Surfaces intersections

Let S be any simply connected surface in nuclear configuration space, bounded by a closed-loop L. Then, if 4>(r,R) changes sign when transported adiabatically round L, there must be at least one point on S at which (r, R) is discontinuous, implying that its potential energy surface intersects that of another electronic state. 

The ratio E/ps, calculated for different phases below the bifurcation, is shown in Fig. 15. In the special case of the C phase the surface intersects itself therefore, in the computation of S/p we have subtracted the volume occupied along the lines of intersection, since it would be counted twice otherwise. The surface area per volume is an increasing function of the surfactant volume fraction and it determines the sequence of phases. Moreover, we have found that the effect of broadening of the interface on the value S/p in different phases is different, and we have a quantitative... 

In Figure 6.9B, the extended surface intersects the sphere in such a way that a point on that surface will see only a portion of the sphere. 

When, on extension, the receiving surface intersects with the sphere (0 > it/ 2 - ), the receiver can not see the total emitter. The view factor F is then given as... 

Semimetals show metallic conductivity due to the overlap of a filled and an empty band. In this case electrons spill over from the filled band into the bottom of the empty band until the Fermi surface intersects both sets of bands. In semimetals holes and electrons coexist even at 0 K. 

The assumption of weak electronic coupling may not be valid for vibrational levels near the region where the reactant and product surfaces intersect. If the extent of electronic coupling is sufficient (tens of cm ), the timescale for electron transfer for vibrational levels near the intersectional region will approach the vibrational timescale, electronic and nuclear motions are coupled, and the Born-Oppenheimer approximation is no longer valid. 

One way of handling this —according to O. Stern —is to divide the aqueous part of the double layer by a hypothetical boundary known as the Stern surface. The Stern surface is situated a distance 6 from the actual surface. Figure 11.9 schematically illustrates the way this surface intersects the double layer potential and how it divides the charge density of the double layer. 

Fig. 2.11 The stress tensor describes the stress state at a point in space. It involves nine components, which are interpreted as components of the stress vectors on three orthogonal surfaces at the point where the three surfaces intersect.

Figure 3.45 Schematic free energy curves in the solvent coordinate z for the discussion of the equilibrium solvation location of the Cl seam in Figure 3.42 Solid curves are the adiabatic curves for very small but finite electronic coupling, while the dashed curves are diabatic curves for zero coupling, (a) The symmetric case, where the filled circle represents the location of the minimum free energy in the upper adiabatic state in the presence of finite electronic coupling, while the open circle represents a free energy minimum when the electronic coupling vanishes exactly (6 = 90°). (b) An asymmetric case where the two surfaces intersect for z > 1 and the equilibrium location of the Cl seam fails.

This figure shows schematically the P-T-composition surfaces Which represi equilibrium states of saturated vapor and saturated liquid for a binary syst The under surface represents saturated-vapor states it is the PTy surface, upper surface represents saturated-liquid states it is the PTx surface, surfaces intersect along the lines UBHCt and KAClt which represent the v pressure-vs.-T curves for pure species 1 and 2. Moreover, the under and u surfaces form a continuous rounded surface across the top of the diagram betw C and C2 the critical points of pure species 1 and 2 the critical points of... 

In the case of the Cr(CO)5 or Fe(CO)4 fragments a Jahn-Teller induced surface intersection between the Sj and S0 states explains the rapid transition to the S state following excitation of the parent. However for the Ni(CO)3 fragment the ground state is the 1A state and this has a planar structure (D3A). The first excited state Sj is the degenerate lE" state which splits by planar distortion producing 1Bl and A, states but not an Al state at the C2v limit. Consequently such a distortion does not provide a deactivation route to the S0 surface of Ni(CO)3. The Ni(CO)3 fragment remains in its Sl state which explains the luminescence observed in this system. 

The super-exchange electronic coupling term describes the coupling of the P Ba" state to the P and P Ha" states at the position of the intersection between the potential surfaces of the P and P Ha" states. When the P and P Ha" potential surfaces intersect at the minimum of the P potential surface (i.e. assuming that electron transfer from P to P Ha" is activationless), the super-exchange coupling (Vsuper) is given by ... 

Let us explore the thermocline distribution first. As large-scale oceanic transport occurs primarily along surfaces of equal potential density, it is instructive to inspect variations along such surfaces. Figure 1.4b shows the NO distribution along the potential density surface Gg = 26.80, which represents Sub-Polar Mode Water (SPMW) in the northern hemisphere and Sub-Antarctic Mode Water (SAMW) in the southern hemisphere (Hanawa and TaUey, 2001). Nitrate concentrations near the outcrops, i.e., where the isopycnal surface intersects the surface of the ocean are near zero, but concentrations increase rapidly as one moves away from the outcrops into the ocean s interior. 

This method has the advantage of not requiring a knowledge of the foil thickness t, but it becomes very difficult to count surface intersections for dislocation densities higher than about 10 cm". Clearly, measurements of the number-density of small dislocation loops or small inclusions (such as bubbles or voids) requires a knowledge of thickness t. 

Spin-coupled calculations revealed crossing seams between potential energy surfaces arising from the I + 02(a) and I + 02 X) asymptotes. In a fully adiabatic treatment of this problem the surface crossings were avoided and somewhat difficult to locate. To facilitate interpretation of the adiabatic results, diabatic calculations were preformed with coupling between states from different asymptotes suppressed. Surface intersections were allowed and readily located using the diabatic treatment. 

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