Profile shape functions

Young RA, Wiles DB (1982) Profile shape functions in Rietveld refinements. J Appl Cryst 15 430-438... 

Mainly Gaussian and Lorentzian profile shape functions, and intermediates between these two shapes, are used. The following functions are all... 

The function 7ord accounts for the particle disorder, and its value is one for perfectly ordered films and zero for an isotropic sample. P(a) represents the profile-shape function used to model the distribution of c axes off the film normal. 

X-ray powder diffraction (XRD) was carried out using a Phillips MPD 1880 diffractometer, equipped with a Cu Ka source (A = 0.15418 nm), at 40 kV and 40 mA. The profiles were recorded at 0.02° (26) and step recording time of 5 s. The Ni(200) peak was fitted to a Pearson-VII profile shape function, as described elsewhere [5], with a residual error always lower than 1%. The method of Bertaut-Warren-Averbach (BWA) [11] was used to obtain CSDs and surface-average crystallite sizes, from which estimates of metal dispersion were obtained considering 0.065 nmVnickel atom and spherical crystallites [1]. Transmission electron microscopy (TEM) was carried out on a Jeol 200C working at 100 kV. The... 

The Voigt function is a convolution product ( ) between L and G. As the convolution is expensive from a computational point of view, the pseudo-Voigt form is more often used. The pseudo-Voigt is characterized by a mixing parameter r], representing the fraction of Lorentzian contribution, i.e. r] = 1(0) means pure Lorentzian (Gaussian) profile shape. Gaussian and Lorentzian breadths can be treated as independent parameters in some expressions. 

A Practical Hint20. In order to most accurately determine PI in Eq. (7.21), a mathematical theorem concerning convolution of a function with a shape function are helpful. The measured primary beam profile of the Kratky camera... 

In the standard setup W (y) is the profile of the primary beam in horizontal direction. In order to solve the smearing integral, the orientation distribution of the layer normals, g (), is approximated by a Poisson kernel121 and W (y) is approximated by a shape function with the integral breadth 2ymax of the primary beam perpendicular to the plane of incidence. In the simplified result... 

The hne-shape function gives the profile of the optical absorption (and emission) band and contains important information about the photon-system interaction. Let us briefly discuss the different mechanisms that contribute to this function, or the different line-broadening mechanisms. 

The calculated intensity z Xi, yj) at any point x, y, of a dififaction pattern is expressed as a function of the integrated intensity h of the reflections contained in the pattern and a normalized analytical peak shape function PS x, y) is used to model the individual profiles. It is given by... 

The run-to-run reproducibility of the profile shape of the FOCS fluorescence-intensity signal is good however, the reproducibility of the absolute intensity values is unsatisfactory. The run-to-run variations in the fluorescence intensities are caused by the differences in resin thickness at the small area "viewed" by the optrode. In addition, substantial resin flow takes place during cure, causing the resin thickness to vary as a function of cure time. However, since this variation in resin thickness mi t be reproducible from run-to-run (if other cure parameters remain unchanged), it may be possible to develop a suitable... 

The complete powder XRD profile (either for an experimental pattern or a calculated pattern) is described in terms of the following components (1) the peak positions, (2) the background intensity distribution, (3) the peak widths, (4) the peak shapes, and (5) the peak intensities. The peak shape depends on characteristics of both the instrument and the sample, and different peak shape functions are appropriate under different circumstances. The most common peak shape for powder XRD is the pseudo-Voigt function, which represents a hybrid of Gaussian and Lorentzian character, although several other types of peak shape function may be applicable in different situations. These peak shape functions and the types of function commonly used to describe the 20-dependence of the peak width are described in detail elsewhere [22]. 

In other words, for tetrahedral molecules, these relationships differ from the ones used for the linear molecules, especially Eq. 4.18. As a consequence, we must rederive the relationships for the spectral line shape and spectral moments. If the intermolecular interaction potential may be assumed to be isotropic, the line shape function Vg(a> T), Eq. 6.49, which appears in the expression for the absorption coefficient a, Eq. 6.50, may still be written as a superposition of individual profiles,... 

Broad bands in optical absorption and emission spectra originating from one or several closely lying electronic transitions are indications of strong vibronic coupling. In the case when the vibrational structure is resolved into a progression of individual bands, a vibronic analysis can be carried out which compares a theoretical line shape function with the intensity profile measured in the... 

With this the band profile (line shape) function Eq. (17) is... 

For establishing the context to the experiment, we compare the line shape function of a transition, Eq. (29), with the corresponding band profile measured in absorption or emission at low temperature T - 0. The parameters Aa, f a,... 

To gain quantitative information on the profile characteristics, the profile shape must be evaluated mathematically. The parameter Dt (D, diffusion constant t, exposure time) that describes the depth of the diffusion front that penetrated into the sample was determined by fitting the data with an error function (erf). The resulting curve describes the result of an undisturbed diffusion process. If the exposure time t is known, e.g. by radiocarbon dating, the diffusion constant D, a material constant, can be derived from this data. 

Figures 5.4 and 5.5 summarize results of a recent study of P. versicolor laccase electrochemistry based on cyclic and rotating disk voltammetry [60]. Figure 5.4 shows unequivocally that this laccase is voltammetrically active and gives a kinetically controlled, unpromoted four-electron peak at edge-plane pyrolytic graphite. Electrochemical reduction of 02 catalyzed by an immobilized laccase monolayer is close to reversible, and unrestricted by mass transport. The electrocatalysis follows, moreover, a Michaelis-Menten pattern (Fig. 5.5). Finally, there is a characteristic bell-shaped functional pH-profile with a pronounced maximum at pH 3.1.

The introduction of Gaussian function in Equation (3) is based on the prediction by Pake23 for crystals of small molecules, such as 1,2-dichloroethane, where the dipole pairs are isolated from each other. However, in the polymeric materials, not only the dipole interactions between the nearest neighbours but also those between the proton pairs at a longer distance contribute to the FID profile shape. Therefore, the distribution of the dipole interaction may be distorted. This is expressed by the introduction of Weibullian form,24 as follows ... 

The line shapes are described by Voigt functions, which reflect the Lorentzian line profiles due to natural line width and Gaussian profiles due to Doppler broadening. The instrumental broadening by the rocking curve of the crystal, de-focusing and the finite resolution of the detector is described well by a Voigt profile shape too [3[. 

Although approximate peak positions could be obtained from many different types of software, programs which use peak shape functions to fit the powder diffraction peak profiles are more appropriate to determine the peak position more precisely. Some of the common programs are Xfit,29 TOPAS, etc. 

As the name indicates, fitting the complete powder diffraction profile as a function of intensities, background and a peak shape function is full-profile fitting. ... 

Fig. 2.21. (a) Time-resolved LIF decay profiles for two closely spaced rotational levels of vibrationally excited CH3O (X). The solid line is an exponential fit for the decay convoluted with the dump laser pulse shape function, (b) Measured state specific unimolecular dissociation rate constants for CH3O (X) compared to calculated k E, J) curves without and with tunneling corrections. 

Because of the landmark nature of van der Waals work we shall now discuss some important aspects of his theory. In doing so a selection has to be made (the German version of van der Waals paper runs to over a hundred pages ). We shall use FICS-nomenclature and follow as much as possible van der Waals own arguments and derivations, although parts of the latter can nowadays be carried out more efficiently. For instance, the minimization of the Helmholtz energy as a function of the profile shape can nowadays be elegantly done by variational calculus, the principles of which will be outlined in appendix 3. 

In general, three different approaches to the description of peak shapes can be used. The first employs empirical peak shape functions, which fit the profile without attempting to associate their parameters with physical quantities. The second is a semi-empirical approach that describes instrumental and wavelength dispersion functions using empirical functions, while specimen properties are modeled using realistic physical parameters. In the third, the so-called fundamental parameters approach, all three components of the peak shape function (Eq. 2.45) are modeled using rational physical quantities. 

Since as5mimetry cannot be completely eliminated, it should be addressed in the profile fitting procedure. Generally, there are three ways of treating the asymmetry of Bragg peaks, all achieved by various modifications of the selected peak shape function ... 

In Eq. 2.61 a is a free variable, i.e. the asymmetry parameter, which is refined during profile fitting and z,- is the distance fi om the maximum of the symmetric peak to the corresponding point of the peak profile, i.e. z,-= 20yfc - 20 . This modification is applied separately to every individual Bragg peak, including Kaj and Ka2 components. Since Eq. 2.61 is a simple intensity multiplier, it may be easily incorporated into any of the peak shape functions considered above. Additionally, in the case of the Pearson-VII function, asymmetry may be treated differently. It works nearly identical to Eq. 2.61 and all variables have the same meaning as in this equation but the expression itself is different ... 

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