Birch's law

Birch's law is an empirical relationship in geophysics that describes the linear dependence of the compressional wave velocity in rocks and minerals on their density. Formulated by Francis Birch (geophysicist) in the early 1960s, it provides a crucial bridge between seismic observations from the Earth's interior and the likely composition of deep planetary interiors. The law is foundational for interpreting data from networks like the Global Seismographic Network and for modeling the structure of other bodies such as the Moon and Mars.

Definition and statement

The law states that for most silicate and oxide minerals of the Earth's mantle and crust (geology), the velocity of P-waves increases linearly with increasing density. It is commonly expressed as \( V_p = a(\rho) + b \), where \( V_p \) is the compressional wave velocity, \( \rho \) is the density, and \( a \) and \( b \) are constants that depend on the mean atomic weight of the material. This relationship was derived from laboratory measurements on olivine, pyroxene, garnet, and feldspar under conditions simulating the shallow upper mantle. The linearity implies that changes in seismic velocity with depth, detected by instruments like those deployed during the Project Mohole, can be primarily attributed to changes in density due to pressure rather than major compositional shifts.

Experimental basis

Birch established the law through pioneering high-pressure experiments using apparatus like the piston-cylinder press at Harvard University. He measured the acoustic velocity and density of numerous rock-forming minerals and igneous rocks such as basalt and granite at pressures up to 10 GPa, corresponding to depths of about 300 kilometers. The data, later supplemented by studies at institutions like the Carnegie Institution for Science, showed a remarkably consistent trend across different mineral groups. These laboratory results provided the first quantitative calibration for converting the travel-time tables from global earthquakes, analyzed by pioneers like Beno Gutenberg and Harold Jeffreys, into estimates of in situ density.

Applications in geophysics

The primary application is in estimating the composition of the Earth's mantle from seismic tomography models. By comparing the observed velocity-depth profiles, such as those from the PREM model, with velocities predicted for candidate minerals like ringwoodite or bridgmanite, geophysicists can infer mineralogy. It is also used to interpret the seismic discontinuity at the Moho, the nature of low-velocity zones, and the composition of the D″ layer. In planetary science, Birch's law has been applied to data from missions like Apollo program lunar seismometry and the InSight mission to Mars to constrain the interiors of these bodies, comparing them to terrestrial analogs like the Kaapvaal Craton.

Limitations and deviations

Significant deviations occur for materials with high iron content, such as iron-nickel alloys in the Earth's core, or for minerals with strong anisotropy, like those in the subduction slabs. The law also breaks down for highly porous or fractured rock in the shallow crust (geology), where factors like pore fluid pressure dominate. Furthermore, at very high pressures corresponding to the lower mantle, the relationship may become nonlinear due to phase transitions to post-perovskite or changes in electronic structure. Studies of metamorphic rocks like eclogite also show scatter, indicating the influence of factors beyond mean atomic weight.

Relationship to other equations of state

Birch's law is conceptually related to, but distinct from, theoretical equations of state like the Murnaghan equation of state or the Birch–Murnaghan equation of state, which describe pressure-volume relationships. It provides a seismic velocity-centric complement to these models. The law is also connected to the Dulong–Petit law through their shared dependence on mean atomic weight, and it informs the calculation of the Bullen parameter, used to assess the adiabatic nature of the mantle. Its empirical form underpins many inverse problem methodologies in global seismology practiced at institutions like the California Institute of Technology and the University of Tokyo.

Category:Geophysics Category:Seismology Category:Empirical laws Category:Equations of state