Diamagnetism & Paramagnetism

The two faces of weak magnetism

Every substance responds to a magnetic field — but most respond so faintly that you need a sensitive balance to notice. There are two basic weak responses, and they pull in opposite directions.

Diamagnetism is universal. When you switch on a field, the flux through each electron's orbital changes, and by Lenz's law the electron cloud reacts to oppose that change. The result is a small induced moment pointing against the field. Because the moment opposes B, a diamagnet is pushed toward weaker field — it flees the magnet. Every atom has this response, but it is overwhelmed whenever the atom also carries permanent moments.

Paramagnetism appears when atoms or molecules have unpaired spins (or unpaired orbital angular momentum). Each unpaired electron is a tiny permanent bar magnet of about one Bohr magneton, μ_B = 9.274×10⁻²⁴ J/T. With no field these moments point every which way and cancel; switch on a field and they partially align with it. The net magnetization follows the field, so a paramagnet is weakly pulled toward stronger field — it follows the magnet.

Magnetic susceptibility: the one number that decides

The whole story is captured by the volume magnetic susceptibility χ, the dimensionless constant relating the induced magnetization M (moment per unit volume) to the applied field H:

M = χ H        B = μ₀(H + M) = μ₀(1 + χ) H = μ₀ μ_r H

So the relative permeability is μ_r = 1 + χ, sitting a hair below 1 for diamagnets and a hair above 1 for paramagnets. The sign of χ is the entire diagnosis:

Property	Diamagnetism	Paramagnetism
Sign of χ	Negative (χ < 0)	Positive (χ > 0)
Typical |χ|	~10⁻⁵ (10⁻⁶ to 10⁻⁴)	~10⁻³ to 10⁻⁵
Origin	Induced orbital moment (Lenz's law)	Alignment of permanent unpaired spins
Force in a field gradient	Repelled (toward weaker B)	Attracted (toward stronger B)
Temperature dependence	Essentially none	χ = C/T (Curie's law)
Present in…	Every material	Atoms/ions with unpaired electrons
Examples	Bismuth, water, copper, graphite, N₂	Aluminium, O₂, Gd³⁺, most transition-metal salts

A point that trips people up: diamagnetism is present in everything, including paramagnets. The reason a material reads as paramagnetic is simply that its permanent-moment contribution is larger than its always-present diamagnetic background, so the sum is positive.

Diamagnetism, quantitatively

The classical Langevin model treats each electron orbit as a current loop. An applied field B induces an extra precession (the Larmor frequency ω_L = eB/2m_e), which is equivalent to a small extra circulating current and hence an opposing moment. Summing over Z electrons with mean-square orbital radius ⟨r²⟩ gives the molar susceptibility:

χ_dia = − (μ₀ N e²)/(6 m_e) · Σ⟨rᵢ²⟩

Three things fall straight out of this formula. The susceptibility is negative (the minus sign from Lenz's law). It scales with the electron-cloud size ⟨r²⟩, so big, fluffy atoms like bismuth or the delocalized π-electrons of graphite give the biggest diamagnetism. And there is no temperature in the equation — diamagnetism barely changes from liquid helium to red heat. Quantum mechanics reproduces the same form; this orbital diamagnetism is sometimes called Larmor or Landau diamagnetism depending on whether the electrons are bound or itinerant.

Paramagnetism and Curie's law

For N independent moments of magnitude μ in a field B at temperature T, statistical mechanics gives the magnetization as the Langevin function (classical) or a Brillouin function (quantum):

M = N μ [ coth(x) − 1/x ],   x = μB / k_B T      (Langevin)

In any everyday field the alignment energy μB is far smaller than the thermal energy k_BT, so x ≪ 1 and the Langevin function linearizes to x/3. That yields Curie's law:

χ = (N μ₀ μ²) / (3 k_B T) = C / T,   C = Curie constant
μ_eff = g √(J(J+1)) μ_B  ≈  √(n(n+2)) μ_B  (spin-only, n unpaired electrons)

The 1/T dependence is the signature of paramagnetism: cool a paramagnet and its susceptibility rises, because thermal tumbling that randomizes the spins weakens. The spin-only effective moment μ_eff = √(n(n+2)) μ_B lets you count unpaired electrons from a measured susceptibility — a workhorse of inorganic chemistry. Below are real numbers at room temperature:

Material	Type	Volume χ (SI, ≈ 293 K)
Bismuth	Diamagnetic	−1.66×10⁻⁴
Pyrolytic graphite (⊥ planes)	Diamagnetic	−4.5×10⁻⁴
Water	Diamagnetic	−9.0×10⁻⁶
Copper	Diamagnetic	−9.6×10⁻⁶
Gold	Diamagnetic	−3.4×10⁻⁵
Aluminium	Paramagnetic	+2.2×10⁻⁵
Tungsten	Paramagnetic	+6.8×10⁻⁵
Liquid oxygen (90 K)	Paramagnetic	+3.5×10⁻³
Iron (for contrast)	Ferromagnetic	~+2×10⁵

Concrete examples you can see

• Liquid oxygen on a magnet. Pour liquid O₂ between the poles of a strong magnet and it bridges the gap and clings — O₂ has two unpaired electrons in its π* orbitals. Liquid nitrogen, all-paired, just runs off.
• Pyrolytic graphite levitation. A thin flake of pyrolytic graphite floats stably above a checkerboard of small neodymium magnets at room temperature, because its in-plane diamagnetism is unusually strong and stable levitation of a diamagnet over a static field is allowed (Earnshaw's theorem doesn't forbid it).
• The flying frog. Andre Geim and Michael Berry floated a live frog inside a 16 T solenoid in 1997. Water's diamagnetism (χ ≈ −9×10⁻⁶) is enough when the field-gradient product B·dB/dz reaches ≈ 1400 T²/m. Geim later shared the 2010 Nobel Prize for graphene; the frog earned him the 2000 Ig Nobel.
• Gouy and Faraday balances. Chemists weigh how hard a sample is pulled into or pushed out of a field to read χ, then back out the number of unpaired electrons via μ_eff — routine for characterizing transition-metal complexes.
• MRI contrast. Gadolinium contrast agents work because Gd³⁺ has seven unpaired f-electrons (μ_eff ≈ 7.9 μ_B), making them strongly paramagnetic and shortening the relaxation time of nearby water protons.

The force in a field gradient

A uniform field exerts no net force on a small sample — only a torque on permanent moments. The translational force comes from the gradient. For a small volume V of susceptibility χ:

F = (χ V) / (2 μ₀) · ∇(B²) = (χ V / μ₀) · B (dB/dz)   (1D)

The sign of χ sets the direction: paramagnets (χ > 0) are pulled up the gradient toward stronger field, diamagnets (χ < 0) are pushed down it toward weaker field. To levitate water against gravity you need χ B (dB/dz)/μ₀ ≈ ρg, which works out to that ~1400 T²/m the frog experiment had to deliver.

JavaScript — susceptibility and forces

// Physical constants (SI)
const MU0 = 4 * Math.PI * 1e-7;   // T·m/A
const MU_B = 9.274e-24;           // J/T (Bohr magneton)
const K_B = 1.381e-23;            // J/K

// Curie-law susceptibility for N moments per m^3 with mu_eff (in mu_B)
function curieChi(N, muEffBohr, T) {
  const mu = muEffBohr * MU_B;
  return (N * MU0 * mu * mu) / (3 * K_B * T);
}

// Spin-only effective moment from number of unpaired electrons
function spinOnlyMu(n) {
  return Math.sqrt(n * (n + 2)); // in units of mu_B
}

console.log(`Fe3+ (5 unpaired): mu_eff = ${spinOnlyMu(5).toFixed(2)} mu_B`); // 5.92
console.log(`Gd3+ (7 unpaired): mu_eff = ${spinOnlyMu(7).toFixed(2)} mu_B`); // 7.94

// Curie 1/T behaviour: doubling T halves chi
const N = 6e27; // moments per m^3 (order of magnitude for a dense salt)
console.log(`chi at 100 K: ${curieChi(N, spinOnlyMu(5), 100).toExponential(2)}`);
console.log(`chi at 300 K: ${curieChi(N, spinOnlyMu(5), 300).toExponential(2)}`); // ~1/3 of above

// Force per unit volume in a field gradient: F/V = chi/mu0 * B * dBdz
function forceDensity(chi, B, dBdz) {
  return (chi / MU0) * B * dBdz; // N/m^3 (positive = toward stronger field)
}

// Diamagnetic water (chi = -9e-6): what B*dBdz levitates it? Need |F/V| = rho*g
const rhoWater = 1000, g = 9.81;
const needed = rhoWater * g * MU0 / Math.abs(-9e-6);
console.log(`Water levitation needs B*dBdz = ${needed.toFixed(0)} T^2/m`); // ~1370

// Sign check: paramagnet pulled in, diamagnet pushed out
console.log(`Al  (chi>0): ${forceDensity(2.2e-5, 5, 50) > 0 ? 'pulled IN' : 'pushed OUT'}`);
console.log(`H2O (chi<0): ${forceDensity(-9e-6, 5, 50) > 0 ? 'pulled IN' : 'pushed OUT'}`);

Where diamagnetism and paramagnetism matter

• Materials characterization. Magnetic susceptibility tells you the number of unpaired electrons, oxidation state, and spin state of a metal complex.
• Maglev and frictionless bearings. Diamagnetic levitation gives passive, power-free stable suspension; superconductors are the extreme case (perfect diamagnets, χ = −1).
• NMR and MRI. Chemical shifts arise from the diamagnetic shielding of nuclei by surrounding electrons; paramagnetic centres shift and broaden signals (used as contrast agents and shift reagents).
• Magnetic separation. High-gradient magnetic separation pulls weakly paramagnetic particles out of slurries — used in mineral processing and water treatment.
• Cryogenics. Adiabatic demagnetization of a paramagnetic salt cools to millikelvin temperatures — direct use of Curie's law.
• Geology and biology. Whole-rock susceptibility maps mineralogy; oxygen's paramagnetism underlies BOLD functional MRI, where oxygenated vs. deoxygenated haemoglobin differ magnetically.

Common mistakes

• Calling everything with unpaired spins ferromagnetic. Paramagnets have independent moments that align only while the field is on and randomize the instant it is removed. Ferromagnets have moments coupled by exchange interaction, retaining magnetization. χ ≈ 10⁻³ vs. 10⁵.
• Forgetting diamagnetism is always there. A measured χ of a paramagnet is the sum of a positive Curie term and a small negative diamagnetic background; precise work subtracts the diamagnetic correction.
• Thinking a uniform field attracts a paramagnet. A uniform field gives zero net force — you need a gradient. The pull is toward higher |B|, which is why samples climb toward a pole, not toward field lines in general.
• Mixing up sign conventions. χ < 0 is diamagnetic (repelled); χ > 0 is paramagnetic (attracted). The induced moment in a diamagnet is antiparallel to B.
• Applying Curie's law to diamagnets. Diamagnetism has no 1/T dependence — if you see susceptibility falling as 1/T, you are looking at the paramagnetic contribution.
• Ignoring Pauli paramagnetism in metals. Conduction electrons give a weak, nearly temperature-independent Pauli paramagnetism (and Landau diamagnetism) that does not obey Curie's law — distinct from local-moment paramagnetism.