Birch–Murnaghan equation of state

In continuum mechanics, an equation of state suitable for modeling solids is naturally rather different from the ideal gas law. A solid has a certain equilibrium volume $V_{0}$, and the energy increases quadratically as volume is increased or decreased a small amount from that value. The simplest plausible dependence of energy on volume would be a harmonic solid, with

$E=E_{0}+{\frac {1}{2}}B_{0}{\frac {(V-V_{0})^{2}}{V_{0}}}.$

The next simplest reasonable model would be with a constant bulk modulus

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

Integrating gives

$P=B_{0}\ln(V_{0}/V).\,$

$V=V_{0}\exp(-P/B_{0}).\,$

$E=E_{0}+B_{0}\left(V_{0}-V+V\ln(V/V_{0})\right).\,$

Murnaghan equation of state

A more sophisticated equation of state was derived by Francis D. Murnaghan of Johns Hopkins University in 1944^[1]. To begin with, we consider the pressure

$P=-\left({\frac {\partial E}{\partial V}}\right)_{S}\qquad (1)$

and the bulk modulus

$B=-V\left({\frac {\partial P}{\partial V}}\right)_{T}.\qquad (2)$

Experimentally, the bulk modulus pressure derivative

$B'=\left({\frac {\partial B}{\partial P}}\right)_{T}\qquad (3)$

is found to change little with pressure. If we take $B'=B'_{0}$ to be a constant, then

$B=B_{0}+B'_{0}P\qquad (4)$

where $B_{0}$ is the value of $B$ when $P=0.$ We may equate this with (2) and rearrange as

${\frac {dV}{V}}=-{\frac {dP}{B_{0}+B'_{0}P}}.\qquad (5)$

Integrating this results in

$P(V)={\frac {B_{0}}{B'_{0}}}\left(\left({\frac {V_{0}}{V}}\right)^{B'_{0}}-1\right)\qquad (6)$

or equivalently

$V(P)=V_{0}\left(1+B'_{0}{\frac {P}{B_{0}}}\right)^{-1/B'_{0}}.\qquad (7)$

Substituting (6) into $E=E_{0}-\int P\,dV$ then results in the equation of state for energy.

$E(V)=E_{0}+{\frac {B_{0}V}{B_{0}'}}\left({\frac {(V_{0}/V)^{B_{0}'}}{B_{0}'-1}}+1\right)-{\frac {B_{0}V_{0}}{B_{0}'-1}}.\qquad (8)$

Many substances have a fairly constant $B'_{0}$ of about 3.5.

The third-order Birch–Murnaghan isothermal equation of state, published in 1947 by Francis Birch of Harvard^[2], is given by:

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

Again, E(V) is found by integration of the pressure:

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

See also

• Albert Francis Birch
• Francis Dominic Murnaghan

References

• ^ 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.
• ^ 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.

External links

• Equation of State Codes and Scripts This webpage provides a list of available codes and scripts used to fit energy and volume data from electronic structure calculations to equations of state such as the Birch–Murnaghan. These can be used to determine material properties such as equilibrium volume, minimum energy, and bulk modulus.