Pure exponential relaxation

There has been, from time to time, considerable interest in the possibility of pure exponential decay of various observable quantities in a relaxing system the necessary and sufficient conditions for exponential decay have been formulated in the form of a set of sum rules [63.S2 64.A], constraints to which the elements of the relaxation matrix must conform, given the energy levels of the system and their degeneracies. If 

6 It would appear that this would be a useful property when discussing the relaxation of mixtures of gases in which one component is a large polyatomic. 

If we substitute equation (226) into equation (2.10) to obtain the symmetrised form of A, we arrive at a very simple form which may be expressed as 

and the ability to recast equation (226) in this form provides us with some extremely powerful analytical capabilities. 

It is also possible that the use of equation (2.25) to describe the internal relaxation may have some value for certain unimolecular reaction problems - for example, in cases where it is not admissible to assume exponential decay of the populations in the unperturbed relaxation -but I have not yet had time to examine such a reformulation to see whether any tractable results ensue. 

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Exponential decay. The pure exponential relaxation of the models just considered results from two simplifications. The more important is our assumption that the jump process is negli bly rapid. This is already inapplicable to the transverse relaxation of the Frohlich model, which is obviously a resonant process dominated by the oscillation frequency v. The other is the restriction to only two sites. 

For small values of single-stress relaxation processes are observed. However, for larger than unity, clear departures occur. These deviations can easily be represented in a Cole-Cole plot, where the loss modrdus G"((Si) is plotted as a function of the storage modulus G ((n). A pure exponential relaxation process appears in such a diagram as a semicircle passing through the origin ... 

Since the velocity relaxation time, m/J, is typically 0.1 ps, t is rather shorter than that estimated from the decay of the velocity autocorrelation function. As an operational convenience, rrel — mjl can be deduced from the decay time re of the velocity autocorrelation functions. However, this procedure still does not entirely adequately describe the details of Brownian motion of particles over short times. The velocity relaxes in a purely exponential manner characteristic of a Markovian process. Further comments on the reduction of the Fokker—Planck equation to the diffusion equation have been made by Harris [526] and Tituiaer [527]. 

Fig. 3.18. Exponential recovery (A) of Mz(t) of a nuclear spin / dipole coupled to a paramagnetic metal ion. When I is also coupled to another nuclear spin J, the latter also coupled to the metal ion, non-exponentiality occurs. If J relaxes slower than /, curves B and C are obtained for a selective and a non-selective experiment respectively. If J relaxes slower than /, curves D (selective) and E (non-selective) are obtained. If J relaxes at the same rate as /, a selective experiment gives an intermediate behavior between curves B and D (not shown), while a non-selective experiment gives pure exponential recovery (A). It is apparent that in all cases non-selective experiments perform better than selective experiments, as they are less sensitive to the non-exponentiality introduced by I-J coupling. Conditions R m = 10 s l R M = 20 s l (B,C), 5 s l (D,E) and 10 s l (A). The I-J cross-relaxation rate ou (Chapter 7) is —20 s"1.

When Tj, one is in the fast modulation limit (homogeneous case) and C(t) can be adequately represented over its full range by a pure exponential. In the opposite, slow modulation, limit (inhomogeneous case) exponential relaxation is only reached in the tail of the decay (or never), and T2 is of little significance. These two limits were considered in the earliest work on this subject, and the more general intermediate case was treated shortly afterward. 

Table 1 also shows that both pure PVP and PVPh-co-PMMA/PEO blends exhibit only single-exponential relaxation through out all of the blends at... 

Harris et described a numerical methodology to obtain more efficient relaxation filters to selectively retain or remove components based on relaxation times. The procedure uses linear combinations of spectra with various recycle or filter delays to obtain components that are both quantitative and pure. Modulation profiles are calculated assuming exponential relaxation behavior. The method is general and can be applied to a wide range of solution or solid-state NMR experiments including direct-polarization (DP), or filtered cross-polariz-... 

Strong collision behaviour is nothing more than a mathematical convenience which is never attainable in practice there are two precise requirements for such behaviour, that the internal relaxation is pure exponential, equation (2.27), and that the rate of interchange between reactive and unreactive states above threshold is infinite. However, many thermal unimolecular reactions give the appearance of being strong collision processes, a fact which we can rationalise as follows. The internal relaxation is obviously not a pure exponential, but it mimics one moderately closely for this to be so, there would not have to be any bottleneck in the relaxation process, which could happen if the rotational... 

In pure water, the exponential relaxation rate is described by r =2.5 0.2ps for the 0-D stretch vibration in H O, somewhat shorter than the 3.0 0.2ps for the O-H relaxation in D O. This is because the relaxation depends on the collective motion of neighboring water molecules and the somewhat larger viscosity of the heavier isotopic form of water [120]. 

Figure 3. Relaxation pseudorates (1,17) 0 and Of (eg. 5) against c for PSL probes (legend. Figure 3b). a) Slow-mode relaxation pseudorate and fits to 0jexp(-ac ). Dashed lines fit to all measurements. Solid lines separate fits for c < c and c > c. b) Fast mode relaxation pseudorate. Dashed lines are simple exponentials. Solid lines are 1) pure exponentials (50. 87nm probes) or 2) a power law (189nm probes). Units of 0 are (pS). ...

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