Engineering & Science
Lindemann Melting Criterion & Thermal RMS Vibration Ratio
Estimate crystalline melting temperature T_m = 0.00229054 * M * Θ_D^2 * c_L^2 * d^2 (K) from Debye temperature Θ_D, atomic mass M (g/mol), nearest-neighbor spacing d (Å), and Lindemann constant c_L (~0.10–0.12).
Why crystals melt as soon as atomic vibrations reach ~11% of bond spacing!
Frederick Lindemann (1910) proposed a remarkably physical criterion for solid-liquid melting: a crystal lattice loses shear rigidity and melts when the root-mean-square thermal displacement SQRT(<u^2>) of atoms around their equilibrium sites reaches a critical fraction c_L (~0.10–0.12 for FCC/BCC metals) of nearest-neighbor interatomic spacing d! Combining high-temperature Debye harmonic vibrations <u^2> = 9 * ħ^2 * T / (m * k_B * Θ_D^2) with SQRT(<u^2>) = c_L * d yields the closed-form Lindemann Melting Law!
Collecting fundamental CODATA constants (k_B, ħ, N_A, and Å^2 = 10^-20 m^2) into a single prefactor K_L = 0.00229054 gives T_m = 0.00229054 * M * Θ_D^2 * c_L^2 * d^2! For FCC copper (M = 63.546 g/mol, Θ_D = 343 K, nearest-neighbor d = 2.556 Å, Lindemann ratio c_L = 0.110), the formula predicts T_m = 1,353.7 K (within 0.3% of copper’s experimental 1,357.8 K melting point!); for aluminum (M = 26.982, Θ_D = 428 K, d = 2.863 Å, c_L = 0.105), T_m = 1,023.1 K!
| FCC Metal Crystal | Debye ΘD (K) & Spacing d (Å) | Lindemann Fraction cL | Predicted Melting Temp Tm (K) |
|---|---|---|---|
| Copper (M = 63.546 g/mol) | ΘD = 343 K, d = 2.556 Å | 0.110 (11.0% of d) | 1,353.7 K (Actual 1,357.8 K) |
| Copper at Lower cL = 0.108 | ΘD = 343 K, d = 2.556 Å | 0.108 (10.8% of d) | 1,304.9 K |
| Aluminum (M = 26.982 g/mol) | ΘD = 428 K, d = 2.863 Å | 0.105 (10.5% of d) | 1,023.1 K |
Lindemann Predicted Melting Temperature T_m (K)
Evaluates 0.00229054 * M * Θ_D^2 * c_L^2 * d^2 from molar mass B2 (g/mol), Debye temperature B3 (K), nearest-neighbor spacing B4 (Å), and Lindemann constant B5.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B5 >= 0.5), "Invalid Crystal Inputs", ROUND(0.00229054 * B2 * (B3 ^ 2) * (B5 ^ 2) * (B4 ^ 2), 1))Critical RMS Thermal Vibration Amplitude at Melting u_rms = c_L * d (Å)
Returns the critical RMS atomic excursion (in Ångströms) that triggers lattice instability.
=IF(OR(B4 <= 0, B5 <= 0, B5 >= 0.5), "Invalid Crystal Inputs", ROUND(B4 * B5, 4))Try the example
Use the original values or change them to test your understanding.