The relaxation equations - R_i'(θ)

The relaxation values of the set R'(θ) include the spin-lattice, spin-spin, and cross-relaxation rates at all field strengths. These rates are respectively (Abragam, 1961)

where J(ω) is the power spectral density function and R_ex is the relaxation due to chemical exchange. The dipolar and CSA constants are defined in SI units as

$${\frac{{1}}{{4}}} \left(\vphantom{\frac{\mu_0}{4\pi}}\right. {\frac{{\mu_0}}{{4\pi}}} \left.\vphantom{\frac{\mu_0}{4\pi}}\right)^{2}_{} {\frac{{(\gamma_{\scriptscriptstyle H}\gamma_{\scriptscriptstyle X}\hbar)^2}}{{\langle r^6 \rangle}}}$$ (7.4)

$${\frac{{(\omega_X \Delta\sigma)^2}}{{3}}}$$ (7.5)

where μ₀ is the permeability of free space, γ_H and γ_X are the gyromagnetic ratios of the H and X spins respectively, $\hbar$ is Plank's constant divided by 2π, r is the bond length, and Δσ is the chemical shift anisotropy measured in ppm. The cross-relaxation rate σ_NOE is related to the steady state NOE by the equation

$${\frac{{\gamma_{\scriptscriptstyle H}}}{{\gamma_{\scriptscriptstyle X}}}} {\frac{{\sigma_{\scriptscriptstyle \mathrm{NOE}}(\theta)}}{{\mathrm{R}_1(\theta)}}}$$ (7.6)