Before exploring Rietveld refinement, it is necessary to establish a greater understanding of powder diffraction data and what information is encoded therein in order to establish a notion of how to create a model of a diffraction pattern, which is of course necessary in Rietveld refinement. A typical diffraction pattern can be described by the positions, shapes, and intensities of multiple Bragg reflections. Each of the three mentioned properties encodes some information relating to the crystal structure, the properties of the sample, and the properties of the instrumentation. Some of these contributions are shown in Table 1, below. The structure of a powder pattern is essentially defined by instrumental parameters and two crystallographic parameters: unit cell dimensions, and atomic content and coordination. So, a powder pattern model can be constructed as follows: • Establish peak positions: Bragg peak positions are established from Bragg's law using the wavelength and d-spacing for a given unit cell. • Determine peak intensity: Intensity depends on the structure factor, and can be calculated from the structural model for individual peaks. This requires knowledge of the specific atomic coordination in the unit cell and geometrical parameters. • Peak shape for individual Bragg peaks: Represented by functions of the FWHM (which vary with Bragg angle) called the peak shape functions. Realistically ab initio modelling is difficult, and so empirically selected peak shape functions and parameters are used for modelling. • Sum: The individual peak shape functions are summed and added to a background function, leaving behind the resultant powder pattern. It is easy to model a powder pattern given the crystal structure of a material. The opposite, determining the crystal structure from a powder pattern, is much more complicated. A brief explanation of the process follows, though it is not the focus of this article. To determine structure from a powder diffraction pattern the following steps should be taken. First, Bragg peak positions and intensities should be found by fitting to a peak shape function including background. Next, peak positions should be indexed and used to determine unit cell parameters, symmetry, and content. Third, peak intensities determine space group symmetry and atomic coordination. Finally, the model is used to refine all crystallographic and peak shape function parameters. To do this successfully, there is a requirement for excellent data which means good resolution, low background, and a large angular range.
Peak shape functions For general application of the Rietveld method, irrespective of the software used, the observed Bragg peaks in a powder diffraction pattern are best described by the so-called peak shape function (PSF). The PSF is a convolution of three functions: the instrumental broadening \Omega(\theta), wavelength dispersion \Lambda(\theta), and the specimen function \Psi(\theta), with the addition of a background function, b(\theta). It is represented as follows: :PSF(\theta) = \Omega(\theta)\otimes\Lambda(\theta)\otimes\Psi(\theta) + b(\theta), where \otimes denotes convolution, which is defined for two functions f and g as an integral: :f(t)\otimes g(t) = \int_{-\infty}^{\infty} f(\tau)g(t-\tau)d\tau = \int_{-\infty}^{\infty} g(\tau)f(t-\tau)d\tau The instrumental function depends on the location and geometry of the source, monochromator, and sample. Wavelength function accounts for the distribution of the wavelengths in the source, and varies with the nature of the source and monochromatizing technique. The specimen function depends on several things. First is dynamic scattering, and secondly the physical properties of the sample such as crystallite size, and microstrain. A short aside: unlike the other contributions, those from the specimen function can be interesting in materials characterization. As such, average crystallite size, \tau, and microstrain, \varepsilon, effects on Bragg peak broadening, \beta (in radians), can be described as follows, where k is a constant: :\beta = \frac{\lambda}{\tau \cdot \cos\theta} and \beta = \kappa \cdot \varepsilon \cdot \tan\theta . Returning to the peak shape function, the goal is to correctly model the Bragg peaks which exist in the observed powder diffraction data. In the most general form, the intensity, Y(i), of the i^\text{th} point (1\leq i \leq n, where n is the number of measured points) is the sum of the contributions y_k from the m overlapped Bragg peaks (1 \leq k \leq m), and the background, b(i), and can be described as follows: :Y(i) = b(i)+ \sum_{k=1}^{m}{I_k[y_k(x_k)]} where I_k is the intensity of the k^\text{th} Bragg peak, and x_i = 2\theta_i - 2\theta_k . Since I_k is a multiplier, it is possible to analyze the behaviour of different normalized peak functions y(x) independently of peak intensity, under the condition that the integral over infinity of the PSF is unity. There are various functions that can be chosen to do this with varying degrees of complexity. The most basic functions used in this way to represent Bragg reflections are the Gauss, and Lorentzian functions. Most commonly though, is the pseudo-Voigt function, a weighted sum of the former two (the full Voigt profile is a convolution of the two, but is computationally more demanding). The pseudo-Voigt profile is the most common and is the basis for most other PSF's. The pseudo-Voigt function can be represented as: : y(x) = V_p(x) = n * G(x) + (1-n) * L(x) , where : G(x) = \frac{C_G^\frac{1}{2}}{\sqrt{\pi}H}e^{-C_Gx^2} and :L(x) = \frac{C_L^\frac{1}{2}}{\sqrt{\pi}H'} \left(1 + C_L x^2\right)^{-1} are the Gaussian and Lorentzian contributions, respectively. Thus, : V_p(x) = \eta \frac{C_G^{\frac{1}{2}}}{\sqrt{\pi}H}e^{-C_Gx^2} + (1-\eta) \frac{C_L^\frac{1}{2}}{\sqrt{\pi}H'}(1+C_Lx^2)^{-1}. where: • H and H' are the full widths at half maximum (FWHM) • x = \frac{2\theta_i - 2\theta_k}{H_k} is essentially the Bragg angle of the i^\text{th} point in the powder pattern with its origin in the position of the k^\text{th} peak divided by the peak's FWHM. • C_G = 4\ln2, C_L = 4 and \frac{C_G^{\frac{1}{2}}}{\sqrt{\pi}H} and \frac{C_L^{\frac{1}{2}}}{\sqrt{\pi}H'} are normalization factors such that \int_{-\infty}^{\infty}G(x)dx = 1 and \int_{-\infty}^{\infty}L(x)dx = 1 respectively. • H^2 = U\tan^2\theta + V\tan\theta + W , known as the Caglioti formula, is the FWHM as a function of \theta for Gauss, and pseudo-Voigt profiles. U, V, and W are free parameters. • H' = \frac{X}{\cos\theta} + Y\tan\theta is the FWHM vs. 2\theta for the Lorentz function. X and Y are free variables • \eta = \eta_0 + \eta_1 2\theta + \eta_2 \theta^2 , where 0 \leq \eta \leq 1 is the pseudo-Voigt mixing parameter, and \eta_{0,1,2} are free variables. The pseudo-Voigt function, like the Gaussian and Lorentz functions, is a centrosymmetric function, and as such does not model asymmetry. This can be problematic for non-ideal powder XRD data, such as those collected at synchrotron radiation sources, which generally exhibit asymmetry due to the use of multiple focusing optics. The Finger–Cox–Jephcoat function is similar to the pseudo-Voigt, but has better handling of asymmetry, which is treated in terms of axial divergence. The function is a convolution of pseudo-Voigt with the intersection of the diffraction cone and a finite receiving slit length using two geometrical parameters, S/L, and H/L, where S and H are the sample and the detector slit dimensions in the direction parallel to the goniometer axis, and L is the goniometer radius.
Peak shape as described in Rietveld's paper The shape of a
powder diffraction reflection is influenced by the characteristics of the beam, the experimental arrangement, and the sample size and shape. In the case of monochromatic neutron sources the convolution of the various effects has been found to result in a reflex almost exactly Gaussian in shape. If this distribution is assumed then the contribution of a given reflection to the profile y_i at position 2 \theta_i is: : y_i = I_k \exp \left [ \frac{-4 \ln \left (2 \right )}{H_k^2} \left (2\theta_i - 2\theta_k \right )^2 \right ] where H_k is the full width at half peak height (full-width half-maximum), 2\theta_k is the center of the reflex, and I_k is the calculated intensity of the reflex (determined from the
structure factor, the Lorentz factor, and
multiplicity of the reflection). At very low diffraction angles the reflections may acquire an asymmetry due to the vertical divergence of the beam. Rietveld used a semi-empirical correction factor, A_s to account for this asymmetry: : A_s = 1 - \left [ \frac {P \left (2\theta_i - 2\theta_k \right )^2}{\tan \theta_k} \right ] where P is the asymmetry factor and s is +1, 0, or -1 depending on the difference 2\theta_i-2\theta_k being positive, zero, or negative respectively. At a given position more than one diffraction peak may contribute to the profile. The intensity is simply the sum of all reflections contributing at the point 2\theta_i.
Integrated intensity For a Bragg peak (hkl), the observed integrated intensity, I_{hkl}, as determined from numerical integration is :I_{hkl} = \sum_{i=1}^j(Y_i^{obs}-b_i), where j is the total number of data points in the range of the Bragg peak. The integrated intensity depends on multiple factors, and can be expressed as the following product: :I_{hkl} = K \times p_{hkl} \times L_\theta \times P_\theta \times A_\theta \times T_{hkl} \times E_{hkl} \times |F_{hkl}|^2 where: • K: scale factor • p_{hkl}: multiplicity factor, which accounts for symmetrically equivalent points in the reciprocal lattice • L_\theta: Lorentz multiplier, defined by diffraction geometry • P_\theta: polarization factor • A_\theta: absorption multiplier • T_{hkl}: preferred orientation factor • E_{hkl}: extinction factor (often neglected as it is usually insignificant in powders) • F_{hkl}: structure factor as determined by the crystal structure of the material
Peak width as described in Rietveld's paper The width of the diffraction peaks are found to broaden at higher Bragg angles. This angular dependency was originally represented by : H_k^2 = U \tan^2 \theta_k + V \tan \theta_k + W where U, V, and W are the half-width parameters and may be refined during the fit. ==Preferred orientation==