Torsional spectroscopy from nuclear magnetization time evolution in the rotating frame

It was recently observed that in a nonequilibrium situation the nuclear magnetization in the rotating frame oscillates during the time interval of several T₂. Spectroscopic information about torsional-dipolar energies (SPOTS) can be obtained by Fourier transforming this oscillatory evolution. In this paper the time evolution of the density matrix due to the off-diagonal part of the dipolar interaction is calculated using the perturbation theory. It is shown that, to second order, the total energy of the Zeeman, secular torsional-dipolar and nonsecular torsional-dipolar systems is a constant, while the energy of each system oscillates. The frequencies of these oscillations which are observed in SPOTS spectroscopy are determined by selection rules of the matrix elements of the off-diagonal dipolar interaction. The spectrum line shape is calculated for a 3-spin-{} system with a tunneling split ground state which appears in theory as a parameter. The theory is then applied to simulate the observed proton SPOTS spectrum of CH₃CD₂I. Agreement is satisfactory.