Rietveld refinement is a technique described by Hugo Rietveld for use in the characterisation of crystalline materials. The neutron and X-ray diffraction of powder samples results in a pattern characterised by reflections (peaks in intensity) at certain positions. The height, width and position of these reflections can be used to determine many aspects of the material's structure.
The Rietveld method uses a least squares approach to refine a theoretical line profile until it matches the measured profile. The introduction of this technique was a significant step forward in the diffraction analysis of powder samples as, unlike other techniques at that time, it was able to deal reliably with strongly overlapping reflections.
The method was first implemented in 1967, and reported in 1969 for the diffraction of monochromatic neutrons where the reflection-position is reported in terms of the Bragg angle, 2θ. This terminology will be used here although the technique is equally applicable to alternative scales such as x-ray energy or neutron time-of-flight. The only wavelength and technique independent scale is in reciprocal space units or momentum transfer Q, which is historically rarely used in powder diffraction but very common in all other diffraction and optics techniques. The relation is
Introduction The most common powder X-ray diffraction (XRD) refinement technique used today is based on the method proposed in the 1960s by Hugo Rietveld. The Rietveld method fits a calculated profile (including all structural and instrumental parameters) to experimental data. It employs the non-linear least squares method, and requires the reasonable initial approximation of many free parameters, including peak shape, unit cell dimensions and coordinates of all atoms in the crystal structure. Other parameters can be guessed while still being reasonably refined. In this way one can refine the crystal structure of a powder material from PXRD data. The successful outcome of the refinement is directly related to the quality of the data, the quality of the model (including initial approximations), and the experience of the user.
The Rietveld method is a powerful technique which began a new era for powder XRD and materials science in general. Powder XRD is a fairly basic experimental technique with diverse applications and experimental options. Despite being slightly limited by the one-dimensionality of PXRD data and limited resolution, powder XRD's power is still notable. It is possible to determine the accuracy of a crystal structure model by fitting a profile to a 1D plot of observed intensity against the angle. Rietveld refinement requires a crystal structure model, and offers no way to come up with such a model on its own. However, it can be used to find structural details missing from a partial or complete ab initio structure solution, such as unit cell dimensions, phase quantities, crystallite sizes/shapes, atomic coordinates/bond lengths, micro strain in crystal lattice, texture, and vacancies.Pecharsky and Zavalij chapters 2, 6 and 7
Powder diffraction profiles: peak positions and shapes 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.
+Powder diffraction pattern as a function of various crystal structure, specimen, and instrumental parametersPattern componentCrystal structureSpecimen propertyInstrumental parameterPeak positionUnit cell parameters (a, b, c, α, β, γ) Absorption Porosity Radiation (wavelength), Instrument/sample alignment Axial divergence of the beamPeak intensityAtomic parameters (x, y, z, B, etc.) Preferred orientation Absorption Porosity Geometry and configuration Radiation (Lorentz polarization)Peak shape Crystallinity Disorder Defects Grain size Strain Stress Radiation (spectral purity) Geometry Beam Conditioning 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 , wavelength dispersion , and the specimen function , with the addition of a background function, . It is represented as follows:
,
where denotes convolution, which is defined for two functions and as an integral:
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, , and microstrain, , effects on Bragg peak broadening, (in radians), can be described as follows, where is a constant:
and .
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, , of the point (, where is the number of measured points) is the sum of the contributions from the overlapped Bragg peaks (), and the background, , and can be described as follows:
where is the intensity of the Bragg peak, and . Since is a multiplier, it is possible to analyze the behaviour of different normalized peak functions 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:
,
where
and
are the Gaussian and Lorentzian contributions, respectively.
Thus,
where:
and are the full widths at half maximum (FWHM) is essentially the Bragg angle of the point in the powder pattern with its origin in the position of the peak divided by the peak's FWHM. , and and are normalization factors such that and respectively. , known as the Caglioti formula, is the FWHM as a function of for Gauss, and pseudo-Voigt profiles. , , and are free parameters. is the FWHM vs. for the Lorentz function. and are free variables , where is the pseudo-Voigt mixing parameter, and 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, , and , where and are the sample and the detector slit dimensions in the direction parallel to the goniometer axis, and 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 at position is:
where is the full width at half peak height (full-width half-maximum), is the center of the reflex, and 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, to account for this asymmetry:
where is the asymmetry factor and is , , or depending on the difference 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 .
Integrated intensity For a Bragg peak , the observed integrated intensity, , as determined from numerical integration is
,
where 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:
where:
: scale factor : multiplicity factor, which accounts for symmetrically equivalent points in the reciprocal lattice : Lorentz multiplier, defined by diffraction geometry : polarization factor : absorption multiplier : preferred orientation factor : extinction factor (often neglected as it is usually insignificant in powders)