Slowly varying continuum

The simplest (though approximate) solution of Eq. (10.11) is obtained by assuming that all the continua are flat, that is, that the bound-continuum matrix elements vary slowly with energy and can be replaced by their value at some average energy, say El = Ei + fi(oh 

This approximation, called the flat continuum or slowly varying continuum approx- imation (SVCA) [191, 325, 326] localizes the autocorrelation function in time, since by Eqs. (10.9) and (10.12) 

The factor of relative to Eq. (10.13) arises because the integration over f fh (10.11) is carried out over the [-oo, t] range and not over the usual [-c , range. 

It follows from Eq. (10.15) that a slowly varying continuum acts as an irreversible perfect absorber since in this approximation bx(t) decreases monotoni-cally (though not necessarily as a simple exponential) with time. 

In many cases the continuum may have structures that are narrower than the bandwidth of the pulse. Such structures may be due to either the natural spectrum of the molecular Hamiltonian [327, 328] or to the interaction with the strong external field [195, 197-199, 329]. Under such circumstances we expect the SVCA approximation to break down, yielding nonmonotonic decay dynamics. 

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It is often the case that the Franck-Condon factors contained in dq(ij) vary sufficiently slowly over the range of E encompassed by the dump pulse to be regarded as constant [191], (This assumption, called the slowly varying continuum approximation is discussed in detail in Chapter 10). Under these circumstances we can use the following generalized Parseval s equality to show that P(q) is independent of the coherence properties of the dump pulse. Specifically we have... 

Numerical studies allow us to explore aspects of these models for a number df molecular continua and pulse configurations. Consider first the effect of the puUe intensity on transition probabilities to a slowly varying continuum by considering continuum composed of single broad Lorentzian [Eq. (10.21)] of wid r, = 2000 cm-1, excited by a 120 cm-1 wide pulse (i.e., a pulse of 80fs durp tion). The central frequency of the pulse is tuned to the center of the continuutfi) (A, = 0) and the pulse peaks at t = 0. 

By invoking the slowly varying continuum approximation (SVCA) [Eq. (10.13)1 WB v. obtain the two-photon analog of Eq. (10.15) ... 

The derivation of Eqs. (102-104) invokes the slowly varying continuum approximation (SVGA) [253] ("flat" continuum approximation), according to which the Rabi frequencies are approximated as = i k,e E) ... 

Equations 8.13 can be further simplified using the flat continuum or slowly varying continuum approximation (SVGA). This approximation assumes that the variation of the continuum-bound dipole matrix elements with energy over the pump laser bandwidth is negligible. With this assumption, and after changing the lower limit of integration over each continuum channel to —oo, we can rewrite the second term in Equation 8.14 as... 

For other situations, especially when the material continuum is slowly varying and tiny changes in the laser frequencies in the one photon transition have absolutely no effect on the product ratios, our method allows for control, via the optical induction of resonances, in complete generality. 

Because /lisa relatively slowly varying function of Reynolds number, the efficiency varies approximately as which means that fine fibers are more efficient aerosol collectors than coarse ones. Because Pe = // ), tj/f ond for the continuum and... 

It should be noted that as t becomes large the lowest order term in the coefficient of the K-term is just 60, that is one half the zero-shear-rate value of the primary normal stress function. A similar result was obtained by Bird and Marsh (7) and by Carreau (14) from the slowly varying flow expansions of two continuum models. Hence the time-dependent behavior of the shear stress is related to the steady-state primary normal stress difference in the limit of vanishingly small shear rate. 

The initial state is discrete, but the final state is in the continuum and transitions are described by a spectral density, slowly varying function of the energy or frequency. 

For compounds X other than O2 (or NO) for which the absorption cross section varies slowly with wavelength (continuum) over the Schumann-Runge region (e.g., H20, CH4, CO2), the photodissociation frequency over interval AAj can be derived from... 

The rate at which population is transferred into the continuum by a coherent field is determined by the Rabi frequency ftQoe(t) =< 0 V(t) e >, where the time dependence must now be included explicitly. The exciting laser is a coherent source of light, so we suppose that the field V(t) = Vo(t) cos (u t + (t)) where both Vo(t) and (t) vary only slowly with time. We then have... 

Thus, interference effects between bound-continuum matrix elements will play an important role in determining the magnitude of the susceptibility. However, since these terms vary slowly with energy, the tuning range for harmonic generation will still be very large. 

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