zero temperature (see figure \[fig1\]). On the other hand, at low temperatures each exponential term in equation (\[Mt2\]) is in general very small unless $\mu$ is close to $\varepsilon_{m}$. Thus we can expand each $\varepsilon_{m}$ around $B_{m}$, where $B^{-1}_m=n_m/\omega+s_m\Delta$ and $\varepsilon_{m}\left(B_{m}\right)=\mu$, so $\left(\mu-\varepsilon_{m}\right)\simeq-\mu\left(B-B_{m}\right)/2B_{m}$. Moreover, $\left(\partial\varepsilon_{m}/\partial B\right)\simeq\left[\mu-s_{m}\mu_{\mathrm{B}}B\right]/2B$, so from equation (\[amp\]) we get $\mathcal{A}^{-1}\left(\partial\varepsilon_{m}/\partial B\right)\simeq-A_{s_{m}}/D.$ Finally, if we take into account the valence contribution to the magnetization (which we show it is not altered by the temperature), we have the total magnetization $M_{T}=M_{V,T}+M_{C,T}=M_{V}+M_{C,T}$. Thus from equation (\[Mt2\]) we get

![\[fig4\]Magnetization given by equation (\[MT\]), for $\mu=250$ meV. The red dashed line corresponds to the zero temperature case, while the solid blue line corresponds to (a) $T=0.1$ K and (b) $T=1$ K.](fig4.eps)

$$M_{T} \simeq M-\sum_{m=0}^{f}\frac{A_{s_{m}}}{1+e^{-\beta\mu\left(B-B_{m}\right)/2B_{m}}}+\sum_{m=f+1}^{\infty}\frac{A_{s_{m}}}{1+e^{\beta\mu\left(B-B_{m}\right)/2B_{m}}}.\label{Mt aprox}$$

where $M=M_{V}+M_{C}$ is the net magnetization at zero temperature, given by equation (\[M0\]). Equation (\[Mt aprox\]) corresponds to the case $\varepsilon_{f}<\mu\leq\varepsilon_{f+1}$, which implies $B_{f+1}\leq B<B_{f}$. Thus the low temperature effect is to introduce a factor proportional to $\left[1+e^{-\beta\mu\left(B-B_{m}\right)/2B_{m}}\right]^{-1}$ if $B<B_{m}$, or proportional to $\left[1+e^{\beta\mu\left(B-B_{m}\right)/2B_{m}}\right]^{-1}$ if $B>B_{m}$. In this way we can generalize equation (\[Mt aprox\]) to all $B$ and get the result (see the appendix for details)

$$\fl M_{T}=\sum_{s}\frac{A_{s}}{\pi}\arctan\left\{ \cot\left[\pi\omega\left(\frac{1}{B}+s\Delta\right)+\sum_{n}\frac{\pi}{1+e^{-\beta\mu\left(B-B_{n,s}\right)/2B_{n,s}}}\right]\right\},\label{MT}$$

where $B_{n,s}^{-1}=n/\omega-s\Delta$. In the practice it 