Weiss Molecular Field Theory

A block of iron below 1043 K magnetizes itself. No external field is applied, yet the electron spins line up spontaneously — billions of them agreeing on a single direction. How does a spin "decide" to align when the fields from its neighbours are, in any laboratory sense, absurdly weak? Pierre Weiss answered this in 1907 with one audacious idea: imagine that each spin feels an internal "molecular field" proportional to the average magnetization of the whole sample, $B_{\text{eff}}=B+\lambda M$ . Because $M$ itself depends on the field that produces it, the problem closes on itself into a self-consistency loop, and that loop has a runaway solution below a critical temperature — spontaneous order out of nothing. The number Weiss needed is staggering. To reproduce iron's Curie temperature the molecular field must be on the order of $10^{3}$ tesla — a thousand times stronger than the strongest steady magnet ever built, and utterly impossible as a real magnetostatic field from neighbouring dipoles. That very impossibility was the clue: the molecular field is not magnetism at all. It is the quantum exchange interaction, an electrostatic effect dressed up as a field, which only Heisenberg would explain twenty-one years later. Weiss guessed the phenomenology decades before its microscopic origin was understood. This article owns the Weiss construction: the effective-field ansatz, the self-consistency equation, the spontaneous magnetization below $T_c$ , and — above $T_c$ — the experimentally celebrated Curie–Weiss law $\chi=C/(T-T_c)$ whose straight-line $1/\chi$ plot is still how experimentalists read off a Curie point today. Weiss theory is the cleanest instance of the "average out your neighbours" strategy that later powers van der Waals fluids, Bragg–Williams alloys, and BCS superconductivity. It gets the mechanism exactly right and the critical exponents famously wrong — a productive failure that seeds the whole modern theory of phase transitions.

Key takeaways

Weiss replaces the intractable interaction between a spin and all its neighbours by a single self-consistent effective field $B_{\text{eff}}=B+\lambda M$ , where $\lambda$ is the molecular-field constant packaging the exchange coupling.
Solving the one-spin paramagnet in that field and demanding the magnetization out equal the magnetization in gives the self-consistency equation $m=\tanh(T_c m/T + \mu B/k_BT)$ for spin-½.
Below the Curie temperature $T_c$ a spontaneous magnetization appears even at $B=0$ , rising as $m\propto(T_c-T)^{1/2}$ near $T_c$ (mean-field exponent $\beta=\tfrac12$ ).
Above $T_c$ the susceptibility follows the Curie–Weiss law $\chi=C/(T-T_c)$ : a $1/\chi$ -versus- $T$ plot is a straight line intercepting the temperature axis exactly at $T_c$ .
The molecular field is enormous ( $\sim10^{3}$ T for iron) because its origin is the quantum exchange interaction, not magnetostatics — treating $\lambda M$ as a real magnetic field misses $T_c$ by orders of magnitude.
Key idea: assume every spin feels the average magnetization of its neighbours as an extra field $\lambda M$ , solve the resulting one-spin paramagnet exactly, then demand the magnetization you get out equals the one you put in — the self-consistency loop spontaneously orders below $T_c=\lambda C$ .

The core insight

Assume every spin feels the average magnetization of its neighbours as an extra internal field $\lambda M$ ; solve the resulting single-spin paramagnet exactly, then close the loop by demanding the magnetization it produces equal the magnetization that produced the field. Nonzero self-consistent solutions appear below $T_c=\lambda C$ — spontaneous order.

Where this lives today

The Curie–Weiss law remains a daily tool of experimental condensed-matter physics: fitting a straight line to $1/\chi$ versus $T$ in the paramagnetic phase yields the Curie temperature from the intercept and the effective magnetic moment from the slope, a routine first characterization of any new magnetic material. A negative intercept flags antiferromagnetic exchange (the Curie–Weiss $\theta$ is then negative), and the deviation of the measured moment from the spin-only value probes orbital contributions and crystal-field effects. Mean-field ideas descended from Weiss underpin the design of permanent magnets, magnetic-refrigeration materials, and spintronic devices. Its deeper legacy is conceptual. The self-consistent-field method Weiss invented is the direct ancestor of the Hartree–Fock approximation in electronic structure, the Bardeen–Cooper–Schrieffer gap equation in superconductivity, and the dynamical mean-field theory used today to attack strongly correlated electrons. Every time a physicist "decouples" an interaction by replacing a fluctuating quantity with its average and solving self-consistently, they are using Weiss's 1907 idea. And its very failure — mean-field critical exponents that ignore fluctuations and get the wrong power laws below four dimensions — was the puzzle that drove Landau's order-parameter theory, the Ginzburg criterion, and ultimately Wilson's renormalization group, the framework that finally computes critical exponents correctly.

The physical picture

Picture one spin buried in a crowd of neighbours. Instead of tracking every neighbour's instantaneous flicker, Weiss makes a bold averaging assumption: each neighbour points, on average, in the same net direction as the whole sample. Our spin therefore feels a single steady "consensus" field proportional to the overall magnetization $M$ . If that consensus field is strong enough to hold the spin aligned, the alignment reinforces itself — a cooperative instability. Cool the sample and the response steepens; at a sharp temperature $T_c$ the feedback becomes self-sustaining, and the magnetization locks in even with no external field. That runaway is ferromagnetism. The mechanism is a feedback loop, and the cleanest way to see the transition is graphically. The magnetization the spin produces in response to a field $\lambda M$ is a saturating curve, $m=\tanh(T_c m/T)$ at $B=0$ . Plot the straight line $y=m$ against the curve $y=\tanh(T_c m/T)$ : they always cross at the origin, but a second, nonzero crossing exists only when the initial slope $T_c/T$ of the tanh exceeds 1. Above $T_c$ the slope is too shallow — thermal agitation wins, $m=0$ is the only solution. Below $T_c$ the slope steepens past 1 and two symmetric nonzero solutions $\pm m$ appear. The tangency condition $T_c/T=1$ is the Curie temperature.

Figure

Graphical solution of the self-consistency equation m = tanh(T_c m/T). The line y = m always crosses the tanh at the origin; a second nonzero crossing appears only when the tanh's initial slope T_c/T exceeds 1 — that threshold defines T_c.

Why is the averaging trustworthy at all? Each spin couples to $z$ neighbours, and for large coordination number $z$ the summed neighbour field is sharply peaked about its mean — fluctuations scale as $1/\sqrt{z}$ and are suppressed. Mean-field theory is exactly the leading term of an expansion in $1/z$ (or in $1/d$ , inverse dimension), becoming exact for infinite-range interactions and above the upper critical dimension $d_c=4$ . The second figure shows the self-consistency as a closed loop: the spins set the magnetization, the magnetization sets the molecular field, and the molecular field acts back on the spins — a circuit that either dies out (paramagnet) or locks in (ferromagnet).

Figure

The Weiss self-consistency loop: the spins set the magnetization M, M sets the molecular field B_eff = B + λM, and that field acts back on the spins. Below T_c the loop gain exceeds 1 and the magnetization locks in.

The same "average out the neighbours" logic recurs across physics. The van der Waals fluid replaces pair attraction by an averaged background pressure $-a/v^2$ ; Bragg–Williams theory replaces the alloy's ordering interaction by an average occupation field; BCS superconductivity replaces the pairing interaction by a self-consistent gap $\Delta$ . Weiss theory is the pedagogical prototype of all of them — one spin, one field, one loop.

The core equations

Everything starts from the exchange Hamiltonian, in which spins prefer to align with their neighbours through a coupling $J$ (positive for a ferromagnet). This is the interaction Weiss must tame.

\hat{\mathcal{H}} = -\tfrac12\sum_{i\neq j} J_{ij}\,\mathbf{S}_i\!\cdot\!\mathbf{S}_j - g\mu_B B\sum_i S_i^z \qquad (1)

where: $J_{ij}$ — exchange coupling ( $>0$ ferromagnetic); $\mathbf{S}_i$ — spin at site $i$ ; $g$ — Landé factor; $\mu_B$ — Bohr magneton; $B$ — applied field. The first term is the many-body knot: the energy of spin $i$ depends on every neighbour's orientation, so the partition function will not factorize. The microscopic decoupling that unties it is owned by aq-sm-mean-field-theory; here we recall its result and build the magnetic thermodynamics. The mean-field move replaces each neighbour spin by its thermal average $\langle\mathbf{S}\rangle$ , keeping only the terms linear in the fluctuations and dropping the fluctuation–fluctuation product. That is the one approximation in the whole theory.

\mathbf{S}_i\!\cdot\!\mathbf{S}_j \;\approx\; \mathbf{S}_i\!\cdot\!\langle\mathbf{S}\rangle + \langle\mathbf{S}\rangle\!\cdot\!\mathbf{S}_j - \langle\mathbf{S}\rangle^2 \qquad (2)

where: $\langle\mathbf{S}\rangle$ — the thermal-average spin, identical on every site by translational symmetry. The subtracted $\langle\mathbf{S}\rangle^2$ corrects for double counting the interaction energy; the discarded term is exactly the spin–spin correlation $\delta\mathbf{S}_i\!\cdot\!\delta\mathbf{S}_j$ , which is what mean-field theory throws away. Summed over neighbours, the linearized interaction turns into an effective field seen by a single spin. This is the Weiss molecular field — the heart of the construction.

B_{\text{eff}} = B + \lambda M \qquad (3)

where: $B_{\text{eff}}$ — total field on one spin; $\lambda$ — Weiss molecular-field constant, set by the exchange strength and coordination number $z$ (schematically $\lambda\sim zJ/g^2\mu_B^2 n$ with $n$ the spin density); $M$ — magnetization. Read it physically: the field a spin feels is the applied field plus a self-generated field proportional to how magnetized the sample already is — the origin of the feedback. Solving the single-spin paramagnet in the field $B_{\text{eff}}$ and demanding self-consistency gives, for general spin $J$ , the Brillouin self-consistency equation. For spin-½ the Brillouin function collapses to a tangent, giving the reduced form used throughout this article.

M = Ng\mu_B J\,\mathcal{B}_J\!\big(\beta g\mu_B J\,B_{\text{eff}}\big)\ \xrightarrow{\;J=1/2\;}\ m=\tanh\!\Big(\frac{T_c}{T}\,m+\frac{\mu B}{k_BT}\Big) \qquad (4)

where: $\mathcal{B}_J$ — Brillouin function (owned by the paramagnetism article; we recall, not re-derive it); $m=M/M_0$ — reduced magnetization, $M_0=N\mu$ the saturation value; $\mu=g\mu_B/2$ for spin-½. The identity $\mathcal{B}_{1/2}(x)=\tanh(x)$ is what reduces the general result to the clean tanh loop. This transcendental equation is the entire theory: its solutions are the magnetic states. Linearizing the loop above $T_c$ produces the experimentally famous susceptibility. The bare Curie response $\chi_0=C/T$ is amplified by the molecular field into a divergence at $T_c$ .

\chi=\frac{C}{T-T_c},\qquad T_c=\lambda C,\qquad C=\frac{N\mu_0 g^2\mu_B^2 J(J+1)}{3k_B} \qquad (5)

where: $\chi$ — susceptibility; $C$ — Curie constant; $T_c=\lambda C$ — Curie temperature, where $\chi$ diverges. The Curie–Weiss law is the workhorse of experimental magnetism: a plot of $1/\chi$ versus $T$ is a straight line whose intercept on the temperature axis reads off $T_c$ and whose slope gives the effective moment through $C$ .

Derivation ACurie–Weiss law and the divergence at T_c

The goal is to derive the paramagnetic susceptibility above the Curie temperature directly from the self-consistency loop, and to show it diverges as $(T-T_c)^{-1}$ with the Curie temperature identified as $T_c=\lambda C$ . The strategy is to linearize the response for small magnetization (valid above $T_c$ , where $M\to0$ as $B\to0$ ), feed the molecular field back into the bare Curie law, and solve the resulting algebraic feedback equation for $\chi=M/B$ . The one subtlety to watch is that the spin responds not to the applied field $B$ but to the total field $B_{\text{eff}}=B+\lambda M$ , so $M$ appears on both sides — that self-reference is exactly what turns the finite Curie response into a divergence.

Step 1. Write the bare Curie response to the total field. A single paramagnetic spin obeys the Curie law $M=\chi_0 B_{\text{eff}}$ with $\chi_0=C/T$ , but the field is the molecular field: $M=\dfrac{C}{T}\big(B+\lambda M\big)$ . Why: above $T_c$ the magnetization is small, so the tanh in equation (4) linearizes to its argument; the ordinary Curie law then applies, but to $B_{\text{eff}}$ , not $B$ .

Step 2. Collect the magnetization terms. Move the $\lambda M$ feedback to the left: $M\Big(1-\dfrac{\lambda C}{T}\Big)=\dfrac{C}{T}\,B$ . Why: the molecular field makes the response self-amplifying; the factor $(1-\lambda C/T)$ measures how close the loop gain is to runaway.

Step 3. Form the susceptibility. Divide through by $B$ : $\chi=\dfrac{M}{B}=\dfrac{C/T}{1-\lambda C/T}=\dfrac{C}{T-\lambda C}$ . Why: the bare Curie susceptibility $C/T$ is divided by $(1-\lambda C/T)$ , the loop-gain deficit; as $T$ falls toward $\lambda C$ the denominator vanishes.

Step 4. Identify T_c and read the divergence. Define $T_c\equiv\lambda C$ , so $\chi=\dfrac{C}{T-T_c}$ , diverging as $(T-T_c)^{-1}$ . Why: the susceptibility blows up precisely where the loop gain reaches 1 — the onset of spontaneous order. Plotting $1/\chi=(T-T_c)/C$ gives a straight line crossing zero at $T_c$ , the standard experimental fingerprint, with critical exponent $\gamma=1$ . *Verified: the algebraic solution $\chi=C/(T-\lambda C)$ and the identification $T_c=\lambda C$ are checked symbolically in verify_weiss-molecular-field.py, matching Pathria & Beale §12.5 and Reif §10.7.*

Go deeper · Derivation B · 1 minSpontaneous magnetization below T_c and the exponent β = 1/2

Derivation BSpontaneous magnetization below T_c and the exponent β = 1/2

Here the goal is to solve the zero-field self-consistency equation just below $T_c$ and extract how the spontaneous magnetization rises from zero — the order-parameter onset and its critical exponent $\beta$ . The route is to set $B=0$ in the spin-½ loop $m=\tanh(T_c m/T)$ , expand the tanh to cubic order (legitimate because $m$ is small near $T_c$ ), cancel the trivial root, and solve for $m^2$ . What to watch is the sign structure: the cubic term must have the right sign to give a real, growing solution below $T_c$ and none above, and the leading behaviour near $T_c$ fixes $\beta=\tfrac12$ — the mean-field value that is famously too large compared with the exact 2D result $\beta=\tfrac18$ .

Step 1. Expand the tanh to cubic order. With $x=T_c m/T$ , use $\tanh x=x-\tfrac13x^3+O(x^5)$ , so $m=\dfrac{T_c}{T}m-\dfrac13\Big(\dfrac{T_c}{T}\Big)^3 m^3+\cdots$ . Why: near $T_c$ the spontaneous $m$ is small, so only the first two terms of the tanh matter; the cubic term is what stabilizes a finite $m$ .

Step 2. Cancel the trivial root. The solution $m=0$ always exists; divide the remaining equation by $m\neq0$ to get $1=\dfrac{T_c}{T}-\dfrac13\Big(\dfrac{T_c}{T}\Big)^3 m^2$ . Why: we want the ordered branch, not the disordered one; dividing out $m$ isolates the equation for the nonzero magnetization.

Step 3. Solve for m². Rearranging, $m^2=3\Big(\dfrac{T}{T_c}\Big)^3\Big(\dfrac{T_c}{T}-1\Big)=3\Big(\dfrac{T}{T_c}\Big)^2\Big(\dfrac{T_c-T}{T_c}\Big)$ . Why: a real positive $m^2$ requires $T<T_c$ — no ordered solution exists above $T_c$ , exactly as the graphical picture demanded.

Step 4. Take the limit near T_c. As $T\to T_c^-$ , $(T/T_c)^2\to1$ , leaving $m^2\simeq 3\big(1-T/T_c\big)$ , hence $m\simeq\sqrt{3}\,\big(1-T/T_c\big)^{1/2}$ . Why: the order parameter rises as a square root, so the critical exponent is $\beta=\tfrac12$ . This overshoots the true onset (exact 2D Ising $\beta=\tfrac18$ ) because mean-field theory ignores the fluctuations that soften real ordering — a limitation stated honestly, and quantified by the Ginzburg criterion downstream. *Verified: the expansion $m^2=3(r-1)/r^3$ with $r=T_c/T$ and its near- $T_c$ limit $m^2\to3(1-T/T_c)$ (prefactor $\sqrt3$ , $\beta=\tfrac12$ ) are checked symbolically in verify_weiss-molecular-field.py; the full mean-field exponent set is owned by aq-sm-mean-field-critical-exponents.*

Limiting cases

The spin-½ self-consistency equation $m=\tanh(T_c m/T+\mu B/k_BT)$ has clean behaviour in each regime.

Case	Result	Meaning
High temperature, $T\gg T_c$	$\chi\simeq C/(T-T_c)$	Curie–Weiss law; reduces to ordinary Curie $C/T$ when $T_c\to0$ (no exchange).
Just below $T_c$	$m\simeq\sqrt{3}\,(1-T/T_c)^{1/2}$	Spontaneous magnetization onsets continuously with mean-field exponent $\beta=\tfrac12$ .
At $T_c$ , small field	$m\simeq(3\,\mu B/k_BT_c)^{1/3}$	Critical isotherm, exponent $\delta=3$ : magnetization grows as the cube root of the field.
Zero temperature	$m\to1$ , $M\to M_0=N\mu$	Complete saturation: every spin aligned, magnetization at its maximum.
Infinite range / $d\to\infty$	mean field becomes exact	For infinite-range (Curie–Weiss) interactions and above $d_c=4$ , the mean-field exponents are correct; fluctuation corrections vanish.

Common trapThe molecular field is a real magnetic field

It is not — and mistaking it for one is the classic error. The Weiss constant $\lambda$ is far too large to come from magnetic dipole–dipole interactions: reproducing iron's Curie point demands a molecular field of order $10^{3}$ tesla, thousands of times any real magnetostatic field from neighbouring moments (which are $\sim10^{-1}$ T at best). The true origin is the quantum exchange interaction — an electrostatic effect rooted in the Pauli principle and Coulomb repulsion, with an energy scale of order an electron-volt per pair, thousands of times the $\sim10^{-4}$ eV magnetostatic coupling. Treat $\lambda M$ as a literal magnetic field and you underestimate $T_c$ by orders of magnitude. Weiss himself did not know its origin in 1907; Heisenberg supplied it in 1928.

Worked examples

1 · The molecular field of iron (confidence builder)

We estimate the magnitude of Weiss's internal field for iron and confront its impossibility as a real magnetic field.

Step 1. Iron orders ferromagnetically up to $T_c=1043$ K, with an effective moment per atom $\mu\approx2.2\,\mu_B=2.2\times9.274\times10^{-24}=2.04\times10^{-23}\ \text{J/T}$ . At the transition, thermal energy $k_BT_c$ balances the magnetic energy $\mu B_{\text{mf}}$ that keeps a spin aligned in the molecular field, so $B_{\text{mf}}\sim k_BT_c/\mu$ .

Step 2. Compute: $B_{\text{mf}}=\dfrac{k_BT_c}{\mu}=\dfrac{(1.381\times10^{-23})(1043)}{2.04\times10^{-23}}=\mathbf{706\ T}$ — a molecular field of order $10^{3}$ tesla.

Step 3. For comparison, the strongest continuous laboratory magnets reach $\sim45$ T; the dipolar field from a neighbouring atomic moment at $\sim2.5$ Å is only $\sim0.1$ T. The Weiss field exceeds both by three to four orders of magnitude. No magnetostatic mechanism can produce it.

Step 4. The resolution: $\lambda M$ is not magnetism but the exchange interaction in disguise. Cross-checking through the saturation magnetization $M_{\text{sat}}\approx1.7\times10^{6}$ A/m gives a molecular-field constant $\lambda=B_{\text{mf}}/M_{\text{sat}}\approx4.2\times10^{-4}$ T·m/A, and $\lambda M_{\text{sat}}$ returns the same $\mathbf{706\ T}$ — internally consistent. The enormity of $\lambda$ is precisely the signature of exchange, not dipole coupling.

2 · Reading a Curie point off a susceptibility plot (exam level)

A paramagnetic sample above its Curie temperature has measured inverse susceptibilities $1/\chi=2.0$ at $T=400$ K and $1/\chi=4.0$ (same arbitrary units) at $T=600$ K. Find $T_c$ and the Curie constant.

Step 1. The Curie–Weiss law $\chi=C/(T-T_c)$ rearranges to $1/\chi=(T-T_c)/C$ , a straight line in $T$ with slope $1/C$ and intercept $-T_c/C$ . Two data points fix both.

Step 2. The slope is $\dfrac{\Delta(1/\chi)}{\Delta T}=\dfrac{4.0-2.0}{600-400}=\dfrac{2.0}{200}=0.010\ \text{units/K}=1/C$ , so $C=100$ K (in these units).

Step 3. Extrapolate to $1/\chi=0$ : using $1/\chi=0.010\,(T-T_c)$ and the point $(400,2.0)$ , $2.0=0.010\,(400-T_c)$ , so $400-T_c=200$ , giving $T_c=\mathbf{200\ K}$ .

Step 4. The straight-line $1/\chi$ plot has thus delivered both the Curie temperature (the axis intercept, $\mathbf{200\ K}$ ) and the Curie constant ( $\mathbf{100\ K}$ , from which the effective moment follows via $C\propto\mu_{\text{eff}}^2$ ). This extrapolation is exactly how Curie temperatures are extracted from paramagnetic-phase data in practice — and a nonzero intercept, versus the pure-Curie intercept at the origin, is the direct evidence of an internal exchange field.

Origins · 1 minPierre Weiss and how this was discovered

Pierre Weiss

1865–1940, French physicist, ETH Zürich and Strasbourg · molecular-field theory of ferromagnetism, 1907

Pierre-Ernest Weiss was the architect of the first successful theory of ferromagnetism. Building on Pierre Curie's 1895 measurements of the temperature dependence of magnetization, Weiss proposed in 1907 that each atomic magnet experiences an internal "molecular field" ( $champ moléculaire$ ) proportional to the sample's own magnetization — a self-consistent effective field that could hold spins aligned even without an applied field. The idea explained the existence of a sharp Curie temperature, the spontaneous magnetization below it, and the Curie–Weiss law above it, all at a stroke. Weiss recognized that the required field was enormous — thousands of tesla — and therefore could not be ordinary magnetism, but he could not identify its origin; that had to wait for Heisenberg's 1928 exchange interaction. Weiss also introduced the notion of magnetic domains to reconcile spontaneous magnetization with the observation that a bulk iron bar is often unmagnetized, and his name attaches to the "Weiss magneton," an early (and ultimately superseded) unit of magnetic moment. His molecular-field method became the template for every mean-field theory that followed, from van der Waals fluids to BCS superconductivity, making him one of the founders of the modern theory of collective phenomena.

History

1895

— Pierre Curie measures the temperature dependence of paramagnetic susceptibility, establishing Curie's law χ ∝ 1/T.

1907

— Pierre Weiss proposes the molecular-field theory of ferromagnetism and the Curie–Weiss law, and introduces magnetic domains.

1911

— Weiss defines the "Weiss magneton" as an empirical unit of atomic magnetic moment.

1928

— Heisenberg identifies the quantum exchange interaction as the true origin of the huge molecular field.

1944

— Onsager's exact solution of the 2D Ising model shows mean-field critical exponents are quantitatively wrong (β = 1/8, not 1/2).

Going further

Texts. Pathria & Beale, Statistical Mechanics, 3rd ed., §12.5 (Ising model, mean-field / Weiss theory). Reif, Fundamentals of Statistical and Thermal Physics, §10.7 (ferromagnetism, molecular field). Kardar, Statistical Physics of Fields, Ch. 1–3 (saddle point / mean field). Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 8–12. Landau & Lifshitz, Statistical Physics Part 1, §§146–148. Blundell, Magnetism in Condensed Matter, Ch. 5–6.

Provenance. Base article: AllQFT track `sm`, "Weiss Molecular Field Theory" (STAT.INTER.WEISS.1255); source Pathria & Beale, Statistical Mechanics, 3rd ed., §12.5. All symbolic and numeric claims re-verified here in verify_weiss-molecular-field.py (the mean-field decoupling identity, the Curie–Weiss law $\chi=C/(T-\lambda C)$ with $T_c=\lambda C$ , the spontaneous magnetization $m\simeq\sqrt3(1-T/T_c)^{1/2}$ giving $\beta=\tfrac12$ , the spin-½ Curie temperature $k_BT_c=zJ/4$ , the Brillouin reduction $\mathcal{B}_{1/2}(x)=\tanh x$ , and the iron molecular field $\sim706$ T); no base physics error found. The microscopic decoupling is owned by aq-sm-mean-field-theory; the full mean-field exponent set by aq-sm-mean-field-critical-exponents; the Brillouin function by the paramagnetism article, recalled here by name.

Prerequisites. Partition Function and Free Energy · Paramagnetism and the Brillouin Function · Ising and Heisenberg Models · Configuration integral.

Leads to. Landau Theory of Phase Transitions · Critical Exponents and Universality · Van der Waals Equation of State · Maxwell Construction and Coexistence.

If the molecular field is not a real magnetic field, why call it a field at all?

Because mathematically it acts exactly like one: in the mean-field decoupling, the interaction term for a single spin becomes $-\mu B_{\text{eff}}S^z$ with $B_{\text{eff}}=B+\lambda M$ , identical in form to a Zeeman coupling to an external field. So a spin in the molecular field behaves precisely like a paramagnet in a real field of that strength — same tanh response, same statistics. The word "field" captures how it enters the equations. What it is not is a magnetostatic field you could measure with a compass; its physical origin is the exchange interaction, an electrostatic effect. The name is a faithful description of its role, not a claim about its source.

Why does the self-consistency equation have a nonzero solution only below T_c?

Graphically, the equation $m=\tanh(T_c m/T)$ asks where the straight line $y=m$ meets the tanh curve. Both always cross at the origin. A second crossing at nonzero $m$ exists only if the tanh starts out steeper than the line — that is, if its initial slope $T_c/T$ exceeds 1. That happens exactly when $T<T_c$ . Above $T_c$ the tanh is too shallow and the only intersection is $m=0$ (disorder). At $T=T_c$ the slope equals 1, the two solutions merge, and below it they split into $\pm m$ . So the transition is the moment the feedback loop's gain crosses unity.

Why does Weiss theory predict the same critical exponents in one, two, and three dimensions?

Because it replaces every neighbour spin by the same average, dimensionality enters only through the coordination number $z$ inside $T_c$ — never through the exponents. All spatial structure has been averaged away, so the theory cannot distinguish 1D from 3D near criticality. This is wrong: real systems have fluctuations that grow near $T_c$ and depend strongly on dimension. The exact 1D Ising model has no finite-temperature transition at all; the exact 2D model has $\beta=\tfrac18$ , not $\tfrac12$ . Mean-field exponents become correct only at and above the upper critical dimension $d_c=4$ , where fluctuations are irrelevant.

How is the Weiss constant λ related to the microscopic exchange coupling J?

Summing the mean-field interaction over a spin's $z$ nearest neighbours, each contributing an averaged exchange energy $\sim J\langle S\rangle$ , produces an effective field proportional to the magnetization. Matching the resulting Curie temperature to $k_BT_c\sim zJ$ fixes $\lambda$ in terms of the exchange integral $J$ , the coordination number $z$ , and the moment per spin. So Weiss's phenomenological $\lambda$ and the microscopic Ising/Heisenberg $J$ are two descriptions of the same physics: $\lambda$ is $J$ repackaged as a magnetic field. This equivalence is why Weiss theory and the Ising mean-field theory (owned by aq-sm-mean-field-theory) give identical results — same loop, different variables.

What exactly does mean-field theory throw away, and why does that matter?

It discards the correlation term $\delta\mathbf{S}_i\!\cdot\!\delta\mathbf{S}_j$ — the joint fluctuation of two neighbouring spins around their average. Keeping only how each spin correlates with the average magnetization, it ignores how spins correlate with each other. Near $T_c$ those neglected correlations become long-ranged and enormous (the correlation length diverges), which is exactly where mean-field theory fails and real critical behaviour sets in. Far from $T_c$ , or in high dimensions where each spin has many neighbours and fluctuations self-average, the discarded term is genuinely small and mean-field theory is accurate.

Does Weiss theory correctly predict that iron loses its magnetism when heated?

Yes, qualitatively and with the right mechanism. It predicts a sharp Curie temperature above which spontaneous magnetization vanishes — iron does demagnetize above 1043 K, exactly as observed. It also correctly gives a continuous (second-order) transition and the Curie–Weiss susceptibility above it. What it gets wrong is quantitative: it overestimates $T_c$ (because it ignores the fluctuations that destabilize order), and near the transition the magnetization falls off with the wrong power ( $\beta=\tfrac12$ instead of the measured $\approx0.33$ for a 3D ferromagnet). The mechanism and the phase diagram are right; the critical exponents are not.