Frequency modulation theory

Actually, however, at least for an isolated XH Y system, the change in length of the H-bonds occurs rhythmically with the frequency of the v(XH Y) vibration. In effect the vKK vibration is frequency modulated by the v(XB. Y) vibration. From a consideration of this more precise classical picture Batuev [29, 30] has shown that the broad band should actually consist of a series of sub-bands of frequencies rXH (XH Y). This is the frequency modulation theory of the origin of the broad vXH bands. This explanation also implies that the band should become narrow at low temperatures when the amplitude of the H-bond stretching vibration is small. 

Prior to the introduction of the frequency modulation theory, a quantum mechanical energy level scheme had been proposed by Stepanov [31], and subsequently developed by this author and his colleagues [32, 33], which also led to the conclusion that the broad vXH bands should, under certain conditions, be resolvable into a... 

These various consequences parallel closely the analogous ones of the fluctuation and frequency modulation theories. There is, however, one important point of difference between the classical and quantum viewpoints which does not seem to have been emphasized previously, namely that transitions from the lowest level of the ground state can occur to several levels of the upper curve. This means that even at very low temperatures, when all the molecules are initially in this lowest energy level, a band of considerable breadth with frequencies rXH + m>(XH Y) will still persist. The temperature independent residual band width is a direct result of the perturbations of the system (in particular the finite change in the distance rxymin) caused by the absorption of a large quantum of radiation of frequency vXH. The same type of explanation may apply to other vibrational bands which remain of finite width at low temperatures the occurrence of such bands have been the cause of considerable discussion [34]. 

Batuev, attempting to marshal every argument in favor of his own frequency modulation theory in preference to the predissociation theory, observes that the predissociation theory ignores the fundamental principles of Marxism-Leninism (155). Neither Marx nor Lenin seems to have published any work on the H bond. 

Chapter 3 is devoted to pressure transformation of the unresolved isotropic Raman scattering spectrum which consists of a single Q-branch much narrower than other branches (shaded in Fig. 0.2(a)). Therefore rotational collapse of the Q-branch is accomplished much earlier than that of the IR spectrum as a whole (e.g. in the gas phase). Attention is concentrated on the isotropic Q-branch of N2, which is significantly narrowed before the broadening produced by weak vibrational dephasing becomes dominant. It is remarkable that isotropic Q-branch collapse is indifferent to orientational relaxation. It is affected solely by rotational energy relaxation. This is an exceptional case of pure frequency modulation similar to the Dicke effect in atomic spectroscopy [13]. The only difference is that the frequency in the Q-branch is quadratic in J whereas in the Doppler contour it is linear in translational velocity v. Consequently the rotational frequency modulation is not Gaussian but is still Markovian and therefore subject to the impact theory. The Keilson-... 

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

Theories based on the Enskog collision time (84) or other solid-like approaches do not have a strongly temperature-dependent frequency correlation time. But they do have a temperature-dependent factor resulting from the need to create the solvent fluctuations in the first place. Thus, all fast-modulation theories predict that the dephasing rate will go to zero at 0 K. 

In pure liquids, short-range repulsive forces are responsible for most of the dephasing. The viscoelastic theory describes the interaction of these forces with the diffusive dynamics of the liquid (Section IV.D). The resulting frequency modulation is in the fast limit in low-viscosity liquids but can reach the slow-modulation limit at higher viscosities. This type of dephasing was seen in supercooled toluene (Section IV.C). 

Silver, J.A. Frequency-modulation spectroscopy for trace species detection theory and comparison among experimental methods. Appl. Opt. 1992, 31 (6), IQl-lYl. 

An important consequence of the lineshape theory discussed above concerns the effect of the bath dynamics on the linewidths of spectral lines. We have already seen this in the discussion of Section 7.5.4, where a Gaussian power spectrum has evolved into a Lorentzian when the timescale associated with random frequency modulations became fast. Let us see how this effect appears in the context of our present discussion based on the Bloch-Redfield theory. 

Appendix C. The limit T -> oo has, of course, special interest since it is used in standard line shape theories, and does not depend on an assumption of whether the frequency modulations are slow. The exact expression for Q in the limit of T oo is given in Eqs. (A.47) and (A.48). These equations are one of the main results of this chapter. It turns out that Q is not a simple function of the model parameters however, as we show below, in certain limits, simple behaviors are found. 

Frequency-modulated (FM) pulses like the hyperbolic secant (HS) pulse are not commonly used for multislice spin-echo MRI because of a nonlinear phase of the transverse magnetization. A general theory and methods are described for conventional spin-echo imaging using a hyperbolic secant (HS) pulse for refocusing.Phase profiles produced by the HS pulse are analytically described. 

M. Ducloy and M. Fichet, General theory of frequency modulated selective reflection. Influence of atom surface interactions. /. Phys. II (France) 1991,1, 1429. 

Introduction and experimental techniques. In previous sections we drew attention to the fact that, in both the classical and quantum theories, expressions derived for the intensity of resonance fluorescence from atoms subjected to an external magnetic field, equations (15,3) and (15.23) respectively, contain terms which may lead to a modulation of the intensity at the Larmor frequency or its second harmonic. This radio-frequency modulation has been observed in several different kinds of experiment, the simplest of which makes use of pulsed excitation and time-resolved detection of the fluorescent light. 

The quantum theory must describe not only the shape of a resolved rotational structure of the Q-branch but its transformation with increase of pressure to a collapsed and well-narrowed spectrum as well. A good example of such a transformation is shown in Fig. 4.6. The limiting cases of very low and very high pressures are relatively easy to treat as they relate to slow modulation and fast modulation limits of frequency exchange. 

Jablonski (48-49) developed a theory in 1935 in which he presented the now standard Jablonski diagram" of singlet and triplet state energy levels that is used to explain excitation and emission processes in luminescence. He also related the fluorescence lifetimes of the perpendicular and parallel polarization components of emission to the fluorophore emission lifetime and rate of rotation. In the same year, Szymanowski (50) measured apparent lifetimes for the perpendicular and parallel polarization components of fluorescein in viscous solutions with a phase fluorometer. It was shown later by Spencer and Weber (51) that phase shift methods do not give correct values for polarized lifetimes because the theory does not include the dependence on modulation frequency. 

Theory. If two or more fluorophores with different emission lifetimes contribute to the same broad, unresolved emission spectrum, their separate emission spectra often can be resolved by the technique of phase-resolved fluorometry. In this method the excitation light is modulated sinusoidally, usually in the radio-frequency range, and the emission is analyzed with a phase sensitive detector. The emission appears as a sinusoidally modulated signal, shifted in phase from the excitation modulation and partially demodulated by an amount dependent on the lifetime of the fluorophore excited state (5, Chapter 4). The detector phase can be adjusted to be exactly out-of-phase with the emission from any one fluorophore, so that the contribution to the total spectrum from that fluorophore is suppressed. For a sample with two fluorophores, suppressing the emission from one fluorophore leaves a spectrum caused only by the other, which then can be directly recorded. With more than two flurophores the problem is more complicated but a number of techniques for deconvoluting the complex emission curve have been developed making use of several modulation frequencies and measurement phase angles (79). 

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