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

Singlet-states |S〉=(|αβ〉-|βα〉)/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 IzSz and then by applying a coherent radio-frequency (RF) irradiation with a carrier frequency ωrf = (ΩI + ΩS)/2 that lies half-way between the chemical shifts of the two spins involved. The life-times TS can be much longer than the spin-lattice relaxation time T1 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 Δω = ωrf − (ΩI + ΩS)/2 or an insufficient amplitude of the RF irradiation that fails to fulfill the condition ω1 ≫ ΔΩ = (ΩI − ΩS). 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 Δω and on the difference ΔΩ of the chemical shifts of the two spins are illustrated by comparison with numerical simulations.