Paramagnetism

Pour liquid oxygen between the poles of a strong electromagnet and it does something no other common liquid will do: it leaps into the gap and hangs there, a pale-blue bridge suspended by its own electrons, until it boils away. Water would be pushed feebly out; oxygen is pulled in, thousands of times harder. The difference is that each O $_2$ molecule carries a permanent magnetic moment — two unpaired electron spins — and permanent moments do not need to be manufactured by the field the way diamagnetic ones are. They already exist; the field merely invites them to align. But the invitation is almost entirely declined. At room temperature in a 1 tesla field, the alignment energy $\mu_B B$ is smaller than the thermal energy $k_BT$ by a factor of about four hundred — the moments are compass needles on a table shaken so violently that only two in a thousand, on statistical average, point north. That ratio, $\mu_B B/k_BT = 2.24\times10^{-3}$ in the lab's best ordinary conditions, runs the whole subject. This article owns the cleanest version of the statistics — the two-level spin- $\tfrac12$ Boltzmann ledger, worked end to end to $M = n\mu\tanh(\mu B/k_BT)$ — and exhibits the celebrated $1/T$ Curie tail that falls out of its small-argument limit. Along the way it keeps the books honest: why paired electrons contribute nothing, why the torque of Forces and Torques on Magnetic Dipoles aligns nothing by itself, why the classical version of the average was already computed in the electric twin Permanent Dipoles and Orientational Polarization, and why — courtesy of the Bohr–van Leeuwen theorem prosecuted in Diamagnetism — a permanent quantized moment is itself quantum mechanics' escape hatch.

Key takeaways

Paramagnetism is the alignment response: atoms with unpaired electrons carry permanent moments $\mu \sim \mu_B$ that partially orient along an applied field, giving $\chi_m > 0$ — matter weakly attracted into strong field, opposite to diamagnets.
The competition is $\mu B$ (ordering) versus $k_BT$ (disordering); at 1 T and 300 K the ratio is $x = \mu_B B/k_B T = 2.24\times10^{-3}$ — lab paramagnets live deep in the linear regime, aligned to about 0.2%.
The two-level spin- $\tfrac12$ ledger gives the exact law $M = n\mu\tanh(\mu B/k_BT)$ : linear Curie tail $M \approx n\mu^2B/k_BT$ at small $x$ , saturation $M_s = n\mu$ at large $x$ .
The Curie tail means $\chi \propto 1/T$ — cooling a paramagnet strengthens its response, the fingerprint that separates it from temperature-flat diamagnetism in one measurement run.
Saturation is honest but remote: at 300 K you would need hundreds of teslas; at 1 T you need sub-kelvin temperatures — which is exactly how adiabatic-demagnetization refrigerators exploit these spins.
Paired electrons contribute nothing (Pauli pairs them moment-against-moment), and in most solids the orbital moment is quenched — spin does nearly all the paramagnetic work.
KEY IDEA: permanent moments plus Boltzmann statistics give $M = n\mu\tanh(\mu B/k_BT)$ — a thermally diluted alignment that is small, positive, and proportional to $1/T$ in every ordinary laboratory condition, saturating only when $\mu B$ finally beats $k_BT$ .

The core insight

Paramagnetism is a competition between field-driven alignment of permanent atomic moments (energy $\sim \mu B$ ) and thermal randomization (energy $\sim k_BT$ ); in the common limit $\mu B \ll k_BT$ this gives a small, positive, $1/T$ susceptibility — Curie's law — with saturation $M_s = n\mu$ waiting at the far end of the tanh.

Where this lives today

The Curie tail is a working instrument across modern physics and medicine. Gadolinium(III), with seven unpaired $f$ electrons, is the payload of MRI contrast agents: its huge paramagnetic moment shortens the relaxation time of nearby water protons, lighting up blood vessels and lesions — millions of clinical doses a year of applied Curie physics. Electron paramagnetic resonance (EPR) spectroscopy drives transitions between the very $\pm\mu B$ levels of this article's ledger, reading out unpaired electrons in chemistry, radiation dosimetry, and quantum-materials labs. And paramagnetic oxygen analyzers — exploiting exactly the fact that O $_2$ is drawn into field gradients while N $_2$ and CO $_2$ are not — meter breathing-gas mixtures in anesthesia machines and combustion trains, a bedside deployment of worked example 2. At the cold frontier, the physics runs in reverse: instead of the field aligning spins, aligned spins are used to pump heat. Adiabatic demagnetization refrigerators magnetize a paramagnetic salt at a few kelvin (pushing $x$ toward 1 and squeezing the spin entropy out), isolate it, then ramp the field down; the moments re-randomize by drinking heat from the sample stage. ADRs cooled the detectors of space X-ray observatories — Hitomi and XRISM fly them — to tens of millikelvin, and laboratory versions reach microkelvin territory with nuclear moments. The same tanh curve that reads as a feeble 0.2% at room temperature becomes, at 50 mK, a fully saturated spin system whose entropy ledger is the most precise refrigerant known.

Origins · 1 minPierre Curie and how this was discovered

Pierre Curie

1859–1906 · Paris — ESPCI and the Sorbonne

Curie measured magnetism before he ever touched radium. Through the early 1890s at the École de Physique et Chimie Industrielles, he built torsion balances of legendary delicacy — instruments descending from his own piezoelectric quartz work with brother Jacques — and put some twenty substances through fields and furnaces up to 1370 °C. His 1895 doctoral thesis distilled the survey into the sharpest empirical statement magnetism-in-matter had yet produced: diamagnetic susceptibility is essentially independent of temperature, while paramagnetic susceptibility falls as $1/T$ — Curie's law, the tail this article derived from the Boltzmann ledger — and iron loses its ferromagnetism entirely above a critical temperature, now called the Curie point. That three-way classification by temperature fingerprint is still how a magnetometer run is read today. Langevin, his former student, supplied the statistical mechanism in 1905; Curie by then had redirected to radioactivity, shared the 1903 Nobel Prize with Marie and Becquerel, and was killed by a horse-drawn cart on the rue Dauphine in 1906. The kelvin-denominated constant in $\chi = C/T$ keeps his name on every susceptibility fit ever published.

History

1845

— Faraday's survey sorts matter drawn into fields from matter repelled; with Whewell he coins paramagnetic and diamagnetic

1895

— Pierre Curie's torsion-balance thesis establishes chi proportional to 1/T for paramagnets and the loss of ferromagnetism at the Curie point

1905

— Langevin derives Curie's law from Boltzmann statistics of permanent molecular moments

1927

— Pauli explains the weak temperature-independent paramagnetism of metals via Fermi statistics

1927

— Brillouin generalizes the alignment law to arbitrary quantized angular momentum J

1933

— Giauque and MacDougall demonstrate adiabatic demagnetization cooling, reaching 0.25 K with a paramagnetic salt

The physical picture

Picture a box of tiny compass needles being shaken. The applied field is a gentle pull toward "north"; the shaking is temperature. With a weak pull and vigorous shaking, only a slight statistical excess points north — that small excess is the magnetization. Strengthen the pull (raise $B$ ) or calm the shaking (lower $T$ ) and the alignment grows. This one image, with the ratio $\mu B/k_BT$ as its only dial, contains the entire subject; everything below is the image made quantitative. The needles are real. An atom with an unpaired electron carries a permanent magnetic moment of order one Bohr magneton, $\mu_B = 9.274\times10^{-24}\ \mathrm{J/T}$ — the natural atomic unit built in Magnetic Dipole Moment of Atoms, along with the spin g-factor $g\approx2$ that makes the electron's spin moment almost exactly $\mu_B$ . That article also settles where the moments come from; here we take them as given hardware and ask only how a warm crowd of them responds. And the field's handle on each needle is the torque $\mathbf{N} = \mathbf{m}\times\mathbf{B}$ , owned by Forces and Torques on Magnetic Dipoles — we spend it, we do not re-derive it. The torque tilts the energy ledger: a moment aligned with $\mathbf{B}$ sits at energy $-\mu B$ , an anti-aligned one at $+\mu B$ , and for a quantum spin- $\tfrac12$ those two entries are the entire menu — no intermediate angles exist to be counted.

Figure

The two-level ledger — a field splits the energies, temperature nearly equalizes the populations

Two honesty clauses before the statistics, both cheap to state and expensive to ignore. First: the torque aligns nothing by itself. An isolated moment in a field does not swing into line like a compass needle in honey — it precesses about $\mathbf{B}$ at the Larmor rate, holding its angle forever, because the torque is always perpendicular to the angular momentum (the dynamics belongs to Forces and Torques on Magnetic Dipoles). Alignment requires the moment to shed energy, and the only agent available is the thermal bath — lattice vibrations, collisions. Temperature therefore plays both sides: it is the disorder that dilutes the alignment and the mechanism that permits any alignment at all. Equilibrium statistical mechanics, which never asks how the energy got exchanged, is exactly the right tool. Second: most electrons are disqualified. Electrons in filled shells and bonding pairs come Pauli-paired, spin-up locked to spin-down, moments cancelling exactly — which is why water, quartz, and nearly everything organic have no permanent moment at all and show only the diamagnetic undertow of the previous article. Paramagnetism is the physics of the unpaired: transition-metal ions with open $d$ shells, rare earths with open $f$ shells, O $_2$ with its two lone $\pi^*$ electrons, conduction electrons in metals (a special quantum-statistical case, named and deferred below). One more flag for solids: the orbital contribution to the moment, so prominent in the free-atom bookkeeping of Magnetic Dipole Moment of Atoms, is largely quenched in crystals — the crystal's electric field locks the orbital motion, and the measured moments of most $3d$ salts come from spin alone. That quenching is why the two-level spin- $\tfrac12$ model below is not a toy: it is the honest skeleton of real magnetochemistry. And a recall that frames the whole enterprise: the Bohr–van Leeuwen theorem, prosecuted in Diamagnetism, proves that no strictly classical ensemble of charges can magnetize at all. The permanent moment we start from — quantized in magnitude, quantized even in its allowed directions — is not classical hardware. Paramagnetism escapes the theorem the same way diamagnetism did: by smuggling in a quantum rigidity (there, frozen orbit radii; here, a fixed moment with two allowed projections) and only then averaging classically. The Boltzmann arithmetic looks nineteenth-century; its legitimacy is pure quantum mechanics.

The response, quantified

The macroscopic dial is the same one the chapter always uses: the magnetization $\mathbf{M}$ (moment per volume, owned by Magnetization) responding linearly to the field,

\mathbf{M} = \chi_m \mathbf{H}, \qquad \chi_m > 0, \qquad \mu_r = 1 + \chi_m > 1 \qquad (1)

where: $\chi_m$ — the magnetic susceptibility, positive for paramagnets and typically $10^{-5}$ to $10^{-3}$ ; $\mathbf{H}$ — the auxiliary field of The Auxiliary Field H; $\mu_r$ — the relative permeability. The convention fine print — why $\chi_m$ is defined against $\mathbf{H}$ rather than $\mathbf{B}$ , and the dictionary between the choices — is owned by Magnetic Susceptibility; at $\chi_m \lesssim 10^{-3}$ the two definitions differ by parts in $10^3$ at worst, so this article writes $\chi \simeq \mu_0 M/B$ where convenient, with the crime confessed. A permeability above $\mu_0$ means the material welcomes flux: paramagnets are drawn toward strong field, the opposite verdict from the diamagnet's retreat. The microscopic energy ledger is the dipole orientation energy, recalled from Forces and Torques on Magnetic Dipoles:

U = -\mathbf{m}\cdot\mathbf{B} \;\;\xrightarrow{\text{spin-}\tfrac12}\;\; U_\pm = \mp\,\mu B \qquad (2)

where: $\mu$ — the magnitude of the moment's projection along $\mathbf{B}$ (for an electron spin, $\mu \approx \mu_B$ ); the two signs are the two allowed quantum orientations. The classical formula $-mB\cos\theta$ with its continuum of angles $\theta$ still governs molecules that tumble freely, and its thermal average was already computed — for electric dipoles — in Permanent Dipoles and Orientational Polarization; the mathematics transfers wholesale with the dictionary $pE \to \mu B$ . Here the quantum two-state case gets the full treatment instead, both because it is cleaner and because quenched-spin solids obey it. Everything is then run by one dimensionless number, the alignment parameter

x \equiv \frac{\mu B}{k_B T} \qquad (3)

where: $k_B = 1.381\times10^{-23}\ \mathrm{J/K}$ — Boltzmann's constant; $T$ — absolute temperature. Put in the lab's ordinary best: $\mu = \mu_B$ , $B = 1$ T, $T = 300$ K gives $x = 2.24\times10^{-3}$ . The level splitting $2\mu_B B$ at 1 T is worth only 1.34 K of temperature — room temperature out-shouts it by a factor of 450. Every paramagnet you will ever handle without a cryostat sits this deep in the linear regime, and the small- $x$ expansion is not an approximation so much as a description of life. How the responses stack up in the real world, in SI volume susceptibility: aluminium $+2.2\times10^{-5}$ , platinum $+2.65\times10^{-4}$ , liquid oxygen at 90 K $+3.4\times10^{-3}$ — against the diamagnetic band near $-10^{-5}$ from the previous article, and dwarfed by ferromagnetic iron's effective $10^3$ – $10^5$ two articles hence. Positive sign, $1/T$ growth on cooling, no memory: that triple fingerprint identifies a paramagnet in one measurement, and the full diagnostic decision tree is owned by Magnetic Susceptibility.

Derivation AThe two-level Boltzmann ledger and the tanh law

The goal is the exact equilibrium magnetization of $n$ non-interacting spin- $\tfrac12$ moments per unit volume at temperature $T$ in field $B$ — the result $M = n\mu\tanh(\mu B/k_BT)$ , owned here end to end. The strategy is the shortest honest route in statistical mechanics: two states, one Boltzmann factor each, a two-term partition function, and a population-weighted average. Watch two things: the average is over populations, not over a torque dynamics (the bath does the aligning, as the physical picture insisted), and no step will assume $x$ small — the law we get is exact for all fields and temperatures, which is what makes its two limits trustworthy.

Step 1. Enumerate the states and their weights. Each moment has two available orientations, along or against $\mathbf{B}$ , with energies $U_\pm = \mp\mu B$ from equation (2). In thermal equilibrium the canonical probabilities are Boltzmann factors, normalized by the partition function:

Z = e^{+\mu B/k_BT} + e^{-\mu B/k_BT} = 2\cosh x, \qquad p_\pm = \frac{e^{\pm x}}{2\cosh x},

with $x = \mu B/k_BT$ as in equation (3). Why Boltzmann factors and nothing else? Because the moments are in weak thermal contact with the lattice and exchange energy with it; equilibrium then dictates the canonical distribution regardless of the microscopic relaxation mechanism — the statistical mechanics needs the bath to exist, not to be modeled.

Step 2. Average the moment. Each aligned moment contributes $+\mu$ to the projection along $\mathbf{B}$ , each anti-aligned one $-\mu$ :

\langle \mu_z \rangle = \mu\,p_+ - \mu\,p_- = \mu\,\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} = \mu \tanh x.

The tanh is the population imbalance — the fraction of net-aligned moments. At $x = 2.24\times10^{-3}$ the populations are 50.11% versus 49.89%: two moments per thousand doing all the magnetic work.

Step 3. Sum to the magnetization. With $n$ moments per unit volume, all statistically identical and non-interacting,

M = n\,\langle\mu_z\rangle = n\,\mu \tanh\!\left(\frac{\mu B}{k_B T}\right).

This is the article's central result — exact, valid from zero field to full saturation. Why is non-interacting a fair assumption? Because in dilute paramagnetic salts and gases the dipole–dipole coupling between neighbors is microkelvins to millikelvins in temperature units — utterly negligible against lab temperatures. When interactions do matter, the subject changes name: cooperation is owned by Ferromagnetism, and the interacting correction to Curie's law by Curie Law and Curie Temperature.

Step 4. Read the two ends of the tanh. For $x \ll 1$ , $\tanh x \approx x$ and $M \approx n\mu^2B/k_BT$ — linear in $B$ , inversely proportional to $T$ : the Curie tail, promoted to a full derivation next. For $x \gg 1$ , $\tanh x \to 1$ and $M \to M_s = n\mu$ — every moment aligned, the ceiling no field can exceed. The crossover sits at $x \sim 1$ , i.e. $\mu B \sim k_BT$ : at 300 K that demands $B \approx 450$ T (no laboratory does this in steady state; even 100 T pulsed fields reach only $x = 0.22$ , a 22% alignment), while at $B = 1$ T it demands $T \lesssim 0.7$ K. Saturating a paramagnet is a cryogenic project, not a field project.

Step 5. Extract the susceptibility in the linear regime. With $M = n\mu^2B/k_BT$ and $B \simeq \mu_0 H$ (legal to parts in $10^3$ since $\chi_m \ll 1$ ),

\chi_m = \frac{M}{H} \simeq \frac{\mu_0 n \mu^2}{k_B T}.

Positive, and falling as $1/T$ . For $\mu = \mu_B$ and a solid-density $n = 2\times10^{28}\ \mathrm{m^{-3}}$ at 300 K this gives $\chi_m = 5.2\times10^{-4}$ — right in the platinum-to-oxygen band of the measured landscape. Verified: reproduces the standard spin-half Brillouin result (Ashcroft and Mermin Ch. 31, Kittel Ch. 11) and Griffiths's §6.1.3 discussion of paramagnetic alignment; the populations, the 450 T saturation-field estimate, and the χ value were recomputed for this edition.

The full curve deserves its portrait, because both of its ends carry physics:

Figure

The tanh law — Curie tail at small x, saturation ceiling at large x, and where the lab actually lives

Go deeper · Derivation B · 2 minThe Curie tail exhibited and the classical dictionary settled

Derivation BThe Curie tail exhibited and the classical dictionary settled

The goal is Curie's $1/T$ law as an honest limit of the exact tanh result, plus the settlement of accounts with the classical calculation — which this chapter refuses to re-derive, because its mathematics already exists, worked in full, in the electric twin Permanent Dipoles and Orientational Polarization. The strategy: expand the tanh, name the resulting law and its constant, then place the classical Langevin-function result beside it via the twin dictionary and reconcile the mismatched numerical factors. Watch for the factor of 3 — it is geometry, not error.

Step 1. Expand the exact law for small argument. With $\tanh x = x - x^3/3 + \cdots$ , the leading term gives

M \;=\; n\mu\tanh x \;\approx\; n\mu\,x \;=\; \frac{n\,\mu^2\,B}{k_B T}, \qquad \chi_m \simeq \frac{\mu_0 n \mu^2}{k_B T} \equiv \frac{C}{T},

with the (spin- $\tfrac12$ ) Curie constant $C = \mu_0 n\mu^2/k_B$ . Susceptibility inversely proportional to absolute temperature is Curie's law — found empirically by Pierre Curie in 1895, mechanism supplied by Langevin in 1905. The next correction is $-x^3/3$ , a part in $10^{-6}$ at lab conditions: the linear tail is, for practical purposes, exact. Why exhibit the law here but not develop it? Chapter memory: the law's own article-level treatment — the general-J Brillouin function it descends from, the Curie–Weiss modification $C/(T-\theta)$ that interactions force on it, and its use as a thermometer — is owned by Curie Law and Curie Temperature. This article establishes the phenomenon and its cleanest derivation; that one builds the full machinery.

Step 2. Recall the classical result — by name, not by re-derivation. A classical moment free to point at any angle, thermally averaged over orientations, yields the Langevin function: $\langle\cos\theta\rangle = L(x) = \coth x - 1/x$ , hence $M = nmL(x)$ . That average — the Boltzmann integral over the sphere of directions, its $u = \cos\theta$ substitution, and the limits $L(x) \to x/3$ and $L \to 1$ — was performed in full for electric dipoles in Permanent Dipoles and Orientational Polarization, and the mathematics is identical under the dictionary

pE \;\longrightarrow\; \mu B, \qquad x = \frac{pE}{k_BT} \;\longrightarrow\; \frac{\mu B}{k_BT}, \qquad P = np\,L(x) \;\longrightarrow\; M = nm\,L(x).

One derivation, two chapters — the deliberate E3 mirror at its most literal.

Step 3. Take the classical small- $x$ limit and read off the classical Curie constant. $L(x) \approx x/3$ gives

M \approx \frac{n m^2 B}{3 k_B T}, \qquad \chi_m \simeq \frac{\mu_0 n m^2}{3 k_B T}, \qquad C = \frac{\mu_0 n m^2}{3 k_B}.

Same $1/T$ anatomy, but a factor $\tfrac13$ where the two-level result had 1.

Step 4. Reconcile the factors — geometry, not contradiction. The classical moment wastes most of its orientations: directions near the equator contribute almost nothing to $\langle\cos\theta\rangle$ , and the average over the full sphere dilutes the response by exactly 3. The spin- $\tfrac12$ moment is spared the dilution — its two allowed states are both fully projective, $\pm\mu$ , nothing in between — so each moment works three times harder. The general quantized case (angular momentum $J$ , with $2J+1$ allowed projections) interpolates between the two: its Brillouin function reduces to our tanh at $J = \tfrac12$ and to the Langevin function as $J \to \infty$ , where many closely spaced projections mimic a classical continuum. That family portrait is owned by Curie Law and Curie Temperature; what matters here is that every member shares the $1/T$ tail — Curie's law is the universal small- $x$ signature of thermally contested permanent moments, whatever their quantum pedigree. Verified: the expansion coefficients and both Curie constants match Griffiths §6.1.3/§6.4 register, Kittel Ch. 11 and Ashcroft and Mermin Ch. 31; the Langevin limits are quoted from the E3 twin, not re-derived; the tanh-to-Brillouin-to-Langevin reduction chain is the standard result.

Limiting cases

Case	Result	Meaning
weak field / high $T$ ( $x \ll 1$ )	$M \approx n\mu^2B/k_BT$ , $\chi = C/T$	Curie's law — the permanent lab regime, 0.2% alignment at 1 T, 300 K
strong field / low $T$ ( $x \gg 1$ )	$M \to M_s = n\mu$	saturation: every moment aligned; needs $\sim$ 450 T at 300 K or sub-kelvin at 1 T
classical continuum of angles	$M = nm\,L(x)$ , $L(x) \to x/3$	Langevin function, recalled from Permanent Dipoles and Orientational Polarization — the $\tfrac13$ is orientation dilution
general quantized $J$	Brillouin function $B_J(x)$	tanh at $J=\tfrac12$ , Langevin at $J\to\infty$ ; owned by Curie Law and Curie Temperature
diamagnetic undertow included	$\chi_{\rm total} = C/T + \chi_{\rm dia}$	the negative, $T$ -flat term of Diamagnetism always rides along; net sign depends on which wins
conduction electrons in metals	Pauli paramagnetism, $T$ -independent	quantum statistics caps the aligning population; named here, owned by the solid-state track
interacting moments near ordering	$\chi = C/(T - \theta)$ diverging	Curie–Weiss behavior announcing cooperation — the on-ramp to Ferromagnetism

Common trapThe torque does not do the aligning — and your paramagnet is nowhere near saturated

Two standard misreadings of this physics, both fatal on exams. First: "the field's torque rotates the moments into alignment." No — an undamped moment in a field precesses at fixed angle forever; $\mathbf{N} = \mathbf{m}\times\mathbf{B}$ changes the direction of the angular momentum, never the energy. Alignment happens only because the thermal bath trades energy with the moments, and the equilibrium bookkeeping is Boltzmann's, not Newton's. The correct mental model is populations relaxing, not needles swinging. Second: treating $M = \chi H$ as if respectable alignments were in play. At 1 T and 300 K the population excess is $\tanh(2.24\times10^{-3}) \approx 0.22\%$ — the magnetization is a whisper of the saturation value $n\mu$ , and doubling the field simply doubles the whisper. If your calculation of a room-temperature paramagnet returns $M$ within an order of magnitude of $M_s$ , you have misplaced a factor of $k_BT$ . Saturation is real physics — the tanh ceiling — but reaching it is a millikelvin-cryostat or hundred-tesla project, and between here and there the linear Curie tail is the whole story.

Worked examples

1 · The 0.2% army — a room-temperature paramagnetic solid (confidence builder)

A crystal contains $n = 2.0\times10^{28}$ spin- $\tfrac12$ moments per m³, each with $\mu = \mu_B = 9.274\times10^{-24}$ J/T, at $T = 300$ K in $B = 1.0$ T. Find the alignment parameter, the level populations, the magnetization, and the fraction of saturation.

Step 1. The alignment parameter: $x = \dfrac{\mu_B B}{k_B T} = \dfrac{(9.274\times10^{-24})(1.0)}{(1.381\times10^{-23})(300)} = 2.24\times10^{-3}$ . The splitting $2\mu_B B$ is worth $2\mu_B B/k_B = 1.34$ K — thermally, a rounding error at room temperature.

Step 2. Populations from the Boltzmann ledger: $p_\pm = e^{\pm x}/2\cosh x$ gives $p_+ = 50.11\%$ along the field, $p_- = 49.89\%$ against. The working excess is $\tanh x = 2.24\times10^{-3}$ — about two moments per thousand.

Step 3. Magnetization: $M = n\mu_B\tanh x = (2.0\times10^{28})(9.274\times10^{-24})(2.24\times10^{-3}) = 415\ \mathrm{A/m}$ . The susceptibility follows as $\chi_m \simeq \mu_0 M/B = 5.2\times10^{-4}$ — platinum-class, sensible.

Step 4. Perspective against the ceiling and against iron. Saturation would be $M_s = n\mu_B = 1.85\times10^{5}$ A/m; we are at 0.22% of it. And ferromagnetic iron, without any applied field, sustains $M_s = 1.7\times10^{6}$ A/m (the number owned by Magnetization) — four thousand times our field-begged 415 A/m. That gulf, between what statistics extracts from non-interacting moments and what cooperation delivers unbidden, is the cliffhanger the next article resolves.

2 · Liquid oxygen, the show-off paramagnet (exam level)

Liquid O $_2$ at 90 K has density $1141\ \mathrm{kg/m^3}$ ; each molecule (molar mass 32.0 g/mol) carries two unpaired electron spins, total spin $S = 1$ , spin-only effective moment $m_{\rm eff} = g\sqrt{S(S+1)}\,\mu_B = 2\sqrt{2}\,\mu_B$ . (a) Predict the Curie susceptibility at 90 K and compare with the measured $\chi = +3.4\times10^{-3}$ . (b) Compute the magnetic force per unit volume in the fringe field of an electromagnet with $B = 1.0$ T and $\mathrm{d}B/\mathrm{d}z = 100$ T/m, and compare with gravity.

Step 1. Number density: $n = \dfrac{\rho}{M_{\rm mol}}N_A = \dfrac{1141}{0.0320}\times6.022\times10^{23} = 2.15\times10^{28}\ \mathrm{m^{-3}}$ . Effective moment: $m_{\rm eff} = 2\sqrt2\,(9.274\times10^{-24}) = 2.62\times10^{-23}$ J/T $= 2.83\,\mu_B$ .

Step 2. Curie estimate (general-moment form with the classical $\tfrac13$ , appropriate for a moment averaged over orientations): $\chi = \dfrac{\mu_0 n\, m_{\rm eff}^2}{3k_BT} = \dfrac{(1.257\times10^{-6})(2.15\times10^{28})(2.62\times10^{-23})^2}{3(1.381\times10^{-23})(90)} = 5.0\times10^{-3}$ .

Step 3. Compare honestly. Measured: $+3.4\times10^{-3}$ — the free-moment estimate overshoots by about 45%. Inverting the Curie law on the measured value gives an apparent moment of $2.34\,\mu_B$ , noticeably below the spin-only $2.83\,\mu_B$ . The shortfall is real physics, not sloppiness: in the dense liquid, neighboring O $_2$ moments couple antiferromagnetically, partially frustrating the alignment — the first hint that interacting moments deviate from ideal Curie behavior (the Curie–Weiss bookkeeping of Curie Law and Curie Temperature). The dilute-gas check vindicates the theory where its assumptions hold: for O $_2$ gas at 293 K and 1 atm ( $n = P/k_BT = 2.5\times10^{25}\ \mathrm{m^{-3}}$ ), the same formula predicts $\chi = 1.8\times10^{-6}$ against a measured $\approx 1.9\times10^{-6}$ — agreement at the few-percent level.

Step 4. The force. Per unit volume, the dipole force machinery of Forces and Torques on Magnetic Dipoles — $\mathbf{F} = \nabla(\mathbf{m}\cdot\mathbf{B})$ , spent, not re-derived — applied with $\mathbf{M} = \chi\mathbf{B}/\mu_0$ gives $f_z = (\chi/\mu_0)B\,\mathrm{d}B/\mathrm{d}z = \dfrac{3.4\times10^{-3}}{1.257\times10^{-6}}(1.0)(100) = 2.7\times10^{5}\ \mathrm{N/m^3}$ . Gravity on the liquid is $\rho g = 1.12\times10^{4}\ \mathrm{N/m^3}$ : the magnetic pull is 24 times the liquid's weight, directed up the gradient toward the pole gap — which is why the demo works, the pale-blue liquid climbing to the poles and clinging there until it boils. Supporting liquid oxygen against gravity needs only $B\,\mathrm{d}B/\mathrm{d}z = \mu_0\rho g/\chi = 4.1\ \mathrm{T^2/m}$ — compare the $1400\ \mathrm{T^2/m}$ that diamagnetic water demanded in Diamagnetism: the permanent-moment response, even thermally diluted, beats the induced one by more than two orders of magnitude.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.1 (Magnetization) and §6.4 (Linear and Nonlinear Media) · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11 (Magnetic Fields in Matter) · Jackson, Classical Electrodynamics, 3rd ed., §5.7–5.8 · Zangwill, Modern Electrodynamics, Ch. 13 (Magnetic Matter) · Kittel, Introduction to Solid State Physics, 8th ed., Ch. 11 (Diamagnetism and Paramagnetism) · Ashcroft & Mermin, Solid State Physics, Ch. 31

Provenance. Equations and derivations are standard textbook results. A verifier should: (1) reproduce the Langevin integral $\int_0^\pi\cos\theta\,e^{x\cos\theta}\sin\theta\,\mathrm{d}\theta/\int_0^\pi e^{x\cos\theta}\sin\theta\,\mathrm{d}\theta = \coth x - 1/x$ (Griffiths §6.4, Kittel Ch. 11); (2) verify the spin-1/2 partition function $Z=2\cosh(\mu B/k_BT)$ and $\langle\mu_z\rangle=\mu\tanh(\mu B/k_BT)$ (Ashcroft & Mermin Ch. 31); (3) check the Brillouin-function limits $B_J\to L(x)$ as $J\to\infty$ and $B_{1/2}=\tanh$ ; (4) confirm physical constants $k_B=1.381\times10^{-23}\,\mathrm{J/K}$ , $\mu_B=9.274\times10^{-24}\,\mathrm{J/T}$ , $\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m/A}$ (CODATA). Curie constant form $C=\mu_0 nm^2/3k_B$ matches Griffiths and Jackson §5.7–5.8. Worked numbers recomputed for this edition: $x = \mu_B B/k_BT = 2.24\times10^{-3}$ at 1 T, 300 K (populations 50.11%/49.89%, $M = 415$ A/m vs $M_s = 1.85\times10^5$ A/m for $n = 2\times10^{28}\ \mathrm{m^{-3}}$ ); liquid O $_2$ ( $n = 2.15\times10^{28}\ \mathrm{m^{-3}}$ , $m_{\rm eff} = 2\sqrt2\,\mu_B$ ): Curie estimate $\chi = 5.0\times10^{-3}$ at 90 K vs measured $+3.4\times10^{-3}$ (apparent moment $2.34\,\mu_B$ ), gas-phase check $1.8\times10^{-6}$ vs $\approx1.9\times10^{-6}$ , force density $2.7\times10^5\ \mathrm{N/m^3}$ at $B\,\mathrm{d}B/\mathrm{d}z = 100\ \mathrm{T^2/m}$ (24 times gravity), support threshold $4.1\ \mathrm{T^2/m}$ .

Builds on: Magnetic Dipole Moment of Atoms · Magnetization · The Auxiliary Field H · Magnetic Susceptibility

Leads to: Diamagnetism · Ferromagnetism · Magnetic Permeability · Linear and Nonlinear Magnetic Media

If the field exerts a torque on every moment, why does the moment not just swing into alignment like a compass needle?

Because a compass needle is damped and a bare moment is not. The torque $\mathbf{m}\times\mathbf{B}$ is always perpendicular to the angular momentum it acts on, so an isolated moment precesses around the field at constant angle — its energy never changes, and no alignment accumulates. A compass needle aligns because friction at the pivot drains the oscillation energy away. In matter, the "friction" is the thermal bath: lattice vibrations and collisions let a moment drop into the lower energy state, at the price of also kicking aligned moments back out. Equilibrium is the standoff between the two, and that is why the answer comes from Boltzmann factors rather than from integrating the torque equation. The precession dynamics itself is owned by Forces and Torques on Magnetic Dipoles.

When do I use the tanh law and when the Langevin function?

Count the allowed orientations. A quantum spin- $\tfrac12$ has exactly two projections along the field, so its thermal average is a two-term sum — the tanh. A moment free to point in any direction (a classical dipole, or in practice a large- $J$ moment whose many projections blur into a continuum) requires the integral over the sphere — the Langevin function $L(x) = \coth x - 1/x$ , whose derivation lives in Permanent Dipoles and Orientational Polarization and transfers here via $pE \to \mu B$ . In between sits the general quantized case, the Brillouin function of Curie Law and Curie Temperature. For most $3d$ transition-metal solids the honest choice is spin-only and low- $J$ — quenching kills the orbital part — which is why the two-level model is the workhorse. All three choices share the same small- $x$ behavior up to a geometric factor: the $1/T$ tail is universal.

Why exactly do paired electrons contribute nothing — is the cancellation really perfect?

For the spin part, yes, and it is enforced rather than accidental. The Pauli principle requires two electrons sharing an orbital to have opposite spin projections; their spin moments are equal and opposite and sum to zero identically — not approximately. A filled shell goes further: both its total spin and its total orbital angular momentum vanish, so the atom's permanent moment is exactly zero and only the induced diamagnetic response of Diamagnetism remains. This is why chemistry so cleanly dictates magnetism: count the unpaired electrons (two in O $_2$ , five in Fe³⁺, seven in Gd³⁺) and you have counted the paramagnetic hardware. The pairing energy is electron-volts — tens of thousands of kelvin — so no laboratory field or temperature un-pairs them; the ledger of who is paired is fixed by chemistry long before magnetism gets a vote.

Aluminium is a metallic paramagnet, but its susceptibility barely changes on cooling. Did Curie's law fail?

Curie's law assumes every moment is free to flip independently, with Boltzmann statistics governing the populations. Conduction electrons in a metal violate that assumption spectacularly: they form a degenerate Fermi sea, and the Pauli principle forbids most of them from flipping — the states they would flip into are already occupied. Only the thin shell of electrons within about $k_BT$ of the Fermi energy can respond, and that fraction shrinks as $T$ falls in exactly the way that cancels the $1/T$ of the free-moment law. The result — Pauli paramagnetism — is weak and nearly temperature-independent, which is what aluminium shows. It is named here for honest bookkeeping and owned by the solid-state track; the Curie-law physics of this article applies to localized moments: insulating salts, dilute ions, molecular gases like O $_2$ .

The liquid-oxygen example under-shot the spin-only prediction. Should I distrust the Curie estimate at that level?

Trust it exactly as far as its assumptions. The Curie derivation assumes non-interacting moments, and in a liquid at $2\times10^{28}$ molecules per m³ the O $_2$ moments are close enough to feel one another — exchange and dipolar couplings with an antiferromagnetic bias partially frustrate the alignment, depressing $\chi$ below the free-moment value (measured $3.4\times10^{-3}$ versus predicted $5.0\times10^{-3}$ , an apparent moment of $2.34\,\mu_B$ against the spin-only $2.83\,\mu_B$ ). The same formula applied to dilute O $_2$ gas — where the assumption is true — lands within a few percent of experiment. The systematic repair is the Curie–Weiss form $\chi = C/(T-\theta)$ with a negative $\theta$ for antiferromagnetic bias, owned by Curie Law and Curie Temperature. The lesson generalizes: a Curie fit that misbehaves is not a broken law but a measurement of the interactions it neglected.

Why is the saturation magnetization of a paramagnet so much smaller than iron's, when iron's atoms carry similar moments?

It is not smaller — that is the shock. Saturated, our example paramagnet reaches $M_s = n\mu \approx 2\times10^5$ A/m, within an order of magnitude of iron's $1.7\times10^6$ A/m (iron packs more moments per volume, each worth $\approx 2.2\,\mu_B$ ). The difference is who pays for the alignment. The paramagnet needs $\mu B \gtrsim k_BT$ — hundreds of teslas at room temperature — because each moment fights the bath alone. Iron's moments get the alignment free: a quantum interaction locks neighboring spins parallel with an effective strength of order 1000 K, so the material saturates locally with no applied field at all. Same hardware, radically different organization — the cooperation is the entire subject of Ferromagnetism, and the cliffhanger of this chapter.

Cooling a paramagnet increases its susceptibility. Can I cool it into becoming a ferromagnet?

Sometimes — and watching for it is a standard laboratory diagnostic. As $T$ falls, $\chi = C/T$ grows, and if the moments interact even weakly, the measured susceptibility follows $C/(T-\theta)$ instead: extrapolate $1/\chi$ versus $T$ and the intercept $\theta$ reveals the interaction's sign and strength. A positive $\theta$ means neighboring moments prefer parallel alignment, and at $T \approx \theta$ the susceptibility diverges — the material spontaneously orders and becomes a ferromagnet below its Curie temperature. Negative $\theta$ signals antiparallel bias, heading toward antiferromagnetic order instead (liquid oxygen leans this way). A truly ideal paramagnet, with $\theta = 0$ , stays paramagnetic to absolute zero. The quantitative machinery — and what happens at the transition — is owned by Curie Law and Curie Temperature and Ferromagnetism.