Magnetization for Paramagnetic Materials

Question

If we consider the magnetization for paramagnetic materials, then we obtain $$M = -nfrac{partial F}{partial B} propto B_J(x),$$ where $$x equiv gleft( JLSright) cdot J cdot frac{mu_Bcdot B}{k_B T}$$ is an auxiliary variable and $B_J(  . )$ the Brillouin function. Now, what exactly is $J$ in this context? I thought that $hat J = hat L  oplus  hat S $, s.t. $J$ would be its quantum number. But then, how can $J$ be an integer, as is shown in the plot in our lecture:

daydreamer · Answer

Well, I can't see why it bothers you. Mathematically we have all these combinations, with different degeneracies, of S (1/2, 1,...)
Physically, integer S can be achieved either by decimating pairs of spin-1/2 particles (where a m=0 sector is accessible) or by considering a bosonic paramagnet, obtained by analyzing the unit cell of some crystal lattice, that sometimes has bosonic character; if this guy has spin-1 and its ground-state is singlet-like, there is a physical realization.
I'd like people to add concrete examples of materials that behave like this in nature.

Magnetization for Paramagnetic Materials

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