Q4Electrical Materials

Curie law: for an ideal paramagnetic material whose permanent atomic moments do not interact with each other, the magnetic susceptibility varies inversely with absolute temperature:

where C is the Curie constant (proportional to the square of the atomic moment and the moment density). Physically, the applied field tends to align the moments while thermal agitation randomizes them; the competition yields net alignment proportional to 1/T. Materials obeying the Curie law (non-interacting moments): gaseous/liquid oxygen (O₂) and dilute paramagnetic salts such as rare-earth salts (e.g., gadolinium sulphate), as well as platinum and aluminium approximately.

Curie-Weiss law: when the atomic moments interact with one another (through quantum-mechanical exchange forces), the mutual interaction assists alignment, and the susceptibility above the magnetic ordering temperature follows the modified relation:

where θ (the Curie-Weiss temperature, approximately equal to the Curie temperature Tc for ferromagnets) measures the strength of the internal interaction. As T decreases toward θ, χ grows without bound — signaling the spontaneous magnetic ordering that sets in at the Curie point, below which the material is ferromagnetic with spontaneous magnetization even at zero field. For antiferromagnets the same form applies with negative θ. Materials obeying the Curie-Weiss law (interacting moments, in their paramagnetic state above Tc): iron (Tc ≈ 770°C) and nickel (Tc ≈ 358°C) — likewise cobalt and gadolinium — whose susceptibilities above their Curie temperatures follow C/(T−θ) accurately.

Relationship and significance: the Curie law is the special case θ = 0 (no interactions) of the Curie-Weiss law; measuring χ(T) and plotting 1/χ versus T yields a straight line whose intercept on the temperature axis gives θ directly, distinguishing experimentally between ideal paramagnetism (intercept at origin), ferromagnetic interactions (positive intercept) and antiferromagnetic interactions (negative intercept) — one of the classical experimental methods of magnetic materials characterization.

Microscopic origin — Weiss molecular field theory: the empirical Curie-Weiss law is explained by Weiss's molecular-field model, in which each atomic moment experiences, in addition to the externally applied field H, an internal 'molecular field' Hm proportional to the bulk magnetization itself, Hm = λM, where λ is the molecular-field (exchange) coefficient representing the quantum-mechanical exchange interaction between neighbouring spins. Treating the total effective field (H + λM) in the ordinary Curie-law expression and solving self-consistently reproduces exactly the Curie-Weiss form χ = C/(T−θ) with θ = Cλ, so the Curie-Weiss temperature θ is a direct measure of the strength of the exchange interaction: larger λ (stronger coupling between neighbouring moments) gives a larger θ and a higher ordering (Curie) temperature. This same self-consistent field, pushed to its extreme below Tc, is what sustains spontaneous magnetization even with zero external field — the defining feature of the ferromagnetic state — because each moment continues to feel the aligning effect of its already-aligned neighbours' molecular field. The practical value of the Curie-Weiss framework is that a single susceptibility-versus-temperature measurement, extrapolated via its 1/χ-vs-T intercept, yields both the sign and approximate strength of the underlying exchange interaction without requiring any direct microscopic probe, which is why it remains a standard first characterization step for any newly studied magnetic material.