Engineering & Science
Vinet (Rose-Smith-Ferrante) Universal Equation of State Formula (GPa)
Compute ultra-high-compression crystal pressure P(V) = 3 * K_0 * ((1 - x) / x^2) * EXP(1.5 * (K′_0 - 1) * (1 - x)) where linear lattice ratio x = (V / V_0)^(1/3).
Why Vinet’s Rydberg cohesive-energy EOS outperforms polynomial strain series above 30% compression!
At extreme multi-megabar compressions (V / V_0 < 0.65 in soft metals, noble-gas solids, or giant-planet cores), truncated Birch-Murnaghan polynomials can diverge unphysically when K′_0 differs markedly from 4! Pascal Vinet, John Ferrante, John R. Smith, and James H. Rose (Journal of Physics C, 1986; Physical Review B, 1987) derived a universal isothermal EOS directly from universal Rydberg interatomic binding-energy scaling in linear lattice compression x = (V / V_0)^(1/3): P(x) = 3 * K_0 * ((1 - x) / x^2) * EXP(1.5 * (K′_0 - 1) * (1 - x))!
Check cubic diamond (K_0 = 442.0 GPa, K′_0 = 3.65, ambient unit-cell volume V_0 = 45.38 Å^3): moderate compression to V = 42.00 Å^3 (x = 0.9745) requires P = 39.35 GPa, while deep compression to V = 38.00 Å^3 (x = 0.9426) requires P = 107.74 GPa (> 1 Megabar)!
| Diamond Unit-Cell Volume V (V0 = 45.38 Å^3) | Linear Ratio x = (V/V0)^(1/3) | Vinet Pressure P (GPa) |
|---|---|---|
| V = 42.00 Å^3 (7.4% Volume Compression) | 0.9745 | 39.35 GPa |
| V = 38.00 Å^3 (16.3% Volume Compression) | 0.9426 | 107.74 GPa (1.08 Mbar) |
| Invalid V <= 0 or K_0 <= 0 | Undefined | "Invalid Vinet EOS Inputs" |
Vinet Universal Isothermal Pressure P(V) (GPa)
Evaluates 3 * K_0 * ((1 - x) / x^2) * EXP(1.5 * (K′_0 - 1) * (1 - x)) with x = (B5 / B4)^(1/3).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Vinet EOS Inputs", ROUND(3 * B2 * ((1 - ((B5 / B4) ^ (1 / 3))) / (((B5 / B4) ^ (1 / 3)) ^ 2)) * EXP(1.5 * (B3 - 1) * (1 - ((B5 / B4) ^ (1 / 3)))), 2))Dimensionless Linear Lattice Compression Ratio x = (V / V_0)^(1/3)
Computes the cube-root Wigner-Seitz radius ratio a / a_0 = (V / V_0)^(1/3).
=IF(OR(B4 <= 0, B5 <= 0), "Invalid Vinet EOS Inputs", ROUND((B5 / B4) ^ (1 / 3), 4))Try the example
Use the original values or change them to test your understanding.