Redfield regime

The problem of a strong coupling between the electron spin system and the classical degrees of freedom (rotation in the first place), as expressed in Eq. (29), can also be treated in another way. This class of methods to circumvent the limitations of the Redfield regime is the subject of this section. 

The considerations above apply to fast dynamic processes in the sense that the amplitude of the modulation of the resonance frequency Amq (induced by modulation of the local field) multiplied with the correlation time Tc is much smaller than unity. This Redfield regime [27] is usually attained in solutions with low viscosity [2], but may also apply to small-amplitude libration in solids [28]. For slower reorientation in solutions with high viscosity or in soft matter above the glass transition temperature (slow tumbling), spectral lineshapes are directly influenced by exchange between different orientations of the molecule (Section 4.1). Relaxation times in solids outside the Redfield regime carmot be predicted from first principles except for a few crystalline systems with very simple structure and few defects [29]. In such systems, qualitative or semi-quantitative analysis of relaxation data can still provide some information on dynamics. 

The Cu(ii) ions were also considered as possible RE agents. For the cases of Fe(iii) and Cu(ii) it is important to verify the applicabilify of the Redfield regime in the whole measured temperature range. The relaxation of these ions is typically slower than the one of Dy(iii) and deviations from the Redfield regime are possible at low temperatures. 

The last assumption is very fundamental. It results in time-independent transition probabilities and makes a clean theory possible. It requires that the product of the time scale of the decay time for the tcf (called the correlation time and denoted x ) and the strength of the perturbation (in angular frequency units) has to be much smaller than unity (17-20). This range is sometimes denoted as the Redfield limit or the perturbation regime. 

Westlund developed also a theory for PRE in the ZFS-dominated limit for S = 1, which included a stringent Redfield-limit approach to the electron spin relaxation in this regime (118). Equations (35) and (38) were used as the starting point also in this case. Again, the correlation function in the integrand of Eq. (38) was expressed as a product of a rotational part and the spin part. However, since it is in this case appropriate to work in the principal frame of the static ZFS, the rotational part becomes proportional to exp(—t/3tb) (if Tfl is the correlation time for reorientation of rank two spherical harmonics, then 3t is the correlation time for rank one spherical... 

The relaxation equation derived so far for electrons and nuclei share a common assumption usually called the perturbation regime or Redfield limit [54]. The... 

Second, we should keep in mind that between the two extreme limits discussed above there exists a regime of intermediate behavior, where dephasing/decoherence and molecular response occur on comparable timescales. In this case the scattering process may exhibit partial coherence. Detailed description of such situations requires treatment of optical response within a formalism that explicitly includes thermal interactions between the system and its environment. In Section 18.5 we will address these issues using the Bloch-Redfield theory of Section 10.5.2. 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

Zheng et al derived a Redfield-type theory for magnetic field gradient-induced relaxation of spins undergoing restricted diffusion. The theory covered both transverse and longitudinal relaxation and the approach was valid in all diffusion regimes. The theory can be useful for MRI in structured media and was illustrated with experiments on gaseous, polarized He. 

Fig. 45a, b. Frequency dependence of the deuteron spin-lattice relaxation time of perdeuterated PEG confined in 10-nm pores of solid PHEMA at 80 °C (a) and in bulk melts (b) [95, 185]. The dispersion of the confined polymers verifies the law Ti (X M° ft)° at high frequencies as predicted for limit (II)de of the tube/reptation model (see Table 1). The low-frequency plateau observed with the confined polymers indicates that the correlation function implies components decaying more slowly than the magnetization relaxation curves, so that the Bloch/Wangsness/Redfield relaxation theory [2] is no longer valid in this regime. The plateau value corresponds to the transverse relaxation time, T2, for deuterons extrapolated from the high-field value measured at 9.4 T... 

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