R dependence

A measurement procedure has been developed that allows to determine the mass of the inclusions as well as their locations with respect to radius, angle, and depth (2). For the depth determination use is made of the approximate 1/R dependence of the magnetic field strength from the distance R to the inclusion When in a first measurement at a small lift off an inclusion is detected, the measurement is repeated at an increased lift off From the signal ratio the depth can be calculated or seen from a diagram like fig. 5a which was generated experimentally. After that, from calibration curves like fig. 5b the absolute value of the signal leads to the mass of the inclusion. 

Note the r dependence of these tenns the charge-indiiced-dipole interaction varies as r, the dipole-indiiced-dipole as and the quadnipole-mduced-dipole as In general, the interaction between a pennanent 2 -pole moment and an induced I -pole moment varies as + L + l) gQ enough r, only the leading tenn is important, with higher tenns increasing in importance as r decreases. The induction forces are clearly nonadditive because a third molecule will induce another set of miiltipole moments in tlie first two, and these will then interact. Induction forces are almost never dominant since dispersion is usually more important. 

It is the relationship between the bound potential energy surface of an adsorbate and the vibrational states of the molecule that detemiine whether an adsorbate remains on the surface, or whether it desorbs after a period of time. The lifetime of the adsorbed state, r, depends on the size of the well relative to the vibrational energy inlierent in the system, and can be written as... 

For the consistency of the scaling fonn, equation (A3.3.92), R dependence should drop out from equation... 

Now select out the r)-dependent part to define vibrational fimctions at some specific p which we call... 

In principle, one can do beder by allowing for R-dependence to U and T. If we allow them to vary linearly with R, then we have Gordon s method [42]. However, the higher order evaluation in this case leads to a much more cumbersome theory that is often less efficient even though larger steps can be used. 

The polarization P is given in tenns of E by the constitutive relation of the material. For the present discussion, we assume that the polarization P r) depends only on the field E evaluated at the same position r. This is the so-called dipole approximation. In later discussions, however, we will consider, in some specific cases, the contribution of a polarization that has a non-local spatial dependence on the optical field. Once we have augmented the system of equation B 1.5.16. equation B 1.5.17. equation B 1.5.18. equation B 1.5.19 and equation B 1.5.20 with the constitutive relation for the dependence of Pon E, we may solve for the radiation fields. This relation is generally characterized tlirough the use of linear and nonlinear susceptibility tensors, the subject to which we now turn. 

Since the potential depends only upon the scalar r, this equation, in spherical coordinates, can be separated into two equations, one depending only on r and one depending on 9 and ( ). The wave equation for the r-dependent part of the solution, R(r), is... 

By expressing the mean-field interaction of an electron at r with the N- 1 other electrons in temis of a probability density pyy r ) that is independent of the fact that another electron resides at r, the mean-field models ignore spatial correlations among the electrons. In reality, as shown in figure B3.T5 the conditional probability density for finding one ofA - 1 electrons at r, given that one electron is at r depends on r. The absence of a spatial correlation is a direct consequence of the spin-orbital product nature of the mean-field wavefiinctions... 

This r dependence is also known as a Yukawa potential. This type of potential has been used to describe the behaviour of latex suspensions at low ionic strength. 

A next step is to consider the surface-particle distance z and curvature (interfacial radius R) dependence of the interactions [1341, for which approximate expressions are ... 

Note that the exact adiabatic functions are used on the right-hand side, which in practical calculations must be evaluated by the full derivative on the left of Eq. (24) rather than the Hellmann-Feynman forces. This forai has the advantage that the R dependence of the coefficients, c, does not have to be considered. Using the relationship Eq. (78) for the off-diagonal matrix elements of the right-hand side then leads directly to... 

As usually indicated by the semicolon, both the wave functions and eigenvalues [ (R)] depend parametrically on the internal nuclear coordinates. 

The fact that the set of hi is, in principle, complete in r-space allows the full (electronic and nuclear) wavefunction h to have its r-dependence expanded in terms of the hp... 

The Si(R) functions, carry the remaining R-dependence of h and are determined by insisting that h as expressed here obey the full Schrodinger equation ... 

Beeause the potential Ej (R) depends on R but not on 0 or ( ), the V/R funetion ean be written as a produet of an angular part and an R-dependent part moreover, beeause E2 eontains the full angle-dependenee of Ty/R, can be written as... 

The energy "gap" between the a and a orbitals at R = depends on the eleetronegativity differenee between the groups X and Y. If this gap is small, it is expeeted that the behavior of this (slightly) heteronuelear system should approaeh that of the homonuelear X2 and Y2 systems. Sueh similarities are demonstrated in the next seetion. 

Practically, the NPSH required for operation without cavitation and vibration in the pump is somewhat greater than the theoretical. The actual (NPSH)r depends on the characteristics of the liquid, the total head, the pump speed, the capacity, and impeller design. Any suction condition which reduces (NPSH ) below that required to prevent cavitation at the desired capacity will produce an unsatisfactoiy installation and can lead to mechanical dimculty. 

Fig. 13.5. Schematic representation of the potential energy surfaces of the ground state (S ,) and the excited state (.5,) of a nonadiabatic photoreaction of reactant R. Depending on the way the classical trajectories enter the conical intersection region, different ground-state valleys, which lead to products P and can be reached. Reproduced from Angew. Chem. Int. Ed. Engl. 34 549 (1995) by permission of Wiley-VCH.

The reactor volume is calculated from Mj and the bulk density of the catalyst material, (-r ) depends not only on composition and temperature, but also on the nature and size of the catalyst pellets and the flow velocity of the mixture. In a heterogeneous reaction where a solid catalyst is used, the reactor load is often determined by the term space velocity, SV. This is defined as the volumetric flow at the inlet of the reactor divided by the reaction volume (or the total mass of catalyst), that is... 

View Factor. The view factor of a point on a plane surface located at a distance L from the center of a sphere (fireball) with radius r depends not only on L and r, but also on the orientation of the surface with respect to the fireball. If 2 is the view angle, and 0 is the angle between the normal vector to the surface and the line connecting the target point and the center of the sphere (see Figure 6.9), the view factor (F) is given by... 

In Steps 2 and 3, the vessel s nondimensional radius and the blast wave s nondimen-sional peak pressure at that radius were calculated. As a blast wave travels outward, its pressure decreases rapidly. The relationship between the peak pressure and the distance R depends upon initial conditions. Accordingly, Figure 6.21 contains several curves. Locate the correct curve by plotting (R, P ) in the figure, as illustrated in Figure 6.28. 

From Table 2.5 it is clearly seen that becomes small (less than 0.001 kcal/ mol) beyond a distance of 10 A. The electrostatic interaction reaches the same level of importance at a distance of 30 A. The Table also shows that the interaction between point charges behaves much like a dipole-dipole interaction, i.e. an R dependence. However, the interaction between net charges is very long range even at 100 A separation, there is a 0.34kcal/mol energy contribution. The cut-off distance corresponding to a contribution of 0.001 kcal/mol is of the order of 3000 A ... 

The R depends on the amine R NH2 used in the reaction. When ammonia was used a bicyclic product (7) was formed from nitro-ethane. ... 

Verlass, m. trust, reliance, verlassen, v.t. leave, give up, abandon. — v.r-depend, rely. — p.a. abandoned, forsaken. verlMsslg, verlasslich, a. reliable, dependable. Verlaub, m. permission. 

A different approach is adopted here. Within the LMTO-ASA method, it is possible to vary the atomic radii in such a way that the net charges are non-random while preserving the total volume of the system . The basic assumption of a single-site theory of electronic structure of disordered alloys, namely that the potential at any site R depends only on the occupation of this site by atom A or B, and is completely independent of the occupation of other sites, is fulfilled, if the net charges... 

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