Bulk modulus pressure dependence

The total pressure also has a parabolic variation with atomic number, which, according to eq. (48), may be attributed to the increasing filling of the bonding 5f states. If the bulk modulus were independent of atomic number and volume, the atomic volumes (radii) could be obtained directly from the total-pressure curve. Although the bulk modulus does depend upon atomic number and volume, it is still correct to conclude that the parabolic trend observed in the atomic volumes of the light actinides, Th-Pu, is determined mainly by the properties of the 5f electrons. 

The bulk modulus of a mineral, k, represents the pressure-dependence of its molar volume, V, defined by... 

Figure 6. Pressure dependencies of the bulk modulus obtained by the direct numerical differentiation of the in situ volumetric measurements of the glassy B203 under pressure ( relaxed modulus) in the two different runs of compression (solid symbols) and decompression (open symbols). The significant jumps of the effective bulk modulus between the final of compression and onset of decompression for both runs correspond to the jumps between relaxed and almost unrelaxed values. The inset shows pressure dependences of the first coordination number for B from the recent X-ray diffraction data. Both data are from Ref. [129].

The determination of the pressure dependence of the lattice parameters gives the experimental equation of state, and hence directly the bulk modulus. Various techniques may be used for this purpose [8], using either laboratory X-ray sources (X-ray tube or rotating anode), or a synchrotron radiation source. 

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 ... 

Straightforward measurements of elastic properties of materials can be made via high-pressure static compression experiments, in which X-ray diffraction (XRD) is used to measure the molar volume (V), or equivalently the density (p), of a material as a function of pressure (P). The pressure dependence of volume is expressed by the incompressibility or isothermal bulk modulus (Kt), where Kp = —V(bP/bV)p. 

Second, the character of the transition may change as a result of the additional contribution to the coefficient of (. Note that this contribution is negative, so the application of pressure will generally drive transitions towards first-order behavior. Whether the renormalisation becomes stronger or weaker as pressure is increased depends on the balance between the strength of the coupling constant A, and the bulk modulus, both of which are expected to increase with increasing pressure. The evolution... 

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... 

The volume—pressure relationship for e-Fe has been determined up to 300 GPa at room temperature (Figure 3 Mao et al., 1990). Determining the temperature dependence of the bulk modulus of iron is crucial for accurate comparison between the density of iron and that of the iimer core. Several in situ X-ray diffraction studies in the diamond-anvil cell (Huangeta/., 1987 Dubrovinsky eta/., 1998, 2000) and in the multi-anvil apparatus (Funamori et al., 1996 Uchida et al., 2001) have provided limited data on the thermal expansion of e-Fe at high pressures. The data of Dubrovinsky et al. 

Williams, Q. and Knittle, E., 1997. Constraints on core chemistry from the pressure dependence of the bulk modulus. Phys. Earth Planet. Interior, 100, 49-59. 

This form is derived by assuming a linear pressure dependence for the bulk modulus ... 

Density (pj,), bulk modulus (K, ), viscosity (r ), sound velocity (Vp), sound attenuation coefficient (a) and electrical resistivity (Rj.) of sea water depend on salinity, temperature and pressure. Table 2A-2 summarizes their values at laboratory... 

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