Zeeman relaxation time

While Tp the Zeeman relaxation time, is sensitive to motions occurring at the Larmor speed in the large applied field, the dipolar relaxation time is sensitive to fluctuations occurring at the Larmor speed in the local field which is three to five orders of magnitude smaller. The former might be in the range of 5-300 MHz while the latter will be in the few kHz range. 

Ihese FC Txd results lead to two important conclusions by contrasting the data with pertinent theoretical expressions. Making use of the well-established finding that order fluctuations (OF) of the nematic director dominate the Zeeman relaxation time, Tiz, in the kHz region up to typically 1 MHz, one can simplify equations (lb), (3a) and (4a) for frequendes <10 kHz to the spedal forms ... 

Quadrupole coupling, isomer shift Quadrupole tensor, nuclear Zeeman splitting, g values, coupling constants, relaxation times... 

To study dipole-dipole relaxation, one must distinguish between homonuclear and heteronuclear (unlike) spin-1 pairs. The latter gives rise to the so-called 3/2 effect.29 For an isolated pair of like spin-i nuclei (/= 1) separated by an intemuclear distance r, the treatment of spin relaxation is identical to that for a spin-1 quadrupole system. The Zeeman spin-lattice relaxation time T1Z and spin-spin relaxation time T2 are given, respectively, by... 

For heteronuclear dipolar relaxation, the dipole-dipole coupling between two unlike spin- nuclei / and S (e.g., 13C-H pair) separated by an internuclear distance rIS is considered. The Zeeman spin-lattice (7jz) and spin-spin (T2) relaxation times for the / spin are given, respectively, by... 

The second step of the evolution towards equilibrium is the Zeeman dipole-dipole relaxation. Hartmann and Anderson estimated this time using the hypothesis that p at any time is of the form (22). As a consequence of the shortness of the dipole-dipole relaxation time we may assume that the dipole-dipole system always remains in equilibrium we are thus led to treat the evolution of the Zeeman system as the Brownian motion of a collective coordinate in the dipole-dipole heat bath. We assume that the diagonal elements of p have the form... 

It follows that the spin-spin relaxation time (exactly the Zeeman, dipole-dipole relaxation time) is not r12 but... 

Caspers relation r of Eq. (62) is in fact equal to the r12 of Eq. (64). But this author essentially looks for a closed equation for Mt without going into the details of the description of the energy exchange between the Zeeman coordinate and the dipole-dipole system. He therefore confuses r12 with the spin-spin relaxation time. 

Recapitulating, the SBM theory is based on two fundamental assumptions. The first one is that the electron relaxation (which is a motion in the electron spin space) is uncorrelated with molecular reorientation (which is a spatial motion infiuencing the dipole coupling). The second assumption is that the electron spin system is dominated hy the electronic Zeeman interaction. Other interactions lead to relaxation, which can be described in terms of the longitudinal and transverse relaxation times Tie and T g. This point will be elaborated on later. In this sense, one can call the modified Solomon Bloembergen equations a Zeeman-limit theory. The validity of both the above assumptions is questionable in many cases of practical importance. 

Let us discuss first the case in which only the first term is present. In the Solomon and Bloembeigen equations for / , (i = 1, 2) there is the cos parameter at the denominator of a Lorentzian function. Up to now cos has been taken equal to that of the free electron. However, in the presence of orbital contributions, the Zeeman splitting of the Ms levels changes its value and cos equals xs / o or (g/h)pBBo- When g is anisotropic (see Fig. 1.16), the value of cos is different from that of the free electron and is orientation dependent. The principal consequence is that another parameter (at least) is needed, i.e. the 0 angle between the metal-nucleus vector and the z direction of the g tensor (see Section 1.4). A second consequence is that the cos fluctuations in solution must be taken into account when integrating over all the orientations. Appropriate equations for nuclear relaxation have been derived for both the cases in which rotation is faster [40,41] or slower [42,43] than the electronic relaxation time. In practical cases, the deviations from the Solomon profile are within 10-20% (see for example Fig. 3.14). 

The H NMRD profile of the diferric transferrin solution (Fig. 5.5) [3] is also instructive for the case of a macromolecule containing a Fe(III) atom. The profile shows four inflections the first is ascribed to the cos dispersion, the second one to the transition from the dominant ZFS limit to the dominant Zeeman limit (see Section 3.7.1), the following increase is due to the field dependent electron relaxation time (see Section 3.7.2) and finally the coj dispersion appears. The best fit analysis provides the presence of a rhombic ZFS with D = 0.2, E/D = 1 /3, in accordance with EPR spectra [9]. The analysis suggests that two sets of electron relaxation times must be considered, in the range 0.3-1 x 10 9 s. In fact, Eqs. (3.11) and (3.12) are inadequate to describe the field dependence of the electron relaxation over the whole range of frequencies due to the presence of static ZFS [10]. 

Low-spin Fe(iii) porphyrins have been the subject of a number of studies. (638-650) The favourably short electronic spin-lattice relaxation time and appreciable anisotropic magnetic properties of low-spin Fe(iii) make it highly suited for NMR studies. Horrocks and Greenberg (638) have shown that both contact and dipolar shifts vary linearly with inverse temperature and have assessed the importance of second-order Zeeman (SOZ) effects and thermal population of excited states when evaluating the dipolar shifts in such systems. Estimation of dipolar shifts directly from g-tensor anisotropy without allowing for SOZ effects can lead to errors of up to 30% in either direction. Appreciable population of the excited orbital state(s) produces temperature dependent hyperfine splitting parameters. Such an explanation has been used to explain deviations between the measured and calculated shifts in bis-(l-methylimidazole) (641) and pyridine complexes (642) of ferriporphyrins. In the former complexes the contact shifts are considered to involve directly delocalized 7r-spin density... 

One key aspect of ENDOR spectroscopy is the nuclear relaxation time, which is generally governed by the dipolar coupling between nucleus and electron. Another key aspect is the ENDOR enhancement factor, as discussed by Geschwind [294]. The radiofrequency frequency field as experienced by the nucleus is enhanced by the ratio of the nuclear hyperfine field to the nuclear Zeeman interaction. Still another point is the selection of orientation concept introduced by Rist and Hyde [276]. In ENDOR of unordered solids, the ESR resonance condition selects molecules in a particular orientation, leading to single crystal type ENDOR. Triple resonance is also possible, irradiating simultaneously two nuclear transitions, as shown by Mobius et al. [295]. 

Figure 5.26 (a) Zeeman diagram for [Fe802(0H)i2(tacn)6]Br8 calculated with D = -0.2, E/D — 0.19 cm" and with applied field along the easy z) axis, (b) Temperature dependence of relaxation time measured in different applied fields. Reprinted with permission from Sangregorio et al., 1997 [52]. Copyright (1997) American Physical Society... 

One such analysis in the review period involves the characterization of the rotation of the methyl groups in pyridoxine (vitamin B6).49 The temperature dependencies of the 1H spin-lattice relaxation time T, and Tld (the relaxation time constant characterizing the relaxation of dipolar order, a population distribution over the Zeeman spin levels, which corresponds to a density operator component T20, he. I z - Ii. I2, to equilibrium) at three different applied field strengths and for a variety of temperatures were determined, yielding the curves in Fig. 30. The only motion that could affect... 

The recovery of the Zeeman polarization to its equilibrium value is characterized by the longitudinal, or spin-lattice relaxation time constant Tiz. Spin-lattice relaxation occurs through dissipation of the excess energy of the spins to the surrounding lattice, brought about by fluctuating fields of the appropriate frequencies, i.e., close to the Larmor frequency cdq and to 2(Oq. 

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