Magnetism — formula sheet

Diamagnetism: the field-induced circulation of paired electrons, giving a moment opposed to the applied field. χ is negative, small (of order −10⁻⁶ to −10⁻⁴ cm³ mol⁻¹ as a molar quantity), and independent of temperature. Every substance is diamagnetic; in most transition-metal complexes the effect is simply swamped by something larger.

Paramagnetism: the partial alignment of permanent moments, carried by unpaired electrons, against thermal randomisation. χ is positive, typically 10⁻³ to 10⁻² cm³ mol⁻¹ at room temperature, and — crucially — it falls as temperature rises, because hotter molecules are harder to align.

Ferromagnetism: spontaneous parallel alignment of neighbouring moments below the Curie temperature T_C. χ is positive and very large (10² to 10⁶), it depends on the applied field, and the magnetisation shows hysteresis. Above T_C the ordering is destroyed and the material becomes an ordinary paramagnet.

Quenching of orbital angular momentum: the suppression of the orbital contribution to the magnetic moment by the ligand field. In a complex the metal is no longer spherically symmetric, so L is generally no longer a good quantum number and the orbital moment is largely or entirely lost.

Sign of λ: λ is positive for a shell that is less than half filled (d¹–d⁴) and negative for a shell that is more than half filled (d⁶–d⁹). For the half-filled d⁵ case L = 0 and λ does not arise.

Pascal’s constants: a set of empirical, additive, per-atom and per-ion diamagnetic increments, with small ‘constitutive’ corrections for multiple bonds and rings, from which the diamagnetism of a whole complex is estimated by summation. They work because diamagnetism, being proportional to Σ⟨r²⟩, is very nearly a sum over local electron distributions.

Sign of the Weiss constant. θ > 0 ⇒ neighbouring moments tend to align parallel — ferromagnetic coupling; χ is larger than Curie would predict, and diverges as T approaches θ from above. θ < 0 ⇒ neighbouring moments tend to align antiparallel — antiferromagnetic coupling; χ is smaller than Curie would predict. θ = 0 ⇒ no interaction: strict Curie behaviour.

Temperature-independent paramagnetism (TIP): a positive, temperature-independent contribution to χ arising from the magnetic field mixing a low-lying excited state into the ground state. It is the second-order Zeeman term of the Van Vleck equation, and because it is a field-induced admixture rather than an orientational effect, no Boltzmann factor and hence no 1/T appears.

Spin crossover (SCO): the behaviour of a complex for which Δ_o and P are so nearly equal that the high-spin and low-spin states differ in energy by only a few kJ mol⁻¹ — comparable to kT — so that both are thermally accessible and the complex can be switched from one to the other by changing the temperature, applying pressure or shining light on it.

LIESST (Light-Induced Excited Spin-State Trapping): irradiation of a low-spin spin-crossover complex at low temperature (typically below about 50 K) into a spin-allowed d–d or charge-transfer band, followed by a cascade of intersystem crossings that deposits the molecule in the high-spin state. Because the high-spin state is separated from the low-spin ground state by a substantial activation barrier — the two states differ by 0.2 Å per bond, so their potential wells are strongly displaced — the molecule is kinetically trapped there. Warming above T(LIESST) restores the low-spin ground state.

Sign of J (in the convention Ĥ = −2J Ŝ₁·Ŝ₂): J > 0 ⇒ ferromagnetic — parallel spins are stabilised, and the state of highest total spin lies lowest. J < 0 ⇒ antiferromagnetic — antiparallel spins are stabilised, and the state of lowest total spin lies lowest.

Direct exchange: coupling arising from direct overlap of the two metals’ magnetic orbitals. Requires a short metal–metal contact and is rare in coordination compounds.Superexchange: coupling transmitted through a diamagnetic bridging ligand — oxide, hydroxide, halide, cyanide, azide, carboxylate — by partial delocalisation of the metal magnetic orbitals onto the bridge. This is the dominant mechanism in polynuclear complexes and in oxide magnets.

Curie temperature T_C: the temperature above which ferromagnetic or ferrimagnetic ordering is destroyed by thermal motion and the solid becomes an ordinary paramagnet.Néel temperature T_N: the same thing for an antiferromagnet — the temperature of the susceptibility maximum, above which the sublattices disorder.

Single-molecule magnet (SMM): a discrete molecule that retains its magnetisation after the field is removed, showing hysteresis of purely molecular origin — not from long-range order, but from a large energy barrier to reversing the direction of its own spin. The barrier requires both a large ground-state spin S and a large, negative axial zero-field-splitting parameter D (easy-axis anisotropy).