Relative phases

We will explore the effect of three parameters 2 -and < )> that is, the time delay between the pulses, the tuning or detuning of the carrier frequency from resonance with an excited-state vibrational transition and the relative phase of the two pulses. We follow closely the development of [22]. Using equation (Al.6.73). 

Figure Al.6,8 shows the experimental results of Scherer et al of excitation of I2 using pairs of phase locked pulses. By the use of heterodyne detection, those authors were able to measure just the mterference contribution to the total excited-state fluorescence (i.e. the difference in excited-state population from the two units of population which would be prepared if there were no interference). The basic qualitative dependence on time delay and phase is the same as that predicted by the hannonic model significant interference is observed only at multiples of the excited-state vibrational frequency, and the relative phase of the two pulses detennines whether that interference is constructive or destructive.

A connnon teclmique used to enliance the signal-to-noise ratio for weak modes is to inject a local oscillator field polarized parallel to the RIKE field at the detector. This local oscillator field is derived from the probe laser and will add coherently to the RIKE field [96]. The relative phase of the local oscillator and the RIKE field is an important parameter in describing the optical heterodyne detected (OHD)-RIKES spectrum. If the local oscillator at the detector is in phase with the probe wave, the heterodyne mtensity is proportional to... 

Depending on the relative phase difference between these temis, one may observe various experimental spectra, as illustrated in figure Bl.5.14. This type of behaviour, while potentially a source of confiision, is familiar for other types of nonlinear spectroscopy, such as CARS (coherent anti-Stokes Raman scattering) [30. 31] and can be readily incorporated mto modelling of measured spectral features. 

This section attempts a brief review of several areas of research on the significance of phases, mainly for quantum phenomena in molecular systems. Evidently, due to limitation of space, one cannot do justice to the breadth of the subject and numerous important works will go unmentioned. It is hoped that the several cited papers (some of which have been chosen from quite recent publications) will lead the reader to other, related and earlier, publications. It is essential to state at the outset that the overall phase of the wave function is arbitrary and only the relative phases of its components are observable in any meaningful sense. Throughout, we concentrate on the relative phases of the components. (In a coordinate representation of the state function, the phases of the components are none other than the coordinate-dependent parts of the phase, so it is also true that this part is susceptible to measurement. Similar statements can be made in momentum, energy, etc., representations.)... 

Coherent states and diverse semiclassical approximations to molecular wavepackets are essentially dependent on the relative phases between the wave components. Due to the need to keep this chapter to a reasonable size, we can mention here only a sample of original works (e.g., [202-205]) and some summaries [206-208]. In these, the reader will come across the Maslov index [209], which we pause to mention here, since it links up in a natural way to the modulus-phase relations described in Section III and with the phase-fiacing method in Section IV. The Maslov index relates to the phase acquired when the semiclassical wave function haverses a zero (or a singularity, if there be one) and it (and, particularly, its sign) is the consequence of the analytic behavior of the wave function in the complex time plane. 

The diffraction space of ropes of parallel SWCNT can similarly be computed by summing the complex amplitudes of the individual SWCNTs taking into account the relative phase shifts resulting from the lattice arrangement at... 

Now suppose that an additional small magnetic field is applied perpendicular to Ho in the plane formed by pi and Hq, call this field (see Fig. 4-4B). Field Hi will act upon pi to increase the angle 6. If field Hy is caused to rotate around Ho at the Larmor precessional frequency of Wq, the torque produced will steadily act to change the angle 6. On the other hand, if the frequency of rotation of Hi is not the same as the precessional frequency, the torque will vary depending upon the relative phases of the two motions, and no sustained effect will be produced. 

Exploratory solid state synthesis seems to be the only workable route to new phases because of a general inability to predict relative phase stabilities and thence structures or compositions , published in K4La6li40s A new Structure Type for Rare-Earth-Metal Cluster Compounds that Contain Discrete Tetrahedral K4l Units. S. Uma, J.D. Corbett, Inorg. Chem. 1999, 38, 3831-3835. 

First to be considered is the isotropic Pq coefficient. The parameter is proportional to the integrated cross-section, a [Eq. (11)]. In fact, the preceding arguments show that when j = 0, I m = Im, and so there are no interference cross terms in this case. Consequently, as is already widely recognized, the integrated cross-section displays no dependence on the relative phase of the final continuum channels. 

The summation for the coefficient of the P2 term likewise includes phase insensitive contributions where I m = Im, but also now one has terms for which I = I 2, which introduce a partial dependence on relative phase shifts— specifically on the cosine of the relative phase shift. Again, this conclusion has long been recognized for example, by an explicit factor cos(ri j — ri i) in one term of the Cooper-Zare formula for the photoelectron (3 parameter in a central potential model [43]. 

It was shown in the preceding section that PECD can be anticipated to have an enhanced sensitivity (compared to the cross-section or p anisotropy parameter) to any small variations in the photoelectron scattering phase shifts. This is because the chiral parameter is structured from electric dipole operator interference terms between adjacent -waves, each of which depends on the sine of the associated channels relative phase shifts. In contrast, the cross-section has no phase dependence, and the p parameter has only a partial dependence on the cosine of the relative phase. The distinction between the sine... 

Figure 5.8 Potential dependence of relative phase difference between %nr arid Xr-...

Figure 5.8 shows the potential dependence of the relative phase difference between and X . The relative phase was changed by about 180° at 200 mV, which is close to the pzc for a Pt electrode in HCIO4 electrolyte solution [52,5 3]. This orientation change is most probably associated with a change in sign of the charge at the Pt surface. This clearly demonstrates that the orientation of water dipoles flips by 180° at the pzc. 

Contimious liquid extraction techniques are used when the sample volume is large, the distribution constant is small, or the rate of extraction is slow. The efficiency of extraction depends on many factors including the viscosity of the phases, the magnitude of the distribution constant, the relative phase volumes, the interfacial surface area, and the relative velocity of the phases. Numerous continuous extractors using llghter-than-water and heavier-than-water solvents vee been described [3,2 7,42,73,74]. Generally, either the ligi Pr or heavier density... 

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