Alpha decay is one example type of radioactive decay, in which an atomic nucleus emits an alpha particle, and thereby transforms (or 'decays') into an atom with a mass number 4 less and atomic number 2 less. Many other types of decays are possible.

Radioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles (ionizing radiation). There are many different types of radioactive decay (see table below). A decay, or loss of energy, results when an atom with one type of nucleus, called the parent radionuclide, transforms to an atom with a nucleus in a different state, or to a different nucleus containing different numbers of protons and neutrons. Either of these products is named the daughter nuclide. In some decays the parent and daughter are different chemical elements, and thus the decay process results in nuclear transmutation (creation of an atom of a new element).

The first decay processes to be discovered were alpha decay, beta decay, and gamma decay. Alpha decay occurs when the nucleus ejects an alpha particle (helium nucleus). This is the most common process of emitting nucleons, but in rarer types of decays, nuclei can eject protons, or specific nuclei of other elements (in the process called cluster decay). Beta decay occurs when the nucleus emits an electron or positron and a type of neutrino, in a process that changes a proton to a neutron or the other way around. The nucleus may capture an orbiting electron, converting a proton into an neutron (electron capture). All of these processes result in nuclear transmutation.

By contrast, there exist radioactive decay processes that do not result in transmutation. The energy of an excited nucleus may be emitted as a gamma ray in gamma decay, or used to eject an orbital electron by interaction with the excited nucleus in a process called internal conversion. Radioisotopes occasionally emit neutrons, and this results in a change in an element from one isotope to another.

One type of radioactive decay results in products which are not defined, but appear in a range of "pieces" of the original nucleus. This decay is called spontaneous fission. This decay happens when a large unstable nucleus spontaneously splits into two (and occasionally three) smaller daughter nuclei, and usually emits gamma rays, neutrons, or other particles as a consequence.

Radioactive decay is a stochastic (i.e., random) process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a particular atom will decay.[1] However, the chance that a given atom will decay is constant over time. For a large number of atoms, the decay rate for the collection is computable from the measured decay constants of the nuclides (or equivalently from the half-lifes).

## Natural origin

Primordial nuclides found in the earth are residues from ancient supernova explosions which occurred before the formation of the solar system. They are the long-lived fraction of radionuclides surviving in the primordial solar nebula through planet accretion until the present. The naturally occurring short-lived radiogenic radionuclides found in rocks are the daughters of these radiactive primordial nuclides. Another minor source of naturally occurring radioactive nuclides are cosmogenic nuclides, formed by cosmic ray bombardment of material in the Earth's atmosphere or crust. For a summary table showing the number of stable nuclides and of radioactive nuclides in each category, see radionuclide. Radionuclides can also be produced artificially e.g. using particle accelerators or nuclear reactors.

## Decay phenomena

The trefoil symbol is used to indicate radioactive material.
The new radioactive danger symbol is used to indicate dangerous ionizing radioactive material.

The neutrons and protons that constitute nuclei, as well as other particles that approach close enough to them, are governed by several interactions. The strong nuclear force, not observed at the familiar macroscopic scale, is the most powerful force over subatomic distances. The electrostatic force is almost always significant, and, in the case of beta decay, the weak nuclear force is also involved.

The interplay of these forces produces a number of different phenomena in which energy may be released by rearrangement of particles in the nucleus, or else the change of one type of particle into others. These rearrangements and transformations may be hindered energetically, so that they do not occur immediately. In certain cases, random quantum vacuum fluctuations are theorized to promote relaxation to a lower energy state (the "decay") in a phenomenon known as quantum tunneling. Radioactive decay half-life of nuclides has been measured over timescales of 55 orders of magnitude, from 2.3 x 10-23 second (for hydrogen-7) to 6.9 x 1031 seconds (for tellurium-128)[2]. The limits of these timescales are set by the sensitivity of instrumentation only, and there are no known natural limits to how brief or long a decay half life for radioactive decay of a radionuclide may be.

The decay process, like all hindered energy transformations, may be analogized by a snowfield on a mountain. While friction between the ice crystals may be supporting the snow's weight, the system is inherently unstable with regard to a state of lower potential energy. A disturbance would thus facilitate the path to a state of greater entropy: The system will move towards the ground state, producing heat, and the total energy will be distributable over a larger number of quantum states. Thus, an avalanche results. The total energy does not change in this process, but, because of the law of entropy, avalanches happen only in one direction and that is toward the "ground state" — the state with the largest number of ways in which the available energy could be distributed.

Such a collapse (a decay event) requires a specific activation energy. For a snow avalanche, this energy comes as a disturbance from outside the system, although such disturbances can be arbitrarily small. In the case of an excited atomic nucleus, the arbitrarily small disturbance comes from quantum vacuum fluctuations. A radioactive nucleus (or any excited system in quantum mechanics) is unstable, and can, thus, spontaneously stabilize to a less-excited system. The resulting transformation alters the structure of the nucleus and results in the emission of either a photon or a high-velocity particle that has mass (such as an electron, alpha particle, or other type).

## Discovery

Radioactivity was discovered in 1896 by the French scientist Henri Becquerel, while working on phosphorescent materials. These materials glow in the dark after exposure to light, and he suspected that the glow produced in cathode ray tubes by X-rays might be associated with phosphorescence. He wrapped a photographic plate in black paper and placed various phosphorescent salts on it. All results were negative until he used uranium salts. The result with these compounds was a blackening of the plate. These radiations were called Becquerel Rays.

Pierre and Marie Curie in their Paris laboratory, before 1907

It soon became clear that the blackening of the plate had nothing to do with phosphorescence, because the plate blackened when the mineral was in the dark. Non-phosphorescent salts of uranium and metallic uranium also blackened the plate. It was clear that there is a form of radiation that could pass through paper that was causing the plate to become black.

At first it seemed that the new radiation was similar to the then recently discovered X-rays. Further research by Becquerel, Ernest Rutherford, Paul Villard, Pierre Curie, Marie Curie, and others discovered that this form of radioactivity was significantly more complicated. Different types of decay can occur, producing very different types of radiation. Rutherford was the first to realize that they all occur with the same mathematical exponential formula (see below), and Rutherford and his student Frederick Soddy were first to realize that many decay processes resulted in the transmutation of one element to another. Subsequently, the radioactive displacement law of Fajans and Soddy was formulated to describe the products of alpha and beta decay.

The early researchers also discovered that many other chemical elements besides uranium have radioactive isotopes. A systematic search for the total radioactivity in uranium ores also guided Marie Curie to isolate a new element polonium and to separate a new element radium from barium. The two elements' chemical similarity would otherwise have made them difficult to distinguish.

## Danger of radioactive substances

The danger classification sign of radioactive materials
Alpha particles may be completely stopped by a sheet of paper, beta particles by aluminum shielding. Gamma rays can only be reduced by much more substantial mass, such as a very thick layer of lead.
Different types of decay of a radionuclide. Vertical: atomic number Z, Horizontal: neutron number N

The dangers of radioactivity and radiation were not immediately recognized. Acute effects of radiation were first observed in the use of X-rays when electrical engineer and physicist Nikola Tesla intentionally subjected his fingers to X-rays in 1896.[3] He published his observations concerning the burns that developed, though he attributed them to ozone rather than to X-rays. His injuries later healed.

The genetic effects of radiation, including the effect of cancer risk, were recognized much later. In 1927, Hermann Joseph Muller published research showing genetic effects, and in 1946 was awarded the Nobel prize for his findings.

Before the biological effects of radiation were known, many physicians and corporations began marketing radioactive substances as patent medicine, glow-in-the-dark pigments. Examples were radium enema treatments, and radium-containing waters to be drunk as tonics. Marie Curie protested this sort of treatment, warning that the effects of radiation on the human body were not well understood. Curie later died from aplastic anemia, likely caused by exposure to ionizing radiation. By the 1930s, after a number of cases of bone necrosis and death of enthusiasts, radium-containing medicinal products had been largely removed from the market (radioactive quackery).

## Types of decay

Types of radioactive decay related to N and Z numbers

As for types of radioactive radiation, it was found that an electric or magnetic field could split such emissions into three types of beams. The rays were given the alphabetic names alpha, beta, and gamma, in order of their ability to penetrate matter. While alpha decay was seen only in heavier elements (atomic number 52, tellurium, and greater), the other two types of decay were seen in all of the elements. Spontaneous decay is evident in elements of atomic number ninety or greater.

In analyzing the nature of the decay products, it was obvious from the direction of electromagnetic forces induced upon the radiations by external magnetic and electric fields that alpha particles carried a positive charge, beta particles carried a negative charge, and gamma rays were neutral. From the magnitude of deflection, it was clear that alpha particles were much more massive than beta particles. Passing alpha particles through a very thin glass window and trapping them in a discharge tube allowed researchers to study the emission spectrum of the resulting gas, and ultimately prove that alpha particles are helium nuclei. Other experiments showed the similarity between classical beta radiation and cathode rays: They are both streams of electrons. Likewise gamma radiation and X-rays were found to be similar high-energy electromagnetic radiation.

The relationship between the types of decays also began to be examined: For example, gamma decay was almost always found associated with other types of decay, and occurred at about the same time, or afterward. Gamma decay as a separate phenomenon (with its own half-life, now termed isomeric transition), was found in natural radioactivity to be a result of the gamma decay of excited metastable nuclear isomers, which were in turn created from other types of decay.

Although alpha, beta, and gamma radiations were found most commonly, other types of decay were eventually discovered. Shortly after the discovery of the positron in cosmic ray products, it was realized that the same process that operates in classical beta decay can also produce positrons (positron emission). In an analogous process, instead of emitting positrons and neutrinos, some proton-rich nuclides were found to capture their own atomic electrons (electron capture), and emit only a neutrino (and usually also a gamma ray). Each of these types of decay involves the capture or emission of nuclear electrons or positrons, and acts to move a nucleus toward the ratio of neutrons to protons that has the least energy for a given total number of nucleons (neutrons plus protons).

A theoretical process of positron capture (analogous to electron capture) is possible in antimatter atoms, but has not been observed since the complex antimatter atoms are not available.[4] This would required antimatter atoms at least as complex as beryllium-7, which is the lightest known isotope of normal matter to undergo decay by electron capture.

Shortly after the discovery of the neutron in 1932, Enrico Fermi realized that certain rare decay reactions yield neutrons as a decay particle (neutron emission). Isolated proton emission was eventually observed in some elements. It was also found that some heavy elements may undergo spontaneous fission into products that vary in composition. In a phenomenon called cluster decay, specific combinations of neutrons and protons other than alpha particles (helium nuclei) were found to be spontaneously emitted from atoms.

Other types of radioactive decay that emit previously seen particles were found, but by different mechanisms. An example is internal conversion, which results in electron and sometimes high-energy photon emission, even though it involves neither beta nor gamma decay. This type of decay (like isomeric transition gamma decay) does not transmute one element to another.

Rare events that involve a combination of two beta-decay type events happening simultaneously (see below) are known. Any decay process that does not violate conservation of energy or momentum laws (and perhaps other particle conservation laws) is permitted to happen, although not all have been detected. An interesting example (discussed in a final section) is bound state beta decay of rhenium-187. In this process, an inverse of electron capture, beta electron-decay of the parent nuclide is not accompanied by beta electron emission, because the beta particle has been captured into the K-shell of the emitting atom. An antineutrino, however, is emitted.

## Decay modes in table form

Radionuclides can undergo a number of different reactions. These are summarized in the following table. A nucleus with mass number A and atomic number Z is represented as (A, Z). The column "Daughter nucleus" indicates the difference between the new nucleus and the original nucleus. Thus, (A − 1, Z) means that the mass number is one less than before, but the atomic number is the same as before.

Mode of decayParticipating particlesDaughter nucleus
Decays with emission of nucleons:
Alpha decayAn alpha particle (A = 4, Z = 2) emitted from nucleus(A − 4, Z − 2)
Proton emissionA proton ejected from nucleus(A − 1, Z − 1)
Neutron emissionA neutron ejected from nucleus(A − 1, Z)
Double proton emissionTwo protons ejected from nucleus simultaneously(A − 2, Z − 2)
Spontaneous fissionNucleus disintegrates into two or more smaller nuclei and other particles
Cluster decayNucleus emits a specific type of smaller nucleus (A1, Z1) smaller than, or larger than, an alpha particle(A − A1, Z − Z1) + (A1, Z1)
Different modes of beta decay:
β decayA nucleus emits an electron and an electron antineutrino(A, Z + 1)
Positron emission (β+ decay)A nucleus emits a positron and an electron neutrino(A, Z − 1)
Electron captureA nucleus captures an orbiting electron and emits a neutrino; the daughter nucleus is left in an excited unstable state(A, Z − 1)
Bound state beta decayA nucleus beta decays to electron and antineutrino, but the electron is not emitted, as it is captured into an empty K-shell; the daughter nucleus is left in an excited and unstable state. This process is suppressed except in ionized atoms that have K-shell vacancies.(A, Z + 1)
Double beta decayA nucleus emits two electrons and two antineutrinos(A, Z + 2)
Double electron captureA nucleus absorbs two orbital electrons and emits two neutrinos – the daughter nucleus is left in an excited and unstable state(A, Z − 2)
Electron capture with positron emissionA nucleus absorbs one orbital electron, emits one positron and two neutrinos(A, Z − 2)
Double positron emissionA nucleus emits two positrons and two neutrinos(A, Z − 2)
Transitions between states of the same nucleus:
Isomeric transitionExcited nucleus releases a high-energy photon (gamma ray)(A, Z)
Internal conversionExcited nucleus transfers energy to an orbital electron, which is subsequently ejected from the atom(A, Z)

Radioactive decay results in a reduction of summed rest mass, once the released energy (the disintegration energy) has escaped in some way (for example, the products might be captured and cooled, and the heat allowed to escape). Although decay energy is sometimes defined as associated with the difference between the mass of the parent nuclide products and the mass of the decay products, this is true only of rest mass measurements, where some energy has been removed from the product system. This is true because the decay energy must always carry mass with it, wherever it appears (see mass in special relativity) according to the formula E = mc2. The decay energy is initially released as the energy of emitted photons plus the kinetic energy of massive emitted particles (that is, particles that have rest mass). If these particles come to thermal equilibrium with their surroundings and photons are absorbed, then the decay energy is transformed to thermal energy, which retains its mass.

Decay energy therefore remains associated with a certain measure of mass of the decay system invariant mass. The energy of photons, kinetic energy of emitted particles, and, later, the thermal energy of the surrounding matter, all contribute to calculations of invariant mass of systems. Thus, while the sum of rest masses of particles is not conserved in radioactive decay, the system mass and system invariant mass (and also the system total energy) is conserved throughout any decay process.

## Decay chains and multiple modes

The daughter nuclide of a decay event may also be unstable (radioactive). In this case, it will also decay, producing radiation. The resulting second daughter nuclide may also be radioactive. This can lead to a sequence of several decay events. Eventually, a stable nuclide is produced. This is called a decay chain (see this article for specific details of important natural decay chains).

Gamma-ray energy spectrum of uranium ore (inset). Gamma-rays are emitted by decaying nuclides, and the gamma-ray energy can be used to characterize the decay (which nuclide is decaying to which). Here, using the gamma-ray spectrum, several nuclides that are typical of the decay chain of 238U have been identified: 226Ra, 214Pb, 214Bi.

An example is the natural decay chain of 238U, which is as follows:

• decays, through alpha-emission, with a half-life of 4.5 billion years to thorium-234
• which decays, through beta-emission, with a half-life of 24 days to protactinium-234
• which decays, through beta-emission, with a half-life of 1.2 minutes to uranium-234
• which decays, through alpha-emission, with a half-life of 240 thousand years to thorium-230
• which decays, through alpha-emission, with a half-life of 77 thousand years to radium-226
• which decays, through alpha-emission, with a half-life of 1.6 thousand years to radon-222
• which decays, through alpha-emission, with a half-life of 3.8 days to polonium-218
• which decays, through alpha-emission, with a half-life of 3.1 minutes to lead-214
• which decays, through beta-emission, with a half-life of 27 minutes to bismuth-214
• which decays, through beta-emission, with a half-life of 20 minutes to polonium-214
• which decays, through alpha-emission, with a half-life of 160 microseconds to lead-210
• which decays, through beta-emission, with a half-life of 22 years to bismuth-210
• which decays, through beta-emission, with a half-life of 5 days to polonium-210
• which decays, through alpha-emission, with a half-life of 140 days to lead-206, which is a stable nuclide.

Some radionuclides may have several different paths of decay. For example, approximately 36% of bismuth-212 decays, through alpha-emission, to thallium-208 while approximately 64% of bismuth-212 decays, through beta-emission, to polonium-212. Both thallium-208 and polonium-212 are radioactive daughter products of bismuth-212, and both decay directly to stable lead-208.

## Occurrence and applications

According to the Big Bang theory, stable isotopes of the lightest five elements (H, He, and traces of Li, Be, and B) were produced very shortly after the emergence of the universe, in a process called Big Bang nucleosynthesis. These lightest stable nuclides (including deuterium) survive to today, but any radioactive isotopes of the light elements produced in the Big Bang (such as tritium) have long since decayed. Isotopes of elements heavier than boron were not produced at all in the Big Bang, and these first five elements do not have any long-lived radioisotopes. Thus, all radioactive nuclei are, therefore, relatively young with respect to the birth of the universe, having formed later in various other types of nucleosynthesis in stars (in particular, supernovae), and also during ongoing interactions between stable isotopes and energetic particles. For example, carbon-14, a radioactive nuclide with a half-life of only 5730 years, is constantly produced in Earth's upper atmosphere due to interactions between cosmic rays and nitrogen.

Nuclides that are produced by radioactive decay are called radiogenic nuclides, whether they themselves are stable or not. There exist stable radiogenic nuclides that were formed from short-lived extinct radionuclides in the early solar system.[5][6] The extra presence of these stable radiogenic nuclides (such as Xe-129 from primordial I-129) against the background of primordial stable nuclides can be inferred by various means.

Radioactive decay has been put to use in the technique of radioisotopic labeling, which is used to track the passage of a chemical substance through a complex system (such as a living organism). A sample of the substance is synthesized with a high concentration of unstable atoms. The presence of the substance in one or another part of the system is determined by detecting the locations of decay events.

On the premise that radioactive decay is truly random (rather than merely chaotic), it has been used in hardware random-number generators. Because the process is not thought to vary significantly in mechanism over time, it is also a valuable tool in estimating the absolute ages of certain materials. For geological materials, the radioisotopes and some of their decay products become trapped when a rock solidifies, and can then later be used (subject to many well-known qualifications) to estimate the date of the solidification. These include checking the results of several simultaneous processes and their products against each other, within the same sample. In a similar fashion, and also subject to qualification, the rate of formation of carbon-14 in various eras, the date of formation of organic matter within a certain period related to the isotope's half-life may be estimated, because the carbon-14 becomes trapped when the organic matter grows and incorporates the new carbon-14 from the air. Thereafter, the amount of carbon-14 in organic matter decreases according to decay processes that may also be independently cross-checked by other means (such as checking the carbon-14 in individual tree rings, for example).

## Radioactive decay rates

The decay rate, or activity, of a radioactive substance are characterized by:

Constant quantities:

• The half-lifet1/2, is the time taken for the activity of a given amount of a radioactive substance to decay to half of its initial value; see List of nuclides.
• The mean lifetimeτ, "tau" the average lifetime of a radioactive particle before decay.
• The decay constantλ, "lambda" the inverse of the mean lifetime.

Although these are constants, they are associated with statistically random behavior of populations of atoms. In consequence predictions using these constants are less accurate for small number of atoms.

In principle the reciprocal of any number greater than one— a half-life, a third-life, or even a (1/2)-life—can be used in exactly the same way as half-life; but the half-life t1/2 is adopted as the standard time associated with exponential decay.

Time-variable quantities:

• Total activityA, is number of decays per unit time of a radioactive sample.
• Number of particlesN, is the total number of particles in the sample.
• Specific activitySA, number of decays per unit time per amount of substance of the sample at time set to zero (t = 0). "Amount of substance" can be the mass, volume or moles of the initial sample.

These are related as follows:

$t_{1/2} = \frac{\ln(2)}{\lambda} = \tau \ln(2)$
$A = - \frac{\mathrm{d}N}{\mathrm{d}t} = \lambda N$
$S_A a_0 = - \frac{\mathrm{d}N}{\mathrm{d}t}\bigg|_{t=0} = \lambda N_0$

where N0 is the initial amount of active substance — substance that has the same percentage of unstable particles as when the substance was formed.

### Units of radioactivity measurements

The SI unit of radioactive activity is the becquerel (Bq), in honor of the scientist Henri Becquerel. One Bq is defined as one transformation (or decay or disintegration) per second. Since sensible sizes of radioactive material contains many atoms, a Bq is a tiny measure of activity; amounts giving activities on the order of GBq (gigabecquerel, 1 x 109 decays per second) or TBq (terabecquerel, 1 x 1012 decays per second) are commonly used.

Another unit of radioactivity is the curie, Ci, which was originally defined as the amount of radium emanation (radon-222) in equilibrium with one gram of pure radium, isotope Ra-226. At present it is equal, by definition, to the activity of any radionuclide decaying with a disintegration rate of 3.7 × 1010 Bq, so that 1 curie (Ci) = 3.7 × 1010 Bq. The use of Ci is currently discouraged by the SI. Low activities are also measured in disintegrations per minute (dpm).

## Mathematics of radioactive decay

### Universal law of radioactive decay

Radioactivity is one very frequent example of exponential decay. The law describes the statistical behavior of a large number of nuclides, rather than individual ones. In the following formalism, the number of nuclides or nuclide population N, is of course a discrete variable (a natural number) - but for any physical sample N is so large (amounts of L = 1023, avagadro's constant) that it can be treated as a continuous variable. Differential calculus is needed to set up differential equations for modelling the behaviour of the nuclear decay.

#### One-decay process

Consider the case of a nuclide A decaying into another B by some process A → B (emission of other particles, like electron neutrinos ν
e
and electrons e in beta decay, are irrelevant in what follows). The decay of an unstable nucleus is entirely random and it is impossible to predict when a particular atom will decay.[1] However, it is equally likely to decay at any time. Therefore, given a sample of a particular radioisotope, the number of decay events −dN expected to occur in a small interval of time dt is proportional to the number of atoms present N, that is

$-\frac{\mathrm{d}N}{\mathrm{d}t} \propto N.$

Particular radionuclides decay at different rates, so each has its own decay constant λ. The expected decay −dN/N is proportional to an increment of time, dt:

 $-\frac{\mathrm{d}N}{N} = \lambda \mathrm{d}t$

The negative sign indicates that N decreases as time increases, as each decay event follows one after another. The solution to this first-order differential equation is the function:

$N(t) = N_0\,e^{-{\lambda}t} = N_0\,e^{-t/ \tau}, \,\!$

where N0 is the value of N at time t = 0.

This equation is of particular interest; the behaviour of numerous important quantities can be found from it (see below). Although the parent decay distribution follows an exponential, observations of decay times will be limited by a finite integer number of N atoms and follow Poisson statistics as a consequence of the random nature of the process.

We have for all time t:

$N_A + N_B = N_\mathrm{total} = N_{A0},$

where Ntotal is the constant number of particles throughout the decay process, clearly equal to the initial number of A nuclides since this is the initial substance.

If the number of non-decayed A nuclei is:

$N_A = N_{A0}e^{-{\lambda}t} \,\!$

then the number of nuclei of B, i.e. number of decayed A nuclei, is

$N_B = N_{A0} - N_A = N_{A0} - N_{A0}e^{-{\lambda}t} = N_{A0} \left ( 1 - e^{-{\lambda}t} \right ).$

#### Chain-decay processes

Chain of two decays

Now consider the case of a chain of two decays: one nuclide A decaying into another B by one process, then B decaying into another C by a second process, i.e. A → B → C. The previous equation cannot be applied to a decay chain, but can be generalized as follows. The decay rate of B is proportional to the number of nuclides of B present, so again we have:

$-\frac{\mathrm{d}N_B}{\mathrm{d}t} \propto N_B,$

but care must be taken. Since A decays into B, then B decays into C, the activity of A adds to the total number of B nuclides in the present sample, before those B nuclides decay and reduce the number of nuclides leading to the later sample. In other words, the number of second generation nuclei B increases as a result of the first generation nuclei decay of A, and decreases as a result of its own decay into the third generation nuclei C.[7] The proportionality becomes an equation:

$\frac{\mathrm{d}N_B}{\mathrm{d}t} = - \lambda_B N_B,$

adding the increasing (and correcting) term obtains the law for a decay chain for two nuclides:

$\frac{\mathrm{d}N_B}{\mathrm{d}t} = -\lambda_B N_B + \lambda_A N_A.$

The equation is not

$\frac{\mathrm{d}N_B}{\mathrm{d}t} = - \left ( \lambda_B N_B + \lambda_A N_A \right ) ,$

since this implies the number of atoms of B is only decreasing as time increases, which is not the case. The rate of change of NB, that is dNB/dt, is related to the changes in the amounts of A and B, NB can increase as B is produced from A and decrease as B produces C.

Re-writing using the previous results:

 $\frac{\mathrm{d}N_B}{\mathrm{d}t} = - \lambda_B N_B + \lambda_A N_{A0} e^{-\lambda_A t}$

The subscripts simply refer to the respective nuclides, i.e. NA is the number of nuclides of type A, NA0 is the initial number of nuclides of type A, λA is the decay constant for A - and similarly for nuclide B. Solving this equation for NB gives:

$N_B = \frac{N_{A0}\lambda_A}{\lambda_B - \lambda_A} \left ( e^{-\lambda_A t} - e^{-\lambda_B t}\right ) .$

Naturally this equation reduces to the previous solution, in the case B is a stable nuclide (λB = 0):

$\lim_{\lambda_B\rightarrow 0} \left [ \frac{N_{A0}\lambda_A}{\lambda_B - \lambda_A} \left ( e^{-\lambda_A t} - e^{-\lambda_B t}\right ) \right ] = \frac{N_{A0}\lambda_A}{0 - \lambda_A} \left ( e^{-\lambda_A t} - 1 \right ) = N_{A0} \left ( 1- e^{-\lambda_A t} \right ),$

as shown above for one decay. The solution can be found by the integration factor method, where the integrating factor is eλBt. This case is perhaps the most useful, since it can derive both the one-decay equation (above) and the equation for multi-decay chains (below) more directly.

Chain of any number of decays

For the general case of any number of consecutive decays in a decay chain, i.e. A1 → A2 ··· → Ai ··· → AD, where D is the number of decays and i is a dummy index (i = 1, 2, 3, ...D), each nuclide population can be found in terms of the previous population. In this case N2 = 0, N3 = 0,..., ND = 0. Using the above result in a recursive form:

$\frac{\mathrm{d}N_j}{\mathrm{d}t} = - \lambda_j N_j + \lambda_{j-1} N_{(j-1)0} e^{-\lambda_{j-1} t}.$

The general solution to the recursive problem are given by Bateman's equations[8]:

 Bateman's equations$N_D = \frac{N_1(0)}{\lambda_D} \sum_{i=1}^D \lambda_i c_i e^{-\lambda_i t}$$c_i = \prod_{j=1, i\neq j}^D \frac{\lambda_j}{\lambda_j - \lambda_i}$

#### Alternative decay modes

In all of the above examples, the initial nuclide decays into only one product.[9] Consider the case of one initial nuclide which can decay into two products, that is A → B + C. We have for all time t:

$N_A = N_B + N_C$

which is constant since the total number of nuclides remains constant. Differentiating with respect to time:

\begin{align} \frac{\mathrm{d}N_A}{\mathrm{d}t} & = \frac{\mathrm{d}N_B}{\mathrm{d}t} + \frac{\mathrm{d}N_C}{\mathrm{d}t} \\ - \lambda_A N_A & = - N_A \left ( \lambda_B + \lambda_C \right ) \\ \end{align}

indicating that the total decay constant λA is the sum of partial decay constants λB and λC:

$\lambda_A = \lambda_B + \lambda_C .$

Solving this equation for NA:

$N_A = N_{A0} e^{-\lambda_A t} .$

where NA0 is the initial number of nuclide A. When measuring the production of one nuclide, one can only observe the total decay constant λA. The decay constants λB and λC determine the probability for the decay to result in products B or C as follows:

$N_B = \frac{\lambda_B}{\lambda_A} N_{A0} \left ( 1 - e^{-\lambda_A t} \right ),$
$N_C = \frac{\lambda_C}{\lambda_A} N_{A0} \left ( 1 - e^{-\lambda_A t} \right ).$

because the fraction λB/λA of nuclei decay into B while the fraction λC/λA of nuclei decay into C.

### Corollaries of the decay laws

The above equations can also be written using quantities related to the number of nuclide particles N in a sample;

where L = 6.023×1023 is Avogadro's constant, Ar is the relative atomic mass number, and the amount of the substance is in moles.

### Decay timing: definitions and relations

#### Time constant and mean-life

For the one-decay solution A → B:

$N = N_0\,e^{-{\lambda}t} = N_0\,e^{-t/ \tau}, \,\!$

the equation indicates that the decay constant λ has units of t-1, and can thus also be represented as 1/τ, where τ is a characteristic time of the process called the time constant.

In a radioactive decay process, this time constant is also the mean lifetime for decaying atoms. Each atom "lives" for a finite amount of time before it decays, and it may be shown that this mean lifetime is the arithmetic mean of all the atoms' lifetimes, and that it is τ, which again is related to the decay constant as follows:

$\tau = \frac{1}{\lambda}.$

This form is also true for two-decay processes simultaneously A → B + C, inserting the equivalent values of decay constants (as given above)

$\lambda = \lambda_B + \lambda_C \,$

into the decay solution leads to:

$\frac{1}{\tau} = \lambda = \lambda_B + \lambda_C = \frac{1}{\tau_B} + \frac{1}{\tau_C}\,$
Simulation of many identical atoms undergoing radioactive decay, starting with either 4 atoms (left) or 400 (right). The number at the top indicates how many half-lives have elapsed. Note the law of large numbers: with more atoms, the overall decay is less random.

#### Half-life

A more commonly used parameter is the half-life. Given a sample of a particular radionuclide, the half-life is the time taken for half the radionuclide's atoms to decay. For the case of one-decay nuclear reactions:

$N = N_0\,e^{-{\lambda}t} = N_0\,e^{-t/ \tau}, \,\!$

the half-life is related to the decay constant as follows: set N = N0/2 and t = T1/2 to obtain

$t_{1/2} = \frac{\ln 2}{\lambda} = \tau \ln 2.$

This relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent, while those that radiate weakly endure longer. Half-lives of known radionuclides vary widely, from more than 1019 years, such as for the very nearly stable nuclide 209Bi, to 10−23 seconds for highly unstable ones.

The factor of ln(2) in the above relations results from the fact that concept of "half-life" is merely a way of selecting a different base other than the natural base e for the lifetime expression. The time constant τ is the e -1 -life, the time until only 1/e remains, about 36.8%, rather than the 50% in the half-life of a radionuclide. Thus, τ is longer than t1/2. The following equation can be shown to be valid:

$N(t) = N_0\,e^{-t/ \tau} =N_0\,2^{-t/t_{1/2}}. \,\!$

Since radioactive decay is exponential with a constant probability, each process could as easily be described with a different constant time period that (for example) gave its "(1/3)-life" (how long until only 1/3 is left) or "(1/10)-life" (a time period until only 10% is left), and so on. Thus, the choice of τ and t1/2 for marker-times, are only for convenience, and from convention. They reflect a fundamental principle only in so much as they show that the same proportion of a given radioactive substance will decay, during any time-period that one chooses.

Mathematically, the nth life for the above situation would be found in the same way as above—by setting N = N0/n, and substituting into the decay solution to obtain

$t_{1/n} = \frac{\ln n}{\lambda} = \tau \ln n.$

### Example

A sample of 14C, whose half-life is 5730 years, has a decay rate of 14 disintegration per minute (dpm) per gram of natural carbon. An artefact is found to have radioactivity of 4 dpm per gram of its present C, how old is the artefact?

Using the above equation, we have:

$N = N_0\,e^{-t/ \tau},$

where: $\frac{N}{ N_0} = 4/14 \approx 0.286,$

$\tau = \frac{T_{1/2}}{\ln 2} \approx 8267$ years,
$t = -\tau\,\ln\frac{N}{ N_0} \approx 10360$ years.

## Changing decay rates

The radioactive decay modes of electron capture and internal conversion are known to be slightly sensitive to chemical and environmental effects which change the electronic structure of the atom, which in turn affects the presence of 1s and 2s electrons that participate in the decay process. A small number of mostly light nuclides are affected. For example, chemical bonds can affect the rate of electron capture to a small degree (in general, less than 1%) depending on the proximity of electrons to the nucleus in beryllium. In 7Be, a difference of 0.9% has been observed between half-lives in metallic and insulating environments.[10] This relatively large effect is because beryllium is a small atom whose valence electrons are in 2s atomic orbitals, which are subject to electron capture in 7Be because (like all s atomic orbitals in all atoms) they naturally penetrate into the nucleus.

Rhenium-187 is a more spectacular example. 187Re normally beta decays to 187Os with a half-life of 41.6 × 109 y,[11] but studies using fully ionised 187Re atoms (bare nuclei) have found that this can decrease to only 33 y. This is attributed to "bound-state β- decay" of the fully ionised atom – the electron is emitted into the "K-shell" (1s atomic orbital), which cannot occur for neutral atoms in which all low-lying bound states are occupied.[12]

A number of experiments have found that decay rates of other modes of artificial and naturally occurring radioisotopes are, to a high degree of precision, unaffected by external conditions such as temperature, pressure, the chemical environment, and electric, magnetic, or gravitational fields.[13] Comparison of laboratory experiments over the last century, studies of the Oklo natural nuclear reactor (which exemplified the effects of thermal neutrons on nuclear decay), and astrophysical observations of the luminosity decays of distant supernovae (which occurred far away so the light has taken a great deal of time to reach us), for example, strongly indicate that decay rates have been constant (at least to within the limitations of small experimental errors) as a function of time as well.

Recent results suggest the possibility that decay rates might have a weak dependence (0.5% or less) on environmental factors. It has been suggested that measurements of decay rates of silicon-32, manganese-54, and radium-226 exhibit small seasonal variations (of the order of 0.1%), proposed to be related to either solar flare activity or distance from the Sun.[14][15][16] However, such measurements are highly susceptible to systematic errors, and a subsequent paper[17] has found no evidence for such correlations in six other isotopes, and sets upper limits on the size of any such effects.

## Notes

1. ^ a b "Decay and Half Life". Retrieved 2009-12-14.
2. ^ NUBASE evaluation of nuclear and decay properties
3. ^ Hrabak, M. et al (July 2008). "Nikola Tesla and the Discovery of X-rays". RadioGraphics 28 (4): 1189–92. doi:10.1148/rg.284075206. PMID 18635636.
4. ^ http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch23/modes.php#fission
5. ^ Clayton, Donald D. (1983). Principles of Stellar Evolution and Nucleosynthesis (2nd ed.). University of Chicago Press. pp. 75. ISBN 0-226-10953-4.
6. ^ Bolt, B. A.; Packard, R. E.; Price, P. B. (2007). "John H. Reynolds, Physics: Berkeley". The University of California, Berkeley. Retrieved 2007-10-01.
7. ^ Introductory Nuclear Physics, K.S. Krane, 1988, John Wiley & Sons Inc, ISBN 978-0-471-80553-3
8. ^ Cetnar, Jerzy (May 2006). "General solution of Bateman equations for nuclear transmutations". Annals of Nuclear Energy 33 (7): 640-645.
9. ^ K.S. Krane (1988). Introductory Nuclear Physics. John Wiley & Sons Inc. p. 164. ISBN 978-0-471-80553-3.
10. ^ B.Wang et al., Euro. Phys. J. A 28, 375-377 (2006) Change of the 7Be electron capture half-life in metallic environments
11. ^ Smoliar, M.I.; Walker, R.J.; Morgan, J.W. (1996). "Re-Os ages of group IIA, IIIA, IVA, and IVB iron meteorites". Science 271 (5252): 1099–1102. Bibcode 1996Sci...271.1099S. doi:10.1126/science.271.5252.1099.
12. ^ Bosch, F.; Faestermann, T.; Friese, J.; Heine, F.; Kienle, P.; Wefers, E.; Zeitelhack, K.; Beckert, K. et al. (1996). "Observation of bound-state β– decay of fully ionized 187Re:187Re-187Os Cosmochronometry". Physical Review Letters 77 (26): 5190–5193. Bibcode 1996PhRvL..77.5190B. doi:10.1103/PhysRevLett.77.5190. PMID 10062738.
13. ^ Emery, G.T. (1972). "﻿Perturbation of Nuclear Decay Rates". Annual Review of Nuclear Science (ACS Publications) 22: 165-202. Retrieved 6 August 2012.
14. ^ "The mystery of varying nuclear decay". Physics World. October 2, 2008.
15. ^ Jenkins, Jere H.; Fischbach, Ephraim (2009). "Perturbation of Nuclear Decay Rates During the Solar Flare of 13 December 2006". Astroparticle Physics 31 (6): 407–411. arXiv:0808.3156. Bibcode 2009APh....31..407J. doi:10.1016/j.astropartphys.2009.04.005.
16. ^ Jenkins, J. H.; et al., Ephraim; Buncher, John B.; Gruenwald, John T.; Krause, Dennis E.; Mattes, Joshua J. (2009). "Evidence of correlations between nuclear decay rates and Earth–Sun distance". Astroparticle Physics 32 (1): 42–46. arXiv:0808.3283. Bibcode 2009APh....32...42J. doi:10.1016/j.astropartphys.2009.05.004.
17. ^ Norman, E. B.; et al., Edgardo; Shugart, Howard A.; Joshi, Tenzing H.; Firestone, Richard B. (2009). "Evidence against correlations between nuclear decay rates and Earth–Sun distance". Astroparticle Physics 31 (2): 135–137. arXiv:0810.3265. Bibcode 2009APh....31..135N. doi:10.1016/j.astropartphys.2008.12.004.