Radioactive decay

In electron capture, some proton-rich nuclides were found to capture their own atomic electrons instead of emitting positrons, and subsequently, these nuclides emit only a neutrino and a gamma ray from the excited nucleus (and often also

Shortly after the discovery of the neutron in 1932, Enrico Fermi realized that certain rare beta-decay reactions immediately yield neutrons as an additional decay particle, so called beta-delayed neutron emission. Neutron emission usually happens from nuclei that are in an excited state, such as the excited ¹⁷O* produced from the beta decay of ¹⁷N. The neutron emission process itself is controlled by the nuclear force and therefore is extremely fast, sometimes referred to as "nearly instantaneous". 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.

More exotic types of decay

Other types of radioactive decay were found to emit previously seen particles but via different mechanisms. An example is

Rare events that involve a combination of two beta-decay-type events happening simultaneously are known (see below). Any decay process that does not violate the 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

If energy circumstances are favorable, a given radionuclide may undergo many competing types of decay, with some atoms decaying by one route, and others decaying by another. An example is

List of decay modes

Decay chains and multiple modes

The daughter nuclide of a decay event may also be unstable (radioactive). In this case, it too will decay, producing radiation. The resulting second daughter nuclide may also be radioactive. This can lead to a sequence of several decay events called a decay chain (see this article for specific details of important natural decay chains). Eventually, a stable nuclide is produced. Any decay daughters that are the result of an alpha decay will also result in helium atoms being created.

Some radionuclides may have several different paths of decay. For example, 35.94(6)%

Occurrence and applications

According to the

The Szilard–Chalmers effect is the breaking of a chemical bond as a result of a kinetic energy imparted from radioactive decay. It operates by the absorption of neutrons by an atom and subsequent emission of gamma rays, often with significant amounts of kinetic energy. This kinetic energy, by Newton's third law, pushes back on the decaying atom, which causes it to move with enough speed to break a chemical bond.^[30] This effect can be used to separate isotopes by chemical means.

The Szilard–Chalmers effect was discovered in 1934 by

Radioactive

• The half-life, t_1/2, is the time taken for the activity of a given amount of a radioactive substance to decay to half of its initial value.
• The
  decay constant, λ "lambda
  ", the reciprocal of the mean lifetime (in s⁻¹), sometimes referred to as simply decay rate.
• The
  mean lifetime, τ "tau", the average lifetime (1/e
  life) of a radioactive particle before decay.

Although these are constants, they are associated with the statistical behavior of populations of atoms. In consequence, predictions using these constants are less accurate for minuscule samples of atoms.

In principle a half-life, a third-life, or even a (1/√2)-life, can be used in exactly the same way as half-life; but the mean life and half-life t_1/2 have been adopted as standard times associated with exponential decay.

Those parameters can be related to the following time-dependent parameters:

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

These are related as follows:

${\displaystyle {\begin{aligned}t_{1/2}&={\frac {\ln(2)}{\lambda }}=\tau \ln(2)\\[2pt]A&=-{\frac {\mathrm {d} N}{\mathrm {d} t}}=\lambda N={\frac {\ln(2)}{t_{1/2}}}N\\[2pt]S_{A}a_{0}&=-{\frac {\mathrm {d} N}{\mathrm {d} t}}{\bigg |}_{t=0}=\lambda N_{0}\end{aligned}}}$

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

Mathematics

Universal law

The mathematics of radioactive decay depend on a key assumption that a nucleus of a radionuclide has no "memory" or way of translating its history into its present behavior. A nucleus does not "age" with the passage of time. Thus, the probability of its breaking down does not increase with time but stays constant, no matter how long the nucleus has existed. This constant probability may differ greatly between one type of nucleus and another, leading to the many different observed decay rates. However, whatever the probability is, it does not change over time. This is in marked contrast to complex objects that do show aging, such as automobiles and humans. These aging systems do have a chance of breakdown per unit of time that increases from the moment they begin their existence.

Aggregate processes, like the radioactive decay of a lump of atoms, for which the single-event probability of realization is very small but in which the number of time-slices is so large that there is nevertheless a reasonable rate of events, are modelled by the Poisson distribution, which is discrete. Radioactive decay and nuclear particle reactions are two examples of such aggregate processes.^[33] The mathematics of Poisson processes reduce to the law of exponential decay, which describes the statistical behaviour of a large number of nuclei, rather than one individual nucleus. In the following formalism, the number of nuclei or the nuclei population N, is of course a discrete variable (a natural number)—but for any physical sample N is so large that it can be treated as a continuous variable. Differential calculus is used to model the behaviour of nuclear decay.

One-decay process

Consider the case of a nuclide A that decays into another B by some process A → B (emission of other particles, like electron neutrinos
ν
_e and electrons e⁻ as in beta decay, are irrelevant in what follows). The decay of an unstable nucleus is entirely random in time so it is impossible to predict when a particular atom will decay. However, it is equally likely to decay at any instant in time. Therefore, given a sample of a particular radioisotope, the number of decay events −dN expected to occur in a small interval of time dt is proportional to the number of atoms present N, that is^[34]

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

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

The negative sign indicates that N decreases as time increases, as the decay events follow one after another. The solution to this first-order

${\displaystyle N(t)=N_{0}\,e^{-{\lambda }t}}$

where N₀ is the value of N at time t = 0, with the decay constant expressed as λ^[34]

We have for all time t:

${\displaystyle N_{A}+N_{B}=N_{\text{total}}=N_{A0},}$

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

If the number of non-decayed A nuclei is:

${\displaystyle N_{A}=N_{A0}e^{-\lambda t}}$

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

${\displaystyle N_{B}=N_{A0}-N_{A}=N_{A0}-N_{A0}e^{-\lambda t}=N_{A0}\left(1-e^{-\lambda t}\right).}$

The number of decays observed over a given interval obeys

Now consider the case of a chain of two decays: one nuclide A decaying into another B by one process, then B decaying into another C by a second process, i.e. A → B → C. The previous equation cannot be applied to the decay chain, but can be generalized as follows. Since A decays into B, then B decays into C, the activity of A adds to the total number of B nuclides in the present sample, before those B nuclides decay and reduce the number of nuclides leading to the later sample. In other words, the number of second generation nuclei B increases as a result of the first generation nuclei decay of A, and decreases as a result of its own decay into the third generation nuclei C.[35] The sum of these two terms gives the law for a decay chain for two nuclides:

${\displaystyle {\frac {\mathrm {d} N_{B}}{\mathrm {d} t}}=-\lambda _{B}N_{B}+\lambda _{A}N_{A}.}$

The rate of change of N_B, that is dN_B/dt, is related to the changes in the amounts of A and B, N_B can increase as B is produced from A and decrease as B produces C.

Re-writing using the previous results:

The subscripts simply refer to the respective nuclides, i.e. N_A is the number of nuclides of type A; N_A0 is the initial number of nuclides of type A; λ_A is the decay constant for A – and similarly for nuclide B. Solving this equation for N_B gives:

${\displaystyle N_{B}={\frac {N_{A0}\lambda _{A}}{\lambda _{B}-\lambda _{A}}}\left(e^{-\lambda _{A}t}-e^{-\lambda _{B}t}\right).}$

In the case where B is a stable nuclide (λ_B = 0), this equation reduces to the previous solution:

${\displaystyle \lim _{\lambda _{B}\rightarrow 0}\left[{\frac {N_{A0}\lambda _{A}}{\lambda _{B}-\lambda _{A}}}\left(e^{-\lambda _{A}t}-e^{-\lambda _{B}t}\right)\right]={\frac {N_{A0}\lambda _{A}}{0-\lambda _{A}}}\left(e^{-\lambda _{A}t}-1\right)=N_{A0}\left(1-e^{-\lambda _{A}t}\right),}$

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

Chain of any number of decays

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

${\displaystyle {\frac {\mathrm {d} N_{j}}{\mathrm {d} t}}=-\lambda _{j}N_{j}+\lambda _{j-1}N_{(j-1)0}e^{-\lambda _{j-1}t}.}$

The general solution to the recursive problem is given by Bateman's equations:^[36]

Alternative modes

In all of the above examples, the initial nuclide decays into just one product.

${\displaystyle N=N_{A}+N_{B}+N_{C}}$

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

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} N_{A}}{\mathrm {d} t}}&=-\left({\frac {\mathrm {d} N_{B}}{\mathrm {d} t}}+{\frac {\mathrm {d} N_{C}}{\mathrm {d} t}}\right)\\-\lambda N_{A}&=-N_{A}\left(\lambda _{B}+\lambda _{C}\right)\\\end{aligned}}}$

defining the total decay constant λ in terms of the sum of partial decay constants λ_B and λ_C:

${\displaystyle \lambda =\lambda _{B}+\lambda _{C}.}$

Solving this equation for N_A:

${\displaystyle N_{A}=N_{A0}e^{-\lambda t}.}$

where N_A0 is the initial number of nuclide A. When measuring the production of one nuclide, one can only observe the total decay constant λ. The decay constants λ_B and λ_C determine the probability for the decay to result in products B or C as follows:

${\displaystyle N_{B}={\frac {\lambda _{B}}{\lambda }}N_{A0}\left(1-e^{-\lambda t}\right),}$

${\displaystyle N_{C}={\frac {\lambda _{C}}{\lambda }}N_{A0}\left(1-e^{-\lambda t}\right).}$

because the fraction λ_B/λ of nuclei decay into B while the fraction λ_C/λ of nuclei decay into C.

Corollaries of laws

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

• The activity: A = λN.
• The amount of substance: n = N/N_A.
• The mass: m = Mn = MN/N_A.

where N_A = 6.02214076×10²³ mol^−1[38] is the Avogadro constant, M is the molar mass of the substance in kg/mol, and the amount of the substance n is in moles.

Decay timing: definitions and relations

Time constant and mean-life

For the one-decay solution A → B:

${\displaystyle N=N_{0}\,e^{-{\lambda }t}=N_{0}\,e^{-t/\tau },\,\!}$

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

In a radioactive decay process, this time constant is also the

Half-life

A more commonly used parameter is the half-life T_1/2. 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:

${\displaystyle N=N_{0}\,e^{-{\lambda }t}=N_{0}\,e^{-t/\tau },\,\!}$

the half-life is related to the decay constant as follows: set N = N₀/2 and t = T_1/2 to obtain

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

This relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent, while those that radiate weakly endure longer. Half-lives of known radionuclides vary by almost 54 orders of magnitude, from more than 2.25(9)×10²⁴ years (6.9×10³¹ sec) for the very nearly stable nuclide ¹²⁸Te, to 8.6(6)×10⁻²³ seconds for the highly unstable nuclide ⁵H.^[26]

The factor of ln(2) in the above relations results from the fact that the concept of "half-life" is merely a way of selecting a different base other than the natural base e for the lifetime expression. The time constant τ is the e −1 -life, the time until only 1/e remains, about 36.8%, rather than the 50% in the half-life of a radionuclide. Thus, τ is longer than t_1/2. The following equation can be shown to be valid:

${\displaystyle N(t)=N_{0}\,e^{-t/\tau }=N_{0}\,2^{-t/t_{1/2}}.\,\!}$

Since radioactive decay is exponential with a constant probability, each process could as easily be described with a different constant time period that (for example) gave its "(1/3)-life" (how long until only 1/3 is left) or "(1/10)-life" (a time period until only 10% is left), and so on. Thus, the choice of τ and t_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.

Mathematically, the n^th life for the above situation would be found in the same way as above—by setting N = N₀/n, t = T_1/n and substituting into the decay solution to obtain

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

Example for carbon-14

Carbon-14 has a half-life of 5700(30) years^[26] and a decay rate of 14 disintegrations per minute (dpm) per gram of natural carbon.

If an artifact is found to have radioactivity of 4 dpm per gram of its present C, we can find the approximate age of the object using the above equation:

${\displaystyle N=N_{0}\,e^{-t/\tau },}$

where:

${\displaystyle {\begin{aligned}{\frac {N}{N_{0}}}&=4/14\approx 0.286,\\\tau &={\frac {T_{1/2}}{\ln 2}}\approx 8267{\text{ years}},\\t&=-\tau \,\ln {\frac {N}{N_{0}}}\approx 10356{\text{ years}}.\end{aligned}}}$

Changing rates

The radioactive decay modes of electron capture and internal conversion are known to be slightly sensitive to chemical and environmental effects that 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 nuclides are affected.

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.^[45] Comparison of laboratory experiments over the last century, studies of the Oklo natural nuclear reactor (which exemplified the effects of thermal neutrons on nuclear decay), and astrophysical observations of the luminosity decays of distant supernovae (which occurred far away so the light has taken a great deal of time to reach us), for example, strongly indicate that unperturbed decay rates have been constant (at least to within the limitations of small experimental errors) as a function of time as well.^{[citation needed]}

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

An unexpected series of experimental results for the rate of decay of heavy highly charged radioactive ions circulating in a storage ring has provoked theoretical activity in an effort to find a convincing explanation. The rates of weak decay of two radioactive species with half lives of about 40 s and 200 s are found to have a significant oscillatory modulation, with a period of about 7 s.^[52] The observed phenomenon is known as the

Such a collapse (a gamma-ray 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 decaying by gamma radiation in a spontaneous emission of electromagnetic radiation, the arbitrarily small disturbance is theorized to come from quantum vacuum fluctuations.^[55]

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).^[56]

Hazard warning signs

• 
  The trefoil symbol used to warn of presence of radioactive material or ionising radiation
• 
  2007 ISO radioactivity hazard symbol intended for IAEA Category 1, 2 and 3 sources defined as dangerous sources capable of death or serious injury^[57]
• 
  One of several dangerous goods transport classification signs for radioactive materials

See also

 Nuclear technology portal  Physics portal

Notes

References

Inline

^[1][2]

General

• "Radioactivity", Encyclopædia Britannica. 2006.
  Encyclopædia Britannica Online
  . December 18, 2006
• Radio-activity by Ernest Rutherford Phd, Encyclopædia Britannica Eleventh Edition

External links

The Wikibook Historical Geology has a page on the topic of: Radioactive decay

Look up radioactivity in Wiktionary, the free dictionary.

• The Lund/LBNL Nuclear Data Search – Contains tabulated information on radioactive decay types and energies.
• Nomenclature of nuclear chemistry
• Specific activity and related topics.
• The Live Chart of Nuclides – IAEA
• Interactive Chart of Nuclides Archived 10 October 2018 at the Wayback Machine
• Health Physics Society Public Education Website
• Beach, Chandler B., ed. (1914). "Becquerel Rays" . The New Student's Reference Work . Chicago: F. E. Compton and Co.
• Annotated bibliography for radioactivity from the Alsos Digital Library for Nuclear Issues Archived 7 October 2010 at the Wayback Machine
• Stochastic Java applet on the decay of radioactive atoms Archived 13 August 2011 at the Wayback Machine by Wolfgang Bauer
• Stochastic Flash simulation on the decay of radioactive atoms by David M. Harrison
• "Henri Becquerel: The Discovery of Radioactivity", Becquerel's 1896 articles online and analyzed on BibNum [click 'à télécharger' for English version].
• "Radioactive change", Rutherford & Soddy article (1903), online and analyzed on Bibnum [click 'à télécharger' for English version]

1. ISSN 0449-3060
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2. ISSN 2322-3480
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