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The Curie constant is the coefficient $C$ in Curie's law $\chi =C/T$ for an appropriate dilute or high-temperature paramagnet. It sets the strength of the weak-field magnetic susceptibility, rather than an ordering temperature. In an SI volume-susceptibility convention for number density $n$ of identical noninteracting moments,Here ${\mu }_{\mathrm{eff}}$ is the effective magnetic moment and ${\mu }_0$ the vacuum permeability. A mole-based susceptibility uses a correspondingly mole-based constant.

Plotting magnetic susceptibility against inverse temperature can estimate $C$ in the linear weak-field regime. Exchange interactions or thermal occupation of additional levels can change the dependence, so a fitted $C$ is meaningful only with its range and normalization stated. The factor $1/3$ reflects orientational averaging in the isotropic moment model; an anisotropic response requires more information than one scalar constant.

The magnetic moment inferred from the weak-field Curie constant of a localized angular-momentum multiplet. Uniform zero-field occupation of its $2J+1$ substates gives $⟨m_J^2⟩=J(J+1)/3$, yielding ${\mu }_{\mathrm{eff}}=|g_J|{\mu }_B\sqrt{J(J+1)}$. Here $g_J$ is the Landé g-factor and ${\mu }_B$ the Bohr magneton. The dimensionless number $m_{\mathrm{eff}}={\mu }_{\mathrm{eff}}/{\mu }_B$ differs from the saturated projection $|g_J|J$.

For independent localized angular-momentum-$J$ ions at fixed field, the weak-polarization mean moment is $⟨{\mu }_z⟩=C_{\mu }/T$. Unlike the volume-susceptibility Curie constant, this coefficient includes the magnetic field and no ion number density. The expansion requires $g{\mu }_B|B|J\ll k_{B}T$; the moment saturates toward $g{\mu }_{B}J$ beyond that regime.

At high temperature, $\chi T$ tends towards the nearly constant Curie constant of weakly correlated $S=\frac{1}{2}$ centers. On cooling, $\chi T$ falls because neighboring spins increasingly correlate antiparallel. The broad maximum in $\chi$ marks the crossover from the high-temperature Curie-like regime to substantial short-range order.

At still lower temperature, the Magnetic susceptibility becomes small because the antiferromagnetic ladder favors a singlet-like ground state and has few thermally accessible magnetic excitations. An even-leg spin ladder can have a spin gap, which reinforces this suppression. A broad Magnetic susceptibility maximum and falling $\chi T$ support short-range antiferromagnetic correlations; they do not establish a three-dimensional Néel transition. Such a transition would require independent evidence, rather than identifying the broad maximum itself as the ordering temperature.

At high temperature, approximately independent paramagnetic ions give the appropriate Curie constant. The negative Weiss temperatures and the reduction of $\chi T$ on cooling initially indicate predominantly antiferromagnetic correlations. The sharp rise near the ordering temperature nevertheless reveals a strongly enhanced low-field magnetic response.

A net moment could arise from a canted antiferromagnet, a ferrimagnet, or a ferromagnet with competing antiferromagnetic interactions. Differences between field-cooled magnetization and zero-field-cooled magnetization suggest history-dependent domain selection or slow relaxation, but are not alone proof of a particular microscopic spin arrangement. An ordinary compensated antiferromagnet would not account for a large ordering-related ferromagnetic-like low-field contribution without an additional mechanism.

Let $x=g{\mu }_{B}B/(k_{B}T)$. The single-ion partition function and mean projected magnetic moment areFor $J>0$, the geometric sum also givesIn the high-temperature, weak-polarization regime $|x|J\ll 1$, expand the Boltzmann factors. The numerator's zero-order term vanishes because $\sum m_J=0$, and the first-order term uses $\sum m_J^2=J(J+1)(2J+1)/3$. The denominator is $2J+1+O(x^2)$. ThereforeThis fixed-field per-ion Curie coefficient describes mean moment per ion at fixed field, so it includes $B$. It is not the field-independent volume-susceptibility Curie constant. For number density $n$ and $B\simeq {\mu }_{0}H$, the latter is $C_{\chi }=n{\mu }_0(g{\mu }_B{)}^{2}J(J+1)/(3k_B)$, giving ${\chi }_H=C_{\chi }/T$. At $J=0$, the specified moment is zero and both expressions vanish.

Within the stipulated localized, one-moment-per-atom spin model, the number density follows directly from mass density and molar mass:The face-centered cubic lattice is consistent with four atoms per conventional unit cell; no additional factor four is needed once $n$ has been obtained from the density. The Curie constant is $C=0.71549\mathrm{K}$. Identifying the model's $\theta =\lambda C$ with the specified Curie temperature givesAt very low temperature, a fully ordered single domain has $M_s=n{\mu }_B=8.4764\times 10^5\mathrm{A}{\mathrm{m}}^{-1}$. Its Weiss molecular field isThe cancellation of $n$ in the middle expression provides a useful check. The ordinary magnetostatic scale is only ${\mu }_0{M}_s\simeq 1.065\mathrm{T}$, smaller by the factor $\lambda$. Thus $B_{\mathrm{eff}}$ is not a plausible ordinary self-generated dipolar magnetic field of the nickel moments. It represents the exchange interaction energy in field units, not an additional Maxwell field of that magnitude. This does not assert that such large externally generated fields are forbidden. Nickel is an itinerant ferromagnet; the imposed one-${\mu }_B$-per-atom model is an estimate rather than its measured microscopic moment.

Assume thermal equilibrium, a three-dimensional nondegenerate semiconductor, one parabolic conduction extremum and one parabolic valence extremum, with two spin states in each. Use the supplied effective masses as scalar density-of-states parameters. With conduction energy $E=E_c+\epsilon$, state counting gives the per-volume density of statesSince $E_c-\mu \gg k_{B}T$, replace the Fermi-Dirac distribution by its Boltzmann factor. Using $x=\epsilon /(k_{B}T)$ and the supplied integral,The electron hole occupation is $1-f(E)$ in the valence band. Applying the same integration below $E_v$ givesTheir product cancels the chemical potential, giving the semiconductor mass-action lawThus, for masses expressed in ordinary mass units,The labels $B,C$ in this last box are exponents, unrelated to the magnetic field or Curie constant above. Valley degeneracy or multiple valence bands must be included in the effective density of states, or absorbed into suitable density-of-states effective masses, not counted twice. Real silicon has anisotropic multiple valleys, so its conductivity mass is not automatically its density-of-states mass; the numerical answers below implement the paper's supplied scalar-mass model.^[ref]

The formula is not an identity for arbitrarily degenerate carriers. For a concrete counterexample, take $\mu -E_c=10k_{B}T$ while holes remain dilute. Fermi occupation gives $n/N_c\simeq 24.1$, whereas the Boltzmann replacement would give $e^{10}\simeq 22026$; the predicted product then differs by about three orders of magnitude. Separate electron and hole quasi-Fermi levels out of equilibrium would also prevent the cancellation used above.

The Curie constant measures the squared magnetic moment averaged over all orientations of the $J$ multiplet, givingFor the usual positive Landé g-factor of the ion here, the absolute-value sign can be omitted. This effective paramagnetic moment differs from the maximum projected or saturation magnetic moment, ${\mu }_{\mathrm{sat}}=|\tilde{g}|J{\mu }_B$.