the temperature at which the spin splitting would not be observable in the MO can be obtained from equation (\[MT\]). We shall define $x\equiv1/B$, and consider two peaks at $x_{1}$ and $x_{2}$, corresponding to the same LL, with different spin, so in general $x_{1}=l/\omega-\Delta$ for the spin up, and $x_{2}=l/\omega+\Delta$ for the spin down (see equation (\[1/Bi\])). Then, between the two peaks, only the exponential terms corresponding to $x_{1}$ and $x_{2}$ would be appreciable, so for $x_{1}<x<x_{2}$ equation (\[MT\]) becomes

$$M_{T}=M-\frac{A_{1}}{1+e^{\beta\mu\left(x-x_{1}\right)/2x}}+\frac{A_{-1}}{1+e^{-\beta\mu\left(x-x_{2}\right)/2x}},\label{Mt spin}$$

where $M$ is given by equation (\[M0\]), and we have rewrite $\left(B-B_{m}\right)/2B_{m}=-\left(x-x_{m}\right)/2x$.

![\[fig5\]Spin splitting effect in the MO. In the left it is shown magnetization at zero temperature (red dashed line) and at different temperatures (blue solid line). In the right it is shown the first (orange solid line) and second (green dashed line) exponential in equation (\[Mt spin\]). The four cases correspond to the exponentials width being (a) $<\Delta$, (b) $=\Delta$, (c) $>\Delta$, (d) $=2\Delta$, where $2\Delta$ is the spin peaks separation.](fig5.eps)

From equation (\[Mt spin\]) we can see that the temperature effect behaves like a Fermi-Dirac distribution. The difference is that in this case our parameter is $x=1/B$, instead of the energy, and each exponential is 