Ferromagnetism

Ferromagnetism is the phenomenon by which certain materials — most famously iron, nickel, and cobalt — can be permanently magnetised and exhibit strong attractive or repulsive forces with other magnets. Unlike the weak effects seen in paramagnetic or diamagnetic materials, ferromagnetic materials can retain a net magnetic moment long after an external magnetic field has been removed, making them the basis of permanent magnets, electric motors, hard drives, and countless other technologies. The key to ferromagnetism lies at the atomic scale. Each iron atom, for example, behaves like a tiny magnetic dipole due to the spin of its unpaired electrons. In most materials these dipoles are randomly oriented and cancel out. In ferromagnetic materials, quantum mechanical exchange interactions cause neighbouring atomic dipoles to align parallel to one another, forming microscopic regions called magnetic domains. Within each domain the magnetisation is nearly saturated, but domains themselves can point in different directions, so an unmagnetised piece of iron has no net field. When an external magnetic field is applied, domains whose magnetisation is already close to the field direction grow at the expense of others (domain wall motion), and eventually all domains rotate to align with the field, producing a large net magnetisation. Because this process is not fully reversible — some domain walls become pinned at crystal defects — the material retains a residual magnetisation called remanence after the field is removed. This history-dependence is called magnetic hysteresis and is one of the defining signatures of ferromagnetism.

Key takeaways

Atomic Magnetic Moments
Exchange Interaction and Spontaneous Alignment
Magnetic Domains
Magnetisation Process and Hysteresis
Curie–Weiss Law Above $T_C$

Atomic Magnetic Moments

Every electron carries an intrinsic angular momentum called spin, quantised as $s = \pm\frac{1}{2}\hbar$ . Associated with this spin is a magnetic moment $\mu_s = -g_s \mu_B m_s$ , where $\mu_B = 9.274 \times 10^{-24}$ J T $^{-1}$ is the Bohr magneton and $g_s \approx 2$ is the electron g-factor. In iron, five of the 3d electrons have unpaired spins, giving each atom a net magnetic moment of about $2.2\,\mu_B$ after accounting for orbital contributions. This makes iron a strong candidate for ferromagnetism.

Exchange Interaction and Spontaneous Alignment

The alignment of neighbouring spins is not due to classical dipole–dipole forces (which are too weak at atomic separations) but to the quantum mechanical exchange interaction. For ferromagnetic materials, the exchange integral $J_{ex} > 0$ , meaning the lowest-energy state has parallel spins. This leads to spontaneous magnetisation: below the Curie temperature $T_C$ , adjacent atoms spontaneously align, creating a domain. For iron, $T_C = 1043$ K; above this temperature thermal energy disrupts the alignment and the material becomes paramagnetic.

Magnetic Domains

A single large domain would have a very high magnetostatic energy (energy stored in the external stray field). Nature minimises total energy by breaking the material into many domains separated by domain walls (Bloch walls), each wall being roughly 100–300 nm thick. Within a domain, all atomic moments are parallel; across a Bloch wall, the direction rotates gradually from one domain's orientation to the next. An unmagnetised sample has domains arranged so that stray fields cancel, giving zero net magnetisation: $\mathbf{M}_{net} = \mathbf{0}$ .

Magnetisation Process and Hysteresis

Applying an external field $\mathbf{H}$ first causes domain walls to move reversibly (low field), then irreversibly as walls jump past pinning sites (Barkhausen jumps), and finally causes remaining domains to rotate into alignment at saturation magnetisation $M_s$ . Removing $\mathbf{H}$ leaves a remanent magnetisation $M_r$ . A reverse field of magnitude $H_c$ (the coercive field) is needed to reduce $M$ to zero. The area enclosed by the $M$ – $H$ hysteresis loop equals the energy dissipated per unit volume per cycle: $W = \mu_0 \oint H\, dM$ (in J m $^{-3}$ ).

Curie–Weiss Law Above $T_C$

Above the Curie temperature, ferromagnets become paramagnetic and their susceptibility follows the Curie–Weiss law: $\chi = \frac{C}{T - T_C}$ , where $C$ is the Curie constant and $T$ is absolute temperature. As $T \to T_C$ from above, $\chi \to \infty$ , signalling the onset of long-range order — a second-order phase transition. This law is extremely useful for experimentally determining $T_C$ by extrapolating a plot of $1/\chi$ versus $T$ to zero.

The Key Equation

$$\chi = \frac{C}{T - T_C}

- **\chi** — Magnetic susceptibility (dimensionless): Ratio of magnetisation to applied field, $\chi = M/H$; measures how easily a material is magnetised. - **C** — Curie constant (K): Material-specific constant proportional to the square of the atomic magnetic moment per unit volume. - **T** — Absolute temperature (K): Temperature of the material in kelvin; must be greater than $T_C$ for this law to apply. - **T_C** — Curie temperature (K): Critical temperature below which spontaneous ferromagnetic order appears; for iron $T_C = 1043$ K. ## Worked Example **Problem.** A sample of nickel has a Curie temperature of $T_C = 631$ K and a Curie constant of $C = 0.50$ K. Calculate the magnetic susceptibility of the nickel at a temperature of $T = 700$ K. Is the material ferromagnetic or paramagnetic at this temperature? Also, what applied magnetic field $H = 1.2 \times 10^4$ A m$^{-1}$ would produce a magnetisation $M$ in this state?

\chi = 7.25 \times 10^{-3},\quad M = 87\text{ A m}^{-1}

**Answer.** Magnetic susceptibility $\chi \approx 7.25 \times 10^{-3}$; the sample is paramagnetic at 700 K; magnetisation $M \approx 87$ A m$^{-1}$.

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Diagram

Origins · 1 minPierre Curie and how this was discovered

Pierre Curie

1859–1906 · French

Pierre Curie systematically investigated the magnetic properties of materials as a function of temperature, discovering in 1895 that ferromagnets lose their magnetisation above a critical temperature — now called the Curie temperature in his honour. He also formulated Curie's law ( $M = C H / T$ ) for paramagnets and, with his student Paul Langevin, laid the groundwork for understanding the microscopic origin of magnetism. His work established that magnetism is a thermal phenomenon, paving the way for the modern domain theory developed by Weiss. Pierre Curie was famously absent-minded; he was killed in 1906 when he stepped in front of a horse-drawn cart in Paris while distracted in thought — a tragic end for one of science's most brilliant experimentalists.

Magnetic Fields and Flux — Understanding magnetic field lines and flux density $B = \mu_0(H + M)$ is essential before analysing how ferromagnets amplify and modify magnetic fields.
Paramagnetism and Diamagnetism — Comparing the weak, temperature-dependent susceptibility of paramagnets and the tiny negative susceptibility of diamagnets with ferromagnetism clarifies what makes ferromagnets uniquely powerful magnetic materials.
Electromagnetic Induction — Ferromagnetic cores in transformers and inductors exploit hysteresis and high permeability; understanding Faraday's law shows why the hysteresis loop area represents wasted energy as heat in AC devices.
Electron Spin and Magnetic Moment — Ferromagnetism ultimately arises from electron spin and the Bohr magneton; a solid grasp of spin quantum numbers and the exchange interaction is the microscopic foundation of the whole phenomenon.