Correspondence relations

The flux N (a,w) is the Sum of contributions from a gaseous phase flux and a flux due to surface diffusion. The surface diffusion contribution is given by equation (7.7) or, more generally, by the corresponding relation which follows from equation (7.5). For the gaseous phase contribution Feng and Stewart assume flux relations of the dusty gas form, (5.1)- ... 

In this subsection we establish the exact form of boundary conditions holding on for the solution 9,x) to problem (3.98). The arguments are formal in the sense that the solution is assumed sufficiently smooth. For brevity, hereafter we denote the quantities IF, w, and 0 by IF, w, and 9, indicating each time the value of the variable t at which the corresponding relations hold. Moreover, we assume that 5 = 1. 

Note that if the problem of accurate graphical representation occurs in the rectification end of the diagram, the corresponding relation to use to calculate the balance of the trays, assuming straight line operating and equilibrium lines in the region is [59] ... 

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A corresponding relation exists for the equilibrium constant of (150) in the hydrolysis of an ammonium salt. 

In non-aqueous solution the process corresponding to hydrolysis is known as solvolysis and corresponding relations must hold for the solvolysis constants. 

Curve B describes the corresponding relation for the oxidation of divalent lead ions Pb2+ -> Pb4+, which implies the formation of lead dioxide since Pb4+ does not exist alone ... 

Obtain the Taylor-Prandtl modification of the Reynolds Analogy for momentum and heat transfer, and give the corresponding relation for mass transfer (no bulk flow). 

The corresponding relation for a real linear chain may be expressed in terms of the equivalent freely jointed chain comprising n bonds each of length V as follows ... 

Corresponding relations for arbitrary space group elements show that if the orbitals r) which make up the density transform asthe irreducible representations of the space group, the density is invariant under all the operations of that group. 

The corresponding relations for 62 and b- follow by cyclic permutations of the subscripts (see Chapter 8). 

In combination with other theorems of Fourier transformation theory many of the fundamental structural parameters in the field of scattering are readily established. Because the corresponding relations are not easily accessible in textbooks, a synopsis of the most important tools is presented in the sequel. 

Equations. For a ID two-phase structure Porod s law is easily deduced. Then the corresponding relations for 2D- and 3D-structures follow from the result. The ID structure is of practical relevance in the study of fibers [16,139], because it reflects size and correlation of domains in fiber direction . Therefore this basic relation is presented here. Let er be50 the direction of interest (e.g., the fiber direction), then the linear series expansion of the slice r7(r)]er of the corresponding correlation function is considered. After double derivation the ID Fourier transform converts the slice into a projection / Cr of the scattering intensity and Porod s law... 

Fig. 3.14. Curve-of-growth shifts for individual lines of individual neutral elements in /u. Cas, plotted against Ax. The solid line shows the corresponding relation for Fe i, so that the offsets give directly the differential abundance changes [M/Fe]. Adapted from Catchpole, Pagel and Powell (1967).

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This equation, which is similar to equation 11.35, gives the corresponding relation between the compositions of the vapour rising to a plate and the liquid on the plate, for the section below the feed plate. These two equations are the equations of the operating lines. 

The most important property of a liquid-gas interface is its surface energy. Surface tension arises at the boundary because of the grossly unequal attractive forces of the liquid subphase for molecules at its surface relative to their attraction by the molecules of the gas phase. These forces tend to pull the surface molecules into the interior of the liquid phase and, as a consequence, cause liquids to minimize their surface area. If equilibrium thermodynamics apply, the surface tension 7 is the partial derivative of the Helmholtz free energy of the system with respect to the area of the interface—when all other conditions are held constant. For a phase surface, the corresponding relation of 7 to Gibbs free energy G and surface area A is shown in eq. [ 1 ]. 

Where Jo and S are the current density of the primary beam and the area of the irradiated sample, Z is the wave length, Ohki the structure factor amplitude, Q the volume cell, Z a factor that takes the microstructure of sample into account (Zm - for a mosaic single crystalline film, Zt - for a texture film), t is the sample thickness, dhu the interplanar spacing, a represents the mean angular distribution of the microcrystallites in the film, p is a multiplicity factor (accounts for the number of reflections of coincidence), R is a horizontal coordinate of a particular reflection in DP from textures and (p is the tilt angle of the sample. In the case of polycrystalline films, a local intensity is usually measured and the corresponding relation is ... 

The corresponding relation including spin is simply Eq. (5.10) with the spin-traced density matrices replaced by their counterparts, including spin. 

Examining our previous results we see that a corresponding relation for the three-electron case may be constructed ... 

The effect of the C2 operation is easily determined since C2 = OxzOyz- There is, of course, a completely parallel set of relations for the 2>py set of orbitals. Writing out the corresponding relations for the 2>d orbitals is left to the interested reader. 

The firing characteristics of commercial electric detonators are shown in Ref 36, p 64, Fig 11, given here as Fig 13. The curve APB of Fig shows the relation between current t and "minimum lag time" for the most rapid detonator in the series, while the curve CPD shows the corresponding relation between current and "maximum excitation time " for the least sensitive detonator of the series. From the curve it can be seen that with current ij, the most rapid detonator will break the circuit in T4 milliseconds, whereas the least sensitive detonator requires at least T5 millisecs of current flow to enable it to fire. At a higher current 12, however, the excitation time T, is less than time T2 which is allowed by the most rapid detonator, and hence the least... 

Figure 7.13. Misfit ratio, (a) One-to-one corresponding relation. Open circles indicate the unit cell of the host phase, and the solid circles show that of the guest phase.

The corresponding relation between the host and guest crystals when evaluating the misfit ratio may be a one-to-one lattice relation in the same direction (a X b to a xb axes), or in different axial directions (aX b axes versus aX <110> axes), or on the basis of one unit cell versus a few unit cell sizes (see Fig. 7.13). Royer s misfit ratio is generally a two-dimensional correspondence, but Hartman [13] extended this relation to the misfit ratio in PBCs (see Section 4.2), which is a one-dimensional correspondence. Royer s epitaxial relations correspond to a relation between the F faces of the host and guest crystals containing more than two PBCs, and an epitaxial relation is not allowed between S faces or K faces. In Hartman s analysis, rela-... 

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