We observe that for $\mu\sim10$ meV the MO are practically seen as one unique oscillation, while for $\mu\sim100$ meV the spin splitting becomes noticeable. Moreover, we observe that at low $\mu$ there is a small curvature in the MO, which disappears at high $\mu$. This behavior can be explained by taking the ratio of amplitudes of both series, given by $A_{s}^{cos}=B\left(-\mu+3s\mu_{\mathrm{B}}B/2\right)/\omega\phi$ and $A_{s}^{sin}=\left(-\mu+s\mu_{\mathrm{B}}B\right)/\phi$. Considering that usually $B\mu_{\mathrm{B}}/\mu\ll1$, we get $A_{s}^{cos}/A_{s}^{sin}\simeq B/\omega$. Thus, if $\mu>0.1$ eV we have $\omega=\mu^{2}/\alpha^{2}>10$ T and $A_{s}^{cos}/A_{s}^{sin}<0.1B\left[\unit{T}\right]$, so unless $B$ is very high we have $A_{s}^{cos}/A_{s}^{sin}\ll1$. For smaller values of $\mu$ this would not be case, and for big $B$ one may even have $A_{s}^{cos}/A_{s}^{sin}>1$. This is seen in figure \[fig1\](a), where at low $\mu$ the cosine series in equation (\[Mosc\]) produces the small curvature in the MO. Nevertheless, we will not take $\mu$ this low for then the Zeeman effect becomes unobservable. Therefore, given that we will be mainly interested with the spin splitting in the MO, which is seen at high $\mu$, we shall directly neglect the first term in equation (\[Mosc\]). Then the MO reduce to equation (\[M sawtooth\]) and the sine series can be evaluated to obtain

$$M=\sum_{s=\pm1}\frac{A_{s}}{\pi}\arctan\left\{ \cot\left[\pi\omega\left(\frac{1}{B}+s\Delta\right)\right]\right\} .\label{M0}$$

![\[fig1\]Magnetization given 