A paramagnet's pull weakens as it warms. Its susceptibility falls inversely with temperature — moments straining to align against the growing thermal disorder that scatters them. Rendered, not quoted: the engine computes χ = C/T and the Langevin magnetization curve live.
source Curie, P. — Propriétés magnétiques des corps à diverses températures, Ann. Chim. Phys. 7th ser., vol. 5, pp. 289–405 (1895) · AMBER: pre-DOI thesis, cited by author/title/year (archive.org scan).
A paramagnet is a lattice of independent magnetic moments μ (unpaired spins). A field B tilts them toward alignment; temperature T shakes them back to chaos. The balance is the Langevin average.
Curie constant C = Msat·μ / (3kB) — the small-field slope of the Langevin curve is C/T.
Small field: L(x) ≈ x/3 ⇒ M ∝ B/T (linear). Large μB/kT: L → 1 ⇒ M → Msat (saturation).
Magnetism fades with heat. Curie 1895: χ = C/T, inverse in temperature, moments aligning against thermal disorder (the Langevin picture).
Cool an interacting paramagnet far enough and the moments lock each other in — below a Curie point it becomes the-ferromagnetism, its susceptibility diverging as Curie–Weiss χ = C/(T−Tc).
Live re-check of the panel's laws. Confirms green; flips red the instant window 6 tampers with the sign of the temperature dependence.
witness: booting…A field B applied to N moments at temperature T. Constants are exact: kB=1.380649×10−23 J/K, μ=μB=eħ/2me.
Left: moments jittering against B (hotter → more random). Right: χ=C/T falloff and the Langevin M–B curve, computed live.
at T=120 K: χ = C/T = — · M(B=2T)/Msat = —
Proven result: susceptibility is inverse in T — halve T and χ exactly doubles; a plot of 1/χ vs T is a straight line through the origin.
result: pending…"Curie's law is universal — every material obeys χ=C/T." False. It holds only for non-interacting, dilute moments. Real solids show exchange coupling (Curie–Weiss offset), crystal-field quenching, and Pauli paramagnetism in metals (nearly T-independent).
The Langevin derivation is a CLASSICAL idealization — moments as free classical vectors. The true answer is the quantum Brillouin function BJ(x); Langevin is its J→∞ limit. Both give the same C/T small-field law, which is why the linear regime survives.
"Cooling a paramagnet forever grows its magnetization without bound." → No. M saturates at Msat once μB ≫ kT; only the slope C/T diverges as T→0.
"Susceptibility grows with temperature." → Inverted. χ ∝ 1/T — it shrinks as heat scatters the moments. (This is exactly what window 6 breaks.)
"1/χ vs T passes through the origin for all magnets." → Only ideal paramagnets. Ferromagnets extrapolate to +Tc, antiferromagnets to −Θ.
Planted void (disclosed): swap the law to χ = C·T — proportional, not inverse — so warming a paramagnet wrongly strengthens it. The witness in 7 catches it live.