Effective relaxation time

For all known cases of iron-sulfur proteins, J > 0, meaning that the system is antiferromagnetically coupled through the Fe-S-Fe moiety. Equation (4) produces a series of levels, each characterized by a total spin S, with an associated energy, which are populated according to the Boltzmann distribution. Note that for each S level there is in principle an electron relaxation time. For most purposes it is convenient to refer to an effective relaxation time for the whole cluster. 

The alternative method is continuous-flow , in which the reactants flow through the detection coil during data acquisition. Continuous-flow NMR techniques have been used for the direct observation of short-lived species in chemical reactions [4—6]. The main difference between stopped- and continuous-flow NMR is that in the latter the sample remains inside the detection coil only for a short time period, termed the residence time, x [7], which is determined by the volume of the detection cell and the flow rate. The residence time alters the effective relaxation times according to the relationship in Eq. (2.5.1) ... 

Both, longitudinal and transverse relaxation of protons in tissue depend on the microstructure and on the chemical composition of several microscopic compartments. Relaxation properties are not necessarily constant for the diiferent compartments inside the cells (cytosol and cavities in cell organella) and in the extracellular space (interstitium and vessels). However, water exchange processes between the compartments are often fast enough to generate one effective relaxation time, which can be assessed by monoexponential fitting of the relaxation dependent data. 

Mainly the water resonance can often be used as an indicator for inflammations or tumours, because effective relaxation times tend to be prolonged in tissue with increasing interstitial or intracellular water content. As reported in Section 3.2, lipids (mainly consisting of triglycerides) reveal more stable relaxation characteristics in all tissues and no significant alterations have been reported so far. 

Longitudinal relaxation measurements do not allow to establish whether the electron relaxation is constant or field-dependent. Transverse relaxation measurements, on the contrary, prove that the effective relaxation time is indeed field-dependent (see Sections I.A.l and II.A.4). The occurrence of a field-dependent electron relaxation time is confirmed by the longitudinal relaxation measurements in glycerol solution (62), as the typical high field relaxivity peak appears, with Af of 0.11 cm and of 7 x 10 s at 298 K. 

In organic solids the determination of rotating frame relaxation is severely complicated by the presence of the strongly interacting proton spin system. Spin-spin fluctuations compete with spin-lattice fluctuations to produce an effective relaxation time large rf field amplitudes are mandated to discriminate against the spin-spin event. The burden of proof lies with the experimenter to establish that a rotating frame relaxation rate actually reflects a motional effect seen by the carbon nuclei. 

These questions were resolved with the use of the same relatively simple epoxy system. All C-13 nuclei in contact with the proton bath were counted when moderate spinning rates were used and in spin-lock cross polarization in rf fields not close to any Tle minimum. The molecular motion determines the relaxation rate, under the Hartmann-Hahn condition when T, = T2. The spin-spin effects determine relaxation when Tle does not equal T2 under the same conditions 62). The spin-spin fluctuations are in competition with the spin-lattice fluctuations in producing an effective relaxation time. To discriminate against the spin-spin fluctuations large rf fields are mandatory. It was pointed out that, with great care, C-13 NMR spectra can reflect molecular motion. 

Schultze pointed out (18) that whenever the enhanced catalytic activity of the catalyst is due to so-called active sites, that is, exposed crystal defects or dislocations, these sites will only be active long term if processes that would lead to healing or recrystallization and accordingly to deactivation have an activation energy in excess of 100 kJ/mol. Such high activation energies would yield at 100°C a retardation of any surface relaxation processes to effective relaxation times in excess of several years. 

The last expression justifies the treatment of magnetic SR in the 0 = 0 limit as a process characterized by just one effective relaxation time xeff. As has been shown in Refs. 29 and 58, the time reff is very close to X in a wide range of a. 

On the other hand, one can try to approximate the magnetization damping process by a single effective relaxation time as... 

For high temperatures ( a. c 1), the effective relaxation time defined by Eq. (4.270) reduces to... 

Let us give some concluding remarks on the concept of the effective relaxation time ... 

The virtual presence of all the spectral terms in the effective time makes this approach more adequate than the superparamagnetic blocking model. In the latter, the effective relaxation time of magnetization is identified just with the inverse of the decrement X], which is the smallest at , 0. 

The results for SNR evaluated with the effective relaxation time approximation in the same linear response framework [i.e., by Eq. (4.281)], are shown in Figure 4.24. One observes a good agreement with respect to the main cusp of the function R(T). However, in the low-temperature range the existing deviations from the exact solution [they do not resolve in the susceptibility graphs of Figure 4.21(a)] become noticeable see the relative positions of the dots and curve 3 for 1/a < 0.1. 

In this equation, h is Planck s constant divided by 2tt, V is the crystal volume, T is temperature, fej, is Boltzmann s constant, phonon frequency, is the wave packet, or phonon group velocity, t is the effective relaxation time, n is the Bose-Einstein distribution function, and q and s are the phonon wave vector and polarization index, respectively. 

Equation (159), which involves the integral relaxation time x(nf, the effective relaxation time xef, and the smallest nonvanishing eigenvalue /., correctly predicts /Jot) both at low (co —> 0) and high (co ocj frequencies. Moreover, for a particular form of the potential V, x( ) ma> be determined in the entire frequency range 0 < co < oo as we shall presently see for a double-well periodic potential representing the internal field due to neighboring molecules. 

Now, we shall demonstrate that the characteristic times of the normal diffusion process, namely, the inverse of the smallest nonvanishing eigenvalue 1 //.], the integral and effective relaxation times xint and xef obtained in [8,62,63], also allow us to evaluate the dielectric response of the system for anomalous diffusion using the two-mode approximation just as normal diffusion (Ref. 8, Section 2.13). Here, we can use known equations for xint, x,f, and X for the normal diffusion in the potential Eq. (163) [8,62,63] these equations are... 

Thus the anomalous relaxation in a double-well potential is effectively determined by the bimodal approximation, Eq. (159) the characteristic times of the normal diffusion process—namely, the inverse of the smallest nonvanishing eigenvalue, the integral, and effective relaxation times—appear as time... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

In the foregoing equations x nr xyef, and A, are calculated as follows. The effective relaxation times and xyef are given in terms of equilibrium averages (P ), and (/J2)o only (Ref- 8, Chapter 8) ... 

For small values of cv, this formula is inadequate. However, we can then use an approximate equation for the effective relaxation time x because for Cy -c 1 Aj 1 /xej [8]. Also, for small values of one can use the exact Taylor series expansion ([8], Section 8.3.2)... 

We note that both the effective relaxation times and the zero-frequency susceptibilities may be written in terms of Dawson s integral. The detailed expressions are given in Ref. 55. 

Thus the effective relaxation time of the dipole mode is modified to... 

It remains to discuss the influence of Hq on the mechanical relaxation modes. An estimate of this may be made by recalling that Eq. (95) is basically the equation of motion of a rigid dipole in a strong constant external field. Moreover, if inertial effects are neglected, it has been shown in Refs. 21 and 61 that the longitudinal and transverse effective relaxation times decrease monotonically with field strength from Xg, having asymptotic behavior... 

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