Natural isotopes are either stable isotopes, radioactive isotopes that have a sufficiently long half-life to allow them to exist in substantial concentrations in the Earth (e.g. bismuth-209, with a half-life of 1.9 * 1019 years, potassium-40 with a half-life of 1.251(3)×109 y), daughter products of those isotopes (e.g. 234Th, with a half-life of 24 days) or cosmogenic elements. The heaviest stable isotope is lead-208, but the heaviest 'natural' isotope is U-238.
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Many elements have both natural and artificial isotopes (e.g. hydrogen has three natural isotopes and another four known artificial isotopes). A further distinction among stable natural isotopes is division into primordial (existed when the Solar System formed) and cosmogenic elements ( created by cosmic ray bombardment or other similar processes).
What defines a natural isotope
Natural isotopes must be either stable, have a half-life exceeding about 7*107 years (there are 34 isotopes in this category, see stable isotope for more details) or are generated in large amounts cosmogenically (e.g. 14C, which has a half life of only 6000 years but is made by cosmic rays colliding with 14N)
Naturally occurring radioisotopes
Some radioisotopes occur in nature with a half-life less than 7*107 years (carbon-14 5,730 ± 40 years, tritium 12.32 years etc.). They are synthesed all the time because of cosmic radiation. For carbon-14 this fact has practice value because it is used for radiocarbon dating.