text: Free evolution of nuclear spins is governed by a Hamiltonian (), which in the case of liquid-state nuclear magnetic resonance may be split into two major terms. The first term () corresponds to the Zeeman interaction between spins and the external magnetic field, which includes chemical shift (). The second term () corresponds to the indirect spin-spin, or J-coupling, interaction.
  , where:
  , and
  Here the summation is taken over the whole system of coupled spins;  denotes the reduced Planck constant;  denotes the gyromagnetic ratio of spin a;  denotes the isotropic part of the chemical shift for the a-th spin;  denotes the spin operator of the a-th spin;  is the external magnetic field experienced by all considered spins, and;  is the J-coupling constant between spins a and b.
  Importantly, the relative strength of  and  (and therefore the spin dynamics behavior of such a system) depends on the magnetic field. For example, in conventional NMR,  is typically larger than 1 T, so the Larmor frequency  of H exceeds tens of MHz. This is much larger than -coupling values which are typically Hz to hundreds of Hz. In this limit,  is a perturbation to . In contrast, at nanotesla fields, Larmor frequencies can be much smaller than -couplings, and  dominates.