Paramagnetism ($\chi >0$)

In a linear isotropic regime, a paramagnetic material has positive magnetic susceptibility: its induced magnetization is aligned with the applied magnetic field intensity. Molecular oxygen supplies an example discussed through molecular orbital theory in the covalent bond article.

A material exhibiting paramagnetism. Its induced magnetization points along the applied magnetic field in the linear-response regime, giving positive magnetic susceptibility. Unlike a ferromagnet, it has no spontaneous ordered moment in that phase. The inverse-temperature response of independent localized moments is one model, rather than a definition that includes all microscopic mechanisms of paramagnetism.

For independent magnetic moments constrained to ${\mu }_z=\pm \mu$, the one-site partition function is $2\cosh [\mu {\mu }_{0}H/(k_{B}T)]$. The weak-field magnetization is $M={\mu }_{0}n{\mu }^{2}H/(k_{B}T)$, so the SI magnetic susceptibility obeys $\chi =C/T$ with $C={\mu }_{0}n{\mu }^2/k_B$. If $\mu =gS{\mu }_B$, this involves $S^2$, not the unrestricted-spin factor $S(S+1)/3$. The constraint on allowed moment orientations must be specified before using a Curie law coefficient.

At fixed positive temperature and weak field, independent spin-$J$ moments have total moment $M=N(g{\mu }_B{)}^{2}J(J+1)B/(3k_{B}T)$. This is the Curie law; the corresponding density divides by the sample volume.

For independent distinguishable moments with angular momentum $J$, define $y=g{\mu }_{B}B/(k_{B}T)$. The one-site partition function is $Z=\sinh [(J+1/2)y]/\sinh (y/2)$. Its logarithmic derivative gives $⟨{\mu }_z⟩=g{\mu }_B[(J+1/2)\coth ((J+1/2)y)-\frac{1}{2}\coth (y/2)]$. At $J=1/2$ this reduces to $(g{\mu }_B/2)\tanh (y/2)$.

The Brillouin function gives the normalized magnetization of independent angular-momentum-$J$ moments. It is odd, increasing, has slope $(J+1)/(3J)$ at zero and tends to $\pm 1$ at large positive or negative argument. Explicitly,It follows by differentiating the logarithm of ${\sum }_{m=-J}^J\exp (mx/J)$, the one-ion partition function. Its limiting value gives magnetic saturation, and its initial slope gives the Curie law.

Paramagnetism arising when thermal excitation populates a magnetic manifold above a less magnetic or nonmagnetic ground manifold. Competing thermal population and orientational randomization can produce a maximum in susceptibility at an intermediate temperature.

The Curie–Weiss law approximates the magnetic susceptibility of interacting localized moments by $\chi =C/(T-\Theta )$ in a suitable paramagnetic temperature range. A positive Weiss temperature often reflects predominantly ferromagnetic interactions, while a negative value often reflects antiferromagnetic interactions. It is not a universal exact law through an ordering transition, and crystal-field effects or itinerant carriers can require another description.

The fitted temperature $\Theta$ in $\chi =C/(T-\Theta )$. It summarizes the leading high-temperature influence of interactions and need not equal the ordering temperature.

For independent localized two-state moments of projections $\pm \mu$ along a field, the one-ion partition function is $Z_1=2\cosh [\mu B/(k_{B}T)]$. Its thermal mean gives $M=n\mu \tanh [\mu B/(k_{B}T)]$. A $J=1/2$ ion has $\mu =g_J{\mu }_B/2$, so the special value $\mu ={\mu }_B$ additionally assumes $g_J=2$. At weak field the SI volume susceptibility is ${\chi }_H={\mu }_{0}n{\mu }^2/(k_{B}T)$, neglecting demagnetizing corrections.

For moment projections $\pm \mu$, $p_{+}=(1+\tanh x)/2$ but $M/M_s=\tanh x$, with $x=\mu B/(k_{B}T)$. At zero field half the moments occupy each projection while the net magnetization is zero. An effective Curie moment is also distinct from the saturated projection used in this two-state model.

For independent moments with magnetic energies $\mp \mu B$, $Z=(2\cosh x{)}^N$ and $S=N{k}_B[\ln (2\cosh x)-x\tanh x]$, where $x=\mu B/(k_{B}T)$. This counts the magnetic orientations only, excluding translational and lattice entropy.