The exchange interaction J responsible for ferromagnetism has an electrostatic origin (Coulomb repulsion + Pauli exclusion), not a magnetic one. Why is this distinction important?
AIt means ferromagnetism is not really a magnetic phenomenon
BMagnetic dipole-dipole interactions between atomic moments give energies of order μ_B²/a³ ~ 0.1 K, far too small to explain Curie temperatures of ~1000 K. Exchange interactions are electrostatic (eV scale) and arise because the Pauli principle correlates spatial and spin wavefunctions — electrons with parallel spins must be spatially antisymmetric, reducing Coulomb repulsion
CIt means that ferromagnetism only occurs in metals
DThe exchange interaction is weaker than the dipole interaction but acts over longer range
If ferromagnetism were due to magnetic dipole forces, the ordering temperature would be ~0.1 K, not ~1000 K. The actual mechanism is exchange: the interplay between the Pauli exclusion principle (which correlates spin and spatial wavefunctions) and the Coulomb interaction (which depends on the spatial configuration). For two electrons, the triplet (parallel spin) state has an antisymmetric spatial wavefunction that keeps electrons apart, reducing Coulomb repulsion by an energy J. This exchange energy J is of order 0.01-0.1 eV, corresponding to temperatures of 100-1000 K — matching observed Curie temperatures.
Question 2 Multiple Choice
Mean-field theory predicts the critical exponent β = 1/2 for the spontaneous magnetization near T_C (M ∝ (T_C - T)^{1/2}). Experiment gives β ≈ 0.33 for real ferromagnets. What causes this discrepancy?
AMean-field theory neglects spin-orbit coupling
BMean-field theory replaces the actual fluctuating local environment with a single effective field, ignoring correlations and critical fluctuations near T_C. Near the critical point, fluctuations on all length scales become important, and the true critical behavior is determined by universality class (dimension and symmetry), not by mean-field theory
CThe Heisenberg model is fundamentally incorrect for real materials
DMean-field theory uses the wrong value of S for iron
Mean-field theory is exact in infinite dimensions but becomes increasingly inaccurate near T_C in real (3D) systems because it ignores the correlated fluctuations that dominate near critical points. The renormalization group shows that critical exponents depend only on the spatial dimension and symmetry of the order parameter (universality class), not on microscopic details. 3D Heisenberg models have β ≈ 0.365, 3D Ising β ≈ 0.326, both far from the mean-field β = 0.5. Mean-field theory remains useful far from T_C and for estimating T_C itself.
Question 3 True / False
The Heisenberg model H = -J Σ S_i · S_j treats all spin components (S^x, S^y, S^z) symmetrically. The Ising model keeps only H = -J Σ S_i^z S_j^z. How does this symmetry difference affect the physics?
TTrue
FFalse
Answer: True
This is not a true-false question as stated, but the key point is: the Heisenberg model has full SU(2) (continuous rotational) symmetry, meaning the magnetization can point in any direction. This allows Goldstone modes (spin waves/magnons) — low-energy excitations where the magnetization direction rotates smoothly. The Ising model has only Z₂ (discrete up/down) symmetry and has no such gapless excitations. In 2D, the Mermin-Wagner theorem forbids spontaneous breaking of continuous symmetry at finite T (no 2D Heisenberg ferromagnet), but the 2D Ising model can order. The symmetry of the order parameter determines the universality class and qualitative physics.
Question 4 Short Answer
Why are iron, cobalt, and nickel ferromagnetic while most other transition metals are not?
Think about your answer, then reveal below.
Model answer: Ferromagnetism requires that the exchange interaction J > 0 (favoring parallel spins) AND that the gain from exchange alignment exceeds the kinetic energy cost of spin polarization (Stoner criterion). In Fe, Co, Ni, the 3d band is narrow (high density of states at E_F) and less than half-filled or positioned such that the Stoner criterion I·g(E_F) > 1 is satisfied, where I is the exchange integral. Other transition metals like Cr, Mn, V have different band fillings where antiferromagnetic coupling or the Stoner criterion not being met prevents ferromagnetism. The specific band structure — not just the existence of d electrons — determines the magnetic ground state.
The Stoner criterion I·g(E_F) > 1 captures the competition: I measures the exchange energy gained by polarizing spins, g(E_F) measures how many states can be polarized at low energy cost. High g(E_F) (narrow bands, large density of states) favors ferromagnetism. This is why ferromagnetism is rare — most metals fail the criterion.