--- title: Diamagnetism and Paramagnetism module: Magnetism in Solids moduleNumber: 9 lessonNumber: 1 order: 901 summary: > Every solid responds to a magnetic field. Filled shells give a small negative diamagnetic susceptibility from induced Larmor currents; localized moments give a positive Curie paramagnetism described by the Brillouin function, with the ground-state moment fixed by Hund's rules. The conduction electrons add a temperature-independent Pauli paramagnetism from the thermal shell near the Fermi surface, partly cancelled by Landau diamagnetism of their orbital motion. topics: [Magnetism in Solids] draft: false sources: - book: Kittel ref: "Ch. 11 — Diamagnetism and Paramagnetism" - book: Ashcroft & Mermin ref: "Ch. 31 — Diamagnetism and Paramagnetism" - book: Hook & Hall ref: "Ch. 7 — Magnetic Properties of Materials" --- 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** $$ \chi = \frac{M}{H} $$ 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** $$ \mu_B = \frac{e\hbar}{2m_e} = 9.274\times10^{-24}\ \text{J T}^{-1}, $$ the moment of one electron's orbital angular momentum quantum.[^codata] An energy $\mu_B B$ at a laboratory field of $1\ \text{T}$ is $9.3\times10^{-24}\ \text{J}$, 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** $$ \omega_L = \frac{eB}{2m_e}. $$ 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 $$ \mu = -\frac{e^2 B}{4 m_e}\langle \rho^2\rangle, \qquad \langle \rho^2\rangle = \langle x^2 + y^2\rangle, $$ directed opposite to $\vec B$. For a spherically symmetric shell $\langle x^2\rangle = \langle y^2\rangle = \langle z^2\rangle$, so $\langle\rho^2\rangle = \tfrac{2}{3}\langle r^2\rangle$. Summing over the $Z$ electrons of each atom and multiplying by the number density $n$ of atoms gives the diamagnetic susceptibility $$ \chi_{\text{dia}} = -\frac{\mu_0 n e^2}{6 m_e}\sum_{i=1}^{Z}\langle r_i^2\rangle. $$ The result is negative, temperature-independent, and small: with $\langle r^2\rangle \sim (0.1\ \text{nm})^2$ and $n \sim 5\times10^{28}\ \text{m}^{-3}$ it gives $\chi \sim -10^{-5}$, matching the inert gases and simple ionic solids. The same $\langle r^2\rangle$ that sets an ion's size sets its diamagnetism, so the two correlate across the periodic table. $$ % caption: Switching on B induces a Larmor circulation whose moment opposes the % field (Lenz's law); the induced current loop gives every filled shell a small % negative susceptibility. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % field lines up \foreach \x in {-2.4,-1.6,-0.8}{ \draw[black, ->] (\x,-1.7) -- (\x,1.9); } \node[black, anchor=south] at (-1.6,1.9) {applied $B$}; % nucleus and orbit \fill[black] (1.6,0) circle (2pt); \draw[acc, very thick] (1.6,0) ellipse (1.15 and 0.55); % current direction arrows on the loop \draw[acc, ->, very thick] (2.75,0.02) -- (2.75,-0.05); \draw[acc, ->, very thick] (0.45,-0.02) -- (0.45,0.05); \node[acc, anchor=west] at (2.85,0.35) {induced current}; % induced moment down \draw[acc, ->, very thick] (1.6,0) -- (1.6,-1.5); \node[acc, anchor=north] at (1.6,-1.55) {moment opposes $B$}; \end{tikzpicture} $$ ## Curie paramagnetism of localized moments An atom or ion with an incomplete shell carries a permanent moment $\vec\mu = -g \mu_B \vec J/\hbar$, 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 $$ E_{m} = m\, g\mu_B B, \qquad m = -J, -J+1, \dots, J. $$ 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 $x = g\mu_B J B/k_B T$, the sum is a finite geometric series, and the thermal-average moment per ion is $$ \langle \mu_z\rangle = g\mu_B J\, B_J(x), $$ where the **Brillouin function** is $$ B_J(x) = \frac{2J+1}{2J}\coth\!\left(\frac{2J+1}{2J}x\right) - \frac{1}{2J}\coth\!\left(\frac{x}{2J}\right). $$ The magnetization of $n$ ions per unit volume is $M = n g\mu_B J\, B_J(x)$. Two limits fix its shape. At large $x$ (strong field or low temperature) $B_J \to 1$ and the moments saturate at $M_s = n g\mu_B J$: every ion points along the field. At small $x$ (the usual laboratory case) $\coth u \approx 1/u + u/3$ gives $$ B_J(x) \approx \frac{J+1}{3J}\,x, $$ so the magnetization is linear in $B$ and the susceptibility follows the **Curie law** $$ \chi = \frac{\mu_0 n\, p^2 \mu_B^2}{3 k_B T} \equiv \frac{C}{T}, \qquad p = g\sqrt{J(J+1)}. $$ 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 $p$. $$ % caption: The Brillouin magnetization M/M_s versus B/T saturates at unity; larger % J rises more steeply and the classical Langevin limit (J to infinity) is the % smoothest curve. All share the same small-field Curie slope. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.6,0) node[below right] {$\frac{B}{T}$}; \draw[->, black] (0,0) -- (0,3.6) node[left] {magnetization}; \draw[black, dashed] (0,3.0) -- (6.4,3.0); \node[black, anchor=south east] at (6.4,3.0) {saturation}; % J = 5/2 : steepest saturating rise \draw[black, thick, domain=0:6.3, samples=120, variable=\x] plot ({\x},{3.0*(1.1*\x)/sqrt(1 + 1.21*\x*\x)}); \node[black, anchor=west] at (2.0,2.9) {$J=\frac{5}{2}$}; % J = 1/2 : intermediate \draw[acc, very thick, domain=0:6.3, samples=120, variable=\x] plot ({\x},{3.0*(0.62*\x)/sqrt(1 + 0.3844*\x*\x)}); \node[acc, anchor=west] at (4.3,2.35) {$J=\frac{1}{2}$}; % classical Langevin: smoothest, slowest approach \draw[black, thick, densely dotted, domain=0:6.3, samples=120, variable=\x] plot ({\x},{3.0*(0.34*\x)/sqrt(1 + 0.1156*\x*\x)}); \node[black, anchor=north west] at (3.2,1.05) {classical}; \end{tikzpicture} $$ The classical case is recovered as $J\to\infty$ with $g\mu_B J = \mu$ fixed. The Brillouin function then becomes the **Langevin function** $$ L(x) = \coth x - \frac{1}{x}, $$ 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. $$ % caption: The Curie law plotted as inverse susceptibility against temperature is % a straight line through the origin; its slope is the reciprocal Curie constant % and fixes the effective moment number p. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.4,0) node[below right] {temperature}; \draw[->, black] (0,0) -- (0,3.8) node[left] {inverse susceptibility}; \draw[acc, very thick] (0,0) -- (5.8,3.4); \node[acc, anchor=west] at (4.5,2.35) {slope $=\frac{1}{C}$}; \fill[black] (2.0,1.17) circle (1.6pt); \fill[black] (3.5,2.05) circle (1.6pt); \fill[black] (5.0,2.93) circle (1.6pt); \end{tikzpicture} $$ ## 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 $L = \lvert\sum m_\ell\rvert$. Electrons orbiting the same way avoid one another and again lower the repulsion. - **Spin–orbit coupling.** Set $J = \lvert L - S\rvert$ 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 $S = \tfrac{5}{2}$, the orbital contributions cancel to $L = 0$, and $J = S = \tfrac{5}{2}$ 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. $$ % caption: Hund-rule filling of a d-shell (five orbitals): the first five % electrons go in parallel to maximize spin before any orbital is doubly occupied. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \foreach \i in {0,1,2,3,4}{ \draw[black] (\i*1.05,0) rectangle (\i*1.05+0.7,0.7); % up spin in each box \draw[acc, very thick, ->] (\i*1.05+0.35,0.12) -- (\i*1.05+0.35,0.58); } \node[black, anchor=north] at (2.1,-0.15) {all spins parallel: $S=\frac{5}{2}$, $L=0$}; \end{tikzpicture} $$ 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** $$ \chi_{\text{Pauli}} = \mu_0\,\mu_B^2\, g(E_F). $$ Using the free-electron $g(E_F) = 3n/2E_F = 3n/2k_B T_F$, $$ \chi_{\text{Pauli}} = \frac{3 n \mu_0 \mu_B^2}{2 k_B T_F}, $$ 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. $$ % caption: In a field the spin-up density of states drops by mu_B B and spin-down % rises; electrons refill to a common Fermi level, leaving a net excess of aligned % spins proportional to g(E_F). Only the shaded Fermi shell participates. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % spin-up (left, shifted down): parabola opening right, filled to E_F \begin{scope} \draw[black, ->] (0,0) -- (0,3.4) node[left] {energy}; \draw[acc, very thick, domain=0:2.9, samples=60, variable=\y] plot ({-1.1*sqrt(\y+0.3)},{\y}); \fill[acc!12] plot[domain=0:2.6, samples=60, variable=\y] ({-1.1*sqrt(\y+0.3)},{\y}) -- (0,2.6) -- (0,0) -- cycle; \draw[black, dashed] (-1.9,2.6) -- (0,2.6); \node[acc, anchor=north] at (-1.1,-0.15) {spin up}; \end{scope} % spin-down (right, shifted up) \begin{scope} \draw[black, very thick, domain=0:2.9, samples=60, variable=\y] plot ({1.1*sqrt(\y+0.3)},{\y}); \draw[black, dashed] (0,2.2) -- (1.9,2.2); \node[black, anchor=north] at (1.1,-0.15) {spin down}; \end{scope} \node[black, anchor=west] at (2.0,2.6) {common $E_F$}; \end{tikzpicture} $$ 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, $$ \chi_{\text{Landau}} = -\tfrac{1}{3}\,\chi_{\text{Pauli}} \quad(\text{free electrons}), $$ so the net conduction-electron susceptibility is $\tfrac{2}{3}\chi_{\text{Pauli}}$ 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](/condensed-matter/magnetism/exchange-and-ferromagnetism) that drives that ordering is the next lesson. | Mechanism | Sign of $\chi$ | Size | $T$-dependence | | --- | --- | --- | --- | | Larmor (closed shell) | $\chi < 0$ | $\sim 10^{-5}$ | independent | | Curie (local moments) | $\chi > 0$ | $10^{-3}$–$10^{-5}$ | $\propto 1/T$ | | Pauli (conduction spin) | $\chi > 0$ | $\sim 10^{-6}$ | independent | | Landau (conduction orbit) | $\chi < 0$ | $\tfrac{1}{3}$ of Pauli | independent | [^codata]: CODATA / NIST recommended value of the Bohr magneton, $\mu_B = 9.2740100783\times10^{-24}\ \text{J T}^{-1}$: [physics.nist.gov](https://physics.nist.gov/cgi-bin/cuu/Value?mub).