Lab 04 — Magnetism

Spins & the Magnetic Phase Transition

Each cell is a microscopic magnet (a spin, ↑ = green / ↓ = gold). Neighbours lower their energy by aligning; heat scrambles them — the 2D Ising model. Cool below the critical temperature Tc ≈ 2.27 and the disordered lattice suddenly locks into one orientation: spontaneous magnetization, the reason iron can be a magnet at all. An external field H tilts the balance either way.

E = -J \sum_{\langle ij \rangle} s_i s_j \; - \; H \sum_i s_i \qquad P(\mathrm{flip}) = \min\!\left(1,\; e^{-\Delta E / T}\right) \qquad T_c = \dfrac{2}{\ln\!\left(1 + \sqrt{2}\right)} \approx 2.269

Drop the temperature below Tc and watch domains grow

TemperatureT = 3.50

External fieldH = 0.00

MagnetizationM = 0.00

The chart tracks M(t) from −1 (all ↓) to +1 (all ↑). Just below Tc, green and gold islands — magnetic domains — wrestle and slowly grow, exactly as inside a real magnet. The faint glow along their edges marks the domain walls: the places where opposite spins meet and pay an extra energy cost. At H = 0, which orientation wins is pure chance: spontaneous symmetry breaking.