Frequency renormalized

Formally, this procedure is correct only for spectra that are linear in the frequency, that is, spectra whose line positions are caused by the Zeeman interaction only, and whose linewidths are caused by a distribution in the Zeeman interaction (in g-values) only. Such spectra do exist low-spin heme spectra (e.g., cytochrome c cf. Figure 5.4F) fall in this category. But there are many more spectra that also carry contributions from field-independent interactions such as hyperfine splittings. Our frequency-renormalization procedure will still be applicable, as long as two spectra do not differ too much in frequency. In practice, this means that they should at least be taken at frequencies in the same band. For a counter-example, in Figure 5.6 we plotted the X-band and Q-band spectra of cobalamin (dominated by hyperfine interactions) normalized to a single frequency. To construct difference spectra from these two arrays obviously will generate nonsensical results. 

Taking into consideration relations (3.2.4), expressions (3.2.6) for absorption lines allowed in infrared spectra (A/ 0) are the same as those obtained formerly by an alternative method which involves no sublattice concept.81 The determination of Davydov splittings as squared frequency differences (3.2.6) results in their independence from the static frequency renormalization. For structures with [Pg.62]

As described in sect. 3 phonon Raman scattering in IV rare-earth compounds has been concerned quite intensively with phonon frequency renormalizations due... 

The situation simplifies when V Q) is a parabola, since the mean position of the particle now behaves as a classical coordinate. For the parabolic barrier (1.5) the total system consisting of particle and bath is represented by a multidimensional harmonic potential, and all one should do is diagonalize it. On doing so, one finds a single unstable mode with imaginary frequency iA and a spectrum of normal modes orthogonal to this coordinate. The quantity A is the renormalized parabolic barrier frequency which replaces in a. multidimensional theory. In order to calculate... 

Here e is the new value of the energy splitting, the co, are the ripplon frequencies, and the A,- are tunneling amplitudes of transitions that excite the corresponding vibrational mode of the domain wall. Those amplitudes will be discussed in due time for now, we repeat, the expression above will be correct in the limit Ai/Ha>i —> 0. Finally, the renormalized value e was used in the denominator. While, according to Feenberg s expansion [118], including e in the resolvent is actually more accurate, we do it here mostly for convenience. 

Our renormalization procedure is internally consistent in that the physical value of the tunneling amplitude depends on the scaling variable—the bare coupling Aq—only logarithmically. This bare coupling must scale with the only quantum scale in the problem—the Debye frequency, as pointed out in the first section. 

We have seen in Chapter 2 that the frequency of an EPR spectrum is not a choice for the operator (once the spectrometer has been built or bought) as it is determined by the combined fixed dimensions of the resonator, the dewar cooling system, and the sample. Even if standardized sample tubes are used and all the samples have the same dielectric constant (e.g., frozen dilute aqueous solutions of metalloproteins), the frequency will still slightly vary over time over a series of consecutive measurements, due to thermal instabilities of the setup. By consequence, two spectra generally do not have the same frequency value, which means that we have to renormalize before we can compare them. This also applies to difference spectra and to spectra... 

Here it is our intention to show that for a system constituted by substrate phonons and laterally interacting low-frequency adsorbate vibrations which are harmonically coupled with the substrate, the states can be subclassified into independent groups by die wave vector K referring to the first Brillouin zone of the adsorbate lattice.138 As the phonon state density of a substrate many-fold exceeds the vibrational mode density of an adsorbate, for each adsorption mode there is a quasicontinuous phonon spectrum in every group of states determined by K (see Fig. 4.1). Consequently, we can regard the low-frequency collectivized mode of the adsorbate, t /(K), as a resonance vibration with the renormalized frequency and the reciprocal lifetime 7k-... 

The real and imaginary parts of the pole in expression (A3.12) define the renormalized frequency and the inverse lifetime t]K ([Pg.178]

Silicon is a model for the fundamental electronic and mechanical properties of Group IV crystals and the basic material for electronic device technology. Coherent optical phonons in Si revealed the ultrafast formation of renormalized quasiparticles in time-frequency space [47]. The anisotropic transient reflectivity of n-doped Si(001) featured the coherent optical phonon oscillation with a frequency of 15.3 THz, when the [110] crystalline axis was parallel to the pump polarization (Fig. 2.11). Rotation of the sample by 45° led to disappearance of the coherent oscillation, which confirmed the ISRS generation,... 

The situation is quite different for physisorbed molecules. In that case, there is no transfer of charge, the mechanical renormalization is weaker due to a much weaker metal-molecule bond and also the image interaction is smaller as the molecule probably is adsorbed further out from the surface. In a recent IRS investigation of CO physisorbed on Al(100) the measured frequency is only shifted down a few cm from the gasphase value. However, there is for this system also a short range intermolecular interaction that certainly will affect the vibrational frequency. As yet there exist no theoretical calculations for the van der Waals interaction between a CO molecule and a metal. 

Using the self-consistently obtained solutions of Eq. 19, the calculated chemical shift (Jiso = (To + a(T) is calculated and compared to the experimental data in Fig. 4. Even though the experimentally observed transition is broader than the calculated one, the agreement between theory and experiment is good. As the discontinuity in the lattice-related mode is small at Tc, where Tc corresponds to a = 0, the chemical shift does not show a discontinuity at Tc within numerical accuracy. It is important to note here that the S-shape in the cf T) data is a direct consequence of using the renormalized frequencies as defined in Eq. 19. 

An approximate analytical solution (see above) yields that Tc is nearly independent of the tunnel mode frequency. Experimentally, it is found that in H2-xDxSQ substantial variations in take place with variations in x [54]. This fact does not contradict the above results since the tunnel mode is renormalized through the coupling C as -/q (S ) - (S > /g4T). As C... 

MCT can be best viewed as a synthesis of two formidable theoretical approaches, namely the renormalized kinetic theory [5-9] and the extended hydrodynamic theory [10]. While the former provides the method to treat both the very short and the very long time responses, it often becomes intractable in the intermediate times. This is best seen in the calculation of the velocity time correlation function of a tagged atom or a molecule. The extended hydrodynamic theory provides the simplicity in terms of the wavenumber-dependent hydrodynamic modes. The decay of these modes are expressed in terms of the wavenumber- and frequency-dependent transport coefficients. This hydrodynamic description is often valid from intermediate to long times, although it breaks down both at very short and at very long times, for different reasons. None of these two approaches provides a self-consistent description. The self-consistency enters in the determination of the time correlation functions of the hydrodynamic modes in terms of the... 

Finally, note that the relaxation equation [Eq. (76)] is usually written in terms of the hydrodynamic modes. In many problems of chemical interest, nonhydrodynamic modes such as intramolecular vibration, play an important role [50]. Presence of such coupling creates an extra channel for dissipation. Thus, the memory kernel, T, gets renormalized and acquires an additional frequency-dependent term [16, 43]. 

Thus we have reduced this problem to the form of conventional SR (see Section II.C) with only a renormalized effective amplitude for the input signal Aeff [cf Eq. 9)] and the function M() replaced by its derivative M (< >) in the first term on the right hand side. By analogy with standard SR, the SNR for heterodyning can be characterized by the ratio R of the low-frequency signal in the intensity of the transmitted radiation, given by 4 (T x(H) 2, to the value of the power spectrum Q(0 (ll) [with Q (Q) given by (7)—(8)]. The susceptibility of the system can be easily calculated and takes the form... 

High-frequency noise renormalization of the transverse H-field... 

Thus the role of the high-frequency oscillators is to suppress the transverse field component (in other words, the transverse g-factor). If we are interested only in the contribution to the level spacing (the Lamb shift), one should consider only the longitudinal ( B) part of the renormalization, i.e. multiply the result by sin0, to obtain Eq. (13). 

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