Temperature-Dependent Equations of State of Solids

Abstract

An isothermal equation of state of a solid is considered, which contains as special cases the equations of Birch, Murnaghan, Bardeen, and others. The equation is generalized to arbitrary temperature by replacing two constants of the equation by temperature-dependent parameters, whose functional form is determined by considerations of thermodynamic consistency. The thermal properties of the solid implied by this equation of state are examined. It is shown that the generalized equation is consistent with the Mie-Grüneisen relation for the thermal pressure of the lattice, and that the corresponding Grüneisen parameter is only slightly dependent on temperature, in general. The form of the generalized equation of state at low temperature is exhibited as an explicit function of volume and temperature for a solid whose heat capacity obeys the Debye law. A comparison with pressure-volume data of Swenson for potassium at low temperatures shows excellent agreement of the generalized equation of state with experiment.

High-Pressure Calibration: A Critical Review

“…Bernardes and Swenson (1963) observe that the experimental data for the alkali metals at low temperatures fit the Birch equation with small values of ~, but that slight deviations from this equation appear along higher temperature isotherms. Gilvarry (1957) notes that equation (3) requires th'at the initial value of the Grunei~en constant at zero pre5~ure must equal 11/6, which, of course, is not satisfied by all solids. This indicates that the two parameter equation (4) is necessary to insure that the P-V relation has the correct initial curvature at zero pressure.…”

“…The temperature-dependent equation of state has been discussed by Gilvarry (1957) and Bernardes and Swensen (1963). Gilvarry concludes that the above isothermal equations should be quite good along any isotherm as long as the appropriate parameters are chosen for that isotherm.…”

“…But an equation of four parameters is not good enough for the alkali metals (Bridgman, 1958). Gilvarry (1957Gilvarry ( , 1956 has given an equation of state which will generate many of the proposed isothermal equations of state for solids (ll) where nand m are constants. There is no theoretical basis for this equation but it can be made to fit a wide class of P-V relations by varying the parameters m and n and is equivalent to equations (2) and (3) for given choices of m and n.…”

Temperature and pressure derivatives of elastic constants with application to the mantle

“…The parameter {K}p played a prominent role in early discussions of the thermal properties of solids [e.g., Griineisen, 1926;Fiirth, 1944;Brillouin, 1964;Gilvarry, 1957]. In recent literature it is given the symbol Yf [Gilvarry, 1957] Anderson, 1967;Birch, 1968] and called the Griineisen-Anderson [e.g., Sumino and Anderson, 1984] or Anderson-Griineisen [e.g., Suzuki and Anderson, 1983] Anderson, 1967bAnderson, , 1987a and in temperature-dependent equations of state.…”

“…In recent literature it is given the symbol Yf [Gilvarry, 1957] Anderson, 1967;Birch, 1968] and called the Griineisen-Anderson [e.g., Sumino and Anderson, 1984] or Anderson-Griineisen [e.g., Suzuki and Anderson, 1983] Anderson, 1967bAnderson, , 1987a and in temperature-dependent equations of state. They express the variation of elastic moduli in terms of volume and with other external parameters at constant volume.…”