--- title: "Paramagnetism, Two-Level Systems, and the Schottky Anomaly" module: The Canonical Ensemble moduleNumber: 4 lessonNumber: 5 order: 405 summary: > A magnetic moment in a field is a two-level system whose partition function is a hyperbolic cosine. The magnetization of a spin-$\tfrac12$ paramagnet is $N\mu\tanh(\mu B/k_BT)$, generalizing to the Brillouin function for spin $J$; it gives Curie's law $\chi\propto 1/T$ at high temperature and saturates at low temperature. A finite level gap produces the Schottky heat-capacity peak, and the temperature dependence of the entropy on the field is the basis of adiabatic demagnetization cooling. topics: [The Canonical Ensemble] sources: - book: Schroeder ref: "Ch. 3 §3.3 Paramagnetism, and Ch. 6 §6.2 Average Values" - book: Reif ref: "Ch. 7 — Simple Applications of Statistical Mechanics; §7.8 Magnetism" - book: Pathria & Beale ref: "Ch. 3 — The Canonical Ensemble; §3.9–3.10 Paramagnetism and the Brillouin Function" - book: Kardar (Statistical Physics of Particles) ref: "Ch. 4 — Classical Statistical Mechanics; §4.6 (two-level systems)" draft: false --- The two-level system is the simplest nontrivial application of the canonical ensemble, and a magnetic moment in an applied field realizes it exactly. The partition function is a sum of two terms, the thermodynamics follows in a few lines, and the results are directly measurable: the magnetization of a paramagnet, its susceptibility, and a characteristic bump in its heat capacity. The same two-level structure describes nuclear spins, defect levels in crystals, and any system with an isolated pair of energy states, so the paramagnet doubles as the generic finite-gap system. ## The spin-½ paramagnet A particle with magnetic moment of magnitude $\mu$ and spin $\tfrac12$ in a magnetic field $B$ has two orientations. The moment aligned with the field has energy $-\mu B$; the moment opposed has energy $+\mu B$. The single-spin partition function sums these two Boltzmann factors, $$ z = e^{+\beta\mu B} + e^{-\beta\mu B} = 2\cosh(\beta\mu B). $$ The probabilities of the two states are $p_\pm = e^{\pm\beta\mu B}/z$, and the mean moment along the field is the difference weighted by $\mu$, $$ \langle\mu_z\rangle = \mu\,\frac{e^{\beta\mu B} - e^{-\beta\mu B}} {e^{\beta\mu B} + e^{-\beta\mu B}} = \mu\tanh(\beta\mu B). $$ For $N$ independent spins the total magnetization is $M = N\langle\mu_z\rangle$, so[^schroeder-para] $$ M = N\mu\tanh\!\left(\frac{\mu B}{k_B T}\right). $$ $$ % caption: A magnetic moment has two states in a field: aligned (lower energy) and opposed (higher energy). The Boltzmann factors set the populations, and the excess in the lower state is the net magnetization. \begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[very thick] (0.5,0.7)--(3.0,0.7); \node[left,font=\scriptsize] at (0.4,0.7) {aligned}; \draw[acc,very thick,fill=acc!14] (3.3,0.55) rectangle (6.3,0.85); \node[acc,right,font=\scriptsize] at (6.4,0.7) {larger population}; \draw[black,very thick] (0.5,3.2)--(3.0,3.2); \node[black,left,font=\scriptsize] at (0.4,3.2) {opposed}; \draw[black,very thick,fill=black!8] (3.3,3.05) rectangle (4.5,3.35); \node[black,right,font=\scriptsize] at (4.6,3.2) {smaller population}; \draw[<->,black] (7.4,0.7)--(7.4,3.2) node[midway,right,font=\scriptsize,black!70]{energy gap}; \node[black!70] at (2.0,1.95) {applied $B$}; \end{tikzpicture} $$ The magnetization is an odd, saturating function of the ratio $\mu B/k_B T$. At small argument the moment grows linearly with field; at large argument it saturates as every spin aligns. $$ % caption: The spin-½ magnetization $M=N\mu\tanh(\mu B/k_BT)$ against $x=\mu B/k_BT$: linear in the field for $x\ll1$ (the Curie regime) and saturating at $N\mu$ for $x\gg1$ (full alignment). \begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (0,0)--(7.2,0) node[right,black]{$x$}; \draw[->,black] (0,0)--(0,4.0) node[above,black]{$M$}; \draw[black,very thick,dashed] (0,3.4)--(7.0,3.4); \node[black,right,font=\scriptsize] at (5.4,3.66) {saturation}; \draw[acc,very thick] plot[domain=0:6.6,samples=90] (\x,{3.4*(1-exp(-2*\x))/(1+exp(-2*\x))}); \node[acc] at (4.6,2.4) {$\tanh$ curve}; \draw[black,dashed] (0,0)--(2.0,3.06); \node[black!70,font=\scriptsize] at (1.9,0.55) {linear at small $x$}; \end{tikzpicture} $$ ## Curie's law and saturation The two limits of the $\tanh$ are the two experimental regimes.[^reif-para] - **High temperature or weak field**, $x = \mu B/k_B T \ll 1$. Then $\tanh x\approx x$, so $M\approx N\mu^2 B/k_B T$. The magnetization is linear in the field, and the magnetic susceptibility $\chi = \partial M/\partial B$ obeys **Curie's law**, $$\chi = \frac{N\mu^2}{k_B T} = \frac{C}{T}, \qquad C = \frac{N\mu^2}{k_B}.$$ The susceptibility diverges as $T\to 0$ because thermal agitation, which randomizes the moments, weakens as the temperature falls. - **Low temperature or strong field**, $x \gg 1$. Then $\tanh x\to 1$ and $M\to N\mu$: every spin is aligned and the magnetization saturates. Further increase of the field or decrease of the temperature has no effect, because there is no state below the aligned one. Curie's law is the signature of independent permanent moments and its measured constant $C$ fixes $\mu$. The linear-in-field response and the $1/T$ divergence are the paramagnetic analogue of the energy fluctuation–response relation of the previous lesson: the susceptibility is proportional to the equilibrium variance of the magnetization, $\chi = \beta\langle\Delta M^2\rangle$. ## The Brillouin function A moment of general angular momentum quantum number $J$ has $2J+1$ orientations, with energies $-g\mu_B m B$ for $m = -J, -J+1, \dots, J$, where $g$ is the Landé factor and $\mu_B$ the Bohr magneton. The partition function is a finite geometric series, and the magnetization is $M = N g\mu_B J\,B_J(x)$ with $x = g\mu_B J B/k_B T$ and the **Brillouin function**[^pathria-brillouin] $$ B_J(x) = \frac{2J+1}{2J}\coth\!\left(\frac{2J+1}{2J}\,x\right) - \frac{1}{2J}\coth\!\left(\frac{x}{2J}\right). $$ Two special cases recover known results. For $J = \tfrac12$ the Brillouin function reduces to $B_{1/2}(x) = \tanh x$, the spin-½ result above. In the classical limit $J\to\infty$ it becomes the **Langevin function** $L(x) = \coth x - 1/x$, the result for a freely orienting classical dipole. Every $B_J$ rises linearly from the origin with slope $(J+1)/3J$ and saturates at unity, and expanding at small $x$ gives the general Curie law $\chi = N g^2\mu_B^2 J(J+1)/3k_B T$. $$ % caption: Brillouin functions $B_J(x)$ for increasing $J$: all rise linearly near the origin and saturate at unity, with $J=\tfrac12$ giving $\tanh x$ and the large-$J$ limit giving the classical Langevin function. \begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (0,0)--(7.2,0) node[right,black]{$x$}; \draw[->,black] (0,0)--(0,4.0) node[above,black]{$B_J$}; \draw[black,dashed] (0,3.4)--(7.0,3.4); \draw[acc,very thick] plot[domain=0.1:6.6,samples=90] (\x,{3.4*(1-exp(-2*\x))/(1+exp(-2*\x))}); \node[acc,font=\scriptsize] at (2.1,3.1) {$J=\frac{1}{2}$}; \draw[black,very thick,dashed] plot[domain=0.15:6.6,samples=90] (\x,{3.4*((4/3)*(1+exp(-8*\x/3))/(1-exp(-8*\x/3))-(1/3)*(1+exp(-2*\x/3))/(1-exp(-2*\x/3)))}); \node[black,font=\scriptsize] at (3.4,2.35) {$J=\frac{3}{2}$}; \draw[black,very thick,densely dotted] plot[domain=0.2:6.6,samples=90] (\x,{3.4*((1+exp(-2*\x))/(1-exp(-2*\x))-1/\x)}); \node[black,font=\scriptsize] at (5.2,1.65) {large $J$}; \end{tikzpicture} $$ ## The Schottky anomaly The heat capacity of a two-level system has a distinctive shape. Take levels at $0$ and $\Delta$ (the same structure as the paramagnet with $\Delta = 2\mu B$). The partition function is $z = 1 + e^{-\beta\Delta}$, and the mean energy is $$ U = \frac{\Delta\, e^{-\beta\Delta}}{1 + e^{-\beta\Delta}} = \frac{\Delta}{e^{\beta\Delta} + 1}. $$ Differentiating gives the heat capacity per system, $$ C = \frac{\partial U}{\partial T} = k_B\left(\frac{\Delta}{k_B T}\right)^2 \frac{e^{\Delta/k_B T}}{\big(e^{\Delta/k_B T} + 1\big)^2}. $$ The $+1$ in the denominator, rather than the $-1$ of the oscillator, marks the finite number of levels. The behavior is non-monotonic — a peak rather than a step — and is called the **Schottky anomaly**.[^reif-schottky] - **Low temperature**, $k_B T\ll\Delta$. The gap cannot be bridged, $U\to 0$, and $C\to k_B(\Delta/k_B T)^2 e^{-\Delta/k_B T}\to 0$ exponentially. - **High temperature**, $k_B T\gg\Delta$. Both levels are equally populated, $U\to\Delta/2$ saturates, and $C\to k_B(\Delta/2k_B T)^2\to 0$ as a power law. Because $C$ vanishes at both ends it must have a maximum in between, located near $k_B T\approx 0.42\,\Delta$. The peak is a direct measure of a level gap: a bump in the heat capacity at temperature $T_{\rm peak}$ reveals a pair of states split by $\Delta\approx 2.4\,k_B T_{\rm peak}$. In a real solid this Schottky term sits on top of the lattice (Debye) and, in a metal, the electronic contributions, and its excess above them isolates the two-level degrees of freedom. $$ % caption: The Schottky heat capacity of a two-level system rises from zero, peaks near $k_BT\approx0.42\,\Delta$, and falls again; the peak position measures the level gap $\Delta$. \begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (0,0)--(7.4,0) node[right,black]{$T$}; \draw[->,black] (0,0)--(0,4.0) node[above,black]{$C$}; \draw[acc,very thick] plot[domain=0.28:7.0,samples=130] (\x,{7.5*(2.0/\x)^2*exp(-2.0/\x)/(1+exp(-2.0/\x))^2}); \draw[black,dashed] (0.84,0)--(0.84,3.3); \node[black!70,font=\scriptsize,below] at (0.9,0) {peak near gap}; \node[acc] at (3.6,2.3) {Schottky peak}; \end{tikzpicture} $$ The peak location follows from extremizing $C$ with respect to temperature. > **Worked example.** Locate the Schottky maximum. With $u = \Delta/k_B T$, the > heat capacity is $C = k_B\,u^2 e^{u}/(e^{u}+1)^2$. Since $u$ is a monotonic > function of $T$, the maximum in $T$ occurs where $\d C/\d u = 0$. Taking the > logarithm, $\ln C = \text{const} + 2\ln u + u - 2\ln(e^{u}+1)$, and > $$ > \frac{\d\ln C}{\d u} = \frac{2}{u} + 1 - \frac{2e^{u}}{e^{u}+1} > = \frac{2}{u} - \tanh\!\frac{u}{2}. > $$ > Setting this to zero gives the transcendental condition > $$u\tanh\frac{u}{2} = 2,$$ > solved numerically by $u \approx 2.40$. The peak therefore sits at > $k_B T_{\rm peak} = \Delta/u \approx 0.417\,\Delta$, and its height is > $$ > C_{\rm peak} = k_B\,\frac{(2.40)^2 e^{2.40}}{(e^{2.40}+1)^2} > \approx 0.44\,k_B > $$ > per two-level system. A measured heat-capacity bump at $T_{\rm peak}$ thus > reports a gap $\Delta \approx 2.40\,k_B T_{\rm peak}$ and, through its height, > the number of two-level systems contributing. ## Adiabatic demagnetization The entropy of a paramagnet depends on the field only through the ratio $B/T$, because that ratio is the sole argument of the Boltzmann factors. At fixed $T$, raising the field aligns the spins and lowers their entropy; at fixed field, lowering the temperature does the same. This dependence is the basis of a cooling method that reaches millikelvin temperatures.[^schroeder-demag] The cycle has two strokes. - **Isothermal magnetization.** With the sample in thermal contact with a bath at temperature $T_i$, apply a strong field. The spins align, the spin entropy falls, and the released heat flows to the bath. The sample ends at $T_i$ with low spin entropy. - **Adiabatic demagnetization.** Isolate the sample and reduce the field slowly. With no heat exchange the total entropy is constant, and since the spin entropy depends only on $B/T$, holding it fixed while $B$ falls requires $T$ to fall in proportion. The sample cools. The temperature reached is limited by residual interactions among the moments, which set a small effective field that survives when the applied field is removed; below that scale the spins order and the method stops. Nuclear moments, with far weaker interactions, extend the technique to microkelvin temperatures. $$ % caption: Spin entropy against temperature at zero field and at high field; isothermal magnetization drops the system from the upper curve to the lower at fixed $T$, then adiabatic demagnetization moves left along the constant-entropy line to a lower temperature. \begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (0,0)--(7.4,0) node[right,black]{$T$}; \draw[->,black] (0,0)--(0,4.2) node[above,black]{$S$}; \draw[black,very thick] plot[domain=0.2:7.0,samples=80] (\x,{3.4*(1-exp(-1.6*\x))/(1+exp(-1.6*\x))}); \node[black!70,right,font=\scriptsize] at (6.2,3.35) {$B=0$}; \draw[acc,very thick] plot[domain=0.2:7.0,samples=80] (\x,{3.4*(1-exp(-0.42*\x))/(1+exp(-0.42*\x))}); \node[acc,right,font=\scriptsize] at (6.2,1.75) {$B$ large}; \draw[black,very thick,->] (5.0,3.13)--(5.0,1.42); \node[black!70,right,font=\scriptsize] at (5.05,2.4) {isothermal}; \draw[black,very thick,->] (5.0,1.42)--(1.55,1.42); \node[black!70,above,font=\scriptsize] at (3.2,1.45) {adiabatic (cools)}; \end{tikzpicture} $$ ## Summary - A magnetic moment in a field is a two-level system with $z = 2\cosh(\beta\mu B)$; $N$ spins give magnetization $M = N\mu\tanh(\mu B/k_B T)$, linear in the field at high $T$ and saturating at $N\mu$ at low $T$. - The linear regime is **Curie's law** $\chi = N\mu^2/k_B T = C/T$, the paramagnetic fluctuation–response relation $\chi = \beta\langle\Delta M^2\rangle$; general spin $J$ replaces $\tanh$ by the **Brillouin function** $B_J(x)$, which becomes the Langevin function as $J\to\infty$. - A finite gap $\Delta$ gives the **Schottky anomaly**, a heat-capacity peak near $k_B T\approx 0.42\,\Delta$ whose position measures the gap. - The spin entropy depends on the field only through $B/T$, so **adiabatic demagnetization** — isothermal magnetization then constant-entropy field reduction — cools the sample, reaching millikelvin (electronic) or microkelvin (nuclear) temperatures before residual interactions intervene. [^schroeder-para]: **Schroeder**, _An Introduction to Thermal Physics_, §3.3 — the two-state paramagnet, its magnetization, and the $\tanh$ dependence on $\mu B/k_B T$. Companion material at . [^reif-para]: **Reif**, _Fundamentals of Statistical and Thermal Physics_, §7.8 — the mean magnetic moment of a spin system, Curie's law, and the approach to saturation. [^pathria-brillouin]: **Pathria & Beale**, _Statistical Mechanics_ (4th ed.), §3.9–3.10 — the general-$J$ paramagnet, the Brillouin function, its Langevin ($J\to\infty$) and $\tanh$ ($J=\tfrac12$) limits, and the Curie constant $Ng^2\mu_B^2 J(J+1)/3k_B$. [^reif-schottky]: **Reif**, _Fundamentals of Statistical and Thermal Physics_, §7.8, and **Pathria & Beale**, §3.10 — the heat capacity of a two-level system and the Schottky peak near $k_B T\approx 0.42\,\Delta$. [^schroeder-demag]: **Schroeder**, _An Introduction to Thermal Physics_, §3.3 — adiabatic demagnetization as a cooling method, and the dependence of the spin entropy on $B/T$. **Reif**, §7.8, treats the residual-interaction limit.