Radioactive decay

 ( Exponentials ) 

Other experimenters, including Elihu Thomson and Nikola Tesla, also reported burns. Thomson deliberately exposed a finger to an X-ray tube over a period of time and suffered pain, swelling, and blistering. Other effects, including ultraviolet rays and ozone, were sometimes blamed for the damage, and many physicians still claimed that there were no effects from X-ray exposure at all.

Despite this, there were some early systematic hazard investigations, and as early as 1902 William Herbert Rollins wrote almost despairingly that his warnings about the dangers involved in the careless use of X-rays were not being heeded, either by industry or by his colleagues. By this time, Rollins had proved that X-rays could kill experimental animals, could cause a pregnant guinea pig to abort, and that they could kill a foetus. He also stressed that "animals vary in susceptibility to the external action of X-light" and warned that these differences be considered when patients were treated by means of X-rays.

The second ICR was held in Stockholm in 1928 and proposed the adoption of the röntgen unit, and the International X-ray and Radium Protection Committee (IXRPC) was formed. Rolf Sievert was named chairman, but a driving force was George Kaye of the British National Physical Laboratory. The committee met in 1931, 1934, and 1937.

After World War II, the increased range and quantity of radioactive substances being handled as a result of military and civil nuclear programs led to large groups of occupational workers and the public being potentially exposed to harmful levels of ionising radiation. This was considered at the first post-war ICR convened in London in 1950, when the present International Commission on Radiological Protection (ICRP) was born. Since then the ICRP has developed the present international system of radiation protection, covering all aspects of radiation hazards.

In 2020, Hauptmann and another 15 international researchers from eight nations (among them: Institutes of Biostatistics, Registry Research, Centers of Cancer Epidemiology, Radiation Epidemiology, and also the U.S. National Cancer Institute (NCI), International Agency for Research on Cancer (IARC) and the Radiation Effects Research Foundation of Hiroshima) studied definitively through meta-analysis the damage resulting from the "low doses" that have afflicted survivors of the atomic bombings of Hiroshima and Nagasaki and also in numerous accidents at nuclear plants that have occurred. These scientists reported, in JNCI Monographs: Epidemiological Studies of Low Dose Ionizing Radiation and Cancer Risk, that the new epidemiological studies directly support excess cancer risks from low-dose ionizing radiation. In 2021, Italian researcher Sebastiano Venturi reported the first correlations between radio-caesium and pancreatic cancer with the role of caesium in biology, in pancreatitis and in diabetes of pancreatic origin. Text was copied from this source, which is available under a Creative Commons Attribution 4.0 International License.

An older unit of radioactivity is the curie, Ci, which was originally defined as "the quantity or mass of radium emanation in equilibrium with one gram of radium (element)". Today, the curie is defined as disintegrations per second, so that 1 curie (Ci) = . For radiological protection purposes, although the United States Nuclear Regulatory Commission permits the use of the unit curie alongside SI units, the European Union European units of measurement directives required that its use for "public health ... purposes" be phased out by 31 December 1985.

The effects of ionizing radiation are often measured in units of gray for mechanical or sievert for damage to tissue.

Decay energy, therefore, remains associated with a certain measure of the mass of the decay system, called invariant mass, which does not change during the decay, even though the energy of decay is distributed among decay particles. The energy of photons, the kinetic energy of emitted particles, and, later, the thermal energy of the surrounding matter, all contribute to the invariant mass of the system. Thus, while the sum of the rest masses of the 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.

The relationship between the types of decays also began to be examined: For example, gamma decay was almost always found to be associated with other types of decay, and occurred at about the same time, or afterwards. 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 , which were in turn created from other types of decay. Although alpha, beta, and gamma radiations were most commonly found, other types of emission 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), along with (classical beta decay produces antineutrinos).

A hypothetical process of positron capture, analogous to electron capture, is theoretically possible in antimatter atoms, but has not been observed, as complex antimatter atoms beyond antihelium are not experimentally available. Such a decay 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.

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 bound state beta decay of rhenium-187. In this process, the 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 is emitted, as in all negative beta decays.

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 copper-64, which has 29 protons, and 35 neutrons, which decays with a half-life of hours. This isotope has one unpaired proton and one unpaired neutron, so either the proton or the neutron can decay to the other particle, which has opposite isospin. This particular nuclide (though not all nuclides in this situation) is more likely to decay through beta plus decay (%) than through electron capture (%). The excited energy states resulting from these decays which fail to end in a ground energy state, also produce later internal conversion and gamma decay in almost 0.5% of the time.

Some radionuclides may have several different paths of decay. For example, % of bismuth-212 decays, through alpha-emission, to thallium-208 while % 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.

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

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

The Szilard–Chalmers effect was discovered in 1934 by Leó Szilárd and Thomas A. Chalmers. They observed that after bombardment by neutrons, the breaking of a bond in liquid ethyl iodide allowed radioactive iodine to be removed.

• The half-life, , 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, "lambda", the reciprocal of the mean lifetime (in ), sometimes referred to as simply decay rate.
• The mean lifetime, "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, could be used in exactly the same way as half-life; but the mean life and half-life 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), , is the number of decays per unit time of a radioactive sample.
• Number of particles, , in the sample.
• Specific activity, , is the number of decays per unit time per amount of substance of the sample at time set to zero (). "Amount of substance" can be the mass, volume or moles of the initial sample.

These are related as follows:

$\backslash begin\{align\}$

  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
     

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 reaction are two examples of such aggregate processes. 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.

$-\; \backslash frac\{\backslash mathrm\{d\}N\}\{\backslash mathrm\{d\}t\}\; \backslash propto\; N$

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

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

$N(t)\; =\; N\_0\backslash ,e^\{-\{\backslash lambda\}t\}$

where is the value of at time = 0, with the decay constant expressed as

We have for all time :

$N\_A\; +\; N\_B\; =\; N\_\backslash text\{total\}\; =\; N\_\{A0\},$

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

If the number of non-decayed nuclei is:

$N\_A\; =\; N\_\{A0\}e^\{-\backslash lambda\; t\}$

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

$N\_B\; =\; N\_\{A0\}\; -\; N\_A\; =\; N\_\{A0\}\; -\; N\_\{A0\}e^\{-\backslash lambda\; t\}\; =\; N\_\{A0\}\; \backslash left(\; 1\; -\; e^\{-\backslash lambda\; t\}\; \backslash right).$

The number of decays observed over a given interval obeys Poisson statistics. If the average number of decays is , the probability of a given number of decays is

$P(N)\; =\; \backslash frac\{\backslash langle\; N\; \backslash rangle^N\; \backslash exp(-\backslash langle\; N\backslash rangle)\}\{N!\}\; .$

$\backslash frac\{\backslash mathrm\{d\}N\_B\}\{\backslash mathrm\{d\}t\}\; =\; -\backslash lambda\_B\; N\_B\; +\; \backslash lambda\_A\; N\_A.$

The rate of change of , that is , is related to the changes in the amounts of and , can increase as is produced from and decrease as produces .

Re-writing using the previous results:

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

$N\_B\; =\; \backslash frac\{N\_\{A0\}\backslash lambda\_A\}\{\backslash lambda\_B\; -\; \backslash lambda\_A\}\; \backslash left(\; e^\{-\backslash lambda\_A\; t\}\; -\; e^\{-\backslash lambda\_B\; t\}\backslash right)\; .$

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

 \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 . 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.

$\backslash frac\{\backslash mathrm\{d\}N\_j\}\{\backslash mathrm\{d\}t\}\; =\; -\; \backslash lambda\_j\; N\_j\; +\; \backslash lambda\_\{j-1\}\; N\_\{(j-1)0\}\; e^\{-\backslash lambda\_\{j-1\}\; t\}.$

The general solution to the recursive problem is given by Bateman's equations:

$N\; =\; N\_A\; +\; N\_B\; +\; N\_C$

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

$\backslash begin\{align\}$

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

$\backslash lambda\; =\; \backslash lambda\_B\; +\; \backslash lambda\_C\; .$

Solving this equation for :

$N\_A\; =\; N\_\{A0\}\; e^\{-\backslash lambda\; t\}\; .$

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

$N\_B\; =\; \backslash frac\{\backslash lambda\_B\}\{\backslash lambda\}\; N\_\{A0\}\; \backslash left\; (\; 1\; -\; e^\{-\backslash lambda\; t\}\; \backslash right\; ),$

$N\_C\; =\; \backslash frac\{\backslash lambda\_C\}\{\backslash lambda\}\; N\_\{A0\}\; \backslash left\; (\; 1\; -\; e^\{-\backslash lambda\; t\}\; \backslash right\; ).$

because the fraction of nuclei decay into while the fraction of nuclei decay into .

• The activity: .
• The amount of substance: .
• The mass: .

where = is the Avogadro constant, is the molar mass of the substance in kg/mol, and the amount of the substance is in moles.

$N\; =\; N\_0\backslash ,e^\{-\{\backslash lambda\}t\}\; =\; N\_0\backslash ,e^\{-t/\; \backslash tau\},\; \backslash ,\backslash !$

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

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

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

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

$\backslash lambda\; =\; \backslash lambda\_B\; +\; \backslash lambda\_C\; \backslash ,$

into the decay solution leads to:

$\backslash frac\{1\}\{\backslash tau\}\; =\; \backslash lambda\; =\; \backslash lambda\_B\; +\; \backslash lambda\_C\; =\; \backslash frac\{1\}\{\backslash tau\_B\}\; +\; \backslash frac\{1\}\{\backslash tau\_C\}\backslash ,$

$N\; =\; N\_0\backslash ,e^\{-\{\backslash lambda\}t\}\; =\; N\_0\backslash ,e^\{-t/\backslash tau\},\; \backslash ,\backslash !$

the half-life is related to the decay constant as follows: set and = to obtain

$t\_\{1/2\}\; =\; \backslash frac\{\backslash ln\; 2\}\{\backslash lambda\}\; =\; \backslash tau\; \backslash ln\; 2.$

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 years ( sec) for the very nearly stable nuclide ¹²⁸Te, to seconds for the highly unstable nuclide ⁵H.

The factor of 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 for the lifetime expression. The time constant is the -life, the time until only 1/ e remains, about 36.8%, rather than the 50% in the half-life of a radionuclide. Thus, is longer than . The following equation can be shown to be valid:

$N(t)\; =\; N\_0\backslash ,e^\{-t/\backslash tau\}\; =\; N\_0\backslash ,2^\{-t/t\_\{1/2\}\}.\; \backslash ,\backslash !$

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 and 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 life for the above situation would be found in the same way as aboveby setting , and substituting into the decay solution to obtain

$t\_\{1/n\}\; =\; \backslash frac\{\backslash ln\; n\}\{\backslash lambda\}\; =\; \backslash tau\; \backslash ln\; n.$

 \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}.
     

• alpha-decay -> strong interaction,
• beta-decay -> weak interaction,
• gamma-decay -> electromagnetism.