Mumaghan equation of state

Figure 2.3 Total energy, E, of Cu in the fee crystal structure as a function of the lattice para meter, a. Data points are from DFT calculations and the dashed curve is the Birch Mumaghan equation of state.

Bulk moduli and pressure derivatives. Results for the bulk modulus and its pressure derivative for all three HMX polymorphs obtained from fitting simulation-predicted isotherms to the equations of state discussed above are summarized in Table 7. For all data sets, we include fits to the Us-Up form (Eq. 18) and both weighting schemes for the third-order Birch-Mumaghan equation of state (Eqs. 20 and 21). In the case of the experimental data for /THMX, values for the moduli based on Eqs. 18 and 20 were taken from the re-analysis of Menikoff and Sewell. Two sets of results are included in the case of Yoo and Cynn, since they reported on the basis of shifts in the Raman spectra a phase transition with zero volume change at 12 GPa. Simulation data of the /T HMX isotherm due to Sorescu et al. were extracted by hand from Fig. 3b of their work. 

Fig. 6. Pressure-volume relations for NO NOs and other molecular systems. NO NOs determined from the present energy-dispersive x-ray diffraction ( ) and that from previous angle-dispersive x-ray diffraction with refined cell parameters ( ), and that from C.S. Yoo et al. ( ) (Ref. [81]), compared with a third-order Birch-Mumaghan (—) and Vinet et al. EOS fits For O2 ( ) data, below 5.5 GPa are for fluid O2 (Ref. [123]) above 5.5 GPa for the solid (Ref. [124]). Experimental data for O2 (o) at several pressures performed from Ref. [125] are also plotted. For N2 ( ), experimentally determined EOS is from Ref [126], for N2O ( ) from Ref. [127]. Volumes for N2O4 ( ) determined in the present study is fitted by the Birch-Mumaghan equation of state (—) tentatively. Also shown are the corresponding volumes of stoichometrically equivalent assemblages of N2 + 2O2 (—) and N2O+ 3/2 O2 (—).

We used the Mumaghan equation of state model described in Ref. [24] for all condensed materials except for carbon. The parameters for the equation of states of these materials is given in Ref. [35]. 

The bulk modulus obtained by fitting the calculated unit eell volumes in the range of 15 to 30 GPa to a third-order Birch-Mumaghan equation of state (EOS) (but excluding the result at 8 GPa) is 180 1 GPa, with K o = 2.3 0.5 The experimental data for C2/c-enstatite[51] is insufficient to allow a similar fit, but we note that Ko calculated here is significantly larger than the values of 123 17 GPa and 111 3 GPa of other pyroxene polymorphs of MgSi03[51]. 

FIGURE 31 Volume per atom as a function of pressure for dysprosium the Birch-Mumaghan equation of state fits from Samudrala and Vohra (2012). 

Fig. 4.10 a V/Vq plot for BagSi4 as a function of pressure (points) along with a third-order Birch-Mumaghan equation of state (line). The macroscopic order parameter corresponds to spontaneous i c., the Variation of the volume cmrected from the compressibility, b Aspo ,a eous as a function of pressure fits with the theoretical analysis based on the Landau theory of phase transition null before the transition pressure, a jump at the transition and a square-root evolution after the transition. Such a behaviOT is also observed for Si atomic displacement parameters that can be used as microscopic order parameter [82]. The correlation between these physical quantities underlines the relationships between the isostructural transition and disordering of the Si sub-lattice... 

The precise manner in which material reduces its volume in response to applied pressure is described by the equation of state, which relates volume reduction with the bulk modulus of the material and its first derivative with respect to pressure Bo. The most frequently used equation is universal the Birch-Mumaghan equation of state (EOS) [3-5] which can be expressed as follows ... 

Fig. 12.17 The pressure-volume equation of state for GeSe2 glass under compression where V is the volume at pressure P and Vo is the volume under ambient conditions. The measured data from Mei et al. [79] blackfilled circles with vertical error bars) are compared to the results obtained from FPMD in the present work (filled red squares) [61] and in the work by Durandurdu and Drabold [83] blue diamond). The measured and simulated data are fitted to a second-order Birch-Mumaghan equation of state solid green and dashed blue curve respectively)...

Several empirical equations of state (EOS), representing correlations between pressure and molar volume data have been defined, one of which is the Birch-Mumaghan EOS,... 

Once the lattice parameters are measured at various pressures, the pressure dependence of the unit cell volume is fitted with an equation of state (EOS). The simplest, and most used, is the Mumaghan [9] EOS based on the assumption that the bulk modulus has a linear dependence with the pressure ... 

Another simple model that can explain how the excess volume in deformations can lead to a reduction of the enthalpy of formation in non-metallic materials is the Birch-Mumaghan (BM) equations of state that give the molar energy as a function of the equilibrium molar energy Eg, the equilibrium molar volume Vo, the actual molar volume V, and the bulk modulus at equilibrium Bo [12]. At OK, the enthalpy of formation coincides with the molar energy of formation between the two deformed materials. 

EXP6 supports a wide range of elements and condensed detonation products. We have applied a Mumaghan[24] equation of state (EOS) form to a variety of metals, metal oxides and other solids. We have also matched phase transition data for many of these solids. For example, this form has recently been applied to the EOS of carbon[31]. Thermal effects in the EOS are included through the dependence of the coefficient of thermal expansion on temperature, which can be directly compared to experiment. We find that we can replicate shock Hugoniot and isothermal compression data for a wide variety of solids with this simple form. 

Figure 67 shows the relative volume V/V asa. function of pressure up to 12.4 GPa at room temperature, where V and Vq are the volumes at high and ambient pressures, respectively. The volume shows a smooth decrease with increasing pressure without any discontinuities. This fact indicates that there is no structural phase transition, that is, the C15 structure is stable up to 12.4 GPa at room temperature. The bulk modulus was estimated by using first-order Mumaghan s equation of state (see Eq. (1) in Section 2). The solid curve in Figure 67 shows the result of least-square fitting, and we obtained the bulk modulus to be 75.3 GPa. 

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