of is: → + + By emitting a beta particle (an electron, e−) and an electron antineutrino (), one of the neutrons in the nucleus changes to a proton and the nucleus reverts to the stable (non-radioactive) isotope . Principles During its life, a plant or animal is in equilibrium with its surroundings by exchanging carbon either with the atmosphere or through its diet. It will, therefore, have the same proportion of as the atmosphere, or in the case of marine animals or plants, with the ocean. Once it dies, it ceases to acquire , but the within its biological material at that time will continue to decay, and so the ratio of to in its remains will gradually decrease. Because decays at a known rate, the proportion of radiocarbon can be used to determine how long it has been since a given sample stopped exchanging carbon – the older the sample, the less will be left. The equation governing the decay of a radioactive isotope is: where N0 is the number of atoms of the isotope in the original sample (at time t = 0, when the organism from which the sample was taken died), and N is the number of atoms left after time t. λ is a constant that depends on the particular isotope; for a given isotope it is equal to the reciprocal of the mean-life – i.e. the average or expected time a given atom will survive before undergoing radioactive decay. The mean-life, denoted by τ, of is 8,267 years, so the equation above can be rewritten as: The sample is assumed to have originally had the same / ratio as the ratio in the atmosphere, and since the size of the sample is known, the total number of atoms in the sample can be calculated, yielding N0, the number of atoms in the original sample. Measurement of N, the number of atoms currently in the sample, allows the calculation of t, the age of the sample, using the equation above. The half-life of a radioactive isotope (usually denoted by t1/2) is a more familiar concept than the mean-life, so although the equations above are expressed in terms of the mean-life, it is more usual to quote the value of 's half-life than its mean-life. The currently accepted value for the half-life of is 5,700 ± 30 years. This means that after 5,700 years, only