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| {{about||particle decay in a more general context|Particle decay|more information on hazards of various kinds of radiation from decay|Ionizing radiation}}
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| {{Redirect|Radioactive}}
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| {{Redirect|Radioactivity}}
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| {{Refimprove|date=April 2010}}
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| [[File:Alpha Decay.svg|thumb|[[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]] decreased by 4 and [[atomic number]] decreased by 2. Many other types of decays are possible.]]
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| {{Nuclear physics}}
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| '''Radioactive decay''', also known as '''nuclear decay''' or '''radioactivity''', is the process by which a [[atomic nucleus|nucleus]] of an unstable [[atom]] loses energy by emitting particles of [[ionizing radiation]]. A material that spontaneously emits this kind of radiation—which includes the emission of energetic [[alpha particle]]s, [[beta particle]]s, and [[gamma ray]]s—is considered '''radioactive'''.
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| Radioactive decay is a [[stochastic]] (i.e., random) process at the level of single atoms, in that, according to [[quantum mechanics|quantum theory]], it is impossible to predict when a particular atom will decay.<ref name="not-predict">{{cite web|url=http://www.iem-inc.com/prhlfr.html|title=Decay and Half Life|accessdate= 2009-12-14}}</ref> 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 constant]]s of the nuclides (or equivalently from the [[half-life]]s).
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| 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]]'' (or ''parent radioisotope''<ref group=note>Radionuclide is the more correct term but radioisotope is used also. The difference between isotope and nuclide is explained at [[Isotope#Isotope vs. nuclide]].</ref>), transforms to an atom with a nucleus in a different state, or to a different nucleus containing different numbers of [[proton]]s and [[neutron]]s. Either of these products is named the ''daughter nuclide''. In some decays the parent and daughter are different [[chemical element]]s, and thus the decay process results in [[nuclear transmutation]] (creation of an atom of a different element).
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| 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 [[nucleon]]s, but in rarer types of decays, nuclei can eject [[proton]]s, 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 a neutron ([[electron capture]]). All of these processes result in nuclear transmutation.
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| 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]]. Highly excited neutron-rich radioisotopes (formed as the product of other types of decay) occasionally lose energy by [[neutron emission|emitting neutrons]], and this results in a change in an element from one [[isotope]] to another. Another 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 [[nuclear fission|fission]]. This decay happens when a large unstable nucleus spontaneously splits into two (and occasionally three) smaller daughter nuclei, and generally immediately emits gamma rays, neutrons, or other particles as a consequence.
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| For a summary table showing the number of stable nuclides and of radioactive nuclides in each category, see [[radionuclide]]. There exist 34 mildly radioactive elements on Earth that are primordial nuclides, still decaying from the formation of the solar system (well known examples are uranium and thorium). Another 50 or so radionuclides can be detected in decay chains resulting from the primordial nuclides (such as radium and radon), and also new cosmogenic processes (for example carbon-14). Radionuclides can also be [[Synthetic element|produced artificially]] e.g. using [[particle accelerators]] or [[nuclear reactors]], with about 650 of these characterized with half-lives over an hour, and several thousand more characterized with even shorter half lives. See [[list of nuclides]] for a list by half life.
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| ==Discovery and history==
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| [[File:Pierre and Marie Curie.jpg|thumb|200px|Pierre and Marie Curie in their Paris laboratory, before 1907]]
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| It has been claimed that radioactivity was discovered by [[Abel Niepce de Saint-Victor]] in 1857. But it was not well publicised and was soon forgotten.{{cn|date=September 2013}}
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| Radioactivity was rediscovered in 1896 by the [[France|French]] scientist [[Henri Becquerel]], while working on [[Phosphorescence|phosphorescent]] materials. These materials glow in the dark after exposure to light, and he suspected that the glow produced in [[cathode ray tube]]s by [[X-ray]]s might be associated with phosphorescence. He wrapped a photographic plate in black paper and placed various phosphorescent [[salt (chemistry)|salt]]s on it. All results were negative until he used [[Uranium|uranium salts]]. The result with these compounds was a blackening of the plate. These radiations were called Becquerel Rays.
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| 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.
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| 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 in accordance 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 [[nuclear transmutation|transmutation]] of one element to another. Subsequently, the [[radioactive displacement law of Fajans and Soddy]] was formulated to describe the products of [[alpha decay|alpha]] and [[beta decay]].
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| The early researchers also discovered that many other [[chemical element]]s besides uranium have [[Radionuclide|radioactive isotope]]s. A systematic search for the total radioactivity in uranium ores also guided [[Pierre Curie]] and [[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.
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| The discovery of radioactive elements in the 1890s opened the way for new [[Radioactive dating|dating techniques]] that suggested an [[Age of the Earth|age for Earth]] of several billion years.
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| ==Types of decay==
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| As for types of radioactive radiation, it was found that an [[Electric field|electric]] or [[magnetic field]] could split such emissions into three types of beams. The rays were given the [[Greek alphabet|alphabetic]] names [[Alpha particle|alpha]], [[Beta particle|beta]], and [[Gamma ray|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. Lead ([[atomic number]] 82) is the heaviest element to have any isotopes stable (to the limit of measurement) to radioactive decay. Radioactive decay is seen in all isotopes of all elements of atomic number 83 ([[bismuth]]) or greater (bismuth, however, is only very slightly radioactive).
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| [[File:Radioactive decay modes.svg|201px|thumb|Transition diagram for decay modes of a [[radionuclide]], with neutron number ''N'' and [[atomic number]] ''Z'' (shown are [[alpha particle|α]], [[electron|β<sup>±</sup>]], [[proton|p<sup>+</sup>]], and [[neutron|n<sup>0</sup>]] emissions, EC denotes [[electron capture]]).]]
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| [[File:Table isotopes en.svg|thumb|right|Types of radioactive decay related to N and Z numbers]]
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| In analyzing the nature of the decay products, it was obvious from the direction of [[electromagnetic force]]s induced upon the radiations by external magnetic and electric fields that [[alpha particle]]s from decay carried a positive charge, [[beta decay|beta particles]] carried a negative charge, and [[gamma ray]]s 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 [[neon lamp|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 ray]]s: They are both streams of high-speed [[electrons]]. Likewise, gamma radiation and X-rays were found to be similar high-energy [[electromagnetic radiation]].
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| 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 isomer]]s, which were, in turn, created from other types of decay.
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| 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 [[nucleon]]s (neutrons plus protons).
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| 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.<ref>[http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch23/modes.php#fission Radioactive Decay<!-- Bot generated title -->]</ref> This would require antimatter atoms at least as complex as beryllium-7, which is the lightest known isotope of normal matter to undergo decay by electron capture.
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| Shortly after the discovery of the [[neutron]] in 1932, [[Enrico Fermi]] realized that certain rare beta-decay reactions immediately 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.
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| 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 (a neutrino is not emitted, and neither the electron nor photon originate in the nucleus). Internal conversion decay (like [[isomeric transition]] gamma decay and neutron emission) involves an excited nuclide releasing energy, without transmutation of one element into another.
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| 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 [[Beta decay#Bound-state β- decay|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.
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| ==Danger of radioactive substances==
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| {{main|Ionizing radiation}}
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| [[File:Dangclass7.svg|thumb|right|150px|The danger classification [[Signage|sign]] of radioactive materials]]
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| [[File:Alfa beta gamma radiation.svg|150px|thumb|[[Alpha particle]]s may be completely stopped by a sheet of paper, [[beta particle]]s by aluminum shielding. [[Gamma ray]]s can only be reduced by much more substantial mass, such as a very thick layer of [[lead]].]]
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| The dangers of radioactivity and radiation were not immediately recognized. The discovery of x‑rays in 1895 led to wide spread experimentation by scientists, physicians, and inventors. Many people began recounting stories of burns, hair loss and worse in technical journals as early as 1896. In February of that year, Professor Daniel and Dr. Dudley of [[Vanderbilt University]] performed an experiment involving x-raying Dudley's head that resulted in him losing hair under where the tube was placed (reported in the ''The X-rays Science'' news supplement). A report by Dr. H.D. Hawks, a graduate of Columbia College, of his suffering severe hand and chest burns in an x-ray demonstration, was the first of many other reports in ''Electrical Review''.<ref name="SansareKhanna2011">{{cite journal |last1=Sansare |first1=K. |last2=Khanna |first2=V. |last3=Karjodkar |first3=F. |title=Early victims of X-rays: a tribute and current perception |journal=Dentomaxillofacial Radiology |volume=40 |issue=2 |year=2011 |pages=123–125 |issn=0250-832X |doi=10.1259/dmfr/73488299 |pmc=3520298}}</ref>
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| Many experimenters including [[Elihu Thomson]] at [[Thomas Edison]]'s lab, [[William J. Morton]], and [[Nikola Tesla]] also reported burns. Elihu Thomson
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| deliberately exposed a finger to an x-ray tube over a period of time and suffered pain, swelling, and blistering.<ref>[http://www.physics.isu.edu/radinf/50yrs.htm Ronald L. Kathern and Paul L. Ziemer, he First Fifty Years of Radiation Protection, physics.isu.edu]</ref> Other effects were sometimes blamed for the damage including ultraviolet rays and (according to Tesla) ozone.<ref>{{Cite journal|title=Nikola Tesla and the Discovery of X-rays|author=Hrabak, M. et al|journal=RadioGraphics|date=July 2008 |volume=28|issue=4|pmid=18635636|pages=1189–92|doi=10.1148/rg.284075206|last2=Padovan|first2=R. S.|last3=Kralik|first3=M.|last4=Ozretic|first4=D.|last5=Potocki|first5=K.}}</ref> Many physicians claimed there were no effects from x-ray exposure at all.<ref>[http://www.physics.isu.edu/radinf/50yrs.htm Ronald L. Kathern and Paul L. Ziemer, he First Fifty Years of Radiation Protection, physics.isu.edu]</ref>
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| 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.
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| Before the biological effects of radiation were known, many physicians and corporations began marketing radioactive substances as [[patent medicine]] in the form of 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]]).
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| ==Radioactive decay rates==
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| <!-- "Neutron activation analysis" links here -->
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| The ''decay rate'', or ''activity'', of a radioactive substance is characterized by:
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| '''Constant quantities''':
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| *The ''[[half-life]]''{{mdash}}{{math|<big>''t''</big><sub>1/2</sub>}}, 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]].
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| *The ''[[decay constant]]''{{mdash}} {{bigmath|''λ''}}, "[[lambda]]" the inverse of the mean lifetime.
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| *The ''[[mean lifetime]]''{{mdash}} {{bigmath|''τ''}}, "[[tau]]" the average lifetime of a radioactive particle before decay.
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| 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.
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| In principle the reciprocal of any number greater than one{{mdash}} a half-life, a third-life, or even a (1/{{sqrt|2}})-life{{mdash}}can be used in exactly the same way as half-life;
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| but the half-life {{math|<big>''t''</big><sub>1/2</sub>}} is adopted as the standard time associated with exponential decay.
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| '''Time-variable quantities''':
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| *''Total activity''{{mdash}}{{bigmath| ''A''}}, is number of decays per unit time of a radioactive sample.
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| *''Number of particles''{{mdash}}{{bigmath|''N''}}, is the total [[number of particles]] in the sample.
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| *''Specific activity''{{mdash}}{{bigmath|''S<sub>A</sub>''}}, 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.
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| These are related as follows:
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| :<math> t_{1/2} = \frac{\ln(2)}{\lambda} = \tau \ln(2)</math>
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| :<math> A = - \frac{\mathrm{d}N}{\mathrm{d}t} = \lambda N </math>
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| :<math> S_A a_0 = - \frac{\mathrm{d}N}{\mathrm{d}t}\bigg|_{t=0} = \lambda N_0 </math>
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| where ''N''<sub>0</sub> is the initial amount of active substance — substance that has the same percentage of unstable particles as when the substance was formed.
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| ===Units of radioactivity measurements===
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| The [[International System of Units|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 10<sup>9</sup> decays per second) or TBq (terabecquerel, 1 x 10<sup>12</sup> decays per second) are commonly used.
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| 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 × 10<sup>10</sup> Bq, so that 1 [[curie]] (Ci) = 3.7 × 10<sup>10</sup> Bq. The use of Ci is currently discouraged by the SI. Low activities are also measured in '''disintegrations per minute''' (dpm).
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| ==Mathematics of radioactive decay==
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| {{hatnote|For the mathematical details of exponential decay in general context, see [[exponential decay]].}}
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| {{hatnote|For related derivations with some further details, see [[half-life]].}}
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| {{hatnote|For the analogous mathematics in 1st order chemical reactions, see [[Rate equation#Consecutive reactions|Consecutive reactions]].}}
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| ===Universal law of radioactive decay===
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| 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'' = 10<sup>23,</sup> [[Avogadro'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.
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| ====One-decay process====
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| Consider the case of a nuclide {{math|''A''}} decaying into another {{math|''B''}} by some process {{math|''A → B''}} (emission of other particles, like [[electron neutrino]]s {{SubatomicParticle|Electron antineutrino}} and [[electron]]s e<sup>–</sup> 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.<ref name="not-predict"/> However, it is equally likely to decay at any time. Therefore, given a sample of a particular radioisotope, the number of decay events {{math|−d''N''}} expected to occur in a small interval of time {{math|d''t''}} is proportional to the number of atoms present {{math|''N''}}, that is<ref name=Patel>{{cite book|last=Patel|first=S.B.|title=Nuclear physics : an introduction|year=2000|publisher=New Age International|location=New Delhi|isbn=9788122401257|pages=62–72}}</ref>
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| :<math>-\frac{\mathrm{d}N}{\mathrm{d}t} \propto N.</math>
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| Particular radionuclides decay at different rates, so each has its own decay constant {{bigmath|λ}}. The expected decay {{math|−d''N''/''N''}} is proportional to an increment of time, {{math|d''t''}}:
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| {{Equation box 1
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| |indent=:
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| |title=
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| |equation=<math> -\frac{\mathrm{d}N}{N} = \lambda \mathrm{d}t</math>
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| |cellpadding
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| |border
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| The negative sign indicates that {{math|''N''}} decreases as time increases, as each decay event follows one after another. The solution to this first-order [[differential equation]] is the [[mathematical function|function]]:
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| :<math>N(t) = N_0\,e^{-{\lambda}t} = N_0\,e^{-t/ \tau}, \,\!</math>
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| where {{math|''N''<sub>0</sub>}} is the value of {{math|''N''}} at time {{math|''t''}} = 0.<ref name=Patel/>
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| We have for all time {{math|''t''}}:
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| :<math> N_A + N_B = N_\mathrm{total} = N_{A0}, </math>
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| where {{math|''N''{{sub|total}}}} is the constant number of particles throughout the decay process, clearly equal to the initial number of {{math|''A''}} nuclides since this is the initial substance.
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| If the number of non-decayed {{math|''A''}} nuclei is:
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| :<math>N_A = N_{A0}e^{-{\lambda}t} \,\!</math>
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| then the number of nuclei of {{math|''B''}}, i.e. number of decayed {{math|''A''}} nuclei, is
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| :<math> N_B = N_{A0} - N_A = N_{A0} - N_{A0}e^{-{\lambda}t} = N_{A0} \left ( 1 - e^{-{\lambda}t} \right ). </math>
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| The number of decays observed over a given interval obeys [[Poisson statistics]]. If the average number of decays is {{math|''<N>''}}, the probability of a given number of decays {{math|''N''}} is<ref name=Patel/>
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| :<math> P(N) = \frac{\langle N \rangle^N \exp(-\langle N\rangle)}{N!} .</math>
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| ====Chain-decay processes====
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| '''Chain of two decays'''
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| Now consider the case of a chain of two decays: one nuclide {{math|''A''}} decaying into another {{math|''B''}} by one process, then {{math|''B''}} decaying into another {{math|''C''}} by a second process, i.e. ''{{math|A → B → C}}''. The previous equation cannot be applied to a decay chain, but can be generalized as follows. Since {{math|''A''}} decays into {{math|''B''}}, ''then'' {{math|''B''}} decays into {{math|''C''}}, the activity of {{math|''A''}} adds to the total number of {{math|''B''}} nuclides in the present sample, ''before'' those {{math|''B''}} nuclides decay and reduce the number of nuclides leading to the later sample. In other words, the number of second generation nuclei {{math|''B''}} increases as a result of the first generation nuclei decay of {{math|''A''}}, and decreases as a result of its own decay into the third generation nuclei {{math|''C''}}.<ref name="Introductory Nuclear Physics 1988">Introductory Nuclear Physics, K.S. Krane, 1988, John Wiley & Sons Inc, ISBN 978-0-471-80553-3</ref> The sum of these two terms gives the law for a decay chain for two nuclides:
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| :<math>\frac{\mathrm{d}N_B}{\mathrm{d}t} = -\lambda_B N_B + \lambda_A N_A.</math>
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| The rate of change of {{math|''N<sub>B</sub>''}}, that is {{math|d''N{{sub|B}}''/d''t''}}, is related to the changes in the amounts of {{math|''A''}} and {{math|''B''}}, {{math|''N<sub>B</sub>''}} can increase as {{math|''B''}} is produced from {{math|''A''}} and decrease as {{math|''B''}} produces {{math|''C''}}.
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| Re-writing using the previous results:
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| {{Equation box 1
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| |indent=:
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| |title=
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| |equation=<math> \frac{\mathrm{d}N_B}{\mathrm{d}t} = - \lambda_B N_B + \lambda_A N_{A0} e^{-\lambda_A t} </math>
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| |cellpadding
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| |border
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| The subscripts simply refer to the respective nuclides, i.e. {{math|''N{{sub|A}}''}} is the number of nuclides of type {{math|''A''}}, {{math|''N''{{sub|''A''0}}}} is the initial number of nuclides of type {{math|''A''}}, {{math|''λ{{sub|A}}''}} is the decay constant for {{math|''A''}} - and similarly for nuclide {{math|''B''}}. Solving this equation for {{math|''N<sub>B</sub>''}} gives:
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| :<math> N_B = \frac{N_{A0}\lambda_A}{\lambda_B - \lambda_A} \left ( e^{-\lambda_A t} - e^{-\lambda_B t}\right ) . </math>
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| Naturally this equation reduces to the previous solution, in the case {{math|''B''}} is a stable nuclide ({{math|''λ{{sub|B}}''}} = 0):
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| :<math> \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 ), </math>
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| as shown above for one decay. The solution can be found by the [[integrating factor|integration factor]] method, where the integrating factor is {{math|''e''{{sup|λ{{sub|''B''}}''t''}}}}. 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.
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| '''Chain of any number of decays'''
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| For the general case of any number of consecutive decays in a decay chain, i.e. {{math|A<sub>1</sub> → A<sub>2</sub> ··· → A<sub>''i''</sub> ··· → A<sub>''D''</sub>}}, where {{math|''D''}} is the number of decays and {{math|''i''}} is a dummy index ({{math|''i'' {{=}} {{sub|1, 2, 3, ...''D''}}}}), each nuclide population can be found in terms of the previous population. In this case {{math|''N''<sub>2</sub> {{=}} 0}}, {{math|''N''<sub>3</sub> {{=}} 0}},..., {{math|''N<sub>D</sub>'' {{=}} 0}}. Using the above result in a recursive form:
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| :<math> \frac{\mathrm{d}N_j}{\mathrm{d}t} = - \lambda_j N_j + \lambda_{j-1} N_{(j-1)0} e^{-\lambda_{j-1} t}. </math>
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| The general solution to the recursive problem are given by '''''Bateman's equations''''':<ref name="general solution of Bateman">{{cite journal|last=Cetnar|first=Jerzy|title=General solution of Bateman equations for nuclear transmutations|journal=Annals of Nuclear Energy|date=May 2006|year=2006|volume=33|issue=7|pages=640–645|url=http://www.sciencedirect.com/science/article/pii/S0306454906000284|doi=10.1016/j.anucene.2006.02.004}}</ref>
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| {{Equation box 1
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| |indent=:
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| |title='''Bateman's equations'''
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| |equation=<math> N_D = \frac{N_1(0)}{\lambda_D} \sum_{i=1}^D \lambda_i c_i e^{-\lambda_i t} </math>
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| <math> c_i = \prod_{j=1, i\neq j}^D \frac{\lambda_j}{\lambda_j - \lambda_i} </math>
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| |cellpadding
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| |border
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| |border colour = #0073CF
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| |background colour=#F5FFFA}}
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| ====Alternative decay modes====
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| In all of the above examples, the initial nuclide decays into only one product.<ref>{{cite book|title=Introductory Nuclear Physics|author=K.S. Krane|year=1988|publisher=John Wiley & Sons Inc|page=164|isbn=978-0-471-80553-3}}</ref> Consider the case of one initial nuclide which can decay into either of two products, that is ''{{math|A → B}}'' and ''{{math|A → C}}'' in parallel. For example in a sample of [[potassium-40]], 89.3% of the nuclei decay to [[calcium-40]] and 10.7% to [[argon-40]]. We have for all time {{math|''t''}}:
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| :<math> N = N_A + N_B + N_C </math>
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| which is constant since the total number of nuclides remains constant. Differentiating with respect to time:
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| :<math> \begin{align}
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| \frac{\mathrm{d}N_A}{\mathrm{d}t} & = - \left(\frac{\mathrm{d}N_B}{\mathrm{d}t} + \frac{\mathrm{d}N_C}{\mathrm{d}t} \right) \\
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| - \lambda N_A & = - N_A \left ( \lambda_B + \lambda_C \right ) \\
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| \end{align}</math>
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| defining the ''total decay constant'' {{math|λ}} in terms of the sum of ''partial decay constants'' {{math|λ<sub>''B''</sub>}} and {{math|λ<sub>''C''</sub>}}:
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| :<math> \lambda = \lambda_B + \lambda_C . </math>
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| Notice that
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| :<math> \frac{\mathrm{d}N_A}{\mathrm{d}t} < 0,\frac{\mathrm{d}N_B}{\mathrm{d}t} > 0,
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| \frac{\mathrm{d}N_C}{\mathrm{d}t} > 0.
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| </math>
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| Solving this equation for {{math|''N{{sub|A}}''}}:
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| :<math> N_A = N_{A0} e^{-\lambda t} .</math>
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| where {{math|''N''<sub>''A''0</sub>}} is the initial number of nuclide A. When measuring the production of one nuclide, one can only observe the total decay constant {{math|''λ''}}. The decay constants {{math|''λ{{sub|B}}''}} and {{math|''λ{{sub|C}}''}} determine the probability for the decay to result in products {{math|''B''}} or {{math|''C''}} as follows:
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| :<math> N_B = \frac{\lambda_B}{\lambda} N_{A0} \left ( 1 - e^{-\lambda t} \right ),</math>
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| :<math> N_C = \frac{\lambda_C}{\lambda} N_{A0} \left ( 1 - e^{-\lambda t} \right ).</math>
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| because the fraction {{math|''λ{{sub|B}}''/''λ''}} of nuclei decay into {{math|''B''}} while the fraction {{math|''λ{{sub|C}}''/''λ''}} of nuclei decay into {{math|''C''}}.
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| ===Corollaries of the decay laws===
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| The above equations can also be written using quantities related to the number of nuclide particles {{math|''N''}} in a sample;
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| * The activity: ''{{math|A {{=}} λN.}}''
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| * The [[amount of substance]]: ''{{math|n {{=}} N/L}}''.
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| * The [[mass]]: ''{{math|1= M = A{{sub|r}}n = A{{sub|r}}N/L}}''.
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| where ''L'' = {{val|6.022|e=23}} is [[Avogadro's constant]], ''{{math|A{{sub|r}}}}'' is the relative atomic mass number, and the amount of the substance is in [[Mole (unit)|mole]]s.
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| ===Decay timing: definitions and relations===
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| ====Time constant and mean-life====
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| For the one-decay solution ''{{math|A → B}}'':
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| :<math>N = N_0\,e^{-{\lambda}t} = N_0\,e^{-t/ \tau}, \,\!</math>
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| the equation indicates that the [[decay constant]] {{math|''λ''}} has units of ''{{math|t<sup>-1</sup>}}'', and can thus also be represented as 1/{{bigmath|''τ''}}, where {{bigmath|''τ''}} is a characteristic time of the process called the ''[[time constant]]''.
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| 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 {{math|''τ''}}, which again is related to the decay constant as follows:
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| :<math>\tau = \frac{1}{\lambda}.</math>
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| This form is also true for two-decay processes simultaneously ''{{math|A → B + C}}'', inserting the equivalent values of decay constants (as given above)
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| :<math> \lambda = \lambda_B + \lambda_C \,</math>
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| into the decay solution leads to:
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| :<math>\frac{1}{\tau} = \lambda = \lambda_B + \lambda_C = \frac{1}{\tau_B} + \frac{1}{\tau_C}\,</math>
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| [[File:Halflife-sim.gif|thumb|right|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-life|half-lives]] have elapsed. Note the [[law of large numbers]]: with more atoms, the overall decay is less random.]]
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| ====Half-life====
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| 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:
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| :<math>N = N_0\,e^{-{\lambda}t} = N_0\,e^{-t/ \tau}, \,\!</math>
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| the half-life is related to the decay constant as follows: set ''{{math|1=N = N{{sub|0}}/2}}'' and {{math|''t''}} = {{math|''T''<sub>1/2</sub>}} to obtain
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| :<math>t_{1/2} = \frac{\ln 2}{\lambda} = \tau \ln 2. </math>
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| 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 [[1 E19 s and more|10<sup>19</sup> years]], such as for the very nearly stable nuclide <sup>209</sup>Bi, to 10<sup>−23</sup> seconds for highly unstable ones.
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| The factor of {{math|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 {{math|''e''}} for the lifetime expression. The time constant {{bigmath|''τ''}} is the {{subSup|''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, {{bigmath|''τ''}} is longer than {{math|''t''{{sub|1/2}}}}. The following equation can be shown to be valid:
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| :<math>N(t) = N_0\,e^{-t/ \tau} =N_0\,2^{-t/t_{1/2}}. \,\!</math>
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| 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 {{bigmath|''τ''}} and ''{{math|t{{sub|1/2}}}}'' 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.
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| Mathematically, the {{math|''n''{{sup|th}}}} life for the above situation would be found in the same way as above{{mdash}}by setting ''{{math|1=N = N{{sub|0}}/n}}'', {{math|''t'' = ''T''{{sub|1/''n''}}}} and substituting into the decay solution to obtain
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| :<math>t_{1/n} = \frac{\ln n}{\lambda} = \tau \ln n. </math>
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| ===Example===
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| A sample of <sup>14</sup>C, 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?
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| Using the above equation, we have:
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| :<math> N = N_0\,e^{-t/ \tau}, </math>
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| where: <math> \frac{N}{ N_0} = 4/14 \approx 0.286, </math>
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| :<math> \tau = \frac{T_{1/2}}{\ln 2} \approx 8267 </math> years,
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| :<math> t = -\tau\,\ln\frac{N}{ N_0} \approx 10360</math> years.
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| ==Changing decay rates==
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| 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 <sup>7</sup>Be, a difference of 0.9% has been observed between half-lives in metallic and insulating environments.<ref>[http://www.springerlink.com/content/6159nj734576136u/ B.Wang et al., Euro. Phys. J. A 28, 375-377 (2006) Change of the <sup>7</sup>Be electron capture half-life in metallic environments]</ref> This relatively large effect is because beryllium is a small atom whose valence electrons are in '''2s''' [[atomic orbital]]s, which are subject to electron capture in <sup>7</sup>Be because (like all '''s''' atomic orbitals in all atoms) they naturally penetrate into the nucleus.
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| In 1992, Jung et al. of the Darmstadt Heavy-Ion Research group observed an accelerated β decay of <sup>163</sup>Dy<sup>66+</sup>. Although neutral <sup>163</sup>Dy is a stable isotope, the fully ionized <sup>163</sup>Dy<sup>66+</sup> undergoes β decay into the K and L shells with a half-life of 47 days.<ref>M. Jung et al., Phys. Rev. Lett. 69, 2164 (1992) First observation of bound-state beta minus decay.</ref>
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| [[Rhenium-187]] is another spectacular example. <sup>187</sup>Re normally [[beta decay]]s to <sup>187</sup>Os with a [[half-life]] of 41.6 × 10<sup>9</sup> years,<ref>{{cite journal | doi=10.1126/science.271.5252.1099 | last1=Smoliar | first1=M.I. | last2=Walker | first2=R.J. | last3=Morgan | first3=J.W. | year=1996 | title=Re-Os ages of group IIA, IIIA, IVA, and IVB iron meteorites | journal=Science | volume=271 | pages=1099–1102 |bibcode = 1996Sci...271.1099S | issue=5252}}</ref> but studies using fully ionised <sup>187</sup>[[rhenium|Re]] atoms (bare nuclei) have found that this can decrease to only 33 years. This is attributed to "[[Beta decay#Bound-state β- decay|bound-state β<sup>−</sup> 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.<ref name=Bosch1996>{{cite journal | doi = 10.1103/PhysRevLett.77.5190 | title = Observation of bound-state β– decay of fully ionized <sup>187</sup>Re:<sup>187</sup>Re-<sup>187</sup>Os Cosmochronometry | last1=Bosch | first1=F. | year = 1996 | journal = Physical Review Letters | volume = 77 | issue = 26 | pages = 5190–5193 | pmid = 10062738 | last2 = Faestermann | first2 = T. | last3 = Friese | first3 = J. | last4 = Heine | first4 = F. | last5 = Kienle | first5 = P. | last6 = Wefers | first6 = E. | last7 = Zeitelhack | first7 = K. | last8 = Beckert | first8 = K. | last9 = Franzke | first9 = B. | bibcode=1996PhRvL..77.5190B}}</ref>
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| [[File:DecayRate vs Solar Time.png|thumb|right|160px|Decay Rate of Radon-222 as a function of date and time of day. The color-bar gives the power of the observed signal and represents ~4% seasonal decay rate variation.]]
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| 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.<ref>{{cite journal |last1= Emery|first1= G.T.|year= 1972|title= Perturbation of Nuclear Decay Rates|journal= Annual Review of Nuclear Science|volume= 22|pages= 165–202|publisher= ACS Publications|url= http://www.whoi.edu/cms/files/1972AnRevNucSci22p165_68424.pdf|accessdate= 6 August 2012 |doi= 10.1146/annurev.ns.22.120172.001121|bibcode = 1972ARNPS..22..165E }}</ref> Comparison of laboratory experiments over the last century, studies of the Oklo [[Natural nuclear fission reactor|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.
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| Recent results suggest the possibility that decay rates might have a weak dependence 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%),<ref>{{cite web |title=The mystery of varying nuclear decay |work=Physics World |date=October 2, 2008 |url=http://physicsworld.com/cws/article/news/36108 }}</ref><ref>{{cite journal |title=Perturbation of Nuclear Decay Rates During the Solar Flare of 13 December 2006 |journal=Astroparticle Physics |volume=31 |issue=6 |year=2009 |pages=407–411 |arxiv=0808.3156 |bibcode = 2009APh....31..407J |doi = 10.1016/j.astropartphys.2009.04.005 |last1=Jenkins |first1=Jere H. |last2=Fischbach |first2=Ephraim }}</ref><ref>{{cite journal |first=J. H. |last=Jenkins |last2=''et al.'' |title=Evidence of correlations between nuclear decay rates and Earth–Sun distance |journal=Astroparticle Physics |volume=32 |issue=1 |pages=42–46 |year=2009 |arxiv=0808.3283 |bibcode = 2009APh....32...42J |doi = 10.1016/j.astropartphys.2009.05.004 |first2=Ephraim |last3=Buncher |first3=John B. |last4=Gruenwald |first4=John T. |last5=Krause |first5=Dennis E. |last6=Mattes |first6=Joshua J. }}</ref> while the decay of [[Radon-222]] exhibit large 4% peak-to-peak seasonal variations,<ref>[http://arxiv.org/pdf/1205.0205v1.pdf Peter A. Sturrock, Gideon Steinitz, Ephraim Fischbach, Daniel Javorsek, II, Jere H. Jenkins, Analysis of Gamma Radiation from a Radon Source: Indications of a Solar Influence], Accessed on line September 2, 2012.</ref> proposed to be related to either [[solar flare]] activity or distance from the Sun. However, such measurements are highly susceptible to systematic errors, and a subsequent paper<ref>{{cite journal |first=E. B. |last=Norman |last2=''et al.'' |title=Evidence against correlations between nuclear decay rates and Earth–Sun distance |journal=Astroparticle Physics |volume=31 |issue=2 |year=2009 |pages=135–137 |url=http://donuts.berkeley.edu/papers/EarthSun.pdf |bibcode = 2009APh....31..135N |doi = 10.1016/j.astropartphys.2008.12.004 |first2=Edgardo |last3=Shugart |first3=Howard A. |last4=Joshi |first4=Tenzing H. |last5=Firestone |first5=Richard B. |arxiv = 0810.3265 }}</ref> has found no evidence for such correlations in seven other isotopes (<sup>22</sup>Na, <sup>44</sup>Ti, <sup>108</sup>Ag, <sup>121</sup>Sn, <sup>133</sup>Ba, <sup>241</sup>Am, <sup>238</sup>Pu), and sets upper limits on the size of any such effects.
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| ==Theoretical basis of decay phenomena==
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| [[File:Radioactive.svg|thumb|right|150px|The [[Hazard symbol#Radioactive sign|trefoil symbol]] is used to indicate radioactive material.<!-- The [[Unicode]] encoding of this symbol is U+2622 ({{unicode|☢}}) -->]]
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| [[File:New radiation symbol ISO 21482.svg|thumb|right|150px|The [[Ionizing radiation|radioactive danger symbol]] is used to indicate dangerous ionizing radioactive material, in use since 2007.]]
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| The [[neutron]]s and [[proton]]s that constitute nuclei, as well as other particles that approach close enough to them, are governed by several interactions. The [[Nuclear force|strong nuclear force]], not observed at the familiar [[macroscopic]] scale, is the most powerful force over subatomic distances. The [[Coulomb's law|electrostatic force]] is almost always significant, and, in the case of [[beta decay]], the [[Weak interaction|weak nuclear force]] is also involved.
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| 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 fluctuation|quantum vacuum fluctuation]]s 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<sup>−23</sup> seconds (for [[hydrogen-7]]) to 6.9 x 10<sup>31</sup> seconds (for [[tellurium-128]]).<ref>[http://amdc.in2p3.fr/nubase/Nubase2003.pdf NUBASE evaluation of nuclear and decay properties]</ref> 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.
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| 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 [[second law of thermodynamics]], avalanches have only been observed 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.
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| 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 fluctuation|quantum vacuum fluctuation]]s. 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).
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| ==Occurrence and applications==
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| According to the [[Big Bang theory]], stable isotopes of the lightest five elements ([[hydrogen|H]], [[helium|He]], and traces of [[lithium|Li]], [[beryllium|Be]], and [[boron|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 [[star]]s (in particular, [[supernova]]e), 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.
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| Nuclides that are produced by radioactive decay are called [[radiogenic nuclide]]s, whether they themselves are [[Stable isotope|stable]] or not. There exist stable radiogenic nuclides that were formed from short-lived [[extinct radionuclide]]s in the early solar system.<ref>{{cite book
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| | first=Donald D. | last=Clayton | year=1983
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| | title=Principles of Stellar Evolution and Nucleosynthesis
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| | page= 75 | edition=2nd
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| | publisher=University of Chicago Press
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| | isbn=0-226-10953-4 }}
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| </ref><ref>{{cite web
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| | author=Bolt, B. A.; Packard, R. E.; Price, P. B. | year=2007
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| | url=http://content.cdlib.org/xtf/view?docId=hb1r29n709&doc.view=content&chunk.id=div00061&toc.depth=1&brand=oac&anchor.id=0
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| | title=John H. Reynolds, Physics: Berkeley
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| | publisher=The University of California, Berkeley
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| | accessdate=2007-10-01 }}</ref> The extra presence of these stable radiogenic nuclides (such as Xe-129 from primordial I-129) against the background of primordial [[stable nuclide]]s can be inferred by various means.
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| 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.
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| On the premise that radioactive decay is truly [[random]] (rather than merely [[chaos theory|chaotic]]), it has been used in [[hardware random-number generator]]s. 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).
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| ==Origins of radioactive nuclides==
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| {{Main|nucleosynthesis}}
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| Radioactive [[primordial nuclide]]s found in the [[Earth]] are residues from ancient [[supernova nucleosynthesis|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 (astrophysics)|accretion]] until the present. The naturally occurring short-lived [[radiogenic]] [[radionuclide]]s found in [[rocks]] are the daughters of these radioactive [[primordial nuclide]]s. Another minor source of naturally occurring radioactive nuclides are [[cosmogenic nuclide]]s, formed by cosmic ray bombardment of material in the Earth's [[atmosphere]] or [[crust (geology)|crust]]. The radioactive decay of these radionuclides in rocks within Earth's [[mantle (geology)|mantle]] and [[crust (geology)|crust]] contribute significantly to [[Earth's internal heat budget]].
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| ==Decay modes in table form==
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| 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.
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| {|class="wikitable"
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| |- style="background:#eee0e0; white-space:nowrap;"
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| !Mode of decay !! Participating particles!!Daughter nucleus
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| |-
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| | style="background:#ccc;" colspan="3"|'''Decays with emission of nucleons:'''
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| |-
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| |[[Alpha decay]] || An [[alpha particle]] (''A'' = 4, ''Z'' = 2) emitted from nucleus || (''A'' − 4, ''Z'' − 2)
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| |-
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| |[[Proton emission]] || A [[proton]] ejected from nucleus || (''A'' − 1, ''Z'' − 1)
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| |-
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| |[[Neutron emission]] || A [[neutron]] ejected from nucleus || (''A'' − 1, ''Z'')
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| |-
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| |[[Proton emission|Double proton emission]] || Two protons ejected from nucleus simultaneously|| (''A'' − 2, ''Z'' − 2)
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| |-
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| |[[Spontaneous fission]] || Nucleus disintegrates into two or more smaller nuclei and other particles || —
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| |-
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| |[[Cluster decay]] || Nucleus emits a specific type of smaller nucleus (''A''<sub>1</sub>, ''Z''<sub>1</sub>) smaller than, or larger than, an alpha particle || (''A'' − ''A''<sub>1</sub>, ''Z'' − ''Z''<sub>1</sub>) + (''A''<sub>1</sub>, ''Z''<sub>1</sub>)
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| |-
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| | style="background:#ccc;" colspan="3"| '''Different modes of beta decay:'''
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| |-
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| |[[Beta decay|β<sup>−</sup> decay]] || A nucleus emits an [[electron]] and an [[electron antineutrino]] || (''A'', ''Z'' + 1)
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| |-
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| |[[Positron emission]] ([[beta decay|β<sup>+</sup> decay]]) || A nucleus emits a [[positron]] and an [[electron neutrino]] || (''A'', ''Z'' − 1)
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| |-
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| |[[Electron capture]] || A nucleus captures an orbiting electron and emits a neutrino; the daughter nucleus is left in an excited unstable state || (''A'', ''Z'' − 1)
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| |-
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| |[[Beta decay#Bound-state β- decay|Bound state beta decay]] || A 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)
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| |-
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| |[[Double beta decay]] || A nucleus emits two electrons and two antineutrinos || (''A'', ''Z'' + 2)
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| |-
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| |[[Double electron capture]] || A nucleus absorbs two orbital electrons and emits two neutrinos – the daughter nucleus is left in an excited and unstable state || (''A'', ''Z'' − 2)
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| |-
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| |[[Electron capture]] with [[positron emission]] || A nucleus absorbs one orbital electron, emits one positron and two neutrinos || (''A'', ''Z'' − 2)
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| |-
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| |[[Double positron emission]] || A nucleus emits two positrons and two neutrinos || (''A'', ''Z'' − 2)
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| |-
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| | style="background:#ccc;" colspan="3"| '''Transitions between states of the same nucleus:
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| |-
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| |[[Isomeric transition]] || Excited nucleus releases a high-energy [[photon]] ([[gamma ray]]) || (''A'', ''Z'')
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| |-
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| |[[Internal conversion]] || Excited nucleus transfers energy to an orbital electron, which is subsequently ejected from the atom || (''A'', ''Z'')
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| |}
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| 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'' = ''mc''<sup>2</sup>. 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.
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| Decay energy therefore remains associated with a certain measure of mass of the decay system, called [[invariant mass]], which does not change in the decay, even though the energy of decay is distributed among decay particles. The energy of photons, kinetic energy of emitted particles, and, later, the thermal energy of the surrounding matter, all contribute to [[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. This is a restatement of the equivalent laws of [[conservation of energy]] and [[conservation of mass]].
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| ==Decay chains and multiple modes==
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| 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).
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| [[File:Gammaspektrum Uranerz.jpg|thumb|250px|[[Gamma spectroscopy|Gamma-ray energy spectrum]] of uranium ore (inset). Gamma-rays are emitted by decaying [[nuclide]]s, 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 <sup>238</sup>U have been identified: <sup>226</sup>Ra, <sup>214</sup>Pb, <sup>214</sup>Bi.]]
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| An example is the natural decay chain of <sup>238</sup>U, which is as follows:
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| *decays, through alpha-emission, with a [[half-life]] of 4.5 billion years to [[thorium-234]]
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| *which decays, through beta-emission, with a half-life of 24 days to [[protactinium-234]]
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| *which decays, through beta-emission, with a half-life of 1.2 minutes to [[uranium-234]]
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| *which decays, through alpha-emission, with a half-life of 240 thousand years to [[thorium-230]]
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| *which decays, through alpha-emission, with a half-life of 77 thousand years to [[radium-226]]
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| *which decays, through alpha-emission, with a half-life of 1.6 thousand years to [[radon-222]]
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| *which decays, through alpha-emission, with a half-life of 3.8 days to [[polonium-218]]
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| *which decays, through alpha-emission, with a half-life of 3.1 minutes to [[lead-214]]
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| *which decays, through beta-emission, with a half-life of 27 minutes to [[bismuth-214]]
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| *which decays, through beta-emission, with a half-life of 20 minutes to [[polonium-214]]
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| *which decays, through alpha-emission, with a half-life of 160 microseconds to [[lead-210]]
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| *which decays, through beta-emission, with a half-life of 22 years to [[bismuth-210]]
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| *which decays, through beta-emission, with a half-life of 5 days to [[polonium-210]]
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| *which decays, through alpha-emission, with a half-life of 140 days to [[lead-206]], which is a stable nuclide.
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| 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]].
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| ==See also==
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| {{cmn|3|
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| * [[Actinides in the environment]]
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| * [[Background radiation]]
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| * [[Chernobyl disaster]]
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| * [[Crimes involving radioactive substances]]
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| * [[Decay chain]]
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| * [[Fallout shelter]]
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| * [[Half-life]]
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| * [[Lists of nuclear disasters and radioactive incidents]]
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| * [[National Council on Radiation Protection and Measurements]]
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| * [[Nuclear engineering]]
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| * [[Nuclear medicine]]
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| * [[Nuclear pharmacy]]
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| * [[Nuclear physics]]
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| * [[Nuclear power]]
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| * [[Particle decay]]
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| * [[Poisson process]]
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| * [[Radiation]]
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| * [[Radiation therapy]]
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| * [[Radioactive contamination]]
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| * [[Radioactivity in biology]]
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| * [[Radiometric dating]]
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| * [[Radionuclide]] a.k.a. "radio-isotope"
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| * [[Secular equilibrium]]
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| * [[Transient equilibrium]]
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| }}
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| ==Notes==
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| {{reflist|group=note}}
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| ==References==
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| ===Inline===
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| {{reflist|2}}
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| ===General===
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| *[http://search.eb.com/eb/article-9110413 "Radioactivity"]{{dead link|date=October 2012}}, Encyclopædia Britannica. 2006. [[Encyclopædia Britannica Online]]. December 18, 2006
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| *Radio-activity by Ernest Rutherford Phd, [[Encyclopædia Britannica Eleventh Edition]]
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| ==External links==
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| {{Wiktionary|radioactivity}}
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| *[http://ie.lbl.gov/toi/abouttoi.htm The Lund/LBNL Nuclear Data Search] – Contains tabulated information on radioactive decay types and energies.
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| *[http://www.radiochemistry.org/nomenclature/ Nomenclature of nuclear chemistry]
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| *[http://www.iem-inc.com/prhlfr.html Specific activity and related topics].
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| *[http://www-nds.iaea.org/livechart The Live Chart of Nuclides – IAEA]
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| *[http://www.radiationanswers.org/ Health Physics Society Public Education Website]
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| *{{Cite NSRW|wstitle=Becquerel Rays}}
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| *[http://alsos.wlu.edu/qsearch.aspx?browse=science/Radioactivity Annotated bibliography for radioactivity from the Alsos Digital Library for Nuclear Issues]
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| *[http://chair.pa.msu.edu/applets/decay/a.htm Stochastic Java applet on the decay of radioactive atoms] by Wolfgang Bauer
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| *[http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/Nuclear/Decay/NuclearDecay.html Stochastic Flash simulation on the decay of radioactive atoms] by David M. Harrison
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| {{Radiation|state=uncollapsed}}
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| {{DEFAULTSORT:Radioactive Decay}}
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| [[Category:Radioactivity| ]]
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| [[Category:Exponentials]]
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| [[Category:Poisson processes]]
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