Engineering & Science
Langevin Function L(x) = coth(x) - 1/x for Superparamagnetic Nanoparticles
Evaluate Paul Langevin’s magnetization curve M / M_sat = coth(x) - 1/x as a function of magnetic-to-thermal energy ratio x = (mu * B) / (k_B * T) for MRI iron-oxide ferrofluids and dielectric dipole alignment.
Model hysteresis-free MRI ferrofluid & nanoparticle saturation from weak field to full alignment!
Single-domain magnetite (Fe3O4) nanoparticles (~10 nm) in biomedical MRI contrast agents and ferrofluids carry giant cluster magnetic moments (mu ~ 10^4 mu_B) that thermally flip without remanent hysteresis! Paul Langevin (1905) derived the exact statistical-mechanics alignment fraction M / M_sat across all field strengths as a function of dimensionless Zeeman-to-thermal ratio x = (mu * B) / (k_B * T): the Langevin Function L(x) = coth(x) - 1 / x!
Using the exact exponential hyperbolic cotangent identity coth(x) = (EXP(x) + EXP(-x)) / (EXP(x) - EXP(-x)) makes L(x) portable to every spreadsheet engine! In weak fields (x = 0.50), L(x) ~ x / 3 gives 16.40% alignment (0.1640); at moderate MRI pre-polarizing ratio x = 2.0, alignment reaches 53.73% (0.5373) while saturation magnetization M = M_sat * L(x) for M_sat = 446 kA/m reaches 239.64 kA/m; and at strong field x = 6.0, L(x) approaches the 1 - 1/x asymptote at 83.33% (0.8333)!
| Zeeman-to-Thermal Ratio x = mu*B / (kB*T) | Langevin Alignment Fraction L(x) | Magnetite M (kA/m, Msat = 446) | Physical Alignment Regime |
|---|---|---|---|
| x = 0.50 (Weak Field Linear) | 0.1640 (~x / 3 = 0.1667) | 73.13 kA/m | Linear Curie Susceptibility |
| x = 2.00 (Intermediate Field) | 0.5373 | 239.64 kA/m | Nonlinear Knee Region |
| x = 6.00 (High Field Saturation) | 0.8333 (~1 - 1/x) | 371.65 kA/m | Near-Saturation Ferrofluid |
Exact Langevin Alignment Function L(x) = coth(x) - 1/x
Evaluates (EXP(x) + EXP(-x)) / (EXP(x) - EXP(-x)) - 1/x for x > 0 (returning 0 at x = 0 via limit guard).
=IF(B2 < 0, "Invalid Langevin Ratio", IF(B2 = 0, 0, ROUND(((EXP(B2) + EXP(-B2)) / (EXP(B2) - EXP(-B2))) - (1 / B2), 4)))Superparamagnetic Magnetization M = M_sat * L(x) (kA/m)
Multiplies saturation magnetization M_sat (kA/m, cell B3) by exact unrounded L(x).
=IF(OR(B2 < 0, B3 <= 0), "Invalid Langevin Ratio", IF(B2 = 0, 0, ROUND(B3 * (((EXP(B2) + EXP(-B2)) / (EXP(B2) - EXP(-B2))) - (1 / B2)), 2)))Try the example
Use the original values or change them to test your understanding.