Magnetic Materials

Intuition

Place any material in an external magnetic field and ask: which way does its magnetization point, and how strong is it? The answer divides every material into one of three families.

• Diamagnets push back gently — their induced magnetization points against the field. Every material does this, but in most cases the effect is so small you’d never notice.
• Paramagnets have permanent atomic moments that don’t talk to each other; an external field nudges them into partial alignment, producing a weak magnetization parallel to the field.
• Ferromagnets have permanent atomic moments that do talk to each other (via the exchange interaction); below a critical temperature they align spontaneously, with or without an applied field, and the rest of this wiki is devoted to their physics.

The single dimensionless number that distinguishes them is the magnetic susceptibility $\chi$ — the slope of $\vec{M}$ vs $\vec{H}$ in the linear-response regime.

Formal Definition

In the linear-response regime,

$$\vec{M}=\chi \vec{H},{\mu }_r=1+\chi .$$

The magnitude and sign of $\chi$ define the three families:

Family	Sign of $\chi$	Typical magnitude	Spontaneous $M$?	Microscopic origin	Examples
Diamagnetic	$\chi <0$	$\sim -10^{-5}$	no	Lenz response of paired electron orbits	Cu, Bi, water, superconductors ($\chi =-1$)
Paramagnetic	$\chi >0$	$10^{-5}$–$10^{-3}$	no	Unpaired atomic moments, randomized by $k_{B}T$	Al, Pt, rare-earth salts
Ferromagnetic	effective $\chi \gg 1$	$\to \infty$ at $T=T_C$	yes ($T<T_C$)	Exchange-locked parallel moments	Fe, Co, Ni, Gd, alloys
Antiferromagnetic	$\chi >0$, small	weak	no (compensated)	Exchange-locked antiparallel moments	Cr, MnO, NiO
Ferrimagnetic	effective $\chi \gg 1$	strong	yes ($T<T_C$)	Antiparallel unequal sub-lattices	Magnetite Fe${}_3$O${}_4$, ferrites

The sign of $\chi$ already separates the diamagnets (negative) from everything else; the presence or absence of spontaneous magnetization then separates the ordered families (ferro / ferri) from the disordered ones (para / antiferro).

Key Results

1. Diamagnetism — universal and Lenz-like

When an external field $\vec{B}$ is turned on, it slightly deforms the orbital motion of bound electrons. By Lenz’s law, the induced orbital current generates a magnetic moment that opposes the applied field, so ${\chi }_{\text{dia}}<0$.

Two facts to keep in mind:

• Every material is diamagnetic. The orbital response exists for any bound electron and gives a small negative $\chi$ that is always present. In materials with unpaired spins (paramagnets, ferromagnets) it is swamped by larger paramagnetic or ferromagnetic contributions and effectively hidden.
• A diamagnet has mostly paired electrons in its outer shells — no permanent atomic moment to align with the field.

The extreme limit is a superconductor, which expels the field entirely (Meissner effect): $\chi =-1$, ${\mu }_r=0$ — perfect diamagnetism.

2. Paramagnetism — permanent moments, no order

A paramagnetic solid contains atoms with permanent magnetic moments (unpaired electrons in incomplete shells), but the moments do not interact strongly with one another. In zero applied field, thermal agitation randomizes their orientations and the macroscopic magnetization averages to zero.

A field $\vec{H}$ creates a partial alignment, giving a small positive $\chi$. Two physical regimes — and two laws for $\chi$ — coexist:

Curie paramagnetism (localized moments)

For localized atomic moments (e.g. rare-earth ions in an insulator), balancing Zeeman energy against $k_{B}T$ gives the Curie law:

$${\chi }_{\text{Curie}}(T)=\frac{C}{T},C=\frac{n{\mu }_0{\mu }_{\text{eff}}^2}{3k_B},$$

where $n$ is the moment density and ${\mu }_{\text{eff}}$ the effective atomic moment. The susceptibility diverges as $T\to 0$ — moments freeze into alignment.

Pauli paramagnetism (itinerant electrons)

The Curie law predicts $\chi \propto 1/T$ for any metal with permanent moments — and would give susceptibilities $\sim 10^{-3}$ at room temperature. Experiment, however, finds normal metals to have $\chi \approx 10^{-5}$, roughly one hundred times smaller than Curie predicts, and essentially independent of temperature. The resolution, due to Pauli (1927), is a purely quantum-statistical effect: conduction electrons form a degenerate Fermi gas, and only a thin shell of states within $\sim k_{B}T$ of the Fermi level can actually respond to a field. The rest are blocked by the exclusion principle.

The derivation goes through the rigid-band picture. Without a field, the two spin sub-bands have equal DOS $g_{\uparrow }=g_{\downarrow }=g(\epsilon )/2$. A Zeeman shift $\pm {\mu }_{B}B$ moves them in opposite directions; refilling to a common Fermi level transfers $\Delta N\approx g({\epsilon }_F){\mu }_{B}B/2$ electrons from spin-↓ to spin-↑. The net magnetization is $M={\mu }_B(N_{\uparrow }-N_{\downarrow })/V$ and yields

$$\boxed{{\chi }_{\text{Pauli}}={\mu }_0{\mu }_B^{2}g({\epsilon }_F)/V.}$$

Comparison to the Curie law makes the physics quantitative. For the same number of electrons,

$$\frac{{\chi }_{\text{Pauli}}}{{\chi }_{\text{Curie}}}\approx \frac{T}{T_F}\approx \frac{300\text{K}}{10^4\text{K}}\approx 10^{-2},$$

— i.e. only a fraction $T/T_F$ of the electrons are thermally active; the rest are frozen by Pauli. Pauli paramagnetism is the dominant magnetic response of “ordinary” metals (Na, Al, Cu), partially cancelled by their orbital diamagnetism. It also provides a direct experimental window on $g({\epsilon }_F)$ — the band structure of a metal can be probed by measuring its susceptibility. Pauli enhanced to within a hair of instability is precisely the regime of the Stoner criterion.

3. Ferromagnetism — exchange wins over $k_{B}T$

In a ferromagnet, the exchange interaction makes neighboring atomic moments prefer to align with each other, independent of any external field. Below the Curie temperature $T_C$ the alignment sets in spontaneously, producing the domain structure that the rest of this wiki is built on.

The susceptibility diverges at $T_C$ (Curie–Weiss law):

$$\chi (T)=\frac{C}{T-T_C},T>T_C.$$

The shift from Curie ($1/T$) to Curie–Weiss ($1/(T-T_C)$) is the fingerprint of the exchange interaction: at $T=T_C$ the inverse susceptibility extrapolates to zero, which is the signature of a second-order phase transition to a spontaneously magnetized state. The Curie temperatures of the elemental ferromagnets are large by microscopic standards — far above what dipole–dipole interactions could ever produce, requiring the much stronger exchange interaction:

Element	$T_C$ (K)	${\mu }_{\text{atom}}({\mu }_B)$	${\mu }_0{M}_s$ (T)
Fe (bcc)	1043	2.22	2.15
Co (hcp)	1388	1.72	1.80
Ni (fcc)	627	0.61	0.64
Gd (hcp)	292	7.55	2.49

The non-integer atomic moments of Fe, Co, Ni are themselves a window on itinerant magnetism — see stoner-model for why they cannot be explained by localized spins. Below $T_C$ the response becomes nonlinear and hysteretic: $\chi$ ceases to be a meaningful single-valued number, and one switches to the language of magnetization curves, domain walls, and dynamics.

Summary

A material’s magnetic class is fixed by what its atoms look like and by how strongly the atoms talk to each other:

• No unpaired spins → only the Lenz response remains → diamagnetic.
• Unpaired spins, weakly interacting → field-driven partial alignment → paramagnetic (Curie for localized, Pauli for itinerant).
• Unpaired spins, strongly exchange-coupled → spontaneous alignment below $T_C$ → ferromagnetic (the subject of micromagnetism).

Connections

• magnetization — defines $\vec{M}$, $\chi$, ${\mu }_r$
• magnetic-atom — paired vs unpaired electrons; sets the available family
• magnetic-moment — the atomic-scale building block
• micromagnetic-energy — exchange, the term that turns a paramagnet into a ferromagnet
• magnetic-domains — what spontaneous order looks like in a ferromagnet

References

• S. Blundell, Magnetism in Condensed Matter (Oxford, 2001), Chs. 1–3.
• J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge, 2010), Chs. 4–5.
• N. W. Ashcroft & N. D. Mermin, Solid State Physics, Ch. 31 — Pauli paramagnetism.