Radiometric Dating EENS 211 Tulane University Earth Materials Prof Stephen A Nelson Radiometric Dating This document last updated on 29 Sep 2004 Prior to 1905 the best and most accepted age of the Earth was that proposed by Lord Kelvin based on the amount of time necessary for the Earth to cool to its present temperature from a completely liquid state Although we now recognize lots of problems with that calculation the age of 25 my was accepted by most physicists but considered too short by most geologists Then in 1896 radioactivity was discovered Recognition that radioactive decay of atoms occurs in the Earth was important in two respects 1 It provided another source of heat not considered by Kelvin which would mean that the cooling time would have to be much longer 2 It provided a means by which the age of the Earth could be determined independently Principles of Radiometric Dating Radioactive decay is described in terms of the probability that a constituent particle of the nucleus of an atom will escape through the potential Energy barrier which bonds them to the nucleus The energies involved are so large and the nucleus is so small that physical conditions in the Earth i e T and P cannot affect the rate of decay The rate of decay or rate of change of the number N of particles is proportional to the number present at any time i e Note that dN dt must be negative The proportionality constant is the decay constant So we can write Rearranging and integrating we get or ln N No t to Page 1 of 14 9 29 2004 Radiometric Dating If we let to 0 i e the time the process started then 1 We next define the half life 1 2 the time necessary for 1 2 of the atoms present to decay This is where N No 2 Thus or ln 2 t so that The half life is the amount of time it takes for one half of the initial amount of the parent radioactive isotope to decay to the daughter isotope Thus if we start out with 1 gram of the parent isotope after the passage of 1 half life there will be 0 5 gram of the parent isotope left After the passage of two half lives only 0 25 gram will remain and after 3 half lives only 0 125 will remain etc Knowledge of 1 2 or would then allow us to calculate the age of the material if we knew the amount of original isotope and its amount today This can only be done for 14C since we know N0 from the atmospheric ratio assumed to be constant through time For other systems we have to proceed further Page 2 of 14 9 29 2004 Radiometric Dating Some examples of isotope systems used to date geologic materials Parent Daughter 1 2 238U 206Pb 4 47 b y 235U 207Pb 707 m y 232Th 208Pb 14 b y 40K 40Ar 40Ca 1 28 b y 10 000 years 87Rb 87Sr 48 b y 10 million years 147Sm 143Nd 106 b y 14C 14N 5 730 y Useful Range Type of Material 10 million years Igneous sometimes metamorphic rocks and minerals 100 70 000 years Organic Material To see how we actually use this information to date rocks consider the following Usually we know the amount N of an isotope present today and the amount of a daughter element produced by decay D By definition D N0 N from equation 1 So D Ne t N N e t 1 2 Now we can calculate the age if we know the number of daughter atoms produced by decay D and the number of parent atoms now present N The only problem is that we only know the number of daughter atoms now present and some of those may have been present prior to the start of our clock We can see how do deal with this if we take a particular case First we ll look at the Rb Sr system Page 3 of 14 9 29 2004 Radiometric Dating The Rb Sr System by decay The neutron emits an electron to become a proton For this decay reaction 1 42 x 10 11 yr 1 2 4 8 x 1010 yr at present 27 85 of natural Rb is 87Rb If we use this system to plug into equation 2 then 87Sr 87Rb e t 1 3 but 87Sr t 87Sr0 87Sr or 87Sr 87Srt 87Sr0 Plugging this into equation 3 87Sr t 87Sr0 87Rb 4 We still don t know 87Sr0 the amount of 87Sr daughter element initially present To account for this we first note that there is an isotope of Sr 86Sr that is 1 non radiogenic not produced by another radioactive decay process 2 non radioactive does not decay to anything else Thus 86Sr is a stable isotope and the amount of 86Sr does not change through time If we divide equation 4 through by the amount of 86Sr then we get 5 This is known as the isochron equation We can measure the present ratios of 87Sr 86Sr t and 87Rb 86Sr t with a mass spectrometer thus these quantities are known The only unknowns are thus 87Sr 86Sr 0 and t Page 4 of 14 9 29 2004 Radiometric Dating Note also that equation 5 has the form of a linear equation i e y mx b where b the y intercept is 87Sr 86Sr 0 and m the slope is e t 1 How can we use this First note that the time t 0 is the time when Sr was isotopically homogeneous i e 87Sr 86Sr was the same in every mineral in the rock such as at the time of crystallization of an igneous rock Inn nature however each mineral in the rock is likely to have a different amount of 87Rb So that each mineral will also have a different 87Rb 86Sr ratio at the time of crystallization Thus once the rock has cooled to the point where diffusion of elements does not occur the 87Rb in each mineral will decay to 87Sr and each mineral will have a different 87Rb and 87Sr after passage of time We can simplify our Isochron equation somewhat by noting that if x is small so that e t 1 t when t is small So applying this simplification 6 and solving for t Page 5 of 14 9 29 2004 Radiometric Dating The initial ratio 87Sr 86Sr 0 is useful as a geochemical tracer The reason for this is that Rb has become distributed unequally through the Earth over time For example the amount of Rb in mantle rocks is generally low i e less than 0 1 ppm The mantle thus has a low 87Rb 86Sr ratio and would not change its 87Sr 86Sr ratio very much with time Crustal rocks on the other hand generally have higher amounts of Rb usually greater than 20 ppm and thus start out with a relatively high 87Rb 86Sr ratio Over time this results in crustal rocks having a much higher …