User talk:Williamborg/Geophysics

Geophysics Sandbox

Liquids in porous solids

 * Porosity
 * Permeability
 * Capillary Pressure

Radioactive decay

 * Radioactive Decay Processes
 * Alpha decay
 * Beta Decay
 * Spontaneous fission

Measurement of radioactivity

 * Ionization

Cosmic radiation
Cosmogenic nuclides (or cosmogenic isotopes) are rare isotopes created when a high-energy cosmic ray interacts with the nucleus of an atom, causing cosmic ray spallation. These isotopes are produced within earth materials such as rocks, soil or atmosphere. By measuring cosmogenic isotopes, we are able to gain insight into a range of geologic processes.

Cosmogenic isotopes display different physical and chemical properties that make it possible to apply surface exposure dating methods on rock surfaces of varying lithology at any latitude and altitude, for exposures ranging from 102 to 107 years. The primary cosmogenic isotopes of interest are: The reaction 14N (n, 12C)³H produces tritium (³H), which has a 12.32 year half life. Tritium decays to ³He, which is stable. Since ³He is found in nature, the ratios must correct for this to separate that which is cosmogenic, and that which was captured during the original magmatic formation of the rock.
 * 3He

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Be-10 is produced by apallation (N and O)
 * 10Be

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 * 21Ne

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 * 26Al

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At depths greater than 2-3 meters, 36Cl production in rocks such as calcite is primarily the result of cosmic ray muon interactions: a) direct negative muon capture by Calcium-40 (40Ca) with subsequent alpha decay to produce 36Cl, and (2) radiative capture of secondary neutrons by 35Cl (secondary neutrons are produced by muon capture and muon-induced photo-disintegration reactions). Rates from these reactions are calculated based on the specific materials involved. The 36Cl found in the sample can be compared directly with the stable 35Cl and 37Cl isotopic concentrations.
 * 36Cl

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81Kr

III.D - Application of radioactivity to geologic processes
Radioactivity contributes both to geological processes and to our ability to date these processes. The decay of uranium, thorium and potassium in earth’s interior provides a continuing heat source. Further, geochronology often uses one of several decay systems that are found in several different types of minerals and deposits:
 * The (U–Th)/He system, primarily involves measuring accumulation of helium-4 from the alpha decay of uranium-238, uranium-235, thorium-232, and associated daughter isotopes.
 * Fission tracks are produced by spontaneous fission of 238U, leaving microscopically visible crystallographic damage zones in the crystal.
 * K–Ar system, yields argon-40 by electron capture from potassium-40. Only about 11% of 40K decays to 40Ar by electron capture; the majority decays to calcium-40 by β+ emission.

Uranium-Thorium geochronology
Uranium-Thorium geochronology involves the measurement of daughter products such as Thorium isotopes and comparison with the parent nuclides such as Uranium isotopes which generated them by decaying.

Parent and daughter abundances are related to time by the radioactive decay equation:
 * $$\frac{dN(t)}{dt} = -\lambda N(t)$$

or, by rearranging,
 * $$\frac{dN(t)}{N(t)} = -\lambda dt.$$

Integrating, we have
 * $$\ln N(t) = -\lambda t + C \,$$

where C is the constant of integration, and hence
 * $$N(t) = e^C e^{-\lambda t} = N_0 e^{-\lambda t} \,$$

where the final substitution, $$N_0 = e^C$$, is obtained by evaluating the equation at $$t=0$$, as $$N_0$$ is defined as being the quantity at $$t=0$$.

Using this relationship, the number of daughter nuclides (or alternately the number of decay or fission related damage tracks) can be calculated. For geochronological applications the abundance of daughter nuclides reflects nuclear decay which takes place after the mineral crystalizes and the daughter products can be captured.

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The 4n+2 chain of U-238 is commonly called the "radium series" (sometimes "uranium series"). Beginning with naturally occurring uranium-238, this series includes the following elements: astatine, bismuth, lead, polonium, protactinium, radium, radon, thallium, and thorium. All are present, at least transiently, in any natural uranium-containing sample, whether metal, compound, or mineral. The series terminates with lead-206.



Cosmogenic isotopes
Cosmogenic radionuclide dating relies on cosmogenic radionuclides to date how long a particular surface has been exposed, how long a certain piece of material has been buried, or how quickly a location or drainage basin is eroding. The basic principle is that these radionuclides are produced at a known rate, and also decay at a known rate.

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Cosmogenic nuclides are produced by chains of spallation reactions from cosmic rays indicent upon the earth. The production rate for a particular nuclide is a function of latitude, the amount of sky that can be seen from the point that is sampled, elevation, sample depth, and density of the material in which the sample is embedded. Decay rates are given by the decay constants of the nuclides. These equations can be combined to give the total concentration of cosmogenic radionuclides in a sample as a function of age.

Common radionuclides used in geomorphology include beryllium-10, aluminium-26,, chlorine-36 , and Carbon-14

Geochronology using -  dating
The cosmogenic radioisotopes (half life $$\tau_{Al-26}$$ = 1.02 ± 0.02 x106 years) and  ($$\tau_{Be-10}$$  = 1.93 ± 0.10 x106 years) are produced when quartz crystals (SiO2) are bombarded by secondary cosmic-ray nucleons and muons near the ground surface. Quartz grains near the surface accumulate an inventory of these radionuclides, whose concentrations depend on the mineral's exposure time to cosmic rays, which in turn depends on the erosion rate of the host rock. Some of the oxygen found in the quartz is transformed into 10Be and the silicon is transformed into 26Al. Although each of these nuclides is produced at a different rate, they can be used individually to date how long the material has been exposed at the surface. Because there are two radionuclides both being produced and decaying, the ratio of concentrations of these two nuclides can be used in combination with other knowledge to determine an age. Because cosmic rays are significantly attenuated as tehy pass through several meters of soil, this is also useful in identifying the date at which a sample was buried below the production depth (typically 2–10 meters).

At the Earth’s surface, and   production in quartz is mostly by cosmic ray spallation reactions with Oxygen and Silicon, with some supplemental production through negative muon capture and fast muon interactions. The total surface production rates of these nuclides in quartz at sea level and high latitude are approximately 5.1 atoms per gram per year for and 31.1 atoms per gram per year. Although the production rates vary predictably with latitude, altitude, and magnetic field variation, the ratio of to  production rate is fixed at 6.1. Hence, for short exposure times (< 50,000 year) or for high erosion rates (>2 meters in 50,000 years or >0.02-0.04 kilograms per square meter per year) the radioactive decay rate is negligible. This initial ratio provides a valuable benchmark for determining both the exposure duration and time of burial after initial exposure.


 * Appreciation of the physics behind 10Be - 26Al dating

Let N be the general symbol for atomic concentration expressed in atoms per cm3. Using this formulation, we specify $${N_0}^{Al-26}$$ as the initial atomic concentration of and $${N_0}^{Be-10}$$ as the initial atomic concentration of, while $${N_1}^{Al-26}$$ and $${N_1}^{Be-10}$$ represent the concentrations at a later time.

The decay constants for and  can be found by the general relationship $$\displaystyle \lambda = {0.693/\tau  }$$, or  $$\displaystyle \lambda_{Al-26} = {0.693/ \tau_{Al-26} }$$ and  $$\displaystyle  \lambda_{Be-10}= {0.693/  \tau_{Be-10} }$$. We know that after exposure to cosmic rays for periods relatively short compared to the decay constants, the ratio of the atomic concentrations, which we will symbolize by $$\beta$$, is 6.1.


 * $$\displaystyle {\beta_0} = {{{N_0}^{Al-26}} \over {{N_0}^{Be-10}} } = 6.1 $$

We know that should the sample be covered over by 10s of meters of sediment, it will no longer be exposed to cosmic rays. As a result the and  will decay, but at different rates. Hence the ratio will change.


 * $$\displaystyle {\beta} = {{{N_0}^{Al-26}e^{-{\lambda_{Al-26}}t}} \over {{N_0}^{Be-10}}e^{-{\lambda_{Be-10}}t} }  $$

Which simplifies to:


 * $$\displaystyle {\beta} = {\beta_0} e^{({\lambda_{Be-10}}-{\lambda_{Al-26}})t}  $$

Solving for the time, t, yields:


 * $$\displaystyle

t = {1 \over {\lambda_{Be-10}}-{\lambda_{Al-26}}} ln({{\beta} \over {\beta_0}}) $$

Carbon-14 dating


Carbon has two stable, nonradioactive isotopes: carbon-12, and carbon-13. In addition, there are trace amounts of the unstable isotope carbon-14 on Earth. Carbon-14 has a relatively short half-life of 5730 years, meaning that the amount of carbon-14 in a sample is halved over the course of 5730 years due to radioactive decay. Carbon-14 would have long ago vanished from Earth were it not for cosmic ray flux interactions with the Earth's atmosphere, which create more of the isotope. The neutrons resulting from the cosmic ray interactions participate in the following nuclear reaction on the atoms of nitrogen molecules in the atmosphere:


 * $$n + \mathrm{^{14}_{7}N} \rightarrow \mathrm{^{14}_{6}C} + p$$

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The highest rate of carbon-14 production takes place at altitudes of 9 to 15 km (30,000 to 50,000 ft), and at high geomagnetic latitudes, but the carbon-14 spreads evenly throughout the atmosphere and reacts with oxygen to form carbon dioxide. Carbon dioxide also permeates the oceans, dissolving in the water. For approximate analysis it is assumed that the cosmic ray flux is constant over long periods of time; thus carbon-14 is produced at a constant rate and the proportion of radioactive to non-radioactive carbon is constant: ca. 1 part per trillion (600 billion atoms/mole). In 1958 Hessel de Vries showed that the concentration of carbon-14 in the atmosphere varies with time and locality. For the most accurate work, these variations are compensated by means of calibration curves. When these curves are used, their accuracy and shape are the factors that determine the accuracy of the age obtained for a given sample. can also be produced at ground level at a rate of 1 x 10−4 g−1s−1, which is not significant enough to impact dating calculations.

Plants take up atmospheric carbon dioxide by photosynthesis, and are ingested by animals, so every living thing is constantly exchanging carbon-14 with its environment as long as it lives. Once it dies, however, this exchange stops, and the amount of carbon-14 gradually decreases through radioactive beta decay with a half-life of 5,730 ± 40 years. Carbon-14 is stored in different amounts in different reservoirs because there is a dynamic equilibrium between production and decay.

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 * $$\mathrm{~^{14}_{6}C}\rightarrow\mathrm{~^{14}_{7}N}+ e^{-} + \bar{\nu}_e$$

Once organic material is dead and has been captured by a geologic process, decay will decrease the fraction. For relatively recent events can provide meaningful age data.

Thermoluminescence dating
Thermoluminescence dating dating of geological materials are useful  for dating volcanic and related materials and events over timescales ranging from 102 to 106 years. Thermoluminescence methods are less precise than 40Ar/39Ar and 14C methods, they offer a useful alternative approach with advantages in terms of age range and scope of material.

When quartz, feldspar and other minerals are exposed to ionizing radiation the decelerating particle detaches electrons from nuclei in the crystal lattice. Some fraction of these electrons are captured in defects in the lattice. For a given type of radiation, The trapped electrons density increases proportionally with the radiation exposure fluence. Heating, illumination or shock of a crystal can free these trapped electrons. Some fraction of these freed electrons return to a vacant position (a ‘hole’) left from a previously displaced electron, with concurrent emission of energy as a photon.

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As an example of this use for dating, Feldspars contain a radioactive isotope of potassium (40K). 40K decays by β− (i.e., electron) decay and has a half-life of 1.248 years, or about 39.38 seconds. When formed during solidification of basalt, the feldspar has no detached electrons. As the feldspar is exposed to the β− decay, it accumulates electrons essentially linearly with time for times significantly shorter than 1.248 years.


 * $$\displaystyle Age = (accumulated_.dose_.of_.ambient_.radiation) / (dose_.accumulated_.per_.year)$$

Neutrino measurement
Radiogenic heating due to decay can be estimated from the flux of neutrinos that routinely accompany beta decay which accompanies uranium, thorium and potassium decays. These electrically neutral particles are emitted during beta decay and pass through the Earth virtually unaffected until detected by specially designed detectors. Current estimates of heat production from radiogenic sources are 20±9 TerraWatts. This makes up approximately half or the 44.2±1 TerraWatt heat flux which has been measured as earth's emission to space.

Gravitational acceleration/force

 * Newton's Law of Gravitation
 * Gravitational Acceleration

Gravitational Potential

 * Potential
 * Potential field equations

V - Material Properties

 * Density of materials
 * Atomic concentrations

VI.D - Buckling

 * Columnar buckling
 * Plate buckling
 * Shell buckling in shperical coordinates
 * Composite shell buckling in shperical coordinates
 * Torsional buckling

VII.B -Wave propagation in solids

 * P waves
 * S waves
 * Rayleigh waves

VII.D -Field methods using shock waves

 * Reflection
 * Refraction

IX.A - Magnetic force

 * Biot-Savart Law

IX.H - Measurement of the earth's magnetic field
The compass is a familiar magnetometer that indicates the horizontal direction of the magnetic-field at the earth's surface.

A wide variety of magnetometers, usually coupled to digital data-acquisition systems, can be used for geological measurements:
 * A fluxgate magnetometer measures vector field components
 * A proton-precession magnetometer measures field intensity

X.F - Methods for use of EM Fields

 * Ground penetrating radar