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| {{Confusion|Murnaghan equation of state}}
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| The '''Birch–Murnaghan isothermal equation of state''', published in 1947 by [[Francis Birch (geophysicist)|Francis Birch]] of [[Harvard]]{{ref|Birch}}, is a relationship between the volume of a body and the pressure to which it is subjected. This equation is named after [[Albert Francis Birch]] and [[Francis Dominic Murnaghan (mathematician)|Francis Dominic Murnaghan]]. Birch proposed this equation in a publication in 1947, based on the work of the Murnaghan of [[Johns Hopkins University]] published in 1944{{ref|Murnaghan}}.
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| == Expressions for the equation of state ==
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| The third-order Birch–Murnaghan isothermal equation of state is given by: | |
| :<math>
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| P(V)=\frac{3B_0}{2}
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| \left[\left(\frac{V_0}{V}\right)^\frac{7}{3} -
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| \left(\frac{V_0}{V}\right)^\frac{5}{3}\right]
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| \left\{1+\frac{3}{4}\left(B_0^\prime-4\right)
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| \left[\left(\frac{V_0}{V}\right)^\frac{2}{3} - 1\right]\right\}.
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| </math>
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| 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
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| :<math> B_0 = -V \left(\frac{\partial P}{\partial V}\right)_{P = 0}</math>
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| and
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| :<math>B_0' = \left(\frac{\partial B}{\partial P}\right)_{P = 0}</math>
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| The expression for the equation of state is obtained by expanding the free energy ''f'' in the form of a series:
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| :<math> | |
| f = \frac{1}{2}\left[\left(\frac{V}{V_0}\right)^{-\frac{2}{3}} - 1\right] \,.
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| </math>
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| The internal energy, ''E''(''V''), is found by integration of the pressure:
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| :<math>
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| E(V) = E_0 + \frac{9V_0B_0}{16}
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| \left\{
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| \left[\left(\frac{V_0}{V}\right)^\frac{2}{3}-1\right]^3B_0^\prime +
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| \left[\left(\frac{V_0}{V}\right)^\frac{2}{3}-1\right]^2
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| \left[6-4\left(\frac{V_0}{V}\right)^\frac{2}{3}\right]\right\}.
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| </math>
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| ==See also==
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| *[[Albert Francis Birch]]
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| *[[Francis Dominic Murnaghan (mathematician)|Francis Dominic Murnaghan]]
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| *[[Murnaghan equation of state]]
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| == References ==
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| {{reflist}}
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| *{{note|Murnaghan}} {{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 }}
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| *{{note|Birch}} {{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 }}
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| ==External links==
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| * [http://courses.cit.cornell.edu/das248/equation_of_state.html 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.
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| {{Statistical mechanics topics}}
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| {{DEFAULTSORT:Birch-Murnaghan equation of state}}
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| [[Category:Continuum mechanics]]
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| [[Category:Equations of state]]
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