Birch-Murnaghan equation of state

An extension, or rather a generalization, of the Murnaghan equation of state was presented by Albert F. Birch in 1947. 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, where the first order approximation reduces to the Murnaghan equation of state.

Since finite strain is represented as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=\frac{1}{2}\left[\left(\frac{V_0}{V}\right)^{2/3}-1\right]}

The internal energy, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} , for the strain is defined as a Taylor expansion:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U=a+bf+cf^2+df^3...}

The pressure, then is the derivative of this equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p=-\left(\frac{\partial U}{\partial f}\right)\left(\frac{\partial f}{\partial V}\right)}

The second order form is thus:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p=\frac{3B_0}{2}\left[\left(\frac{V_0}{V}\right)^{7/3}-\left(\frac{V_0}{V}\right)^{5/3}\right] }

Where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_0} is the isothermal (or calibration) bulk modulus. However, since this form is not dependent on the bulk modulus derivative, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_0'} , 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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]}