Transmutation describes a process by which the nucleus of a radioactive atom undergoes decay into an atom with a different number of protons, until such time as a stable nucleus is produced.
An alpha particle (i.e., a helium nucleus) is released during alpha decay of a radioactive substance. An element with a lower mass is formed.
i.e.
In the above example uranium-238 releases an alpha particle from the nucleus, producing thorium-234. Mass is not conserved. Atomic mass number (or nucleon number, or baryon number) is conserved.
Beta decay (beta negative decay) occurs when a beta (negative) particle is released from the nucleus (i.e., electron).
i.e.
In the above example thorium-234 releases a beta particle, forming protactinium-234. Mass is also not conserved in beta decay. Nucleon number is conserved.
In beta decay, the beta particle released originated in the nucleus of the atom, not in the electron orbital. A neutron is lost, and in its place a proton and an electron are formed.
In general, if X is the parent nucleus and Y is the daughter nucleus:
For alpha decay:
Alpha decay can only occur if Mx > My + MHe. The atomic masses of He and Y are less than the mass of the parent atom, X. This "lost" mass is converted into energy (E = mc2) which appears as kinetic energy of the alpha particle.
For beta decay:
(v is a neutrino)
(Actually an antineutrino is produced for beta emission.)
My < Mx The mass deficit appears as kinetic energy of the electron (and energy of the neutrino).
A neutrino was "invented" to maintain conservation of energy, linear momentum, and angular momentum in beta decay. It has no mass, no charge, and virtually no interaction with matter. It travels at the speed of light and carries off energy and momentum.
During a nuclear transmutation, energy is released
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A series of nuclear transmutations occurs until a stable nucleus results. The series of steps in the transmutations is called a disintegration series (or decay series).
Gamma decay is the release of excess stored energy from the nucleus. No transmutation occurs. However, gamma decay often accompanies alpha and beta negative decay in a disintegration series.
Gamma decay occurs when an excited nucleus (excited by photon or particle bombardment, or it may be a decay product in an excited state) returns to the ground state. An excited nucleus is heavier than the ground state, by a mass equal to the mass/energy equivalent of the energy of the emitted gamma ray.
i.e.
(The asterisk indicates an "excited" nucleus.)
Nuclide charts, with atomic number plotted against neutron number, are used in nuclear physics to illustrate a disintegration series.
Background radiation comes from a variety of radioactive sources. Cosmic rays penetrating the Earth's atmosphere from outer space usually account for less than 25% of background radiation (but this depends on altitude).
Minute quantities of naturally occurring radioactive sources in the surroundings (e.g., soil, air, drinking water, building materials, food, etc.) also contribute to background radiation.
Each radioactive nuclide emits radioactivity at its characteristic rate, different from that of other nuclides. The rate of radioactive decay is related to the energy change that accompanies the transformation, but it is not a direct relationship.
The rate of radioactive emissions of a radioactive nuclide is directly proportional to the amount of radioactive material present.
where A is the activity in Bq, N is the number of radioactive nuclei, No is the initial number of radioactive nuclei at t = 0, 
is the decay constant, and t is time.
The decay constant is a measure of the rate at which the nuclide releases radioactive emissions.
The mean, or average lifetime of a nuclide, 
.
The rate of decay of a radioactive nuclide is measured by its half-life.
Half-life (T½) is the time required for one half of the atoms in any starting sample of a radioisotope to decay.
If the half-life of a radioactive nuclide is known, its decay constant can be calculated by:
where No is the starting number of nuclei, Ni is the number of nuclei remaining after time t, is the decay constant, and
e 
2.718.
The units for the decay constant would be s-1 (or sometimes expressed in disintegrations per second) if the half-life is expressed in seconds.
This relationship expresses radioactive decay based on statistics and probability, from an examination of the behaviour of a large number of individual situations. Note that it does not give any indication when a particular nucleus will undergo decay, but only the amount of time needed for a certain proportion of the nuclei in the sample to decay.
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Z represents the number of protons. A is the number of nucleons (the number of protons plus the number of neutrons). X is the chemical symbol of the element.
from earler, exponentials can be avoided using:
Repeat the test with fresh kernels. Is there any way to predict the exact time at which a coloured kernel will pop? Is it possible to determine the time it will take most of the kernels to pop? Can the "half-life" or the "decay constant" for popcorn be determined? Do these vary depending on the brand of popcorn used?
where No is the number of incident rays on the absorber, Nx is the number of rays penetrating the absorber, x is the absorber thickness, and 
is the absorption coefficient for a particular
material with respect to a particular type of radiation at a
specific energy.
Some simple calculations show that (to the surprise of many) if a given thickness stopped 50% of the radiation, doubling the thickness does not stop the other 50%. Half of the remaining 50%, or 25%, still penetrate.