The component of the total spin in the direction of H

[Another]

i don't understand,
Why The component of the total spin in the direction of H is $M_s = \frac{1}{2}(N_{up}-N_{down})$

 

[DrClaude]

Why The component of the total spin in the direction of H is $M_s = \frac{1}{2}(N_{up}-N_{down})$

Because $N_{\uparrow}$ particles contribute $+1/2$ and $N_{\downarrow}$ particles contribute $-1/2$ to the total magnetisation.