Magnetic Properties of Solids

• Magnetically dilute substance: the paramagnetic centres (ions with unpaired electrons) are far apart, separated by diamagnetic ligands or a diamagnetic host lattice, so there is no significant magnetic interaction between neighbouring centres. Each centre behaves like an isolated magnet, and the susceptibility follows the simple Curie law, χ = C/T. Example: [Mn(H₂O)₆]SO₄, or a trace of Cu²⁺ doped into diamagnetic ZnSO₄.
• Magnetically concentrated substance: the magnetic centres sit close together (as in metal oxides or the metals themselves), so neighbouring spins interact through exchange coupling. The susceptibility follows the Curie–Weiss law, χ = C/(T − θ), and below a critical temperature the solid orders ferro- or antiferromagnetically. Examples: Fe, Co, Ni metals; MnO, Fe₃O₄.

Superexchange: in ionic solids (oxides, fluorides) the magnetic metal ions are too far apart for direct overlap of their d-orbitals, yet they still couple magnetically. The coupling happens indirectly through the filled p-orbitals of the bridging diamagnetic ion (usually O²⁻) sitting between them — the bridge “carries” the exchange by virtual electron transfer.

Example — MnO (rock-salt structure): Mn²⁺–O²⁻–Mn²⁺ links are linear (180°). The O 2p orbital overlaps a d-orbital on each Mn²⁺. By the Pauli principle the electron transferred virtually from O²⁻ to one Mn²⁺ must be antiparallel to that ion's d-electrons, forcing the second Mn²⁺ to align antiparallel to the first. Result: antiferromagnetic coupling below the Néel temperature (116 K for MnO). NiO behaves the same way.

Goodenough–Kanamori rule of thumb: 180° M–O–M superexchange is usually antiferromagnetic; 90° superexchange is usually ferromagnetic.

The coupling between two neighbouring spins is described by the Heisenberg exchange Hamiltonian:

The sign of J — the spin exchange coupling constant — decides the magnetic order:

• J > 0 → ferromagnetism: parallel alignment of neighbouring spins is lower in energy, so the solid develops a spontaneous magnetisation below the Curie temperature T_C. The susceptibility is very large and follows the Curie–Weiss law with a positive Weiss constant θ. Examples: Fe, Co, Ni (Fe: T_C = 1043 K).
• J < 0 → antiferromagnetism: antiparallel alignment is lower in energy; neighbouring spins cancel, so the net magnetisation is zero below the Néel temperature T_N. The susceptibility passes through a maximum at T_N and follows the Curie–Weiss law with a negative θ. Examples: MnO (T_N = 116 K), NiO.

The magnitude |J| measures the strength of the coupling and sets the ordering temperature — larger |J| means a higher T_C or T_N.

In antiferromagnetism, the magnetic moments of neighbouring ions in the solid align antiparallel to each other — up, down, up, down — so that they cancel out and the crystal has no net magnetisation in the absence of an external field.

• This antiparallel alignment comes from a negative exchange coupling constant (J < 0), usually transmitted by superexchange through a bridging oxide ion (for example, Mn–O–Mn in MnO).
• Below the Néel temperature (T_N) the antiparallel order is stable; above T_N thermal agitation destroys it and the solid becomes paramagnetic.
• The magnetic susceptibility rises with temperature, passes through a maximum at T_N, and then falls — this cusp is the experimental fingerprint of antiferromagnetism.

Example: MnO (T_N = 116 K) — Mn²⁺ ions (d⁵) couple antiferromagnetically through O²⁻ by superexchange; NiO (T_N = 523 K) behaves the same way.

• Curie point (T_C): the temperature above which a ferromagnetic material loses its spontaneous magnetisation and becomes an ordinary paramagnet. Below T_C the spins are parallel-aligned; thermal agitation destroys this order above T_C. Example: Fe, T_C = 1043 K.
• Néel point (T_N): the temperature above which an antiferromagnetic material loses its antiparallel spin order and becomes paramagnetic. The susceptibility passes through a maximum at T_N. Example: MnO, T_N = 116 K.

From the formula, AgO looks like it should contain Ag²⁺ (d⁹, one unpaired electron → paramagnetic). In reality solid “AgO” is diamagnetic — because it is actually a mixed-valence compound, Ag^IAg^IIIO₂:

• Ag⁺ is d¹⁰ — all electrons paired, diamagnetic.
• Ag³⁺ is low-spin d⁸ (square-planar, t_2g⁶e_g⁰) — all electrons paired, diamagnetic.

With no unpaired electrons anywhere, the solid shows only weak diamagnetism, not the paramagnetism naively expected for Ag(II).

The magnetic moment of a free ion has both spin and orbital contributions. In a solid or complex, the surrounding ligands create a crystal (ligand) field that splits the d-orbitals and “pins” the electrons into fixed, real (non-degenerate) orbitals. The electrons can no longer circulate freely around the nucleus, so the orbital angular momentum is largely destroyed — it is “quenched”. The observed moment then follows the spin-only formula:

Example: [Fe(H₂O)₆]³⁺ (high-spin d⁵, n = 5) shows μ ≈ 5.92 BM, exactly the spin-only value — the orbital part is quenched by the octahedral field.

When is quenching incomplete? For ions with orbitally degenerate ground terms (e.g. high-spin Co²⁺) or the heavy 4f ions, where spin–orbit coupling is stronger than the ligand field, part of the orbital contribution survives — their moments exceed the spin-only value (Co²⁺ complexes show μ ≈ 4.4–5.2 BM instead of 3.87 BM).

Note: the volume susceptibility χ (= M/H) itself is dimensionless in SI; multiplying by the molar volume gives χ_m in m³·mol⁻¹.

Double exchange (Zener) occurs in mixed-valence systems containing the same metal in two oxidation states — the classic case is Mn³⁺–O–Mn⁴⁺ in manganites such as La_1−xSr_xMnO₃.

An e_g electron hops in a real (not virtual) fashion from Mn³⁺ (d⁴) to the neighbouring Mn⁴⁺ (d³) through the bridging O 2p orbital — and it does so without flipping its spin. Because of strong Hund coupling to the t_2g core spins, this hopping is easiest when the core spins on the two Mn ions are parallel. The system therefore adopts ferromagnetic alignment, and the mobile electrons make it metallically conducting.