The Bateman equations for the radioactive decay of linear chain n-nuklide series, which describe nuclide concentrations, are shown in the figure as follows. Other types of radioactive decay have been found to emit previously seen particles, but via different mechanisms. An example is internal conversion, which results in the initial emission of electrons and then often other characteristic X-ray and Auger electron emissions, although the internal conversion process does not involve beta decay or gamma decay. A neutrino is not emitted, and none of the emitted electrons and photons come from the nucleus, although the energy that emits them all comes from there. Internal transformation decay, such as isomeric transition gamma decay and neutron emission, involves the release of energy by an excited nuclide, without transmutation from one element to another. About 50 other short-lived radionuclides, such as radium-226 and radon-222, found on Earth are the products of decay chains that began with urnuclides, or are the product of ongoing cosmogenic processes, such as the production of carbon-14 from nitrogen-14 in the atmosphere by cosmic rays. Radionuclides can also be produced artificially in particle accelerators or nuclear reactors, resulting in 650 of them with half-lives of more than an hour and several thousand others with even shorter half-lives. (See List of nuclides for a list of nuclides sorted by half-life.) because the λB/λ fraction of the nuclei decays into B, while the λC/λ fraction of the nuclei decays into C. Consider the case of a nuclide A, which decays into another B by a process A → B (the emission of other particles, such as electron neutrinos νe and electrons E− as in beta decay, are not relevant below).

The decay of an unstable nucleus is completely random over time, making it impossible to predict when a particular atom will decay. However, it is also likely to disintegrate at some point. Therefore, for a sample of a particular radioisotope, the number of decay events −dN expected in a small time interval dt is proportional to the number of atoms present, i.e. [29], where λ is the proportionality constant (or radioactive decay constant or decay constant). The negative sign indicates that N decreases over time as decay events follow one another. The solution to this first-order differential equation is function: radioactive decay results in a reduction in the total mass at rest once the released energy (the decay energy) has somehow escaped. Although decay energy is sometimes defined as associated with the difference between the mass of parent nuclide products and the mass of decay products, this only applies to rest mass measurements where some of the energy has been removed from the product system. This is true because the decay energy must always carry the mass with it wherever it appears (see mass in the special theory of relativity) according to the formula E = mc2. The decay energy is first released as the energy of the emitted photons plus the kinetic energy of the massively emitted particles (i.e. particles that have a mass at rest). When these particles enter into thermal equilibrium with their environment and the photons are absorbed, the decay energy is converted into thermal energy that retains its mass. The nuclides formed by radioactive decay are called radiogenic nuclides, whether they are themselves stable or not.

There are stable radiogenic nuclides formed in the early solar system from short-lived extinct radionuclides. [52] [53] The additional presence of these stable radiogenic nuclides (such as xenon-129 from extinct iodine-129) in the context of primordial stable nuclides can be derived by various means. Shortly after the discovery of the neutron in 1932, Enrico Fermi realized that some rare beta decay reactions immediately produce neutrons in the form of decay particles (neutron emission). An isolated emission of protons was finally observed in some elements. It was also found that some heavy elements can undergo spontaneous cleavage into products of different composition. In a phenomenon called cluster decay, specific combinations of neutrons and protons that are not alpha particles (helium nuclei) have been found that are emitted spontaneously by atoms. Radioactive decay law: The number of nuclei undergoing decay per unit of time is proportional to the number of unchanged nuclei present at that time. The index characters simply refer to the respective nuclides, i.e. NA is the number of type A nuclides; NA0 is the initial number of type A nuclides; λA is the decay constant for A – and similar for nuclide B. The solution of this equation for NB is as follows: where N (number of particles) is the total number of particles in the sample, A (total activity) is the number of decays per unit of time of a radioactive sample, m is the mass of the remaining radioactive material.

Table with examples of half-lives and decay constants. Note that short half-lives are associated with large decay constants. Short-lived radioactive materials are much more radioactive, but obviously lose their radioactivity quickly. More common in severe nuclides is the competition between alpha and beta decay. The daughter nuclides then usually decay by beta or alpha to land in the same place. Radioactive decay is a stochastic (i.e. random) process at the level of individual atoms. According to quantum theory, it is impossible to predict when a particular atom will decay, regardless of how long it lasts. [2] [3] [4] However, for a significant number of identical atoms, the total decay rate can be expressed as a decay constant or half-life. The half-lives of radioactive atoms have a long range; from almost instantaneous to much longer than the age of the universe. The equation indicates that the decay constant λ has units of T−1 and can therefore also be represented as 1/τ, where τ is a time characteristic of the process called the time constant.

For the general case of any number of successive decays in a decay chain, i.e. A1 → A2 ··· → Ai ··· → AD, where D is the number of decays and i is a dummy index (i = 1, 2, 3, . D), any population of nuclides can be found in relation to the previous population. In this case, N2 = 0, N3 = 0,…, ND = 0. Where, N: the total number of nuclei in the sample Δ N: Number of decaying nuclei Δt: Unit of time The calculations of the decay of radioactive nuclei are relatively simple, because there is only one Basic Law that regulates all decay processes. When an alpha particle emits its nucleus, the process is called alpha decay. The alpha decay formula is given as follows: Let us now consider the case of a chain of two decays: a nuclide A decays by one process into another B, then B decays by a second process, that is, A → B → C. The previous equation cannot be applied to the decay chain, but can be generalized as follows. Since A decays to B, B decays to C, A`s activity increases to the total number of B nuclides in the current sample before these B nuclides decay and reduce the number of nuclides leading to the subsequent sample. In other words, the number of nuclei of the second generation B increases as a result of the decay of the nuclei of the first generation of A and decreases as a result of their own decay in the nuclei of the third generation C. [30] The sum of these two terms gives the law of a decay chain for two nuclides: The process of decay, like all energy conversions hindered, can be analogous by a snowfield on a mountain.

While the friction between the ice crystals can support the weight of the snow, the system is inherently unstable in terms of its lower potential energy state. A disturbance would thus facilitate the path to a state of greater entropy; The system moves to the ground state, generating heat, and the total energy will be distributed over a greater number of quantum states, resulting in an avalanche. Total energy does not change in this process, but due to the second law of thermodynamics, avalanches have only been observed in one direction, that is, in the direction of the “ground state” – the state with the greatest number of ways in which the available energy could be distributed. In 1992, Jung et al. of the Darmstadt Heavy Ion Research Group observed an accelerated decay of β of 163Dy66+.