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In his article Light Attenuation and Exponential Laws in the last issue of Plus, Ian Garbett discussed the phenomenon of light attenuation, one of the many physical phenomena in which the exponential function crops up.In this second article he describes the phenomenon of radioactive decay, which also obeys an exponential law, and explains how this information allows us to carbon-date artefacts such as the Dead Sea Scrolls.We end up with a solution known as the "Law of Radioactive Decay", which mathematically is merely the same solution that we saw in the case of light attenuation.We get an expression for the number of atoms remaining, N, as a proportion of the number of atoms N, where the quantity l, known as the "radioactive decay constant", depends on the particular radioactive substance.
Plotting t against F with a value of l=1 gives the graph on the right. The equivalent thickness for the medium in radiation attenuation is known as "half-value thickness".Here isotopes with longer half lives are used, which enables dating of geological formations and rocks. For example, in lava form, molten lead and Uranium-238 (standard isotope) are constantly mixed in a certain ratio of their natural abundance.Once solidified, the lead is "locked" in place and since the uranium decays to lead, the lead-to-uranium ratio increases with time.This is because there is carbon dioxide (CO exchange, and so the ratio of C-14 to the far more common carbon isotope, C-12, will begin to decrease as the C-14 atoms decay, yielding nitrogen (N-14) with the emission of an electron (or "beta particle") plus an anti-neutrino.The ratio of C-14 to C-12 in the atmosphere's carbon dioxide molecules is about 1.3×10, and this value is assumed constant for the main part of archaeological history since the formation of the earth's atmosphere.