the anisotropic $z$ axis for all magnetic ions. The parameters for DTN and its crystal symmetry then yield magnetic properties that can be studied within this picture.

In DTN the high single-ion anisotropy (energy gap) is responsible for the splitting of the $S = 1$ spin triplet of the Ni$^{2+}$ in a $S^{z} = 0$ ground state and an $S^{z} = \pm1$ excited doublet considerably higher in energy. The excited doublet can hop to a neighboring doublet via the transverse component of the magnetic exchange interaction $J$, resulting in delocalized states analogous to the delocalized electronic states in crystals. At zero field the small exchange interaction between the spins is not enough to induce long-range ordering, even at zero temperature. The application of a magnetic field $H$ leads to a splitting of the doublet, which brings down the component parallel to the field linearly with $H$ until the gap between the excited and ground states closes at $H=H_{c}$. At this point magnetic excitations, i. e., bosons, are created (Fig. 4a).

The more realistic view of the energy levels that takes the interaction into account is shown in Fig. 4b. The exchange interaction produces a dispersion of the energy levels, and 