Nuclear spin-lattice relaxation in a multi-spin system. A spin 1/2 particle dipolar coupled to a spin 1 and spin 3/2 particle. ¹³C spin relaxation in urea

Abstract

Density matrix theory is used to show that for S = 1/2, 1, 3/2 the decay of the I-magnetization (I = 4) in a two-spin IS dipolar coupled system is governed by the differential equation

                d<I_z>/dt= -k₁{J(ω_I-ω_S)+3J(ω_I)+6J(ω_I+ω_S)}{<I_z>-I₀}

                         -k₂{6J(ω_I+ω_S)-J(ω_I-ω_S)}{<S_Z>-S₀}

where k₁ and k₂ are positive constants dependent on S and the dipolar interaction constants. It follows that if <S₂> = 0, as arises on S- irradiation, or <S_Z> = S₀, as in paramagnetic transition-metal complexes, the decay of the I-magnetization is exponential. 

The theory is used to analyse ¹⁴N,²H dipolar-induced ¹³C spin relaxation in deuterated urea. It is shown that at low temperatures dipolar interactions dominate the relaxation whereas at high temperatures spin rotation effects become important. The relaxation times are very long; at 300 T₁ is 114s.