The general slow-motion theory

We now come back to the simplest possible nuclear spin system, containing only one kind of nuclei 7, hyperfine-coupled to electron spin S. In the Solomon-Bloembergen-Morgan theory, both spins constitute the spin system with the unperturbed Hamiltonian containing the two Zeeman interactions. The dipole-dipole interaction and the interactions leading to the electron spin relaxation constitute the perturbation, treated by means of the Redfield theory. In this section, we deal with a situation where the electron spin is allowed to be so strongly coupled to the other degrees of freedom that the Redfield treatment of the combined IS spin system is not possible. In Section V, we will be faced with a situation where the electron spin is in 

Consider a general system described by the Hamiltonian of Eq. (5), where = Huif) describes the interaction between the spin system (7) and its environment (the lattice, L). The interaction is characterized by a strength parameter co/i- When deriving the WBR (or the Redfield relaxation theory), the time-dependence of the density operator is expressed as a kind of power expansion in Huif) or (17-20). The first (linear) term in the expansion vanishes if the ensemble average of HiL(t) is zero. If oo/z,Tc 5c 1, where the correlation time, t, describes the decay rate of the time correlation functions of Huif), the expansion is convergent and it is sufficient to retain the first non-zero term corresponding to oo/l. This leads to the Redfield equation of motion as stated in Eq. (18) or (19). In the other limit, 1 the expan- 

The stochastic Liouville equation, in the form relevant for the ESR line shape calculation, can be written in a form reminiscent of the Redfield equation in the superoperator formulation, Eq. (19) (70-73)  

The basic idea of the slow-motion theory is to treat the electron spin as a part of the lattice and limit the spin part of the problem to the nuclear spin rather than the IS system. The difficult part of the problem is to treat, in an appropriate way, the combined lattice, now containing the classical degrees of freedom (such as rotation in condensed matter) as well as quantized degrees of freedom (such as the electron Zeeman interaction). The Liouville superoperator formalism is very well suited for treating this type of problems. 

The whole system, nuclear spin and the composite lattice, is described by the equation of motion  

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B. Paramagnetic cross-correlation and interference phenomena TV. The general (slow-motion) theory... 

In this approach, the diffusion constant, Di, is related to the corresponding characteristic time, x, describing the distortions of the normal coordinate, Westlund et al. (85) used the framework of the general slow-motion theory to incorporate the classical vibrational dynamics of the ZFS tensor, governed by the Smoluchowski equation with a harmonic oscillator potential. They introduced an appropriate Liouville superoperator ... 

The quantum alternative for the description of the vibrational degrees of freedom has been commented by Westlund et al. (85). The comments indicate that, to get a reasonable description of the field-dependent electron spin relaxation caused by the quantum vibrations, one needs to consider the first as well as the second order coupling between the spin and the vibrational modes in the ZFS interaction, and to take into account the lifetime of a vibrational state, Tw, as well as the time constant,T2V, associated with a width of vibrational transitions. A model of nuclear spin relaxation, including the electron spin subsystem coupled to a quantum vibrational bath, has been proposed (7d5). The contributions of the T2V and Tw vibrational relaxation (associated with the linear and the quadratic term in the Taylor expansion of the ZFS tensor, respectively) to the electron spin relaxation was considered. The description of the electron spin dynamics was included in the calculations of the PRE by the SBM approach, as well as in the framework of the general slow-motion theory, with appropriate modifications. The theoretical predictions were compared once again with the experimental PRE values for the Ni(H20)g complex in aqueous solution. This work can be treated as a quantum-mechanical counterpart of the classical approach presented in the paper by Kruk and Kowalewski (161). 

The slow-motion theory describes the electron relaxation processes implicitly, through a combined effect of static and transient ZFS, and reorienta-tional and pseudorotational dynamics. This is necessary under very general conditions, but simpler descriptions, appropriate in certain physical limits, can also be useful. In this chapter, we review some work of this type. 

Fig. 8. NMRD profiles calculated for a given set of parameters and different angles 0 between the principal axes of the dipole-dipole interaction and the static ZFS. (a) S = 1 (reproduced with permission from Nilsson, T Svoboda, J. Westlund, P.O. Kowalewski, J. J. Chem. Phys. 1998, 109, 6364-6375. Copyright 1998 American Institute of Physics) (b) S = 3/2. (Reprinted from J. Magn. Reson. vol. 146, Nilsson, T Kowalewski, J., Slow-motion theory of nuclear spin relaxation in paramagnetic low-symmetry complexes A generalization to high electron spin , pp. 345-358, Copyright 2000, with permission from Elsevier.)...

At first glance a review of Onsager s theory might appear to have little or no relevance to the topics of this chapter. This, however, is not the case. This is because Onsager s theory, as extended to include memory by Mori [5] is the most general slow variable theory of irreversible motion. Thus, our examination of Onsager s work exposes limitations inherent in all slow variable models. Especially, it shows that the limitations of the Kramers-type models for reactions merely reflect the macroscopic scope of the general theory of irreversible processes [1,3-5]. 

It should be noted that the Nekhoroshev s theory is not limited to two-dimensional systems, but rather it holds in general dimensions. Therefore, in the system with many degrees of freedom, the Nekhoroshev s theorem may explain sticky motions. This is also the case with the KAM theory. As given before, the statement of the KAM theorem is not limited to the Hamiltonian with few degrees of freedom. In Section IV, we will discuss to what extend these perturbation theories have capability to predict the slow motions in manydimensional systems. 

The T2 values for the network polymers of gel are generally very short and the motion of main chains especially lies in the slow motion region of the BPP theory of NMR relaxation [12]. In this region, the T2 value is shown to be almost constant. Therefore, the Ti measurements are performed in general to get information on the motion of polymer gel systems. 

In contrast, the particle s fast variable equation of motion is nonclassical and a potential V S-,x) that is very different from VF(S x) drives the dynamics. (Similar problems occur in the general theory of irreversible processes. There they are resolved by assuming slow variable relaxation [1-5], see Appendix.)... 

It is instructive to compare the high-temperature limits (78. Ill) and (91.III) of the two classical (semiclassical) or quantum-mechanical formulations of the rate theory, based on a collisional and a statistical approach, respectively. In the general case, in which the reaction coordinate is non-separable, these equations are not identical, since they correspond to the extreme conditions of a very fast and a very slow motion along the reaction path, as expressed by the opposed inequalities (72.Ill) and (82.Ill), or, equivalently, by (74. Ill) and (87.Ill), respectively. 

The conventional, and very convenient, index to describe the random motion associated with thermal processes is the correlation time, r. This index measures the time scale over which noticeable motion occurs. In the limit of fast motion, i.e., short correlation times, such as occur in normal motionally averaged liquids, the well known theory of Bloembergen, Purcell and Pound (BPP) allows calculation of the correlation time when a minimum is observed in a plot of relaxation time (inverse) temperature. However, the motions relevant to the region of a glass-to-rubber transition are definitely not of the fast or motionally averaged variety, so that BPP-type theories are not applicable. Recently, Lee and Tang developed an analytical theory for the slow orientational dynamic behavior of anisotropic ESR hyperfine and fine-structure centers. The theory holds for slow correlation times and is therefore applicable to the onset of polymer chain motions. Lee s theory was generalized to enable calculation of slow motion orientational correlation times from resolved NMR quadrupole spectra, as reported by Lee and Shet and it has now been expressed in terms of resolved NMR chemical shift anisotropy. It is this latter formulation of Lee s theory that shall be used to analyze our experimental results in what follows. The results of the theory are summarized below for the case of axially symmetric chemical shift anisotropy. 

This model has been found to be useful in studying the slow motion of viscoelastic fluids around particles, droplets and bubbles, and in predicting the directions of secondary flows in rotating systems. Furthermore it is very helpful in providing a framework for the presentation of kinetic theory results obtained by perturbation theories. The retarded motion expansion is not, however, useful for most industrial flow problems, in which large velocity gradients or rapid time responses are generally encountered. 

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