Murnaghan equation

Moreover, upon comparing (4.32) with (4.14), it can be seen that (Jeanloz and Grover, 1988) the Birch-Murnaghan equation (4.32) with a2 = 0 describes the isentropic equation of state provided the linear shock-particle velocity relation (4.5) describes the Hugoniot. In combination, these require that... 

The two most usual equations of state for representation of experimental data at high pressure are the Murnaghan and Birch-Murnaghan equations of state. Both models are based on finite strain theory, the Birch-Murnaghan or Eulerian strain [26], The main assumption in finite strain theory is the formal relationship between compression and strain [27] ... 

Figure 2.15 Pressure-volume data for diamond, SiC>2-stishovite, MgSiC>3 and 8102-quartz based on third order Birch-Murnaghan equation of state descriptions. The isothermal bulk modulus at 1 bar and 298 K are given in the figure.

Volumetric data for four different substances represented by the third-order Birch-Murnaghan equation of state are shown in Figure 2.15. 

Macdonald (144) analyzed several equations of state which had a variety of mathematical forms including the Tammann equation and the secant bulk modulus equation chosen by Hayward. (In his statistical analysis, Macdonald used the PVT data of Kell and Whalley (26) which has been shown to be in error (29) Thus, the conclusions of Macdonald may be questionable.) He disagreed with Hayward and selected the Murnaghan equation to be superior to either the Tammann equation or the linear secant modulus equation chosen by Hayward. If, however, the Tammann equation and the Murnaghan equation were both expanded to second order in pressure, then Macdonald found that the results obtained from both equations would agree. As shown earlier, the expansion of the Tammann equation to second order is equivalent to the bulk modulus form of the original Tait equation. 

Pq describes the pressure-dependent change of the isothermal bulk modulus Pq, i.e. the stiffening of a material upon application of pressure. If only the first two terms of the expansion series are taken into consideration, integration for constant temperature leads to the well-known Murnaghan equation of... 

Solving the equation for P results in the frequently-used inverse Murnaghan equation ... 

Fig. 5.12 The pressure dependence of the relative volume V/Vq of the naphthalene crystal N-dg at room temperature. The measured values with different symbols are from different, independent experiments. The solid curve is a fit to the so called Murnaghan equation (Eq. (5.10)). The dashed line is a fit to a modified Murnaghan equation. From [12].

The dependence of the energy on volume, if interpolated by a suitable analytical formula, yields all relevant information on the equations of state of magnesite at 0 K. The Murnaghan equation[26], widely used in solid state thermodynamics, is based on the assumption of a linear dependence of the elastic bulk modulus on pressure B = Bo + B p) ... 

TABLES. Hartree-Fock energies (including relaxation contribution), and relaxed c/a ratios and oxygen fractional coordinates vs. V/Vo for magnesite and calcite. The Vq/6 (least-energy volume per formula unit), Ef) = E(Vo), Bq (zero-pressure bulk modulus) and B = (dB/dp)o- 0 quantities have been fitted to a Murnaghan equation of state. 

Table 3.5 Parameters of the third-order Birch-Murnaghan equations of states of a three unit-ceii-based atomic modeisofthe spinei-structured Ca303N that have the lowestenergy calculated within the LDA >.

Fig. 4.16 More detailed examination of the V(P) relations for Sijss clathrate using a third-order Birch-Murnaghan equation of state and linearized Eulerian strain variables (F-f plot) gave a bulk modulus Kq = 90(3) GPa and its pressure derivative Kd = 5.2(8) (from [96])...

If the calculations shown schematically in Fig. 3.1 were to extend over a large range of lattice constants, the simple linear relation for p(a) would have to replaced by some p(V) equation of state the Murnaghan equation... 

The first applications were to the semiconductors. However, for the purpose of illustration here, we first discuss the results for diamond. The calculation was carried out using the LCAO basis with three Gaussian exponents for each of the s, p, Py, and pz orbitals totaling 12 basis functions per carbon atom. The calculated total energy as a function of volume E(V) is present in Fig. 4. The points are the computed values, and the curve is a fit of the results to the Murnaghan equation of state. 

The continuous curve is the Murnaghan equation of state fit to the calculated points. (from Ref. 16)... 

The theoretical study of phase transitions is made as follows. For each crystal phase, the total energy E is computed at a number of unit-cell volumes V at each volume, the structure parameters (nuclear coordinates and lattice parameters) that minimize E are determined. An analytical representation of E vs. V is obtained by using a polynomial expression or the Murnaghan equation of state (or any other fitting function). The Murnaghan function, by far the most universally adopted, is as follows ... 

Table 6.3. Comparison of some bnlk properties of silicon obtained with the LDA and the GGA equilibrinm lattice constant (a), bulk modulus (B) and cohesive energy (Ec). Bnlk modnli were obtained by fitting the Murnaghan equation of state [84] to the calculated total energy vs. volume curve. The experimental results (expt.) are those cited in [85]...

Fig. 130. Relative volume versus pressure of PuTe. Experimental values (solid curve) from Dabos-Seignon et al. (1990). Dashed curve trivalent Murnaghan equation with a fit at 140kbar. Inset relative volume vs. pressure, same as the main figure, but with an extension towards negative pressure and inclusion of a divalent Murnaghan equation, (After Wachter et al. 1991.)...

Fig. 12.13 Pressure-volume equation of state for amorphous GeSe2. Experimental compression in a hydrostatic medium, a 4 1 methanoEethanol mixture [79] solid black circles)-, simulation data from [83] red open diamond)-, third-order isothermal Birch-Murnaghan equation of state fit to the experimental compression data solid black line)...

The Murnaghan equation [31] of state was used to relate the lattice constant a and pressure P,... 

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