Double-quantum cross-polarization NMR in solids

Abstract

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 $S=1$ 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-\textonehalf{} formalism. The spin $S=1$ Hamiltonian can always be factored into two commuting parts (independent thermodynamic reservoirs), one of which behaves as a fictitious spin \textonehalf{} 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.