Rotating-Frame Nuclear-Double-Resonance Dynamics: Dipolar Fluctuation Spectrum in CaF2

Abstract

A series of rotating-frame nuclear-double-resonance experiments is reported in which the NMR of the rare isotope $^{43}\mathrm{Ca}$ (natural abundance 0.13%) in Ca${\mathrm{F}}_{2}$ is detected via the observable resonance signal of the abundant $^{19}\mathrm{F}$ nuclei. The technique consists of (1) cooling the $^{19}\mathrm{F}$ dipolar-interaction energy reservoir by means of adiabatic demagnetization in the rotating frame, (2) coupling the rare $^{43}\mathrm{Ca}$ to this cooled reservoir by means of a rotating magnetic field at or near the $^{43}\mathrm{Ca}$ nuclear Larmor frequency, and (3) examining the final state of this reservoir by measuring the "dipolar" free-induction signal following a \textonequarter{}\ensuremath{\pi} rf pulse at the $^{19}\mathrm{F}$ Larmor frequency. With this basic format, results have been obtained and are reported here for (a) the relative heat capacities of the $^{43}\mathrm{Ca}$ rotating-frame Zeeman Hamiltonian and the $^{19}\mathrm{F}$ dipolar Hamiltonian terms, (b) the cross-relaxation time for the $^{43}\mathrm{Ca}$-$^{19}\mathrm{F}$ energy-transfer process as a function of rotating-frame effective field strength and orientation, (c) transient oscillations of the $^{19}\mathrm{F}$ dipolar spin temperature upon application of an rf pulse at the $^{43}\mathrm{Ca}$ resonance frequency, (d) the transverse (${T}_{2}$) relaxation process of the $^{43}\mathrm{Ca}$ nuclei, and (e) the spin-lattice relaxation time (${T}_{1}$) of the $^{43}\mathrm{Ca}$ nuclei. Theoretical expressions for results (a) and (b) are obtained using the well-known thermodynamic model to describe the state of the $^{19}\mathrm{F}$ and $^{43}\mathrm{Ca}$ spin systems. The $^{19}\mathrm{F}$-$^{43}\mathrm{Ca}$ dipolar coupling is treated as a perturbation to obtain a simple "Fermi Golden rule" expression for the cross-relaxation rate. This model, first applied to nuclear double resonance by Hartmann and Hahn, is combined here with the experimentally determined fluctuation spectrum of the $^{19}\mathrm{F}$-$^{43}\mathrm{Ca}$ coupling operator to yield calculated cross-relaxation rates in excellent agreement with the experimental results. The above-mentioned fluctuation spectrum is found to be a very nearly exponential function of the rotating-frame $^{43}\mathrm{Ca}$ Zeeman splitting over a range of cross-relaxation rates spanning nearly three orders of magnitude for two distinct orientations of the applied field relative to the crystal axes, and to be essentially independent of the orientation of the rotating-frame effective field. This rather surprising spectral form is crucial to the successful calculation of the results we find, and also leads to satisfactory agreement between theory and experiment for results (c) and (d). The transient oscillation phenomena (c) are the rotating-frame analog of the pulsed dc-field experiments carried out by Strombotne and Hahn. A density-matrix perturbation-expansion technique similar to that used by those authors is employed here to explain our results. The form of the $^{43}\mathrm{Ca}$ transverse decay process (d) is calculated by means of the Anderson-Weiss-model theory; the exponential decay time is closely approximated by a calculated result using the exponential fluctuation spectrum noted above, and the initial region of nonexponential behavior predicted by this model theory is clearly evident in our results. $^{43}\mathrm{Ca}$ ${T}_{1}$ values (e) were obtained at 300 and 355\ifmmode^\circ\else\textdegree\fi{}K with a variety of field orientations in the (100) plane of the crystal axes. The ${T}_{1}$ process is found to be isotropic and to have a temperature dependence suggesting that it is quadrupolar in origin. The temperature dependence of ${T}_{1}$ is in reasonable accord with the Van Kranendonk theory.