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. 6069

The basic experimental procedure in powder diffraction is the measurement of intensity as a function of scattering angle. The profile shapes and 2θ angles are derived from the observed intensities and hence the counting statistical accuracy has an important role. There is a wide range of precision requirements depending on the application and many factors are involved: instrument factors, counting statistics, profile shape, and particlesize statistics of the specimen. The quality of the specimen preparation is often the most important factor in determining the precision of powder diffraction data.
D. K. Smith and colleagues (see, for example, Borg & Smith, 1969; see also Yvon, Jeitschko & Parthé, 1977) developed a method for calculating theoretical powder patterns from well determined singlecrystal structures and have made available a Fortran program (Smith, Nichols & Zolensky, 1983). This has important uses in powder diffraction studies because it provides reference data with correct I's and d's, free of sample defects, preferred orientation, statistical errors, and other factors. The data can be displayed as recorded patterns by using plot parameters corresponding to the experimental conditions (Subsection 2.3.3.9). Calculated patterns have been used in a large variety of studies such as identification standards, computing intermediate members of an isomorphous series, testing structure models, ordered and disordered structures, and others. Many experiments can be performed with simulated patterns to plan and guide research. The method must be used with some care because it is based on the small single crystal used in the crystalstructure determination and the large powder samples of minerals and ceramics, for example, may have a different composition. Errors in the structure analysis are magnified because the powder intensities are based on the squares of the structure factors.
The Lorentz and polarization factors for diffractometry geometry have been discussed by Ladell (1961) and Pike & Ladell (1961).
Smith & Snyder (1979) have developed a criterion for rating the quality of powder patterns; see also de Wolff (1968a).
Ideally, the specimen should contain a large number of small equalsized randomly oriented particles. The surface must be flat and smooth to avoid microabsorption effects, i.e. particle interferences which reduce the intensities of the incident and reflected beams and can lead to significant errors (Cline & Snyder, 1983). The specimen should be homogeneous, particularly if it is a mixture or if a standard has been added. Low packing density and specimensurface displacement (§2.3.1.1.6) may cause significant errors. It is recommended that the powder and the prepared specimen be examined with a lowpower binocular optical microscope. Smith & Barrett (1979), Jenkins, Fawcett, Smith, Visser, Morris & Frevel (1986), and Bish & Reynolds (1989) have surveyed methods of specimen preparation and they include bibliographies on special handling problems. Powder diffraction standards for angle and intensity calibration are described in Section 5.2.9 .
Preferred orientation changes the relative intensities from those obtained with a randomly oriented powder sample. It occurs in materials that have good cleavage or a morphology that is platy, acicular or any special shape in which the particles tend to orient themselves in specimen preparation. The micas and clay minerals are examples of materials that exhibit very strong preferred orientation. When they are prepared as reflection specimens, the basal reflections dominate the pattern. It is common in prepared thin films where preferred orientation occurs frequently or may be purposely induced to enhance certain optical, electrical, or magnetic properties for electronic devices. By comparison of the relative intensities with the random powder pattern, the degree of preferred orientation can be observed.
Powder reflections take place from crystallites oriented in different ways in the instrument geometries as shown in Fig. 2.3.1.2. In reflection specimen geometry with θ–2θ scanning, reflections can occur only from lattice planes parallel to the surface and in the transmission mode they must be normal to the surface. In the Seemann–Bohlin and fixed specimen with 2θ scanning methods, the orientation varies from parallel to about 45° inclination to the surface. The effect of preferred orientation can be seen in diffraction patterns obtained by using the same specimen in the different geometries.
The effect is illustrated in Fig. 2.3.3.1 for mchlorobenzoic acid, C_{7}H_{5}ClO_{2}, with reflection and transmission patterns and the pattern calculated from the crystal structure. The degree of preferred orientation is shown by comparing the peak intensities of four reflections in the three patterns:
Care is required to make certain the differences are not caused by a few fortuitously oriented large particles.
Various methods have been used to minimize preferred orientation in the specimen preparation (Calvert, Sirianni, Gainsford & Hubbard, 1983; Smith & Barrett, 1979; Jenkins et al., 1986; Bish & Reynolds, 1989). These include using small particles, loading the powder from the back or side of the specimen holder, and cutting shallow grooves to roughen the surface. The powder has also been sifted directly on the surface of a microscope slide or singlecrystal plate that has been wetted with the binder or petroleum jelly. Another method is to mix the powder with an inert amorphous powder such as Lindemann glass or rice starch, or add gum arabic, which after setting can be reground to obtain irregular particles. Any additive reduces the intensity and the peaktobackground ratio of the pattern. A promising method that requires a considerable amount of powder is to mix it with a binder and to use spray drying to encapsulate the particles into small spheres which are then used to prepare the specimen (Smith, Snyder & Brownell, 1979).
Preferred orientation would not cause a serious problem in routine identification providing the reference standard had a similar preferred orientation and both patterns were obtained with the same diffractometer geometry. However, when accurate values of the relative intensities are required, as in crystalstructure refinement and quantitative analysis, it may be the major factor limiting the precision.
In practice, it is very difficult to prepare specimens that have a completely random orientation. Even materials that do not have good cleavage or special morphological forms, such as quartz and silicon, show small deviations from a completely random orientation. These show up as errors in the structure refinement and a correction factor is required.
An empirical correction factor determined by the acute angle between the preferredorientation plane and the diffracting plane (hkl) can be used (Will et al., 1988). Three functions have been used to represent and the term GP is the variable refined: (Rietveld, 1969) for transmission specimens; for reflection specimens; and (Dollase, 1986).
These functions are quite similar for small amounts of nonrandomness. The preferredorientation plane is selected by trial and error. For example, a modified fast routine of the powder leastsquares refinement program with only seven cycles of refinement on each plane for the first dozen allowed Miller indices can be used to find the plane that gives the lowest R(Bragg) value as shown in Table 2.3.3.1. All three functions improve the R(Bragg) value as shown in Table 2.3.3.2 but the evidence is not conclusive as to which is the best. More research is required in this area. Several specimens made of the same material may show different preferredorientation planes, and in some cases the preferredorientation plane never occurred in the crystal morphology. A more complicated method examines the polaraxis density distribution using a cubic harmonic expansion to describe the crystallite orientation of a rotating sample (Järvinen, Merisalo, Pesonen & Inkinen 1970; Ahtee, Nurmela, Suortti & Järvinen, 1989; Järvinen, 1993).
^{†}Selected preferred orientation plane.


In addition to profile broadening, which begins to appear when the crystallite sizes are 1–2 μm, the sizes have a strong effect on the absolute and relative intensities (de Wolff, Taylor & Parrish, 1959; Parrish & Huang, 1983). The particle sizes have to be less than about 5 μm to achieve 1% reproducible relative intensities from a stationary specimen in conventional diffractometer geometry (Klug & Alexander, 1974). The statistical errors arising from the number of particles irradiated can be greatly reduced by using smaller particles and rotating the specimen around the diffraction vector. This brings many more particles into reflecting orientations.
The particlesize effect is illustrated in Fig. 2.3.3.2 for specimens of NIST silicon standard powder 640 sifted to different size fractions. The powders were packed in a 1 mm deep cavity in a 25.4 mm diameter Al holder using 5% collodion/amyl acetate binder. They were rotated by a synchronous motor (a stepper motor can also be used) around the axis normal to the centre of the specimen surface with the detector arm fixed at the peak position and the intensity recorded with a stripchart. Rapid rotation, ∼60 r min^{−1}, gives the average peak intensity for all azimuths of the specimen and the small variations result only from the counting statistics. Scaling the intensities to (111) = 100% for the 5–10 μm fraction, the 10–20 μm fraction is 94%, 20–30 μm 88% and μm 59%. The decrease is probably due to lower particlepacking density and increasing interparticle microabsorption. The μm fraction = 95% may be due to the larger ratio of oxide coating around the particles to the mass of the particles.

Effect of specimen rotation and particle size on Si powder intensity using a conventional diffractometer (Fig. 2.3.1.3) and Cu Kα. Numbers below fast rotation are the average intensities. 
Slow rotation, ∼1/7 r min^{−1}, shows the variation of the peak intensity with azimuth angle . The pattern repeats after 360° rotation and the magnitude of the fluctuations increases with increasing particle sizes and resolution. There is no correlation between the fluctuations of different reflections, as can be seen by comparing the 111, 220 and 311 reflections of the 10–20 μm specimen (lower left side) for which the incidentbeam intensity was adjusted to give the same average amplitude. The horizontal lines are ±10% of the average. This shows the magnitude of errors that could occur using stationary specimens. Similar particlesize effects were found using the integrated intensities derived from profile fitting. The above discussion and Fig. 2.3.3.2 refer to a continuous scan. If the stepscan mode is used to collect data, it is clearly not necessary to rotate the specimen through more than one revolution at each step.
The rotating specimen also averages the inplane preferred orientation but has virtually no effect on the planes oriented parallel to the specimen surface. The slow rotation method is useful in testing the grinding and sifting stages in specimen preparation. When calibrated with known size fractions, it can be used as a rough qualitative measure of the particle sizes.
A common source of error arises from the Kα doublet which produces a pair of peaks for each reflection. The separation of the Cu Kα_{1}, Kα_{2} peaks increases from 0.05° at 20°2θ to 1.08° at 150°2θ. The overlapping is also dependent on the instrument resolution and may cause errors in the peak angles and intensities when stripchart recording or peaksearch methods (described below) are used. The Kα_{1} wavelength is generally used to calculate all the d's even when the lowangle peaks are unresolved. In the region where the doublet is only slightly resolved, the apparent Kα_{1} peak angle is shifted to higher angles because of the overlapping Kα_{2} tail and similarly the peak intensities will be in error. The relative peak intensities of a reflection with superposed doublet compared to a resolved doublet could have an error as large as 50%. Relative peak intensities are used in the ICDD standards file and cause no problem because the unknowns are measured in the same way. The integrated intensity avoids this difficulty but is impractical to use in routine identification.
Rachinger (1948) described a simple graphical procedure for removing Kα_{2} peaks. The method causes errors because it makes the incorrect assumption that Kα_{2} is the exact halfscale version of Kα_{1}. Ladell, Zagofsky & Pearlman (1975) developed an exact algorithm using the actual mathematical shapes observed with the user's diffractometer but, with lineprofilefitting programs now available, the Kα_{2} component can be modelled precisely along with the Kα_{1}.
It is possible to isolate the Kα_{1} line when using a highquality incidentbeam focusing monochromator as described in Subsection 2.3.1.2, Fig. 2.3.1.12(b), but there may be a loss of intensity. The source size must be narrow and the focal length long enough to separate the components.
The most obvious and commonly used measure of the reflection angle of a profile is the position of maximum intensities (Fig. 2.3.3.3 ). The midpoints of chords at various heights have often been used but their values vary with the profile asymmetry. Another method is to connect the midpoints of chords near the top of the profile and extrapolate to the peak. The computer methods using derivatives are the most accurate and fastest as described in Subsection 2.3.3.7.
A more fundamental measure that uses the entire intensity distribution is the centre of gravity (or centroid) defined as The variance (meansquare deviation of the mean) is defined as
The use of the centroid and variance has two important advantages: (1) most of the aberrations (§2.3.1.1.6) were derived in terms of the centroid and variance; and (2) they are additive, making it easy to determine the composite effect of a number of aberrations. Mathematically, the integration extends from to but the aberrations have a finite range. However, the practical use of these measures causes some difficulty. If the profile shapes are Lorentzian, the tails decay slowly. A very wide range would be required to reach points where the signal could no longer be separated from the background and the profiles must be truncated for the calculation. Truncation limits that have been used are 90% ordinate heights of Kα_{1} (Ladell, Parrish & Taylor, 1959), and equal 2θ or limits from the centroid (Taylor, Mack & Parrish, 1964; Langford, 1982). The limits such as and in Fig. 2.3.3.3 must be carefully chosen to avoid errors and this involves the correct determination of the background level. It is not practical to use centroids for overlapping peak clusters unless the profile fitting can accurately resolve the individuals with their correct positions and intensities. Their use has, therefore, been confined to simple patterns with small unit cells in which the profiles were well separated.
The difference between the angle derived from the peak and the centroid depends on the asymmetry of the profile, which in turn varies with the Kαdoublet separation and the aberration broadening. Tournarie (1958) found that the centre of a horizontal chord at 60.6% of the Kα_{1} peak height corresponds well to the centroid of that line in fairly well resolved doublets. The number, of course, depends on the profile shape. There is also the basic problem that most of the Xray wavelengths were probably determined from the spectral peaks and, if the centroids are measured for the powder pattern, the Bragg equation becomes nonlinear in the sense that the 1:1 correspondence between λ and is lost.
Formerly, the most common method of obtaining diffractometer data was by using a ratemeter and stripchart recorder with the paper moving synchronously with the constant angular velocity of the scan. This simple analogue method is still used and a large fraction of the JCPDS (ICDD) file prior to about 1982 was obtained in this way.
The method has several limitations: the data are not in the digital form required for computers, and are distorted; manual measurement of the chart takes a long time and has low accuracy. The output of the strip chart lags behind the input by an amount determined by the product of the scanning speed and the time constant of the ratemeter, including the speed of the recorder pen. The peak height is decreased and shifted in the direction of the scan causing asymmetric broadening with loss of resolution. The profile shape, Kαdoublet separation, and scan direction also contribute to distortion. When the product of the scan speed and time constant have the same value, the profile shapes are the same even though the total count is determined by the scan speed, Figs. 2.3.3.4 (a) and (b). If the product is large, the distortion is severe (c), and very weak peaks may be lost.
Most diffractometers are now sold with computer automation. Older instruments can be easily upgraded by adding a stepping motor to the geardrive shaft. A large variety of computers and programs is available, and it is not easy to make the best selection. Continuing improvements in computer technology have been made to handle expanded programs with increased speed and storage capabilities. The collected data are displayed on a VDU screen and/or computer printer and stored on hard disk or diskette for later use and analysis. Microprocessors are often used to select the Xraygenerator operating conditions, shutter control, specimen change, and similar tasks that were formerly performed manually. Aside from the elimination of much of the manual labour, automation provides far better control of the datacollection and datareduction procedures. However, computers do not preclude the necessity of precise alignment and calibration. Smith (1989) has written a detailed description of computer analysis for phase identification and also includes related programs and their sources.
Personal computers are widely used for powderdiffraction automation and a typical arrangement is shown in Fig. 2.3.3.5(a) . The automation may provide for step scanning, continuous scanning with readout on the fly, or slewing to selected angles to read particular points. Step scanning is the method most frequently used. It is essential that absolute registration and step tracking be reliably maintained for all experimental conditions.
The step size or angular increment Δ2θ and count time t at each step, and the beginning and ending angles are selectable. For a given total time available for the experiment, it usually makes no difference in the counting statistical accuracy if a combination of small or large Δ2θ and t (within reasonable limits) is used. A minimal number of steps of the order of to 0.2 FWHM is required for profile fitting isolated peaks. It is clear that the greater the number of steps, the better the definition of the profile shape. The step size becomes important when using profile fitting to resolve patterns containing overlapped reflections and to detect closely spaced overlaps from the width and small changes in slopes of the profiles. A preliminary fast run to determine the nature of the pattern may be made to select the best run conditions for the final pattern. Will et al. (1988) recorded a quartz pattern with 1.28 Å synchrotron Xrays and 0.01° steps to test the stepsize role. The profile fitting was done using all points and repeated with the omission of every second, third, and fourth point corresponding to Δ2θ = 0.02, 0.03 and 0.04°. The R(Bragg) values were virtually the same (except for 0.04° where it increased), indicating the experimental time could have been reduced by a factor of three with little loss of precision; see also Hill & Madsen (1984). Patterns with more overlapping would require smaller steps. Ideally, the steps could be larger in the background but this also requires a prior knowledge of the pattern and special programming.
A typical VDU screen menu for diffractometeroperation control is shown in Fig. 2.3.3.5(b). A number of runs can be defined with the same or different experimental parameters to run consecutively. The run log number, date, and time are usually automatically entered and together with the comment and parameters are carried forward and recorded on the printouts and graphics to make certain the runs are completely identified. The menu is designed to prompt the operator to enter all the required information before a run can be started. Error messages appear if omissions or entry mistakes are made. There are, of course, many variations to the one shown.
Xray quanta arrive at the detector at random and varying rates and hence the rules of statistics govern the accuracy of the intensity measurements. The general problems in achieving maximum accuracy in minimum time and in assessing the accuracy are described in books on mathematical statistics. Chapter 7.5 reviews the pertinent theory; see also Wilson (1980). In this section, only the fixedtime method is described because the fixedcount method takes too long for most practical applications.
Let be the average of N, the number of counts in a given time t, over a very large number of determinations. The spread is given by a Poisson probability distribution (if is large) with standard deviation Any individual determination of N or the corresponding counting rate n (= N/t) will be subject to a proportionate error which is also a function of the confidence level, i.e. the probability that the result deviates less than a certain percentage from the true value. If Q is the constant determined by the confidence level, then where Q = 0.67 for the probable relative error (50% confidence level) and Q = 1.64 and 2.58 for the 90 and 99% confidence levels respectively. For a 1% error, N = 4500, 27 000, 67 000 for , , , respectively. Fig. 2.3.3.6 shows various percentage errors as a function of N for several confidence levels.
In practice, there is usually a background count . The net peak count is dependent on the P/B ratio as well as on and separately. The relative error of the net peak count is which shows that is similarly influenced by both absolute errors and . The absolute standard deviation of the net peak height is and expressed as the per cent standard deviation is The accuracy of the net peak measurement decreases rapidly as the peaktobackground ratio falls below 1. For example, with N_{B} = 50, the dependence of on P/B is It is obviously desirable to minimize the background using the best possible experimental methods.
The accurate location of the 2θ angle corresponding to the peak of the profile has been discussed in many papers (see, for example, Wilson, 1965). Computers are now widely used for data reduction, thereby greatly decreasing the labour, improving the accuracy, and making possible the use of specially designed algorithms. It is not possible to present a description of the large number of private and commercial programs. The peaksearch and profilefitting methods described below have been successfully used for a number of years and are representative of the results that can now be obtained. They have greatly improved the results in phase identification, integrated intensity measurement, and analyses requiring precise profileshape determination. It is likely that even better programs and methods will be developed in this rapidly changing field.
There are two levels of the types of data reduction that may be done. The easiest and most frequently used method is usually called `peak search'. It computes the 2θ angles and intensities of the peaks. The results have good precision for isolated peaks but give the values of the composite overlapping reflections as they appear, for example, on a stripchart recording. The calculation is virtually instantaneous and is often all that is needed for phase identification, latticeparameter determination, and similar analyses. The second, profile fitting, described below, is a more advanced procedure that can resolve overlapping peaks into individual reflections and determines the profile shape, width, peak and integrated intensities, and reflection angle of each resolved peak. This method requires a prior knowledge of the profilefitting function. It is used to determine the integrated intensities for analyses requiring higher precision such as crystalstructure refinement and quantitative analysis, and profileshape parameters for small crystallite size, microstrain and similar studies.
To measure weak peaks, the counting statistical accuracy must be sufficient to delineate the peak from the background. When the intensity and peaktobackground ratio are low, the computing time is much increased. Since powder patterns often contain a number of weak peaks that may not be required for the analysis, computer programs often permit the user to select a minimum peak height (MPH) and a standard deviation (SD) that the peak must exceed to be included in the data reduction. For example, MPH = 1 would reject peaks less than 1% of the highest peak in the recorded pattern, and SD = 4 requires the intensity to exceed the background adjacent to the peak by . The number of peaks rejected depends on the intensity and peaktobackground ratio as illustrated in Fig. 2.3.3.7 , where the cutoff level was set at for two recordings of the same pattern with about a 40 times difference in intensities. All visible peaks are included in the highintensity recording and several are rejected by the cutoff level selected in the lowerintensity pattern.

Effect of 4σ maximum peak height (horizontal line) on dropping weak peaks from inclusion in computer calculation. Step scan with (a) t = 5 s and (b) t = 0.1 s. Fivecompound mixture, Cu Kα. 
Before carrying out the computer calculations, it may be desirable to subtract unusual background such as is caused by a glass substrate in a thinfilm pattern.
The following method was developed using computergenerated profiles having the same shapes as conventional diffractometer (Fig. 2.3.1.3) profiles and adding random counting statistical noise (Huang & Parrish, 1984; Huang, 1988). The best results were obtained using the first derivative (dx/dy = 0) of a leastsquaresfitted cubic polynomial to locate the peaks, combined with the second derivative (d^{2}y/dx^{2} = minimum) of a quadratic/cubic polynomial to resolve overlapped reflections (Fig. 2.3.3.8 ). Overlaps with a separate 0.5 FWHM can be resolved and measured and the accuracy of the peak position is 0.001° for noisefree profiles. Real profiles with statistical noise have a precision of ±0.003 to 0.02° depending on the noise level. The Savitzky & Golay (1964) method (see also Ateiner, Termonia & Deltour, 1974; Edwards & Willson, 1974) was used for smoothing and differentiation of the data by least squares in which the values of the derivatives can be calculated using a set of tabulated integers. The convolution range CR expressed as a multiple of the FWHM of the peak can be selected. A minimum of five points is required. For asymmetric peaks, such as occur at small 2θ's, a CR 0.5 FWHM gives the best precision. The larger the CR the larger the intrinsic error but the smaller the random error, and the smaller the number of peaks identified in overlapping patterns. The larger CR also avoids false peaks in patterns with poor counting statistics. Fig. 2.3.3.8(c) shows the dependence of the accuracy of the peak determination on P/ σ. The computer results list the 2θ's, d's, absolute and relative intensities (scaled to 100) of the identified peaks. The calculation is made with a selected wavelength such as and the possible peaks are flagged.
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 leastsquares 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 peakmaximum 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 leastsquares 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 leastsquares parameters, which are based on the counting statistics.
The experimental profiles are a convolution of the Xray 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 profilefitting function. In conventional Xray tube focusing methods, the profiles are asymmetric and the shapes change continually across the scatteringangle 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 profilefitting 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. Xray 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 singleelement 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 pseudoVoigt (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 pseudoVoigt 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 synchrotronradiation profiles. In the particular case shown in Figs. 2.3.3.9(c)–(f), the pseudoVoigt has the best fit as shown by the difference curve at halfheight and the lowest R_{p} [= 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[2^{1/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 splittype function, consisting of two profile functions, each of which defines onehalf the total peak, i.e. the low or highangle 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 lowresolution synchrotron data (Will, Masciocchi, Parrish & Hart, 1987), a Gaussian with shifted Lorentzian component to account for the asymmetry on the low2θ 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 analyticaltype functions, some empirical functions have been developed. They are the `learned' (experimental) peakshape 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 Xray tube focusing profiles (Parrish & Huang, 1980; Taupin, 1973). The instrument function is determined (see below) by a sum of Lorentzian curves, three each for Kα_{1} and Kα_{2} and one for the weak Kα_{3} 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, halfwidth at halfheight, 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 setup. 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 Kα_{1} 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 profilefitting 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 Xray 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 pseudoVoigt function as the true data function and the convolution of exponential and pseudoVoigt 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 secondorder 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 leastsquares 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 highintensity peaks and peak clusters is usually higher and should be avoided.
In the leastsquares 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 wholepowderpattern fitting technique without using the structural model was proposed for analysing neutron powder data (Pawley, 1981) and then extended to Xray 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 T_{j} in equation (2.3.3.12) is a function of unitcell parameters, and the unitcell 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 profileshape dependency on 2θ is ignored when fitting a small 2θ range, but it must be taken into account in the wholepowderpattern fitting in both focusing and parallelbeam geometries. The leastsquares ill conditioning is handled by imposing the constraints on the peak positions in the profilefitting procedure.
Approximate unitcell parameters are required to start the refinement. Advantages of this technique are: (1) the unitcell 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 pseudoVoigt or Pearson VII has adjustable parameters of crystallite size and strain (Toraya, 1989). The anisotropic crystallite size assuming cylindrical shape has been determined by wholepowderpattern 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 Kα_{1} peaks are 81:97:100, which differ from the profilefitted 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 peaksearch (or stripchart) intensities that are not corrected for overlaps are more likely to correspond to the ICDD powder file than the profilefitted 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 .
An interactive graphics display program is a very important asset for interpreting and analysing powder diffraction data. If a colour graphics station is used, the display can be enhanced by using various colours. The simplest form is the VDU display of experimental points connected with straight lines, which appears similar to a stripchart recording but has no timeconstant error and is printed on pagesize paper. It avoids storing large numbers of charts because hundreds of patterns can be stored on a diskette and displayed and printed at any time.
The basic parameters required in one of the published methods (Parrish, Huang & Ayers, 1984) are the d's and I's of the reflections, the wavelength and profile shapes instrument function). This makes it possible to produce a pattern exactly as it would appear on the user's diffractometer, aside from contributions arising from sample microstructure. The step size can be included if experimental patterns are to be reproduced or if patterns are to be subtracted. A section of the pattern can be enlarged to the full screen size by entering the desired angular range and highest peak intensity. A linear background can be added by entries at the low and high2θ points. Nonlinear background, e.g. from an amorphous substrate, can be transferred from a stored file. Counting statistical noise can be added to a simulated pattern by using a normally distributed random number with a standard deviation scaled to the calculated . Noise in an experimental pattern can be smoothed. A profile broadening factor can be added to the function. Quantitative synthesis of a mixture can be simulated by entering the relative weight percentage and reference intensity, the ratio of the intensities of the strongest lines of each pattern in a 50–50 mixture, or the ICDD values compared to αAl_{2}O_{3} (de Wolff & Visser, 1988; Davis & Smith, 1988). The program has access to the ICDD file stored on disk so that any card can be reproduced as a pattern using any wavelength. Some examples are shown in Fig. 2.3.3.12 .
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