Zero-approximation

The methodical elaboration is included for estimation of random and systematic errors by using of single factor dispersion analysis. For this aim the set of reference samples is used. X-ray analyses of reference samples are performed with followed calculation of mass parts of components and comparison of results with real chemical compositions. Metrological characteristics of x-ray fluorescence silicate analysis are established both for a-correction method and simplified fundamental parameter method. It is established, that systematic error of simplified FPM is less than a-correction method, if the correction of zero approximation for simplified FPM is used by preliminary established correlation between theoretical and experimental set data. 

Here the index 0 at Z) refers to the diffusion coefficient in a system at rest (no bias), and Vq denotes the zero approximation i.e., this relation is applicable for slow drift in dilute medium only. 

The conversion of energy is a difficult and inefficient transformation because 100% efficiency is only possible if the exhaust gas is received at absolute zero (approximately -273 °C or -459 °F). 

The nonlinear part of the susceptibility was introduced into the quasi-linear finite-difference scheme via iterations, so that at any longitudinal point, the magnitude of E calculated at the previous longitudinal point was used as a zero approximation. This approach is better than the split-step method since it allows one to jointly simulate both the mode field diffraction on irregular sections of the waveguide and the self-action effect by introducing the nonlinear permittivity into the implicit finite-difference scheme which describes the propagation of the total field. 

Fermi resonance physchem In a polyatomic molecule, the relationship of two vibrational levels that have In zero approximation nearly the same energy they repel each other, and the eigenfunctions of the two states mix. fer-me, rez-3n-3ns fermium chem Asynthetic radioactive element, symbol Fm, with atomic number 100 discovered in debris of the 1952 hydrogen bomb explosion, and now made in nuclear reactors. fer-me-3m )... 

The TMCs electronic wave function formalizing the CFT ionic model is one with a fixed number of electrons in the d-shell. In the EHCF method it is used as a zero approximation. The interactions responsible for electron transfers between the d-shell and the ligands are treated as perturbations. Following the standards semiempirical setting we restrict the AO basis for all atoms of the TMC by the valence orbitals. All the AOs of the TMC are... 

Equation (92) gives the negative of the slope of AGM with respect to x at x = and constant T. The equation is valid for all degrees of dissociation and shows that the slope at x = j depends on fiA, / 13, and/ 14 — 034 and is zero if these are all zero. Approximate treatments for the case of complete association (z = 1) have sometimes been given that incorrectly yield a V shape to the AGM isotherms and hence a discontinuity in jil — fi3 at x =... 

Let us start with zero approximation states of H0 consisting of the discrete states (Xx, X2), 2(Xls X2),..., n( -i, X2) and continuum states time-independent) eigenstates of the physical system are obtained by diagonalizing the total Hamiltonian in this representation, and can be displayed as a superposition of these zero-order states. For the sake of simplicity we consider just one zero-order... 

There are some fundamental investigations devoted to analysis of the flow in tubular polymerization reactors where the viscosity of the final product has a limit (viscosity < >) i.e., the reactive mass is fluid up to the end of the process. As a zero approximation, flow can be considered to be one-dimensional, for which it is assumed that the velocity is constant across the tube cross-section. This is a model of an ideal plug reactor, and it is very far from reality. A model with a Poiseuille velocity profile (parabolic for a Newtonian liquid) at each cross-section is a first approximation, but again this is a very rough model, which does not reflect the inherent interactions between the kinetics of the chemical reaction, the changes in viscosity of the reactive liquid, and the changes in temperature and velocity profiles along the reactor. 

The initial system can be constructed as a series with respect to powers of [39], A zero approximation here is a solution of the degenerated system. This approach is, however, very rarely used since the increase of accuracy results in a significant complication of calculations. 

At fast diffusion the recombination is kinetic and so weak that in the zero approximation we obtain the following equation from (3.170) (p(ro) = 1. However, in the next (first-order) approximation it follows from the same equation that [147]... 

The set of relaxation equations in the single-mode approach (9.46) and (9.47) can be written in different approximations. One can see, that in zero approximation (ip = 0), the relaxation equations (9.46) and (9.47) appear to be independent. The expansion of the quantity ujk in powers of velocity gradient begins with terms of the second order (see equation (7.39)), so that, according to equation (9.47), the variable ujk is not perturbed in the first and second approximations at all and, consequently, can be omitted at tp = 0. In virtue of ip -C 1, the second variable has to considered to be small in any case and can be neglected with comparison to the first variable, so that the system of equations can be written in a simpler way. In the simplest case, relaxation equation (9.46) in terms of the new variables jk can be rewritten as... 

That does not mean that a valid semiempirical parametrization based on the HFR MO LCAO scheme cannot be built for a certain narrow class of compounds or even for a specific purpose. It is done for example in [86] even for iron(II) porphyrins. But in a more general case there is no way to arrive at any definite conclusion [118] about the validity of a semi-empirical parametrization relying on the HFR approximation. On the other hand we have to mention that the semiempirical method ZINDO/1 [119] which allows for some true correlation by taking into account the configuration interaction may be considered a prospective setting for further parametrization, provided the HFR solution required by this method as a zero approximation can be obtained. This will be discussed in more detail later. 

All these statements, although correct in principle, are not precise from the technical point of view. For example, the zero approximate wave function in the PCILO method is a one-electron approximate function constructed from the bond wave functions determined by an a posteriori localization procedure from an HFR function. Thus the bond orbitals appear after a unitary transformation of the canonical MOs, which correspond to some more or less arbitrary localization criteria [123-125]. 

The zero approximate operator Vo projecting on the ionic limit ground state corresponds to u = 1 z = v = 0. To ensure the correct (ionic) limit of a general one-dimensional projection operator in a three-dimensional space, we apply the prescription of eqs. (1.107) and (1.104) with the notion that dim Im Vo = 1 and... 

The maximum repulsion generated by the asymmetric distribution of ions is however, small when the surface charge is zero (approximately one order of magnitude smaller than a typical van der Waals attraction) and therefore the neutral physiological colloids will coagulate, in agreement with the DLVO theory. 

Abstract, The solution of the Debye-Huckel equation for a system of spheres with arbitrary radii and surface charge in electrolyte solutions is described. The general theoretical approach to describe such systems is elaborated. The practically important case of two spheres is considered in detail. Finite closed formulae to calculate the interaction energy of two spherical particles with constant surface charges are obtained from general expressions in zero approximation. Known relationships follow from our formulae in limiting cases. 

Now we assume that the surface charge density is constant. The simplest case is to take only one term in the expansions of potentials, i.e. if l = l = 0 (a zero approximation). It should be noted that in the zero approximation solutions do not depend on sphere permittivity, as it follows from Eq. (12) for a(k). If terms of higher order in comparison with the quantity k2 (kc ) are neglected, the formula for the potential energy of interaction V (d) can be derived... 

Figure 2. Interactions energy E = V / V0 of two identical spheres with the constant charges versus kH at Ka = 1 1 -Derjaguin approximation 2 - Ohshima approximation 3 - our simplest zero approximation.

Results obtained from the improved zero approximation and the simplest zero approximation begin to differ at values kH 1. The difference depends on the value parameter kg. For example, at Ka = 10 the maximum difference is about 30% when charges have the same signs at H = 0 and q = 5. At large q and Ka values it is necessary to use the improved zero approximation. In the... 

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