Stationary phase point

If the photodissociation starts from a / 0 level, the a E) will be an oscillatory curve with v" + 1 lobes (v" nodes) (see Fig. 7.5). The possibility of destructive interference arising from multiple stationary phase points (e.g. i i,i 2) (Tellinghuisen, 1984) is negligible for direct dissociation because the difference potential V R) — V" R) = AV(R) will be single valued (i.e. AV(-Ri) AV(R2)) over the Franck-Condon region, R in < R < R max-... 

The important exception occurs when the phase reaches a minimum — at this stationary phase point the contributions from similar paths no longer cancel. Consequently, the most important contributions to the integral come from paths close to the one for which... 

According to Landau [2], the first order probability p 2ci,fo proportional to an exponential fimction which contains the absolute value of the difference in the classical actions for the initial and final states, calculated from the classical turning points qti and qff to the stationary phase point q tph which in oiu- case lies in the classically forbidden region of motion. Explicitly,... 

If the instantaneous collisional energies cross to produce a stationary phase point in (3.10), this occurs twice during a collision, once as the molecule and perturber approach each other and once as they recede. As the transition can occur either on the inward or outward portion of the trajectory, the transition probability is given by the expression ... 

As an illustration of the kinds of / that may occur, consider a simple model where the long-range behavior of the potentials f ss[R(0] and F7.7.[R(t)] arise from van der Waals interactions and vary as t.[R(0] Given the existence of a stationary phase point, (3.16), the stationary phase approximation to (3.10) yields the general result... 

In some ceises the influence functional F may be teiken out of the integrals over the imit cell in (2.2.2). It is possible either when Pi(Rc) and Gqq(Ro) depend on Rq weakly compared to GUq is, when the crystal corrugation is much smaller than the phonon relief), or when S Ro) oscillates strongly enough for the evaluation of the integrals by the stationary phase method (the classical limit). In the latter case F is treated as a pre-exponential factor and may be replaced with its value at the stationary phase point. Then the general structural factor Hq can be factorized into the product of the form factor (diffi action intensities) and the dynamic structural factor ... 

Apolar stationary phases having no dipolar moments, that is their center of gravities of their positive and negative electric charges coincide. With this type of compound, the components elute as a function of their increasing boiiing points. The time difference between the moment of injection and the moment the component leaves the column is called the retention time. 

The resolution of capillary columns enables the separation of all principal components of a straight-run gasoline. The most frequently used stationary phases are silicone-based, giving an order of hydrocarbon elution times close to the order of increasing boiling point. 

The probes are assumed to be of contact type but are otherwise quite arbitrary. To model the probe the traction beneath it is prescribed and the resulting boundary value problem is first solved exactly by way of a double Fourier transform. To get managable expressions a far field approximation is then performed using the stationary phase method. As to not be too restrictive the probe is if necessary divided into elements which are each treated separately. Keeping the elements small enough the far field restriction becomes very week so that it is in fact enough if the separation between the probe and defect is one or two wavelengths. As each element can be controlled separately it is possible to have phased arrays and also point or line focussed probes. 

The last modification in the Rayleigh integral concerns the delay of propagation between the two points which is simply the time taken by the energy to propagate along the path of stationary phase. It is denoted T and given by,... 

Recalling the plate theory, it must be emphasized that (Vm) is not the same as (Vm)-(Vm) is the moving phase and a significant amount of (Vm) will be static (e.g., that contained in the pores). It should also be pointed out that the same applies to the volume of stationary phase, (Vs), which is not the same as (Vs), which may include material that is unavailable to the solute due to exclusion. 

The isotherm will be close to linear at very low concentrations of solute in the mobile phase and thus, again, all concentrations in the peak will travel at the same speed and the resulting peak is symmetrical. However, when the points y, x and y", x" are reached, the concentration of solute in the stationary phase is increasing more rapidly with the solute concentration in the mobile phase and thus the distribution coefficient at the higher concentrations is larger. It follows that, as the peak velocity is inversely proportional to the distribution coefficient, then the higher concentrations in the peak will migrate more slowly than the lower concentrations. 

In a packed column, however, the situation is quite different and more complicated. Only point contact is made between particles and, consequently, the film of stationary phase is largely discontinuous. It follows that, as solute transfer between particles can only take place at the points of contact, diffusion will be severely impeded. In practice the throttling effect of the limited contact area between particles renders the dispersion due to diffusion in the stationary phase insignificant. This is true even in packed LC columns where the solute diffusivity in both phases are of the same order of magnitude. The negligible effect of dispersion due to diffusion in the stationary phase is also supported by experimental evidence which will be included later in the chapter. 

Equation (14), although derived from the approximate random walk theory, is rigorously correct and applies to heterogeneous surfaces containing wide variations in properties and to perfectly uniform surfaces. It can also be used as the starting point for the random walk treatment of diffusion controlled mass transfer similar to that which takes place in the stationary phase in GC and LC columns. 

It should be pointed out, however, that the diffusivity of the solute in the mobile phase can be changed in two ways. The solute that is chromatographed can be changed, in which case the above assumptions are clearly valid, as (Ds) is likely to change linearly with (Dm)- However, the solute diffusivity can also be changed by the employing a different mobile phase. In this case, (Dm) will be changed but (Ds) will remain the same. In the second case, the above assumptions are not likely to be precisely correct. Nevertheless, if the resistance to mass transfer in the stationary phase makes only a small contribution to the overall value of (H) (i.e., because df dp (see equation (l)),then the assumption Dm = eDg will still be approximately... 

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