Interaction with nuclear spins
So far, the interaction of the electron spin with the external magnetic field yields an EPR spectrum with one single line which may be broadened due to the g anisotropy. The information content of the EPR spectrum is considerably increased if nuclei with a spin quantum number I > 0 are found in the environment of the electron spin, as e.g. protons (I = 1/2) or nitrogen atoms (I = 1). The intrinsic spin angular momentum of these nuclei is associated with a magnetic moment. In addition to the external magnetic field B0 the electron spin will now experience the local magnetic fields built up by the nuclei close to the unpaired electron. The interaction of the electron spin with these nuclear magnetic moments is called hyperfine interaction (HFI).
In analogy to the electron spin the nuclear spin I is quantized in a magnetic field resulting in the nuclear Zeeman splitting of the nuclear spin states. The nuclear Zeeman interaction (NZI) is expressed by
$$ \mathcal{H}_n ={-g}_n\frac{\beta_n}{h}\mathbf{I B}_0 \qquad (9) $$
Here gn is the dimensionless nuclear g-factor, a constant depending on the type of nucleus, and βn the nuclear magneton (J·T-1). For a given nuclear quantum number I a splitting into (2I+1) energy levels is observed, each level being characterized by a nuclear magnetic spin quantum number mI = I, I-1,..., -I. The NZI is much smaller than the EZI (e.g. the ratio gβe/gnβn = 660 for an unpaired electron with g = 2 and a proton with gn = 5.58) and the small anisotropy is usually neglected in EPR. The local magnetic fields set up by the nuclear spins through the HFI at the electron spin depend on the orientation of the nuclear magnetic moment with respect to the external magnetic field1. This results in a splitting of each electron spin state into (2I+1) levels.
1 Note that the hyperfine fields exerted by the electron spin at the nucleus are much stronger than the nuclear hyperfine fields. They may easily exceed the NZI. In this case the nuclear spin is not quantized along the external magnetic field B0, but along an effective field set up mainly by the electron spin through the HFI.