at the low-temperature limit [@1; @7; @9; @32] and letting the chemical potential be $\mu_{eff}$: $$\rho(T)\approx
\zeta(3/2)\left(\frac{mk_{B}T}{2\pi\hbar^{2}}\right)^{3/2},$$ since the density of bosons per magnetic ion is $n_{c}(T)$= $\rho(T)V$, where $V$ is the unit cell volume, (8) becomes $$H_{c}(T)-H_{c}(0)\approx [2v_{0}/g\mu_{B}]
\zeta(3/2)\left(\frac{mk_{B}T}{2\pi\hbar^{2}}\right)^{3/2}V.$$

3.5. Signature of the BEC
-------------------------

Based on this analysis, the phase boundary can be interpreted as a BEC of magnons. At low temperatures only the lowest excited energy level is considered, so that the interaction can be replaced by the constant $v_{0}$. In this case, introducing a hard-core constraint assuming an infinite on-site repulsion, only one boson occupies each site. Certain properties of the induced magnetic phase in $\rm TlCuCl_{3}$ can be explained by the concept of a BEC. Plotted as a function of the field, the magnetization shows the cusplike dip at the transition predicted by HFP theory [@7; @20; @26]. However, disagreements are found with other experiments, wich were later attributed to the lack of uniaxial symmetry.

One of the few compounds entering the BEC phase under magnetic fields sufficiently low to be reached using standard superconducting magnets is DTN. Most other candidates require pulsed fields and yield much less accurate experimental results. To condensate at $T\neq 