DIAMAGNETISM AND PARAMAGNETISM

Bring a magnet near a piece of copper. Nothing happens. Bring it near a drop of water. Nothing. Near a chunk of aluminium — still nothing a finger can feel. The last topic (FIG.17) set up the bookkeeping of magnetisation and the H field, and it closed on a promise: every material sits somewhere on a spectrum of the dimensionless number χ_m. Iron's χ is enormous, a hundred thousand or more; that story is the next topic. This one covers everyone else. The two quiet regimes — diamagnetism and paramagnetism — are the small, honest responses every ordinary solid, liquid, and gas gives a magnetic field. Small enough that you mostly don't notice; consistent enough that they explain why magnets do, or don't, interact with almost anything you pick up.

Start with an atom that has no net magnetic moment of its own — every electron paired up, spins cancelling, closed shells all round. Noble gases are like this. So is water, bismuth, graphite, diamond, and most organic chemistry. Classically, such an atom is a nucleus ringed by electrons on tiny closed orbits. Each electron is a current loop; two electrons on the same shell orbit in opposite senses so their magnetic moments cancel. The atom's total moment is zero.

Now switch on an external B field. Each electron orbit feels a change in the magnetic flux threading it, and by Lenz's law — the bookkeeping rule of §05 — every current loop resists that change by generating a current whose own field opposes the new flux. Classically, this means the counter-circulating electrons don't cancel anymore: one is pushed to orbit a bit faster, the other a bit slower. The imbalance is tiny but real, and its induced magnetic moment points against B.

Summed over all the electrons in a cubic metre of material, this gives a magnetisation M pointing opposite to the applied H. In the linear regime M = χ·H, so this means χ_m is negative for a diamagnet. A schematic estimate, summing the classical Larmor response for every bound electron, comes out as:

where n is the number density of electrons, ⟨r²⟩ is the mean-square orbital radius, and m_e is the electron mass. The minus sign is the Lenz signature. The numerical value is parts-per-million and universal — every material has a diamagnetic contribution at this order. Michael Faraday, in the 1840s, was the first to classify materials by their sign-of-response to a magnet, and coined the word diamagnetic for the ones that were repelled. He ran a lit candle flame through a strong field; the flame bent away from the poles. That was diamagnetism.

Now give the atom a net moment of its own — an unpaired electron spin, or a partially-filled d- or f-shell where the spins don't fully cancel. Free oxygen molecules, aluminium atoms in metal, rare-earth ions in salt crystals, the Cu²⁺ ion in copper sulphate: all of these carry a permanent magnetic moment, roughly one Bohr magneton in size, attached to each atom.

With no applied field, these moments point every direction at once — thermal motion jostles them so that the average over the whole sample is zero. Turn on B, and each moment feels a torque pulling it toward the field. A fully-aligned sample would give a huge M — every moment contributing — but thermal motion fights the alignment. The balance is what Pierre Curie measured in his 1895 doctoral thesis, and the classical derivation was done shortly after by Paul Langevin.

Let μ be the moment per atom and B the applied field. The energy of a moment aligned with B is −μB; against B, +μB. A Boltzmann-weighted average over all orientations gives the Langevin function:

At small x — weak field or warm sample — the bracket is just x/3, and the magnetisation is proportional to B with a 1/T prefactor. That is Curie's law:

Cold samples respond more strongly. Hot ones fight the alignment harder and respond less. Curie published the law in 1895 after weighing rare-earth salts in temperature-controlled magnets, one of the most-cited doctoral theses in physics. His student Pierre Weiss would extend it a decade later to cover the ferromagnetic case (next topic).

At very large x — huge field, very cold — the Langevin function saturates at 1 and every moment is fully aligned. This is called magnetic saturation, and in a paramagnetic salt you can reach it at about 1 K in a few tesla. Below saturation, M is linear in B; above, it plateaus. The entire non-trivial shape lives inside one function: L(x) = coth x − 1/x.

Plotted on a symmetric log axis, every ordinary material is a tight cluster near zero. Bismuth is the standard-bearer among diamagnets at χ ≈ −1.7 × 10⁻⁴ — strong enough that a bismuth disc suspended above a strong magnet shows visible repulsion. Water, copper, and graphite hover in the low 10⁻⁵ range, all negative. The one odd ball at the negative extreme is pyrolytic graphite, which is anisotropic and strongly diamagnetic along its c-axis: the "levitating frog" experiments of the 1990s used exactly this effect, and a frog's water is diamagnetic enough to float in a 16-tesla field.

On the positive side, aluminium sits at +2 × 10⁻⁵. Liquid oxygen climbs to several parts per thousand, which is enough to demonstrate: pour it between the poles of a strong magnet and it visibly sticks. Gadolinium salts reach the tenths. Everything above 10⁻³ is a specialty material; everything above 10³ is a ferromagnet, a different animal, covered next.

Diamagnetism is what makes superconductors "float" magnets — a superconductor is the ultimate diamagnet, with χ_m = −1 exactly (FIG.20 covers the full Meissner story). Less dramatically, it's why water-filled biological tissue shows up at all in MRI contrast: the subtle χ of each tissue type slightly shifts the local Larmor frequency, and the scanner measures those shifts.

Paramagnetism is the basis of electron paramagnetic resonance spectroscopy, the electronic cousin of NMR, which probes unpaired spins in transition-metal complexes. Clinically, paramagnetic gadolinium chelates are injected as MRI contrast agents — gadolinium's seven unpaired f-electrons make Gd³⁺ the most paramagnetic stable ion, and a shot of it sharpens vascular images by shortening the water relaxation time. Liquid-oxygen demonstrations are a physics-teacher staple: pour it between the poles of a Neodymium magnet and it hangs there, held by Curie's law. Neither diamagnetism nor paramagnetism does much heavy lifting on its own. Their importance is as the baseline — the quiet background that the next topic's cooperative iron atoms stand out against.