The relaxation equations - Ri'(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)

\begin{subequations}\begin{align}\mathrm{R}_1(\theta) &= d \Big( J(\omega_H - \o...
...omega_H + \omega_X) - J(\omega_H - \omega_X) \Big),\end{align}\end{subequations}

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

d = $\displaystyle {\frac{{1}}{{4}}}$$\displaystyle \left(\vphantom{\frac{\mu_0}{4\pi}}\right.$$\displaystyle {\frac{{\mu_0}}{{4\pi}}}$$\displaystyle \left.\vphantom{\frac{\mu_0}{4\pi}}\right)^{2}_{}$$\displaystyle {\frac{{(\gamma_{\scriptscriptstyle H}\gamma_{\scriptscriptstyle X}\hbar)^2}}{{\langle r^6 \rangle}}}$, (7.4)
c = $\displaystyle {\frac{{(\omega_X \Delta\sigma)^2}}{{3}}}$, (7.5)

where μ0 is the permeability of free space, γH and γX 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 σNOE is related to the steady state NOE by the equation

NOE(θ) = 1 + $\displaystyle {\frac{{\gamma_{\scriptscriptstyle H}}}{{\gamma_{\scriptscriptstyle X}}}}$$\displaystyle {\frac{{\sigma_{\scriptscriptstyle \mathrm{NOE}}(\theta)}}{{\mathrm{R}_1(\theta)}}}$. (7.6)

The relax user manual (PDF), created 2016-10-28.