ALBERT

Notes: Expressions are derived for the intermolecular contribution to the nuclear-spin relaxation rate in solutions containing dissolved paramagnetic ions with spin S≥1. The calculation assumes that the electron-spin Hamiltonian is dominated by a large axial zero-field splitting, and it accounts for effects of Zeeman interactions to first order. The expressions are used to analyze proton-spin relaxation of the acetone solvent in solutions of tris-(acetylacetonato)Mn(iii)/ acetone. The main objective was to measure electron-spin relaxation times of Mn(iii), which in this complex is a high-spin, d4 ion with integer spin S=2. Spin-lattice relaxation measurements were conducted over a range of magnetic field strengths (0.28–1.1 T) where the zero-field splitting is large compared to the Zeeman energy. Electron-spin relaxation times of Mn(iii) were found to be 8±2 ps, with little dependence on temperature over the range 215–303 K and on magnetic field strength up to 1.1 T. Use of the assumption that Zeeman splittings dominate zero-field splittings (Solomon–Bloembergen–Morgan theory) resulted in computed electron-spin relaxation times that are too short by a factor of 3–4.

Notes: Expressions are derived describing nuclear spin relaxation in paramagnetic salt solutions under conditions where the electron spin Hamiltonian is dominated by a uniaxial quadratic zero-field splitting (zfs) interaction. In this situation, the electron spin vector is quantized along molecular axes rather than along the external magnetic field. By expressing the time dependence of the electron spin operators, written in the molecular coordinate frame, in the Heisenberg representation and then transforming these expressions to the laboratory coordinate system, simple closed form expressions for the paramagnetic nuclear relaxation increment have been derived. Electron–nuclear dipole–dipole and scalar relaxation mechanisms are considered. The resulting expressions parallel those of Solomon–Bloembergen–Morgan theory, but are valid in the zfs limit rather than the Zeeman limit. Nuclear relaxation rates in the zfs and Zeeman limits exhibit characteristic qualitative differences, some of which have been noted in earlier studies. Of particular note is the fact that the scalar contribution to T−11p is much larger in the zfs than in the Zeeman limit. In most circumstances, T−11p=T−12p in the zfs limit, while in the Zeeman limit, scalar relaxation usually contributes significantly only to T−12p. A vector model of this phenomenon is suggested. The results are valid for arbitrary values of the electron spin quantum number but they assume that electron spin relaxation is in the Redfield limit, i.e., that the correlation times of the coupling between electron spin and the lattice be short on the time scale of electron spin relaxation. This condition is probably satisfied widely when the static zfs is large.

Notes: The NMR (nuclear magnetic resonance) paramagnetic relaxation enhancement (NMR-PRE) that is produced by paramagnetic solutes in solution has been investigated theoretically with respect to the influence of zero field splitting (zfs) interactions in the electron spin Hamiltonian, in particular with respect to the effects of anisotropy in the zfs tensor. These effects are a physical consequence of the influence of the zfs on the motion of the electron spin vector S¯. When the zfs energy is large compared to the Zeeman energy (the zfs limit), the precessional motion of S¯ is quantized in the molecule-fixed coordinate system that diagonalizes the zfs tensor. The uniaxial portion of the zfs tensor influences the NMR-PRE primarily through its influence on the quantization axes of S¯; the characteristic behavior of the NMR-PRE under the influence of a uniaxial zfs has been described in detail previously. Anisotropy in the zfs tensor induces oscillatory motion in Sz. This motion has a profound influence on the NMR-PRE, the major part of which normally arises from low frequency components of the local magnetic field that are associated with Sz, rather than from the rapidly precessing local fields that are associated with the transverse components S±. For this reason, the NMR-PRE is a sensitive function of zfs anisotropy, which acts to lower the NMR-PRE below the value that occurs in the uniaxial situation. The magnitude of this effect depends on the ratio (E/D) of the anisotropic and uniaxial zfs parameters, on the reduced dipolar correlation time, and on the location of the nuclear spin in the molecular coordinate frame. A second physical effect of zfs anisotropy on the NMR-PRE arises from a resonance between the electron spin precessional motion in the transverse plane with the precessional motion that is perpendicular to the transverse plane (the latter due to zfs anisotropy). Resonance of these motions, which occurs spin energy levels crossings, gives rise to low frequency transverse components of S¯ which result in a resonant increase in the NMR-PRE within a restricted range of E/D ratios.

Notes: Effects due to the nonuniaxial part of the zero field splitting (ZFS) tensor on NMR relaxation enhancements produced by paramagnetic species in solution (the NMR PRE) has been studied theoretically and experimentally in the ZFS limit, i.e., in the limit where the ZFS energy is large compared to the Zeeman energy. In the ZFS limit, the precessional motion of the electron spin is quantized with respect to molecule-fixed coordinate axes. The uniaxial part of the ZFS tensor induces precessional motion in the transverse (x,y) components of the electron spin vector S, and x,y anisotropy in the ZFS tensor (i.e., a nonzero ZFS parameter E) induces precessional motion in the z component of S. The NMR-PRE phenomenon is particularly sensitive to the motion of Sz and hence also to ZFS anisotropy in the xy plane. Mathematical expressions have been derived which describe the motion of the spin vector evolving under the influence of a general rhombic ZFS Hamiltonian and the influence of this motion on the NMR PRE in the ZFS limit. It is shown that oscillations in Sz occur at the transition frequencies of the S spin system; the frequencies and amplitudes of the precessional components of Sz can be calculated by diagonalizing the general ZFS Hamiltonian. These motions and their consequences with respect to the behavior of the NMR PRE are described in detail for the S=2 spin system. A parametrization of NMR-PRE data is proposed which gives a clear criterion for the conditions under which rhombic parts of the ZFS tensor significantly affect the relaxation enhancements produced by an S=2 spin system. This criterion is of considerable practical importance for the analysis of NMR-PRE data, since it defines conditions under which data may be analyzed without the need for independent experimental information concerning the magnitude of the ZFS tensor.

Notes: The influence of zero field splitting (zfs) interactions on the magnetic field dispersion profile of the nuclear magnetic resonance–paramagnetic relaxation (NMR–PRE) (i.e., the enhancement of nuclear magnetic relaxation rates that is produced by paramagnetic solute species in solution) has been explored systematically for S=1, 3/2, 2, and 5/2 spin systems using recently developed theory. To facilitate comparison of results for different spin values, the theory was expressed in a reduced form with Larmor frequencies in units of ωD (the uniaxial zfs parameter D in rad s−1), and correlation times and spin relaxation times in units of ωD−1. For S=1, the functional form of the profile can be described in terms of five types of qualitative features. Two of these are characteristic of Zeeman-limit [Solomon, Bloembergen, and Morgan (SBM)] theory and result from the magnetic field dependence of the spin energy level splittings. The remaining three have no analog in Zeeman-limit theory and arise from a change in the quantization axis of the electron spin precessional motion which, in the zfs limit, lies along molecule-fixed coordinate axes, and, in the Zeeman limit, lies along the external field direction.The reduced field dispersion profiles for the integer spin systems S=1 and S=2 were found to be very similar to each other, the principal difference being that the midfield positions of the requantization features (types 2, 3, and 4) are shifted for S=2 relative to S=1, the magnitude and sign of the shift depending on the position of the nuclear spin in the molecular coordinate frame. For half-integer spins, the dispersion profiles exhibit, in addition to the five features characteristic of integer spins, a sixth type of feature, which is centered somewhat to low field of ωSτc=1, where τc is the dipolar correlation time. The type-6 feature results from field-dependent level splitting of the mS=±1/2 Kramers doublet. It is present when ωDτc≥1. These theoretical predictions have been examined by means of reinterpretations of the NMR–PRE data for tris-(acetylacetonato)–metal complexes of V(III) (S=1), Cr(III) (S=3/2), Mo(III) (S=3/2), Mn(III) (S=2), and Fe(III) (S=5/2). As predicted, type-6 features are absent for the integer spin complexes, for which the T1 field dispersion profiles are nearly field independent.The experimental profiles were successfully simulated quantitatively by the generalized theory, but not by Zeeman-limit theory. For the half-integer spin systems, the predicted zfs-related type-6 features appear to be present in the profiles, particularly for Mo(acac)3, for which the data deviate significantly from the Zeeman-limit profile in a manner that is explained by the generalized theory.