Birch-Murnaghan 3rd-Order Isothermal Equation of State (P–V GPa)

Why linear Hooke’s law fails inside Earth’s mantle—and Eulerian finite strain succeeds to 100+ GPa!

At planetary-interior and diamond-anvil-cell pressures (10–150 GPa), crystal unit-cell volume V shrinks by 15–35% relative to zero-pressure volume V_0, causing bulk modulus K to stiffen dramatically! Expanding elastic energy in Eulerian finite strain f_E = ((V_0 / V)^(2/3) - 1) / 2, Francis Birch (Physical Review, 1947) derived the gold-standard mineral physics equation of state: P(V) = 1.5 * K_0 * ((V_0 / V)^(7/3) - (V_0 / V)^(5/3)) * (1 + 0.75 * (K′_0 - 4) * ((V_0 / V)^(2/3) - 1)), where K_0 is zero-pressure isothermal bulk modulus (GPa) and K′_0 = (∂K/∂P)_0 (~4 for most oxides/silicates)!

Calibrate a synchrotron X-ray diffraction run using MgO periclase pressure standard (K_0 = 160.2 GPa, K′_0 = 4.03, V_0 = 74.70 Å^3): compressing unit-cell volume to V = 62.00 Å^3 yields P = 43.49 GPa (lower-mantle top), while compressing further to V = 55.00 Å^3 reaches P = 91.09 GPa!

3rd-Order Birch-Murnaghan Isothermal Pressure P(V) (GPa)

Evaluates P(V) in GPa from zero-pressure bulk modulus K_0 (B2, GPa), pressure derivative K′_0 (B3), ambient volume V_0 (B4, Å^3), and compressed volume V (B5, Å^3).

=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Crystal EOS Inputs", ROUND(1.5 * B2 * (((B4 / B5) ^ (7 / 3)) - ((B4 / B5) ^ (5 / 3))) * (1 + 0.75 * (B3 - 4) * (((B4 / B5) ^ (2 / 3)) - 1)), 2))

Eulerian Finite Strain f_E = 0.5 * ((V_0 / V)^(2/3) - 1)

Computes the dimensionless Eulerian finite compression strain used on Birch F–f linearized plots.

=IF(OR(B4 <= 0, B5 <= 0), "Invalid Crystal EOS Inputs", ROUND(0.5 * (((B4 / B5) ^ (2 / 3)) - 1), 4))

Further reading

• Microsoft Support — ROUND function