Magnetic Domains

2D Ising model — Weiss domains, domain walls & B-H hysteresis loop

Material Presets

Ising Hamiltonian:
H = −J ∑<ij> sisj − μH ∑i si
Metropolis: accept flip if ΔE ≤ 0 or rand < e−ΔE/kT

B-H Loop Control

Live Values


0.00

0.00

300

–

50%

–
About — Magnetic Domains & the Ising Model

The Ising model

The 2D Ising model is the canonical model of ferromagnetism. Each site i on a square lattice has a spin si = ±1. Neighbours interact with coupling constant J. An external magnetic field H adds a Zeeman energy. The Hamiltonian is:

H = −J ∑ sisj − μH ∑ si

The exact 2D Ising model was solved by Onsager (1944). The critical Curie temperature is kBTc = 2J/ln(1+√2) ≈ 2.269J. Below Tc, spontaneous magnetisation appears; above Tc, the system is paramagnetic.

Metropolis algorithm

To simulate equilibrium configurations at temperature T, we use the Metropolis-Hastings Monte Carlo algorithm: pick a random spin, compute the energy change ΔE of flipping it, accept the flip if ΔE ≤ 0 or with probability exp(−ΔE/kBT) if ΔE > 0. This satisfies detailed balance and converges to the Boltzmann distribution.

Weiss domains and domain walls

A real ferromagnet breaks into magnetic domains where spins are all aligned, separated by domain walls (Bloch or Néel walls). Domains form to minimise magnetostatic energy (demagnetising field). The wall width δ ~ π√(A/K) (A = exchange stiffness, K = anisotropy constant). Iron domains are typically 10–300 μm wide.

B-H hysteresis loop

As H sweeps from +Hs to −Hs and back, the magnetisation traces the hysteresis loop. Key parameters:

  • Saturation Ms: maximum magnetisation (all spins aligned)
  • Remanence Mr: magnetisation remaining at H=0 after saturation
  • Coercive field Hc: H needed to reduce M back to zero
  • Energy loss per cycle: proportional to the loop area (Steinmetz law)

Soft magnets (Permalloy, soft ferrite) have low Hc (<10 A/m). Hard magnets (NdFeB, SmCo) have high Hc (>800 kA/m).