Voigt parameter

Voigt parameter (also a parameter associated with exchange narrowing)... 

Fig. 1. The Voigt lineshape is plotted for three different values of the Voigt parameter, a - WL/WG, namely a = 0 (Gaussian, dotted line), a = 1 (solid line, with the component Lorentzian and Gaussian lineshapes having an equal width given by tl/(, W 0.6107 H-) and a x (Lorentzian, dashed line). The frequency scale is given in units of the FWHM of the Voigt lineshape, W.

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,...

Usually for the characterization of the MO effects the MO (Voigt) parameter Q is used. It is defined as... 

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. 

Figure 3.10 Basic mechanical elements for solids and fluids a) dash pot for a viscous response, b) spring for an elastic response, c) Voigt or Kelvin solid, d) Maxwell fluid, and e) the four-parameter viscoelastic fluid...

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. 

The four-parameter model is very simple and often a reasonable first-order model for polymer crystalline solids and polymeric fluids near the transition temperature. The model requires two spring constants, a viscosity for the fluid component and a viscosity for the solid structured component. The time-dependent creep strain is the summation of the three time-dependent elements (the Voigt element acts as a single time-dependent element) ... 

Equation (4) expresses G as a function of temperature and state of applied stress (pressure) (o. Pa), (/(a) is given by the force field for the set of lattice constants a, Vt is the unit cell volume at temperature T, and Oj and are the components of the stress and strain tensors, respectively (in Voigt notation). The equilibrium crystal structure at a specified temperature and stress is determined by minimizing G(r, a) with respect to die lattice parameters, atomic positions, and shell positions, and yields simultaneously the crystal structure and polarization of minimum free energy. 

Summarizing The basic idea, mentioned in chapter 6, that creep of solid polymers could be represented by a simple four-parameter model (the Burgers model), composed of a Maxwell and a Kelvin-Voigt model in series, appears to be inadequate for three reasons ... 

The bulk and grain boundary behaviour of a polycrystalline electroceramic material can be described by a Voigt structure consisting of two RC circuits. This simple Voigt structure is shown in Figure 4.24a. The parameters of this model all have direct physical meanings Rl =Rh, ci = R2 = Rgb, and C2 = Cgb, where b refers to bulk and gb refers to grain boundary. 

FIGURE2.9. VSFspectraoftightlypackedmonolayerscomposedofDLPC(n = 12),DMPC(n = 14),DPPC (n = 16) and DSPC in = 18) where n is the number of carbon atoms in the acyl chains, (a) The CCLi/water interface, (b) The air/water interface. Order parameters appear beneath the phosphochohne acronyms. Spectral features were fit with Voigt profiles, although the solid fines in the figures serve only as guides to the eye. From Ref. [49]. 

The simplest flaws of the Maxwell and Voigt models, the fact that one cannot model creep while the other cannot model stress relaxation, can easily be fixed by combining our basic linear elements in different ways. One such is the so-called four-parameter model (Figure 13-94), which combines a Maxwell model in series with a Voigt model. The four parameters are the Maxwell modulus and viscosity, Eu and and the Voigt modulus and viscosity Ev and r v... 

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... 

Example 3.5 Evaluation of Chi-Squared Statistics Consider that, for a given measurement, regression of a model to real and imaginan/ parts of impedance data yielded = 130. Measurements were conducted at 70 frequencies. The regressed parameters needed to model the data included the solution resistance and 9 Voigt elements, resulting in use of 19 parameters. Under assumption that the variances used in the evaluation ofxf were obtained independently, evaluate the hypothesis that the x value cannot be reduced by refinement of the model. 

Example 19.1 Nonlinear Models Show that the equation for the impedance of a Voigt element is nonlinear with respect to parameters. 

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