Single correlation-time theory

While use of multiple internal rotations does allow prediction of the Tp s at any one field better than a single correlation time theory, it still does not predict a Tp field dependence. 

These relations are together generally referred to as single correlation time theory and used to correlate the relaxation phenomena for monomeric substances in solution to their molecular motion. Nevertheless, in the case of macromolecules, the C-H vectors in the molecular structure are not thought to undergo such isotropic spherical diffusional rotation. In fact, the relaxation phenomena of macromolecules seldom follow these relations and particular modes of motion must be assumed for the internuclear vectors considering the detailed molecular structure. 

As mentioned above, the relaxation phenomena of macromolecules seldom follow the single correlation time theory dictated by eqn (36). In such cases, a wide distribution is usually introduced in the correlation time. However, as discussed elsewhere, the distribution of correlation time not only fails to explain the temperature dependencies of Ti, T2 and the NOE of the non-crystalline components observed by scalar decoupled NMR on linear polyesters and polyethylene, but also overlooks the intrinsic motion of long-chain molecules. On the contrary, the 3r theory dictated by eqn (41) was found to be very effective to describe such temperature dependencies of the relaxation parameters. Irrespectively of whether the motional mode assumed in the 3t model for the C-H vector is really true, the concept that the C-H vector in macromolecules involves plural independent diffusional motions with discretely different correlation times is very useful to explain the magnetic relaxation phenomena of macromolecules, as will be shown later. 

These data may reflect that PDES retains both ordered and disordered phases in the range of -60 to -10°C. Above 25°C, PDES takes only a disordered phase and the molecular motion is in the fast-motion region for the single correlation-time model based on BPP theory [22], because the Si Ti values increase as the temperature is increased from 25 to 125°C. That is to say, the disordered phase (I) is conformationally disordered but shows rudimentary intermolecular packing and reflect a single motional state. 

This implies liquid-like behavior of a dominating water fraction with a liquid-like single correlation time as predicted by the theory of Bloembergen, Purcell, and Pound (BPP) (22). 

According to standard NMR theory, the spin-lattice relaxation is proportional to the spectral density of the relevant spin Hamiltonian fluctuations at the transition frequencies coi. The spectral density is given by the Fourier transform of the auto-correlation fimction of the single particle fluctuations. For an exponentially decaying auto-correlation function with auto-correlation time Tc, the well-known formula for the spectral density reads as ... 

It is evident that the noncrystalline component is distributed in two phases that are associated with the same Tic but different T2C values. What does this mean In order to understand this phenomenon we have to refer to the theory of the relaxation reviewed in Section 2.2 [43]. Provided the internuclear vector between carbon and hydrogen nuclei involves only a single motion, that is, if each term of the correlation function of the dipole-dipole interaction between 13C and H spins evolves exponentially with one correlation time ic (relaxation time of... 

The C Ti values were plotted against inverse absolute-temperature in Fig. 9. 17 together with those of the amorphous region for MQPESL (melt-quenched polyethylene (single C-labeled)) which was melted at 150°C and quenched to -70°C and PESL (polyethylene single C-labeled) which was dissolved at 130°C in xylene at a concentration of 0.03% and crystallized. The Ti curves for these three samples have the minimum at different temperatures. The Ti minimum for polyethylene adsorbed on the surface of silica gel appears at the highest temperature compared with other samples. According to the BPP theory, the correlation time for molecular motion at the Ti minimum corresponds to the resonance frequency. 

The correlation time is 8, where 8 is the single parameter to measure the deviation from the white noise situation. The robustness of white-noise-induced phenomena, at least for small correlation time, are verified by this perturbation expansion method. Slightly different results were obtained by Sancho et al. (1982). They studied the qualitative properties of the stationary distribution as a function of the intensity and correlation time of the noise. Two results of their experiments on an electric circuit with a digital noise generator were not in accordance with the white noise limit theory (i) for higher noise intensity the stationary distribution might be bimodal even for rather small correlation time (ii) the location of the maximum of the stationary distribution depends on the noise characteristics. 

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