Relaxation pathways

Another example of the role played by a nonradiative relaxation pathway is found in the photochemistry of octatetraene. Here, the fluorescence lifetime is found to decrease dramatically with increasing temperature [175]. This can be assigned to the opening up of an efficient nonradiative pathway back to the ground state [6]. In recent years, nonradiative relaxation pathways have been frequently implicated in organic photochemistry, and a number of articles published on this subject [4-8]. 

Thus, identification of all pairwise, interproton relaxation-contribution terms, py (in s ), for a molecule by factorization from the experimentally measured / , values can provide a unique method for calculating interproton distances, which are readily related to molecular structure and conformation. When the concept of pairwise additivity of the relaxation contributions seems to break down, as with a complex molecule having many interconnecting, relaxation pathways, there are reliable separation techniques, such as deuterium substitution in key positions, and a combination of nonselective and selective relaxation-rates, that may be used to distinguish between pairwise, dipolar interactions. Moreover, with the development of the Fourier-transform technique, and the availability of highly sophisticated, n.m.r. spectrometers, it has become possible to measure, routinely, nonselective and selective relaxation-rates of any resonance that can be clearly resolved in a n.m.r. spectrum. 

A single-quantum transition involves one spin only, whereas the zero- and doublequantum transitions involve two spins at the same time. The zero- and double-quantum transitions give rise to cross-relaxation pathways, which provide an efficient mechanism for dipole-dipole relaxation. 

Fig. 4.—The Three Distinct Proton-Proton Relaxation Pathways in a Six-membered Ring in the C, Conformation. [Vicinal-Gauche (vg),vicinal-tranr (vt), and 1,3-diaxial (aa). Geminal relaxation-pathways are not shown.]...

Since the equilibrium state has been disturbed, the system tries to restore equilibrium. For this it can use as the predominant relaxation pathways the double-quantum process (in fast-tumbling, smaller molecules), leading to a positive nOe, or the zero-quantum process 1% (in slower-tumbling macromolecules), leading to a negative nOe. 

Let us first consider the situation in which is the predominant relaxation pathway. The population states immediately after the pulse are represented in Fig. 4.2b. The system tries to reestablish the equilibrium state by transferring some population x from the state, which had a population of — (d/2) to the aa state, which had a population of N + d/ 2). This transfer of population (x), shown in Fig. 4.2d (represented by a half circle), gives level 4 a somewhat reduced population [A — (d/2) — x] and level 1 a correspondingly increased population [A — (d/2) + x]. 

Hence the population difference between the lower and upper energy states of the two I transitions becomes d + x, as compared to the original difference of d at equilibrium. Thus an intensification of the lines for nucleus 1 will be observed by an amount corresponding to this increased difference x. This is the positive nuclear Overhauser effect that is encountered in small, rapidly tumbling molecules, in which Wj is the predominant relaxation pathway. 

In large molecules that tumble slowly, the predominant relaxation pathway is via This is shown schematically in Fig. 4.2c. A part of the population x is now transferred from the /3q state to the a)3 state. This causes an increase in the population of the upper level of one 1 transition (/], level 3) and a decrease in the lower population level of the other I transition (4, level 2). As a result, the population difference between the lower and upper levels of each I transition is reduced to d — x (i.e., level 1 — level 3, or level 2 — level 4, becomes d — x). The reduction in population difference by x as compared to the equilibrium situation (Fig. 4.2a)... 

While the final magnitude of nOe depends, as indicated earlier, on the relaxation pathways Wi, W, and W), the initial rate of buildup of nOe (transient nOe) depends only on the rate of cross-relaxation between the nuclei, and this can provide valuable information about the distance between the nuclei (r). This rate of buildup can be proportional to r" , where r is the distance between the nuclei. Thus, if the proportionality constant is determined, we can calculate an approximate distance between the two nuclei. The best results are obtained in rigid molecules when the nuclei are less than 3 A apart. If only direct nOe s are involved in a two-spin... 

The positive nOe observed in small molecules in nonviscous solution is mainly due to double-quantum relaxation, whereas the negative nOe observed for macromolecules in viscous solution is due to the predominance of the zero-quantum 1% cross-relaxation pathway. 

If only single-quantum transitions (h, I2, S], and S ) were active as relaxation pathways, saturating S would not affect the intensity of I in other words, there will be no nOe at I due to S. This is fairly easy to understand with reference to Fig. 4.2. After saturation of S, the fMjpula-tion difference between levels 1 and 3 and that between levels 2 and 4 will be the same as at thermal equilibrium. At this point or relaxation processes act as the predominant relaxation pathways to restore somewhat the equilibrium population difference between levels 2 and 3 and between levels 1 and 4 leading to a negative or positive nOe respectively. 

Transient nOe represents the rate of nOe buildup. The nOe effect (so-called equilibrium value) itself depends only on the competing balance between various complex relaxation pathways. But the initial rate at which the nOe grows (so-called transient nOe) depends only on the rate of cross-relaxation t, between the relevant dipolarly coupled nuclei, which in turn depends on their internuclear distance (r). 

To measure distances in the wider temperature range, this procedure was modified. Relaxation of the carotenoid occurs through several different mechanisms including the dipolar-dipolar interaction. Assuming that kAA is the rate constant of the dipolar-dipolar interaction and K=(k,l + k2 + k3 +. ..) is the sum of the rate constants of all other relaxation pathways, we can extract kAA from the following equation ... 

One other (very rarely encountered) situation is that of the stabilised free radical. It is possible for certain conjugated multi-ring heterocyclic compounds to support and stabilise a delocalised, free electron in their pi clouds. Such a free electron again provides an extremely efficient relaxation pathway for all... 

A major limitation of CW double resonance methods is the sensitivity of the intensities of the transitions to the relative rates of spin relaxation processes. For that reason the peak intensities often convey little quantitative information about the numbers of spins involved and, in extreme cases, may be undetectable. This limitation can be especially severe for liquid samples where several relaxation pathways may have about the same rates. The situation is somewhat better in solids, especially at low temperatures, where some pathways are effectively frozen out. Fortunately, fewer limitations occur when pulsed radio and microwave fields are employed. In that case one can better adapt the excitation and detection timing to the rates of relaxation that are intrinsic to the sample.50 There are now several versions of pulsed ENDOR and other double resonance methods. Some of these methods also make it possible to separate in the time domain overlapping transitions that have different relaxation behavior, thereby improving the resolution of the spectrum. 

Fig. 4 Relaxation pathways between quadrupole-perturbed Zeeman levels of / = 3/2 nuclear spin. Reprinted from [50]...

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