit-cell parameters, and the unit-cell parameters are refined instead of individual peak positions. Furthermore, the angular dependence of the profile width can be expressed approximately as
where
,
, and
are adjustable parameters (Caglioti, Paoletti & Ricci, 1958
); but see also Louër & Langford (1988
). The profile-shape dependency on 2θ is ignored when fitting a small 2θ range, but it must be taken into account in the whole-powder-pattern fitting in both focusing and parallel-beam geometries. The least-squares ill conditioning is handled by imposing the constraints on the peak positions in the profile-fitting procedure.
Approximate unit-cell parameters are required to start the refinement. Advantages of this technique are: (1) the unit-cell parameters are refined to high precision; (2) the analysis is rapid and straightforward; (3) it is also powerful in analysing complex powder patterns. The output of indices and integrated intensities of all reflections can be used to calculate Patterson and Fourier diagrams, and thus used for ab initio structure determination (McCusker, 1988) and the structure refinement based on the integrated intensities such as used in the POWLS program (Will, 1979
).
The convolution equation is used in place of P(x) in equation (2.3.3.12), in which the true data function represented by a pseudo-Voigt or Pearson VII has adjustable parameters of crystallite size and strain (Toraya, 1989
). The anisotropic crystallite size assuming cylindrical shape has been determined by whole-powder-pattern fitting for complex powder patterns.
The advantage of profile fitting is illustrated in Fig. 2.3.3.10
for the quartz cluster at 68° with Cu Kα where the doublet separation is 0.19° and the FWHM is 0.14°. The relative intensities of the 122, 203, and 301 Kα1 peaks are 81:97:100, which differ from the profile-fitted peaks, 90:100:67, due to the overlapping. The sum of the fitted curves is the solid line which passes through the experimental points. The peak-search (or strip-chart) intensities that are not corrected for overlaps are more likely to correspond to the ICDD powder file than the profile-fitted values. Profile fitting is capable of about ±0.0004°2θ and 0.2% intensity for good experimental data (Parrish & Huang, 1980
). Even in data with poor counting statistical accuracy, it is possible to identify very weak peaks with low P/B as shown in Fig. 2.3.3.11
.
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