Redfield relaxation theory

A good introductory treatment of the density operator formalism and two-dimensional NMR spectroscopy, nice presentation of Redfield relaxation theory. 

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-... 

Redfield limit, and the values for the CH2 protons of his- N,N-diethyldithiocarbamato)iron(iii) iodide, Fe(dtc)2l, a compound for which Te r- When z, rotational reorientation dominates the nuclear relaxation and the Redfield theory can account for the experimental results. When Te Ti values do not increase with Bq as current theory predicts, and non-Redfield relaxation theory (33) has to be employed. By assuming that the spacings of the electron-nuclear spin energy levels are not dominated by Bq but depend on the value of the zero-field splitting parameter, the frequency dependence of the Tj values can be explained. Doddrell et al. (35) have examined the variable temperature and variable field nuclear spin-lattice relaxation times for the protons in Cu(acac)2 and Ru(acac)3. These complexes were chosen since, in the former complex, rotational reorientation appears to be the dominant time-dependent process (36) whereas in the latter complex other time-dependent effects, possibly dynamic Jahn-Teller effects, may be operative. Again current theory will account for the observed Ty values when rotational reorientation dominates the electron and nuclear spin relaxation processes but is inadequate in other situations. More recent studies (37) on the temperature dependence of Ty values of protons of metal acetylacetonate complexes have led to somewhat different conclusions. If rotational reorientation dominates the nuclear and/or electron spin relaxation processes, then a plot of ln( Ty ) against T should be linear with slope Er/R, where r is the activation energy for rotational reorientation. This was found to be the case for Cu, Cr, and Fe complexes with Er 9-2kJ mol" However, for V, Mn, and... 

A.G. Redfield, Relaxation Theory, Density Matrix Formalism. Encyclopedia of Nuclear Magnetic Resonance (Wiley, Chichester, 1996)... 

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... 

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... 

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] ... 

Redfield A G 1996 Relaxation theory density matrix formulation Encyclopedia of Nuclear Magnetic Resonance ed D M Grant and R K Harris (Chichester Wiley) pp 4085-92... 

Interactions between molecules are taken into account phenomenologically by adding linear damping terms to the right-hand side of Equation 28, using some first-order relaxation theory for the density matrix such as Redfield theory.12,13 jhis gives for the complete equation of motion of the system... 

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. 

A possibility to overcome this limitation of the above conical-intersection models, at least in a quahtative manner, is to consider anhar-monic couplings of the active degrees of freedom of the conical intersection with a large manifold of spectroscopically inactive vibrational modes. The effect of such a couphng with an environment has been investigated for the pyrazine model in the weak-coupling limit (Redfield theory) in Ref. 19. The simplest ansatz for the system-bath interaction, which is widely employed in quantum relaxation theory assumes a coupling term which is bilinear in the system and bath operators... 

Let us apply the Redfield theory to a deuteron with its quadrupole moment experiencing a fluctuating electric field gradient arising from anisotropic molecular motions in liquids. When the static average of quadrupole interaction is nonzero, i.e. 0, it can be included in the static Hamiltonian Hq. The density operator matrix for a deuteron spin is of the dimension 3x3 and the corresponding Redfield relaxation supermatrix has the dimension V- x 3. When only nuclear spin-lattice relaxation is considered, the spin precession term in Equation [22] is set to zero and the diagonal elements 2, 3)... 

The function G(t) depends on the system under study but also on the characteristics of the fluctuating physical quantity which is considered. The choice of a correct analytical form for the G(t) function is still a very difficult task. Redfield s comment on this problem (7, p. 28) remains correct, more than 15 years later, when he said that the calculation of spectral densities or correlation functions is "the most difficult problem in any relaxation theory."... 

In this section, Redfield s theory of spin relaxation in the presence of a fluctuating transverse field is formulated in such a way that it can be included in a simulation using stochastic theory which utilises the wavefunction of the chemical system. Although the method of simulating relaxation as a disaete event on f is plausible (as described in the previous section), it is nevertheless important to compare this method with more usual methods to ensure no errors have been intfoduced. 

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... 

We begm tliis section by looking at the Solomon equations, which are the simplest fomuilation of the essential aspects of relaxation as studied by NMR spectroscopy of today. A more general Redfield theory is introduced in the next section, followed by the discussion of the coimections between the relaxation and molecular motions and of physical mechanisms behind the nuclear relaxation. 

A. Highly symmetric systems and the Redfield theory for electron spin relaxation... 

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 ... 

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... 

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