plane (Fig. 2). The first thermodynamic property used to study the transitions is the magnetization, one of the most fundamental quantities observed in experiments, followed by specific heat and magnetocaloric effect. More properties such as magnetostriction, thermal transport, EPR, inelastic neutron scattering, and other techniques are used to identify the induced magnetic phase as BEC.

4.1. Magnetization
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The boundary of the induced phase diagram of DTN can be experimentally determined by curves of magnetization $vs$. temperature at fixed $H$, with a cusp-like minimum at $T_{c}$, or by $H_{c}$ at the inflexion in scans of magnetization at fixed temperature $T$. These critical points originate from the competition of two behaviors of the magnetization, which correspond to the number of magnons: above $T_{c}$, magnetization is determined by the number of thermally excited magnons, which decreases with decreasing $T$; below $T_{c}$ the number of condensed magnons increases with decreasing temperature, overcoming the decrease of thermally excited magnons. As a result of this competition on the origin of the magnons, $M$ has a minimum at $T_{c}$, with the value $M_{c}$ (Fig. 6a). For the same reason, when the phase boundary is crossed by sweeping $H$ at a fixed temperature, the condensation occurs at 