Synchronization of chaos

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https://en.wikipedia.org/wiki/Synchronization_of_chaos

Synchronization of chaos is a phenomenon that may occur when two or more dissipative chaotic systems are coupled.

Because of the exponential divergence of the nearby trajectories of chaotic systems, having two chaotic systems evolving in synchrony might appear surprising. However, synchronization of coupled or driven chaotic oscillators is a phenomenon well established experimentally and reasonably well-understood theoretically.

The stability of synchronization for coupled systems can be analyzed using master stability. Synchronization of chaos is a rich phenomenon and a multi-disciplinary subject with a broad range of applications.

Synchronization may present a variety of forms depending on the nature of the interacting systems and the type of coupling, and the proximity between the systems.

Identical synchronization

This type of synchronization is also known as complete synchronization. It can be observed for identical chaotic systems. The systems are said to be completely synchronized when there is a set of initial conditions so that the systems eventually evolve identically in time. In the simplest case of two diffusively coupled dynamics is described by

${\displaystyle x^{\mathrm{\prime }}=F(x)+\alpha (y-x)}$

${\displaystyle y^{\mathrm{\prime }}=F(y)+\alpha (x-y)}$

where ${\displaystyle F}$ is the vector field modeling the isolated chaotic dynamics and ${\displaystyle \alpha }$ is the coupling parameter. The regime ${\displaystyle x(t)=y(t)}$ defines an invariant subspace of the coupled system, if this subspace ${\displaystyle x(t)=y(t)}$ is locally attractive then the coupled system exhibit identical synchronization.

If the coupling vanishes the oscillators are decoupled, and the chaotic behavior leads to a divergence of nearby trajectories. Complete synchronization occurs due to the interaction, if the coupling parameter is large enough so that the divergence of trajectories of interacting systems due to chaos is suppressed by the diffusive coupling. To find the critical coupling strength we study the behavior of the difference ${\displaystyle v=x-y}$. Assuming that ${\displaystyle v}$ is small we can expand the vector field in series and obtain a linear differential equation - by neglecting the Taylor remainder - governing the behavior of the difference

${\displaystyle v^{\mathrm{\prime }}=DF(x(t))v-2\alpha v}$

where ${\displaystyle DF(x(t))}$ denotes the Jacobian of the vector field along the solution. If ${\displaystyle \alpha =0}$ then we obtain

${\displaystyle u^{\mathrm{\prime }}=DF(x(t))u,}$

and since the dynamics of chaotic we have ${\displaystyle \Vert u(t)\Vert \le \Vert u(0)\Vert e^{\lambda t}}$, where ${\displaystyle \lambda }$ denotes the maximum Lyapunov exponent of the isolated system. Now using the ansatz ${\displaystyle v=u{e}^{-2\alpha t}}$ we pass from the equation for ${\displaystyle v}$ to the equation for ${\displaystyle u}$. Therefore, we obtain

${\displaystyle \Vert v(t)\Vert \le \Vert u(0)\Vert e^{(-2\alpha +\lambda )t}}$

yield a critical coupling strength ${\displaystyle {\alpha }_c=\lambda /2}$, for all ${\displaystyle \alpha >{\alpha }_c}$ the system exhibit complete synchronization. The existence of a critical coupling strength is related to the chaotic nature of the isolated dynamics.

In general, this reasoning leads to the correct critical coupling value for synchronization. However, in some cases one might observe loss of synchronization for coupling strengths larger than the critical value. This occurs because the nonlinear terms neglected in the derivation of the critical coupling value can play an important role and destroy the exponential bound for the behavior of the difference. It is however, possible to give a rigorous treatment to this problem and obtain a critical value so that the nonlinearities will not affect the stability.

Generalized synchronization

This type of synchronization occurs mainly when the coupled chaotic oscillators are different, although it has also been reported between identical oscillators. Given the dynamical variables ${\displaystyle (x_1,x_2,...x_n)}$ and ${\displaystyle (y_1,y_2,...y_m)}$ that determine the state of the oscillators, generalized synchronization occurs when there is a functional, ${\displaystyle \varphi }$, such that, after a transitory evolution from appropriate initial conditions, it is ${\displaystyle [y_1(t),y_2(t),...,y_m(t)=\varphi [x_1(t),x_2(t),...,x_n(t)]}$. This means that the dynamical state of one of the oscillators is completely determined by the state of the other. When the oscillators are mutually coupled this functional has to be invertible, if there is a drive-response configuration the drive determines the evolution of the response, and Φ does not need to be invertible. Identical synchronization is the particular case of generalized synchronization when ${\displaystyle \varphi }$ is the identity.

Phase synchronization

Phase synchronization occurs when the coupled chaotic oscillators keep their phase difference bounded while their amplitudes remain uncorrelated. This phenomenon occurs even if the oscillators are not identical. Observation of phase synchronization requires a previous definition of the phase of a chaotic oscillator. In many practical cases, it is possible to find a plane in phase space in which the projection of the trajectories of the oscillator follows a rotation around a well-defined center. If this is the case, the phase is defined by the angle, φ(t), described by the segment joining the center of rotation and the projection of the trajectory point onto the plane. In other cases it is still possible to define a phase by means of techniques provided by the theory of signal processing, such as the Hilbert transform. In any case, if φ₁(t) and φ₂(t) denote the phases of the two coupled oscillators, synchronization of the phase is given by the relation nφ₁(t)=mφ₂(t) with m and n whole numbers.

Anticipated and lag synchronization

In these cases, the synchronized state is characterized by a time interval τ such that the dynamical variables of the oscillators, ${\displaystyle (x_1,x_2,...x_n)}$ and ${\displaystyle (x_1^{\prime },x_2^{\prime },...x_n^{\prime })}$, are related by ${\displaystyle x_i^{\prime }(t)=x_i(t+\tau )}$; this means that the dynamics of one of the oscillators follows, or anticipates, the dynamics of the other. Anticipated synchronization may occur between chaotic oscillators whose dynamics is described by delay differential equations, coupled in a drive-response configuration. In this case, the response anticipates the dynamics of the drive. Lag synchronization may occur when the strength of the coupling between phase-synchronized oscillators is increased.

Amplitude envelope synchronization

This is a mild form of synchronization that may appear between two weakly coupled chaotic oscillators. In this case, there is no correlation between phases nor amplitudes; instead, the oscillations of the two systems develop a periodic envelope that has the same frequency in the two systems.

This has the same order of magnitude than the difference between the average frequencies of oscillation of the two chaotic oscillator. Often, amplitude envelope synchronization precedes phase synchronization in the sense that when the strength of the coupling between two amplitude envelope synchronized oscillators is increased, phase synchronization develops.

All these forms of synchronization share the property of asymptotic stability. This means that once the synchronized state has been reached, the effect of a small perturbation that destroys synchronization is rapidly damped, and synchronization is recovered again. Mathematically, asymptotic stability is characterized by a positive Lyapunov exponent of the system composed of the two oscillators, which becomes negative when chaotic synchronization is achieved.

Some chaotic systems allow even stronger control of chaos, and both synchronization of chaos and control of chaos constitute parts of what's known as "cybernetical physics".

Causal loop diagram

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https://en.wikipedia.org/wiki/Causal_loop_diagram

Example of a positive reinforcing loop between two values: bank balance and earned interest

A causal loop diagram (CLD) is a causal diagram that aids in visualizing how different variables in a system are causally interrelated. The diagram consists of a set of words and arrows. Causal loop diagrams are accompanied by a narrative which describes the causally closed situation the CLD describes. Closed loops, or causal feedback loops, in the diagram are very important features of CLDs because they may help identify non-obvious vicious circles and virtuous circles.

The words with arrows coming in and out represent variables, or quantities whose value changes over time and the links represent a causal relationship between the two variables (i.e., they do not represent a material flow). A link marked + indicates a positive relation where an increase in the causal variable leads, all else equal, to an increase in the effect variable, or a decrease in the causal variable leads, all else equal, to a decrease in the effect variable. A link marked - indicates a negative relation where an increase in the causal variable leads, all else equal, to a decrease in the effect variable, or a decrease in the causal variable leads, all else equal, to an increase in the effect variable. A positive causal link can be said to lead to a change in the same direction, and an opposite link can be said to lead to change in the opposite direction, i.e. if the variable in which the link starts increases, the other variable decreases and vice versa.

The words without arrows are loop labels. As with the links, feedback loops have either positive (i.e., reinforcing) or negative (i.e., balancing) polarity. CLDs contain labels for these processes, often using numbering (e.g., B1 for the first balancing loop being described in a narrative, B2 for the second one, etc.), and phrases that describe the function of the loop (i.e., "haste makes waste"). A reinforcing loop is a cycle in which the effect of a variation in any variable propagates through the loop and returns to reinforce the initial deviation (i.e. if a variable increases in a reinforcing loop the effect through the cycle will return an increase to the same variable and vice versa). A balancing loop is the cycle in which the effect of a variation in any variable propagates through the loop and returns to the variable a deviation opposite to the initial one (i.e. if a variable increases in a balancing loop the effect through the cycle will return a decrease to the same variable and vice versa). Balancing loops are typically goal-seeking, or error-sensitive, processes and are presented with the variable indicating the goal of the loop. Reinforcing loops are typically vicious or virtuous cycles.

Example of positive reinforcing loop shown in the illustration:

• The amount of the Bank Balance will affect the amount of the Earned Interest, as represented by the top blue arrow, pointing from 'Bank Balance to Earned Interest.
• Since an increase in Bank balance results in an increase in Earned Interest, this link is positive, which is denoted with a +.
• The Earned interest gets added to the Bank balance, also a positive link, represented by the bottom blue arrow.
• The causal effect between these variables forms a positive reinforcing loop, represented by the green arrow, which is denoted with an R. (The terms positive and negative mean "tends to increase" and "tend to reduce", not subjective value judgements.)

History

Main article: System Dynamics

The use of words and arrows (known in network theory as nodes and edges) to construct directed graph models of cause and effect dates back, at least, to the use of path analysis by Sewall Wright in 1918. According to George Richardson's book "Feedback Thought in Social Science and Systems Theory", the first published, formal use of a causal loop diagram to describe a feedback system was Magoroh Maruyama's 1963 article "The Second Cybernetics: Deviation-Amplifying Mutual Causal Processes".

Positive and negative causal links

• Positive causal link means that the two variables change in the same direction, i.e. if the variable in which the link starts decreases, the other variable also decreases. Similarly, if the variable in which the link starts increases, the other variable increases.
• Negative causal link means that the two variables change in opposite directions, i.e. if the variable in which the link starts increases, then the other variable decreases, and vice versa.

Example

Dynamic causal loop diagram: positive and negative links

Reinforcing and balancing loops

To determine if a causal loop is reinforcing or balancing, one can start with an assumption, e.g. "Variable 1 increases" and follow the loop around. The loop is:

• reinforcing if, after going around the loop, one ends up with the same result as the initial assumption.
• balancing if the result contradicts the initial assumption.

Or to put it in other words:

• reinforcing loops have an even number of negative links (zero also is even, see example below)
• balancing loops have an odd number of negative links.

Identifying reinforcing and balancing loops is an important step for identifying Reference Behaviour Patterns, i.e. possible dynamic behaviours of the system.

• Reinforcing loops are associated with exponential increases/decreases.
• Balancing loops are associated with reaching a plateau.

If the system has delays (often denoted by drawing a short line across the causal link), the system might fluctuate.

Example

Causal loop diagram of Adoption model, used to demonstrate systems dynamics

Causal loop diagram of a model examining the growth or decline of a life insurance company

Vicious circle

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Vicious_circle

Depression expressed as a vicious circle

A vicious circle (or cycle) is a complex chain of events that reinforces itself through a feedback loop, with detrimental results. It is a system with no tendency toward equilibrium (social, economic, ecological, etc.), at least in the short run. Each iteration of the cycle reinforces the previous one, in an example of positive feedback. A vicious circle will continue in the direction of its momentum until an external factor intervenes to break the cycle. A well-known example of a vicious circle in economics is hyperinflation.

When the results are not detrimental but beneficial, the term virtuous cycle is used instead.

Examples

Vicious circles in the subprime mortgage crisis

Further information: Subprime mortgage crisis

Vicious cycles in the subprime mortgage crisis

The contemporary subprime mortgage crisis is a complex group of vicious circles, both in its genesis and in its manifold outcomes, most notably the late 2000s recession. A specific example is the circle related to housing. As housing prices decline, more homeowners go "underwater", when the market value of a home drops below that of the mortgage on it. This provides an incentive to walk away from the home, increasing defaults and foreclosures. This, in turn, lowers housing values further from over-supply, reinforcing the cycle.

The foreclosures reduce the cash flowing into banks and the value of mortgage-backed securities (MBS) widely held by banks. Banks incur losses and require additional funds, also called "recapitalization". If banks are not capitalized sufficiently to lend, economic activity slows and unemployment increases, which further increase the number of foreclosures. Economist Nouriel Roubini discussed vicious circles in the housing and financial markets in interviews with Charlie Rose in September and October 2008.

Designing ecological virtuous circles

By involving all stakeholders in managing ecological areas, a virtuous circle can be created where improved ecology encourages the actions that maintain and improve the area.

Other

Other examples include the poverty cycle, sharecropping, and the intensification of drought. In 2021, Austrian Chancellor Alexander Schallenberg described the recurring need for lockdowns in the COVID-19 pandemic as a vicious circle that could only be broken by a legally-required vaccination program.

Radiometric dating

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Radiometric_dating

Radiometric dating, radioactive dating or radioisotope dating is a technique which is used to date materials such as rocks or carbon, in which trace radioactive impurities were selectively incorporated when they were formed. The method compares the abundance of a naturally occurring radioactive isotope within the material to the abundance of its decay products, which form at a known constant rate of decay. The use of radiometric dating was first published in 1907 by Bertram Boltwood and is now the principal source of information about the absolute age of rocks and other geological features, including the age of fossilized life forms or the age of Earth itself, and can also be used to date a wide range of natural and man-made materials.

Together with stratigraphic principles, radiometric dating methods are used in geochronology to establish the geologic time scale. Among the best-known techniques are radiocarbon dating, potassium–argon dating and uranium–lead dating. By allowing the establishment of geological timescales, it provides a significant source of information about the ages of fossils and the deduced rates of evolutionary change. Radiometric dating is also used to date archaeological materials, including ancient artifacts.

Different methods of radiometric dating vary in the timescale over which they are accurate and the materials to which they can be applied.

Fundamentals

Radioactive decay

Example of a radioactive decay chain from lead-212 (²¹²Pb) to lead-208 (²⁰⁸Pb) . Each parent nuclide spontaneously decays into a daughter nuclide (the decay product) via an α decay or a β⁻ decay. The final decay product, lead-208 (²⁰⁸Pb), is stable and can no longer undergo spontaneous radioactive decay.

All ordinary matter is made up of combinations of chemical elements, each with its own atomic number, indicating the number of protons in the atomic nucleus. Additionally, elements may exist in different isotopes, with each isotope of an element differing in the number of neutrons in the nucleus. A particular isotope of a particular element is called a nuclide. Some nuclides are inherently unstable. That is, at some point in time, an atom of such a nuclide will undergo radioactive decay and spontaneously transform into a different nuclide. This transformation may be accomplished in a number of different ways, including alpha decay (emission of alpha particles) and beta decay (electron emission, positron emission, or electron capture). Another possibility is spontaneous fission into two or more nuclides.

While the moment in time at which a particular nucleus decays is unpredictable, a collection of atoms of a radioactive nuclide decays exponentially at a rate described by a parameter known as the half-life, usually given in units of years when discussing dating techniques. After one half-life has elapsed, one half of the atoms of the nuclide in question will have decayed into a "daughter" nuclide or decay product. In many cases, the daughter nuclide itself is radioactive, resulting in a decay chain, eventually ending with the formation of a stable (nonradioactive) daughter nuclide; each step in such a chain is characterized by a distinct half-life. In these cases, usually the half-life of interest in radiometric dating is the longest one in the chain, which is the rate-limiting factor in the ultimate transformation of the radioactive nuclide into its stable daughter. Isotopic systems that have been exploited for radiometric dating have half-lives ranging from only about 10 years (e.g., tritium) to over 100 billion years (e.g., samarium-147).

For most radioactive nuclides, the half-life depends solely on nuclear properties and is essentially constant. This is known because decay constants measured by different techniques give consistent values within analytical errors and the ages of the same materials are consistent from one method to another. It is not affected by external factors such as temperature, pressure, chemical environment, or presence of a magnetic or electric field. The only exceptions are nuclides that decay by the process of electron capture, such as beryllium-7, strontium-85, and zirconium-89, whose decay rate may be affected by local electron density. For all other nuclides, the proportion of the original nuclide to its decay products changes in a predictable way as the original nuclide decays over time. This predictability allows the relative abundances of related nuclides to be used as a clock to measure the time from the incorporation of the original nuclides into a material to the present.

Decay constant determination

See also: Radioactive decay law

The radioactive decay constant, the probability that an atom will decay per year, is the solid foundation of the common measurement of radioactivity. The accuracy and precision of the determination of an age (and a nuclide's half-life) depends on the accuracy and precision of the decay constant measurement. The in-growth method is one way of measuring the decay constant of a system, which involves accumulating daughter nuclides. Unfortunately for nuclides with high decay constants (which are useful for dating very old samples), long periods of time (decades) are required to accumulate enough decay products in a single sample to accurately measure them. A faster method involves using particle counters to determine alpha, beta or gamma activity, and then dividing that by the number of radioactive nuclides. However, it is challenging and expensive to accurately determine the number of radioactive nuclides. Alternatively, decay constants can be determined by comparing isotope data for rocks of known age. This method requires at least one of the isotope systems to be very precisely calibrated, such as the Pb–Pb system.

Accuracy of radiometric dating

Thermal ionization mass spectrometer used in radiometric dating.

The basic equation of radiometric dating requires that neither the parent nuclide nor the daughter product can enter or leave the material after its formation. The possible confounding effects of contamination of parent and daughter isotopes have to be considered, as do the effects of any loss or gain of such isotopes since the sample was created. It is therefore essential to have as much information as possible about the material being dated and to check for possible signs of alteration. Precision is enhanced if measurements are taken on multiple samples from different locations of the rock body. Alternatively, if several different minerals can be dated from the same sample and are assumed to be formed by the same event and were in equilibrium with the reservoir when they formed, they should form an isochron. This can reduce the problem of contamination. In uranium–lead dating, the concordia diagram is used which also decreases the problem of nuclide loss. Finally, correlation between different isotopic dating methods may be required to confirm the age of a sample. For example, the age of the Amitsoq gneisses from western Greenland was determined to be 3.60 ± 0.05 Ga (billion years ago) using uranium–lead dating and 3.56 ± 0.10 Ga (billion years ago) using lead–lead dating, results that are consistent with each other.

Accurate radiometric dating generally requires that the parent has a long enough half-life that it will be present in significant amounts at the time of measurement (except as described below under "Dating with short-lived extinct radionuclides"), the half-life of the parent is accurately known, and enough of the daughter product is produced to be accurately measured and distinguished from the initial amount of the daughter present in the material. The procedures used to isolate and analyze the parent and daughter nuclides must be precise and accurate. This normally involves isotope-ratio mass spectrometry.

The precision of a dating method depends in part on the half-life of the radioactive isotope involved. For instance, carbon-14 has a half-life of 5,730 years. After an organism has been dead for 60,000 years, so little carbon-14 is left that accurate dating cannot be established. On the other hand, the concentration of carbon-14 falls off so steeply that the age of relatively young remains can be determined precisely to within a few decades.

Closure temperature

Main article: Closure temperature

The closure temperature or blocking temperature represents the temperature below which the mineral is a closed system for the studied isotopes. If a material that selectively rejects the daughter nuclide is heated above this temperature, any daughter nuclides that have been accumulated over time will be lost through diffusion, resetting the isotopic "clock" to zero. As the mineral cools, the crystal structure begins to form and diffusion of isotopes is less easy. At a certain temperature, the crystal structure has formed sufficiently to prevent diffusion of isotopes. Thus an igneous or metamorphic rock or melt, which is slowly cooling, does not begin to exhibit measurable radioactive decay until it cools below the closure temperature. The age that can be calculated by radiometric dating is thus the time at which the rock or mineral cooled to closure temperature. This temperature varies for every mineral and isotopic system, so a system can be closed for one mineral but open for another. Dating of different minerals and/or isotope systems (with differing closure temperatures) within the same rock can therefore enable the tracking of the thermal history of the rock in question with time, and thus the history of metamorphic events may become known in detail. These temperatures are experimentally determined in the lab by artificially resetting sample minerals using a high-temperature furnace. This field is known as thermochronology or thermochronometry.

The age equation

Lu-Hf isochrons plotted of meteorite samples. The age is calculated from the slope of the isochron (line) and the original composition from the intercept of the isochron with the y-axis.

The mathematical expression that relates radioactive decay to geologic time is

D* = D₀ + N(t) (e^λt − 1)

where

• t is age of the sample,
• D* is number of atoms of the radiogenic daughter isotope in the sample,
• D₀ is number of atoms of the daughter isotope in the original or initial composition,
• N(t) is number of atoms of the parent isotope in the sample at time t (the present), given by N(t) = N₀e^−λt, and
• λ is the decay constant of the parent isotope, equal to the inverse of the radioactive half-life of the parent isotope times the natural logarithm of 2.

The equation is most conveniently expressed in terms of the measured quantity N(t) rather than the constant initial value N_o.

To calculate the age, it is assumed that the system is closed (neither parent nor daughter isotopes have been lost from system), D₀ either must be negligible or can be accurately estimated, λ is known to high precision, and one has accurate and precise measurements of D* and N(t).

The above equation makes use of information on the composition of parent and daughter isotopes at the time the material being tested cooled below its closure temperature. This is well established for most isotopic systems. However, construction of an isochron does not require information on the original compositions, using merely the present ratios of the parent and daughter isotopes to a standard isotope. An isochron plot is used to solve the age equation graphically and calculate the age of the sample and the original composition.

Modern dating methods

Radiometric dating has been carried out since 1905 when it was invented by Ernest Rutherford as a method by which one might determine the age of the Earth. In the century since then the techniques have been greatly improved and expanded. Dating can now be performed on samples as small as a nanogram using a mass spectrometer. The mass spectrometer was invented in the 1940s and began to be used in radiometric dating in the 1950s. It operates by generating a beam of ionized atoms from the sample under test. The ions then travel through a magnetic field, which diverts them into different sampling sensors, known as "Faraday cups," depending on their mass and level of ionization. On impact in the cups, the ions set up a very weak current that can be measured to determine the rate of impacts and the relative concentrations of different atoms in the beams.

Uranium–lead dating method

Main article: Uranium–lead dating

A concordia diagram as used in uranium–lead dating, with data from the Pfunze Belt, Zimbabwe. All the samples show loss of lead isotopes, but the intercept of the errorchron (straight line through the sample points) and the concordia (curve) shows the correct age of the rock.

Uranium–lead radiometric dating involves using uranium-235 or uranium-238 to date a substance's absolute age. This scheme has been refined to the point that the error margin in dates of rocks can be as low as less than two million years in two-and-a-half billion years. An error margin of 2–5% has been achieved on younger Mesozoic rocks.

Uranium–lead dating is often performed on the mineral zircon (ZrSiO₄), though it can be used on other materials, such as baddeleyite and monazite (see: monazite geochronology). Zircon and baddeleyite incorporate uranium atoms into their crystalline structure as substitutes for zirconium, but strongly reject lead. Zircon has a very high closure temperature, is resistant to mechanical weathering and is very chemically inert. Zircon also forms multiple crystal layers during metamorphic events, which each may record an isotopic age of the event. In situ micro-beam analysis can be achieved via laser ICP-MS or SIMS techniques.

One of its great advantages is that any sample provides two clocks, one based on uranium-235's decay to lead-207 with a half-life of about 700 million years, and one based on uranium-238's decay to lead-206 with a half-life of about 4.5 billion years, providing a built-in crosscheck that allows accurate determination of the age of the sample even if some of the lead has been lost. This can be seen in the concordia diagram, where the samples plot along an errorchron (straight line) which intersects the concordia curve at the age of the sample.

Samarium–neodymium dating method

Main article: Samarium–neodymium dating

This involves the alpha decay of ¹⁴⁷Sm to ¹⁴³Nd with a half-life of 1.06 x 10¹¹ years. Accuracy levels of within twenty million years in ages of two-and-a-half billion years are achievable.

Potassium–argon dating method

Main article: Potassium–argon dating

This involves electron capture or positron decay of potassium-40 to argon-40. Potassium-40 has a half-life of 1.3 billion years, so this method is applicable to the oldest rocks. Radioactive potassium-40 is common in micas, feldspars, and hornblendes, though the closure temperature is fairly low in these materials, about 350 °C (mica) to 500 °C (hornblende).^[

Rubidium–strontium dating method

Main article: Rubidium–strontium dating

This is based on the beta decay of rubidium-87 to strontium-87, with a half-life of 50 billion years. This scheme is used to date old igneous and metamorphic rocks, and has also been used to date lunar samples. Closure temperatures are so high that they are not a concern. Rubidium-strontium dating is not as precise as the uranium–lead method, with errors of 30 to 50 million years for a 3-billion-year-old sample. Application of in situ analysis (Laser-Ablation ICP-MS) within single mineral grains in faults have shown that the Rb-Sr method can be used to decipher episodes of fault movement.

Uranium–thorium dating method

Main article: Uranium–thorium dating

A relatively short-range dating technique is based on the decay of uranium-234 into thorium-230, a substance with a half-life of about 80,000 years. It is accompanied by a sister process, in which uranium-235 decays into protactinium-231, which has a half-life of 32,760 years.

While uranium is water-soluble, thorium and protactinium are not, and so they are selectively precipitated into ocean-floor sediments, from which their ratios are measured. The scheme has a range of several hundred thousand years. A related method is ionium–thorium dating, which measures the ratio of ionium (thorium-230) to thorium-232 in ocean sediment.

Radiocarbon dating method

Main article: Radiocarbon dating

Ale's Stones at Kåseberga, around ten kilometres south east of Ystad, Sweden were dated at 56 CE using the carbon-14 method on organic material found at the site.

Radiocarbon dating is also simply called carbon-14 dating. Carbon-14 is a radioactive isotope of carbon, with a half-life of 5,730 years (which is very short compared with the above isotopes), and decays into nitrogen. In other radiometric dating methods, the heavy parent isotopes were produced by nucleosynthesis in supernovas, meaning that any parent isotope with a short half-life should be extinct by now. Carbon-14, though, is continuously created through collisions of neutrons generated by cosmic rays with nitrogen in the upper atmosphere and thus remains at a near-constant level on Earth. The carbon-14 ends up as a trace component in atmospheric carbon dioxide (CO₂).

A carbon-based life form acquires carbon during its lifetime. Plants acquire it through photosynthesis, and animals acquire it from consumption of plants and other animals. When an organism dies, it ceases to take in new carbon-14, and the existing isotope decays with a characteristic half-life (5730 years). The proportion of carbon-14 left when the remains of the organism are examined provides an indication of the time elapsed since its death. This makes carbon-14 an ideal dating method to date the age of bones or the remains of an organism. The carbon-14 dating limit lies around 58,000 to 62,000 years.

The rate of creation of carbon-14 appears to be roughly constant, as cross-checks of carbon-14 dating with other dating methods show it gives consistent results. However, local eruptions of volcanoes or other events that give off large amounts of carbon dioxide can reduce local concentrations of carbon-14 and give inaccurate dates. The releases of carbon dioxide into the biosphere as a consequence of industrialization have also depressed the proportion of carbon-14 by a few percent; in contrast, the amount of carbon-14 was increased by above-ground nuclear bomb tests that were conducted into the early 1960s. Also, an increase in the solar wind or the Earth's magnetic field above the current value would depress the amount of carbon-14 created in the atmosphere.

Fission track dating method

Main article: fission track dating

Apatite crystals are widely used in fission track dating.

This involves inspection of a polished slice of a material to determine the density of "track" markings left in it by the spontaneous fission of uranium-238 impurities. The uranium content of the sample has to be known, but that can be determined by placing a plastic film over the polished slice of the material, and bombarding it with slow neutrons. This causes induced fission of ²³⁵U, as opposed to the spontaneous fission of ²³⁸U. The fission tracks produced by this process are recorded in the plastic film. The uranium content of the material can then be calculated from the number of tracks and the neutron flux.

This scheme has application over a wide range of geologic dates. For dates up to a few million years, tektites , and meteorites are best used. Older materials can be dated using zircon, apatite, titanite, epidote and garnet which have a variable amount of uranium content. Because the fission tracks are healed by temperatures over about 200 °C the technique has limitations as well as benefits. The technique has potential applications for detailing the thermal history of a deposit.

Chlorine-36 dating method

Large amounts of otherwise rare ³⁶Cl (half-life ~300ky) were produced by irradiation of seawater during atmospheric detonations of nuclear weapons between 1952 and 1958. The residence time of ³⁶Cl in the atmosphere is about 1 week. Thus, as an event marker of 1950s water in soil and ground water, ³⁶Cl is also useful for dating waters less than 50 years before the present. ³⁶Cl has seen use in other areas of the geological sciences, including dating ice and sediments.

Luminescence dating methods

Main article: Luminescence dating

Luminescence dating methods are not radiometric dating methods in that they do not rely on abundances of isotopes to calculate age. Instead, they are a consequence of background radiation on certain minerals. Over time, ionizing radiation is absorbed by mineral grains in sediments and archaeological materials such as quartz and potassium feldspar. The radiation causes charge to remain within the grains in structurally unstable "electron traps". Exposure to sunlight or heat releases these charges, effectively "bleaching" the sample and resetting the clock to zero. The trapped charge accumulates over time at a rate determined by the amount of background radiation at the location where the sample was buried. Stimulating these mineral grains using either light (optically stimulated luminescence or infrared stimulated luminescence dating) or heat (thermoluminescence dating) causes a luminescence signal to be emitted as the stored unstable electron energy is released, the intensity of which varies depending on the amount of radiation absorbed during burial and specific properties of the mineral.

These methods can be used to date the age of a sediment layer, as layers deposited on top would prevent the grains from being "bleached" and reset by sunlight. Pottery shards can be dated to the last time they experienced significant heat, generally when they were fired in a kiln.

Other methods

Other methods include:

• Argon–argon (Ar–Ar)
• Iodine–xenon (I–Xe)
• Lanthanum–barium (La–Ba)
• Lead–lead (Pb–Pb)
• Lutetium–hafnium (Lu–Hf)
• Hafnium–tungsten dating (Hf-W)
• Potassium–calcium (K–Ca)
• Rhenium–osmium (Re–Os)
• Uranium–uranium (U–U)
• Krypton–krypton (Kr–Kr)
• Beryllium (¹⁰Be–⁹Be)

Dating with decay products of short-lived extinct radionuclides

Absolute radiometric dating requires a measurable fraction of parent nucleus to remain in the sample rock. For rocks dating back to the beginning of the solar system, this requires extremely long-lived parent isotopes, making measurement of such rocks' exact ages imprecise. To be able to distinguish the relative ages of rocks from such old material, and to get a better time resolution than that available from long-lived isotopes, short-lived isotopes that are no longer present in the rock can be used.

At the beginning of the solar system, there were several relatively short-lived radionuclides like ²⁶Al, ⁶⁰Fe, ⁵³Mn, and ¹²⁹I present within the solar nebula. These radionuclides—possibly produced by the explosion of a supernova—are extinct today, but their decay products can be detected in very old material, such as that which constitutes meteorites. By measuring the decay products of extinct radionuclides with a mass spectrometer and using isochronplots, it is possible to determine relative ages of different events in the early history of the solar system. Dating methods based on extinct radionuclides can also be calibrated with the U–Pb method to give absolute ages. Thus both the approximate age and a high time resolution can be obtained. Generally a shorter half-life leads to a higher time resolution at the expense of timescale.

The ¹²⁹I – ¹²⁹Xe chronometer

See also: Iodine-129 § Meteorite age dating

¹²⁹
I
beta-decays to ¹²⁹
Xe
with a half-life of 16.14±0.12 million years. The iodine-xenon chronometer is an isochron technique. Samples are exposed to neutrons in a nuclear reactor. This converts the only stable isotope of iodine (¹²⁷
I
) into ¹²⁸
Xe
via neutron capture followed by beta decay (of ¹²⁸
I
). After irradiation, samples are heated in a series of steps and the xenon isotopic signature of the gas evolved in each step is analysed. When a consistent ¹²⁹
Xe
/¹²⁸
Xe
ratio is observed across several consecutive temperature steps, it can be interpreted as corresponding to a time at which the sample stopped losing xenon.

Samples of a meteorite called Shallowater are usually included in the irradiation to monitor the conversion efficiency from ¹²⁷
I
to ¹²⁸
Xe
. The difference between the measured ¹²⁹
Xe
/¹²⁸
Xe
ratios of the sample and Shallowater then corresponds to the different ratios of ¹²⁹
I
/¹²⁷
I
when they each stopped losing xenon. This in turn corresponds to a difference in age of closure in the early solar system.

The ²⁶Al – ²⁶Mg chronometer

Another example of short-lived extinct radionuclide dating is the ²⁶
Al
– ²⁶
Mg
chronometer, which can be used to estimate the relative ages of chondrules. ²⁶
Al
decays to ²⁶
Mg
with a half-life of 720 000 years. The dating is simply a question of finding the deviation from the natural abundance of ²⁶
Mg
(the product of ²⁶
Al
decay) in comparison with the ratio of the stable isotopes ²⁷
Al
/²⁴
Mg
.

The excess of ²⁶
Mg
(often designated ²⁶
Mg
*) is found by comparing the ²⁶
Mg
/²⁷
Mg
ratio to that of other Solar System materials.

The ²⁶
Al
– ²⁶
Mg
chronometer gives an estimate of the time period for formation of primitive meteorites of only a few million years (1.4 million years for Chondrule formation).

A terminology issue

In a July 2022 paper in the journal Applied Geochemistry, the authors proposed that the terms "parent isotope" and "daughter isotope" be avoided in favor of the more descriptive "precursor isotope" and "product isotope", analogous to "precursor ion" and "product ion" in mass spectrometry.