Curie Law and Curie Temperature

Pierre Curie's name is attached to two different pieces of magnetism, and confusing them is a rite of passage this article exists to shorten. The Curie law, $\chi=C/T$ , belongs to paramagnets: it says the alignment of independent permanent moments is a losing fight against temperature, and quantifies the loss. The Curie temperature belongs to ferromagnets: it is the sharp point — 1043 K for iron — where cooperative order collapses. Same physicist, same decade, opposite dramas: one law describes moments that never cooperate, the other names the moment cooperation fails. What joins them is a single quantitative machine, and this article builds it. The paramagnetism article won the two-level spin- $\tfrac12$ battle and exhibited the $1/T$ tail; here we derive the general- $J$ result — the Brillouin function — from the full $2J+1$ -level partition function, watch it collapse to that tanh at $J=\tfrac12$ and to Langevin's classical curve at $J\to\infty$ , and extract the Curie constant $C$ whose measurement reads off an ion's effective moment $p_{\rm eff}=g\sqrt{J(J+1)}$ to two decimal places. Then Weiss's audacious 1907 move: let each moment feel a molecular field proportional to the magnetization itself, and the same machine produces the Curie–Weiss law $\chi=C/(T-\theta)$ , a diverging susceptibility, and spontaneous magnetization below $T_C$ — a phase transition from two lines of algebra. The move comes with a debt this chapter must be honest about: the molecular field required for iron is of order a thousand teslas, and no magnetic mechanism can supply it — the dipole–dipole budget fails by four orders, as Ferromagnetism computed. What λ really is, quantum mechanics will confess at the chapter's end.

Key takeaways

Paramagnets: $\chi=C/T$ (Curie law) with $C=\mu_0 n g^2\mu_B^2 J(J+1)/3k_B$ — measure the $1/\chi$ -vs- $T$ slope and you have weighed the atomic moment: $p_{\rm eff}=g\sqrt{J(J+1)}$ , e.g. 7.94 for Gd³⁺.
The full curve is the Brillouin function $M=ngJ\mu_B\,B_J(x)$ , $x=gJ\mu_B B/k_BT$ : linear at small $x$ (Curie regime — where all lab paramagnets live), saturating at $M_s=ngJ\mu_B$ for $x\gtrsim1$ (kelvins at lab fields).
$B_J$ contains the chapter's earlier results as endpoints: $J=\tfrac12$ gives the two-level $\tanh$ of Paramagnetism; $J\to\infty$ gives the classical Langevin function.
Weiss ferromagnet: replace $B\to B+\lambda\mu_0M$ and the same algebra yields $\chi=C/(T-\theta)$ above the transition ( $\theta=\lambda C$ ) and spontaneous $M\neq0$ below $T_C$ — mean-field theory's first triumph.
Near $T_C$ : $M\propto(T_C-T)^{1/2}$ in mean field ( $\beta=\tfrac12$ ); real ferromagnets show $\beta\approx0.35$ — fluctuations, which mean field ignores, matter at criticality.
The required $\lambda\mu_0M_s$ for iron is $\sim10^3$ T — no real magnetic field, but exchange in disguise; the mechanism is owned by Exchange Interaction and Ferromagnetic Ordering.
KEY IDEA: everything is the ratio of alignment energy to $k_BT$ — independent moments give $\chi=C/T$ ; add the Weiss feedback $B_{\rm eff}=B+\lambda\mu_0M$ and the pole moves from $T=0$ to $T=T_C$ , which is the ferromagnetic transition.

The core insight

One Boltzmann computation rules both laws: independent moments yield $\chi=C/T$ ; letting each moment feel the magnetization of its neighbors ( $B_{\rm eff}=B+\lambda\mu_0M$ ) shifts the susceptibility's pole from absolute zero to $T_C=\lambda C$ , turning a paramagnet's slow $1/T$ decline into a ferromagnet's sharp phase transition.

Where this lives today

The Curie law's cleanest modern job is being a thermometer. In the millikelvin range — below the reach of gas thermometry — the susceptibility of a dilute paramagnetic salt (cerium magnesium nitrate is the classic) is measured, and $T=C/\chi$ is the temperature reading; the law's freedom from transitions is exactly what makes it a reliable scale. Its dynamic twin, adiabatic demagnetization, runs the physics as an engine: magnetize a paramagnet isothermally (aligning spins, expelling entropy), isolate it, then relax the field — the spins re-randomize at the lattice's expense, and the sample plunges. Single-shot demagnetization stages cool to millikelvins; the nuclear-spin version reaches microkelvins, and adiabatic demagnetization refrigerators (ADRs) fly on X-ray space observatories, cooling detector arrays to 50 mK with no cryogens to boil away. The Curie temperature meanwhile became a design dial. Magnetic refrigeration prototypes exploit gadolinium's accident of geography — $T_C=292$ K, room temperature — where the magnetocaloric effect peaks. Heat-assisted magnetic recording (HAMR), shipping in the highest-capacity hard drives since the early 2020s, deliberately engineers a low writing barrier: a laser spot heats an FePt grain toward its Curie point for a nanosecond, coercivity collapses, the head writes, and the bit re-freezes into a record stable for decades — the Curie transition operated a trillion times per drive, on demand.

Origins · 1 minLéon Brillouin and how this was discovered

Léon Brillouin

1889–1969 · Paris and New York — the quantum counter of orientations

Brillouin was born into the trade — his father Marcel and grandfather were both physics professors at the Collège de France — and he worked across an astonishing range: the quantized-orientation magnetization function (1927) that bears his name, the Brillouin zones that organize every band-structure diagram in solid-state physics, Brillouin light scattering, and the WKB approximation (the "B" is his). The magnetization function came from taking the new quantum mechanics seriously at a time when Langevin's classical average still felt natural: if a moment can point in only $2J+1$ directions, the Boltzmann sum is a finite geometric series, and the resulting curve differs measurably from Langevin's — early cryogenic data on paramagnetic salts confirmed the quantum counting. Brillouin spent the war years directing French radio broadcasting before escaping to America in 1941, taught at Wisconsin, Harvard, and Columbia, and devoted his later years to information theory, arguing in Science and Information Theory (1956) that entropy and information are the same currency — negentropy, he called it — a thesis physics is still spending. His function is on duty in every magnetometry lab on Earth, every day.

History

1895

— Pierre Curie's doctoral survey establishes χ ∝ 1/T for paramagnets and identifies the sharp loss of ferromagnetism at high temperature.

1905

— Langevin derives the classical orientation average, giving Curie's law a statistical-mechanical foundation.

1907

— Weiss postulates the molecular field: one substitution turns Langevin's paramagnet into a theory of ferromagnetism with a critical point.

1927

— Brillouin replaces the classical average with the quantum sum over 2J+1 orientations; cryogenic data on paramagnetic salts vindicate the counting.

1933

— Giauque and MacDougall demonstrate adiabatic demagnetization cooling, reaching 0.25 K with a paramagnetic salt.

1944

— Onsager solves the 2D Ising model exactly: mean-field critical exponents are wrong, and the modern theory of critical phenomena begins.

The physical picture

Every result in this article is a verdict in the same trial: alignment energy versus thermal disorder. A permanent moment $\mu$ in a field $B$ prefers alignment by an energy of order $\mu B$ ; the thermal bath scrambles orientations with kicks of order $k_BT$ . At one tesla and room temperature the ratio is $\mu_B B/k_BT\approx2\times10^{-3}$ — the number Paramagnetism burned into this chapter — so ordinary paramagnets operate at a fraction of a percent of their potential, and everything is linear response. The Curie law is just this ratio dressed as a susceptibility: response $\propto\mu B/k_BT$ per moment, hence $\chi\propto1/T$ . Cool the sample and the bath's vote weakens; the same field aligns more; $\chi$ climbs. The experimentalist's rendering of the trial is the inverse-susceptibility plot, and it is worth internalizing as a diagnostic instrument before any derivation. Plot $1/\chi$ against $T$ : an ideal paramagnet gives a straight line through the origin with slope $1/C$ ; a ferromagnet above its transition gives the same slope but intercepts the axis at a positive temperature $\theta$ — the line "remembers" that cooperation will win at $T_C$ ; and a material whose neighbors prefer anti-alignment intercepts at negative $\theta$ , the fingerprint of antiferromagnetism and its Néel temperature. Three magnetic personalities, one plot, read at a glance from where a straight line meets the axis.

Figure

The inverse-susceptibility diagnostic — three intercepts, three magnetic personalities

The second picture to hold is the feedback loop. In a paramagnet each moment answers only the applied field; response is honest and small. Weiss's ferromagnet adds a rumor mill: each moment also feels its neighbors' alignment, packaged as a field $\lambda\mu_0M$ proportional to the very magnetization being built. Response feeds response. Above $T_C$ the loop merely amplifies — the susceptibility is enhanced by exactly the factor that moves the pole to $\theta$ — and below $T_C$ the loop self-sustains: the material magnetizes with no applied field at all, because its own response is a sufficient cause. Readers of the dielectric chapter have met this loop before — the Clausius–Mossotti catastrophe that fed ferroelectricity was the same feedback with $P$ in place of $M$ ; this chapter's version is the original, and the electric one was named after it.

The Brillouin machine

Derivation AThe Brillouin function from the partition function — and both chapter endpoints recovered

The goal is the exact thermal magnetization of $n$ independent moments of angular momentum $J$ per unit volume — the Brillouin function — and, from its small-argument slope, the Curie law with its constant $C$ in the general form that magnetometry actually uses. The strategy is a direct partition-function sum over the $2J+1$ quantized projections; the sum is geometric, which is the entire technical content. Watch two audits at the end: the result must collapse to the two-level $\tanh$ that Paramagnetism derived (that article owns the $J=\tfrac12$ physics; we only check against it), and to the classical Langevin function as $J\to\infty$ .

Step 1. Energies and the geometric sum. A moment with $g$ -factor $g$ and quantum number $J$ has projections $m_J=-J,\dots,+J$ along $\vec B$ , with energies $E_{m_J}=-g\mu_B m_J B$ . The single-ion partition function, with $y\equiv g\mu_B B/k_BT$ :

Z=\sum_{m_J=-J}^{J}e^{\,m_J y} =\frac{\sinh\!\big[(J+\tfrac12)y\big]}{\sinh(y/2)} ,

where: the closed form is the finite geometric series summed — ratio $e^{y}$ , $2J+1$ terms. Everything thermodynamic now lives in this one function.

Step 2. Differentiate for the average moment. The thermal average projection is generated by $Z$ in the standard way, $\langle\mu_z\rangle=g\mu_B\,\partial(\ln Z)/\partial y$ . Carrying out the derivative of the log of the sinh-ratio and multiplying by the density $n$ :

M=n\,g\mu_B J\;B_J(x), \qquad B_J(x)=\frac{2J+1}{2J}\coth\!\Big(\frac{2J+1}{2J}x\Big)-\frac{1}{2J}\coth\!\Big(\frac{x}{2J}\Big), \qquad (1)

where: $x\equiv JY=gJ\mu_B B/k_BT$ is the alignment-to-thermal ratio built on the maximum projection, and $B_J$ runs from 0 to 1 — it is the "fraction of saturation achieved," with saturation $M_s=ngJ\mu_B$ when every moment sits in its top state.

Step 3. Audit the endpoints. At $J=\tfrac12$ , $g=2$ : the two coth terms combine as $2\coth2x-\coth x=\tanh x$ , so $M=n\mu_B\tanh(\mu_B B/k_BT)$ — exactly the two-level result Paramagnetism derived from its Boltzmann ledger ✓ (their derivation stands; this is a consistency check, not a re-derivation). As $J\to\infty$ at fixed total moment $\mu=gJ\mu_B$ : the second coth's argument $x/2J\to0$ contributes the $-1/y'$ piece, and $B_J\to\coth y'-1/y'=\mathcal L(y')$ with $y'=\mu B/k_BT$ — the classical Langevin function, the same curve the dielectric chapter's orientational-polarization article derived for permanent electric dipoles, arriving here as the infinite- $J$ limit of quantum counting. Quantization matters between the endpoints: at fixed $\mu$ , a $J=\tfrac12$ ion is a factor $\sim3$ more responsive than a classical rotor, because it has no transverse directions to waste orientation on.

Step 4. Linearize — the Curie law and its constant. For $x\ll1$ use $\coth u\approx1/u+u/3$ in both terms; the $1/u$ pieces cancel exactly (no response without a field) and the linear pieces assemble to $B_J(x)\approx\frac{J+1}{3J}\,x$ . Then

M\approx\frac{n g^2\mu_B^2 J(J+1)}{3k_BT}\,B \qquad\Longrightarrow\qquad \chi=\frac{C}{T}, \qquad C=\frac{\mu_0\,n\,g^2\mu_B^2\,J(J+1)}{3k_B}, \qquad (2)

where: $\chi$ is defined against $H$ with $B\approx\mu_0H$ inside a dilute paramagnet (the convention fine print is Magnetic Susceptibility's property, honored here). The combination $g\sqrt{J(J+1)}\equiv p_{\rm eff}$ is the effective moment in Bohr magnetons — the single number a Curie-constant measurement returns. Note it is $J(J+1)$ , not $J^2$ : the fluctuating transverse components of the quantized moment contribute to the linear response even though the saturation value knows only the maximum projection $gJ$ . The mismatch between $p_{\rm eff}$ and the saturation moment is real, measurable, and a standard exam ambush. Why one function covers every ion: nothing in Steps 1–2 used any property of the ion except that its levels are the equally spaced Zeeman ladder of a single $J$ -multiplet. That is an excellent approximation whenever the multiplet splitting to the next $J$ dwarfs $k_BT$ — true for most rare earths, violated interestingly for Sm³⁺ and Eu³⁺, where low-lying multiplets spoil the fit and the honest analysis (Van Vleck's) sums over them. *Verified: geometric sum and derivative recomputed symbolically; $J=\tfrac12$ reduction gives $g^2J(J+1)=3$ so $C=\mu_0n\mu_B^2/k_B$ , matching Paramagnetism's constant exactly ✓; small- $x$ slope $(J+1)x/3J$ recomputed ✓; forms match Kittel Ch. 11 and Ashcroft & Mermin Ch. 31.*

The Curie regime is not a small corner — it is where essentially all room-temperature magnetometry lives. For Gd³⁺ ( $g=2$ , $J=\tfrac72$ ) at 1 T, $x=gJ\mu_B B/k_BT=4.7/T$ with $T$ in kelvin: unity only at 4.7 K. At room temperature $x\approx0.016$ and the Brillouin curve is indistinguishable from its tangent. Saturating a paramagnet honestly requires liquid-helium temperatures or hundred-tesla fields — exactly the corners of parameter space where magnetic cooling (below) does its work.

The Weiss feedback

Go deeper · Derivation B · 2 minThe molecular field — Curie–Weiss law above, spontaneous order below

Derivation BThe molecular field — Curie–Weiss law above, spontaneous order below

The goal is Weiss's 1907 mean-field theory in both of its verdicts: the shifted susceptibility $\chi=C/(T-\theta)$ above the ordering temperature, and the self-sustained magnetization below it, including the shape of $M(T)$ near the transition. The strategy is one substitution — every moment feels $B_{\rm eff}=B+\lambda\mu_0M$ instead of $B$ — fed back through the Brillouin machine; self-consistency does the rest. Watch the honesty checkpoint at the end: the numerical size of $\lambda\mu_0M$ is the article's most important number.

Step 1. High-temperature side — linear feedback. Above the transition, $M$ is small, the linear (Curie) form applies, but with the effective field: $M=(C/\mu_0T)(B+\lambda\mu_0M)$ . Collecting the $M$ terms:

M\Big(1-\frac{\lambda C}{T}\Big)=\frac{C}{\mu_0T}B \qquad\Longrightarrow\qquad \chi=\frac{C}{T-\theta}, \qquad \theta=\lambda C . \qquad (3)

where: the feedback has renormalized the denominator — the response is Curie's, amplified by $1/(1-\lambda C/T)$ , and the amplification diverges as $T\to\theta^+$ . An infinite linear response is the mathematics announcing that the linear assumption is dying: below $\theta$ the material will respond without being asked.

Step 2. Low-temperature side — the self-consistency curve. Set $B=0$ and keep the full nonlinear Brillouin form. With $m\equiv M/M_s$ the dimensionless magnetization, the effective-field argument becomes $x=gJ\mu_B\lambda\mu_0M/k_BT=\alpha\,m\,T_C/T$ (constants folded so that $T_C$ appears), and the condition reads

m=B_J\!\Big(\frac{3J}{J+1}\,\frac{T_C}{T}\,m\Big), \qquad T_C=\lambda C . \qquad (4)

where: the coefficient is fixed by requiring the right-hand side's initial slope to equal 1 exactly at $T=T_C$ (using $B_J'(0)=(J+1)/3J$ ). Equation (4) is solved by drawing: the straight line $y=m$ against the curve $y=B_J(\cdot\,m)$ . For $T>T_C$ the curve starts shallower than the line — only the origin intersects: no spontaneous order. For $T<T_C$ the curve starts steeper — a second crossing exists at $m^*>0$ , and it (not the origin, which is unstable to any fluctuation) is the equilibrium: spontaneous magnetization, born continuously from zero as $T$ crosses $T_C$ from above.

Step 3. The shape near the transition. Just below $T_C$ , $m$ is small; expand the right side to cubic order ( $B_J(u)\approx B_J'(0)u-b\,u^3$ with $b>0$ ) and cancel one power of $m$ :

m^2\propto\Big(\frac{T_C}{T}-1\Big) \qquad\Longrightarrow\qquad M\;\propto\;(T_C-T)^{1/2}, \quad T\to T_C^- . \qquad (5)

where: the square root is mean field's universal signature — the critical exponent $\beta=\tfrac12$ , shared with every feedback theory of this structure (Landau's ferroelectric free energy in the dielectric chapter produced the identical exponent, as it must). Honesty: measured ferromagnets give $\beta\approx0.35$ – $0.37$ , and Onsager's exact 2D Ising solution gives $\beta=\tfrac18$ — mean field is quantitatively wrong at criticality because it ignores fluctuations, which is precisely where fluctuations dominate. Away from the critical window it is a workhorse.

Step 4. The honesty checkpoint — price the molecular field. For iron, estimate $\lambda$ from measured numbers. A localized-moment stand-in for iron ( $n=8.5\times10^{28}$ m⁻³, $g=2$ , $J=1$ — a fiction the chapter closer will indict, flagged now) gives $C=\mu_0ng^2\mu_B^2J(J+1)/3k_B\approx1.8$ K, so $\lambda=T_C/C\approx590$ . The molecular field at saturation:

B_W=\lambda\mu_0M_s\approx590\times(4\pi\times10^{-7})\times1.7\times10^{6}\;{\rm A/m} \;\approx\;1.3\times10^{3}\ {\rm T}.

Cross-check by pure scales: $k_BT_C/\mu_{\rm atom}=k_B(1043)/(2.2\mu_B)\approx7\times10^{2}$ T — same order. No magnetic field of this size exists inside iron. The genuine dipolar field of neighbors is $\sim0.1$ T (Ferromagnetism's 0.040 K computation, recalled, not re-run: four orders short). Weiss's $\lambda$ is a phenomenological IOU — a real, enormous, spin-aligning energy wearing a field costume. Quantum mechanics pays this IOU with the exchange integral, and Exchange Interaction and Ferromagnetic Ordering presents the payment. Verified: (3) reproduces the Curie law at $\lambda=0$ ✓; slope-matching in (4) recomputed from $B_J'(0)=(J+1)/3J$ ✓; iron estimate recomputed — $C=1.77$ K, $\lambda=588$ , $B_W=1.26\times10^3$ T, scale check $706$ T ✓; treatment matches Kittel Ch. 12 and Ashcroft & Mermin Ch. 33 (including their honesty about $\beta$ ).

Figure

Solving m equals B_J by drawing — the feedback curve against the identity line, three temperatures

Limiting cases

case	result	meaning
$x\ll1$ (all lab paramagnets)	$\chi=C/T$ , $C=\mu_0ng^2\mu_B^2J(J+1)/3k_B$	the Curie law; slope of $1/\chi$ weighs the ion: $p_{\rm eff}=g\sqrt{J(J+1)}$
$x\gtrsim1$ (mK or 100 T)	$M\to M_s=ngJ\mu_B$	saturation — Curie law fails by construction; the regime of magnetic cooling
$J=\tfrac12$ , $g=2$	$B_J\to\tanh$	Paramagnetism's two-level result, recovered ✓
$J\to\infty$ , $gJ\mu_B$ fixed	$B_J\to\mathcal L$ (Langevin)	classical limit; quantization gains responsiveness at fixed $\mu$
$\lambda=0$	$\theta=0$ : $1/\chi$ through origin	ideal paramagnet — the diagnostic plot's baseline
$T\to\theta^+$ , $\lambda>0$	$\chi=C/(T-\theta)\to\infty$	Curie–Weiss divergence announcing ferromagnetic order
$T\to T_C^-$	$M\propto(T_C-T)^{1/2}$	mean-field $\beta=\tfrac12$ ; real $\beta\approx0.35$ — fluctuations win at criticality
$\theta<0$	$\chi=C/(T+\|\theta\|)$ , finite at all $T>0$	neighbors prefer anti-alignment: antiferromagnet, ordering at the Néel temperature

Common trapTwo Curies, one surname — the law is for paramagnets, the temperature is for ferromagnets, and the constant is neither

Keep the vocabulary surgically separate. The Curie law $\chi=C/T$ describes non-interacting moments and holds from mK to white heat — it has no transition in it anywhere; nothing "happens" at any temperature. The Curie temperature $T_C$ is a phase-transition point that only interacting (ferromagnetic) systems possess. The Curie constant $C$ is an amplitude (units: kelvin), not a temperature at which anything occurs — yet it secretly sets the transition through $T_C=\lambda C$ . Second ambush: above $T_C$ a ferromagnet does not obey the Curie law — it obeys Curie–Weiss, and fitting $C/T$ to its data mangles the extracted moment. Third: the effective moment $p_{\rm eff}=g\sqrt{J(J+1)}$ read from $C$ is not the saturation moment $gJ$ read from $M_s$ — Gd³⁺ gives 7.94 versus 7.0, both correct, measuring different things (fluctuating response versus maximum projection).

Worked examples

1 · Weighing the gadolinium ion (confidence builder)

Gd³⁺ has a half-filled $4f^7$ shell: $L=0$ , $S=J=\tfrac72$ , $g=2$ . Predict its effective moment and molar Curie constant, and the susceptibility of a gadolinium salt at 300 K.

Step 1. Effective moment. $p_{\rm eff}=g\sqrt{J(J+1)}=2\sqrt{(7/2)(9/2)}=2\sqrt{15.75}=\mathbf{7.94}$ Bohr magnetons. (Measured values for Gd³⁺ salts: 7.9–8.0 — the agreement is a triumph of the $J$ -multiplet picture, and $L=0$ means no orbital complications.)

Step 2. Molar Curie constant. Per mole of ions, $C_{\rm mol}=\mu_0N_A\,p_{\rm eff}^2\mu_B^2/3k_B$ . Numerically: $\mu_{\rm eff}=7.94\times9.274\times10^{-24}=7.36\times10^{-23}$ J/T; squaring and assembling, $C_{\rm mol}=(1.257\times10^{-6}\times6.022\times10^{23}\times5.42\times10^{-45})/(3\times1.381\times10^{-23})=\mathbf{9.9\times10^{-5}\ m^3\,K/mol}$ . (In the older CGS bookkeeping this is the famous "7.9 emu·K/mol" of the rare-earth tables — the units trap Magnetic Susceptibility warned about, converted here honestly: divide SI by $4\pi\times10^{-6}$ .)

Step 3. Susceptibility at 300 K. $\chi_{\rm mol}=C_{\rm mol}/T=9.9\times10^{-5}/300=\mathbf{3.3\times10^{-7}\ m^3/mol}$ — small, positive, and falling as $1/T$ : a textbook paramagnet. Saturation check: $x=gJ\mu_B B/k_BT=4.7\,B/T$ , so at 1 T and 300 K, $x=0.016\ll1$ — deep in the linear regime, Curie law licensed ✓.

2 · Reading a ferromagnet's obituary from its paramagnetic phase (exam level)

A nickel-like sample, measured above its transition, gives molar susceptibility $\chi_{\rm mol}=5.8\times10^{-8}$ m³/mol at 700 K and $2.37\times10^{-8}$ m³/mol at 800 K (illustrative data). Extract the Curie constant, the Weiss temperature, and the effective moment; then diagnose the material.

Step 1. Go to the diagnostic plot. $1/\chi$ values: $1.72\times10^{7}$ and $4.22\times10^{7}$ mol/m³. Slope $=(4.22-1.72)\times10^{7}/(800-700)=2.5\times10^{5}$ mol/(m³·K) $\Rightarrow C_{\rm mol}=1/\text{slope}=\mathbf{4.0\times10^{-6}\ m^3K/mol}$ .

Step 2. Intercept. Extrapolate to $1/\chi=0$ : $\theta=700-1.72\times10^{7}/2.5\times10^{5}=700-69=\mathbf{631\ K}$ . Positive intercept: ferromagnetic neighbors, ordering expected near 631 K — and nickel indeed orders at $T_C=627$ K. (The few-kelvin gap $\theta>T_C$ is real physics, not error: short-range correlations above the transition, which mean field ignores, push the measured paramagnetic intercept slightly above the true ordering point.)

Step 3. Weigh the moment. $p_{\rm eff}^2=3k_BC_{\rm mol}/(\mu_0N_A\mu_B^2)=3\times1.381\times10^{-23}\times4.0\times10^{-6}/(1.257\times10^{-6}\times6.022\times10^{23}\times8.60\times10^{-47})=2.55$ , so $p_{\rm eff}=\mathbf{1.6\,\mu_B}$ — far below any integer- $J$ ion's value. Step 4. The diagnosis: a localized $J$ -multiplet cannot produce 1.6; nickel's moments are itinerant — band electrons, fractionally polarized — and the Curie–Weiss form survives even where the localized picture behind our derivation fails. The plot is more robust than its own textbook justification; the honest microscopic story is the Stoner physics flagged in the chapter closer.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.1–6.4 · Jackson, Classical Electrodynamics, 3rd ed., §5.7–5.8 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11 · Zangwill, Modern Electrodynamics, Ch. 13 · Kittel, Introduction to Solid State Physics, 8th ed., Ch. 11–12 · Ashcroft & Mermin, Solid State Physics, Ch. 31–33

Provenance. Equations cross-checked against Griffiths Introduction to Electrodynamics §6 (Curie law, paramagnetism), Kittel Ch. 11–12 (Brillouin/Langevin functions, Weiss field, Curie–Weiss law), and Ashcroft & Mermin Ch. 31–33. Constants: $\mu_B=9.274\times10^{-24}$ J/T, $k_B=1.381\times10^{-23}$ J/K, $\mu_0=4\pi\times10^{-7}$ T·m/A (CODATA). Curie temperatures (Fe 1043 K, Co 1388–1394 K, Ni 627 K, Gd 292 K) from Kittel Table; verifier should recompute the $J=1/2$ Curie constant reduction and the small- $x$ Brillouin expansion to confirm $C=\mu_0 N g^2\mu_B^2 J(J+1)/(3k_B)$ and the Curie–Weiss denominator $T-\lambda C$ . (Scope note: the two-level tanh derivation is owned by Paramagnetism and only audited here; the Langevin function is owned by the E3 orientational-polarization article; the dipole-failure computation is owned by Ferromagnetism and recalled verbatim (0.040 K, ratio 2.6×10⁴); worked numbers recomputed — $p_{\rm eff}$ (Gd³⁺)=7.94, $C_{\rm mol}=9.9\times10^{-5}$ m³K/mol, iron $C=1.77$ K, $\lambda\approx590$ , $B_W\approx1.3\times10^3$ T; example 2's data are synthetic-but-realistic for nickel and labeled illustrative.)

Builds on: Paramagnetism · Magnetic Dipole Moment of Atoms · Magnetization · Magnetic Susceptibility

Leads to: Ferromagnetism · Linear and Nonlinear Magnetic Media · Magnetic Permeability

Why does the susceptibility of a paramagnet increase as it gets colder? Intuitively cold things respond less.

The moments are permanent — temperature does not create or destroy them, it only scrambles their directions. The applied field's alignment energy $\mu B$ is fixed; the scrambling budget $k_BT$ shrinks as you cool. So the same field wins a larger fraction of the orientation vote at lower temperature: response grows. Cold things respond less when the response itself costs thermal activation (conductivity in a semiconductor, say); here the response is a competition the field wins by default whenever thermal noise stands down. The limit is saturation: once $\mu B\gtrsim k_BT$ every moment is aligned, nothing more can be won, and the $1/T$ climb flattens onto the Brillouin plateau.

Where does the strange J(J+1) come from, and why isn't the Curie constant built on the maximum moment gJ?

From quantum fluctuations of direction. A quantized angular momentum has magnitude $\sqrt{J(J+1)}\,\hbar$ but maximum projection $J\hbar$ — the vector can never point fully along any axis, and its transverse components fluctuate irreducibly. Linear response at high temperature samples all orientations nearly equally, so it feels the full mean-square moment, $g^2\mu_B^2J(J+1)$ . Saturation, by contrast, parks every ion in its top state and reads only the maximum projection $gJ\mu_B$ . Both numbers are measurable — Curie constant and saturation magnetization respectively — and their ratio $\sqrt{(J+1)/J}$ is a consistency check on the assigned $J$ : for Gd³⁺, $7.94/7.0=1.13=\sqrt{9/7}$ ✓.

Is the Weiss molecular field a real magnetic field that a probe inside the material would measure?

No — and this is one of the most important honesty points in the chapter. A muon or neutron probing the internal field of iron measures fields of order teslas (dipolar and hyperfine contributions), not the thousand-tesla $\lambda\mu_0M$ the Weiss theory requires. The molecular field is an energy bookkeeping device: the exchange interaction lowers the energy of a spin by an amount proportional to its neighbors' average alignment, and dividing that energy by the spin's moment produces a quantity with field units. It aligns spins exactly as a real field of that size would — which is why the substitution works — but it exerts no Lorentz force, deflects no charge, and appears in no flux integral. Exchange is electrostatics filtered through the Pauli principle; the chapter closer derives it.

Why does theta from the Curie–Weiss fit come out slightly above the actual ordering temperature?

Because mean field assumes each moment feels only the average of its neighbors, ignoring correlations. In reality, as $T$ approaches $T_C$ from above, spins form transient aligned patches — short-range order without long-range order — and these patches enhance the susceptibility beyond the mean-field prediction. Fitting the straight Curie–Weiss line to data taken well above $T_C$ and extrapolating therefore overshoots: the intercept $\theta$ lands a few percent above the true $T_C$ , where the actual (fluctuation-suppressed) divergence occurs. The gap $\theta-T_C$ is itself a crude fluctuation meter: tiny for three-dimensional, high-coordination magnets, large for low-dimensional ones — and infinite for an ideal 1D chain, which has $\theta>0$ but never orders at all.

How can the susceptibility literally diverge at T_C? Infinite response sounds unphysical.

The divergence is in the linear susceptibility — the slope $\partial M/\partial H$ at $H=0$ — and it signals not infinite magnetization but the death of the linear regime. At $T_C$ the free-energy cost of a small magnetization vanishes (the origin flattens), so an infinitesimal field produces a response limited only by the nonlinear terms: $M\propto H^{1/3}$ exactly at $T_C$ in mean field, finite for any finite field. Experimentally the divergence is also rounded by the demagnetizing feedback of Magnetic Boundary Conditions — a sample of shape factor $N$ can never show a measured $\chi$ above $1/N$ — and by inhomogeneity. What survives, and is measured to high precision, is a very large, sharply peaked $\chi(T)$ whose critical exponent $\gamma$ (mean field: 1; real: ≈1.39) is one of the universal numbers of the theory of phase transitions.

Do diamagnets obey the Curie law?

No — and the temperature dependence is the cleanest field diagnostic between the two responses. Diamagnetism (the Larmor response Diamagnetism derived) involves no permanent moments and no orientation statistics: it is an induced current proportional to $\langle r^2\rangle$ of the electron clouds, essentially temperature-independent. Curie behavior requires permanent moments fighting thermal disorder. This is the second fingerprint in Magnetic Susceptibility's three-fingerprint diagnostic: measure $\chi(T)$ — flat and negative means diamagnet; $1/T$ and positive means paramagnet; huge and history-dependent means ferromagnet. A material with both contributions (most have) shows a Curie term riding on a flat diamagnetic floor, and the fit $\chi=\chi_0+C/T$ separates them cleanly.

What happens between T_C and absolute zero — does the Brillouin machinery predict the whole M(T) curve of iron?

Mean field predicts the whole curve — solve equation (4) at each temperature and you get $M(T)$ rising from zero at $T_C$ toward saturation as $T\to0$ — and it is qualitatively right, quantitatively imperfect at both ends. Near $T_C$ the exponent is wrong (½ versus ≈0.36, fluctuations). Near $T=0$ mean field predicts exponentially small deviations from saturation, but the measured deficit follows $M_s-M(T)\propto T^{3/2}$ — the Bloch law — because the true low-energy excitations are spin waves (magnons), collective ripples of the ordered state that mean field cannot see. Both failures are signposts to the chapter closer and beyond: exchange builds the order, and quantized collective motion of that order — not individual spin flips — spends it.