Profile fitting parameters approximate

When all the phases present were identified, we can quantify their volume fraction in the analyzed volume similarly to the way the Rietveld-method is used for phase analysis in XRD. A whole profile fitting is used in ProcessDifraction, modeling background and peak-shapes, and fitting the shape parameters, thermal parameters and volume fractions. Since the kinematic approximation is used for calculating the electron diffraction intensities, the grain size of both phases should be below 10 nm (as a rule of... 

Fig. 36. Typical [241] volume fraction vs depth cf)(z) profile of Pl-dPS (N=893) copolymers in a PS (P=88) matrix, after 24 h of annealing at 180 °C. The NRA profile (Q), corresponding to the brush layer created by diblocks at the free PS surface (z=0), is approximated by the convolution of the method resolution (p=8 nm) with the top hat-like function of Fig. 33b with different width L but constant z. The inset shows the variation of the fit parameter (mean square of residuals) with L. Best fit obtained for L=19(2) nm generates the profile marked by a solid line in the main figure. For comparison profiles calculated for L= 10 and 30 nm are shown by broken lines...

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

As described in further details in Section 5, we analyze the scans using the software DCDT+ (Philo, 2006), which converts the raw concentration profiles into time derivatives (dc/dt) and fits these values to approximate unbounded solutions of the Lamm equation (Philo, 2000 Stafford, 1994). As the rotor speed (ft)) and the concentration of the macromolecules (c) are known, and the time (t) and the radial concentration distribution [c(x, f)] are obtained from the scans of absorbance profiles, the fitting yields values of s and D. As both parameters are dependent on the solvent viscosity and temperature, they are transformed to standard values with reference to a standard temperature (20 °C) and a standard solvent (water) and reported as 52o,w and /92o,w This standardization allows analysis of the changes in the intrinsic properties of solute molecules with changes in solution condition and is a prerequisite in cation-mediated folding studies of RNA molecules. 

Full profile refinement is computationally intense and employs the nonlinear least squares method (section 6.6), which requires a reasonable initial approximation of many fi ee variables. These usually include peak shape parameters, unit cell dimensions and coordinates of all atoms in the model of the crystal structure. Other unknowns (e.g. constant background, scale factor, overall atomic displacement parameter, etc.) may be simply guessed at the beginning and then effectively refined, as the least squares fit converges to a global minimum. When either Le Bail s or Pawley s techniques were employed to perform a full pattern decomposition prior to Rietveld refinement, it only makes sense to use suitably determined relevant parameters (background, peak shape, zero shift or sample displacement, and unit cell dimensions) as the initial approximation. 

A difference Fourier map, calculated at this point, reveals an additional small electron density maximum in the tetrahedral cavity next to the partially occupied V2. Thus, it is reasonable to assume that the V2 site splits into two independent partially occupied positions with the coordinates, which distribute V atoms in a random fashion in two adjacent tetrahedral positions rather than being simply vanadium-deficient. We label these two sites as V2a (corresponding to the former V2) and V2b (corresponding to the Fourier peak). Refinement of this model slightly improves the fit. Subsequently, additional profile parameters (F, F , and sample displacement) were included in the refinement, followed by a typical procedure of refining the porosity in the Suortti approximation with fixed atomic coordinates and Ui o, and then fixing the porosity parameters for the remainder of the refinement. 

The subsequent refinement included profile parameters X, Y, X , Ya, peak asymmetry, sample displacement and transparency shift. Preferred orientation was switched from the March-Dollase to the 8 -order spherical harmonics expansion (6 variables total) and 12 coefficients of the shifted-Chebyshev polynomial background approximation were employed. A reasonably good fit, shown in Figure 7.36, was achieved as a result. 

The distributions of a shown in Figure 27.9 were fitted using the full profile likelihoods derived from the likelihoods given in Section 27.8.2. The likelihood for the lognormal approximations to the distributions of a wctc thus convolution integrals of such profile likelihoods with a lognormal and were evaluated numerically. Mean estimates for a for the three rodent species were obtained from the parameters of the fitted lognormal distributions (Table 27.2). 

Ette, Sun, and Ludden (1998) examined the effect of sample size and between-subject variability using balanced study designs wherein an equal number of samples were collected from each subject. In their study, concentration data from 30 to 1000 subjects were simulated using a 2-compartment model with intravenous administration at steady-state. Sample times were chosen based on D-optimality. Once obtained, the concentration-time profile was divided into three windows and two samples per subject were randomly sampled from each time window, i.e., six samples were collected from each subject. Between-subject variability in all the parameters was systematically varied from 30 to 100% with residual variability fixed at 15%. All models were fit using FO-approximation. 

An alternative approach to extracting an apparent chi-parameter from SANS data is to fit the scattering curves to an IRPA form over the entire measured wavevector range. Using Eqs. (6.12) one obtains for the scattering profile in the effectively incompressible approximation ... 

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