Birch's law

Birch's law, discovered by the geophysicist Francis Birch, establishes a linear relation between compressional wave velocity $v p$ and density $\rho$ of rocks and minerals:

v_{p}=a(M_{{\mathrm {avg}}})+b\rho ,

where $M avg$ is the mean atomic mass in a formula unit and $a(x)$ is an experimentally determined function.^[1]

Example

The mean atomic mass of forsterite (Mg₂SiO₄) is equal to the sum of the atomic masses divided by the number of atoms in the formula:

M_{\mathrm {avg} }=(2\times 24.3+28.3+4\times 16)/7=20.13.

Typical oxides and silicates in the mantle have values close to 20, while in the earth's core it is close to 50.^[1]

Applications

Birch's law applies to rocks that are under pressures of a few tens of gigapascals, enough for most cracks to close]].^[1] 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 of material.^[2]

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 for the increased pressure that would be reached as you look deeper into the mantle near the Transition zone (Earth). In such cases where the assumption was made past the Transition zone (Earth), the solutions may need to be revised. In future cases, other Laws may be needed to determine the velocities at high pressures.^[3]

Solving Birch's law experimentally

The relationship between the density of a material and the velocity of a P wave 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 ceramic (the transducer) which is attached to the 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 other 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.^[4]

Velocity of compressional waves in rocks

The below table shows the velocities for different rocks ranging in pressure from 10 bars to 10,000 bars. It represents the how the change in density, as given in the second column, is related to the velocity of the P wave moving in the material. An increase in the density of the material leads to an increase in the velocity which can be determined using Birch's Law.

Velocities of Compressional Waves in Rocks^[4]
Rock Type	Density	Velocity (P=10)	Velocity (P=500)	Velocity (P=2000)	Velocity (P=10,000)
Serpentinite Thetford, Que.	2.601	5.6		5.73	6.00
Serpentinite Ludlow, Vt.	2.614	4.7	6.33	6.59	6.82
Granite Westerly R. I. “G. I.”	2.619	4.1	5.63	5.97	6.23
Granite Quincy, Mass.	2.621	5.1	6.04	6.20	6.45
Granite Rockport, Mass.	2.624	5.0	5.96	6.29	6.51
Granite Stone Mt., GA	2.625	3.7	5.42	6.16	6.40
Granite Chelmsford, Mass.	2.626	4.2	5.64	6.09	6.35
Gneiss, Pelham, Mass.	2.643	3.4	5.67	6.06	6.31
Quartz monzonite Porterville, Cal.	2.644	5.1		6.07	6.37
Quartzite Montana	2.647	5.6		6.15	6.35
Granite Hyderabad, India	2.654	5.4	6.26	6.38	6.56
Granite Barre, Vt.	2.655	5.1	5.86	6.15	6.39
Sandstone N. Y.	2.659	3.9	5.0	5.44	5.85
Pyrophyllite Granite Sacred Heart, Minn.	2.662	5.9		6.28	6.45
Granite Barriefield, Ont.	2.672	5.7	6.21	6.35	6.51
Gneiss Hell Gate, N. Y.	2.675	5.1	6.06	6.23	6.50
Granite Hyderabad, India	2.676	5.7		6.46	6.61
“Granite” Englehart, Ont.	2.679	6.1	6.28	6.37	6.57
Greywacke, New Zealand	2.679	5.4	5.63	5.87	6.13
“Granite" Larchford, Ont.	2.683	5.7	6.13	6.25	6.41
Albitie Sylmar, Pa.	2.687	6.40		6.65	6.76
Granodiorite Butte, Mont.	2.705	4.4		6.35	6.56
Graywacke Quebec	2.705	5.4		6.04	6.28
Serpentinite Cal.	2.710	5.8		6.08	6.31
Slate Medford, Mass.	2.734	5.49		5.91	6.22
“Charnockite” Pallavaram, India	2.740	6.15		6.30	6.46
Granodiorite gneiss, N. H.	2.758	4.4		6.07	6.30
Tonalite Val Verde, Cal,	2.763	5.1		6.43	6.60
Anorthosite Tahawus, N. Y.	2.768	6.73		6.90	7.02
Anorthosite Stillwater Complex, Mont.	2.770	6.5		7.01	7.10
Augite syenite Ontario	2.780	5.7		6.63	6.79
Mica schist Woodsville, Vt	2.797	5.7		6.48	6.64
Serpentinite Ludlow, Vt.	2.798	6.4		6.57	6.84
Quartz diorite San Luis Rey quad., Cal.	2.798	5.1		6.52	6.71
Anorthosire Bushveld Complex	2.807	5.7	6.92	7.05	7.21
Chlorite schist Chester Quarry, Vt.	2.841	4.8		6.82	7.07
Quartz diorite Dedham, Mass.	2.906	5.5		6.53	6.71
Talc schist Chester, Vt	2.914	4.9		6.50	6.97
Gabbro Mellen, Wis.	2.931	6.8	7.04	7.09	7.21
Diabase Centerville, Va.	2.976	6.14		6.76	6.93
Diabase Holyoke, Mass.	2.977	6.25	6.40	6.47	6.63
Norite Pertoria Transvaal	2.978	6.6	7.02	7.11	7.28
Dunite Webster, N. C.	2.980	6.0		6.46	6.79
Diabase Sudbury, Ont.	3.003	6.4	6.67	6.76	6.91
Diabase Frederick, Md.	3.012	6.76		6.80	6.92
Gabbro French Creek, Pa.	3.054	5.8	6.74	7.02	7.23
Amphibolite Madison Co., Mont.	3.120	6.89		7.12	7.35
Jadeite, Japan	3.180	7.6		8.22	8.28
Actinoliter schist Chester, Vt.	3.194	6.61		7.20	7.54
Dunite Webster, N. C.	3.244	7.0		7.59	7.78
Pyroxenite Sonoma Co., Cal.	3.247	6.8		7.79	8.01
Dunite Mt. Dun, New Zealand	3.258	7.5	7.69	7.80	8.00
Dunite Balsam Gap, N. C.	3.267	7.0	7.82	8.01	8.28
Bronzitite Stillwater Complex	3.279	7.42		7.65	7.83
Dunite Addie N. C.	3.304	7.70		8.05	8.28
Dunite Twin Sisters Peaks, Wash.	3.312	7.7	8.11	8.27	8.42
“Eclogite” Tanganyika	3.328	6.64	7.30	7.46	7.71
Jadeite Burma	3.331	8.45		8.69	8.78
Harzdurgite, Bushveld Complex	3.369	6.9	7.74	7.81	7.95
Eclogite Kimberley	3.376	7.17	7.65	7.73	7.87
Eclogite Sunnmore, Norway	3.376	5.2		7.30	7.69
Eclogite Healdsburg, Cal.	3.441	7.31		7.81	8.01
Garnet Conn.	3.561	6.3		8.55	8.99
Dunite Moonihoek Mine, Transvaal	3.744	6.7	7.13	7.21	7.36

References

1 2 3 Poirier, Jean-Paul (2000). Introduction to the physics of the earth's interior (2nd ed.). Cambridge [u.a.]: Cambridge Univ. Press. pp. 79–80. ISBN 9780521663922.
↑ Liebermann, Robert; Ringwood, A. E. (October 20, 1973). "Birch's Law and Polymorphic Phase Transformations". Journal of Geophysical Research. 78 (29): 6926–6932. Bibcode:1973JGR....78.6926L. doi:10.1029/JB078i029p06926.
↑ Birch, F. (1961). "The velocity of compressional waves in rocks to 10 kilobars. Part 2". Journal of Geophysical Research. 66 (7): 2199–2224. Bibcode:1961JGR....66.2199B. doi:10.1029/JZ066i007p02199.
1 2 Birch, Francis (April 1960). "The Velocity of Compressional Waves in Rocks to 10 Kilobars, Part 1". Journal of Geophysical Research. 65 (4): 1083–1102. Bibcode:1960JGR....65.1083B. doi:10.1029/JZ065i004p01083.

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[Poirier-1] 1 2 3 Poirier, Jean-Paul (2000). Introduction to the physics of the earth's interior (2nd ed.). Cambridge [u.a.]: Cambridge Univ. Press. pp. 79–80. ISBN 9780521663922.

[ReferenceA-2] Liebermann, Robert; Ringwood, A. E. (October 20, 1973). "Birch's Law and Polymorphic Phase Transformations". Journal of Geophysical Research. 78 (29): 6926–6932. Bibcode:1973JGR....78.6926L. doi:10.1029/JB078i029p06926.

[ReferenceB-3] Birch, F. (1961). "The velocity of compressional waves in rocks to 10 kilobars. Part 2". Journal of Geophysical Research. 66 (7): 2199–2224. Bibcode:1961JGR....66.2199B. doi:10.1029/JZ066i007p02199.

[F_Birch-4] 1 2 Birch, Francis (April 1960). "The Velocity of Compressional Waves in Rocks to 10 Kilobars, Part 1". Journal of Geophysical Research. 65 (4): 1083–1102. Bibcode:1960JGR....65.1083B. doi:10.1029/JZ065i004p01083.