Astronomy and Astrophysics – Astronomy
Scientific paper
Dec 2000
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2000geoji.143..621s&link_type=abstract
Geophysical Journal International, Volume 143, Issue 3, pp. 621-628.
Astronomy and Astrophysics
Astronomy
18
Equations Of State, K&Prime, Lower Mantle
Scientific paper
The variation with pressure of the derivative K'≡dK/dP, where K is the bulk modulus and P is the pressure, is more sensitive to the precise form of an equation than are the variations of density, ρ, or of K. Also, it leads more directly to estimates of thermal properties via the Grüneisen parameter, γ. Recent discussions have focused on equations relating K' to P/K to make use of the infinite pressure extrapolation 1/K'∞=(P/K)∞. The fundamental significance of K'∞ is emphasized by a thermodynamic demonstration that, for solids, it has the same value for isothermal and adiabatic moduli, their isothermal and adiabatic derivatives and as the limit for both isothermal and adiabatic equations of state. However, this proof conflicts with the assumption, used to estimate K'∞ for regions of the Earth's deep interior, that a linear relationship between μ/K and P/K, where μ is the rigidity, extrapolates to (μ/K)∞=0. The new relationship requires K'∞>=1+γ∞, where the equality is a special condition, applicable to ideal gases, but the inequality appears to be the normal condition for real materials. For solids the usual theories of γ all converge to the relationship γ∞=K'∞/2-1/6, in which case the thermodynamic inequality becomes K'∞>5/3. Most finite strain theories fail this test. It compels reassessment of K'∞ for the lower mantle of the Earth and of equations of state suitable for the deep Earth. A new empirical relationship now appears to satisfy all requirements: where B=1-K'∞/K'0=- K0K''0/K'2∞, subscripts zero indicating zero-pressure values. Its use avoids the bias introduced by theories, such as that due to Birch, that impose values of K'∞ determined by the forms of the equations and not by the fitted data. It is important also to constrain K'0 by information additional to earth model data.
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