Intensity Sum Rule for NMR

The general NMR intensity sum rule $${\displaystyle \sum _k}{I}_{j_0^k}={\displaystyle \sum _i}{I}_{{\mathrm{ij}}_0}{\text{+2F}}_z{\mathrm{(j}}_0)$$ is proved. The sum on the left is the sum of the intensities of all the transitions to the state j₀ (ν_j0←k), the sum on the right is the sum of the intensities of all the transitions from the state j₀ (ν_i←j0), and the third term is twice the z component of the total spin of this state j₀.