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An extension, or rather a generalization, of the [[Murnaghan equation of state]] was presented by Albert F. Birch in 1947. <ref>[http://dx.doi.org/10.1103/PhysRev.71.809 Francis Birch "Finite Elastic Strain of Cubic Crystals", Physical Review '''71''' pp. 809-824 (1947)]</ref>  It has become known as the '''Birch-Murnaghan equation of state'''.  The generalization followed from the identification that the [[strain]] energy could be approximated as a [[Taylor expansion | Taylor series]] based on the finite strain in the crystal.  Common orders include first, second and third, where  the first order approximation reduces to the Murnaghan equation of state.
An extension, or rather a generalization, of the [[Murnaghan equation of state]] was presented by Albert F. Birch in 1947. <ref> [http://adsabs.harvard.edu/abs/1947PhRv...71..809B Birch, Francis (1947). "Finite Elastic Strain of Cubic Crystals". Physical Review 71 (11): 809–824.] </ref>  It has become known as the '''Birch-Murnaghan equation of state'''.  The generalization followed from the identification that the strain energy could be approximated as a taylor series based on the finite strain in the crystal.  Common orders include first, second and third, with the first reducing to the Murnaghan equation of state.


Since finite strain is represented as:
Since finite strain is represented as:
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:<math> f=\frac{1}{2}\left[\left(\frac{V_0}{V}\right)^{2/3}-1\right]</math>
:<math> f=\frac{1}{2}\left[\left(\frac{V_0}{V}\right)^{2/3}-1\right]</math>


The  [[internal energy]], <math>U</math>, for the strain is defined as a Taylor expansion:
An energy for the strain is defined as a Taylor expansion:


:<math> U=a+bf+cf^2+df^3...</math>
:<math> E=a+bf+cf^2+df^3...</math>


The [[pressure]], then is the derivative of this equation:
A pressure, then is the derivative of this equation:


:<math>p=-\left(\frac{\partial U}{\partial f}\right)\left(\frac{\partial f}{\partial V}\right)</math>
:<math>P=-\left(\frac{\partial E}{\partial f}\right)\left(\frac{\partial f}{\partial V}\right)</math>


The second order form is thus:
The second order form is thus:


:<math> p=\frac{3B_0}{2}\left[\left(\frac{V_0}{V}\right)^{7/3}-\left(\frac{V_0}{V}\right)^{5/3}\right] </math>
:<math> P=\frac{3K_0}{2}\left[\left(\frac{V_0}{V}\right)^{7/3}-\left(\frac{V_0}{V}\right)^{5/3}\right] </math>


Where <math>B_0</math> is the isothermal (or calibration) [[Compressibility |bulk modulus]].  However, since this form is not dependent on the bulk modulus derivative, <math>B_0'</math>, it is rarely used and either the first order or third order form are used.  The third order shows increased accuracy over the Murnaghan equation of state and has a relatively simple analytical form (Eq. 4.42 in <ref>[http://dx.doi.org/10.2277/052166392X Jean-Paul Poirier "Introduction to the Physics of the Earth's Interior", Cambridge University Press, 2nd Edition (2000)] ISBN 9780521663922</ref>):
Where <math>K_0</math> is the isothermal (or calibration) bulk modulus.  However, since this form is not dependent on the bulk modulus derivative, <math>K_0'</math>, it is rarely used and either the first order or third order form are used.  The third order shows increased accuracy over the Murnaghan equation of state and has a relatively simple analytical form:


:<math> p=\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'-4\right)\left(\left(\frac{V_0}{V}\right)^{2/3}-1\right)\right]</math>
:<math> P=\frac{3K_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(K_0'-4\right)\left(\left(\frac{V_0}{V}\right)^{2/3}-1\right)\right]</math>


==References==
==References==
<references/>
<references/>
[[category: equations of state]]
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