Zeeman energy levels of spin systems in a quantizing field B⃑₀ (assumed to point upwards). The double arrows between the levels indicate the (allowed) zero-, single-, and double-quantum transitions relevant for spin relaxation. The spectral densities (ω_k) of the fluctuations inducing these transitions are indicated. In the schemes, all gyromagnetic ratios are assumed to be positive. (a) Dipolar coupled ‘like’ spin pairs with quantum numbers I=½, S=½, magnetic quantum numbers m_I, m_S and a common gyromagnetic ratio γ. The spin eigenstates are symbolized by kets |↑↑〉, |↑↓〉, |↓↑〉 and |↓↓〉 for the diverse combinations of spin-up and spin-down states relative to the vector B⃑₀. The Zeeman eigenenergies are , where ω=γB₀ is the angular Larmor frequency. The angular transition frequencies are ω_k=kω for zero- (k=0), single- (k=1) and double- (k=2) quantum transitions. Typical examples are protons in organic materials. (b) Pairs of dipolar coupled ‘unlike’ spins ½ having different gyromagnetic ratios γ_I≠γ_S and Larmor frequencies ω_I=γ_IB₀ and ω_S=γ_SB₀. The Zeeman eigenenergies are . The angular transition frequencies are ω_k=|ΔE_k|/ħ for zero- (ω₀=|ω_S−ω_I|), single- (ω₁=ω_I) and double- (ω₂=ω_S+ω_I) quantum transitions. Typical examples are protons with spins I coupled to unpaired electrons with spin S. (c) (Single) spins 1 subjected to quadrupole interaction in the high-field limit. Quadrupolar coupled spin-1 particles have three Zeeman eigenstates with the kets |m=1〉, |m=0〉 and |m=−1〉 and energies E_m=−mħω. The angular Larmor frequency is ω=γB₀ as before. The angular transition frequencies are ω_k=|ΔE_k|/ħ=kω for single- (k=1) and double- (k=2) quantum transitions. Typical examples are deuterons.