Bloch-Redfield theory

Szymanski S, Gryff-Keller A M and Binsch G A 1986 Liouville space formulation of Wangsness-Bloch-Redfield theory of nuclear spin irelaxation suitable for machine computation. I. Fundamental aspects J. Magn. Reson. 68 399-432... 

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. 

An important consequence of the lineshape theory discussed above concerns the effect of the bath dynamics on the linewidths of spectral lines. We have already seen this in the discussion of Section 7.5.4, where a Gaussian power spectrum has evolved into a Lorentzian when the timescale associated with random frequency modulations became fast. Let us see how this effect appears in the context of our present discussion based on the Bloch-Redfield theory. 

Our discussion of motional narrowing has focused so far on the homogeneous spectrum described by the Bloch-Redfield theory, which is valid only when the bath is fast relative to the system timescale. In this case we could investigate the Xc 0 limit, but the opposite case, Tc —> oo, cannot be taken. In contrast, the classical... 

Despite the fact that the exact j or TZ can be formally expressed in terms of an infinite series expansion, its evaluation, however, amounts to solve the total composite system of infinite degrees of freedom. In practice, one often has to exploit weak system-bath interaction approximations and the resulting COP [Eq. (1.2)] and POP [Eq. (1.3)] of QDT become nonequivalent due to the different approximation schemes to the partial consideration of higher order contributions. It is further noticed that in many conventional used QDT, such as the generalized quantum master equation, Bloch-Redfield theory and Fokker-Planck equations, there involve not only... 

Like the exact QDT counterpart [cf. Eq. (4.6)], the POP-CS-QDT preserves both the reduced Gaussian dynamics and the effective local field pictinre for the DBO system. Its TZg [Eq. (4.11a)] has the same dissipation superoperator terms as those in ]Zf [Eq. (4.6b)]. The first and the last terms in the right-hand-side of Eq. (4.11a) for TZg or Eq. (4.6b) for are mainly responsible for the energy renormalization (or self-energy) contribution [38] and their dynamics implications are often neglected in phenomenological quantum master equations such as the optical Bloch-Redfield theory [36]. Note that the bath response function relates to the spectral density as [cf. Eq. (2.8)]... 

A more general formulation of relaxation theory, suitable for systems with scalar spin-spin couplings (J-couplings) or for systems with spin quantum numbers higher than 1/2, is known as the Wangsness, Bloch and Redfield (WBR) theory or the Redfield theory 17). In analogy with the Solomon-Bloembergen formulation, the Redfield theory is also based on the second-order perturbation approach, which in certain situations (not uncommon in... 

A more general approach is required to interpret the current experiments, Jean and co-workers have developed multilevel Redfield theory into a versatile tool for describing ultrafast spectroscopic experiments [22-25], In this approach, terms neglected at the Bloch level play an important role for example, coherence transfer terms that transform a coherence between levels i and j into a coherence between levels j and k ( /t - = 2) or between levels k and l ( f - j - 2, k-j = 2) and couplings between populations and coherences. Coherence transfer processes can often compete effectively with vibrational relaxation and dephasing processes, as shown in Fig. 4 for a single harmonic well, initially prepared in a superposition of levels 6 and 7. The lower panel shows the population of levels 6 and 7 as a function of time, whereas the upper panels display off-diagonal density matrix ele-... 

Spin relaxation phenomena are usually described by the semiclassical theory developed by Wangsness, Bloch and Redfield and known as the WBR theory or Redfield theory. The semiclassical nature of the theory implies that the spin system is treated quantum mechanically, while the remaining degrees of freedom (such as molecular rotations) are treated classically. Few years ago, Segnorile and Zamar studied the issue of quantum decoherence (loss of system phase memory) in proton NMR of nematic liquid crystals. The spin dynamics - and the decay of the free induction decay - was found to be governed by several different processes, partly of purely quantum nature. During the period under the present review, the same group reported a related work concerned with the Jeener-Broekaert experiment on liquid crystals. ... 

The theory of nuclear spin relaxation (see monographs by Slichter [4], Abragam [5] and McConnell [6] for comprehensive presentations) is usually formulated in terms of the evolution of the density operator, cr, for the spin system under consideration from some kind of a non-equilibrium state, created normally by one or more radio-frequency pulses, to thermal equilibrium, described by Using the Bloch-Wangsness-Redfield (BWR) theory, usually appropriate for the liquid state, we can write [7, 8] ... 

Several formalisms have been applied to relaxation in exchanging radicals. Principal among these are modifications of the classical Bloch equations (8, l ) and the more rigorous quantum mechanical theory of Redfield al. (8 - IJ ). When applied in their simplest form, as in the present case for K3, both approaches lead to the same result. Since the theory has been elegantly described by many authors (8 - 12 ), only those details which pertain to the particular example of K3 will be presented here. Secular terms contribute to the ESR linewidth (r) and transverse relaxation time (T ) by an amount... 

The Redfield equations that lead to Eqs (18.43) or (18.46) were obtained under the assumption that thennal environment is fast relative to the system and therefore correspond to this homogeneous limit. Consequently the absorption spectrum (18.49) obtained from these equations corresponds to a homogeneous lineshape. In contrast, the classical stochastic theory of lineshape, Section 7.5.4, can account for both limits and the transition between them. We will see in the next section that an equivalent theory can be also constructed as an extension of the Bloch equations (18.43). 

Spin relaxation phenomena in general are usually described by the semi-classical theory developed by Wangsness, Bloch and Redfield. The semi-classical nature of the theory implies that the spin system is treated quantum mechanically, while the remaining degrees of freedom (such as molecular... 

Several theories " have been proposed to explain the DeVoe-Brewer results. Most of these theories are an extension of a method developed by Redfield for modifying the Bloch equations to include the effects of local, magnetically-induced fluctuations of the transition frequency of the radiating spins. 

In the frame of the Bloch/Wangsness/Redfield (BWR) relaxation theory [2, 17], the fluctuations of the spin Hamiltonians are described with the aid of (preferably normalized) autocorrelation functions of the type... 

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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