Magnetism Explained: From Atoms to Hard Drives

Magnetism is a quantum mechanical phenomenon: it arises from the intrinsic spin of electrons and their orbital motion around nuclei. Understanding why iron sticks to a fridge but copper doesn't — and how permanent magnets store energy — requires delving from atomic-scale quantum mechanics to domain-scale classical physics used in billion-dollar technologies.

1. Atomic Origins of Magnetism

Magnetic moment of an electron: Two contributions: 1. Orbital angular momentum L: μ_L = -e/(2m_e) \cdot L (classical analogy: current loop) |μ_L| = μ_B \cdot \sqrt(l(l+1)) where μ_B = eℏ/2m_e = 9.274\times10⁻²⁴ J/T (Bohr magneton) 2. Spin angular momentum S (intrinsic, quantum mechanical — no classical analogue): μ_S = -g_S \cdot e/(2m_e) \cdot S g_S \approx 2.002 (Landé g-factor, corrected by QED) |μ_S| = \sqrt(s(s+1)) \cdot g_S \cdot μ_B For s = 1/2: μ_S \approx 1.73 μ_B (but z-projection = \pm1 μ_B) Hund's rules (filling atomic shells): 1. Maximise total spin S (half-filled shells most magnetic) 2. Then maximise orbital angular momentum L 3. J = |L-S| for less-than-half filled; J = L+S for more-than-half filled Fe atom: [Ar] 3d⁶ 4s² 3d shell has 6 electrons: 5 spin-up ↑↑↑↑↑ + 1 spin-down ↓ Net spin: S = 2 \to 4 unpaired spins \to large magnetic moment (μ = 4 μ_B per atom) Ferromagnetic materials need large uncompensated spin in partially filled d or f shells: Fe (4 μ_B), Co (3 μ_B), Ni (0.6 μ_B), Gd (7 μ_B — highest among elements)

2. Dia-, Para-, and Ferromagnetism

Materials respond to external magnetic fields in fundamentally different ways depending on their electronic structure:

Diamagnetism: All materials exhibit diamagnetism. Applied B-field induces orbital currents opposing the applied field (Lenz's law). Susceptibility χ < 0, very small (χ ~ −10⁻⁵). Water, copper, bismuth (χ = −1.66×10⁻⁴). Levitating frogs (Earnshaw's theorem bypass for diamagnets). No permanent magnetisation.
Paramagnetism: Materials with unpaired electrons but no cooperative ordering. Atomic magnetic moments align partially with applied field. Susceptibility χ > 0, small. Curie's Law: χ = C/T — susceptibility inversely proportional to temperature. Oxygen gas (O₂ has 2 unpaired electrons), aluminium, platinum.
Ferromagnetism: Strong interaction between adjacent spins (exchange interaction) causes spontaneous parallel alignment even without applied field. Below the Curie temperature T_C: permanent magnetisation possible. Fe (T_C = 1044 K), Co (1388 K), Ni (627 K).
Antiferromagnetism: Adjacent spins align antiparallel → zero net moment. MnO, FeO, CrO. Néel temperature T_N. Important for exchange-bias in spin-valves (hard disk read heads).
Ferrimagnetism: Two sublattices with antiparallel alignment but unequal magnitudes → net moment. Ferrites (Fe₃O₄ — lodestone, the original magnet). Enables magnetic insulators crucial for microwave applications.

3. Exchange Interaction

Heisenberg exchange Hamiltonian: H = −2J Σ_{<i,j>} S_i · S_j J = exchange integral (quantum mechanical, depends on orbital overlap) J > 0: parallel alignment favoured → ferromagnetism J < 0: antiparallel alignment favoured → antiferromagnetism Origin: The Pauli exclusion principle + Coulomb repulsion. Two electrons with parallel spins must occupy different spatial orbitals (symmetric spatial function forbidden by antisymmetry requirement). Parallel spins → electrons spatially separated → lower Coulomb energy. Whether this outweighs kinetic energy differences determines the sign of J. Stoner criterion for band ferromagnetism: In metals (band picture, not localised moments): Ferromagnetism occurs when: I(E_F) · D(E_F) > 1 where I = Stoner exchange parameter (eV) D(E_F) = density of states at Fermi level (states/eV) Fe, Co, Ni satisfy this criterion. Cu, Pd come close but don't. High D(E_F) in 3d transition metals → spontaneous spin splitting. Mean-field (Weiss) theory: Effective molecular field: B_eff = B_ext + λM → spontaneous magnetisation at T < T_C = μ₀ λ n μ² / 3k_B → Below T_C, Brillouin function analysis gives M(T) → 0 at T → T_C → Second-order phase transition (order parameter M vanishes continuously)

4. Magnetic Domains and Domain Walls

A ferromagnetic material below T_C is spontaneously magnetised locally, but divides into magnetic domains — regions of uniform magnetisation in different directions — to minimise total energy:

Why domains form: Magnetostatic energy (demagnetising field energy from surface poles) is reduced by closing flux inside the material. Exchange energy prefers parallel spins (wants fewer domains). Anisotropy energy prefers spins along easy axes. Domain structure minimises total free energy.
Domain walls (Bloch walls): A Bloch wall is a narrow region (~10-100 nm) where magnetisation rotates 180° from one domain to the next. Wall width δ_w ∝ √(A/K₁) where A = exchange stiffness, K₁ = magnetocrystalline anisotropy. Thinner walls in high-anisotropy materials (SmCo₅).
Domain wall motion: Under an external field, domains aligned favourably grow at the expense of unfavourable ones. Walls pin at grain boundaries and crystallographic defects. Irreversible pinning → hysteresis.

5. Hysteresis: B–H Curves

B-H relationship: B = μ₀(H + M) = μ₀ μ_r H where H = applied field (A/m), M = magnetisation (A/m), B = flux density (T) Hysteresis loop parameters: Saturation magnetisation M_s: maximum M at high field Fe: M_s = 1.71\times10⁶ A/m (B_s \approx 2.15 T) Nd₂Fe₁₄B: M_s = 1.28\times10⁶ A/m Remanence B_r: B remaining after field removed Coercivity H_c: Field needed to reduce B back to zero Energy product (BH)_max: measure of magnet strength Hard vs soft magnetic materials: Soft magnets (low coercivity H_c): Easy to magnetise AND demagnetise Small hysteresis loop area \to low energy loss per cycle Examples: electrical steel (Si-Fe), permalloy (Ni₈₀Fe₂₀), ferrites Applications: transformer cores, motor laminations, inductors Hard magnets (high coercivity H_c): Resist demagnetisation \to retain magnetisation Large hysteresis loop area Examples: AlNiCo, SmCo₅, Nd₂Fe₁₄B (neodymium magnets) Applications: loudspeakers, generators, MRI, electric motors (EVs) Energy loss per cycle: W = μ₀ \oint H dM = area enclosed by B-H loop (per unit volume) Eddy current losses add at high frequency: W \propto f² \to Why transformer cores use thin laminations (to break eddy current paths)

6. Permanent Magnets

The strongest permanent magnets exploit rare-earth elements that combine large magnetic moment (4f electrons) with high magnetocrystalline anisotropy:

Nd₂Fe₁₄B (Neodymium magnets, 1984): (BH)_max ≈ 400–520 kJ/m³ — the strongest permanent magnets known. Tetragonal crystal structure gives uniaxial anisotropy. T_C = 585 K (use limit ~80°C). Used in hard drives, headphones, electric motors (Tesla uses NdFeB in motors). Rare earth supply chain dominated by China (~85%).
SmCo₅ / Sm₂Co₁₇: (BH)_max ≈ 240–350 kJ/m³. Higher Curie temperature (T_C = 1020 K) → better high-T performance (turbines, aerospace). Cobalt provides excellent corrosion resistance.
AlNiCo (1930s): (BH)_max ≈ 40–80 kJ/m³. Low coercivity (easy to demagnetise), but very high Curie temperature and good corrosion resistance. Used in electric guitar pickups, some motors.

Exchange-spring magnets: Nano-composite structures alternating hard-phase (SmCo, NdFeB) and soft-phase (Fe, FeCo) layers at the nanometre scale. The soft phase's high M_s exchanges its high magnetisation into the hard phase's direction via exchange coupling across the interface. Can theoretically exceed state-of-the-art (BH)_max. Active research area for reducing rare-earth content in EV motors.

7. Applications: Data Storage, MRI, Motors

Hard disk drives (HDD): Data stored as magnetic domains in a CoCrPt thin film. Domain oriented along or against track direction = bit 1 or 0. Read head uses Giant Magnetoresistance (GMR, Nobel Prize 2007): resistance of spin-valve changes with magnetisation direction of free layer. Current areal densities: ~2 Tb/in² using Heat Assisted Magnetic Recording (HAMR).
MRI: Superconducting electromagnets (NbTi coils at 4 K) produce homogeneous B₀ fields of 1.5–7 T. Gradient coils (conventional electromagnets) create controlled field gradients for spatial encoding. Radio-frequency coils perturb nuclear spins (¹H) and detect their precession signals.
Electric motors: NdFeB permanent magnet motors (PMSM) dominate EV drivetrains (Tesla, BMW i3). Rare-earth content: ~1–2 kg per motor. Interior permanent magnet (IPM) designs allow flux weakening at high speed. Efficiency >96% at rated load.
Spintronics: MRAM (Magnetic Random Access Memory) — non-volatile memory using spin states. STT-MRAM uses spin-transfer torque to switch magnetic tunnel junctions. Potential for universal memory (fast, non-volatile, low power). Commercial products from Everspin; emerging in cache memory.