where $n$ is the density of bosons per magnetic ion, given by $n=M/M_{sat}$.

The chemical potential $$\mu=g\mu_{B}(H-H_c)$$ determining the number of bosons in the ground state characterizes an additional direct contribution to the magnon energy from the external field $H$.

Bosons condense when the renormalized effective chemical potential is zero [@3; @7; @30] (Fig. 5): $$\mu_{eff}=\mu-2{v_{0}}{n_{}}=0;$$ the factor of two comes from exchange.

![\[fig5\] Effective chemical potential as a function of temperature for fermions, bosons and classical particles. For decreasing temperatures, $\mu$ for bosons reaches the zero limit at the temperature $T_\mathrm{critical}$. As $T$ becomes lower than $T_\mathrm{critical}$, $\mu$ is pinned at zero [@31]](figure5){width="6cm"}

At this “point of condensation”, where the gap is closed, the applied magnetic field is responsible for driving the system to the BEC phase in which bosons are created at ($H_{c1},T_{c1}$). At $H_{c1}$ the magnetization acquires the value $M_{c}(T)$ and $n_{c}(T)= M_{c}(T)/M_{sat}$ gives the number of bosons created at the transition. Using (6) and (7), the low-field boundary of the phase diagram $H$ $vs$. $T$ can be written as [@3; @20; @21; @25; @26] $$H_{c}(T)-H_{c}(0)= [2v_{0}/g\mu_{B}]n_{c}(T).$$

On the other hand, the density of created bosons is given by Bose-Einstein statistics $$\rho(T)= (1/V) \sum_{{\bf k}}[1/[exp(\epsilon_{k}/T)-1)]$$

That is, 