Diamagnetism and Paramagnetism

A solid placed in an applied field $\vec{H}$ acquires a magnetization $\vec{M}$, the magnetic moment per unit volume. For fields well below saturation the response is linear, and the volume susceptibility

is a dimensionless material constant (SI). The flux density inside is $\vec{B}={\mu }_0(\vec{H}+\vec{M})={\mu }_0(1+\chi )\vec{H}$. The sign and size of $\chi$ sort non-ordered matter into two classes. Diamagnets have $\chi <0$, of order $-10^{-5}$, and are repelled by a field; the response comes from the induced currents of otherwise moment-free filled shells. Paramagnets have $\chi >0$, between $10^{-5}$ and $10^{-2}$, and are drawn into a field; the response comes from permanent atomic moments (or conduction electrons) aligning with it. Ordered magnets — ferromagnets and antiferromagnets — are the subject of the following lessons; here the moments are independent.

The natural unit of atomic moment is the Bohr magneton

the moment of one electron's orbital angular momentum quantum.¹ An energy ${\mu }_{B}B$ at a laboratory field of $1\text{T}$ is $9.3\times 10^{-24}\text{J}$, or $0.67\text{K}$ in temperature units — small against $k_{B}T$ at room temperature, which is why paramagnetic alignment is weak and why the linear regime is the usual one.

Larmor diamagnetism of closed shells

An atom with all shells filled has no net spin or orbital angular momentum and no permanent moment. It still responds to a field, because switching the field on induces electronic currents that, by Lenz's law, oppose the change. This is the Langevin–Larmor diamagnetism.

Consider one electron in an atom subjected to a field $\vec{B}$ along $\hat{z}$. The classical effect of the field is to superimpose on the electron's motion a uniform precession about $\hat{z}$ at the Larmor frequency

This precession is a circulating current $I=-e{\omega }_L/2\pi$ enclosing the projected area of the orbit. A current loop of area $A$ carries moment $IA$, so each electron acquires an induced moment

directed opposite to $\vec{B}$. For a spherically symmetric shell $⟨x^2⟩=⟨y^2⟩=⟨z^2⟩$, so $⟨{\rho }^2⟩=\frac{2}{3}⟨r^2⟩$. Summing over the $Z$ electrons of each atom and multiplying by the number density $n$ of atoms gives the diamagnetic susceptibility

The result is negative, temperature-independent, and small: with $⟨r^2⟩\sim (0.1\text{nm}{)}^2$ and $n\sim 5\times 10^{28}{\text{m}}^{-3}$ it gives $\chi \sim -10^{-5}$, matching the inert gases and simple ionic solids. The same $⟨r^2⟩$ that sets an ion's size sets its diamagnetism, so the two correlate across the periodic table.

Curie paramagnetism of localized moments

An atom or ion with an incomplete shell carries a permanent moment $\vec{\mu }=-g{\mu }_B\vec{J}/\hslash$, where $\vec{J}$ is the total angular momentum and $g$ the Landé factor. In a field $\vec{B}=B\hat{z}$ the $2J+1$ orientations of $\vec{J}$ have energies

At temperature $T$ the states are populated by Boltzmann weights, and the mean moment along the field follows from the partition function $Z={\sum }_m{e}^{-E_m/k_{B}T}$. Writing $x=g{\mu }_{B}JB/k_{B}T$, the sum is a finite geometric series, and the thermal-average moment per ion is

where the Brillouin function is

The magnetization of $n$ ions per unit volume is $M=ng{\mu }_{B}J{B}_J(x)$. Two limits fix its shape.

At large $x$ (strong field or low temperature) $B_J\to 1$ and the moments saturate at $M_s=ng{\mu }_{B}J$: every ion points along the field. At small $x$ (the usual laboratory case) $\coth u\approx 1/u+u/3$ gives

so the magnetization is linear in $B$ and the susceptibility follows the Curie law

The dimensionless $p$ is the effective moment number; $C$ is the Curie constant. The $1/T$ dependence is the fingerprint of independent moments: thermal agitation randomizes them, and the aligning tendency of a fixed field wins in inverse proportion to temperature. Plotting $1/\chi$ against $T$ gives a straight line through the origin whose slope measures $p$.

The classical case is recovered as $J\to \infty$ with $g{\mu }_{B}J=\mu$ fixed. The Brillouin function then becomes the Langevin function

the result for a freely rotating classical dipole. That Langevin's classical theory and the quantum Brillouin theory agree at small field — both give a $1/T$ law — is why paramagnetism was understood before quantum mechanics, and why the measured $p$ (not the classical value) was one of the early confirmations of angular-momentum quantization.

Hund's rules and the ground-state moment

To predict $p$ for a given ion the ground-state values of $S$, $L$, and $J$ of the partly filled shell are needed. For the free ion these follow from Hund's rules, which minimize the electrostatic and spin–orbit energy of the open-shell electrons:

• Maximum spin. Arrange the electron spins to maximize the total $S=\sum m_s$ allowed by the exclusion principle. Parallel spins keep electrons in different orbitals, lowering their mutual Coulomb repulsion.
• Maximum orbital angular momentum. Consistent with that $S$, maximize $L=|\sum m_{\ell }|$. Electrons orbiting the same way avoid one another and again lower the repulsion.
• Spin–orbit coupling. Set $J=|L-S|$ for a shell less than half full, and $J=L+S$ for a shell more than half full. A half-filled shell has $L=0$ and $J=S$.

For example, the ${\text{Fe}}^{3+}$ ion has a half-filled $3d^5$ shell: five parallel spins give $S=\frac{5}{2}$, the orbital contributions cancel to $L=0$, and $J=S=\frac{5}{2}$ with $g=2$. The predicted $p=g\sqrt{J(J+1)}=5.9$ matches the measured value for iron-group salts. The rare-earth ions, whose $4f$ shells are buried inside the closed $5s5p$ shells and so nearly free, agree with the full $p=g\sqrt{J(J+1)}$ across the series.

Two caveats matter in solids. For the iron-group ions the crystal field of the neighboring atoms is stronger than the spin–orbit coupling and quenches the orbital angular momentum, so $L$ is effectively zero and only the spin survives; the measured $p$ agrees with $g=2$, $J=S$ rather than with the free-ion $J$. The rare earths, shielded from the crystal field, keep their free-ion $J$.

Pauli paramagnetism of the conduction electrons

A metal's conduction electrons each carry a spin moment ${\mu }_B$, so a naive Curie estimate would predict a large $1/T$ paramagnetism. The measured spin susceptibility of the alkali metals is instead small and nearly temperature-independent. The exclusion principle is again the reason: only the electrons within about $k_{B}T$ of the Fermi energy can flip their spins into empty states, a fraction $T/T_F$ of the whole.

In a field $\vec{B}$ the spin-up and spin-down bands shift in energy by $\mp {\mu }_{B}B$. Electrons transfer from the higher (spin-antiparallel) band to the lower until the Fermi levels align, producing an excess $\Delta n=g(E_F){\mu }_{B}B$ of aligned spins, where $g(E_F)$ is the density of states at the Fermi energy (both spins). The magnetization $M={\mu }_B\Delta n$ gives the Pauli susceptibility

Using the free-electron $g(E_F)=3n/2E_F=3n/2k_B{T}_F$,

which is the Curie result with $T$ replaced by the far larger Fermi temperature $T_F\sim 10^4\text{K}$ — smaller by two orders of magnitude, and independent of $T$ because the excited shell width and the level spacing both scale with $T$ and cancel.

The orbital motion of the same conduction electrons, quantized into Landau levels by the field, adds a diamagnetic contribution. For free electrons this Landau diamagnetism is exactly one-third of the Pauli term with the opposite sign,

so the net conduction-electron susceptibility is $\frac{2}{3}{\chi }_{\text{Pauli}}$ and still paramagnetic. In a real band the ratio changes because the orbital motion feels the band effective mass $m^{\ast }$ while the spin feels the bare mass; where $m^{\ast }$ is small (as in bismuth) the Landau term can dominate and the metal is net diamagnetic.

The three responses — closed-shell diamagnetism, local-moment Curie paramagnetism, and the conduction-electron Pauli and Landau terms — coexist in any real solid, and their signs and temperature dependences let an experiment separate them. Where the local moments interact strongly enough to order, the independent-moment picture breaks down; the exchange interaction that drives that ordering is the next lesson.