Oscillatory continuum

LIF from Brj excited by 158 nm radiation (F2 laser) has been observed and interpreted. Oscillatory continuum emission, involving three bound-free transitions, dominates in the region 210—440 nm. Collisional interstate transfer in the presence of SF< Nj, and He was found to be efficient and was discussed in terms of a possible optically pumped Bra laser. 

Figure 15.7 Emission from a bound upper state (/ =40) to a repulsive lower state, with the resulting oscillatory continuum emission spectrum (simulated) the inset shows the related potential energy curves. Adapted from Tellinghuisen, Adv. Chem. Phys., 1985, 60 299, with permission of John Wiley Sons Ltd...

Figure 15.8 Oscillatory continuum emission X > 285 )nm from I2(f0+). Below A = 285 nm the emission is to bound states and shows normal ro-vibrational structure. The difference between the two types of emission is difficult toseeatA = 285 nm, due to the limited resolution. However, the difference is clearly ro-seen at the two ends of the spectrum sharp ro-vibrational structure is seen atX— 260 nmand broad oscillatory continuum bands ztX = 340 nm...

It is interesting to note that for each peak in the oscillatory continuum (frequency domain) there will be a corresponding pulse of atoms with a well-defined kinetic energy. Thus, the observation of the TOF spectrum of the atoms produced following oscillatory continuum emission provides complementary information, in the time domain. 

The EOg state of I2 then undergoes fluorescence, producing an oscillatory continuum, which leads to dissociation (see Chapter 15.5). An interesting consequence of multiple-photon excitation is that each step partially and successively aligns the population of the excited states, an extension of the process shown in Figure 15.3. 

Fig.l Radial part of resonance wavefunction clearly showing bound state charatcter in the strong interaction region and oscillatory, continuum character in the non-interacting region. 

Above roughening, the simulations confirm the classical continuum theory. The width of the bumps spreads with time as fwhere b= Ml for evaporation-condensation, and b= 1/4 for surface diffusion. In the latter case, the profile shows an oscillatory behavior away from the foot of the bump, as for pairs of steps and wires. ... 

So-called solvation/structural forces, or (in water) hydration forces, arise in the gap between a pair of particles or surfaces when solvent (water) molecules become ordered by the proximity of the surfaces. When such ordering occurs, there is a breakdown in the classical continuum theories of the van der Waals and electrostatic double-layer forces, with the consequence that the monotonic forces they conventionally predict are replaced (or accompanied) by exponentially decaying oscillatory forces with a periodicity roughly equal to the size of the confined species (Min et al, 2008). In practice, these confined species may be of widely variable structural and chemical types — ranging in size from small solvent molecules (like water) up to macromolecules and nanoparticles. 

As r — oo the wavefunctions are oscillatory sine and cosine functions, as shown by Eqs. (2.14). For small r the wavefunctions of the continuum are functionally identical to the bound wavefunctions, differing only in their normalization. Since continuum waves extend to r = oo they cannot be normalized in the same way as a bound state wavefunction. We shall normalize the continuum waves per unit... 

W. M. Fawzy, R. J. Le Roy, B. Simard, H. Niki, and P. A. Hackctt, ]. Chem. Phys., 98, 140 (1993). Determining Repulsive Potentials of InAr from Oscillatory Bound -> Continuum Emission. 

This ultrasimple classical theory is, of course, too crude for practical applications, especially for highly excited states of the parent molecule. Its usefulness gradually diminishes as the degree of vibrational excitation increases, i.e., as the initial wavefunction becomes more and more oscillatory. If both wavefunctions oscillate rapidly, they can be approximated by semiclassical WKB wavefunctions and the radial overlap integral of the bound and the continuum wavefunctions can subsequently be evaluated by the method of steepest descent. This leads to analytical expressions for the spectrum (Child 1980, 1991 ch.5 Tellinghuisen 1985, 1987). In particular, relation (13.2), which relates the coordinate R to the energy E, is replaced by... 

The generation of attosecond laser pulses in high-harmonic generation is a natural consequence of the physics discussed in Sects. 3.2 and 3.3. As discussed in Sect. 3.3, the ionization that launches the electron into the continuum is a highly non-linear phenomenon that will favor field maxima in the femtosecond driver laser. Following this ionization step, and in the spirit of the results presented in Sect. 3.2, the electrons will be accelerated by the oscillatory field of the laser and move along relatively well-defined trajectories that carry the electron back to the parent ion at well-defined times. Consequently, we expect the electron-parent ion recombination and the XUV production to occur only during a small portion of the optical cycle. 

An underlying question remains in the quest for understanding the hydrophobic effect Can continuum models adequately describe this phenomenon Force measurements between mica surfaces suspended in KC1 solution clearly exhibit an oscillatory behavior with a spatial periodicity approximately the same as the diameter of a water molecule [17], suggesting a challenge to the use of continuum models. 

Although the microscopic motions in a liquid occur on a continuum of time scales, one can still partition this continuum into two relatively distinct portions. The short-time behavior in a liquid is characterized by frustrated inertial motions of the molecules. While an isolated molecule in the gas phase can translate and rotate freely, in a liquid these same motions are interrupted by collisions with other molecules. Liquids are dense enough media that collisions occur very frequently, so that molecules undergo pseudo-oscillatory motion in the local potentials defined by their... 

In their study on solvent effects on the properties of oligothiophenes, Fig. 9, Meng et calculated also the dipole moment for chains consisting of 2-6 thiophene units, either in the gas phase or in either w-hexane or water. The calculations were carried through using the B3LYP density-functional method for the solute and the polarizable continuum method for the solvent. As Table 30 shows, the solvent leads to an increase of the dipole moment, but in all cases a clear even-odd oscillatory pattern is identified. The latter can be related to the zigzag-like structure of the systems, cf. Fig. 9. 

Electronic Feshbach resonances are often very long lived and hence have narrow (often < 0.01 eV) widths. Their lifetimes are determined by the coupling between the quasibound and asymptotic components of their electronic wavefunctions. Because the Feshbach decay process Involves ejection of one electron and deexcltatlon of a second, It proceeds via the two-electron terms e /r.. In the Hamiltonian. For example, the rate of electron loss lil H (2s2p, P°) Is proportional to the square of the two-electron Integral <2s2p(e /r.21 Is kp>, where kp represents the continuum p-wave orbital. This Integral, and hence the decay rate. Is often quite small because of the size difference between the 2s or 2p and Is orbitals and because of the oscillatory nature of the kp orbital. 

The quantities in Eqs. (21.6) and (21.7) are tj (Q) and tj"(Q) where tj and rf are the quantities defined in 7 for the small amplitude oscillatory experiment. Hence when the limit is taken as T->0, xXJ(—QY) - rj and xyJ(— QY)->t)". The same limiting result was obtained for this flow by Bird and Harris (4) using an integral continuum model. Thus, this gives additional support to the argument that the Maxwell orthogonal rheometer flow provides the possibility of measuring the dynamic properties of fluids. 

The quantity in Eq. (22.8) is rj (to) of the small-amplitude oscillatory experiment and is obtained, as expected, in the limit of vanishingly small k. It should be noted that when Act) = kQ and (AOT)=Ak, the expressions for tj of Eq. (22.8) and xxJ(—QY) of Eq. (21.6) are equivalent except for the final term in the series. Bird and Harris (4) found these two series to be equivalent to any order of terms from a calculation made using an integral continuum model. From the above result for the rigid dumbbell model we conclude that Bird and Harris result is a fortuitous one and that, in general, these two series are not equal. The only data to date on the transverse superposition experiment are those of Simmons (69 a), which show that tf —tjs as a function of cw decreases with increasing k, and that the curves of rf" as a function of to go through a maximum. 

Consider the continuum description of an oscillatory medium subject to advection and diffusion... 

Recapitulation will be discussed in more detail in section 3.4. As n increases, the number of nodes also increases and, as n — oo, the inner nodes coincide with those of a continuum functions. In fact, the positions of the nodes determine the phase of the continuum function (which is oscillatory) at threshold. There is a simple relation between the phase shift above threshold and the quantum defect of the bound states, which will be explained in chapter 3. If the eigenfunctions recapitulate, i.e. the positions of the nodes are nearly constant, then it follows that the... 

This property is actually not surprising as the series limit is approached, the bound state wavefunction acquires more and more nodes, and tends to the oscillatory function of the continuum. The position of the nodes is related to the phase in the continuum, and we may expect that the two are connected, since the wavefunction at very high n must change smoothly into the free electron s wavefunction just above the series limit. In QDT, as for H, the wavefunction for r > ro preserves this... 

Recapitulation is an important property because it provides us with an immediate interpretation of the physical meaning of the quantum defect p for large enough n, the bound state wavefunctions possess an oscillatory inner part which defines a phase, and is nearly independent of energy if p is nearly constant in energy. A change in the value of p corresponds to a shift in the radial position of all the nodes. As one tends to the series limit, the oscillatory part grows. Continuum functions, of course, become... 

The last two result in damping terms, while the interatom separation can be determined very accurately, essentially by a Fourier transform of the oscillatory part of the pattern. Although there may be very little atomic structure in the continuum, it cannot be completely neglected. There is one situation in which care is needed in the interpretation doubly-excited configurations may extend far into the continuum above an inner-shell threshold and may coincide in energy with EXAFS for some atoms. They are then not easily separated from each other, which may become a source of error in the interpretation. 

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