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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.

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[Birch47-1] 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.

[Murn44-2] 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.

[Poirier-3] Poirier, J.-P. (2000). "Introduction to the Physics of the Earth's Interior". Cambridge. ISBN 9781139164467.

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[2]

[3]

@@ Line 1: / Line 1: @@
+{{Confusion|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 <math>V_0</math>, 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
+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.
-:<math>
-E = E_0 + \frac{1}{2} B_0 \frac{(V-V_0)^2}{V_0}.
-</math>
-The next simplest reasonable model would be with a constant [[bulk modulus]]
-:<math>
-B_0 = - V \left( \frac{\partial P}{\partial V} \right)_T.
-</math>
-Integrating gives
-:<math>
-P = B_0 \ln(V_0/V).
-</math>
-:<math>
-V = V_0 \exp(-P/B_0).
-</math>
-:<math>
-E = E_0 + B_0 \left( V_0 - V + V \ln(V/V_0) \right).
-</math>
-==Murnaghan equation of state==
-A more sophisticated equation of state was derived by
-[[Francis Dominic Murnaghan (mathematician)|Francis D. Murnaghan]] of [[Johns Hopkins University]] in 1944{{ref|Murnaghan}}. To begin with, we consider the pressure
-:<math>  P = - \left( \frac{\partial E}{\partial V} \right)_S	\qquad (1)
-</math>
-and the bulk modulus
-:<math>
-B = - V \left( \frac{\partial P}{\partial V} \right)_T.	\qquad (2)
-</math>
-Experimentally, the bulk modulus pressure derivative
-:<math>
-B' =  \left( \frac{\partial B}{\partial P} \right)_T	\qquad (3)
-</math>
-is found to change little with pressure.  If
-we take <math>B' = B'_0</math> to be a constant, then
-:<math>
-B = B_0 + B'_0 P	\qquad(4)
-</math>
-where <math>B_0</math> is the value of <math>B</math> when <math>P = 0.</math>
-We may equate this with (2) and rearrange as
-:<math>
-\frac{d V}{V} = -\frac{d P}{B_0 + B'_0 P}.	\qquad (5)
-</math>
-Integrating this results in
-:<math>
-P(V) = \frac{B_0}{B'_0} \left(\left(\frac{V_0}{V}\right)^{B'_0}
-    - 1\right)	\qquad (6)
-</math>
-or equivalently
-:<math>
-V(P) = V_0 \left(1+B'_0
-    \frac{P}{B_0}\right)^{-1/B'_0}.		\qquad (7)
-</math>
-Substituting (6) into <math>E = E_0 - \int P
-dV</math> then results in the equation of state
-for energy.
-:<math>
-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)
-</math>
-Many substances have a fairly constant <math>B'_0</math> of about 3.5.
-==Birch–Murnaghan equation of state==
-The third-order Birch–Murnaghan isothermal equation of state, published in 1947 by [[Francis Birch (geophysicist)|Francis Birch]] of [[Harvard]]{{ref|Birch}}, is given by:
+== Expressions for the equation of state ==
-:<math>
+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)^\frac{7}{3} -
+\left[\left(\frac{V_0}{V}\right)^{7/3} -
-\left(\frac{V_0}{V}\right)^\frac{5}{3}\right]
+\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)^\frac{2}{3} - 1\right]\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>.
-Again, E(V) is found by integration of the pressure:
+The internal energy, {{math|''E''(''V'')}}, is found by integration of the pressure:
-:<math>
+<math display="block">
 E(V) = E_0 + \frac{9V_0B_0}{16}
 \left\{
-\left[\left(\frac{V_0}{V}\right)^\frac{2}{3}-1\right]^3B_0^\prime +
+\left[\left(\frac{V_0}{V}\right)^{2/3} - 1\right]^3 B_0^\prime +
-\left[\left(\frac{V_0}{V}\right)^\frac{2}{3}-1\right]^2
+\left[\left(\frac{V_0}{V}\right)^{2/3} - 1\right]^2
-\left[6-4\left(\frac{V_0}{V}\right)^\frac{2}{3}\right]\right\}.
+\left[6-4\left(\frac{V_0}{V}\right)^{2/3}\right]\right\}.
 </math>
 ==See also==
 *[[Albert Francis Birch]]
-*[[Francis Dominic Murnaghan]]
+*[[Francis Dominic Murnaghan (mathematician)|Francis Dominic Murnaghan]]
+*[[Murnaghan equation of state]]
 == References ==
 {{reflist}}
-*{{note|Murnaghan}} F.D. Murnaghan, 'The Compressibility of Media under Extreme Pressures', in ''Proceedings of the National Academy of Sciences'', vol. 30, pp.&nbsp;244-247, 1944. [http://links.jstor.org/sici?sici=0027-8424%2819440915%2930%3A9%3C244%3ATCOMUE%3E2.0.CO%3B2-P]
-*{{note|Birch}} Francis Birch, 'Finite Elastic Strain of Cubic Crystals', in ''Physical Review'', vol. 71, pp.&nbsp;809-824 (1947). [http://prola.aps.org/abstract/PR/v71/i11/p809_1]
-==External links==
-* [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.
 {{Statistical mechanics topics}}
+{{DEFAULTSORT:Birch-Murnaghan equation of state}}
 [[Category:Continuum mechanics]]
-[[Category:Equations]]
+[[Category:Eponymous equations of physics]]
+[[Category:Equations of state]]
+{{classicalmechanics-stub}}
-[[de:Zustandsgleichung von Birch-Murnaghan]]
+{{statisticalmechanics-stub}}
-[[fr:Équation d'état de Birch-Murnaghan]]

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