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 We were embarrassed to find that there is no other full explanation of carbon isotopes in Palaeos, so here we go. We assume no background.
All ordinary materials are made up of atoms. Atoms contain a very small, very dense nucleus. The nucleus is made up of two kinds of particles: protons, which have a positive charge, and neutrons which are very similar, but have no charge. The electric charges in the nucleus do not cause the nucleus to fly apart because these nucleons (protons & neutrons) are held together by the strong nuclear force, which is stronger than electrical repulsion at very short distances. The nucleus is surrounded by a very much larger, but more diffuse, cloud of negatively-charged electrons. Each electron has exactly the same charge as a proton, although it is much less massive. This supplies overall electrical charge neutrality to the atom.
The chemical behavior of atoms is almost (remember the almost!) completely determined by the electron cloud, and the number of electrons is controlled by the number of protons in the nucleus. So, any atom with 6 protons in the nucleus behaves about the same as any other atom with the same count. Thus elements are defined by reference to the number of protons in the nucleus, which is also the atomic number of the element. Any atom with 6 protons is an atom of carbon.
Notice we have said nothing about neutrons. For chemical purposes, they don't matter. However, only nuclei with certain numbers of neutrons are stable for each element. Thus, we only find carbon atoms with 6, 7, or 8 neutrons. These different varieties of carbon atom are referred to as isotopes. Isotopes are referred to by their total number of nucleons. Thus, the most common type of carbon has 6 protons (by definition) and 6 neutrons and is designated 12C. The isotope 13C is also common and is also stable. Carbon-14, or 14C, is created in very small quantities by nuclear reactions in the upper atmosphere involving nitrogen (element 7) and cosmic rays. 14C is not stable. One of the neutrons tends to flip back to being a proton, with the release of energy in the form of a very energetic electron. That energetic electron is one form of radioactivity. There is a 50% chance that this radioactive decay will happen to any given atom of 14C in 5730 years. Thus, we say that 14C is a radioactive isotope with a half-life of 5730 years.
In the universe as a whole, 12C makes up about 99% of all carbon, while 13C is about 1%. 14C makes up a vanishingly small fraction, which we can ignore. All atoms of carbon ought to behave in almost same way in chemical reactions. Do you still remember the "almost"? It becomes important here because it turns out that the enzymes involved in photosynthesis are a little biased. They are slightly more likely to fix CO2 molecules with 12C than we would expect by chance. Virtually all carbon in living things ultimately derives from photosynthesis. Thus almost all organic (or biogenic) carbon, carbon which is or was part of a living creature, is isotopically light, meaning it has slightly more than the usual 99% 12C. When a great deal of carbon is tied up in organic sediments, the remaining carbon in atmospheric carbon dioxide becomes isotopically heavy. This is called a positive excursion in 13C, or, for those wishing to be particularly obscure, a positive δ13C anomaly. Fortunately, it is remarkably easy to measure the ratios of carbon isotopes to very high precision using mass spectroscopy (= mass spec). By examining ancient inorganic carbon, we can read the isotopic state of the atmosphere at any given time in the past. There can be great debate about the reasons for an excursion, but the existence, and usually the magnitude, of the excursion are easily determined and quite reproducible.
Footnote to footnote: We apologize for the stupid picture. Do not get the idea that electrons orbit about the nucleus like a planet circling a star. That is completely inaccurate. Explaining why would take too long, so we'll take it up another time.
 Chris's essay explains the cap carbonates here. These are calcium magnesium carbonates, or dolomite [CaMg(CO3)2], associated with the end of the major glaciations, and overlying the glacial tillite (unsorted terminal glacial rock) on all continents. The cap carbonates are sometimes found without tillite, presumably deposited in deep water not actually reached by the continental glaciers.
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