Bulk modulus calculated values

The bulk modulus calculated with two-body increments only is 0.132 Mbar, considerably lower than the experimental value, and similar to the result of LDA (in the rhombohedral lattice), 0.187 Mbar. In the hep lattice the LDA bulk modulus does not change much, with a value of 0.190 Mbar. This strong underestimation of the bulk modulus with LDA is in contrast to what would normally be expected for an overbound structure. The two-body increments with only -correlation are still repulsive, as mentioned above. When only the -correlation of the three-body increments is included, the bulk modulus increases to a value of 0.383 Mbar. The final result of the method of increments, with the inclusion of -correlation for... 

The detailed first principles study of the three stable polymorphs has been performed recently using the LCAO technique The main drawback of that work is that no cell optimization was performed for anatase or brookite. The energy-volume curves that were used to calculate the bulk modulus, B, and its pressure derivative, B, have been produced by varying the volume with the c/a ratio and fractional atomic coor nates being fixed at experimental values which makes results unreliable. 

Table 1. Calculated properties of ratile, anatase, brookite, and columbite phases. Relative deviation from experimental values is shown in brackets. Structural experimental data are from 19,20,21,9 respectively. Bulk modulus of ratile extrapolated to 0 K is from 2.

Table 2 gives our calculated results for the equilibrium volume Vq, bulk modulus Bq, and enthalpy of formation AH. Theoretical results refer to T=0, uncorrected for zero point motion, whereas experimental values refer to room temperature. Note that the extensive quantities AH and Vq arc reported per atom in the present paper, i.e., divided by the total number of atoms. As well known the LDA underestimates the volume. Comparing the bulk modulus for T3 and D8s we see that the addition of Si to pure Ti has a large (26 %) effect on the bulk modulus, indicating that p electrons of Si have a strong effect on the bonding in this system. 

The calculated and experimental values of the equilibrium lattice constant, bulk modulus and elastic stiffness constants across the M3X series are listed in Table I. With the exception of NiaGa, the calculated values of the elastic constants agree with the experimental values to within 30 %. The calculated elastic constants of NiaGa show a large discrepancy with the experimental values. Our calculated value of 2.49 for the bulk modulus for NiaGa, which agrees well with the FLAPW result of 2.24 differs substantially from experiment. The error in C44 of NiaGe is... 

The behavior of cristobalite PON has been studied as a function of pressure. No in situ evidence for pressure-induced amorphization was noticed. Whereas cristobalite Si02 displays four crystalline phases up to 50 GPa (195), PON remains in a cristobalite phase (193, 196). By using Raman spectroscopy and synchrotron X-ray diffraction, Kingma et al. (193, 197) observe a displacive transformation below 20 GPa to a high-pressure cristobalite-related structure, which then remains stable to at least 70 GPa. The high value of the calculated bulk modulus (71 GPa) (196) is indicative of the remarkable stiffness of the phase. 

Table 5.22 Bulk modulus and thermal expansion for various compositional terms of the garnet phase. The experimental range for bulk moduli is relative to a common value of K = 4. The calculated values are from Ottonello et al. (1996). References for thermal expansion are as follows (1) Ottonello et al. (1996) (2) Skinner (1956) (3) Suzuki and Anderson (1983) (4) Isaak et al. (1992) (5) Armbruster and Geiger (1993). ...

While the results and conclusions are consistent with the asperity contact model discussed earlier, the data does not unambiguously demonstrate the connection to asperity deformation. One of the complicating assumptions in Ref. [14] was that the shear modulus used in the comparison was a composite modulus calculated from the bulk material properties of each component in a two-pad stack. If asperity deformation is a dominant factor, a more appropriate value is the shear modulus of the contacting member. 

From the calculation in (7), the softer PEB region was shown to have maximum adhesive force in nature with the calculated modulus in the range of 15 1 MPa (Table 2). The harder PS domains found to have modulus in the range of 24 1 MPa in the SEBS/clay nanocomposite. The non attractive clay regions generally did not fit the JKR model. This was the reason for obtaining much less modulus than that of the literature values for clays in the GPa range. The discussion infers that the bulk modulus of the SEBS/clay nanocomposite (26 1 MPa as shown in Table 2) was dictated by the contribution from PS domains in the matrix. 

Table III. Minus the total Si crystal valence electron energy per atom with relaxation energy and pseudopotential corrections included, along with the equilibrium lattice constant, bulk modulus, and cohesive energy calculated with four different exchange-correlation functionals (defined in the caption of Table I) are compared with experimental values. The experimental total energy is the sum of Acoh plus the four-fold ionization energy.

Bulk modulus has been calculated from first principles by a local-density approximation [19] and by a linear muffin-tin orbital method [10], suggesting a value B = 165 GPa. Measurements of bulk modulus in high-pressure induced rock salt phase yielded 170 GPa [20]. 

TABLE 13.6 Bulk modulus of some polymers, experimental versus calculated values... 

Let us then turn to the ionic solids themselves. Kim and Ciordon (1974) recalculated the total energy for the alkali halides that form rocksalt structures and thereby computed from first principles the lattice spacing, the separation energy (the energy per ion pair required to separate the solid into isolated ions -this comes from the theory more naturally than does the cohesive energy, which is relative to isolated neutral atoms), and the bulk modulus. For KCI, the agreement of the values for the three properties with experimental values is typical of the calculations. The calculated (and in parentheses, the experimental) values for KCI are 3.05 (3.15) A, 175 (166) kcal/mole, and 2.3 (1.9) x 10 dync/cmA Again we may say that the interactions are quite well understood in terms of the microscopic theory. We shall return to the interpretation of these properties in terms of simple models in Section 13-D. 

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