Relaxation multi-exponential

In case of slow exchange between phases, the relaxation decay becomes the sum of relaxation of the various phases and thereby multi-exponential ... 

Brownstein and Tarr35 considered if the size of a muscle cell could give rise to non-mono-exponential transverse relaxation by assuming simple diffusion and planar geometry, and found that under these conditions and in the slow diffusion range, multi-exponential transverse relaxation could be expected when the sample size is between 1 and 30 pm. Accordingly, as muscle cells have a diameter of 10-100 pm,36 the calculations performed by Brownstein and Tarr35 indicate that the anatomical features of muscle cells are consistent with the expectation of multi-exponential transverse relaxation. 

Transverse relaxation of musculature is relatively fast compared with many other tissues. Measurements in our volunteers resulted in T2 values of approximately 40 ms, when mono-exponential fits were applied on signal intensities from images recorded with variable TE. More sophisticated approaches for relaxometry revealed a multi-exponential decay of musculature with several T2 values." Normal muscle tissue usually shows lower signal intensity than fat or free water as shown in Fig. 5c. Fatty structures inside the musculature, but also water in the intermuscular septa (Fig. 5f) appear with bright signal in T2-weighted images. 

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... 

In the case of hi- or multi-exponential relaxation curves the treatment involved can be rather complex (119-123). It becomes even more problematic. Needles to say, the same is true for systems with suspected continuous distributions of relaxation rates, whose evaluation by numerical analysis of the decay curves (124-128) represents one of the most arduous mathematical problems (124-128). In general, evaluation tasks of this kind need to be treated off-line, using specific programs and algorithms. 

The type of the distribution of T-values is a much discussed topic. Experience shows that, in a mono-exponential case, the values should spread over an interval of more than 3 x Ti but not much over 4 x Ti, and a linear distribution appears to be slightly better than a logarithmic one. This is probably due to the fact that in a three-parameter exponential fit, the points with large x-values play as crucial a role in determining the relaxation rate as the slope at small x-values, and one needs both to determine R. On the other hand, it is evident that in multi-exponential cases, logarithmic distribution is often better suited for the task, especially when the relaxation rates of different sample components differ by an order of magnitude or more. 

E.g. tryptophane residues of proteins excite at 290-295 mn but they emit photons somewhere between 310 and 350 mn. The missing energy is deposited in the tryptophane molecular enviromuent in the form of vibrational states. While the excitation process is complete in pico-seconds, the relaxation back to the initial state may take nano-seconds. While this period may appear very short, it is actually an extremely relevant time scale for proteins. Due to the inherent thermal energy, proteins move in their (aqueous) solution, they display both translational and rotational diffusion, and for both of these the characteristic time scale is nano-seconds for normal proteins. Thus we may excite the protein at time 0 and recollect some photons some nano seconds later. With the invention of lasers, as well as of very fast detectors, it is completely feasible to follow the protein relax back to its ground state with sub-nano second resolution. The relaxation process may be a simple exponential decay, although tryptophane of reasons we will not dwell on here display a multi-exponential decay. 

The initial steep change in the extinction is ascribed to the temperature equilibration of the sample. Following this, a multi-exponential decay in the extinction is observed, indicating the complex relaxation kinetics of the homogeneous reaction between the various cationic SE s. Finally, the long-term single exponential relaxation... 

In Fig. 10, the transients exhibit quite different behavior from opal A to opal CT. In particular, a bi-exponential decay (Eq. 2) failed to reproduce the kinetics of opal CT. In this material, the emission is red-shifted towards 2.6 eV and the PL is strongly quenched at shorter time delays, with an unusual, non-linear kinetics in semi-log scale, indicating a complex decay channel either involving multi-exponential relaxation or exciton-exciton annihilations. Runge-Kutta integration of Eq. 5 seems to confirm the latter assumption with satisfactory reproduction of the observed decays. The lifetimes and annihilation rates are Tct = 9.3 ns, ta = 13.5 ns, 7ct o = 650 ps-1 and 7 0 = 241 ps-1, for opal CT and opal A, respectively. 

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

For example, considering the energy flow diagram shown in Figure 1, the TG signal is expected to rise with a multi-exponential function, of which rise rates represent the relaxation rates. If the kinetics are roughly separated into... 

Electron Spin Relaxation. Proton relaxivity also depends on electron spin relaxation, which is in general a multi-exponential, magnetic field-dependent phenomenon. Electron spin relaxation in Gd chelates, which are S = 7/2 species, is a complex phenomenon that depends on stochastic zero field splitting (zfs) interactions, which may arise from both (1) collisional distortion of the zfs tensor and (2) Brownian reorientation of the zfs principal axes. The importance of both mechanistic contributions has been emphasized in recent study of electron spin relaxation in Gd(DOTA) and Gd(DPTA) by Borel et al which reanalyses... 

This approach of data processing considers distributions of relaxation rates or other physical parameters rather than a sum of limited and discrete characteristic quantities. Prerequisite is that the decaying fimction is multi-exponential. ILT may not be applied to Gaussian decays, which are observed for instance in dipolar coupled systems (hard polymers). Prominent examples for the ILT are relaxation studies of liquid-like samples and investigations of (restricted) diffusion by varying the gradient amplitude g. Even in case of restricted diffusion, like in droplets size studies, the relation between signal attenuation and is often exponmtial. 

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