Redfield condition

The assumption of a single electron spin and a single T2 holds usually for S = 1/2 and for S > 1 in certain limits. Let us assume that the instantaneous distortions of the solvation sphere of the ion result in a transient ZFS and that the time-dependence of the transient ZFS can be described by the pseudorotation model, with the magnitude of the transient ZFS equal to At and the correlation time t . The simple picture of electron relaxation for S = 1 is valid if the Redfield condition (Att <5c 1) applies. Under the extreme narrowing conditions ((Os v 1), the longitudinal and transverse electron spin relaxation rates are equal to each other and to the low-field limit rate Tgo, occurring in Eqs. (14) and (15). The low field-limit rate is then given by (27,86) ... 

High-symmetry systems discussed in the previous section are scarce. In systems with lower symmetry and S > 1, we must expect a static ZFS, which can have a profound effect on both the electron spin relaxation and the PRE. The treatment of the PRE in systems with static ZFS requires caution. The reorientational motion of the complex modulates the ZFS which can cause the breach of both the Redfield condition for the electron spin relaxation and the assumption that electron spin relaxation and molecular reorientation are statistically independent (the decomposition approximation). One limit where the decomposition approximation is valid is for slowly rotating systems. 

The theory presented above is valid as long as the Redfield condition is satisfied. In the case of susceptibility-induced relaxation, this condition may be written as Acotd < 3.4, where Aco is the equatorial magnetic field at the surface of the particle (expressed in angular frequency units). The Redfield condition defines thus the distinction between small and large particles. 

Bernath (1960) correlation. The transient CHF was also tested by Schrock et al. (1966) in a water velocity of 1 ft/sec (0.3 m/s). They also reported transient CHF values that were higher than those under steady-state conditions. Borishanskiy and Fokin (1969) tested transient CHF in flow boiling of water at atmospheric pressure. They found that the transient CHF in water was approximately the same as the steady-state value. On the basis of Bernath s correlation (Bernath, 1960) and Schrock et al. s (1966) data, Redfield (1965) suggested a transient CHF correlation as follows ... 

Because phosphate is released during remineralization with no decrease in O2, the A02/AP0 produced via denitrification should be lower than that predicted by the aerobic respiration of Redfield-Richards planktonic detritus. To reach the suboxic conditions required for denitrification requires the aerobic respiration of a considerable amount of POM and, hence, release of phosphate. Thus, A02/AP0 ratios less than 138 are most likely to be found in waters with high phosphate concentrations. The prevalence of denitrification in deep waters is suggested by their low (14.7) average N-to-P ratio (Figure 8.3). Areas where the OMZ are pronoimced, such as coastal upwelling areas, have particularly low N-to-P ratios as shown in Figure 10.7. 

Models for the outer-sphere PRE, allowing for faster rotational motion, have been developed, in analogy with the inner sphere approaches discussed in the Section V.C. The outer-sphere counterpart of the work by Kruk et al. 123) was discussed in the same paper. In the limit of very low magnetic field, the expressions for the outer-sphere PRE for slowly rotating systems 96,144) were found to remain valid for an arbitrary rotational correlation time Tr. New, closed-form expressions were developed for outer-sphere relaxation in the high-field limit. The Redfield description of the electron spin relaxation in terms of spectral densities incorporated into that approach, was valid as long as the conditions A t j 1 and 1 were fulfilled. The validity... 

The importance (or not) of the distinction between N and P as a model currency is closely related to the concept of the Redfield ratio (Redfield et al. 1963). Because many models impHcidy or expHcitly enforce fixed elemental ratios, the distinction between N and P hmitation is often of fittle importance in models. In most cases a fixed ratio (e.g., Redfield ) model will behave almost identically in terms of primary production, plankton abundance, and carbon fluxes, whether N or P is specified to be the primary fimiting nutrient (the exception is when initial or boundary conditions are drawn from observations in which deviations from the Redfield N P ratio are present). 

Under optimal conditions, diatoms assimilate Si and N in approximately equimolar ratios (Brzezinski, 1985), essentiaUy extending the Redfield ratio (Redfield et al, 1963) to include Si. However, unlike N, many diatom species exhibit... 

Analysis of the composition of algae is difficult because it is not easily possible to separate the algae from other colloidal or suspended material. A further complication is that metal ions in the water column are complexed to various degrees by natural organic ligands. More data are needed to draw conclusions whether, and under what conditions, Redfield ratios are maintained. 

The Redfield equation, Eq. (10.155) has resulted from combining a weak system-bath coupling approximation, a timescale separation assumption, and the energy state representation. Equivalent time evolution equations valid under similar weak coupling and timescale separation conditions can be obtained in other representations. In particular, the position space representation cr(r, r ) and the phase space representation obtained from it by the Wigner transform... 

A third requirement is less absolute but still provide a useful consistency check for models that reduce to simple Brownian motion in the absence of external potentials The dissipation should be invariant to translation (e.g. the resulting friction coefficient should not depend on position). Although it can be validated only in representations that depend explicitly on the position coordinate, it can be shown that Redfield-type time evolution described in such (position or phase space) representations indeed satisfies this requirement under the required conditions. 

The main shortcoming of the Redfield time evolution is that it does not necessarily conserve the positivity property. In fact, it has been shown by Lindblad that a linear Markovian time evolution that satisfies this condition has to be of the form... 

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