1.1.6 The External Field. Zeeman Energy

Until now, we have treated the case of magnetic body not subject to external field. Therefore, all the energy terms introduced in the previous sections can be regarded as parts of the Helmholtz free energy functional. When the external field is considered, it is convenient to introduce the Gibbs free energy functional. In this respect, the additional term (see Eq. (1.17)) related to the external field $\mathbf{H}_$a, is itself a long-range contribution too. In fact, it can be seen as the potential energy of a continuous magnetic moments distribution [15] subject to external field $\mathbf{H}_$a:

$$G_ =-\int_\Omega \mu_0 \textbf{M}\cdot\mathbf{H}_ \quad.$$ (1.51)

This energy term is referred in literature to as Zeeman energy.