Lorentzian decay

Furthermore, as it is seen from numerous experiments, the far red wing of the absorption profile of chromophores in solutions does not appear to follow a Lorentzian decay, as implemented in the conventional expression for the frequency dependent linear absorption cross section, but rather some kind of fast exponential, Urbach-like, decay [22, 23]. 

The exponential decay of the A population corresponds to a Lorentzian line shape for the absorption (or emission) cross section, a, as a fiinction of energy E. The lineshape is centred around its maximum at E. The fiill-width at half-maximum (F) is proportional to... 

The vibrational echo experiments yielded exponential decays at all temperatures. The Fourier-transfonn of the echo decay gives the homogeneous lineshape, in this case Lorentzian. The echo decay time constant is AT, where is... 

Apparently, the time-domain and frequency-domain signals are interlinked with one another, and the shape of the time-domain decaying exponential will determine the shape of the peaks obtained in the frequency domain after Fourier transformation. A decaying exponential will produce a Lorentzian line at zero frequency after Fourier transformation, while an exponentially decaying cosinusoid will yield a Lorentzian line that is offset from zero by an amount equal to the frequency of oscillation of the cosinusoid (Fig. 1.23). 

Lorentzian line at zero, (b) The FT of an exponentially decaying consinusoid FID gives a Lorentzian line offset from zero frequency. The offset from zero is equal to the frequency of oscillation of the consinusoid. (Reprinted from S. W. Homans, A dictionary of concepts in NMR, copyright 1990, p. 127-129, by permission of Oxford University Press, Walton Street, Oxford 0X2 6DP, U.K.)... 

Matched filter The multiplication of the free induction decay with a sensitivity enhancement function that matches exactly the decay of the raw signal. This results in enhancement of resolution, but broadens the Lorentzian line by a factor of 2 and a Gaussian line by a factor of 2.5. 

Figure 5.1. Representations of double-exponential and bimodal Lorentzian distribution analyses of DPH fluorescent decay lifetimes in liver microsomal membranes. Results (see Table 5.2) are normalized to the major component. The double-exponential analysis, represented by the vertical lines, recovers lifetimes near the centers of the Lorentzian distributions. The width of the distributions represents contributions from a variety of lifetimes. (From Ref. 17.)...

Table 5.2. Double-Exponential and Bimodal Lorentzian Analyses of DPH Fluorescence Decay Lifetimes in Liver Microsomes and Vesicles of Extracted Phospholipids"1 ...

Most methods assume an exponential decay for the resonances in the time domain giving rise to Lorentzian lineshapes in the frequency domain. This assumption is only valid for ideal experimental conditions. Under real experimental circumstances multi-exponential relaxation, imperfect shimming, susceptibility variations and residual eddy current usually lead to non-ideal... 

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 ... 

Proton n.m.r. investigations of coals swollen in deuterated pyridine showed that the free induction decay /FID/ consists of Gaussian and Lorentzian components related to two populations of protons which have widely different degrees of rotational mobility (1-5). The Gaussian component of FID has been unanimously ascribed to the macromolecular part of the coal matter that is supposed to have very limited rotational mobility. These publications as well as the ensuing debates (6-9) however, reflected the controversy regarding the nature of the Lorentzian /mobile/ protons in coals. 

In many of these descriptions of lineshapes, chemical exchange line-shapes are treated as a unique phenomenon, rather than simply another example of relaxation effects on lineshapes. This is especially true for line-shapes in the intermediate time scale, where severe broadening or overlapping of lines may occur. The complete picture of exchange lineshapes can be somewhat simplified, following Reeves and Shaw [13], who showed that for two sites, the lineshape at coalescence can always be described by two NMR lines. This fact can be exploited to produce a clarified picture of exchange effects on lineshapes and to formulate a new method for the calculation of exchange lineshapes [16, 23]. This method makes use of the fact that lineshapes, even near coalescence, retain Lorentzian characteristics [13] (fig. 3). These lines, or coherences, are each defined by an intensity, phase, position, and linewidth, and for each line in the spectrum, the contribution of that particular line to the overall free induction decay (FID) or spectrum can be calculated. 

Adjustment) [3], FIDDLE (Free Induction Decay Deconvolution for Line-shape Enhancement) [6] and QUALITY (QUAntification improvement by converting LIneshapes to the Lorentzian TYpe) [5],... 

In these Lorentzian lines, the parameter x describes the kinetic decay lifetime of the molecule. One says that the spectral lines have been lifetime or Heisenberg broadened by an amount proportional to l lx. The latter terminology arises because the finite lifetime of the molecular states can be viewed as producing, via the Heisenberg uncertainty relation AEAt > -h, states whose energy is "uncertain" to within an amount AE. 

Among them, Li-i-HP can be considered a benchmark model system [29, 30] because its low number of electrons makes possible to calculate accurate PES s. Its electronic spectrum has been meassured by Polanyi and coworkers [22], and has been recently very nicely reproduced using purely adiabatic PES s [31]. In the simulation of the spectrum[31], the transition lines were artificially dressed by lorentzians which widths were fitted to better reproduce the experimental envelop. The physical origin of such widths is the decay of the quasibound states of the excited electronic states through electronic predissociation (EP) towards the ground electronic state. This EP process is the result of the non-adiabatic cou-... 

We illustrate the general expressions (4.189)-(4.191) for a typical non-Markovian Lorentzian bath spectrum, that is, an exponentially decaying correlation function d>(t) = being the correlation (memory) time. 

This coupling means that there is no population decay to the bath and the decoherence is only via the phase. In other pioneering works [39-41], off-diagonal coupling to a Lorentzian bath was considered within the RWA. The Hamiltonians in all of these works should be contrasted with the more general Hamiltonians (4.18), which account for both dephasing and decay into arbitrary baths, without the RWA. 

Yes, our analysis applies to a sum of Fano type resonances as well as to the sum of Lorentzians (giving rise to the exponential decays which I showed) studied here. 

It is very important, in the theory of quantum relaxation processes, to understand how an atomic or molecular excited state is prepared, and to know under what circumstances it is meaningful to consider the time development of such a compound state. It is obvious, but nevertheless important to say, that an atomic or molecular system in a stationary state cannot be induced to make transitions to other states by small terms in the molecular Hamiltonian. A stationary state will undergo transition to other stationary states only by coupling with the radiation field, so that all time-dependent transitions between stationary states are radiative in nature. However, if the system is prepared in a nonstationary state of the total Hamiltonian, nonradiative transitions will occur. Thus, for example, in the theory of molecular predissociation4 it is not justified to prepare the physical system in a pure Born-Oppenheimer bound state and to force transitions to the manifold of continuum dissociative states. If, on the other hand, the excitation process produces the system in a mixed state consisting of a superposition of eigenstates of the total Hamiltonian, a relaxation process will take place. Provided that the absorption line shape is Lorentzian, the relaxation process will follow an exponential decay. 

The free induction decay following 90° pulse has a line shape which generally follows the Weibull functions (Eq. (22)). In the homogeneous sample the FID is described by a single Weibull function, usually exponential (Lorentzian) (p = 1) or Gaussian (p = 2). The FID of heterogeneous systems, such as highly viscous and crosslinked polydimethylsiloxanes (PDMS) 84), hardened unsaturated polyesters 8S), and compatible crosslinked epoxy-rubber systems 52) are actually a sum of three... 

A pseudo solid-like behavior of the T2 relaxation is also observed in i) high Mn fractionated linear polydimethylsiloxanes (PDMS), ii) crosslinked PDMS networks, with a single FID and the line shape follows the Weibull function (p = 1.5)88> and iii) in uncrosslinked c/.s-polyisoprenes with Mn > 30000, when the presence of entanglements produces a transient network structure. Irradiation crosslinking of polyisoprenes having smaller Mn leads to a similar effect91 . The non-Lorentzian free-induction decay can be a consequence of a) anisotropic molecular motion or b) residual dipolar interactions in the viscoelastic state. 

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