Cross-over resonance

In a later experiment the laser was replaced by a cw ring dye laser, the output of which was amplified in four stages, using a XeCl excimer laser as pump. In this way the cw output (30 60 mW) at 486 nm was amplified into 25 mJ, 10 ns pulses with a repetition rate of 50 s . In addition in this later experiment the frequency of the transition was determined with respect to the saturated spectrum of Tc2, a line of which lies within 50 MHz of half the 1 S- 2 S energy interval. Furthermore, an acousto-optic modulator was used to create a laser sideband at 50 MHz higher than the fundamental frequency so that each Tea absorption line consisted of 3 peaks, at vo, vo + 50 MHz and Vo + 25 MHz, the cross-over resonance. 

In Figs.9.45 a so-called cross-over resonance is shown. This is an inherent phenonomenon in saturation spectroscopy and occurs when two lower or upper state sublevels have transitions to a common level in the other state. The two oppositely propagating laser beams can then interact at frequencies half-way between the normal resonances. Then atoms moving with a certain velocity along the laser beams are utilized [9.170,171]. 

Fig. 9.55. Set-up for saturation spectroscopy of the sodium D line [9.345]. A schematic spectrum is also shown where cross-over resonances (Fig. 9.58) have been omitted...

Fig. 1. The Marcus parabolic free energy surfaces corresponding to the reactant electronic state of the system (DA) and to the product electronic state of the system (D A ) cross (become resonant) at the transition state. The curves which cross are computed with zero electronic tunneling interaction and are known as the diabatic curves, and include the Born-Oppenheimer potential energy of the molecular system plus the environmental polarization free energy as a function of the reaction coordinate. Due to the finite electronic coupling between the reactant and charge separated states, a fraction k l of the molecular systems passing through the transition state region will cross over onto the product surface this electronically controlled fraction k l thus enters directly as a factor into the electron transfer rate constant...

As a rule, it is profitable to examine first the nonequivalence behavior of the racemic solute, if it is available. In doing so, one readily determines the resonances for which nonequivalence may be observed and the conditions that are optimal for examining a particular pair of resonances. When different sets of solute resonances are closely spaced, maximum nonequivalence is not always desired on first examination, for resonances may cross over one another, making nonequivalence sense determination ambiguous or an e.e. measurement impossible. In the case of highly enriched samples, this experiment becomes imperative, since one must be certain to correctly identify the signal from the minor enantiomer. In the latter case the signal may also be identified by addition of the racemate or the opposite solute enantiomer. 

The chemical shift 5, defined by Equation (22), was measured at 40.0 and 15.6 Mc./sec. and was found to be —3 2 relative to water for both SA and SG. The derivatives of the resonance absorptions were recorded in the measurements. If the total anisotropy of the chemical shift of protons in the solid is somewhat less than the line width, the cross-over point of the derivative will correspond to the average value of S as for liquids, and will be directly comparable with the shifts for protons in the liquid state. Comparison of the shift value with those of H3O+ (aqueous) (1 1), S = -1-11, OH (aqueous) (121), S = -1-10 dilute solutions of alcoholic-type protons... 

A typical example of a resonance assisted device might be a cell separator designed to discriminate between cells which have only a small difference in dielectric properties. The cross over frequencies from positive to negative DEP will be similar but not identical for the two cells. By working at a frequency between the two crossovers it is possible to separate the cells but the forces produced will be small unless very high fields are used (Fig. 10). Controlled resonance can be used to boost the fields at the working frequency. 

At first sight, instrumentation, even physical instrumentation within organic chemistry, may not appear to be an interdisciplinary area. However, if by interdisciplinary we mean an intellectual zone where scientists from different disciplines meet and interact, no field could be more deserving of the title. Not only did the construction of these instruments draw on new developments in electronics and optics and stimulated further innovation, but the techniques themselves came from outside organic chemistry. Nuclear magnetic resonance and mass spectroscopy, to give just two pertinent examples, crossed over from physics, and organic chemists had to collaborate with chemical physicists to obtain the best results from these new techniques. 

By Eqs. (21), (22), and (26), APG and Asc scale with rpair, and thus decrease with x, following the pairing line in Fig. 6, as has been observed. Since —(k0) is zero for an AF, its value (and thus Eves) is expected to increase with x, distancing from an AF state. However, by Eq. (23) its linewidth cannot remain small if it crosses the value of Asc, which decreases with x. Thus the energy Eres of a sharp resonance mode is expected to cross over from an increase to a decrease with x when it approaches the value of 2 Asc, as has been observed [18], This crossover could be followed by a shift of the resonance wave vector 2kmin from the AF wave vector Q to incommensurate wave vectors. 

Figure 25 (upper plot) A schematic plot of the enantiodiscriminator. The three levels of each enantiomer are resonantly coupled by three fields. The dipole moments of the two enantiomers have opposite signs, (middle plot) The time evolution of the population of the three levels. The D and L enantiomers start in the 1) state. At the end of the process one enantiomer is found in the 3) state and the other in the 1) state, (lower plot) The time-dependence of the eigenvalues of the Hamiltonian of Eq. (73). The population initially follows the E0) dark state. At t rthe population crosses over diabatically to ) for one enantiomer and to E+) for the other. 

Model studies discussed in previous chapters show that the reactivity of cations and alkenes are very strongly affected by inductive and resonance effects in the substituents. Correlation of the rate constants of addition of benzhydryl cation to various styrenes with Hammett substituted benzhydryl cations to a standard alkene (2-methyl-2-pentene) gave also good correlation and p+ = 5.1 [28]. The large p value signals difficult copolymerizations between alkenes, even of similar structures. Thus, in contrast to radical copolymerization which easily provides random copolymers, cationic systems have a tendency to form either mixtures of two homopolymers or block copolymer (if the cross-over reaction is possible). 

The results that were expected in isolated molecules can easily be summarized as follows A molecule initially excited to 0, may fluoresce with some probability pf, or cross-over to 0, with probabilty 1 -pf. Because the coupling vsl between s and 10,1, the levels 0, 0,1 represents a finite set of molecular states which are in resonance, using the traditional chemical sense of the term resonance. Thus, when the molecule is in (0,1, it will eventually cross-back to 0S. (Energy must be conserved in all crossings in truly isolated molecules.) Once in 4>s the molecule may again fluoresce or cross-over with probabilities pf and 1 -pf, respectively, etc. Continued to the limit, this argument implies that the total fluorescence probability, the fluorescence quantum yield, must be unity in an isolated molecule. An identical argument holds in the case of phosphorescence. 

The "single line-double arrow" notation, <— , for reversible steps is employed here with apologies to the organic chemist who likes to see it reserved for resonance structures and prefers "double line-double arrow," = , for reversible reactions. The latter notation, however, causes problems in depiction of reversible catalytic cycles Since the arrowheads along the inner and outer circles in the diagram of a cycle point in opposite directions, either all reactants or all products would have to be crowded into the interior of the circle, or cross-overs would occur. For a book in which the distinction between reversible and irreversible steps of cycles is essential and resonance is not an issue, the notation appears more practical. 

Figure 5. Plot of peak frequencies in the 1H off-resonance selectively decoupled 13C spectra of NAD+ as a function of position of irradiation in the 1H spectrum, expressed in ppm from internal dioxan. The positions of the peaks in the 1H noise decoupled 13C spectrum are shown by lines on the ordinate and the position of the proton peaks by lines on the abscissa. The arrows t indicate the point of collapse of the 1 3C doublet and the connection between a given 1 3C peak and proton peak. The errors in the position of measurements are indicated by the size of the points except near the cross-over positions where the errors are larger ( 0-15 ppm). (Taken from Birdsall et al., 1972a.)...

Fig. 4. Energy variation of reaction cross section for neutrons of energy E incident on a nucleus of radius R (in 10 cm), averaged over resonances (Feshbach and Weisskopf). =0.22 Ea MeV.

If two molecular transitions with a common lower or upper level overlap within their Doppler width, extra resonances, called cross-over signals, occur in the Lamb-dip spectrum. Their generation is explained in Fig. 2.14a. 

In Fig. 2.15 the three Lamb dips (two direct resonances and 1 cross-over signal) are illustrated for the case of two transitions with a common lower level. Although these cross-over signals increase the number of observed Lamb dips and may therefore increase the complexity of the spectrum, they have the great advantage that they allow one to assign pairs of transitions with a common level. This may also facilitate the assignment of the whole spectrum see, for example. Fig. 2.14b and [205, 214-216]. 

---