Lorentzian functions, spectral densities

A common assumption in the relaxation theory is that the time-correlation function decays exponentially, with the above-mentioned correlation time as the time constant (this assumption can be rigorously derived for certain limiting situations (18)). The spectral density function is then Lorentzian and the nuclear spin relaxation rate of Eq. (7) becomes ... 

We will first consider the error bounds for a spectral density broadened by a Lorentzian slit function, Eq. (15), describing the response to an exponentially damped perturbation. In this case the broadened spectrum,... 

Another approach to estimating spectral densities, which has the advantage of guaranteeing that the approximate functions are positive, can be based on the error bounds constructed in Section III-A for the spectral density broadened by a Lorentzian slit function. If we had a sufficient number of moments to make the error bounds very precise, then we could reduce the broadening as much as we like, so that the broadened distribution of spectral density becomes as close as we like to the true distribution. In order to estimate these higher moments, we should need to take advantage of some special feature of the distribution. For example, in the case of the harmonic vibrations of a crystalline solid, the distribution of frequencies lies between limits — co,nax and +comax, and is zero outside... 

As the hot band transitions should exist in both of the v, band of hydrogen-bonded and physisorbed acetonitrile, we must get the accurate v, band without the effect by the hot band transitions. We have to determine the analytical functions for fundamental transitions of v/a and vJp and for hot band transitions upon the least-squares procedure. We assumed that the reorientational-vibrational relaxation processes of physisorbed acetonitrile could be described as a diffusion process like bulk liquid molecule, whose spectral density has a Lorentzian form. Accordingly it is supposed that the v,p band is reproduced as a sum of three Lorentzian curves of the v, p, v," p, and bands. In the previous study [9], it was assumed that the Vj a band has a Gaussian band shape and the hot band transitions could be ignored. In the present study we assumed that the v,a band over the range of PIP, =... 

For a simple exponential correlation function, given in Eq. [18], the corresponding spectral density is a Lorentzian... 

Plot of the spectral density as a function of the dimensionless variable a>tc. The curve is a lorentzian... 

The spectral density function, J(co0), is modeled as a constant, a, plus a sum of Lorentzian terms, each characterized by an amplitude parameter, / , and a correlation time, rc ... 

In order to obtain a more realistic description of reorientational motion of intemuclear axes in real molecules in solution, many improvements of the tcf of equation Bl.13.11 have been proposed [6]. Some of these models are characterized in table Bl.13.1. The entry number of terms refers to the number of exponential functions in the relevant tcf or, correspondingly, the number of Lorentzian terms in the spectral density function. 

For processes with several relaxation times, the analysis of homodyne data is quite difficult. For a one-relaxation time process, however, the homodyne correlation function (or spectral density) is still exponential (Lorentzian), except that the relaxation time that appears is r/2 rather than t. 

Fig. 1 (a) °N-H dipolar and (b) CSA contribution to longitudinal and transverse relaxation rates (/f 1, and R ) as a function of a correlation time. The rates were calculated assuming a simple Lorentzian spectral density function, J((a) = t/(1 + Solid and dotted lines indicate rates... 

The slow motion is a combination of droplet tumbling and the lateral diffusion of surfactant molecules within the surfactant film. For spherical aggregates, this motion is described by a Lorentzian spectral density function,... 

With the assumptions of (1) extreme narrowing conditions for the fast motions and (2) a Lorentzian spectral density function of the slow motion, the expressions for the relaxation rates become... 

The correlation function (Equation 48) corresponds to the spectral density (see Equation 26), which represents a superposition of Lorentzians, i.e. of curves like y = alir x +... 

Another fundamental aspect of semiconductor lasers is that their field spectrum (spectral density function of the laser field) is not Lorentzian but possess side bands. 1 > xhe side bands can be understood by considering first the density of electrons in the conduction band of the laser medium when the input current i(t) is modulated at w... 

The fundamental requirement for longitudinal relaxation of a proton nucleus is a time dependent magnetic field fluctuating at the Larmor frequency of the nuclear spin. In the fast motion limit, the frequency distribution of the fluctuating magnetic fields associated with the molecular motion, i.e. the spectral density function J(a)), has, in the simplest case, a Lorentzian shape, as described by Eq. (1), and is characterized by the correlation time xc ... 

Following Eq. (26), a distribution of relaxational modes, with frequencies equal to Xy, contributes additivity to the macroscopically observed transient property . may be conveniently replaced by the above described correlation functions associated with specific vectorial quantities, depending on the experimental system, or on the investigated quantity. This representation is particularly suitable for taking the Fourier transform and expressing the spectral density of = M2(t), for example, as a sum of Lorentzians... 

M = co, . The usual form of the spectral density function is a simple Lorentzian, characterized by a single correlation time Tc. 

It is essential to know a priori the spectral density function /(co). We assume the spectral density function of Lorentzian form, i.e.. 

Figure 10.6C shows the spectral density function obtained from the correlation function shown in Fig. 10.6B. If the correlation function M mjk decays exponentially with a single time constant t, Jmijkio)) is a Lorentzian peaking at w = 0 (Appendix A3 and Eq. (2.70)). The Lorentzian has a peak amplitude of VnmVjk and a width at half-maximal amplitude of 2/7. ...

Magnetic resonance methods have been used extensively to probe the structure and dynamics of thermotropic nematic liquid crystals both in the bulk and in confined geometry. Soon after de Gennes [27] stressed the importance of long range collective director fluctuations in the nematic phase, a variable frequency proton spin-lattice relaxation Tx) study [32] showed that the usual BPP theory [33] developed for classical liquids does not work in the case of nematic liquid crystals. In contrast to liquids, the spectral density of the autocorrelation function is non-Lorentzian in nematics. As first predicted independently by Pincus [34] and Blinc et al. [35], collective, nematic type director fluctuations should lead to a characteristic square root type dependence of the spin-lattice relaxation rate rf(DF) on the Larmor frequency % ... 

Finally, we give the operational relations for the a and the 7. If the spectral density is fitted by m Lorentzian functions... 

P (o, t (Oo) is the probability density that the chromophore has transition frequency (o at time t given that it had frequency coo at time 0. While the functional form of this spectral diffusion kernel is quite complicated in general, at short times certain simplifications occur. In particular, if the positions of the TLSs occupy a regular lattice in three-dimensional space, all of the relaxation rates Kj are the same, all of the occupation probabilities pj are the same and equal to 1/2 (the high-temperature limit), and the perturbations vj are dipolar, then it was shown by Klauder and Anderson [29] and more recently by Zumofen and Klafter [30] that the spectral diffusion kernel is Lorentzian ... 

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