Crossing of curves

The first special case to be dealt with is the crossing of curves 1 and 2 at about pH 8. Consider the relative positions of curves 1 and 2 for pH > 8. This indicates that S(s) is not stable in this region and will undergo a spontaneous autoredox reaction, S(s) - S04 + HS (not balanced). The consequence is that S(s) is not stable for pH > 8 and the relevant redox couple will be that for SOj /HS" ... 

Four sets of two-point lines obtained from crossings of curves from experiment 1 with those from experiment 2 shown in Figure 6.1. The slopes of all lines joining corresponding pairs of points must be optimized to give the same value. The best common slope then gives N, the exponent of the decay function. 

Figure 15.5 Potential energy curve crossing, exemplified for ionic-covalent crossing of curves with the same symmetry (here E ). Solid lines diabatic curves, l/dia dashed lines adiabatic curves, l id internuclear separation of curve crossing...

The ab initio-computed [12 b, c] four-curve diagram for Eq. (41) is shown in Fig. 17. It is apparent that structures 47 and 48 do not contribute cardinally to the ground- and transition-states, and mix efficiently only with the excited states. Therefore, CH5 arises mainly from the avoided crossing of curves 45 and 46, and remains therefore a high-energy transition state. In this transition state the charge is delocalized mainly in the axial bonds and is virtually prohibited to delocalize into the equatorial bonds. 

Trofimov O.E. To the problem of restoration of functin from three variables by its integrals along the lines, crossing given curve., Avtometriya, N5, 1991, p..29-33. 

However, because of the avoided crossing of the potential energy curves the wave functions of Vq and Fi are mixed, very strongly at r = 6.93 A and less strongly on either side. Consequently, when the wave packet reaches the high r limit of the vibrational level there is a chance that the wave function will take on sufficient of the character of Na + 1 that neutral sodium (or iodine) atoms may be detected. 

Now refer to Figure 28.13(a) to obtain the skin effect ratio Ffac/ffdc- Consider the cross-sectional curves for EIE-M grade of flat busbars at an operating temperature of 85°C for a cross-sectional area of 6.45 cm and determine the Ffac/ dc ratio on the skin effect curve having... 

The cross-section curve a(E) gives the dependence of the nuclear cross-section on the projectile energy, E. The measured energy spectra of emitted particles or the excitation curve N(Eq) wiU depend on the depth profile N(x) of the analyzed isotope and on the cross-section curve (t(E(x)), where E(x) gives the energy of the projectiles at a depth x. Evaluation of the depth profile N (x) from measured energy spectra or excitation curves often requires a tedious evaluation procedure if the cross-section curve has a complex structure. It is simplified for two special types of behavior of the cross-section curve ... 

The production of N02 by charge transfer from either SF6 or SF5 , previously observed by Curran (4), has been confirmed during this work. The attachment cross-section curves for SF6 and for SF5 closely overlapped in the low energy range, and it was impossible to determine whether SF6 or SF5 was the reacting ion. 

The double integral in Equation (8.4) is a fairly general definition of the mixing-cup average. It is applicable to arbitrary velocity profiles and noncircular cross sections but does assume straight streamlines of equal length. Treatment of curved streamlines requires a precise and possibly artificial definition of the system boundaries. See Nauman and Buffham. ... 

One notes that the proportionality constant, a, depends on the reaction energy, AEy. Therefore, Eq. (1.3) is not strictly a linear relation between activation energy change and reaction energy. In the extreme limit of high exothermicity of the reaction energy a = 0, and the crossing point of the two curves is at the minimum of curve Vj. In this case the transition state is called early. Its structure is close to that of the reactant state. 

In the limit of high endothermicity a = and now the crossing point is close to the minimum of curve V2. The transition-state structure is now close to that of the final state. The transition state can now be considered to be late. This analysis is important, since it illustrates why a varies between 0 and 1. Often a is simply assumed to be equal to 0.5. 

In the above numerical examples the held parameter F is taken to be the laser frequency and the nonadiabatic transition used is the Landau-Zener type of curve-crossing. The periodic chirping method, however, can actually be more... 

The 2 angular-distribution parameters show somewhat more molecule-to-molecule variation, but are essentially still quite similar to one another. They also display low energy structures that readily correlate with the noted features in the cross-section curves, and the variations are mainly confined to changes in the relative intensity of these. Much bigger differences are, however, found in the calculated chiral parameters, though here as well there is a common... 

Thus, while it is possible in theory to cross a curved distillation boundary as shown in Figure 12.35, it is generally more straightforward to follow designs that will be feasible over a wide range of reflux ratios and in the presence of uncertainties. Such designs can be readily developed using distillation line and residue curve maps. 

Fig. 2. The singlet-triplet reversion of SiH2, referenced to CH2, leads to the avoidance of the crossing of two potential curves. Thus, the Si=Si bond dissociation energy of Si2H4 is lowered by twice the singlet-triplet gap of SiH2, i.e. 38 kcal mol 1.

If these concepts of curve analysis shall be applied to the anisotropic scattering of polymer fibers, one should choose to study either the longitudinal or the transversal density fluctuations. According to the decision made, the fiber scattering must be projected either on the fiber axis or on the cross-sectional plane. This results in scattering curves with a one- or a two-dimensional Porod s law. Because modern radiation sources always feature a point-focus, the required plots for the separation of fluctuation and transition zone are readily established (cf. Table 8.3). 

Figure 3a shows the mean-field predictions for the polymer phase diagram for a range of values for Ep/Ec and B/Ec. The corresponding simulation results are shown in Fig. 3b. As can be seen from the figure, the mean-field theory captures the essential features of the polymer phase diagram and provides even fair quantitative agreement with the numerical results. A qualitative flaw of the mean-field model is that it fails to reproduce the crossing of the melting curves at 0 = 0.73. It is likely that this discrepancy is due to the neglect of the concentration dependence of XeS Improved estimates for Xeff at high densities can be obtained from series expansions based on the lattice-cluster theory [68,69].

Figure 55-6 Expansions of the first and second derivative curves. Figure 55-6a The region around the zero-crossing of the first derivative can be approximated with a straight line. Figure 55-6b The region around the peak of the second derivative can be approximated with a parabola.

Figures 8c and 8d represent the projection of the reaction pathways on the same plane, namely the front plane in which the dissociative electron transfer step is represented. This two-dimensional representation is easier to decipher than the 3D representation for determining the preferred pathway. They may however be misleading if it is not borne in mind that, in the 2D representation, the crossings between the three curves should not be considered as actual crossings of reaction pathways.

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