Engineering & Science
Brillouin Function B_J(x): Quantum Paramagnetic Magnetization Saturation
Calculate fractional quantum paramagnetic magnetization M / M_sat = B_J(x) for discrete angular momentum J across high-field/cryogenic regimes vs. linear Curie-law approximation.
From linear Curie susceptibility at room temperature to full quantum spin alignment near absolute zero!
Curie’s linear law M / M_sat ≈ ((J + 1) / (3 * J)) * x works only when Zeeman splitting x = g_J * μ_B * J * B / (k_B * T) << 1—at cryogenic SQUID magnetometry temperatures (2–4.2 K) and high Tesla fields, linear Curie diverges above 100% saturation! Léon Brillouin (1927) summed the 2J + 1 quantized m_J = -J..+J Boltzmann levels into the exact quantum Brillouin function: B_J(x) = ((2J + 1) / (2J)) * coth((2J + 1) * x / (2J)) - (1 / (2J)) * coth(x / (2J)), using coth(u) = (EXP(2u) + 1) / (EXP(2u) - 1) for universal spreadsheet portability!
Check Fe^3+ or Mn^2+ high-spin ions (S = J = 2.5) at moderate Zeeman ratio x = 1.20: linear Curie overpredicts M / M_sat = 0.5600, whereas quantum Brillouin gives 0.4935—and at strong cryogenic field x = 5.00, Brillouin smoothly saturates to 0.9374 (< 1.0000)!
| Zeeman Ratio x = g μ_B J B / (k_B T) (J = 2.5) | Brillouin M / M_sat = B_J(x) | Linear Curie Approx ((J+1)/(3J))*x | Physical Regime |
|---|---|---|---|
| x = 1.20 (Moderate Cryogenic Field) | 0.4935 | 0.5600 (+13.5% Overestimate) | Onset of Spin Saturation |
| x = 5.00 (High-Field 4.2 K SQUID) | 0.9374 | 2.3333 (Unphysical > 1!) | Near-Complete m_J = +J Alignment |
| Invalid J <= 0 or x <= 0 | "Invalid Spin or Field Ratio" | Error Guard | Quantum Domain Guard |
Exact Quantum Brillouin Fractional Magnetization B_J(x) = M / M_sat
Evaluates B_J(x) via exponential hyperbolic cotangents coth(u) = (EXP(2u) + 1) / (EXP(2u) - 1) for total angular momentum quantum number J (B2) and dimensionless Zeeman ratio x (B3).
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Spin or Field Ratio", ROUND((((2 * B2 + 1) / (2 * B2)) * ((EXP(2 * ((2 * B2 + 1) * B3 / (2 * B2))) + 1) / (EXP(2 * ((2 * B2 + 1) * B3 / (2 * B2))) - 1))) - ((1 / (2 * B2)) * ((EXP(2 * (B3 / (2 * B2))) + 1) / (EXP(2 * (B3 / (2 * B2))) - 1))), 4))Low-Field Curie Linear Approximation ((J + 1) / (3 * J)) * x
First-order Taylor expansion of B_J(x) valid only for x << 1.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Spin or Field Ratio", ROUND(((B2 + 1) / (3 * B2)) * B3, 4))Try the example
Use the original values or change them to test your understanding.