Double-quantum cross-polarization NMR in solids

Double-quantum NMR is a useful way to obtain spectra of quadrupolar nuclei (${}_{}{}^2{}_{}{}^{}\mathrm{D}$, ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}$,...) in solids. This allows measurements of the chemical shifts for these nuclear spins. The theory of Hartmann-Hahn cross polarization between $I=\frac{1}{2}$ and such $S=1\mathrm{}$ spins is discussed. Particular attention is drawn to the cross polarization of the double-quantum transition. The thermodynamics and the dynamics of the process are evoked in detail using a fictitious spin-½ formalism. The spin $S=1\mathrm{}$ Hamiltonian can always be factored into two commuting parts (independent thermodynamic reservoirs), one of which behaves as a fictitious spin ½ which is cross polarized with the $I=\frac{1}{2}$ spins. Modified Hartmann-Hahn conditions emerge from the theory, and the dependence of cross-polarization times $T_{\mathrm{IS}}$ on rf intensity and frequency for spin locking and adiabatic demagnetization in the rotating-frame experiments are calculated. Measurements on the ${}_{}{}^1{}_{}{}^{}\mathrm{H}$-${}_{}{}^2{}_{}{}^{}\mathrm{D}$ double resonance in dilute solid benzene-$d_1$ are reported, verifying the predictions and indicating that cross polarization provides a sensitive means of detecting the ${}_{}{}^2{}_{}{}^{}\mathrm{D}$ double-quantum transition. Values are reported for the thermodynamic parameters and cross-polarization times as a consequence. Three possible versions of double-resonance detection of double-quantum spectra are possible—direct detection of the cross-polarized double-quantum decay, indirect detection of the frequency spectrum following Hartmann and Hahn, and indirect detection of the free-induction decay following Mansfield and Grannell.