Past exam of the condensed matter physics course of the University of Oxford / 2013 / 4 / d / Solution

Linear Curie behavior fails gradually when the Zeeman energy of a substantial moment is comparable to thermal energy. Its appropriate scale isAt $T=300\mathrm{K}$, $k_{B}T/{\mu }_B=446.6\mathrm{T}$. Thus $B_{\mathrm{nl}}\sim 446.6/(gJ)\mathrm{T}$: about $223\mathrm{T}$ for $g=2,J=1$, or $447\mathrm{T}$ for a spin-one-half moment with $g=2$. Larger moments saturate at smaller fields. The source does not specify numerical $g,J$, so it determines this parametric scale rather than one universal room-temperature threshold. At sufficiently large field the moment tends toward $g{\mu }_{B}J$ and cannot keep increasing linearly with $B$.