Compression/dilation of condensed matter vis-à-vis an ideal symmetric material

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Both the isothermal Helmholtz energy and the adiabatic internal energy of condensed matter exhibit minima at zero pressure. As a consequence, the leading term in a density expansion of these two thermodynamic potentials is quadratic in density displacement. When the quadratic term dominates, the material behaves symmetrically in response to isotropic compressive and tensile forces. From an atomistic viewpoint, compressing or stretching atomic bonds to the same degree in a symmetric material increases a corresponding thermodynamic potential by the same amount. The quadratic term contributes to the pressure as a simple cubic equation of state (EOS). Among 29 metals and 17 inorganic solids surveyed, only metallic gold satisfies this simple EOS to the highest measured compressions (>40%). Other solids such as Pt and NaCl also follow this simple EOS to significant compressions and all materials should follow it at low compressions/dilations. A thermodynamic protocol is proposed to extract the bulk modulus ${B}_{0}$ as well as higher order modulus pressure coefficients,${B}_{1},{B}_{2},\mathrm{and}{B}_{3},$ from least squares smoothed data without taking derivatives or by appealing to an EOS model. Reliable and unbiased experimental values of these higher order modulus coefficients have been obtained for 46 solids. These moduli are strongly correlated and satisfy the relationship ${B}_{0}^{2}{B}_{3}\ensuremath{\simeq}\ensuremath{-}2{B}_{0}{B}_{1}{B}_{2}$. This relation obtains exactly from an adiabatic EOS based on the Mie (m-n) potential and to a high degree of approximation for the well-known Birch-Murnaghan and Vinet EOS. Truncating the thermodynamic expansion at the fifth order term with only the pressure coefficient ${B}_{1}$ as a parameter, this approximate thermodynamic EOS models experimental values of ${B}_{2}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{B}_{3}$ better than either the third order Birch-Murnaghan or Vinet EOS.