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# Decay chain

In nuclear science, the decay chain refers to a series of radioactive decays of different radioactive decay products as a sequential series of transformations. It is also known as a "radioactive cascade". Most radioisotopes do not decay directly to a stable state, but rather undergo a series of decays until eventually a stable isotope is reached.

Decay stages are referred to by their relationship to previous or subsequent stages. A parent isotope is one that undergoes decay to form a daughter isotope. One example of this is uranium (atomic number 92) decaying into thorium (atomic number 90). The daughter isotope may be stable or it may decay to form a daughter isotope of its own. The daughter of a daughter isotope is sometimes called a granddaughter isotope.

The time it takes for a single parent atom to decay to an atom of its daughter isotope can vary widely, not only between different parent-daughter pairs, but also randomly between identical pairings of parent and daughter isotopes. The decay of each single atom occurs spontaneously, and the decay of an initial population of identical atoms over time t, follows a decaying exponential distribution, e−λt, where λ is called a decay constant. One of the properties of an isotope is its half-life, the time by which half of an initial number of identical parent radioisotopes have decayed to their daughters, which is inversely related to λ. Half-lives have been determined in laboratories for many radioisotopes (or radionuclides). These can range from nearly instantaneous (less than 10−21 seconds) to more than 1019 years.

The intermediate stages each emit the same amount of radioactivity as the original radioisotope (i.e. there is a one-to-one relationship between the numbers of decays in successive stages) but each stage releases a different quantity of energy. If and when equilibrium is achieved, each successive daughter isotope is present in direct proportion to its half-life; but since its activity is inversely proportional to its half-life, each nuclide in the decay chain finally contributes as many individual transformations as the head of the chain, though not the same energy. For example, uranium-238 is weakly radioactive, but pitchblende, a uranium ore, is 13 times more radioactive than the pure uranium metal because of the radium and other daughter isotopes it contains. Not only are unstable radium isotopes significant radioactivity emitters, but as the next stage in the decay chain they also generate radon, a heavy, inert, naturally occurring radioactive gas. Rock containing thorium and/or uranium (such as some granites) emits radon gas that can accumulate in enclosed places such as basements or underground mines.[1]

Quantity calculation with the Bateman-Function for 241Pu

The quantity of isotopes in the decay chains at a certain time are calculated with the Bateman equation.

## History

All the elements and isotopes found on Earth, with the exceptions of hydrogen, deuterium, helium, helium-3, and perhaps trace amounts of stable lithium and beryllium isotopes which were created in the Big Bang, were created by the s-process or the r-process in stars, and for those to be today a part of the Earth, must have been created not later than 4.5 billion years ago. All the elements created more than 4.5 billion years ago are termed primordial, meaning they were generated by the universe's stellar processes. At the time when they were created, those that were unstable began decaying immediately. All the isotopes which have half-lives less than 100 million years have been reduced to 2.8×10−12% or less of whatever original amounts were created and captured by Earth's accretion; they are of trace quantity today, or have decayed away altogether. There are only two other methods to create isotopes: artificially, inside a man-made (or perhaps a natural) reactor, or through decay of a parent isotopic species, the process known as the decay chain.

Unstable isotopes decay to their daughter products (which may sometimes be even more unstable) at a given rate; eventually, often after a series of decays, a stable isotope is reached: there are about 200 stable isotopes in the universe. In stable isotopes, light elements typically have a lower ratio of neutrons to protons in their nucleus than heavier elements. Light elements such as helium-4 have close to a 1:1 neutron:proton ratio. The heaviest elements such as lead have close to 1.5 neutrons per proton(e.g. 1.536 in lead-208). No nuclide heavier than lead-208 is stable; these heavier elements have to shed mass to achieve stability, most usually as alpha decay. The other common decay method for isotopes with a high neutron to proton ratio (n/p) is beta decay, in which the nuclide changes elemental identity while keeping the same mass and lowering its n/p ratio. For some isotopes with a relatively low n/p ratio, there is an inverse beta decay, by which a proton is transformed into a neutron, thus moving towards a stable isotope; however, since fission almost always produces products which are neutron heavy, positron emission is relatively rare compared to electron emission. There are many relatively short beta decay chains, at least two (a heavy, beta decay and a light, positron decay) for every discrete weight up to around 207 and some beyond, but for the higher mass elements (isotopes heavier than lead) there are only four pathways which encompass all decay chains. This is because there are just two main decay methods: alpha radiation, which reduces the mass by 4 atomic mass units (amu), and beta, which does not change the atomic mass at all (just the atomic number and the p/n ratio). The four paths are termed 4n, 4n + 1, 4n + 2, and 4n + 3; the remainder from dividing the atomic mass by four gives the chain the isotope will use to decay. There are other decay modes, but they invariably occur at a lower probability than alpha or beta decay. (It should not be supposed that these chains have no branches: the diagram below shows a few branches of chains, and in reality there are many more, because there are many more isotopes possible than are shown in the diagram.) For example, the third atom of nihonium-278 synthesised underwent six alpha decays down to mendelevium-254, followed by an electron capture (a form of beta decay) to fermium-254, and then a seventh alpha to californium-250, upon which it would have followed the 4n + 2 chain as given in this article. However, the heaviest superheavy nuclides synthesised do not reach the four decay chains, because they reach a spontaneously fissioning nuclide after a few alpha decays that terminates the chain: this is what happened to the first two atoms of nihonium-278 synthesised, as well as to all heavier nuclides produced.

Three of those chains have a long-lived isotope (or nuclide) near the top; this long-lived isotope is a bottleneck in the process through which the chain flows very slowly, and keeps the chain below them "alive" with flow. The three long-lived nuclides are uranium-238 (half-life=4.5 billion years), uranium-235 (half-life=700 million years) and thorium-232 (half-life=14 billion years). The fourth chain has no such long lasting bottleneck isotope, so almost all of the isotopes in that chain have long since decayed down to very near the stability at the bottom. Near the end of that chain is bismuth-209, which was long thought to be stable. Recently, however, bismuth-209 was found to be unstable with a half-life of 19 billion billion years; it is the last step before stable thallium-205. In the distant past, around the time that the solar system formed, there were more kinds of unstable high-weight isotopes available, and the four chains were longer with isotopes that have since decayed away. Today we have manufactured extinct isotopes, which again take their former places: plutonium-239, the nuclear bomb fuel, as the major example has a half-life of "only" 24,500 years, and decays by alpha emission into uranium-235. In particular, we have through the large-scale production of neptunium-237 successfully resurrected the hitherto extinct fourth chain.[2] The tables below hence start the four decay chains at isotopes of californium with mass numbers from 249 to 252.

## Types of decay

This diagram illustrates the four decay chains discussed in the text: thorium (4n, in blue), neptunium (4n+1, in pink), radium (4n+2, in red) and actinium (4n+3, in green).

The four most common modes of radioactive decay are: alpha decay, beta decay, inverse beta decay (considered as both positron emission and electron capture), and isomeric transition. Of these decay processes, only alpha decay changes the atomic mass number (A) of the nucleus, and always decreases it by four. Because of this, almost any decay will result in a nucleus whose atomic mass number has the same residue mod 4, dividing all nuclides into four chains. The members of any possible decay chain must be drawn entirely from one of these classes. All four chains also produce helium-4 (alpha particles are helium-4 nuclei).

Three main decay chains (or families) are observed in nature, commonly called the thorium series, the radium or uranium series, and the actinium series, representing three of these four classes, and ending in three different, stable isotopes of lead. The mass number of every isotope in these chains can be represented as A = 4n, A = 4n + 2, and A = 4n + 3, respectively. The long-lived starting isotopes of these three isotopes, respectively thorium-232, uranium-238, and uranium-235, have existed since the formation of the earth, ignoring the artificial isotopes and their decays since the 1940s.

Due to the relatively short half-life of its starting isotope neptunium-237 (2.14 million years), the fourth chain, the neptunium series with A = 4n + 1, is already extinct in nature, except for the final rate-limiting step, decay of bismuth-209. Traces of 237Np and its decay products still do occur in nature, however, as a result of neutron capture in uranium ore.[3] The ending isotope of this chain is now known to be thallium-205. Some older sources give the final isotope as bismuth-209, but it was recently discovered that it is very slightly radioactive, with a half-life of 2.01×1019 years.[4]

There are also non-transuranic decay chains of unstable isotopes of light elements, for example those of magnesium-28 and chlorine-39. On Earth, most of the starting isotopes of these chains before 1945 were generated by cosmic radiation. Since 1945, the testing and use of nuclear weapons has also released numerous radioactive fission products. Almost all such isotopes decay by either β or β+ decay modes, changing from one element to another without changing atomic mass. These later daughter products, being closer to stability, generally have longer half-lives until they finally decay into stability.

## Actinide alpha decay chains

Actinides and fission products by half-life
Actinides[5] by decay chain Half-life
range (a)
Fission products of 235U by yield[6]
4n 4n+1 4n+2 4n+3
4.5–7% 0.04–1.25% <0.001%
228Ra 4–6 a 155Euþ
244Cmƒ 241Puƒ 250Cf 227Ac 10–29 a 90Sr 85Kr 113mCdþ
232Uƒ 238Puƒ 243Cmƒ 29–97 a 137Cs 151Smþ 121mSn
248Bk[7] 249Cfƒ 242mAmƒ 141–351 a

No fission products
have a half-life
in the range of
100–210 ka ...

241Amƒ 251Cfƒ[8] 430–900 a
226Ra 247Bk 1.3–1.6 ka
240Pu 229Th 246Cmƒ 243Amƒ 4.7–7.4 ka
245Cmƒ 250Cm 8.3–8.5 ka
239Puƒ 24.1 ka
230Th 231Pa 32–76 ka
236Npƒ 233Uƒ 234U 150–250 ka 99Tc 126Sn
248Cm 242Pu 327–375 ka 79Se
1.53 Ma 93Zr
237Npƒ 2.1–6.5 Ma 135Cs 107Pd
236U 247Cmƒ 15–24 Ma 129I
244Pu 80 Ma

... nor beyond 15.7 Ma[9]

232Th 238U 235Uƒ№ 0.7–14.1 Ga

Legend for superscript symbols
₡  has thermal neutron capture cross section in the range of 8–50 barns
ƒ  fissile
metastable isomer
№  primarily a naturally occurring radioactive material (NORM)
þ  neutron poison (thermal neutron capture cross section greater than 3k barns)
†  range 4–97 a: Medium-lived fission product
‡  over 200 ka: Long-lived fission product

In the four tables below, the minor branches of decay (with the branching probability of less than 0.0001%) are omitted. The energy release includes the total kinetic energy of all the emitted particles (electrons, alpha particles, gamma quanta, neutrinos, Auger electrons and X-rays) and the recoil nucleus, assuming that the original nucleus was at rest. The letter 'a' represents a year (from the Latin annus).

In the tables below (except neptunium), the historic names of the naturally occurring nuclides are also given. These names were used at the time when the decay chains were first discovered and investigated. From these historical names one can locate the particular chain to which the nuclide belongs, and replace it with its modern name.

The three naturally-occurring actinide alpha decay chains given below—thorium, uranium/radium (from U-238), and actinium (from U-235)—each ends with its own specific lead isotope (Pb-208, Pb-206, and Pb-207 respectively). All these isotopes are stable and are also present in nature as primordial nuclides, but their excess amounts in comparison with lead-204 (which has only a primordial origin) can be used in the technique of uranium-lead dating to date rocks.

### Thorium series

The 4n chain of Th-232 is commonly called the "thorium series" or "thorium cascade". Beginning with naturally occurring thorium-232, this series includes the following elements: actinium, bismuth, lead, polonium, radium, radon and thallium. All are present, at least transiently, in any natural thorium-containing sample, whether metal, compound, or mineral. The series terminates with lead-208.

The total energy released from thorium-232 to lead-208, including the energy lost to neutrinos, is 42.6 MeV.

nuclide historic name (short) historic name (long) decay mode half-life
(a=year)
energy released, MeV product of decay
252Cf α 2.645 a 6.1181 248Cm
248Cm α 3.4×105 a 5.162 244Pu
244Pu α 8×107 a 4.589 240U
240U β 14.1 h .39 240Np
240Np β 1.032 h 2.2 240Pu
240Pu α 6561 a 5.1683 236U
236U Thoruranium[10] α 2.3×107 a 4.494 232Th
232Th Th Thorium α 1.405×1010 a 4.081 228Ra
228Ra MsTh1 Mesothorium 1 β 5.75 a 0.046 228Ac
228Ac MsTh2 Mesothorium 2 β 6.25 h 2.124 228Th
228Th RdTh Radiothorium α 1.9116 a 5.520 224Ra
224Ra ThX Thorium X α 3.6319 d 5.789 220Rn
220Rn Tn Thoron,
Thorium Emanation
α 55.6 s 6.404 216Po
216Po ThA Thorium A α 0.145 s 6.906 212Pb
212Pb ThB Thorium B β 10.64 h 0.570 212Bi
212Bi ThC Thorium C β 64.06%
α 35.94%
60.55 min 2.252
6.208
212Po
208Tl
212Po ThC′ Thorium C′ α 299 ns 8.784 [11] 208Pb
208Tl ThC″ Thorium C″ β 3.053 min 1.803 [11] 208Pb
208Pb ThD Thorium D stable . . .

### Neptunium series

The 4n + 1 chain of 237Np is commonly called the "neptunium series" or "neptunium cascade". In this series, only two of the isotopes involved are found naturally in significant quantities, namely the final two: bismuth-209 and thallium-205. Some of the other isotopes have been detected in nature, originating from trace quantities of 237Np produced by the (n,2n) knockout reaction in primordial 238U.[3] A smoke detector containing an americium-241 ionization chamber accumulates a significant amount of neptunium-237 as its americium decays; the following elements are also present in it, at least transiently, as decay products of the neptunium: actinium, astatine, bismuth, francium, lead, polonium, protactinium, radium, thallium, thorium, and uranium. Since this series was only discovered and studied in 1947–1948,[12] its nuclides do not have historic names. One unique trait of this decay chain is that the noble gas radon is only produced in a rare branch and not the main decay sequence; thus, it does not migrate through rock nearly as much as the other three decay chains.

The total energy released from californium-249 to thallium-205, including the energy lost to neutrinos, is 66.8 MeV.

nuclide decay mode half-life
(a=year)
energy released, MeV product of decay
249Cf α 351 a 5.813+.388 245Cm
245Cm α 8500 a 5.362+.175 241Pu
241Pu β 14.4 a 0.021 241Am
241Am α 432.7 a 5.638 237Np
237Np α 2.14·106 a 4.959 233Pa
233Pa β 27.0 d 0.571 233U
233U α 1.592·105 a 4.909 229Th
229Th α 7340 a 5.168 225Ra
225Ra β 14.9 d 0.36 225Ac
225Ac α 10.0 d 5.935 221Fr
221Fr α 99.9952%
β 0.0048%
4.8 min 6.3
0.314
217At
221Ra
221Ra α 28 s 6.9 217Rn
217At α 99.992%
β 0.008%
32 ms 7.0
0.737
213Bi
217Rn
217Rn α 540 µs 7.9 213Po
213Bi β 97.80%
α 2.20%
46.5 min 1.423
5.87
213Po
209Tl
213Po α 3.72 μs 8.536 209Pb
209Tl β 2.2 min 3.99 209Pb
209Pb β 3.25 h 0.644 209Bi
209Bi α 1.9·1019 a 3.137 205Tl
205Tl . stable . .

### Uranium series

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

The total energy released from uranium-238 to lead-206, including the energy lost to neutrinos, is 51.7 MeV.

parent nuclide historic name (short)[13] historic name (long) decay mode [RS 1] half-life
(a=year)
energy released, MeV [RS 1] product of decay [RS 1]
250Cf α 13.08 a 6.12844 246Cm
246Cm α 4800 a 5.47513 242Pu
242Pu α 3.8·105 a 4.98453 238U
238U UI Uranium I α 4.468·109 a 4.26975 234Th
234Th UX1 Uranium X1 β 24.10 d 0.273088 234mPa
234mPa UX2, Bv Uranium X2, Brevium IT, 0.16%
β, 99.84%
1.159 min 0.07392
2.268205
234Pa
234U
234Pa UZ Uranium Z β 6.70 h 2.194285 234U
234U UII Uranium II α 2.45·105 a 4.8698 230Th
230Th Io Ionium α 7.54·104 a 4.76975 226Ra
226Ra Ra Radium α 1600 a 4.87062 222Rn
218Po RaA Radium A α, 99.980%
β, 0.020%
3.098 min 6.11468
0.259913
214Pb
218At
218At α, 99.9%
β, 0.1%
1.5 s 6.874
2.881314
214Bi
218Rn
218Rn α 35 ms 7.26254 214Po
214Pb RaB Radium B β 26.8 min 1.019237 214Bi
214Bi RaC Radium C β, 99.979%
α, 0.021%
19.9 min 3.269857
5.62119
214Po
210Tl
214Po RaC' Radium C' α 164.3 μs 7.83346 210Pb
210Tl RaC" Radium C" β 1.3 min 5.48213 210Pb
α, 1.9·10−6%
22.20 a 0.063487
3.7923
210Bi
206Hg
210Bi RaE Radium E β, 100%
α, 1.32·10−4%
5.012 d 1.161234
5.03647
210Po
206Tl
210Po RaF Radium F α 138.376 d 5.03647 206Pb
206Hg β 8.32 min 1.307649 206Tl
206Tl RaE Radium E β 4.202 min 1.5322211 206Pb
206Pb RaG Radium G stable - - -
1. ^ a b c "Evaluated Nuclear Structure Data File". National Nuclear Data Center.

### Actinium series

The 4n+3 chain of uranium-235 is commonly called the "actinium series" or "actinium cascade". Beginning with the naturally-occurring isotope U-235, this decay series includes the following elements: actinium, astatine, bismuth, francium, lead, polonium, protactinium, radium, radon, thallium, and thorium. All are present, at least transiently, in any sample containing uranium-235, whether metal, compound, ore, or mineral. This series terminates with the stable isotope lead-207.

The total energy released from uranium-235 to lead-207, including the energy lost to neutrinos, is 46.4 MeV.

nuclide historic name (short) historic name (long) decay mode half-life
(a=year)
energy released, MeV product of decay
251Cf α 900.6 a 6.176 247Cm
247Cm α 1.56·107 a 5.353 243Pu
243Pu β 4.95556 h 0.579 243Am
243Am α 7388 a 5.439 239Np
239Np β 2.3565 d 0.723 239Pu
239Pu α 2.41·104 a 5.244 235U
235U AcU Actin Uranium α 7.04·108 a 4.678 231Th
231Th UY Uranium Y β 25.52 h 0.391 231Pa
231Pa Pa Protactinium α 32760 a 5.150 227Ac
227Ac Ac Actinium β 98.62%
α 1.38%
21.772 a 0.045
5.042
227Th
223Fr
227Th RdAc Radioactinium α 18.68 d 6.147 223Ra
223Fr AcK Actinium K β 99.994%
α 0.006%
22.00 min 1.149
5.340
223Ra
219At
223Ra AcX Actinium X α 11.43 d 5.979 219Rn
219At α 97.00%
β 3.00%
56 s 6.275
1.700
215Bi
219Rn
219Rn An Actinon,
Actinium Emanation
α 3.96 s 6.946 215Po
215Bi β 7.6 min 2.250 215Po
215Po AcA Actinium A α 99.99977%
β 0.00023%
1.781 ms 7.527
0.715
211Pb
215At
215At α 0.1 ms 8.178 211Bi
211Pb AcB Actinium B β 36.1 min 1.367 211Bi
211Bi AcC Actinium C α 99.724%
β 0.276%
2.14 min 6.751
0.575
207Tl
211Po
211Po AcC' Actinium C' α 516 ms 7.595 207Pb
207Tl AcC" Actinium C" β 4.77 min 1.418 207Pb
207Pb AcD Actinium D . stable . .

## Notes

1. ^ "Archived copy". Archived from the original on 2008-09-20. Retrieved 2008-06-26.CS1 maint: archived copy as title (link)
2. ^ Koch, Lothar (2000). Transuranium Elements, in Ullmann's Encyclopedia of Industrial Chemistry. Wiley. doi:10.1002/14356007.a27_167.
3. ^ a b Peppard, D. F.; Mason, G. W.; Gray, P. R.; Mech, J. F. (1952). "Occurrence of the (4n + 1) series in nature" (PDF). Journal of the American Chemical Society. 74 (23): 6081–6084. doi:10.1021/ja01143a074.
4. ^ Audi, G.; Kondev, F. G.; Wang, M.; Huang, W. J.; Naimi, S. (2017). "The NUBASE2016 evaluation of nuclear properties" (PDF). Chinese Physics C. 41 (3): 030001. Bibcode:2017ChPhC..41c0001A. doi:10.1088/1674-1137/41/3/030001.
5. ^ Plus radium (element 88). While actually a sub-actinide, it immediately precedes actinium (89) and follows a three-element gap of instability after polonium (84) where no nuclides have half-lives of at least four years (the longest-lived nuclide in the gap is radon-222 with a half life of less than four days). Radium's longest lived isotope, at 1,600 years, thus merits the element's inclusion here.
6. ^ Specifically from thermal neutron fission of U-235, e.g. in a typical nuclear reactor.
7. ^ Milsted, J.; Friedman, A. M.; Stevens, C. M. (1965). "The alpha half-life of berkelium-247; a new long-lived isomer of berkelium-248". Nuclear Physics. 71 (2): 299. Bibcode:1965NucPh..71..299M. doi:10.1016/0029-5582(65)90719-4.
"The isotopic analyses disclosed a species of mass 248 in constant abundance in three samples analysed over a period of about 10 months. This was ascribed to an isomer of Bk248 with a half-life greater than 9 [years]. No growth of Cf248 was detected, and a lower limit for the β half-life can be set at about 104 [years]. No alpha activity attributable to the new isomer has been detected; the alpha half-life is probably greater than 300 [years]."
8. ^ This is the heaviest nuclide with a half-life of at least four years before the "Sea of Instability".
9. ^ Excluding those "classically stable" nuclides with half-lives significantly in excess of 232Th; e.g., while 113mCd has a half-life of only fourteen years, that of 113Cd is nearly eight quadrillion years.
10. ^ Trenn, Thaddeus J. (1978). "Thoruranium (U-236) as the extinct natural parent of thorium: The premature falsification of an essentially correct theory". Annals of Science. 35 (6): 581–97. doi:10.1080/00033797800200441.
11. ^ a b http://nucleardata.nuclear.lu.se
12. ^ Thoennessen, M. (2016). The Discovery of Isotopes: A Complete Compilation. Springer. p. 20. doi:10.1007/978-3-319-31763-2. ISBN 978-3-319-31761-8. LCCN 2016935977.
13. ^ Thoennessen, M. (2016). The Discovery of Isotopes: A Complete Compilation. Springer. p. 19. doi:10.1007/978-3-319-31763-2. ISBN 978-3-319-31761-8. LCCN 2016935977.

## References

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