Birch–Murnaghan equation of state

The Birch–Murnaghan isothermal equation of state, published in 1947 by Albert Francis Birch of Harvard,^[1] is a relationship between the volume of a body and the pressure to which it is subjected. Birch proposed this equation based on the work of Francis Dominic Murnaghan of Johns Hopkins University published in 1944,^[2] so that the equation is named in honor of both scientists.

Expressions for the equation of state

The third-order Birch–Murnaghan isothermal equation of state is given by

P(V)={\frac {3B_{0}}{2}}\left[\left({\frac {V_{0}}{V}}\right)^{7/3}-\left({\frac {V_{0}}{V}}\right)^{5/3}\right]\left\{1+{\frac {3}{4}}\left(B_{0}^{\prime }-4\right)\left[\left({\frac {V_{0}}{V}}\right)^{2/3}-1\right]\right\}.

where P is the pressure, V₀ is the reference volume, V is the deformed volume, B₀ is the bulk modulus, and B₀' is the derivative of the bulk modulus with respect to pressure. The bulk modulus and its derivative are usually obtained from fits to experimental data and are defined as

B_{0}=-V\left({\frac {\partial P}{\partial V}}\right)_{P=0}

and

B_{0}'=\left({\frac {\partial B}{\partial P}}\right)_{P=0}

The expression for the equation of state is obtained by expanding the Helmholtz free energy in powers of the finite strain parameter f, defined as

f={\frac {1}{2}}\left[\left({\frac {V_{0}}{V}}\right)^{2/3}-1\right]\,,

in the form of a series.^[3]^: 68–69 This is more evident by writing the equation in terms of f. Expanded to third order in finite strain, the equation reads,^[3]^: 72

P(f)=3B_{0}f(1+2f)^{5/2}(1+af+{\mathit {higher~order~terms}})\,,

with

a={\frac {3}{2}}(B_{0}'-4)

The internal energy, E(V), is found by integration of the pressure:

E(V)=E_{0}+{\frac {9V_{0}B_{0}}{16}}\left\{\left[\left({\frac {V_{0}}{V}}\right)^{2/3}-1\right]^{3}B_{0}^{\prime }+\left[\left({\frac {V_{0}}{V}}\right)^{2/3}-1\right]^{2}\left[6-4\left({\frac {V_{0}}{V}}\right)^{2/3}\right]\right\}.

References

^ Birch, Francis (1947). "Finite Elastic Strain of Cubic Crystals". Physical Review. 71 (11): 809–824. Bibcode:1947PhRv...71..809B. doi:10.1103/PhysRev.71.809.
^ Murnaghan, F. D. (1944). "The Compressibility of Media under Extreme Pressures". Proceedings of the National Academy of Sciences of the United States of America. 30 (9): 244–247. Bibcode:1944PNAS...30..244M. doi:10.1073/pnas.30.9.244. JSTOR 87468. PMC 1078704. PMID 16588651.
^ ^a ^b Poirier, J.-P. (2000). "Introduction to the Physics of the Earth's Interior". Cambridge. ISBN 9781139164467.