Electron spin relaxation, theory

Electronic relaxation is a crucial and difficult issue in the analysis of proton relaxivity data. The difficulty resides, on the one hand, in the lack of a theory valid in all real conditions, and on the other hand in the technical problems of independent and direct determination of electronic relaxation parameters. Proton relaxivity is essentially influenced by the longitudinal electron spin relaxation time, Tle, of Gd111. This decay is too fast to be assessed by commonly available techniques, though very recently Tlc values have been directly measured.74 Nevertheless,... 

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

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... 

A. Highly Symmetric Systems and the Redfield Theory for Electron Spin Relaxation... 

Using the case of S = 5/2 as an illustrative example, he demonstrated that it was possible to derive closed-form analytical expressions for the PRE of the form of the SBM equations times (1 + correction term). For typical parameter values, the effect of the correction term was to increase the prediction of the SBM theory by 5-7%. A similar approach was also applied to the S = 7/2 system, such as Gd(III) (101), where the correction terms could be larger. For that case, the estimations of the electron spin relaxations rates, obtained in the solution for PRE, were also used for simulations of ESR lineshapes. 

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... 

Bertini and co-workers 119) and Kruk et al. 96) formulated a theory of electron spin relaxation in slowly-rotating systems valid for arbitrary relation between the static ZFS and the Zeeman interaction. The unperturbed, static Hamiltonian was allowed to contain both these interactions. Such an unperturbed Hamiltonian, Hq, depends on the relative orientation of the molecule-fixed P frame and the laboratory frame. For cylindrically symmetric ZFS, we need only one angle, p, to specify the orientation of the two frames. The eigenstates of Hq(P) were used to define the basis set in which the relaxation superoperator Rzpsi ) expressed. The superoperator M, the projection vectors and the electron-spin spectral densities cf. Eqs. (62-64)), all become dependent on the angle p. The expression in Eq. (61) needs to be modified in two ways first, we need to include the crossterms electron-spin spectral densities, and These terms can be... 

Sharp and Lohr proposed recently a somewhat different point of view on the relation between the electron spin relaxation and the PRE (126). They pointed out that the electron spin relaxation phenomena taking a nonequilibrium ensemble of electron spins (or a perturbed electron spin density operator) back to equilibrium, described in Eqs. (53) and (59) in terms of relaxation superoperators of the Redfield theory, are not really relevant for the PRE. In an NMR experiment, the electron spin density operator remains at, or very close to, thermal equilibrium. The pertinent electron spin relaxation involves instead the thermal decay of time correlation functions such as those given in Eq. (56). The authors show that the decay of the Gr(T) (r denotes the electron spin vector components) is composed of a sum of contributions... 

Bayburt and Sharp 143) formulated a low-field theory (i.e. a theory for the case of ZFS dominating over the electron spin interaction) for the outer-sphere relaxation, treating also the electron spin relaxation in the simplified manner expressed by Eq. (52). That model predicted only a weak dependence of the PRE on the magnitude of the static ZFS and its application to the cases of high static ZFS is problematic. 

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

Kruk and Kowalewski combined the theory allowing for the radial distribution with their Redfield-limit description of the electron spin relaxation (147). Including the g(r) in the theory led to a more complicated form of the function f(x) of Eq. (69), which becomes dependent on the g(r), as well as on the propagator P(ro, 0/ r, t). The rest of the theory remains unchanged with respect to the presentation in sections VII.A-VII.B. The propagator was computed using the Smoluchowski equation ... 

The quantum alternative for the description of the vibrational degrees of freedom has been commented by Westlund et al. (85). The comments indicate that, to get a reasonable description of the field-dependent electron spin relaxation caused by the quantum vibrations, one needs to consider the first as well as the second order coupling between the spin and the vibrational modes in the ZFS interaction, and to take into account the lifetime of a vibrational state, Tw, as well as the time constant,T2V, associated with a width of vibrational transitions. A model of nuclear spin relaxation, including the electron spin subsystem coupled to a quantum vibrational bath, has been proposed (7d5). The contributions of the T2V and Tw vibrational relaxation (associated with the linear and the quadratic term in the Taylor expansion of the ZFS tensor, respectively) to the electron spin relaxation was considered. The description of the electron spin dynamics was included in the calculations of the PRE by the SBM approach, as well as in the framework of the general slow-motion theory, with appropriate modifications. The theoretical predictions were compared once again with the experimental PRE values for the Ni(H20)g complex in aqueous solution. This work can be treated as a quantum-mechanical counterpart of the classical approach presented in the paper by Kruk and Kowalewski (161). 

Following the solid-state approach, equations have been derived [8,9] also for the electron spin relaxation of 5 = V2 ions in solution determined by the aforementioned processes. Instead of phonons, collisions with solvent should be taken into consideration, whose correlation time is usually in the range 10"11 to 10 12 s. However, there is no satisfactory theory that unifies relaxation in the solid state and in solution. The reason for this is that the solid state theory was developed for low temperatures, while solution theories were developed for room temperature. The phonon description is a powerful one when phonons are few. By increasing temperature, the treatment becomes cumbersome, and it is more convenient to use stochastic theory (see Section 3.2) instead of analyzing the countless vibrational transitions that become active. 

The electronic relaxation rates, as described by Bloembergen, Morgan and McLachlan [12], also depend on the magnetic field. For Gd(III) complexes they are usually interpreted in terms of zero field splitting interactions (ZFS). The electronic relaxation rates can be described by the Eqs. (14-16), often called as the Bloembergen-Morgan theory of paramagnetic electron spin relaxation ... 

The combination of the modified Solomon-Bloembergen Eqs. (7-11) with the equations for electron spin relaxation (14-16) constitutes a complete theory to relate the observed paramagnetic relaxation rate enhancement to the microscopic properties, and it is generally referred as to Solomon-Bloembergen-Mor-gan (SBM) theory. Detailed discussions of the relaxation theory have been published [13,14]. 

Edwards, Lusis, and Sienko have recently reported an ESR study (60) of frozen lithium-methylamine solutions which suggests the existence of a compound tetramethylaminelithium(O), Li(CH3NH2)4, bearing all the traits (60) of a highly expanded metal lying extremely close to the metal-nonmetal transition. Specifically, both the nuclear-spin and electron-spin relaxation characteristics of the compound, although nominally metallic, cannot be described in terms of the conventional theories of conduction ESR (6,15, 71) and NMR in pure metals (60, 96, 169). 

B. Gd(III) EPR. The literature on Gd(III)EPR in solution is not extensive. Hudson and Lewis (27) have presented a theory for the electron spin relaxation of S ions (e.g., gadoliniumdll)) in solution. These authors assumed that the dominant line broadening mechanism for an ion is provided by the modulation of the zero-field splitting by a process with a characteristic time X. The transverse relaxation rate is given by... 

---