Voigt approximation

By using the relations between the elastic-stiffness coefficients in the cubic class from Table 10.3 in Eq. 10.19, the Voigt approximation of the Young s modulus is obtained for a material with cubic symmetry ... 

Similarly, the Reuss and Voigt approximations for the bulk moduh of polycrystalline aggregates composed of cubic and orthorhombic crystaUites are given by Eqs. 10.39 through 10.42 ... 

To calculate the peak shift there are two possibilities that should give the same result starting from Equation (67b) as we did for the Voigt approximation or starting from Equation (67a). In the first case Equation (93) is replaced in Equation (73b) and both Equations (73b) and (74b) are replaced in Equation (67b) the peak shift becomes ... 

Table 5 The Integral Breaths of Voigt-Approximated Diffraction Profile, (i and Calculated Values of the Surface-Weighted (Ds) and Volume-Weighted (D J Domain Size and the Gaussian Strain e...

The presence of the central spot (the primary beam) and diffuse rings Idiff from the film support brings significant errors into estimated intensities. The shape of the primary beam feam can be approximated by one of several peak-shape functions such as pseudo-Voigt, Gaussian or Lorentzian [16], The diffuse background can be described by a polynomial function of order 12. Then equation (1) becomes... 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

It is common, however, for liquid-phase systems to include many specific absorbing species. Such species could include isotopic variations, conformational isomers, and solvent-solute interactions resulting in varied-lifetime transient associations between molecules. Distributions resulting from these effects give the Voigt profile utility in studying liquid spectra. We must understand, however, that the functions introduced here are only rough approximations when applied to the spectra of liquids because of the complexities just mentioned and others beyond the scope of this work. 

Gaussian Laser Profile-Voigt Atom Profile. This case turns out to be a better approximation of our experimental situation, i.e., the laser FWHM is fairly broad compared to the absorption line width and the absorption profile of atoms in an atmospheric combustion flame is described by a Voigt profile. Here the laser is assumed to have a Gaussian spectral profile as well as a Gaussian atomic absorption profile. In this case, convolution of two Gaussian functions is still a Gaussian function. Evaluation of the ratio n2/nT, and the fluorescence radiance. Bp, allows determination of the half width of the fluorescence excitation profile, 6X... 

Using a Taylor series expansion in the TD it can be shown that, to a good approximation,78 the Voigt function can be written as V, a linear combination of Lorentzian, g (f) and Gaussian gad) functions having the same width, W = Wi = Wq, namely... 

Fig. 11. The peak area residual between the Voigt function and the approximated Voigt function is shown as a function of the Voigt parameter, a. This diagram was obtained by replotting the information in S. Bruce, J. Higinbotham, I. Marshall and P. H. Beswick,...

The NMR lineshapes obtained in MRS are often assumed to have a Voigt lineshape.32,79 The Voigt lineshape, however, does not have a simple analytical form. None the less, as shown earlier, the Voigt lineshape can conveniently be approximated as a linear sum of Gaussian and Lorentzian lineshapes.33,78... 

Fitting in the frequency domain is readily visualized graphically and the well-established Levenberg-Marquardt method85 is straightforward to implement. This method is applicable to any lineshape function. For example, Marshall et al 2 have used V ( f), an approximation to the Voigt function, when fitting... 

A second approximation of the tensile creep function is the Voigt-Kelvin model... 

The XANES spectrum for P409 80,140) shows a preedge absorption band at 2148.2 eV, which is assigned to an electron transition from the P(III)-ls orbital into an unoccupied MO with main contributions from P(III) and 0. Above 2150 eV, two broad resonances have been observed, which could be fitted by three Voigt profiles of approximately... 

Using the relations between the elastic-stiffness coefficients from Table 10.3 in Eqs. 10.20 and 10.26, one may also derive the Voigt and Reuss approximations for the rigidity modulus of a cubic monocrystal. These are given by Eqs. 10.35 and 10.36, respectively ... 

It should be noted that for a polycrystal composed of cubic crystalhtes, the Voigt and Reuss approximations for the bulk modulus are equal to each other, as they should be since the bulk modulus represents a volume change but not shape change. Therefore, in a cube the deformation along the principal strain directions are the same. Hence, Eqs. 10.39 and 10.40 are equal and these equations also hold for an isotropic body. The... 

Use the relations in Tables 10.3 and 10.4 to derive the Voigt and Reuss approximations for the bulk modulus of an elastically isotropic polycrystalline aggregate composed of tetragonal monocrystals. 

Show, for an isotropic cubic polycrystal for which the Cauchy relations (cn = 3ci2 C44 = C12) hold, that the Voigt and Reuss approximations of the shear modulus reduce to ... 

Wilding and Ward showed that the creep and recovery behaviour of the low molecular weight samples could be represented to a good approximation by the model representation shown in Fig. 35(b), which consists of a Maxwell and Voigt element in series, on the basis that the parameters E, E, r and r), are dependent on the stress level. Data for the creep response of the samples under discussion at a constant applied stress Op were therefore fitted to the equation... 

In atomic and simple molecular liquids, the Kerr effect, electric saturation, etc., are caused not only by non-linearities directly induced in the atoms and molecules by the external electric field (the Voigt effect), but are primarily due to Yvon-Kirkwood fluctuational-statistical processes. One should keep in mind, however, that the intrinsic electric polarizability of non-dipolar molecules is subject to modification by at least three factors (i) the effect of translational fluctuations, (//) intermolecular attractive or repulsive forces, and (in) non-linearities induced by the tenq>oraIly and spatially fluctuating electric fields of neighbouring molecules. Thus, the Yvonr-Kirkwood process discussed in Section 4 leads, in dipolar approximation, to the following variation of the electric polarizability tensor of a molecule immersed in a statistically noit-uniform medium ... 

The Kelvin-Voigt elements are used to describe data from a creep experiment and the retardation time (t2) is the time required for the spring and the dashpot to deform to (1 — 1 /e), or 63.21 % of the total creep. In contrast, the relaxation time is that required for the spring and dashpot to stress relax to 1 /e or 0.368 of a (0) at constant strain. To a first approximation, both z and Z2 indicate a measure of the time to complete about half of the physical or chemical phenomenon being investigated (Sperling, 1986). 

At first glance, it may not be obvious that such an approach should work. It is well known, for example, that the impedance spectrum associated with an electrochemical reaction limited by the rate of diffusion through a stagnant layer (either the Warburg or the finite-layer diffusion impedance) can be approximated by an infinite number of RC circuits in series (the Voigt model). In theory, then, a measurement model based on the Voigt circuit should require an infinite number of parameters to adequately describe the impedance response of any electrochemical system influenced by mass transfer. 

In some advanced implementations of the modified pseudo-Voigt function, an asymmetric peak can be constructed as a convolution of a symmetric peak shape and a certain asymmetrization function, which can be either empirical or based on the real instrumental parameters. For example, as described in section 2.9.1, and using the Simpson s multi-term integration rule this convolution can be approximated using a sum of several (usually 3 or 5) symmetric Bragg peak profiles ... 

Figure 6.19. Full widths at half maximum (FWHM) and mixing parameters (p) of the pseudo-Voigt function used to approximate peak shapes in the x-ray Figure 6.16) and neutron Figure 6.18) powder diffraction patterns collected from the same CeRhQea powder.

Pseudo-Voigt peak shape function. Asymmetry was changed to Finger, Cox and Jephcoat approximation, which better represents peak shapes measured on this powder diffractometer using Mo Ka radiation, see section 6.10. 

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