Decay chain

In nuclear science a decay chain refers to the predictable series of radioactive disintegrations undergone by the nuclei of certain unstable chemical elements.

Radioactive isotopes do not usually decay directly to stable isotopes, but rather into another radioisotope. The isotope produced by this radioactive emission then decays into another, often radioactive isotope. This chain of decays always terminates in a stable isotope, whose nucleus no longer has the surplus of energy necessary to produce another emission of radiation. Such stable isotopes are then said to have nuclei that have reached their ground states.

The stages or steps in a decay chain are referred to by their relationship to previous or subsequent stages. Hence, a parent isotope is one that undergoes decay to form a daughter isotope. For example element 92, uranium, has an isotope with 143 neutrons (²³⁶U) and it decays into an isotope of element 90, thorium, with 142 neutrons (²³²Th). The daughter isotope may be stable or it may itself decay to form another daughter isotope. ²³²Th does this when it decays into radium-228. The daughter of a daughter isotope, such as ²²⁸Ra, is sometimes called a granddaughter isotope.

The time required for an atom of a parent isotope to decay into its daughter is fundamentally unpredictable and varies widely. For individual nuclei the process is not known to have determinable causes and the time at which it occurs is therefore completely random. The only prediction that can be made is statistical and expresses an average rate of decay. This rate can be represented by adjusting the curve of a decaying exponential distribution with a decay constant (λ) particular to the isotope. On this understanding the radioactive decay of an initial population of unstable atoms over time t follows the curve given by e^−λt.

One of the most important properties of any radioactive material follows from this analysis, its half-life. This refers to the time required for half of a given number of radioactive atoms to decay and is inversely related to the isotope's decay constant, λ. Half-lives have been determined in laboratories for many radionuclides, and can range from nearly instantaneous—hydrogen-5 decays in less time than it takes for a photon to go from one end of its nucleus to the other—to fourteen orders of magnitude longer than the age of the universe: tellurium-128 has a half-life of 2.2×10²⁴ years.

Quantity calculation with the Bateman-Function for ²⁴¹Pu

The Bateman equation predicts the relative quantities of all the isotopes that compose a given decay chain once that decay chain has proceeded long enough for some of its daughter products to have reached the stable (i.e., nonradioactive) end of the chain. A decay chain that has reached this state, which may require billions of years, is said to be in equilibrium. A sample of radioactive material in equilibrium produces a steady and steadily decreasing quantity of radioactivity as the isotopes that compose it traverse the decay chain. On the other hand, if a sample of radioactive material has been isotopically enriched, meaning that a radioisotope is present in larger quantities than would exist if a decay chain were the only cause of its presence, that sample is said to be out of equilibrium. An unintuitive consequence of this disequilibrium is that a sample of enriched material may occasionally increase in radioactivity as daughter products that are more highly radioactive than their parents accumulate. Both enriched and depleted uranium provide examples of this phenomenon.

History

The chemical elements came into being in two phases. The first commenced shortly after the Big Bang. From ten seconds to 20 minutes after the beginning of the universe the earliest condensation of light atoms was responsible for the manufacture of the four lightest elements. The vast majority of this primordial production consisted of the three lightest isotopes of hydrogen—protium, deuterium and tritium—and two of the nine known isotopes of helium—helium-3 and helium-4. Trace amounts of lithium-7 and beryllium-7 were likely also produced.

So far as is known, all heavier elements came into being starting around 100 million years later, in a second phase of nucleosynthesis that commenced with the birth of the first stars. The nuclear furnaces that power stellar evolution were necessary to create large quantities of all elements heavier than helium, and the r- and s-processes of neutron capture that occur in stellar cores are thought to have created all such elements up to iron and nickel (atomic numbers 26 and 28). The extreme conditions that attend supernovae explosions are capable of creating the elements between oxygen and rubidium (i.e., atomic numbers 8 through 37). The creation of heavier elements, including those without stable isotopes—all elements with atomic numbers greater than lead's, 82—appears to rely on r-process nucleosynthesis operating amid the immense concentrations of free neutrons released during neutron star mergers.

Most of the isotopes of each chemical element present in the Earth today were formed by such processes no later than the time of our planet's condensation from the solar protoplanetary disc, around 4.5 billion years ago. The exceptions to these so-called primordial elements are those that have resulted from the radioactive disintegration of unstable parent nuclei as they progress down one of several decay chains, each of which terminates with the production of one of the 251 stable isotopes known to exist. Aside from cosmic or stellar nucleosynthesis, and decay chains the only other ways of producing a chemical element rely on atomic weapons, nuclear reactors (natural or manmade) or the laborious atom-by-atom assembly of nuclei with particle accelerators.

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 251 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 uranium have close to 1.5 neutrons per proton (e.g. 1.587 in uranium-238). No nuclide heavier than lead-208 is stable; these heavier elements have to shed mass to achieve stability, mostly by alpha decay. The other common way for isotopes with a high neutron to proton ratio (n/p) to decay is beta decay, in which the nuclide changes elemental identity while keeping the same mass number 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 or electron capture are 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 mass number (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 (radium series) 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 nuclide 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 nuclide near the top, so almost all of the nuclides in that chain have long since decayed down to just before the end: bismuth-209. This nuclide was long thought to be stable, but in 2003 it was found to be unstable, with a very long half-life of 20.1 billion billion years; it is the last step in the chain before stable thallium-205. Because this bottleneck is so long-lived, very small quantities of the final decay product have been produced, and for most practical purposes bismuth-209 is the final decay product.

In the distant past, during the first few million years of the history of the Solar System, there were more kinds of unstable high-mass nuclides in existence, and the four chains were longer, as they included nuclides that have since decayed away. Notably, ²⁴⁴Pu, ²³⁷Np, and ²⁴⁷Cm have half-lives over a million years and would have then been lesser bottlenecks high in the 4n, 4n+1, and 4n+3 chains respectively. (There is no nuclide with a half-life over a million years above ²³⁸U in the 4n+2 chain.) Today some of these formerly extinct isotopes are again in existence as they have been manufactured. Thus they again take their places in the chain: plutonium-239, used in nuclear weapons, is the major example, decaying to uranium-235 via alpha emission with a half-life 24,500 years. There has also been large-scale production of neptunium-237, which has resurrected the hitherto extinct fourth chain. The tables below hence start the four decay chains at isotopes of californium with mass numbers from 249 to 252.

Summary of the four decay chain pathways

Name of series	Thorium	Neptunium	Uranium	Actinium
Mass numbers	4n	4n+1	4n+2	4n+3
Long-lived nuclide	232Th (244Pu)	209Bi (237Np)	238U	235U (247Cm)
Half-life (billions of years)	14 (0.08)	20100000000 (0.00214)	4.5	0.7 (0.0156)
End of chain	208Pb	205Tl	206Pb	207Pb

These four chains are summarised in the chart in the following section.

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 (fission of a helium-4 nucleus) 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. This divides the list of nuclides into four classes. All the members of any possible decay chain must be drawn entirely from one of these classes.

Three main decay chains (or families) are observed in nature. These are 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 created 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 do occur in nature, however, as a result of neutron capture in uranium ore. The ending isotope of this chain is now known to be thallium-205. Some older sources give the final isotope as bismuth-209, but in 2003 it was discovered that it is very slightly radioactive, with a half-life of 2.01×10¹⁹ years.

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 by decay chain				Half-life range (a)	Fission products of 235U by yield
4n	4n + 1	4n + 2	4n + 3		4.5–7%	0.04–1.25%	<0.001%
228Ra№				4–6 a		155Euþ
248Bk				> 9 a
244Cmƒ	241Puƒ	250Cf	227Ac№	10–29 a	90Sr	85Kr	113mCdþ
232Uƒ		238Puƒ	243Cmƒ	29–97 a	137Cs	151Smþ	121mSn
	249Cfƒ	242mAmƒ		141–351 a	No fission products have a half-life in the range of 100 a–210 ka ...
	241Amƒ		251Cfƒ	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₡
				1.33 Ma	135Cs₡
	237Npƒ			1.61–6.5 Ma	93Zr	107Pd
236U			247Cmƒ	15–24 Ma		129I₡
244Pu				80 Ma	... nor beyond 15.7 Ma
232Th№		238U№	235Uƒ№	0.7–14.1 Ga
₡,  has thermal neutron capture cross section in the range of 8–50 barns ƒ,  fissile №,  primarily a naturally occurring radioactive material (NORM) þ,  neutron poison (thermal neutron capture cross section greater than 3k barns)

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 uranium-238), and actinium (from uranium-235)—each ends with its own specific lead isotope (lead-208, lead-206, and lead-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 thorium-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.

Plutonium-244 (which appears several steps above thorium-232 in this chain if one extends it to the transuranics) was present in the early Solar System, and is just long-lived enough that it should still survive in trace quantities today, though it is uncertain if it has been detected.

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

Nuclide	Historic names		Decay mode	Half-life (a = years)	Energy released MeV	Decay product
	Short	Long
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	0.39	240Np
240Np			β−	1.032 h	2.2	240Pu
240Pu			α	6561 a	5.1683	236U
236U		Thoruranium	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	294.4 ns	8.954	208Pb
208Tl	ThC″	Thorium C″	β−	3.053 min	1.803	208Pb
208Pb	ThD	Thorium D	stable	—	—	—

Neptunium series

The 4n + 1 chain of neptunium-237 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. 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, radon, thallium, thorium, and uranium. Since this series was only discovered and studied in 1947–1948, 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 (not shown in the illustration) but not the main decay sequence; thus, radon from this decay chain does not migrate through rock nearly as much as from the other three. Another unique trait of this decay sequence is that it ends in thallium (practically speaking, bismuth) rather than lead. This series terminates with the stable isotope thallium-205.

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 = years)	Energy released MeV	Decay product
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− 99.998% α 0.002%	14.9 d	0.36 5.097	225Ac 221Rn
225Ac	α	10.0 d	5.935	221Fr
221Rn	β− 78% α 22%	25.7 min	1.194 6.163	221Fr 217Po
221Fr	α 99.9952% β− 0.0048%	4.8 min	6.458 0.314	217At 221Ra
221Ra	α	28 s	6.880	217Rn
217Po	α 97.5% β− 2.5%	1.53 s	6.662 1.488	213Pb 217At
217At	α 99.992% β− 0.008%	32 ms	7.201 0.737	213Bi 217Rn
217Rn	α	540 μs	7.887	213Po
213Pb	β−	10.2 min	2.028	213Bi
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	2.01×1019 a	3.137	205Tl
205Tl	.	stable	.	.

Uranium series

(More comprehensive graphic)

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, mercury, 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		Decay mode	Half-life (a= years)	Energy released MeV	Decay product
	Short	Long
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
222Rn	Rn	Radon, Radium Emanation	α	3.8235 d	5.59031	218Po
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
210Pb	RaD	Radium D	β−, 100% 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			β−	4.202 min	1.5322211	206Pb
206Pb	RaG	Radium G	stable	-	-	-

Actinium series

The 4n+3 chain of uranium-235 is commonly called the "actinium series" or "actinium cascade". Beginning with the naturally-occurring isotope uranium-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.

(More detailed graphic)

In the early Solar System this chain went back to ²⁴⁷Cm. This manifests itself today as variations in ²³⁵U/²³⁸U ratios, since curium and uranium have noticeably different chemistries and would have separated differently.

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

Nuclide	Historic name		Decay mode	Half-life (a = years)	Energy released MeV	Decay product
	Short	Long
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	.	.

See also

• Nuclear physics
• Radioactive decay
• Valley of stability
• Decay product
• Radioisotopes (radionuclide)
• Radiometric dating

References

• C.M. Lederer; J.M. Hollander; I. Perlman (1968). Table of Isotopes (6th ed.). New York: John Wiley & Sons.

External links

• Nucleonica nuclear science portal
• Nucleonica's Decay Engine for professional online decay calculations
• EPA – Radioactive Decay
• Government website listing isotopes and decay energies
• National Nuclear Data Center – freely available databases that can be used to check or construct decay chains
• IAEA – Live Chart of Nuclides (with decay chains)
• Decay Chain Finder