Birch's law, discovered by the

geophysicist Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...

Francis Birch, establishes a linear relation between

compressional wave Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal wav ...

velocity and density of rocks and minerals: :

v_\mathrm = a( \bar M ) + b \rho

where

\, \bar M \,

is the mean

atomic mass The atomic mass (''m''a or ''m'') is the mass of an atom. Although the SI unit of mass is the kilogram (symbol: kg), atomic mass is often expressed in the non-SI unit dalton (symbol: Da) – equivalently, unified atomic mass unit (u). 1&nbs ...

formula unit In chemistry, a formula unit is the empirical formula of any ionic or covalent network solid compound used as an independent entity for stoichiometric calculations. It is the lowest whole number ratio of ions represented in an ionic compound. Exa ...

s and

\, a(x) \,

is an empirical function determined by experiment.

Example

The mean

forsterite Forsterite (Mg2SiO4; commonly abbreviated as Fo; also known as white olivine) is the magnesium-rich end-member of the olivine solid solution series. It is isomorphous with the iron-rich end-member, fayalite. Forsterite crystallizes in the orthor ...

(Mg₂SiO₄) is equal to the sum of the

es divided by the number of atoms in the formula: :

\bar = \frac = 20.13 ~.

Typical

oxide An oxide () is a chemical compound that contains at least one oxygen atom and one other element in its chemical formula. "Oxide" itself is the dianion of oxygen, an O2– (molecular) ion. with oxygen in the oxidation state of −2. Most of the E ...

s and

silicate In chemistry, a silicate is any member of a family of polyatomic anions consisting of silicon and oxygen, usually with the general formula , where . The family includes orthosilicate (), metasilicate (), and pyrosilicate (, ). The name is al ...

s in the mantle have values close to 20, while in the

earth's core The internal structure of Earth is the solid portion of the Earth, excluding its atmosphere and hydrosphere. The structure consists of an outer silicate solid crust, a highly viscous asthenosphere and solid mantle, a liquid outer core whose ...

it is close to 50.

Applications

Birch's law applies to rocks that are under pressures of a few tens of

gigapascal The pascal (symbol: Pa) is the unit of pressure in the International System of Units (SI), and is also used to quantify internal pressure, stress, Young's modulus, and ultimate tensile strength. The unit, named after Blaise Pascal, is defined as ...

s, enough for most cracks to close. It can be used in the discussion of geophysical data. The law is used in forming compositional and mineralogical models of the mantle by using the change in the velocity of the seismic wave and its relationship with a change in density of the material the wave is moving in. Birch's law is used in determining chemical similarities in the mantle as well as the discontinuities of the transition zones. Birch's Law can also be employed in the calculation of an increase of velocity due to an increase in the

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

of material.

Shortcomings

It had been previously assumed that the velocity-density relationship is constant. That is, that Birch's law will hold true in any case, but as you look deeper into the mantle, the relationship does not hold true deeper in the mantle for the increased pressures near the transition zone. In cases where Birch's law was applied beyond the transition zone, parts of the formula need to be revised. For higher pressure regimes, different laws may be needed to determine wave velocities.

Determining Birch's law experimentally

The relationship between the density of a material and the velocity of a moving through the material was noted when research was conducted on waves in different materials. In the experiment, a pulse of voltage is applied to a circular plate of polarized

barium titanate Barium titanate (BTO) is an inorganic compound with chemical formula BaTiO3. Barium titanate appears white as a powder and is transparent when prepared as large crystals. It is a Ferroelectricity, ferroelectric, Pyroelectricity, pyroelectric, and ...

ceramic (the transducer) which is attached to the near end of the material sample. The added voltage creates vibrations in the sample. Those vibrations travel through the sample to a second transducer on the far end. The vibrations are then converted into an electrical wave which is viewed on an oscilloscope to determine the travel time. The velocity is the lender of the damper decided by the wave's travel time. The resulting relationship between the density of the material and the discovered velocity is known as Birch's law.

Velocity of compressional waves in rocks

The below table shows the velocities for different rocks ranging in pressure from 10

bar Bar or BAR may refer to: Food and drink * Bar (establishment), selling alcoholic beverages * Candy bar * Chocolate bar Science and technology * Bar (river morphology), a deposit of sediment * Bar (tropical cyclone), a layer of cloud * Bar (u ...

s to 10,000

s. It represents the how the change in

, as given in the second column, is related to the velocity of the moving in the material. An increase in the

of the material leads to an increase in the velocity which can be determined using Birch's Law. :

References