Birch-Murnaghan equation of state - Revision history

Carl McBride: Changed a minus for a plus in the equation.

2011-11-03T14:10:31Z

Changed a minus for a plus in the equation.

← Older revision		Revision as of 16:10, 3 November 2011
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	:<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{3B_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:		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>):

	:<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{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>

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

Carl McBride: Added some internal links + slight tidy.

2011-11-03T10:29:31Z

Added some internal links + slight tidy.

← Older revision		Revision as of 12:29, 3 November 2011
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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://~~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.		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.

	Since finite strain is represented as:		Since finite strain is represented as:
Line 5:		Line 5:
	:<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>

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

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

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

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

	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:		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:

	:<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>		:<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>

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

Josephrpeterson at 02:56, 3 November 2011

2011-11-03T02:56:13Z

← Older revision		Revision as of 04:56, 3 November 2011
Line 19:		Line 19:
	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:		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{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(\frac{V}{~~V_0~~}\right)^{-2/3}-1\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/>

Josephrpeterson: Created page with "An extension, or rather a generalization, of the Murnaghan equation of state was presented by Albert F. Birch in 1947. [http://adsabs.harvard.edu/abs/1947PhRv...71..809..."

2011-11-03T02:55:12Z

Created page with "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..809..."

New page

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:

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

An energy for the strain is defined as a Taylor expansion:

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

A pressure, then is the derivative of this equation:

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

The second order form is thus:

:<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>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{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(\frac{V}{V_0}\right)^{-2/3}-1\right]</math>

==References==
<references/>