Dipolar relaxation, theory

This simple relaxation theory becomes invalid, however, if motional anisotropy, or internal motions, or both, are involved. Then, the rotational correlation-time in Eq. 30 is an effective correlation-time, containing contributions from reorientation about the principal axes of the rotational-diffusion tensor. In order to separate these contributions, a physical model to describe the manner by which a molecule tumbles is required. Complete expressions for intramolecular, dipolar relaxation-rates for the three classes of spherical, axially symmetric, and asymmetric top molecules have been evaluated by Werbelow and Grant, in order to incorporate into the relaxation theory the appropriate rotational-diffusion model developed by Woess-ner. Methyl internal motion has been treated in a few instances, by using the equations of Woessner and coworkers to describe internal rotation superimposed on the overall, molecular tumbling. Nevertheless, if motional anisotropy is present, it is wiser not to attempt a quantitative determination of interproton distances from measured, proton relaxation-rates, although semiquantitative conclusions are probably justified by neglecting motional anisotropy, as will be seen in the following Section. 

Relaxation theory of nuclear spin systems is well documented in several books18-24 and review articles.4-25-26 Therefore, the theory presented in this chapter is limited to a summary of some of the basic concepts crucial for understanding the material in the following sections. Furthermore, the discussion will be focused on dipolar relaxation, which is known to be the dominant relaxation mechanism in most molecules of chemical interest. For a detailed treatment of other mechanisms, the reader is referred to appropriate review articles.4-18"26... 

P. J. W. Debye, Polar Molecules (Dover, New York, reprint of 1929 edition) presents the fundamental theory with stunning clarity. See also, e.g., H. Frohlich, "Theory of dielectrics Dielectric constant and dielectric loss," in Monographs on the Physics and Chemistry of Materials Series, 2nd ed. (Clarendon, Oxford University Press, Oxford, June 1987). Here I have taken the zero-frequency response and multiplied it by the frequency dependence of the simplest dipolar relaxation. I have also put a> = if and taken the sign to follow the convention for poles consistent with the form of derivation of the general Lifshitz formula. This last detail is of no practical importance because in the summation Jf over frequencies fn only the first, n = 0, term counts. The relaxation time r is such that permanent-dipole response is dead by fi anyway. The permanent-dipole response is derived in many standard texts. 

Based on random fields, relaxation theory R2 decreases as molecular tumbling gets faster and more effectively averages the residual dipolar broadening. Both in the fast and slow motion limits, R2 is proportional to tc. ... 

In order to progress further in the interpretation of dipolar relaxation behaviour we must develop a molecular model in still more detail. Many different models, depending on the type of material concerned, have been proposed anji these have led to a large number of theoretical treatments. We shall confine ourselves to two theories, which are particularly relevant to polymers, and provide a suitable basis for general discussion of the main features encountered. First, in this section, we will consider the theory which has been central to the understanding of the temperature dependence of nearly all reaction rate processes thermal activation over a potential-energy barrier. 

Jeener and Broekaert introduced, in 1%7, a three-pulse B,(r) sequence to measure the relaxation time Tm of the dipolar order of / = 1 spin systems in the presence of a conventional high Zeeman field, Bq, which is based on the decay time of the so-called Jeener echo . It was later extended by Spiess and Kemp-Harper and Wimperis to study in a similar way the quadrupolar order in / a 1 systems. The appearance of a Jeener echo depends upon the existence of interactions that are not averaged out by molecular motions on the considered time scale. The method has become of great importance in recent relaxation studies, in particular of liquid crystals because, in standard spin relaxation theories, it provides a power l means to separate and analyse the spectral densities / v) and /2) j. i4,is,2025 ggg... 

According to EM theory, the dielectric loss may be contributed by the processes like natural resonance, Debye dipolar relaxation and electron polarization relaxation, etc. In the Debye dipolar relaxation regime, the relative complex permittivity can be expressed as ... 

Brown, M.E 1984. Theory of spin-lattice relaxation in hpid bUayers and biological membranes. Dipolar relaxation. J. Chem. Phys. 80 2808-2831. 

The direct NMR method for determining translational difiFusion constants in liquid crystals was described previously. The indirect NMR methods involve measurements of spin-lattice relaxation times (Ti,Ti ),Tip) [7.45]. Prom their temperature and frequency dependences, it is hoped to gain information on the self-diflPusion. In favorable cases, where detailed theories of spin relaxation exist, difiFusion constants may be calculated. Such theories, in principle, can be developed [7.16] for translational difiFusion. However, until recently, only a relaxation theory of translational difiFusion in isotropic hquids or cubic solids was available [7.66-7.68]. This has been used to obtain the difiFusion correlation times in nematic and smectic phases [7.69-7.71]. Further, an average translational difiFusion constant may be estimated if the mean square displacement is known. However, accurate determination of the difiFusion correlation times is possible in liquid crystals provided that a proper theory of translational difiFusion is available for liquid crystals, and the contribution of this difiFusion to the overall relaxation rate is known. In practice, all of the other relaxation mechanisms must first be identified and their contributions subtracted from the observed spin relaxation rate so as to isolate the contribution from translational difiFusion. This often requires careful measurements of proton Ti over a very wide frequency range [7.72]. For spin - nuclei, dipolar interactions may be modulated by intramolecular (e.g., collective motion, reorientation) and/or intermolecular (e.g., self-diffusion) processes. Because the intramolecular (Ti ) and intermolecular... 

The theory of Torrey [7.66, 7.67, 7.75] treats relaxation via dipolar interactions between spins on different molecules in an isotropic liquid. An extension of this relaxation theory to the case of translational diffusion in nematic [7.76], smectic A [7.77], and smectic B phases [7.78] has recently been developed. Zumer and Vilfan assumed an elongated cylindrical shape for the molecule with a particular distribution of spins on the cylindrical surface and a perfect ordered ((P2) = 1) system. Their expression for Ti due to self-diffusion in nematics is given by [7.76]... 

Figure 7. (a) The real part of the susceptibility e as a function of frequency a for HAT6. The top curve corresponds to r=362 K, the middle curve to T=384.2 K, and the lower curve to T=311.7 K. Solid lines are data, crosses are fits using simple Debye theory, (b) The frequency dependence of the imaginary part (bottom curve) of the dielectric constant for HAT6. Solid lines are data and crosses are theory fits using simple Debye theory with a weakly temperature dependent relaxation time of =5 x 10" s [38], Top curve r=384.2 K, middle curve T= 362 K, lower curve T=311.7 K (in (a) and (b) the old nomenclature K and D(, instead of Cr and Col is used), (c) The temperature dependence of the real part e(T) as a function of T at two frequencies +, 10 Hz , 10 Hz. Note that despite the typical dipolar relaxation law observed in (a) and (b), the temperature dependence only exhibits small variations which are related to structural order. 

Fig. 4. (A) EPR titration experiment demonstrating electron-electron dipolar relaxation. The highest field line of enzyme-bound Mn - in a sample containing O.S mM glutamine synthetase, 10 mM glutamate, and 0.2 mAf Mn + recorded at 35 GHz and 3°C. (B) a, y-CrATP was titrated into this solution at concentrations from 0 to 2 mM and the peak-to-peak signal amplitude replotted. The Mn signal is diminished in height by enzyme-bound CrATP without broadening as predicted by the Leigh theory.

Distances between the tight Mn site and the Cr of various complexes were determined using EPR and NMR data. From the diminution elicited by paramagnetic Cr on the EPR transitions of bound Mn +, a distance was calculated using the Leigh theory because the results suggested a dipolar relaxation phenomenon similar to that observed with glutamine synthetase. The data that were evaluated were at a Mn PPase ratio of more than 0.26, where 95% of the Mn was in E-Mn complex. 

From SCRP spectra one can always identify the sign of the exchange or dipolar interaction by direct exammation of the phase of the polarization. Often it is possible to quantify the absolute magnitude of D or J by computer simulation. The shape of SCRP spectra are very sensitive to dynamics, so temperature and viscosity dependencies are infonnative when knowledge of relaxation rates of competition between RPM and SCRP mechanisms is desired. Much use of SCRP theory has been made in the field of photosynthesis, where stnicture/fiinction relationships in reaction centres have been connected to their spin physics in considerable detail [, Mj. 

Finally, it can be noted that there also exist dipolar-dipolar crosscorrelation rates which involve two different dipolar interactions. These quantities may play a role, for instance, in the carbon-13 longitudinal relaxation of a CH2 grouping.11,12 Due to the complexity of the relevant theory and to their marginal effect under proton decoupling conditions, they will be disregarded in the following. 

Relaxation processes are probably the most important of the interactions between electric fields and matter. Debye [6] extended the Langevin theory of dipole orientation in a constant field to the case of a varying field. He showed that the Boltzmann factor of the Langevin theory becomes a time-dependent weighting factor. When a steady electric field is applied to a dielectric the distortion polarization, PDisior, will be established very quickly - we can say instantaneously compared with time intervals of interest. But the remaining dipolar part of the polarization (orientation polarization, Porient) takes time to reach its equilibrium value. When the polarization becomes complex, the permittivity must also become complex, as shown by Eq. (5) ... 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

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