= Independent-moment magnetic response
{title2=$M=n\mu\tanh\left(\mu\mu_0H_{\rm app}/(k_BT)\right)$}
Consider a stipulated dilute collection of independent <two-level systems>, each with longitudinal <magnetic moment> $s\mu$, $s=\pm1$, and <energy> $E_s=-s\mu B_0$. The prescribed vacuum/applied <magnetic flux density> is $B_0=\mu_0H_{\rm app}$. Ignore interactions, orbital magnetic response, and the small sample-field correction. This is the magnetic version of the existing <independent spin-one-half two-level system>, with alignment probabilities rather than an interacting <Curie–Weiss model>.
For <temperature> $T>0$, the <canonical ensemble> gives the one-moment <partition function> $Z_1=2\cosh x$, with $x=\mu B_0/(k_BT)$. Taking the <expected value> over its two <Boltzmann weights> gives
$$
\langle s\rangle=\frac{e^x-e^{-x}}{e^x+e^{-x}}=\tanh x,\qquad
M=n\mu\tanh x,\qquad
\chi_{\rm model}\equiv\left(\frac{\partial M}{\partial H_{\rm app}}\right)_T
=\frac{\mu_0n\mu^2}{k_BT}\frac1{\cosh^2x}.
$$
Here $n$ is the <number density>, $k_B$ the <Boltzmann constant>, and $\tanh$ the <hyperbolic tangent>. For $|x|\ll1$, its <small-argument expansion of the hyperbolic tangent> gives $M\simeq\mu_0n\mu^2H_{\rm app}/(k_BT)$, a <Curie's law> response in the applied-$H$ convention. For large $|x|$, $M$ approaches the <saturation magnetization> $n\mu$ with sign set by the field, and its <differential magnetic susceptibility> falls towards zero.
Both plots use stipulated $n=10^{26}\,\mathrm{m^{-3}}$, $\mu=9.2740101\times10^{-24}\,\mathrm{J\,T^{-1}}$, $k_B=1.380649\times10^{-23}\,\mathrm{J\,K^{-1}}$, and $\mu_0=1.25663706\times10^{-6}\,\mathrm{T\,m\,A^{-1}}$. The moment is approximately one <Bohr magneton>; the chosen <vacuum permeability> is a rounded model input. At $T=2\,\mathrm K$, $M_s=927.401\,\mathrm{A\,m^{-1}}$ and $\chi_{\rm model}(0)=3.91410\times10^{-4}$. The linear approximation overestimates $M$ by less than $3\%$ for $|x|\le0.3$, corresponding to $|\mu_0H_{\rm app}|\le0.89324\,\mathrm T$ or $|H_{\rm app}|\le0.71082\,\mathrm{MA\,m^{-1}}$. These parameters keep the modeled response weak; neglected magnetic feedback is of relative order $\chi_{\rm model}(0)$, rather than an explanation of a material's ordered phase.
\Image[/independent-moment-magnetization-linear-range.png]
{title=<Magnetization> approaching <saturation magnetization>, with its small-field linear range}
{description=Original independent-moment calculation at stipulated $T=2\,\mathrm K$. The shaded range has less than $3\%$ error relative to the exact <magnetization>. The second panel enlarges this range. The prescribed field is $B_0=\mu_0H_{\rm app}$; curves are model predictions, not measured data.}
\Image[/independent-moment-susceptibility-field-temperature.png]
{title=Bias-field and <temperature> dependence of the independent-moment <differential magnetic susceptibility>}
{description=Original calculation using the same stipulated moments and <number density>. The left panel compares field sweeps at $2$, $4$ and $8\,\mathrm K$. The right panel compares <temperature> sweeps at prescribed $B_0=0$, $1$ and $4\,\mathrm T$. The zero-field slope follows <Curie's law>; cooling at fixed nonzero field eventually suppresses the slope through <saturation magnetization>.}
At $B_0=0$, cooling increases the zero-field slope as $1/T$. At fixed $B_0\ne0$, cooling instead gives $\chi_{\rm model}\sim4\mu_0n\mu^2e^{-2\mu|B_0|/(k_BT)}/(k_BT)$: a saturated moment can hardly change further. The order of the zero-field and zero-<temperature> limits matters. This independent model has no finite-<temperature> collective <phase transition>; the diverging zero-field slope alone does not establish one.
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