Extrapolation formula for finding the volume of solids at high pressures

Abstract

A formula is proposed for extrapolating from data taken at low or moderate pressures to the high pressures that exist in the interior of the earth and planets. The formula, which predicts the curve of reduced volume v/v0 versus reduced pressure P ≡ p/K0, follows from integration of the following assumed expression for the pressure derivative of the bulk modulus K ≡ −v dp/dv When m = K0′, the formula reduces to the well-known Murnaghan relation, which is itself remarkably successful. In general, there is an improvement on the Murnaghan relation because the above expression allows the derivative to change from its initial value K0′ to a more realistic value m as P → ∞. The Keane equation, d(K/K0)/dP = m + (K0′ − m)/(K/K0), has this same property, but with the disadvantage of behaving unreasonably if K0′<0. To apply our formula, K0′ is determined from low-pressure ultrasonic data (0 to 6 kb), m is fixed at some reasonable value, and the remaining parameter is then determined by trial and error to fit the high-pressure data that are available. Rough estimates of the initial value of the second pressure derivative of the bulk modulus can be obtained in this way. As examples, the formula is fitted to experimental data that are already in the literature on aluminum oxide, α-quartz, magnesium, potassium, sodium, and lead.