Analytical treatment of proton double-quantum NMR intensity build-up: multi-spin couplings and the flip-flop term

A modified Anderson-Weiss approximation for describing double quantum (DQ) NMR experiments in systems with many I = 1/2 spins is proposed, taking inter-spin flip-flop processes into special consideration. In this way, an analytical result is derived for multi-spin systems for the first time. It is shown that in the initial stages of DQ intensity build-up, the probability of flip-flop processes in DQ experiments is half as large as in analogous Hahn-echo or free-induction-decay experiments. Their influence on the experimentally observed DQ NMR signal becomes dominant at times t > (9/2)1/2 T2eff ≈ 2.12T2eff, where T2eff is the effective spin-spin relaxation time measured by the Hahn echo. Calculations and a comparison with spin-dynamics simulations of small spin systems up to 8 spins reveal a satisfactory agreement.