Binary approximation

Storer model used in this theory enables us to describe classically the spectral collapse of the Q-branch for any strength of collisions. The theory generates the canonical relation between the width of the Raman spectrum and the rate of rotational relaxation measured by NMR or acoustic methods. At medium pressures the impact theory overlaps with the non-model perturbation theory which extends the relation to the region where the binary approximation is invalid. The employment of this relation has become a routine procedure which puts in order numerous experimental data from different methods. At low densities it permits us to estimate, roughly, the strength of collisions. 

Using the data given in Fig. 1.17 we consider the deviation of the isothermic dependence of 1 from linear (binary) relationship (1.124). The dependence of l/(xj(v)) on i/ in a liquid is presented in Fig. 1.23. The experimental results practically coincide with a straight line corresponding to a binary approximation up to a critical point. Hence the impact approximation is not too bad even for moderately condensed gases. However, the abrupt increase in 1/tj observed in the cryogen liquid is too sharp to be described even with the hard-sphere correction... 

Bessel function 40, 99, 201, 264 binary approximation 7, 41 binary collisions adiabatic/non-adiabatic 4 angular momentum, computer simulations 40... 

We do not discuss here variational estimates of the for details see the review article [33]. Note here only that radius estimates show that for a typical small-radius electron defect, the activator atom A0 and begin to deviate from the Onsager radius L at very small D values only (low temperature) when the applicability of binary approximation itself is questionable. It comes from the fact that due to small values of tq 0.5 A and e 5 the Onsager... 

However, at still larger concentrations only DET/UT is capable of reaching the kinetic limit of the Stem-Volmer constant and the static limit of the reaction product distribution. On the other hand, this theory is intended for only irreversible reactions and does not have the matrix form adapted for consideration of multistage reactions. The latter is also valid for competing theories based on the superposition approximation or nonequilibrium statistical mechanics. Moreover, most of them address only the contact reactions (either reversible or irreversible). These limitations strongly restrict their application to real transfer reactions, carried out by distant rates, depending on the reactant and solvent parameters. On the other hand, these theories can be applied to reactions in one- and two-dimensional spaces where binary approximation is impossible and encounter theories inapplicable. 

We have found that in most pratical cases, only three terms (n = 2) are required. This usually reduces the error to less than 5%, which makes the numerical results as accurate as the physical quantities (D ) used in computing the diffusion fluxes, and in the original derivation of this form of the diffusion equation (5,7). In addition, for vector lengths of ten (10) or larger and for five diffusion flux term (n v ) is as fast as doing the usual differencing for the binary approximation (V-D Vp ). 

Making use of the elastic constant entering equation (3.1.4) for F, H centres inKBr a = 3eVA3 [69], one can estimate easily that the effective radius of annihilation stimulated by elastic interaction, (equation (4.2.29)) varies from 11 A down to 7 A as the temperature increases from 40 K to 200 K (and then is independent of the annihilation radius R 4 A). On the other hand, the effective radius of tunnelling recombination, equation (4.2.17), decreases from 10 A (at 40 K) down to 5 A (60 K). It coincides with the elastic radius, 7 eh at37 K, where diffusion is very slow and the binary approximation, equation (4.2.19), does not hold any longer. 

Under the binary approximation we can break the whole history of each cluster impact into a series of elementary steps. Each collision can be treated as an isolated one as is the case in the gas-phase, including the... 

In this section we establish simple model expressions that approximately describe narrow EELS resonances near a critical point in the binary approximation [8]. In this analysis we terminate the correlational series, taking into account only the pairwise contributions (the accuracy of this approximation is increased from the band of moderate densities). We use a Lennard-Jones potential... 

The empirical predictions described above for pseudo-binary mixtures are compared with rigorous calculations for multicomponent mixtmes in Section 21-3.3 to justify the pseudo-binary approximation. 

We will discuss the case where the motion of heavy atoms is confined to two dimensions, while the motion of light atoms can be either two- or three-dimensional. It will be shown that the Hamiltonian 10.76 with Ues in 10.44 supports the first-order quantum gas-crystal transition at T = 0 [68], This phase transition resembles the one for the flux lattice melting in superconductors, where the flux lines are mapped onto a system of bosons interacting via a two-dimensional Yukawa potential [73]. In this case Monte Carlo studies [74,75] identified the first-order liquid-crystal transition at zero and finite temperatures. Aside from the difference in the interaction potentials, a distinguished feature of our system is related to its stability. The molecules can undergo collisional relaxation into deeply bound states, or form weakly bound trimers. Another subtle question is how dilute the system should be to enable the use of the binary approximation for the molecule-molecule interaction, leading to Equations 10.76 and 10.44. 

RE1 Rehage, G. and Koningsveld, R., Liquid-liquid phase separation in multicomponent polymer solutions VI. Some errors introduced by the binary approximation, J. Polym. Sci., Polym. Lett, 6, 421, 1968. 

How to make a reasonable binary approximation of a multicomponent column. Such an approximation greatly reduces computation and time. 

Take the exanqtle of a metal carbonate, MCO3. In the ideal crystal, we consider that there are only two types of occupied structure elements the metal ion in cation position and the complex anion carbonate in anion positiow We will thus not distinguish the individual behavior of oxygen or carbon atoms. These compounds are thus regarded as binary ones, from where the name of the pseudo-binary approximation comes. These compounds can have defects, for example, an oxygen ion in the place normally occupied by a carbonate ion (this substitution does not involve any charge deficiency). 

To determine the coexistence curve, two methods using a specially designed centrifuge[10] and a scanning electron microscope(SEM) with image analysis, respectively, were employed. Volume ratios of separated phases for at least two different initial compositions enabled us to determine compositions of coexisting phases under the quasi-binary approximation. 

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