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By Thomas J. Shankland, Jay D. Bass

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Chapter 10 Hydrostatic compression of perovskite?type MgSiO3 (pages 466–474): T. Yagi, H.K. Mao and P.M. Bell

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The other, which we may call the "true" compressibility P, is -(l/V)(dV/dP); here V is the volume at the pressure at which the derivative dV/dP is taken. Both of these are functions of the pressure, and for most solids differ by only a few percent up to pressures of the order of 10 kg/cm . It is the true compressibility which is required in the above relations. Its reciprocal, the true bulk modulus K, may be found from the general expression for small strain about an arbitrary hydrostatic compression by setting T '= — 8 P , ana 6 — 8 s e', where P' is a small increment of pressure.

NOTATION v p T P S ^ = = = = = = specific volume density; p = 1/v absolute temperature pressure entropy Helmholtz free energy a = volume thermal expansion; p = isothermal compressibility; f3 = — ~ (j$p) = ~ (^) a = ~ (fr) — = T T f$ = adiabatic compressibility; (3 S s K K y C C G g V V = K /p Grtineisen's ratio; 7 = a/(pP C ) = a/(pP C ) specific heat at constant pressure specific heat at constant volume gravitational constant acceleration of gravity velocity of waves of compression velocity of waves of rotation or shear Poisson's ratio T ^/^ = p = ~ (fp) T = p T s s T v s P — a/C P INTRODUCTION The interior of the Earth is a problem at once fascinating and baffling, as one may easily judge from the vast literature, and the few established facts, concerning it.

By definition of,K = p, whence it follows that P 7 3 2 P 2 T T s l s 1 - g' d(j>/dr = dK /dP 1 s + afir/g.. (9) Bullen gives simply dK/dP for the right side of (9), in accordance with the assump­ tion of an adiabatic gradient implicit in the Williamson-Adams method. In general, however, a temperature effect is also concealed in dK /dP, since we have s Elastic Properties and Equations of State Copyright American Geophysical Union Special Publications Elastic Properties and Equations of State ELASTICITY AND dKs dP = CONSTITUTION (dK \ \dP Jr^ OF EARTH'S (dK,\ \dT s \ dP J 237 r r P (dKs) INTERIOR dT ) 'dP \dT ) \_pC T Vol.

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