Weiss Mean-Field Theory of Ferromagnetism

Derivation

Weiss Mean-Field Theory of Ferromagnetism

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Statement

For a lattice of quantum spins \(S\) coupled by the isotropic Heisenberg exchange \(-J\,\hat{\vec S}_i\cdot\hat{\vec S}_j\) over \(z\) nearest neighbours, the Weiss mean-field approximation replaces the exchange with a self-consistent molecular field \(B_{\mathrm{mf}}=\Lambda M\). This yields the self-consistency relation \(M = n g\mu_B S\,B_S\!\left(\tfrac{g\mu_B S(B_0+\Lambda M)}{k_B T}\right)\), a continuous ferromagnetic transition at \(k_B T_c = \tfrac{1}{3}zJS(S+1)\), and the Curie–Weiss susceptibility \(\chi = C/(T-T_c)\) for \(T>T_c\).

Why it matters

Weiss mean-field theory is the first-principles link between the microscopic quantum exchange interaction (itself a consequence of antisymmetry, not of any magnetic dipole force) and the macroscopic phenomena of spontaneous magnetisation and the Curie point. It explains why a purely electrostatic effect can drive collective ordering at temperatures of hundreds of kelvin, and it fixes the molecular-field constant that Weiss postulated in 1907 in terms of \(J\) and the coordination number \(z\).

Beyond ferromagnetism, it is the archetype of every mean-field theory of symmetry breaking — the same algebra reappears in BCS superconductivity, the van der Waals gas, and Landau theory — so its successes and its characteristic failures (wrong exponents, no order in low dimension) are the template for understanding when a mean field is trustworthy.

Assumptions

Localised, well-defined spins.If the magnetism is itinerant (band electrons, as in Fe/Ni) the moment is non-integer and \(T\)-dependent; a Heisenberg model with fixed \(S\) misrepresents the amplitude fluctuations and Stoner physics is needed.

Isotropic, nearest-neighbour exchange \(J\) with \(J>0\).If \(J<0\) the same treatment gives antiferromagnetic order (staggered field) and the uniform \(\chi\) develops a Néel, not Curie, temperature; longer-range or frustrated \(J\) shift \(T_c\) and can suppress order entirely.

Each spin feels only the average of its neighbours (mean-field decoupling).Correlated fluctuations \(\delta\hat{\vec S}_i\cdot\delta\hat{\vec S}_j\) are discarded; near \(T_c\) they diverge, so critical exponents and the very existence of order in low \(d\) come out wrong.

The order parameter is spatially uniform and the free energy is analytic in \(M\).If dropped, one must keep a spatially varying \(M(\vec r)\) (Ginzburg–Landau) to capture domains, spin waves, and the non-analytic critical singularities the uniform ansatz misses.

Derivation

\[\hat H=-J\sum_{\langle ij\rangle}\hat{\vec S}_i\cdot\hat{\vec S}_j-g\mu_B B_0\sum_i \hat S_i^z\]

Isotropic Heisenberg model plus a Zeeman coupling to an applied field \(B_0\) along \(z\); \(\langle ij\rangle\) runs once over each nearest-neighbour bond. This is the starting model, taken as given. A

\[\hat{\vec S}_i\cdot\hat{\vec S}_j=\langle\hat{\vec S}\rangle\!\cdot\!\hat{\vec S}_j+\hat{\vec S}_i\!\cdot\!\langle\hat{\vec S}\rangle-\langle\hat{\vec S}\rangle\!\cdot\!\langle\hat{\vec S}\rangle+\underbrace{\delta\hat{\vec S}_i\!\cdot\!\delta\hat{\vec S}_j}_{\text{dropped}}\]

Write \(\hat{\vec S}_i=\langle\hat{\vec S}\rangle+\delta\hat{\vec S}_i\) and discard the product of fluctuations. Legal because in high coordination each spin averages over many neighbours, suppressing the neglected term by \(1/z\). B

\[\hat H\approx\sum_i\hat H_i+\text{const},\qquad \hat H_i=-g\mu_B\,B_{\mathrm{eff}}\,\hat S_i^z,\quad B_{\mathrm{eff}}=B_0+B_{\mathrm{mf}}\]

Summing the linearised bond terms over the \(z\) neighbours of each site collapses the many-body problem to \(N\) identical single-site problems; by symmetry \(\langle\hat{\vec S}\rangle=\langle\hat S^z\rangle\hat z\). The exchange has become an effective internal field. B

\[g\mu_B B_{\mathrm{mf}}=Jz\langle\hat S^z\rangle\;\Rightarrow\; B_{\mathrm{mf}}=\frac{Jz}{n(g\mu_B)^2}\,M\equiv\Lambda M,\qquad M=n g\mu_B\langle\hat S^z\rangle\]

Match the neighbour term \(-Jz\langle\hat S^z\rangle\hat S_i^z\) to the Zeeman form \(-g\mu_B B_{\mathrm{mf}}\hat S_i^z\), then use the definition of the magnetisation (\(n\) spins per unit volume) to express it through \(M\). This identifies the Weiss constant \(\Lambda\). A

\[Z_1=\sum_{m=-S}^{S}e^{\,g\mu_B B_{\mathrm{eff}}m/k_BT},\qquad \langle\hat S^z\rangle=\frac{\partial \ln Z_1}{\partial(g\mu_B B_{\mathrm{eff}}/k_BT)}=S\,B_S(y)\]

Single spin in a field is an exactly solvable \((2S+1)\)-level problem; the geometric sum and its logarithmic derivative give the Brillouin function \(B_S\) with \(y=g\mu_B S B_{\mathrm{eff}}/k_BT\). Uses the Boltzmann distribution over the exact eigenstates \(\hat S^z=m\). A

\[B_S(y)=\frac{2S+1}{2S}\coth\!\left(\frac{2S+1}{2S}y\right)-\frac{1}{2S}\coth\!\left(\frac{y}{2S}\right)\]

\[\boxed{\,M=n g\mu_B S\,B_S\!\left(\frac{g\mu_B S(B_0+\Lambda M)}{k_B T}\right)\,}\]

Insert \(B_{\mathrm{eff}}=B_0+\Lambda M\): the argument now contains \(M\) itself, so this transcendental equation must hold self-consistently — the field that aligns the spins is the field they generate. A

\[\coth x=\frac{1}{x}+\frac{x}{3}-\frac{x^3}{45}+\cdots\;\Rightarrow\; B_S(y)=\frac{S+1}{3S}\,y-\frac{(S+1)\big[(S+1)^2+S^2\big]}{90\,S^3}\,y^3+\cdots\]

Expand both \(\coth\) terms; the \(1/x\) poles cancel and the linear pieces combine to the exact slope \(B_S'(0)=(S+1)/3S\). Retaining the cubic term is what later fixes the mean-field exponent \(\beta=\tfrac12\). Legal as a convergent Laurent expansion for \(|y|<\pi\). C

\[M=\frac{n(g\mu_B)^2S(S+1)}{3k_BT}\,\Lambda M\;\Rightarrow\;1=\frac{n(g\mu_B)^2S(S+1)\Lambda}{3k_BT_c},\qquad \boxed{\,k_BT_c=\tfrac13 zJS(S+1)\,}\]

Set \(B_0=0\) and keep only the linear term of Step 7. A non-zero \(M\) exists exactly when the self-consistency slope equals one; substituting \(\Lambda=Jz/[n(g\mu_B)^2]\) cancels the material prefactors and leaves the Curie temperature in microscopic terms. B

\[M=\frac{C'}{T}(B_0+\Lambda M),\;\; C'=\frac{n(g\mu_B)^2S(S+1)}{3k_B}\;\Rightarrow\; M=\frac{C'B_0}{T-C'\Lambda}=\frac{C'B_0}{T-T_c}\]

\[\chi=\frac{\mu_0 M}{B_0}=\frac{C}{T-T_c},\qquad C=\mu_0 C'=\frac{\mu_0 n(g\mu_B)^2 S(S+1)}{3k_B}\]

For \(T>T_c\) and small \(B_0\), linearise Step 6 and solve for \(M\); since \(C'\Lambda=T_c\) the denominator is \(T-T_c\). Dividing by \(B_0\) (and multiplying by \(\mu_0\) for the dimensionless volume susceptibility \(M/H\)) gives the Curie–Weiss law. B

Result

\[M=n g\mu_B S\,B_S\!\left(\frac{g\mu_B S(B_0+\Lambda M)}{k_B T}\right),\quad k_BT_c=\frac{zJS(S+1)}{3},\quad \chi=\frac{C}{T-T_c}\]

Reading. The magnetisation solves its own field equation: below \(T_c\) the self-consistency curve has a slope steeper than one at the origin, so a stable non-zero \(M\) branches off continuously from zero — a second-order transition. \(T_c\) rises linearly with the exchange \(J\) and the coordination \(z\), and with the "spin size" \(S(S+1)\). Above \(T_c\) the susceptibility diverges as \(1/(T-T_c)\): the Curie \(1/T\) law of independent moments, but shifted so the divergence occurs at \(T_c\) rather than \(0\), the shift measuring the strength of the molecular field.

Units check. \(k_BT_c=\tfrac13 zJS(S+1)\): \([J]\) is energy, \(z,S(S+1)\) dimensionless, so the right side is an energy matching \(k_BT_c\); \(T_c\) in kelvin. In \(\chi=C/(T-T_c)\), \(C=\mu_0 n(g\mu_B)^2S(S+1)/3k_B\) has \(\mu_0(g\mu_B)^2 n/k_B\to (\mathrm{T\,m/A})(\mathrm{J/T})^2(\mathrm{m^{-3}})/(\mathrm{J/K})=\mathrm{K}\), so \(C/(T-T_c)\) is dimensionless — correct for \(M/H\).

Limiting cases

\(S=\tfrac12\): \(B_{1/2}(y)=\tanh y\); the self-consistency becomes \(M=n g\mu_B\tfrac12\tanh\!\big(g\mu_B(B_0+\Lambda M)/2k_BT\big)\), the familiar Ising-like mean field.
\(S\to\infty\) (classical spin): \(B_S(y)\to L(y)=\coth y-1/y\), the Langevin function; recovers classical paramagnetism and the classical molecular-field theory.
\(T\gg T_c\): \(\chi\to C/T\), the pure Curie law of \(n\) non-interacting moments of effective size \(\mu_{\mathrm{eff}}=g\sqrt{S(S+1)}\,\mu_B\).
\(T\to0\): \(B_S\to1\), so \(M\to M_s=ng\mu_B S\), full saturation with every spin maximally aligned.
\(T\to T_c^-\): the cubic term of Step 7 gives \(M\propto(T_c-T)^{1/2}\), the mean-field order-parameter exponent \(\beta=\tfrac12\).

Breaks when

Low dimensions. In \(d\le2\) with continuous (Heisenberg) symmetry the Mermin–Wagner theorem forbids long-range order at any \(T>0\): thermal spin-wave fluctuations, entirely absent from the uniform mean field, destroy the ordered state, yet the theory predicts a finite \(T_c\) regardless of \(d\).
The critical region. Within the Ginzburg window around \(T_c\), correlated fluctuations dominate and the true exponents (\(\beta\approx0.365\), \(\gamma\approx1.39\) for 3D Heisenberg) replace the mean-field values \(\tfrac12,1\); the cusp shape and specific-heat singularity are qualitatively wrong.
Itinerant/weak ferromagnets. When the moment comes from delocalised band electrons, \(|M|\) itself is temperature-dependent and \(T_c\) is set by Stoner/spin-fluctuation physics, not by a fixed-\(S\) Heisenberg exchange.
Frustration or competing interactions. Antiferromagnetic or geometrically frustrated bonds make \(\langle\hat{\vec S}\rangle\) non-uniform or vanishing; a single uniform molecular field cannot represent spiral, glassy, or spin-liquid states.

Failure modes

Dropping \(z\). Writing \(k_BT_c=\tfrac13 JS(S+1)\) forgets that each spin has \(z\) neighbours; \(T_c\) scales with coordination and is off by a factor of 6–12 for real lattices.
\(S^2\) instead of \(S(S+1)\). Using the classical \(\langle S^z\rangle^2\to S^2\) rather than the quantum \(\hat{\vec S}^2\to S(S+1)\) gives the wrong \(T_c\) and wrong effective moment, especially for small \(S\).
Double-counting bonds. Confusing \(\sum_{\langle ij\rangle}\) (each bond once) with \(\tfrac12\sum_{i}\sum_{j}\) leads to a spurious factor of 2 in \(J\) and hence in \(T_c\).
Treating \(\Lambda\) (or \(T_c\)) as an independent fit parameter while also inserting the microscopic \(J\) — the two are tied by \(\Lambda=Jz/n(g\mu_B)^2\); using both independently over-counts the interaction.
Curie law where Curie–Weiss is needed. Fitting \(\chi=C/T\) to data just above \(T_c\) ignores the intercept; the correct \(1/\chi\)-vs-\(T\) line hits zero at \(T_c\), not the origin.
Forgetting \(\mu_0\). Reporting \(\chi=M/B_0\) rather than \(M/H=\mu_0M/B_0\) makes the "dimensionless" susceptibility carry stray units of \(\mathrm{T^{-1}}\).
Sign of \(J\). Plugging \(J<0\) into the ferromagnetic \(T_c\) formula yields a nonsensical negative temperature instead of switching to the antiferromagnetic (Néel) treatment.

Discussion

The essential move is to trade a genuinely many-body Hamiltonian for a single spin in a field that the spins themselves generate. The molecular field \(B_{\mathrm{mf}}=\Lambda M\) is not a magnetic dipole field — its magnitude, hundreds to thousands of tesla for a strong ferromagnet, dwarfs anything dipolar. It is the exchange energy in disguise, and exchange is electrostatic: it comes from the antisymmetry of the electronic wavefunction forcing spatially correlated charge distributions for parallel versus antiparallel spins. Weiss's phenomenological constant of 1907 is thereby given a quantum-mechanical value, \(\Lambda=Jz/n(g\mu_B)^2\).

The transition is a spontaneous breaking of the \(O(3)\) rotational symmetry of the Heisenberg Hamiltonian: above \(T_c\) the free energy is minimised at \(M=0\) and is rotationally invariant; below \(T_c\) a direction is chosen. Expanding the free energy in the uniform order parameter reproduces the Landau form \(F=F_0+a(T-T_c)M^2+bM^4\), from which \(\beta=\tfrac12\), \(\gamma=1\), and \(\delta=3\) follow immediately — the mean-field universality class shared with the van der Waals gas and BCS. Because the broken symmetry is continuous, the ordered phase supports gapless Goldstone modes (spin waves/magnons), which the static mean field cannot see but which control the low-temperature \(M(T)\) through Bloch's \(T^{3/2}\) law.

Mean-field theory is quantitatively reliable only where fluctuations are weak. The Ginzburg criterion compares the fluctuation of the order parameter over a correlation volume to its mean; it fails within a temperature window \(\Delta T/T_c\sim (a_0/\xi_0)^{6-d}\cdots\) and, more sharply, becomes exact only above the upper critical dimension \(d_c=4\). For \(d>4\) the mean-field exponents are correct; for \(d=3\) they are merely a first approximation, and for \(d\le2\) Heisenberg they are qualitatively wrong (no order). This is why the same simple algebra can be a triumph (predicting \(T_c\), the shape of \(M(T)\), the effective moment) and a systematic failure (every critical exponent) simultaneously.

Common misconceptions. The molecular field is not a real magnetic field one could measure with a probe — it is a bookkeeping device for the exchange energy, and it acts on spin, not on orbital moment. Also, \(T_c\) being finite in this theory for all \(d\) is an artefact of neglecting fluctuations, not a physical prediction; the theory's own assumptions are what break down in low dimension.

Worked examples

Curie temperature of a simple-cubic \(S=\tfrac12\) ferromagnet. Given \(z=6\) (simple cubic), \(S=\tfrac12\), and an exchange constant \(J/k_B=10\ \mathrm{K}\), find \(T_c\). A

\[k_BT_c=\frac{zJS(S+1)}{3}\;\Rightarrow\;T_c=\frac{z\,(J/k_B)\,S(S+1)}{3}\]

Symbolic form first; divide through by \(k_B\) so the given \(J/k_B\) enters directly.

\[T_c=\frac{6\times 10\ \mathrm{K}\times \tfrac12\!\cdot\!\tfrac32}{3}=\frac{6\times10\times0.75}{3}\ \mathrm{K}=15\ \mathrm{K}\]

\[T_c\approx 15\ \mathrm{K}\]

Reading. Even a modest \(10\ \mathrm{K}\) exchange, multiplied by six neighbours and the spin factor, orders the lattice at \(15\ \mathrm{K}\); doubling the coordination (bcc-like) would push \(T_c\) up proportionally.

Curie constant and paramagnetic susceptibility. For the same material take \(n=8\times10^{28}\ \mathrm{m^{-3}}\), \(g=2\), \(S=\tfrac12\), \(T_c=15\ \mathrm{K}\). Find the Curie constant \(C\) and the volume susceptibility \(\chi\) at \(T=300\ \mathrm{K}\). B

\[C=\frac{\mu_0 n (g\mu_B)^2 S(S+1)}{3k_B},\qquad \chi=\frac{C}{T-T_c}\]

Symbols first; then insert \(\mu_0=1.257\times10^{-6}\ \mathrm{T\,m/A}\), \(\mu_B=9.274\times10^{-24}\ \mathrm{J/T}\), \(k_B=1.381\times10^{-23}\ \mathrm{J/K}\), \(g\mu_B=1.855\times10^{-23}\ \mathrm{J/T}\).

\[C=\frac{(1.257\times10^{-6})(8\times10^{28})(1.855\times10^{-23})^2(0.75)}{3(1.381\times10^{-23})}\ \mathrm{K}\approx0.63\ \mathrm{K}\]

\[\chi=\frac{0.63\ \mathrm{K}}{(300-15)\ \mathrm{K}}=\frac{0.63}{285}\approx 2.2\times10^{-3}\]

\[C\approx0.63\ \mathrm{K},\qquad \chi(300\,\mathrm{K})\approx2.2\times10^{-3}\]

Reading. The dimensionless \(\chi\sim10^{-3}\) is a typical paramagnetic value well above \(T_c\); plotting \(1/\chi\) versus \(T\) would give a straight line intercepting the axis at \(T_c=15\ \mathrm{K}\), the signature that lets one read \(T_c\) off susceptibility data.

Problems

bcc Curie point. A body-centred-cubic magnet has \(z=8\), \(S=1\), and \(J/k_B=15\ \mathrm{K}\). Find \(T_c\).
Solution
\(k_BT_c=\tfrac13 zJS(S+1)\), with \(S(S+1)=2\). \(T_c=\tfrac13\times8\times15\ \mathrm{K}\times2=\tfrac{240}{3}\ \mathrm{K}=80\ \mathrm{K}\).
Extracting \(J\) from data. An fcc ferromagnet (\(z=12\)) with \(S=\tfrac12\) is measured to order at \(T_c=600\ \mathrm{K}\). Find \(J/k_B\).
Solution
Invert \(k_BT_c=\tfrac13 zJS(S+1)\): \(J/k_B=\dfrac{3T_c}{zS(S+1)}=\dfrac{3\times600}{12\times0.75}=\dfrac{1800}{9}=200\ \mathrm{K}\). So \(J\approx200\,k_B\approx2.8\times10^{-21}\ \mathrm{J}\) (\(\approx17\ \mathrm{meV}\)).
Size of the molecular field. Show that the saturation molecular field is \(B_{\mathrm{mf}}=\dfrac{3k_BT_c}{g\mu_B(S+1)}\) and evaluate it for iron-like parameters \(T_c=1043\ \mathrm{K}\), \(g=2\), \(S=1\).
Solution
At saturation \(\langle\hat S^z\rangle=S\), so \(B_{\mathrm{mf}}=Jz S/(g\mu_B)\) (from \(g\mu_B B_{\mathrm{mf}}=Jz\langle\hat S^z\rangle\)). Using \(Jz=3k_BT_c/[S(S+1)]\) gives \(B_{\mathrm{mf}}=\dfrac{3k_BT_c}{g\mu_B(S+1)}\). Numerically: \(\dfrac{3(1.381\times10^{-23})(1043)}{(2)(9.274\times10^{-24})(2)}\approx\dfrac{4.32\times10^{-20}}{3.71\times10^{-23}}\approx1.2\times10^{3}\ \mathrm{T}\) — of order \(10^3\) T, vastly larger than any dipolar field, confirming the field is exchange in origin.
Susceptibility above \(T_c\). A sample obeys Curie–Weiss with \(C=0.50\ \mathrm{K}\) and \(T_c=200\ \mathrm{K}\). Find \(\chi\) at \(T=250\ \mathrm{K}\) and at \(T=2T_c\).
Solution
\(\chi(250)=\dfrac{0.50}{250-200}=\dfrac{0.50}{50}=1.0\times10^{-2}\). At \(T=2T_c=400\ \mathrm{K}\): \(\chi=\dfrac{C}{2T_c-T_c}=\dfrac{C}{T_c}=\dfrac{0.50}{200}=2.5\times10^{-3}\). The susceptibility falls steeply as \(T\) moves away from \(T_c\).
Mean-field critical exponent. Using the cubic expansion \(B_S(y)\approx\frac{S+1}{3S}y-b\,y^3\) with \(b>0\), show that just below \(T_c\) (at \(B_0=0\)) the spontaneous magnetisation obeys \(M\propto(T_c-T)^{1/2}\), i.e. \(\beta=\tfrac12\).
Solution
Self-consistency at \(B_0=0\): \(\langle\hat S^z\rangle=S B_S(y)\) with \(y=g\mu_B S\Lambda M/k_BT\) and \(M=ng\mu_B\langle\hat S^z\rangle\), so \(y=(T_c/T)\,m\) where \(m\equiv M/M_s\) is the reduced magnetisation and \(M_s=ng\mu_BS\) (using \(C'\Lambda=T_c\) and \(B_S'(0)=(S+1)/3S\)). Then \(m=B_S(y)/S\) linearises to \(m=\frac{T_c}{T}m-\tilde b\left(\frac{T_c}{T}\right)^3 m^3\) for small \(m\), with \(\tilde b>0\). The trivial \(m=0\) aside, divide by \(m\): \(1=\frac{T_c}{T}-\tilde b(T_c/T)^3 m^2\), giving \(m^2=\dfrac{(T_c/T)-1}{\tilde b(T_c/T)^3}\approx\dfrac{T_c-T}{\tilde b\,T_c}\) near \(T_c\). Hence \(m\propto(T_c-T)^{1/2}\), so \(\beta=\tfrac12\) — the mean-field value, larger than the true 3D-Heisenberg \(\beta\approx0.365\).