magnetizationRatio
plain-language theorem explainer
Magnetization ratio supplies the mean-field temperature dependence of spontaneous magnetization in ferromagnets. Researchers working on phase transitions cite it when connecting exchange energy to the Curie point. The definition uses a direct conditional that returns the square-root form below T_C and zero otherwise.
Claim. The mean-field magnetization ratio is defined by $M(T)/M(0) = 0$ if $T ≥ T_C$ or $T_C = 0$, and $M(T)/M(0) = √(1 - (T/T_C)^2)$ otherwise, approximating the Brillouin function near the Curie temperature.
background
The ferromagnetism module derives spontaneous alignment of atomic moments from the exchange interaction, which follows from Pauli exclusion reducing Coulomb repulsion among parallel d-electron spins. Eight-tick coherence enters through Hund's rule coupling that favors maximum spin in degenerate d-orbitals. The Stoner criterion U D(E_F) > 1 sets the threshold for ferromagnetism, with Curie temperature T_C scaling directly with exchange strength.
proof idea
The declaration is a direct definition that evaluates a piecewise conditional on the temperature arguments and applies the square-root expression when the physical regime is below the Curie point.
why it matters
This definition is invoked by the downstream theorems nonzero_below_curie and zero_above_curie that establish the sign change across T_C. It supplies the explicit mean-field step required by the module's RS mechanism for ferromagnetism, linking the 8-tick structure and Stoner criterion to observable magnetization curves. The module lists concrete predictions for Fe, Co, and Ni Curie temperatures that sit inside the broader phi-ladder scaling.
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