Lifetimes of the singlet-states under coherent off-resonance irradiation in NMR spectroscopy

Singlet-states |S=(|alphabeta> - |betaalpha>)/sq.rt.2 can be excited in pairs of coupled spins I and S, first by preparing either a non-vanishing zero-quantum coherence I(+)S(-) or a state of longitudinal two-spin order I(z)S(z) and then by applying a coherent radio-frequency (RF) irradiation with a carrier frequency omega(rf) = (Omega(I) + Omega(S))/2 that lies half-way between the chemical shifts of the two spins involved. The life-times T(S) can be much longer than the spin-lattice relaxation time T(1) of longitudinal magnetization, but singlet-states are ultimately relaxed, not only by dipolar interactions between the active spins or with the external spins, but also as a result of a non-vanishing offset Deltaomega = omega(rf) - (Omega(I) + Omega(S))/2 or an insufficient amplitude of the RF irradiation that fails to fulfill the condition omega(1) > DeltaOmega = (Omega(I) - Omega(S)). In this work, the effect of off-resonance irradiation is explored and an approximate formula for the effective relaxation rate of the singlet population is provided on the basis of perturbation theory. The qualitative features of the dependence of the relaxation rate of the singlet population on the offset Deltaomega and on the difference DeltaOmega of the chemical shifts of the two spins are illustrated by comparison with numerical simulations.