Distribution functions cross-terms

We propose to describe the distribution of the number of fronts crossing x by the Poisson distribution function, discussed in Sec. 1.9. This probability distribution function describes the probability P(F) of a specific number of fronts F in terms of that number and the average number F as follows [Eq. (1.38)] ... 

Sinee the probability distribution functions (1) and (2) are even functions of Xia and Xg, respectively, the cross-term integral containing Xm and Xe to the first power will vanish.)... 

Fig. 10. The degree of association into nearest- and next-nearest-neighbour complexes, p, versus concentration, c, at 500°C for manganese ions and cation vacancies in sodium chloride. Filled circles represent the simple association theory, open circles the Lidiard association theory, and crosses the present theory using Eq. (173) when the first term only has been retained in the virial appearing in the equation for the defect distribution function (Eq. (168)). The point of highest concentration represented by a cross may be in error due to the neglect of higher terms in the virial series, and the dotted curve has not been extended to include it.

If we consider the variation of concentration along the axial direction as a distribution of concentration, we can calculate the moments for each model in terms of their respective parameters, and then compare the moments to find the relationship between parameters. Since we shall consider the concentration as a distribution function along the axis of the tube, the moments are with respect to axial distance, rather than with respect to time as used previously. Since flow in cylindrical vessels is so common, we will discuss only this case in detail. Aris (A6) gives the more general treatment in vessels of arbitrary cross section. 

In this way, (3.93) and (3.94) define the macroscopic rate coefficients in terms of cross sections and distribution functions. 

A necessary condition for the two-term expansion of the distribution function of equation (2) to be valid is that the electron collision frequency for momentum transfer must be larger than the total electron collision frequency for excitation for all values of electron energy. Under these conditions electron-heavy particle momentum-transfer collisions are of major importance in reducing the asymmetry in the distribution function. In many cases as pointed out by Phelps in ref. 34, this condition is not met in the analysis of N2, CO, and C02 transport data primarily because of large vibrational excitation cross sections. The effect on the accuracy of the determination of distribution functions as a result is a factor still remaining to be assessed. 

Here. .. depends on the W-kernels introduced above, but in general stands for terms of the form cross-section, appropriately weighted by factors [1 — cosl(0)], l = 1, 2,. .., if the efficiency of exchange of quantity A in a collision depends on the scattering angle 0 in this way (as, e.g., in case of elastic neutral particle - ion collisions, see Sect. 2.2.4). In case of inelastic collision processes a is simply the total cross-section, denoted weighting exponent l = 1, 2,... is possible) or a1. In principle the detector functions qa must be obtained by numerical integration and tabulated for the parameters of the relevant distribution functions fa. 

Calculation of the cross-term partial structure factor fits using Eq. 25 yields values of oo/( dd oo)° he order of 1.2, which is clearly not possible, since the maximum modulus value of the cosine function is unity. This result implies that there is little separation between the centers of the OA and DPPC distributions. 

For 1 = 0 the monopole structure can be determined completely if Boo(h) is known. In fact, the phase problem of reduces to the determination of the sign of F(,(,. This is usually not too difficult a task for Bqo as plausible arguments can be made concerning the corresponding radial mass distribution v fr) of the resonant label atoms. Once the signs of the sinusoidal function BQQ(h) are known for each peak, the phases of Ao(,(h) can be determined directly by using the cross term. 

We therefore consider a different reaction flow model as our basic targeting model—one that can address temperature manipulation by feed mixing as well as by external heating or cooling. The model consists of a differential sidestream reactor (DSR), shown in Fig. 6, with a sidestream concentration set to the feed concentration and a general exit flow distribution function. (As mentioned in Section II, the boundary of an AR can be defined by DSRs for higher-dimensional (> 3) problems). We term this particular structure a cross-flow reactor. By construction, this model not only allows the manipulation of reactor temperature by feed mixing, but often eliminates the need to check for PFR extensions. 

These MOs would be distinguished from one another by the values of their coefficients. The square of each of these MOs would give the electron density distribution for the electrons in that MO. This distribution function would include many cross terms, each of which represents the interaction between a pair of AOs on the two different atoms. It is difficult to visualize the result as the basis for qualitative arguments. 

D(r,R) is the distance distribution function of a sphere of radius R, and D(r,d , R) is the cross-term distance distribution between the nth and mth sphere with a mutual distance d . ... 

Wang et al. (1979) studied amorphous Gd-Co films by means of electron diffraction and found that the radial distribution functions are significantly different for the perpendicular anisotropic and in-plane anisotropic films. Wang et al. and also Nishihara et al. (1979) conclude from their results that the perpendicular anisotropy is associated with a predominance of Gd-Co pairs. This would agree with the pseudo-dipolar model of Chaudhari and Cronemeyer (1975) where the cross term predominates in their expression for given by... 

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