Diamagnetism

Hold a strong magnet up to almost anything — a splash of water, a wooden bead, a strawberry, your own hand — and the object is pushed away. Feebly, invisibly without instruments, but universally: every atom in every material responds to an applied magnetic field with a small induced moment pointing against it. This is diamagnetism, and its logic is Lenz's law shrunk to atomic scale. Switching on a field threads flux through each electron orbit; the orbit reacts like the current loop it is, adjusting its circulation to oppose the change. Unlike a compass needle, which turns to align, the diamagnetic moment is created backwards and vanishes the instant the field does. Two things make this little effect worth a whole article. First, it is the honest baseline of all magnetic response — present in everything, visible only where nothing louder (paramagnetism, ferromagnetism) shouts over it, and strong enough, at 16 tesla, to levitate a living frog. Second, it hides the deepest embarrassment of classical physics: the Bohr–van Leeuwen theorem proves that a strictly classical gas of charges in thermal equilibrium has zero magnetization — no diamagnetism, no paramagnetism, nothing. The tidy classical derivation this article works through gets the right answer only because quantum mechanics quietly holds the orbit radius fixed. We will derive the induced moment, sum it into the Langevin susceptibility, check it against real water, and then prosecute the theorem that says the whole calculation had no classical right to succeed.

Key takeaways

Diamagnetism is the universal induced response: an applied $\mathbf{B}$ perturbs every electron orbit, and by Lenz's law the induced moment opposes the field — $\chi < 0$ , always and everywhere.
Per electron, $\Delta m = -\dfrac{e^2\langle\rho^2\rangle}{4m_e}B$ : at 1 T this is about $2\times10^{-6}$ Bohr magnetons — two millionths of an atom's natural moment unit, which is why diamagnetism is so feeble.
Summed over matter: the Langevin susceptibility $\chi = -\mu_0 nZe^2\langle r^2\rangle/6m_e \sim -10^{-5}$ , temperature-independent because no thermal alignment is involved.
The response survives only in closed-shell matter (water, copper, bismuth, noble gases); one unpaired electron and paramagnetism, a hundred times larger, buries it.
Diamagnets are repelled from strong field and seek field minima — for water, levitation needs $B\,\mathrm{d}B/\mathrm{d}z \approx 1400\ \mathrm{T^2/m}$ , delivered by a 16 T research magnet (the levitating frog).
The Bohr–van Leeuwen theorem: a full classical thermal average gives $M = 0$ identically — magnetism in matter is quantum mechanical, and the classical Langevin model works only because it freezes $\langle r^2\rangle$ by hand.
KEY IDEA: switching on a field induces, by Lenz's law, a change in every atomic orbit whose moment opposes $\mathbf{B}$ — a weak, negative, temperature-independent response present in all matter, and classically impossible: only quantum rigidity of the orbits lets any of it survive a thermal average.

The core insight

An applied field induces, by Lenz's law, atomic orbital currents whose magnetic moments oppose the field; the net effect is a small negative magnetization present in all matter, captured by $\chi_m<0$ and $\mu_r<1$ .

Where this lives today

Diamagnetism earns its keep as a background you must subtract and a handle you can hold. Every SQUID or vibrating-sample magnetometry measurement of an interesting magnetic material rides on top of the sample holder's and solvent's diamagnetic response, subtracted via tabulated Pascal's constants — a century-old bookkeeping of atomic $\langle r^2\rangle$ contributions still cited in every molecular-magnetism paper. In NMR, the diamagnetic circulation of electron clouds shields each nucleus by parts per million — the chemical shift, arguably the single most information-rich number in chemistry, is your molecule's local diamagnetism read out nucleus by nucleus. And diamagnetic levitation has become a laboratory tool rather than a stunt: containerless crystal growth, microgravity simulation for biological samples in high-field bores, and desktop graphite levitators for force sensing at the nano-newton scale. The frontier cases push the two extremes of the susceptibility landscape. Superconductors realize $\chi = -1$ — perfect diamagnetism, flux fully expelled — and every maglev demonstration, MRI magnet quench-safety analysis, and SQUID pickup loop is engineering built on that endpoint (the mechanism, a macroscopic quantum state rather than atomic Larmor currents, is honestly deferred to the linear-media article's superconductor limit and to superconductivity itself). At the other extreme, bismuth and graphite — the anomalous diamagnets whose delocalized electrons sweep huge effective areas — became the platform where Landau's quantum orbital diamagnetism of band electrons turns into measurable oscillations, the de Haas–van Alphen effect that maps Fermi surfaces to this day.

Origins · 1 minPaul Langevin and how this was discovered

Paul Langevin

1872–1946 · Paris — ESPCI and Collège de France

Langevin turned magnetism-in-matter from a catalogue of curiosities into statistical physics. His 1905 pair of papers built both halves of the classical theory in one stroke: the induced, temperature-flat diamagnetism of electron orbits (this article's equation 3), and the thermally contested alignment of permanent moments that yields Curie's $1/T$ paramagnetism (the next article's). The tools were audacious for the time — electrons inside atoms treated as real current loops, Boltzmann averaging applied to their orientations — and the results matched Pierre Curie's decade-old precision measurements so well that Curie's law acquired a mechanism overnight. Langevin's later life kept the same nerve: he championed Einstein's relativity in France, invented sonar ultrasonics in World War I, and was dismissed by the Vichy government for his antifascism, surviving to be restored in 1944. The irony he lived to see: Bohr and van Leeuwen proved his classical derivations shouldn't have worked — the orbits he froze by physical instinct were being frozen, in truth, by quantum mechanics. His formulas survived the revolution that overturned their foundations, which is its own kind of monument.

History

1778

— Brugmans observes bismuth repelled by a magnet, the first recorded diamagnetic effect

1845

— Faraday shows the repulsion is universal across materials and, with Whewell, coins diamagnetic

1895

— Pierre Curie's balance survey separates temperature-flat diamagnets from Curie-law paramagnets

1905

— Langevin derives both responses from electron theory: induced opposition and thermal alignment

1911

— Bohr's dissertation proves classical equilibrium magnetization vanishes; van Leeuwen completes the theorem by 1921

1933

— Meissner and Ochsenfeld find superconductors expel flux entirely — the perfect diamagnet

1997

— Berry and Geim levitate a living frog in a 16 T magnet at Nijmegen

The physical picture

Picture the electron orbit as what magnetically it is — a current loop (the Magnetic Dipole Moment of Atoms article built this identification and its gyromagnetic dictionary; we spend it here). Now raise a magnetic field through the loop. While the field ramps, the changing flux drives a circulating electric field around the orbit — Faraday's law, borrowed at one-line level from its E6 home — and that tangential push changes the electron's speed. Here is the beautiful part: the sign of the change is rigged. Lenz's law guarantees the orbit speeds up or slows down in exactly the sense that opposes the added flux, whichever way the electron happened to be circulating. Orbits that circulate one way gain moment against $\mathbf{B}$ ; orbits circulating the other way lose moment along $\mathbf{B}$ ; the change is antiparallel to the field in every case. That universality of sign is why the effect never cancels over an atom, and why every material on Earth is at least a little diamagnetic.

Figure

Lenz's law at the atom — the induced change in moment opposes the field for either sense of orbit

The cleaner, frame-free version of the same statement is Larmor's theorem: to first order in $B$ , the entire electron cloud simply precesses about the field axis at $\omega_L = eB/2m_e$ , superimposed on whatever bound motion it already had. A precessing charge distribution is an extra circulation of charge — one shared, coherent current loop wrapped around the field axis — and its moment is antiparallel to $\mathbf{B}$ , proportional to how much charge there is and how far it sits from the axis: $\langle\rho^2\rangle$ . The induced moment costs no thermal bargaining. Nothing has to align against $k_BT$ ; the geometry of the orbit fixes the response outright. That is why diamagnetic susceptibility is temperature-independent, in flat contrast to paramagnetism's Curie $1/T$ (next article) — and why $\chi$ here is a materials constant in the strictest sense, pinned by $\langle r^2\rangle$ alone. How feeble is it? At $B = 1$ T — a serious laboratory field — the induced moment per electron is about $2\times10^{-6}\,\mu_B$ : two millionths of the Bohr magneton, the natural atomic unit of moment (owned by Magnetic Dipole Moment of Atoms). Equivalently, the electron's orbital speed shifts by a few metres per second on top of $\sim 2\times10^{6}$ m/s of orbital motion. Materials whose atoms carry permanent moments — one unpaired spin means a full $\mu_B$ available for thermal alignment — show a paramagnetic response some $10^2$ – $10^3$ times larger, and the diamagnetic undertow becomes an invisible correction. Pure diamagnetism is therefore the signature of closed shells: noble gases, water, NaCl, copper, bismuth, nearly everything organic. It is the magnetic sound of a completed electron ledger.

Figure

The susceptibility landscape — diamagnets sit just below zero, and everything else shouts over them

One more picture before the algebra, because it pays off at the end of the article. A diamagnet's induced moment is antiparallel to $\mathbf{B}$ , and the force on a dipole in a nonuniform field is $\mathbf{F} = \nabla(\mathbf{m}\cdot\mathbf{B})$ (owned by Forces and Torques on Magnetic Dipoles; we spend it, we do not re-derive it). Antiparallel moment means $\mathbf{m}\cdot\mathbf{B} < 0$ , growing more negative where the field is stronger — so the force points downhill in field strength. Diamagnets seek field minima. A free-space minimum of $|\mathbf{B}|$ is allowed (unlike a maximum), so a diamagnetic object can sit in stable levitation above nothing at all — the loophole in Earnshaw's theorem through which frogs fly.

The response, quantified

The macroscopic bookkeeping runs through the magnetization $\mathbf{M}$ (moment per volume — the chapter's opening article, Magnetization, owns the definition and its mesoscopic average). For a diamagnet the response is linear to extraordinary accuracy, and the standard dial is written

\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, a pure number, negative and of order $10^{-5}$ for diamagnets; $\mathbf{H}$ — the auxiliary field (constructed two articles ahead in 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 two choices — is owned by Magnetic Susceptibility; at $|\chi_m|\sim10^{-5}$ the distinction is invisible ( $\mu_0 M/B$ and $M/H$ differ by parts in $10^5$ ), so this article writes $\chi \simeq \mu_0 M/B$ without further apology. A permeability below $\mu_0$ means field lines are weakly expelled: the diamagnet is the material that says "less flux through me, please." The microscopic content sits in two results derived below. The single-electron response, for an orbit of mean-square axial distance $\langle\rho^2\rangle$ ,

\Delta \mathbf{m} = -\frac{e^2 \langle\rho^2\rangle}{4 m_e}\,\mathbf{B} \qquad (2)

where: $e$ , $m_e$ — electron charge and mass; the minus sign is Lenz's law made algebraic. Read the structure: the response is linear in $B$ (an induced effect, not an aligned one), grows with orbit area (more flux to defend), and carries $e^2$ — independent of the sign of the carrier and of the sense of circulation, which is why atoms with electrons orbiting both ways still respond coherently. At 1 T equation (2) delivers $1.97\times10^{-29}\ \mathrm{J/T} \approx 2\times10^{-6}\,\mu_B$ for a hydrogen-like orbit — the number behind "feeble." Summing over $Z$ electrons per atom and $n$ atoms per volume, with the sphere average $\langle\rho^2\rangle = \tfrac{2}{3}\langle r^2\rangle$ converting axial distance to true orbital radius, gives the Langevin (classical) diamagnetic susceptibility:

\chi = -\frac{\mu_0\, n\, Z\, e^2\, \langle r^2\rangle}{6\, m_e} \qquad (3)

where: $\langle r^2\rangle$ — mean-square electron distance from the nucleus, averaged over all $Z$ electrons. Every factor is a physical dial: density of responders ( $nZ$ ), size of each loop ( $\langle r^2\rangle$ ), and no temperature anywhere. Feed in water's numbers and (3) returns $-8\times10^{-6}$ against a measured $-9.05\times10^{-6}$ — worked example 2 does it honestly. The same expression falls out of the quantum-mechanical $A^2$ term $\tfrac{e^2}{8m_e}B^2\langle x^2+y^2\rangle$ in the atomic Hamiltonian, which is the modern reason to trust a formula whose classical pedigree derivation 3 is about to revoke. The kinematic heart of both results is the Larmor frequency,

\omega_L = \frac{eB}{2m_e} \qquad (4)

where: $\omega_L$ — the precession rate of the whole electron distribution about $\mathbf{B}$ , to first order in $B$ . Note the 2: this is half the cyclotron frequency $eB/m_e$ of a free electron, because a bound electron keeps its binding force and only the perturbation rotates the motion. At 1 T, $\omega_L = 8.79\times10^{10}\ \mathrm{rad/s}$ — a 14 GHz precession, the frequency scale that reappears in magnetic resonance (the Larmor precession of moments in Forces and Torques on Magnetic Dipoles is this same theorem wearing its dynamical hat).

Derivation AThe induced moment from Lenz's law at the atom

The goal is equation (2): the change in an orbiting electron's magnetic moment when a field is ramped from $0$ to $B$ perpendicular to the orbit, derived from induction alone. The strategy is Griffiths's own (§6.1.3): during the ramp, Faraday's law supplies a tangential electric field at the orbit; integrating the resulting force over the whole switch-on gives a speed change independent of how fast the ramp was run, and the speed change is a moment change. Watch two things: the Faraday step is quoted at one-line level (its full machinery lives in E6, Faraday's Law of Induction — using it early is standard and honest, since only the integrated flux matters), and the orbit radius $r$ is held fixed throughout — a quiet assumption that derivation 3 will expose as the quantum ingredient.

Step 1. Ramp the field and read off the induced electric field. For a circular loop of radius $r$ threaded by uniform $B(t)$ , Faraday's law $\oint\mathbf{E}\cdot\mathrm{d}\boldsymbol{\ell} = -\mathrm{d}\Phi/\mathrm{d}t$ gives, by symmetry,

E(t)\,(2\pi r) = -\pi r^2\,\frac{\mathrm{d}B}{\mathrm{d}t} \quad\Longrightarrow\quad E = -\frac{r}{2}\frac{\mathrm{d}B}{\mathrm{d}t}.

Why a single loop and not the real smeared orbit? Because only $\langle\rho^2\rangle$ survives the final average; the circular orbit is the bookkeeping-cleanest representative, upgraded to the general average in derivation 2.

Step 2. Integrate the tangential force over the switch-on. The electron (charge $-e$ ) feels $F = (-e)E$ along its direction of motion:

m_e\,\frac{\mathrm{d}v}{\mathrm{d}t} = \frac{er}{2}\,\frac{\mathrm{d}B}{\mathrm{d}t} \quad\Longrightarrow\quad \Delta v = \frac{er}{2m_e}\,B.

Why does the ramp rate drop out? The force integrates the flux history: $\Delta v$ depends only on the total change in $B$ , not on the schedule — switch-on adiabaticity is free here.

Step 3. Convert the speed change to a moment change. An orbiting charge is a current loop of moment magnitude $m = \tfrac{1}{2}evr$ (the current-loop dictionary owned by Magnetic Dipole Moment of Atoms), and the induced piece opposes the field by Lenz's law:

\Delta m = -\frac{e}{2}\,r\,\Delta v = -\frac{e^2 r^2}{4 m_e}\,B.

Run the sign honestly for both senses of circulation: an electron orbiting so its moment initially points along $\mathbf{B}$ is decelerated (its moment shrinks — change against $\mathbf{B}$ ); one orbiting the other way is accelerated (its moment grows — also against $\mathbf{B}$ ). The change is antiparallel to the field in both cases, which is why the atomic sum never cancels.

Step 4. Numbers, to calibrate "feeble". For $r = a_0 = 0.529\ \mathrm{Å}$ and $B = 1$ T: $\Delta v = 4.65$ m/s on top of an orbital speed of order $2.2\times10^6$ m/s (a part in $2\times10^6$ ), and $\Delta m = 1.97\times10^{-29}\ \mathrm{J/T} = 2.1\times10^{-6}\,\mu_B$ . The response is real, rigidly anti-aligned, and two millionths of the natural moment scale. Verified: reproduces Griffiths §6.1.3 (Eq. 6.11) with $\langle\rho^2\rangle\to r^2$ ; the equivalent Larmor route — precession at $\omega_L = eB/2m_e$ constitutes a current $I = -e\omega_L/2\pi$ , moment $I\pi r^2 = -e^2r^2B/4m_e$ — lands on the same coefficient, and the numbers above were recomputed for this edition.

Go deeper · Derivation B · 1 minFrom one orbit to matter — the Langevin susceptibility

Derivation BFrom one orbit to matter — the Langevin susceptibility

The goal is equation (3), the bulk susceptibility, from the single-electron result — a short derivation whose only substantive move is a geometric average, plus the honest bookkeeping of what $\langle r^2\rangle$ means. The strategy: replace the fixed perpendicular orbit of derivation 1 with an atom of spherically distributed electrons, average, multiply by the density of electrons, and convert magnetization to susceptibility.

Step 1. Average the geometry. Equation (2) involves the mean-square distance from the field axis, $\langle\rho^2\rangle = \langle x^2 + y^2\rangle$ . For a spherically symmetric electron cloud, $\langle x^2\rangle = \langle y^2\rangle = \langle z^2\rangle = \tfrac{1}{3}\langle r^2\rangle$ , so

\langle\rho^2\rangle = \frac{2}{3}\,\langle r^2\rangle.

Why insist on the average? Because a real atom offers orbits at every inclination; the sphere average is exact for closed shells, which are precisely the atoms whose diamagnetism is not masked.

Step 2. Sum the responders. Each of the $Z$ electrons in each of the $n$ atoms per unit volume contributes the averaged version of (2), so the magnetization is

M = n Z\,\Delta m = -\frac{n Z e^2 \langle r^2\rangle}{6 m_e}\,B,

with $\langle r^2\rangle$ now understood as the average over all electrons in the atom — inner shells contribute little (small $r^2$ ), valence electrons dominate.

Step 3. Convert to the standard dial. With $|\chi|\ll 1$ , $B \simeq \mu_0 H$ inside the material to parts in $10^5$ , so

\chi = \frac{M}{H} \simeq \frac{\mu_0 M}{B} = -\frac{\mu_0\, n Z e^2\, \langle r^2\rangle}{6 m_e}.

Why is the $H$ -versus- $B$ carelessness legal here and only here? Because the correction is itself of order $\chi$ — a $10^{-5}$ relative error on a $10^{-5}$ effect. The convention that makes this precise is owned by Magnetic Susceptibility.

Step 4. Read the result. Negative — always. Temperature-independent — no $k_BT$ ever entered; nothing aligned, nothing thermalized. Scale: with $n \sim 3\times10^{28}\ \mathrm{m^{-3}}$ , $Z\sim10$ , $\sqrt{\langle r^2\rangle}\sim 0.65\ \mathrm{Å}$ , the formula returns $\chi \sim -8\times10^{-6}$ — squarely the measured band of water, copper, and friends (worked example 2). Quantum mechanics reproduces exactly this expression from the $A^2$ term of the minimal-coupling Hamiltonian evaluated on frozen atomic states, which is why it works — as the next derivation shows, no classical average could have. Verified: matches the Langevin diamagnetism formula of Ashcroft and Mermin Ch. 31 and Griffiths §6.4 register; the water estimate is recomputed in worked example 2 against the CRC value $-9.05\times10^{-6}$ .

Go deeper · Derivation C · 2 minBohr and van Leeuwen — why classical physics forbids the answer we just got

Derivation CBohr and van Leeuwen — why classical physics forbids the answer we just got

The goal is the Bohr–van Leeuwen theorem: in classical statistical equilibrium, the magnetization of any system of charges vanishes identically — and with it the honest accounting of why derivations 1–2 nevertheless produced the observed susceptibility. The strategy is disarmingly short: write the classical partition function of charges in a magnetic field, notice that the field enters only through the vector potential's shift of the canonical momentum, and shift it away inside the momentum integrals. This is the theorem the Magnetic Dipole Moment of Atoms article states on credit; here we pay the proof. Watch for where the fixed-radius assumption of derivation 1 would have to appear — it never does, and that is the point.

Step 1. Set up the classical ensemble. $N$ charges with positions $\mathbf{r}_i$ , canonical momenta $\mathbf{p}_i$ , in a field described by vector potential $\mathbf{A}(\mathbf{r})$ , with any electrostatic interactions $V$ :

\mathcal{H} = \sum_{i}\frac{\left(\mathbf{p}_i - q_i\mathbf{A}(\mathbf{r}_i)\right)^2}{2m_i} + V(\mathbf{r}_1,\dots,\mathbf{r}_N), \qquad Z_{\rm cl} = \int \prod_i \mathrm{d}^3r_i\,\mathrm{d}^3p_i \; e^{-\beta \mathcal{H}}.

Why canonical momenta? Because equilibrium statistical mechanics integrates over phase space $(\mathbf{r}, \mathbf{p})$ , and in a field the kinetic momentum is $\mathbf{p} - q\mathbf{A}$ — the field hides entirely inside that shift.

Step 2. Shift the field away. At each fixed configuration $\{\mathbf{r}_i\}$ , substitute $\mathbf{p}_i' = \mathbf{p}_i - q_i\mathbf{A}(\mathbf{r}_i)$ in the momentum integrals. The Jacobian is 1 (a translation), and the limits remain $(-\infty, \infty)$ :

Z_{\rm cl} = \int \prod_i \mathrm{d}^3r_i\, e^{-\beta V} \int \prod_i \mathrm{d}^3p_i'\; e^{-\beta \sum p_i'^2/2m_i}.

Every trace of $\mathbf{A}$ — hence of $\mathbf{B}$ — is gone. The classical partition function of any charge system is exactly field-independent.

Step 3. Take the thermodynamic verdict. Magnetization is the field derivative of the free energy, $M = -\frac{1}{V}\,\partial F/\partial B$ with $F = -k_BT\ln Z_{\rm cl}$ ; a $Z_{\rm cl}$ independent of $B$ gives

M_{\rm classical} = 0 \qquad \text{— always, at every temperature, for every material.}

No diamagnetism, no paramagnetism, no ferromagnetism. Classical physics cannot do magnetism in matter, full stop.

Step 4. Reconcile with derivation 1 — where was the cheat? The Larmor argument computed the response of orbits with frozen radii: each electron kept its $r$ , its binding, its identity as a loop. A truly classical thermal ensemble grants no such rigidity. The kinetic picture of the cancellation: in a bounded classical sample, the interior electrons execute their little Larmor circulations, but electrons near the wall skip along the boundary in long arcs circulating the opposite way, and the boundary current exactly cancels the bulk one when the phase-space average is done honestly. Quantum mechanics rescues the observed effect by making the orbit radius non-negotiable — $\langle r^2\rangle$ is fixed by the atomic ground state, discrete levels forbid the continuous re-thermalization that classical cancellation requires, and the $A^2$ expectation value on those frozen states reproduces equation (3). So the Langevin formula is right, but its classical costume is borrowed: diamagnetism is a quantum effect that happens to be computable by pretending orbits are rigid. Verified: the momentum-shift argument is the standard Bohr (1911 dissertation) and van Leeuwen (1919/1921) proof as presented in Ashcroft and Mermin Ch. 31 and Van Vleck's 1932 monograph; the boundary-orbit cancellation picture follows Peierls's Surprises in Theoretical Physics discussion. The theorem's scope is equilibrium classical statistical mechanics — it does not forbid transient or driven classical currents.

Limiting cases

Case	Result	Meaning
closed-shell atom, ordinary matter	$\chi \sim -10^{-6}$ to $-10^{-5}$ , flat in $T$	the pure Langevin response, unmasked
bismuth, pyrolytic graphite	$\chi \sim -1.7\times10^{-4}$ , $-4.5\times10^{-4}$	anomalously large: delocalized/band electrons with huge effective $\langle r^2\rangle$
unpaired electrons present	paramagnetic $\chi \sim +10^{-5}$ to $+10^{-3}$ wins	permanent moments align; the diamagnetic undertow becomes a small correction (next article)
superconductor	$\chi = -1$ exactly, $\mu_r = 0$	perfect diamagnetism (Meissner effect) — a cooperative macroscopic quantum state, not atomic Langevin response (the linear-media article and superconductivity own it)
strict classical equilibrium	$M = 0$ identically	Bohr–van Leeuwen: the entire chapter's subject is quantum-mechanical at root
free conduction electrons (quantum)	Landau diamagnetism $= -\tfrac13$ of Pauli paramagnetism	named for completeness; quantum statistics of metals, solid-state track

Common trapA diamagnet does not have a moment that turns — it manufactures one, backwards, on demand

The compass-needle intuition fails here twice, and both failures cost marks. First: there is no pre-existing moment. The diamagnetic moment is created by the field, antiparallel to it, and evaporates when the field is removed — remanence is impossible, and "the moments align against the field" is a contradiction in terms (alignment against the field costs energy; induction is what buys the backwards moment). Second: never let the temperature-independence surprise you into doubt. Alignment physics fights $k_BT$ and inherits a $1/T$ ; induction physics has no thermal contest, so $\chi_{\rm dia}(T)$ is flat — a measurable fingerprint that distinguishes closed-shell materials from paramagnets in one cooling run. And the converse trap: finding a tiny negative $\chi$ does not mean "no magnetism here"; it means every louder mechanism is absent — which is real chemical information (closed shells, paired spins), not an absence of physics.

Worked examples

1 · One electron, one tesla (confidence builder)

An electron orbits at radius $r = a_0 = 0.529\times10^{-10}$ m in a plane perpendicular to a field ramped to $B = 1.0$ T. Find the speed change, the induced moment, and compare with $\mu_B = 9.274\times10^{-24}$ J/T.

Step 1. Speed change from derivation 1: $\Delta v = \dfrac{erB}{2m_e} = \dfrac{(1.602\times10^{-19})(0.529\times10^{-10})(1.0)}{2\,(9.109\times10^{-31})} = 4.65\ \mathrm{m/s}$ — against an orbital speed of order $\alpha c \approx 2.2\times10^{6}$ m/s: a perturbation of two parts per million, justifying every first-order step.

Step 2. Induced moment: $\Delta m = \dfrac{e^2 r^2 B}{4 m_e} = \dfrac{(1.602\times10^{-19})^2 (0.529\times10^{-10})^2 (1.0)}{4\,(9.109\times10^{-31})} = 1.97\times10^{-29}\ \mathrm{J/T}$ , directed against $\mathbf{B}$ .

Step 3. Compare with the atomic unit: $\Delta m/\mu_B = 1.97\times10^{-29}/9.274\times10^{-24} = 2.1\times10^{-6}$ . Two millionths of a Bohr magneton per tesla — whenever a full $\mu_B$ is available for alignment (one unpaired spin), it outshouts this response by six orders at the single-atom level; thermal disorder knocks the aligned average down to $\sim10^{-3}\mu_B$ at room temperature (next article), still a hundred-fold louder. The masking hierarchy is now quantitative.

2 · Water — from the atomic formula to a floating frog (exam level)

Water has $n = 3.34\times10^{28}$ molecules/m³ and $Z = 10$ electrons per molecule with r.m.s. electron radius $\approx 0.65$ Å. (a) Estimate $\chi$ from equation (3) and compare with the measured $-9.05\times10^{-6}$ . (b) Find the field–gradient product needed to levitate water, and the gradient required at $B = 16$ T.

Step 1. Langevin estimate: with $\langle r^2\rangle = (0.65\times10^{-10}\ \mathrm{m})^2 = 4.23\times10^{-21}\ \mathrm{m^2}$ ,

\chi = -\frac{\mu_0 nZe^2\langle r^2\rangle}{6m_e} = -\frac{(1.257\times10^{-6})(3.34\times10^{28})(10)(1.602\times10^{-19})^2(4.23\times10^{-21})}{6\,(9.109\times10^{-31})} = -8.3\times10^{-6}.

Measured: $-9.05\times10^{-6}$ . An atomic-scale model with one geometric input lands within 8% — and the residual honestly reflects that "r.m.s. electron radius of a water molecule" is itself only defined to that sort of precision.

Step 2. Levitation condition. The force density on weakly magnetized matter, from $\mathbf{F} = \nabla(\mathbf{m}\cdot\mathbf{B})$ applied per volume with $\mathbf{M} = \chi\mathbf{B}/\mu_0$ , is $f_z = (\chi/\mu_0)B\,\mathrm{d}B/\mathrm{d}z$ . With $\chi<0$ this pushes toward weaker field; hanging the sample below the field maximum of a vertical-bore solenoid (field decreasing upward... rather, sample above the coil centre where $B\,\mathrm{d}B/\mathrm{d}z < 0$ ) turns the push upward. Balance against gravity $\rho g$ :

\left|B\,\frac{\mathrm{d}B}{\mathrm{d}z}\right| = \frac{\mu_0\,\rho\,g}{|\chi|} = \frac{(1.257\times10^{-6})(1000)(9.81)}{9.05\times10^{-6}} = 1.36\times10^{3}\ \mathrm{T^2/m}.

Step 3. At $B = 16$ T (the Nijmegen Bitter magnet): $\mathrm{d}B/\mathrm{d}z = 1362/16 = 85\ \mathrm{T/m}$ — steep, but exactly what the fringe region of a 16 T bore provides. A frog is mostly water, so a frog floats, every molecule levitated individually; the 1997 Berry–Geim experiment did precisely this. Stability comes free of Earnshaw's theorem because $|\mathbf{B}|$ can have a free-space minimum and diamagnets seek minima. For pyrolytic graphite ( $\chi_\perp = -4.5\times10^{-4}$ , $\rho = 2200\ \mathrm{kg/m^3}$ ) the requirement drops to $B\,\mathrm{d}B/\mathrm{d}z \approx 60\ \mathrm{T^2/m}$ — within reach of NdFeB permanent magnets on a desk, which is why the graphite-wafer levitation demo needs no cryogenics and no power.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.1.3 and §6.4 · Jackson, Classical Electrodynamics, 3rd ed., §5.7 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11 · Zangwill, Modern Electrodynamics, §13.4 · Ashcroft & Mermin, Solid State Physics, Ch. 31 (Larmor diamagnetism, Bohr–van Leeuwen theorem)

Provenance. Equations cross-checked against Griffiths Introduction to Electrodynamics 4th ed. §6.1.3 (induced moment, Lenz's law) and §6.4 (susceptibility, permeability); Jackson 3rd ed. §5.7 and Ashcroft & Mermin Ch. 31 for the Langevin formula and Bohr–van Leeuwen theorem. A verifier should: (1) re-derive $\Delta m=-e^2 r^2 B/4m_e$ from Faraday's law; (2) confirm $\omega_L=eB/2m_e$ ; (3) recompute the order-of-magnitude $\chi_m\sim-10^{-5}$ from the stated atomic numbers; (4) check tabulated susceptibilities (water $-9.0\times10^{-6}$ , copper $-9.6\times10^{-6}$ , bismuth $-1.7\times10^{-4}$ ) against the CRC Handbook / Griffiths Table 6.1; (5) verify $\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m/A}$ and $\mu_r=1+\chi_m$ . Worked numbers recomputed for this edition: $\omega_L(1\,\mathrm{T}) = 8.79\times10^{10}$ rad/s; $\Delta v = 4.65$ m/s and $\Delta m = 1.97\times10^{-29}$ J/T $= 2.1\times10^{-6}\mu_B$ at $r=a_0$ , $B=1$ T; water estimate $\chi = -8.3\times10^{-6}$ vs measured $-9.05\times10^{-6}$ ; levitation $B\,\mathrm{d}B/\mathrm{d}z = 1.36\times10^{3}\ \mathrm{T^2/m}$ (water), $\approx 60\ \mathrm{T^2/m}$ (pyrolytic graphite); note the 2019 SI redefinition makes $\mu_0 = 1.25663706\times10^{-6}$ N/A² a measured quantity.

Builds on: Magnetic Dipole Moment of Atoms · Magnetization · Magnetic Susceptibility · Bound Currents

Leads to: Paramagnetism · Magnetic Permeability · Linear and Nonlinear Magnetic Media · The Auxiliary Field H

If every material is diamagnetic, why do we call only some materials diamagnets?

Because the label goes to the net response, and diamagnetism is the quietest voice in the room. Every atom's orbits contribute the induced negative response, but the moment an atom carries unpaired electrons, its permanent moment participates in paramagnetic alignment roughly a hundred to a thousand times stronger at room temperature — and if those moments couple cooperatively, ferromagnetism wins by many orders more. "Diamagnet" therefore means "material in which nothing louder exists": closed shells, paired spins, no net angular momentum. The underlying diamagnetic response of an iron atom is still there — it is simply a $10^{-5}$ correction on a response of order unity.

The derivation held the orbit radius fixed while the field switched on. Is that actually legitimate?

It is the single most important sleight of hand in the article — legitimate, but not classically. Quantum mechanics fixes the orbital size: $\langle r^2\rangle$ is a property of the atomic ground state, protected by the energy gap to excited states, and a slowly ramped field cannot change it (adiabatic theorem). Classically, nothing protects it — a thermal ensemble would re-distribute orbit sizes and velocities continuously, and the Bohr–van Leeuwen theorem shows the re-thermalized average cancels exactly, boundary-skipping orbits nullifying the interior Larmor currents. So "fixed $r$ " is precisely where quantum mechanics enters the calculation disguised as an innocent idealization.

Why is the Larmor frequency eB over 2m and not the cyclotron eB over m?

The factor 2 is the fingerprint of bound versus free. A free electron's entire centripetal force is magnetic, giving the cyclotron rate $eB/m_e$ . A bound electron keeps its electrostatic binding as the dominant central force; the magnetic force is a perturbation, and Larmor's theorem says its first-order effect is fully absorbed by viewing the original motion from a frame rotating at $eB/2m_e$ — half the cyclotron rate, because the Coriolis force in the rotating frame enters with a factor 2 that must match the magnetic force's single power of $v$ . Use the cyclotron frequency for beams and plasmas; use Larmor's for atoms.

Bismuth and graphite have susceptibilities twenty to fifty times larger than other diamagnets. Did the Langevin formula fail?

The formula is honest about its own limits: $\chi \propto \langle r^2\rangle$ , computed for electrons localized in atoms with $r \sim 1$ Å. In bismuth and in graphite's basal planes, the relevant electrons are not localized — they live in extended band states (in graphite, delocalized $\pi$ orbitals; in bismuth, a tiny-mass semimetal band structure), sweeping effective areas vastly larger than an atom. Large effective $\langle r^2\rangle$ , large $|\chi|$ . The proper treatment is Landau's quantum diamagnetism of band electrons, which for these materials produces exactly the observed anomalies — a solid-state story flagged here and deferred.

Is a superconductor just a very good Langevin diamagnet?

No — same sign, different universe. Atomic diamagnetism is a perturbative response of individual bound electrons, capped at $|\chi| \sim 10^{-4}$ by atomic geometry. A superconductor's $\chi = -1$ (below its critical field) is the Meissner effect: a macroscopic quantum condensate spontaneously generating surface currents that expel all flux from the bulk, not a sum of atomic moments. The distinction is measurable: atomic diamagnetism is linear and unconditional; the Meissner state is a thermodynamic phase with a transition temperature, a critical field, and flux quantization. The linear-media article uses $\chi = -1$ as the formal endpoint of the susceptibility scale; superconductivity owns the mechanism.

Does the induced diamagnetic current dissipate, like an eddy current in copper?

No — and the difference is worth savoring. An eddy current flows through a resistive medium: collisions convert its energy to heat and it decays the moment the driving stops. The atomic Larmor circulation is a shift in the electron's stationary quantum state — there is no resistance inside an atom, nothing to collide with that can degrade the motion, so the induced moment persists undiminished as long as the field does, for years or geologic ages, and unwinds reversibly when the field ramps down. Persistent, dissipationless induced currents are the atomic norm; the superconductor merely scales the same idea up to a wire loop you can hold.

Why do diamagnets seek field minima while iron filings rush to the poles of a magnet?

Both follow the same force law, $\mathbf{F} = \nabla(\mathbf{m}\cdot\mathbf{B})$ , with opposite signs of alignment. Iron's moment aligns along $\mathbf{B}$ , making $\mathbf{m}\cdot\mathbf{B}$ positive and largest where the field is strongest — the energy landscape tilts toward the poles. A diamagnet's induced moment is antiparallel, so $\mathbf{m}\cdot\mathbf{B}$ is negative and grows more negative in strong field — the landscape tilts away, toward minima. Since Maxwell's equations permit free-space minima of $|\mathbf{B}|$ but never maxima, only the diamagnetic sign can be trapped in mid-air without feedback: paramagnets and ferromagnets always crash into the magnet, while frogs float.