Relaxation time, definition

The four spin-lattice relaxation mechanisms that are usually considered for nuclei of spin are (i) dipole-dipole interactions (DD) (ii) spin-rotation interactions (SR) (iii) scalar coupling (SC) and (iv) chemical screening anisotropy (CSA). (5) In Si NMR SC is not generally considered, except in the rare cases when Si is spin-coupled to I or other nuclei with resonance frequencies close to that of Si. Table XXX compares the contribution of the three remaining mechanisms with Si relaxation times. Definitions of these relaxation mechanisms can be found in ref. 229. Since a theoretical discussion of Si relaxation and NOE is detailed elsewhere [(5) and references therein] only general observations are made in this section. 

Suggested modification for the stretch relaxation time definition in the mWMmodel... 

Unfortunately, even for low molecular weight material it is difficult to obtain clear experimental evidence for a roughening transition [71]. This is mainly due to the fact that during growth the interface generally assumes a metastable shape and relaxation times are long and increase with crystal size. Therefore we certainly cannot expect a definitive answer for macromolecules. We shall therefore just make several comments which hopefully will be of use when reading the literature. 

It is evident from Table 2 that the chemical shift data are very similar in both states of aggregation. Only the carbonyl carbon show a small but definite shifts, 2 ppm. In the solution state, in acetone -d6 solution the relaxation times T1 of the pyranose carbon atoms are very similar and only slightly smaller than those of the carbon atom of the methyl group in the acetyl substituent, while the T1-value of the carbon atom of the carbonyl group is considerably higher. 

Let us demonstrate that the tendency to narrowing never manifests itself before the whole spectrum collapses, i.e. that the broadening of its central part is monotonic until Eq. (6.13) becomes valid. Let us consider quantity x j, denoting the orientational relaxation time at ( = 2. If rovibrational interaction is taken into account when calculating Kf(t) it is necessary to make the definition of xg/ given in Chapter 2 more precise. Collapse of the Q-branch rotational structure at T = I/ojqXj 1 shifts the centre of the whole spectrum to frequency cog. It must be eliminated by the definition... 

Limiting ourselves to derivation of the Hubbard relation in the simplest case ( = 1 (for t = 2 see Appendix 9), we write out the definition of orientational relaxation time... 

The response time (relaxation time, adjustment time) of a reservoir is a time scale that characterizes the adjustment to equilibrium after a sudden change in the system. A precise definition is not easy to give except in special circumstances like in the following example. 

In order to have a finite probability that termolecular collisions can occur, we must relax our definition of a collision. We will assume that the approach of rigid spheres to within a distance of one another constitutes a termolecular collision that can lead to reaction if appropriate energy and geometry requirements are met. This approach is often attributed to Tolman (41). The number of ternary collisions per unit volume per unit time between molecules A, B, and C such that A and C are both within a distance of B is given by ZABC. 

From the basic definition of the relaxation time it is evident that... 

The aim of this chapter is to describe approaches of obtaining exact time characteristics of diffusion stochastic processes (Markov processes) that are in fact a generalization of FPT approach and are based on the definition of characteristic timescale of evolution of an observable as integral relaxation time [5,6,30—41]. These approaches allow us to express the required timescales and to obtain almost exactly the evolution of probability and averages of stochastic processes in really wide range of parameters. We will not present the comparison of these methods because all of them lead to the same result due to the utilization of the same basic definition of the characteristic timescales, but we will describe these approaches in detail and outline their advantages in comparison with the FPT approach. 

It should be noted that besides being widely used in the literature definition of characteristic timescale as integral relaxation time, recently intrawell relaxation time has been proposed [42] that represents some effective averaging of the MFPT over steady-state probability distribution and therefore gives the slowest timescale of a transition to a steady state, but a description of this approach is not within the scope of the present review. 

Alternatively, the definition of the mean transition time (5.4) may be obtained on the basis of consideration of optimal estimates [54]. Let us define the transition time i) as the interval between moments of initial state of the system and abrupt change of the function, approximating the evolution of its probability Q(t.X(t) with minimal error. As an approximation consider the following function v /(f,xo, ) = flo(xo) + a (xo)[l(f) — l(f — i (xo))]. In the following we will drop an argument of ao, a, and the relaxation time d, assuming their dependence on coordinates of the considered interval c and d and on initial coordinate x0. Optimal values of parameters of such approximating function satisfy the condition of minimum of functional ... 

In line with these probabilities, we may introduce into consideration two relaxation times 0i and 02 according to a common definition (5.75). From the evident equality Pi (t) + P2W 1, it follows that 0, = 02 for both d > x0 and... 

Fig. 4.15 Momentum transfer (Q)-dependence of the characteristic time r(Q) of the a-relaxation obtained from the slow decay of the incoherent intermediate scattering function of the main chain protons in PI (O) (MD-simulations). The solid lines through the points show the Q-dependencies of z(Q) indicated. The estimated error bars are shown for two Q-values. The Q-dependence of the value of the non-Gaussian parameter at r(Q) is also included (filled triangle) as well as the static structure factor S(Q) on the linear scale in arbitrary units. The horizontal shadowed area marks the range of the characteristic times t mr- The values of the structural relaxation time and are indicated by the dashed-dotted and dotted lines, respectively (see the text for the definitions of the timescales). The temperature is 363 K in all cases. (Reprinted with permission from [105]. Copyright 2002 The American Physical Society)...

In order to extract some more information from the csa contribution to relaxation times, the next step is to switch to a molecular frame (x,y,z) where the shielding tensor is diagonal (x, y, z is called the Principal Axis System i.e., PAS). Owing to the properties reported in (44), the relevant calculations include the transformation of gzz into g x, yy, and g z involving, for the calculation of spectral densities, the correlation function of squares of trigonometric functions such as cos20(t)cos20(O) (see the previous section and more importantly Eq. (29) for the definition of the normalized spectral density J((d)). They yield for an isotropic reorientation (the molecule is supposed to behave as a sphere)... 

Bloembergen et al. (S) have presented a relationship between the correlation time for molecular rotation in liquids and the relaxation times assuming that relaxation takes place via mechanism (i) of Section II,A,3. Although the theory can be at best semiquantitative when applied to the protons of water molecules adsorbed on silica gel, values of the nuclear correlation time have been calculated 18) from the T data. These correlation times as a function of x/m show a definite change of slope near a monolayer coverage. This result, if corroborated by data on other solids, may provide a rather unique method for the determination of monolayer coverage. 

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