Birch-Murnaghan equation of state

The two equations of state after Murnaghan and after Birch (named after Francis Murnaghan and Albert Francis Birch ) describe the relationship between the volume of a solid and the external hydrostatic pressure acting on it . ${\displaystyle V}$ ${\displaystyle p}$

Murnaghan equation of state

The Murnaghan equation of state is:

${\displaystyle p={\frac {K_{0}}{K_{0}'}}\left[\left({\frac {V_{0}}{V}}\right)^{K_{0}'}-1\right]}$

${\displaystyle \Leftrightarrow V=V_{0}\cdot \left[{\frac {K_{0}'}{K_{0}}}p+1\right]^{-{\frac {1}{K_{0}'}}}}$

With

• the volume of the solid at a pressure of 0 G Pa ${\displaystyle V_{0}}$
• the compression modulus at a pressure of 0 GPa: ${\displaystyle K_{0}}$

${\displaystyle K_{0}=-V\left.{\frac {\partial p}{\partial V}}\right|_{p=0\,\mathrm {GPa} }}$

• the first derivative of the compression modulus according to the pressure at a pressure of 0 GPa: ${\displaystyle K_{0}'}$

${\displaystyle K_{0}'=\left.{\frac {\partial K}{\partial p}}\right|_{p=0\,\mathrm {GPa} }}$.

This equation of state is obtained if Murnaghans integrates the following assumptions:

• the compression modulus of a solid increases linearly with the pressure acting on it:

${\displaystyle K(p)=K_{0}+p\,K_{0}'}$

• the size does not depend on the pressure. ${\displaystyle K_{0}'}$

Equation of state according to Birch (-Murnaghan)

Another way of describing the behavior of condensed matter under pressure was taken by Francis Birch. He assumed that according to the Maxwell relations there is a connection between the pressure and the free energy : ${\displaystyle p}$ ${\displaystyle F}$

${\displaystyle p=\left({\frac {\partial F}{\partial V}}\right)_{T}}$

Birch represented the free energy of a solid as a series expansion :

${\displaystyle F=\sum _{n=1}^{\infty }a_{n}\epsilon ^{n}}$

Here are

• ${\displaystyle a_{n}}$ pressure dependent coefficients
• ${\displaystyle \epsilon ^{n}}$is Euler's elongation .

${\displaystyle \epsilon ={\frac {1}{2}}\left[1-\left({\frac {V}{V_{0}}}\right)^{-{\frac {2}{3}}}\right]}$

After a series expansion, the representation of which would lead too far in this context, one obtains the Birch equation of state:

${\displaystyle p={\frac {3}{2}}K_{0}\left[\left({\frac {V}{V_{0}}}\right)^{-{\frac {7}{3}}}-\left({\frac {V}{V_{0}}}\right)^{-{\frac {5}{3}}}\right]\left[1+{\frac {3}{4}}\left(K_{0}'-4\right)\left[\left({\frac {V}{V_{0}}}\right)^{-{\frac {2}{3}}}-1\right]\right]}$

In the meantime it has become common to refer to this equation as the Birch-Murnaghan equation of state , even if Birch's approach has nothing in common with Murnaghan's approach.

literature

• F. Birch: Finite elastic strains of cubic crystals , Phys. Rev. 71, 809 (1947)
• B. Buras and L. Gerward: Application of X-ray energy dispersive diffraction for characterization of materials under high pressure , Prog. Cryst. Growth and Characterization 18, 93 (1989)