This is one of many state equations that have been used in earth sciences and shock physics to model the behavior of matter under conditions of high pressure.
Regression can also be performed on the values of the energy as a function of the volume obtained from ab-initio and molecular dynamics calculations.
Moreover, unlike many proposed equations of state, it gives an explicit expression of the volume as a function of pressure V(P).
The study of the internal structure of the earth through the knowledge of the mechanical properties of the constituents of the inner layers of the planet involves extreme conditions; the pressure can be counted in hundreds of gigapascal and temperatures in thousands of degrees.
The study of the properties of matter under these conditions can be done experimentally through devices such as diamond anvil cell for static pressures, or by subjecting the material to shock waves.
[2] These are empirical relationships, the quality and relevance depend on the use made of it and can be judged by different criteria: the number of independent parameters that are involved, the physical meaning that can be assigned to these parameters, the quality of the experimental data, and the consistency of theoretical assumptions that underlie their ability to extrapolate the behavior of solids at high compression.
The assumption Murnaghan is to assume that the bulk modulus is a linear function of pressure :[1] Murnaghan equation is the result of the integration of the differential equation: We can also express the volume depending on the pressure: This simplified presentation is however criticized by Poirier as lacking rigor.
[4] The same relationship can be shown in a different way from the fact that the incompressibility of the product of the modulus and the thermal expansion coefficient is not dependent on the pressure for a given material.
In some circumstances, particularly in connection with ab initio calculations, the expression of the energy as a function of the volume will be preferred,[7] which can be obtained by integrating the above equation according to the relationship P = −dE/dV .
It can be written to K'0 different from 1, The next simplest reasonable model would be with a constant bulk modulus Integrating gives A more sophisticated equation of state was derived by Francis D. Murnaghan of Johns Hopkins University in 1944[1].
Despite its simplicity, the Murnaghan equation is able to reproduce the experimental data for a range of pressures that can be quite large, on the order of K0/2.
[10] Regardless of this theoretical argument, experience clearly shows that K' decreases with pressure, or in other words that the second derivative of the incompressibility modulus K" is strictly negative.
A second order theory based on the same principle (see next section) can account for this observation, but this approach is still unsatisfactory.
In fact, this is an inevitable contradiction whatever polynomial expansion is chosen because there will always be a dominant term that diverges to infinity.
[3] These important limitations have led to the abandonment of the Murnaghan equation, which W. Holzapfel calls "a useful mathematical form without any physical justification".
[13] Finally, a very general limitation of this type of equation of state is their inability to take into account the phase transitions induced by the pressure and temperature of melting, but also multiple solid-solid transitions that can cause abrupt changes in the density and bulk modulus based on the pressure.
[3] In practice, the Murnaghan equation is used to perform a regression on a data set, where one gets the values of the coefficients K0 and K'0.
The data set is mostly a series of volume measurements for different values of applied pressure, obtained mostly by X-ray diffraction.
It is also possible to work on theoretical data, calculating the energy for different values of volume by ab initio methods, and then regressing these results.
The following table lists some of the results of different materials, with the sole purpose of illustrating some numerical analyses that have been made using the Murnaghan equation, without prejudice to the quality of the models obtained.
Given the criticisms that have been made in the previous section on the physical meaning of the Murnaghan equation, these results should be considered with caution.
To improve the models or avoid criticism outlined above, several generalizations of the Murnaghan equation have been proposed.
Developments to an order greater than 2 are possible in principle,[19] but at the cost of adding an adjustable parameter for each term.