Radioactive decay

# Overview
Radioactive decay is the process in which an unstable  atomic nucleus  loses energy by emitting radiation in the form of particles or electromagnetic waves.  This decay, or loss of energy, results in an atom of one type, called the parent nuclide transforming to an atom of a different type, called the daughter nuclide.  For example: a carbon-14 atom (the "parent") emits radiation and transforms to a nitrogen-14 atom (the "daughter"). This is a random process on the atomic level, in that it is impossible to predict when a particular atom will decay, but given a large number of similar atoms, the decay rate, on average, is predictable.
The SI unit of radioactive decay (the phenomenon of natural and artificial radioactivity) is the becquerel (Bq).  One Bq is defined as one transformation (or decay) per second. Since any reasonably-sized sample of radioactive material contains many atoms, a Bq is a tiny measure of activity; amounts on the order of TBq (terabecquerel) or GBq (gigabecquerel) are commonly used.  Another unit of (radio)activity is the curie, Ci, which was originally defined as the activity of one gram of pure radium, isotope Ra-226. At present it is equal (by definition) to the activity of any radionuclide decaying with a disintegration rate of 3.7  1010 Bq. The use of Ci is presently discouraged by SI.

# Explanation
The neutrons and protons that constitute nuclei, as well as other particles that may approach them, are governed by several interactions. The strong nuclear force, not observed at the familiar macroscopic scale, is the most powerful force over subatomic distances. The electrostatic force is also significant, while the weak nuclear force is responsible for beta decay.
The interplay of these forces is simple.  Some configurations of the particles in a nucleus have the property that, should they shift ever so slightly, the particles could fall into a lower-energy arrangement (with the extra energy moving elsewhere).  One might draw an analogy with a snowfield on a mountain: while friction between the snow crystals can support the snow's weight, the system is inherently unstable with regard to a lower-potential-energy state, and a disturbance may facilitate the path to a greater entropy state (i.e., towards the ground state where heat will be produced, and thus total energy is distributed over a larger number of quantum states). Thus, an avalanche results. The total energy does not change in this process, but because of entropy effects, avalanches only happen in one direction, and the end of this direction, which is dictated by the largest number of chance-mediated ways to distribute available energy, is what we commonly refer to as the "ground state".
Such a collapse (a decay event) requires a specific activation energy.  In the case of a snow avalanche, this energy classically comes as a disturbance from outside the system, although such disturbances can be arbitrarily small.  In the case of an excited atomic nucleus, the arbitrarily small disturbance comes from quantum vacuum fluctuations. A nucleus (or any excited system in quantum mechanics) is unstable, and can thus spontaneously stabilize to a less-excited system. This process is driven by entropy considerations: the energy does not change, but at the end of the process, the total energy is more diffused in spacial volume. The resulting transformation alters the structure of the nucleus. Such a reaction is thus a nuclear reaction, in contrast to chemical reactions, which also are driven by entropy, but which involve changes in the arrangement of the outer electrons of atoms, rather than their nuclei.
Some nuclear reactions do involve external sources of energy, in the form of collisions with outside particles. However, these are not considered decay. Rather, they are examples of induced nuclear reactions. Nuclear fission and fusion are common types of induced nuclear reactions.

# Discovery
Radioactivity was first discovered in 1896 by the French scientist Henri Becquerel while working on phosphorescent materials. These materials glow in the dark after exposure to light, and he thought that the glow produced in cathode ray tubes by X-rays might be connected with phosphorescence. He wrapped a photographic plate in black paper and placed various phosphorescent minerals on it. All results were negative until he used uranium salts. The result with these compounds was a deep blackening of the plate.
It soon became clear that the blackening of the plate had nothing to do with phosphorescence, because the plate blackened when the mineral was in the dark. Non-phosphorescent salts of uranium and metallic uranium also blackened the plate. Clearly there was a form of radiation that could pass through paper that was causing the plate to blacken.
At first it seemed that the new radiation was similar to the then recently discovered X-rays. Further research by Becquerel, Marie Curie, Pierre Curie, Ernest Rutherford and others discovered that radioactivity was significantly more complicated. Different types of decay can occur, but Rutherford was the first to realize that they all occur with the same mathematical approximately exponential formula (see below).
As for types of radioactive radiation, it was found that an electric or magnetic field could split such emissions into three types of beams. For lack of better terms, the rays were given the alphabetic names alpha, beta and gamma, still in use today. It was obvious from the direction of electromagnetic forces that alpha rays carried a positive charge,  beta rays carried a negative charge, and gamma rays were neutral.  From the magnitude of deflection, it was clear that alpha particles were much more massive than beta particles.  Passing alpha particles through a very thin glass window and trapping them in a discharge tube allowed researchers to study the emission spectrum of the resulting gas, and ultimately prove that alpha particles are helium nuclei.  Other experiments showed the similarity between beta radiation and cathode rays; they are both streams of electrons, and between gamma radiation and X-rays, which are both high energy electromagnetic radiation.
Although alpha, beta, and gamma are most common, other types of decay were eventually discovered. Shortly after discovery of the neutron in 1932, it was discovered by Enrico Fermi that certain rare decay reactions yield neutrons as a decay particle. Isolated proton emission was eventually observed in some elements. 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), analogously to negative electrons. Each of the two types of beta decay acts to move a nucleus toward a ratio of neutrons and protons which has the least energy for the combination. Finally, in a phenomenon called cluster decay, specific combinations of neutrons and protons other than alpha particles were spontaneously emitted from atoms on occasion.
Still other types of radioactive decay were found which emit previously seen particles, but by different mechanisms. An example is internal conversion, which results in electron and sometimes high energy photon emission, even though it involves neither beta nor gamma decay.
The early researchers also discovered that many other chemical elements besides uranium have radioactive isotopes.  A systematic search for the total radioactivity in uranium ores also guided Marie Curie to isolate a new element polonium and to separate a new element radium from barium. The two elements' chemical similarity would otherwise have made them difficult to distinguish.
The dangers of radioactivity and of radiation were not immediately recognized. Acute effects of radiation were first observed in the use of X-rays when the Serbo-Croatian-American electric engineer Nikola Tesla intentionally subjected his fingers to X-rays in 1896. He published his observations concerning the burns that developed, though he attributed them to ozone rather than to X-rays. His injuries healed later.
The genetic effects of radiation, including the effects on cancer risk, were recognized much later. In 1927 Hermann Joseph Muller published research showing genetic effects, and in 1946 was awarded the Nobel prize for his findings.
Before the biological effects of radiation were known, many physicians and corporations had begun marketing radioactive substances as patent medicine and radioactive quackery. Examples were radium enema treatments, and radium-containing waters to be drunk as tonics. Marie Curie spoke out against this sort of treatment, warning that the effects of radiation on the human body were not well understood (Curie later died from aplastic anemia assumed due to her work with radium, but later examination of her bones showed that she had been a careful laboratory worker and had a low burden of radium. A more likely cause was her exposure to unshielded X-ray tubes while a volunteer medical worker in WWI). By the 1930s, after a number of cases of bone necrosis and death in enthusiasts, radium-containing medical products had nearly vanished from the market.

# Modes of decay
Radionuclides can undergo a number of different reactions.  These are summarized in the following table. A nucleus with mass number A and atomic number Z is represented as (A, Z). The column "Daughter nucleus" indicates the difference between the new nucleus and the original nucleus.  Thus, (A1, Z) means that the mass number is one less than before, but the atomic number is the same as before.
Radioactive decay results in a reduction of summed rest mass, which is converted to energy (the disintegration energy) according to the formula E = mc^2.  This energy is released as kinetic energy of the emitted particles. The energy remains associated with a measure of mass of the decay system invariant mass, inasmuch the kinetic energy of emitted particles contributes also to the total invariant mass of systems. Thus, the sum of rest masses of particles is not conserved in decay, but the system mass or system invariant mass (as also system total energy) is conserved.

# Decay chains and multiple modes
The daughter nuclide of a decay event may also be unstable (radioactive).  In this case, it will also decay, producing radiation.  The resulting second daughter nuclide may also be radioactive.  This can lead to a sequence of several decay events.  Eventually a stable nuclide is produced.  This is called a decay chain.
An example is the natural decay chain of uranium-238 which is as follows:
- decays, through alpha-emission, with a half-life of 4.5 billion years to thorium-234
- which decays, through beta-emission, with a half-life of 24 days to protactinium-234
- which decays, through beta-emission, with a half-life of 1.2 minutes to uranium-234
- which decays, through alpha-emission, with a half-life of 240 thousand years to thorium-230
- which decays, through alpha-emission, with a half-life of 77 thousand years to radium-226
- which decays, through alpha-emission, with a half-life of 1.6 thousand years to radon-222
- which decays, through alpha-emission, with a half-life of 3.8 days to polonium-218
- which decays, through alpha-emission, with a half-life of 3.1 minutes to lead-214
- which decays, through beta-emission, with a half-life of 27 minutes to bismuth-214
- which decays, through beta-emission, with a half-life of 20 minutes to polonium-214
- which decays, through alpha-emission, with a half-life of 160 microseconds to lead-210
- which decays, through beta-emission, with a half-life of 22 years to bismuth-210
- which decays, through beta-emission, with a half-life of 5 days to polonium-210
- which decays, through alpha-emission, with a half-life of 140 days to lead-206, which is a stable nuclide.
Some radionuclides may have several different paths of decay.  For example, approximately 36% of bismuth-212, decays, through alpha-emission, to thallium-208 while approximately 64% of bismuth-212 decays, through beta-emission, to polonium-212.  Both the thallium-208 and the polonium-212 are radioactive daughter products of bismuth-212, and both decay directly to stable lead-208.

# Occurrence and applications
According to the Big Bang theory, stable isotopes of the lightest five elements (H, He, and traces of Li, Be, and B) were produced very shortly after the emergence of the universe, in a process called Big Bang nucleosynthesis. These lightest stable nuclides (including deuterium) survive to today, but any radioactive isotopes of the light elements produced in the Big Bang (such as tritium) have long since decayed. Isotopes of elements heavier than boron were not produced at all in the Big Bang, and these first five elements do not have any long-lived radioisotopes. Thus, all radioactive nuclei are therefore relatively young with respect to the birth of the universe, having formed later in various other types of nucleosynthesis in stars (particularly supernovae), and also during ongoing interactions between stable isotopes and energetic particles. For example, carbon-14, a radioactive nuclide with a half-life of only 5730 years, is constantly produced in Earth's upper atmosphere due to interactions between cosmic rays and nitrogen.
Radioactive decay has been put to use in the technique of radioisotopic labeling, 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 chaotic), 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-live 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 which may also be independently cross-checked by other means (such as checking the carbon-14 in individual tree rings, for example).

# Radioactive decay rates
The decay rate, or activity, of a radioactive substance  are characterized by:
Constant quantities:
- half life  symbol  t_{1/2}   the time taken for the activity of a given amount of a radioactive substance to decay to half of its initial value.
- mean lifetime  symbol  \tau   the average lifetime of a radioactive particle.
- decay constant  symbol  \lambda   the inverse of the mean lifetime.
Time-variable quantities:
- Total activity  symbol A  number of decays an object undergoes per second.
- Number of particles  symbol N  the total number of particles in the sample.
- Specific activity  symbol S_A  number of decays per second per amount of substance. (The "amount of substance" can be the unit of either mass or volume.)
These are related as follows:

## Activity measurements
The units in which activities are measured are: becquerel  (symbol Bq) = number of disintegrations per second; curie (Ci) = 3.7  1010 disintegrations per second. Low activities are also measured in disintegrations per minute (dpm).

# Decay timing
As discussed above, the decay of an unstable nucleus  is entirely random and it is impossible to predict when a particular atom will decay.  However, it is equally likely to decay at any 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.  If N is the number of atoms, then the probability of decay ( dN/N) is proportional to dt:
Particular radionuclides decay at different rates, each having its own decay constant ().  The negative sign indicates that N decreases with each decay event.  The solution to this first-order differential equation is the following function:
Where N0 is the amount of N at time zero (t = 0). The second equation recognizes that the differential decay constant  has units of 1/time, and can thus also be represented as 1/\tau, where \tau is a characteristic time for the process. This characteristic time is called the time constant of the process. 
In radioactive decay, this process time constant it 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 \tau, which again is related to the decay constant as follows:
The previous exponential function generally represents the result of exponential decay. It is only an approximate solution, for two reasons.  Firstly, the exponential function is continuous, but the physical quantity N can only take non-negative integer values.  Secondly, because it describes a random process, it is only statistically true.  However, in most common cases, N is a very large number and the function is a good approximation.
A more commonly used parameter is the half-life.  Given a sample of a particular radionuclide, the half-life is the time taken for half the radionuclide's atoms to decay. The half life is related to the decay constant as follows:
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 widely, from more than 1019 years (such as for very nearly stable nuclides, e.g. 209Bi), to 10-23 seconds for highly unstable ones.