View source for Birch–Murnaghan equation of state

{{Confusion|Murnaghan equation of state}}
The '''Birch–Murnaghan isothermal equation of state''', published in 1947 by [[Francis Birch (geophysicist)|Albert Francis Birch]] of [[Harvard]],<ref name=Birch47>{{cite journal |doi=10.1103/PhysRev.71.809 |title=Finite Elastic Strain of Cubic Crystals |year=1947 |last1=Birch |first1=Francis |journal=Physical Review |volume=71 |issue=11 |pages=809–824|bibcode = 1947PhRv...71..809B }}</ref> 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 (mathematician)|Francis Dominic Murnaghan]] of [[Johns Hopkins University]] published in 1944,<ref name=Murn44>{{cite journal |jstor=87468 |pages=244–247 |last1=Murnaghan |first1=F. D. |title=The Compressibility of Media under Extreme Pressures |volume=30 |issue=9 |journal=Proceedings of the National Academy of Sciences of the United States of America |year=1944 |pmid=16588651 |pmc=1078704|bibcode = 1944PNAS...30..244M |doi = 10.1073/pnas.30.9.244 |doi-access=free }}</ref> 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
<math display="block">
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\}.
</math>
where ''P'' is the pressure, ''V''<sub>0</sub> is the reference volume, ''V'' is the deformed volume, ''B''<sub>0</sub> is the bulk modulus, and ''B''<sub>0</sub>' 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
<math display="block"> B_0 = -V \left(\frac{\partial P}{\partial V}\right)_{P = 0}</math>
and
<math display="block">B_0' = \left(\frac{\partial B}{\partial P}\right)_{P = 0}</math>
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
<math display="block">f = \frac{1}{2}\left[\left(\frac{V_0}{V}\right)^{{2/3}} - 1\right] \,,</math>
in the form of a series.<ref name=Poirier>{{cite book |last=Poirier |first=J.-P. |title="Introduction to the Physics of the Earth's Interior" |date=2000 | publisher=Cambridge |isbn=9781139164467}}</ref>{{rp|68-69}}
This is more evident by writing the equation in terms of ''f''.  Expanded to third order in finite strain, the equation reads,<ref name=Poirier/>{{rp|72}}
<math display="block">
P(f) = 3 B_0 f (1 + 2 f)^{5/2} ( 1 + a f + \mathit{higher~order ~terms})\,,
</math>
with <math> a = \frac{3}{2}(B_0' - 4) </math>.

The internal energy, {{math|''E''(''V'')}}, is found by integration of the pressure:
<math display="block">
E(V) = E_0 + \frac{9V_0B_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\}.
</math>

==See also==
*[[Albert Francis Birch]]
*[[Francis Dominic Murnaghan (mathematician)|Francis Dominic Murnaghan]]
*[[Murnaghan equation of state]]

== References ==
{{reflist}}

{{Statistical mechanics topics}}

{{DEFAULTSORT:Birch-Murnaghan equation of state}}
[[Category:Continuum mechanics]]
[[Category:Eponymous equations of physics]]
[[Category:Equations of state]]


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