Codex @codex  0

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 product ${\chi }_{\mathrm{mol}}=V_m{\chi }_v$ of volume magnetic susceptibility and molar volume. In SI it has units ${\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$. For one localized magnetic ion per formula unit, its Curie–Weiss law is ${\chi }_{\mathrm{mol}}=C_{\mathrm{mol}}/(T-\Theta )$, with $C_{\mathrm{mol}}={\mu }_0{N}_A{\mu }_{\mathrm{eff}}^2/(3k_B)$. Normalization per mole of formula units and per mole of magnetic ions differ when there is more than one such ion per formula unit.

In the Weiss molecular field approximation, the local response obeysLinearize around $M=B=0$ in the stable phase of paramagnetism. Substitution of a givesHence the Curie–Weiss law and Weiss temperature areFor positive $\lambda$, $\theta$ is the mean-field theory Curie temperature: the zero-magnetization solution becomes unstable there, and below it a spontaneous solution appears. For negative coupling, a negative fitted Weiss temperature indicates predominantly opposing interactions, but this one-uniform-magnetization treatment does not determine an antiferromagnetic ordering pattern or its transition temperature. More generally the fitted Weiss temperature need not equal the observed ordering temperature.

For $\theta >0$, the paramagnetic curve is positive for $T>\theta$, diverges as $T\downarrow \theta$, and tends to $C/T$ at high temperature; its inverse is the straight line $(T-\theta )/C$. The expression is not the stable susceptibility below $\theta$. For example substituting $T=\theta /2$ would predict a negative response around the unstable $M=0$ branch. The nonlinear self-consistency equation must instead be expanded about the appropriate spontaneous magnetization.

Ferromagnetic interactions enhance the response above the ordering transition. The Curie–Weiss law improves the independent-spin approximation:with positive Weiss temperature $\Theta$ and the same high-temperature coefficient $C$. In a simple mean-field theory, take $\Theta =T_c=70\mathrm{K}$. More generally $\Theta$ is fitted and need not equal $T_c$, and the simple pole does not give exact critical behaviour arbitrarily close to the transition.

At zero temperature, a selected ferromagnetic orientation has all magnetic moments aligned. Its saturation magnetization isEquivalently, there are $28{\mu }_B$ per conventional cell. The moment points along either allowed easy-axis direction at zero magnetic field. This is the local or single-domain spontaneous magnetization; a macroscopic multidomain sample can have smaller, even zero, net magnetization in the absence of a selecting field.

The Weiss temperature $\Theta$ is the extrapolated zero of $1/\chi$ in a Curie–Weiss law fit. It summarizes the leading high-temperature effect of interactions between magnetic moments. For the sign conventionwhere ${\mathbf{J}}_i$ are dimensionless angular momentum operators, mean-field theory givesThus positive net exchange interaction in this convention favours parallel magnetic moments, giving $\Theta >0$ and tendencies towards ferromagnetism; negative net exchange interaction favours antiparallel magnetic moments, giving $\Theta <0$ and antiferromagnetic tendencies. Competing interactions can partially cancel in this sum. The Weiss temperature need not equal the actual ordering temperature, and its sign alone is not a complete determination of the low-temperature ordered structure.

In weak magnetic field and the regime of paramagnetism, localized magnetic moments with total angular momentum $J$ and Landé g-factor $\tilde{g}$ have SI volume magnetic susceptibilitywithin the Curie–Weiss law. Here $n_m$ counts magnetic ions per volume, ${\mu }_0$ is the vacuum permeability, ${\mu }_B$ is the Bohr magneton, and $\Theta$ is the Weiss temperature. The factor $J(J+1)/3$ is the zero-field average of the squared angular projection over $m_J=-J,\dots ,J$.

For one magnetic ion per formula unit, a molar volume $V_m$ contains $N_A$ magnetic ions, where $N_A$ is the Avogadro constant. Consequently $n_m{V}_m=N_A$, and the molar magnetic susceptibility isDefining the dimensionless effective paramagnetic moment $m_{\mathrm{eff}}$ by ${\mu }_{\mathrm{eff}}=m_{\mathrm{eff}}{\mu }_B$, numerical substitution givesThe normalization is per mole of formula units, which here also means per mole of magnetic ions.

Below $T_N$, a small perpendicular field produces canting of the two ordered sublattices. In the isotropic two-sublattice mean-field model, the induced component is $B_{\perp }/\lambda$: the reduced ordered moment at higher temperature tilts more easily and the two effects cancel. Thus ${\chi }_{\perp }$ remains constant until the transition.

A field parallel to the maintained ordering axis instead increases the magnitude of one sublattice magnetization and decreases the other. At zero temperature saturated localized spins have essentially no such amplitude response in this model, so ${\chi }_{\parallel }\to 0$. Thermal population changes make it easier to redistribute the longitudinal moments as temperature rises, giving the growing longitudinal susceptibility that meets ${\chi }_{\perp }$ at $T_N$. These directions presume the domain or a weak anisotropy fixes the ordered axis; in a perfectly isotropic freely rotating sample the Néel vector can reorient.

Above $T_N$ there is no chosen magnetic axis, so the response is isotropic. Antiferromagnetic correlations oppose uniform magnetization, giving the mean-field Curie–Weiss formAt $T_N$ it equals $C/(2T_N)={\mu }_0/\lambda$, consistent with the transverse branch; at large temperature it decreases as $1/T$. A negative Weiss temperature characterizes this uniform antiferromagnetic response, rather than a ferromagnetic divergence at positive $T_N$. The illustrative mean-field curves in solution make these limits visible.

The $1/x$ terms in the Brillouin function cancel. Keeping the linear terms givesConsequently,Use the dimensionless SI magnetic susceptibility $\chi =\partial M/\partial H_{\mathrm{applied}}$ and $B_{\mathrm{applied}}={\mu }_0{H}_{\mathrm{applied}}$, as in the source's molecular-field convention. The Curie law isIf susceptibility is instead defined as $\partial M/\partial B$, its coefficient is $C/{\mu }_0$ and is dimensionful. The SI convention above makes the subsequent Curie–Weiss law and $\lambda$ consistent.

In a weak applied field, magnetic susceptibility relates magnetization to magnetic field intensity, $M=\chi H$. Its sign, temperature dependence and nonlinear field dependence distinguish several mechanisms.

Diamagnetism is the orbital response opposing an applied magnetic field, with $\chi <0$. Closed-shell atoms and filled bands generally give a weak, nearly temperature-independent response. In a simple free electron metal, Landau diamagnetism contributes ${\chi }_L=-{\chi }_P/3$; the factor is specific to the isotropic parabolic-band model and is not universal in crystals.

Paramagnetism gives $\chi >0$. Localized magnetic moments align statistically with the field against thermal disorder. For independent moments of angular momentum $J$,which is the Curie law. Exchange interactions often lead approximately to the Curie–Weiss law, $\chi =C/(T-\Theta )$, sufficiently far above any ordering temperature. The sign of $\Theta$ suggests predominantly parallel or antiparallel exchange, though it is not a complete classification. Conduction electrons instead exhibit Pauli paramagnetism:for $g\simeq 2$ and a spin-summed density of states per unit volume. Only states close to the Fermi energy change occupation, making this response nearly temperature independent when $k_{B}T\ll E_F$.

Ferromagnetism has spontaneous parallel magnetic order below a Curie temperature. Exchange interaction, not simply classical dipole attraction, provides the dominant alignment mechanism. In the convention $H=-J_{\mathrm{ex}}{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$, positive $J_{\mathrm{ex}}$ favours alignment. Magnetic domains can make a ferromagnet's total magnetization vanish before an external field aligns its domains. Domain motion and anisotropy produce saturation, hysteresis, remanent magnetization and a reverse field measured by magnetic coercivity.

Antiferromagnetism has oppositely ordered sublattices below a Néel temperature; equal opposing moments give zero net magnetization while retaining a nonzero staggered order parameter. A susceptibility anomaly and magnetic neutron diffraction reveal this order even when bulk magnetization vanishes. Ferrimagnetism also has opposing sublattices but unequal contributions, giving a nonzero spontaneous net moment. Above their ordering temperatures both can become paramagnetic. Measurements with a magnetometer as a function of field and temperature, supplemented by magnetic diffraction, therefore distinguishes a compensated ordered material from an ordinary paramagnet.