Calculations from experimental functions exact

Thermochemical data on the separate phases in equUibrium are needed to constmct accurate phase diagrams. The Gibbs energy of formation for a pure substance as a function of temperature must be calculated from experimentally determined temperature-dependent thermodynamic properties such as enthalpy, entropy, heat capacity, and equihbrium constants. By a pure substance, one generally means a stoichiometric compound in which the atomic constituents ate present in an exact, simple reproducible ratio. 

Instead of an exact calculation, Gouy and Chapman have assumed that (4) can be approximated by combining the Poisson equation with a Boltzmann factor which contains the mean electrical potential existing in the interface. (This approximation will be rederived below). From this approach the distribution of the potential across the interface can be calculated as the function of a and from (2) we get a differential capacitance Cqc- It has been shown by Grahame that Cqc fits very well the measurements in the case of low ionic concentrations [11]. For higher concentrations another capacitance in series, Q, had to be introduced. It is called the inner layer capacitance and it was first considered by Stern [1,2]. Then the experimental capacitance Cexp is analyzed according to ... 

In order to compare calculated and experimentally observed phase portraits it is necessary to know very exactly all the coefficients of the describing nonlinear differential Equation 14.3. Therefore, different methods of determination of the nonlinear coefficient in the Duffing equation have been compared. In the paraelectric phase the value of the nonlinear dielectric coefficient B is determined by measuring the shift of the resonance frequency in dependence on the amplitude of the excitation ( [1], [5]). In the ferroelectric phase three different methods are used in order to determine B. Firstly, the coefficient B is calculated in the framework of the Landau theory from the coefficient of the high temperature phase (e.g. [4]). This means B = const, and B has the same values above and below the phase transition. Secondly, the shift of the resonance frequency of the resonator in the ferroelectric phase as a function of the driving field is used in order to determine the coefficient B. The amplitude of the exciting field is smaller than the coercive field and does not produce polarization reversal during the measurements of the shift of the resonance frequency. In the third method the coefficient B was determined by the values of the spontaneous polarization... 

Clearly, Eq. (268) can be approximated by a Poole-Frenkel-type function with a coefficient a > atheor. The exact value of a varies from sample to sample dependent on the type and extent of disorder—through the disorder-dependent component (Ij, Following an example of electron injection from A1 into Alq3 id = 150 nm) (see Fig. 82b), the experimental a values can be calculated from the slopes of the PF-type plots. As predicted by (204) and (265), they should be temperature... 

In calculating the value of R for a particular diffraction line, various factors should be kept in mind. The unit cell volume v is calculated from the measured lattice parameters, which are a function of carbon and alloy content. When the martensite doublets are unresolved, the structure factor and multiplicity of the martensite are calculated on the basis of a body-centered cubic cell this procedure, in effect, adds together the integrated intensities of the two lines of the doublet, which is exactly what is done experimentally when the integrated intensity of an unresolved doublet is measured. For greatest accuracy in the calculation of F, the atomic scattering factor f should be corrected for anomalous scattering by an amount A/ (see Sec. 13-4), particularly when Co Ka radiation is used. The value of the temperature factor can be taken from the curve of Fig. 4-20. 

For the calculations of the optical properties of polymer films with embedded nanoparticles, two routes can be selected. In the exact route, the extinction cross sections Cact(v) of single particles are calculated. The calculated extinction spectra for single particles—or, better, a summation of various excitation spectra for a particle assembly—can be compared with the experimental spectra of the embedded nanoparticles. In the statistic route, an effective dielectric function e(v) is calculated from the dielectric function of the metal e(T) and of the polymer material po(v) by using a mixing formula, the so-called effective medium theory. The optical extinction spectra calculated from the effective dielectric functions by using the Fresnel formulas can be compared with the experimental spectra. 

The radial distribution function, g(r), is proportional to the density of atoms at the distance R from a certain atom taken for the central atom. The pair correlator is directly comparable to the structural factor obtained from the experiments on the X-ray scattering and it provides only the information on interatomic distances. Figure 6.1 shows the calculated Cu, Ni and Au RDFs in supercooled state in comparison with the experimental data by Waseda [ 14]. It should be noted that there is good correspondence for gold. For copper and nickel the correspondence of the calculated and experimental RDFs is satisfactory. The discrepancy can be explained by the inaccuracy of the used interatomic interaction potentials. Nevertheless, the first RDF peak for all metals under study is very well reproduced, which allows to speak about the adequacy of further analysis of the cluster structure of the melts. Moreover, the discrepancy of the calculated and experimental RDFs allows to clarify the degree of the influence of the accuracy of the interatomic interaction description on the cluster structure by the comparison of the results with those in Refs. [7-9], where the exacter ab-initio methods of simulation were used. 

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