Appendix A: A Brief Introduction to Isotope Geochemistry Radioactive isotope systems are used to determine ages or of minerals and rocks and as tracers of magmatic pro- λt cesses and evolution of the earth. We have seen many D ¼ Do þ Nðe À 1Þ (A.3) examples of the use of Nd, Sr, and Pb isotopes and, in a few cases, He isotopes, throughout the book. It is Equation (A.3) is the basis for geochronology, anticipated that the student will have a proper geo- which is the science of determining ages of minerals chemistry course in which isotope systems are and rocks. discussed in great detail. Therefore, what follows Radioactive decay reactions are generally expressed below is a brief overview of the theoretical under- in terms of half-lives, i.e., when t ¼ t1/2, N ¼ 1/2No. pinnings of these isotopic systems. Substituting this expression into Eq. (A.1) and In any isotope system that involves the breakdown rearranging terms we get of radioactive parent atoms into daughter atoms (which Àλ may or may not be radioactive), the number of atoms of 1=2 ¼ e t1=2 the parent isotope (No) at time (t) ¼ 0 and the number of atoms of the parent isotope (N) remaining after time or t are related by the following relation: ln 2 ¼ λt1=2 Àλt N=No ¼ e (A.1) Rearranging terms, we obtain λ is a constant for any particular isotopic system and is known as the decay constant. The decay constant is t = ¼ ln 2=λ; i:e:; t = ¼ 0:693=λ (A.4) known experimentally for each individual isotopic 1 2 1 2 system. Below we examine some isotopic systems that are Assuming that all of the atoms of a daughter isotope useful in petrology. Table A.1 provides a list of these present in a rock were generated via radioactive decay, isotopes and their half-lives and decay constants. then at any time t, the number of atoms of daughter isotope (D*) in that rock is D* ¼ No À N. Substituting this relationship to Eq. (A.1), we get Rb–Sr System =ð à þ Þ¼ Àλt N D N e In this system, atoms of the radioactive parent isotope 87Rb break down into the atoms of the daughter isotope By rearranging terms we obtain 87Sr and release a β-particle in the process. Based on à λ Eq. (A.3) above, one may write D =N ¼ e t À 1 (A.2) 87 ¼87 þ87 ð λt À Þ In reality, however, some atoms of the daughter Sr Sro Rb e 1 (A.5) isotope (Do) may be originally present in the rock at Scientists are able to determine isotope ratios with t ¼ 0, so that D ¼ D*+Do, and substituting this into Eq. (A.2), we get much greater precision than absolute values of individ- ual isotopes; therefore, Eq. (A.5) is modified into the λt ðD À DoÞ=N ¼ e À 1 following equation by converting the above into ratios G. Sen, Petrology, DOI 10.1007/978-3-642-38800-2, 351 # Springer-Verlag Berlin Heidelberg 2014 352 Appendix A: A Brief Introduction to Isotope Geochemistry by dividing each isotope by a non-radiogenic isotope of a 0.7075 Sr, 86Sr. 86Sr value should be constant in a rock and in 86 its constituent minerals because Sr is not produced by 0.7065 radioactive breakdown reaction. We then get Sr 0.7055 86 λ Sr/ 87 =86 ¼½87 =86 þ87 =86 ð t À Þ 87 0.7045 Sr Sr Sr Sr o Rb Sr e 1 (A.6) 0.7035 This equation has the form of a straight line: λt 0.7025 y ¼ mx þ c, where m ¼ slope ¼ðe À 1Þ, and 0.00 0.05 0.10 0.15 0.20 0.25 c ð¼ ½87Sr=86Sr Þ is the intercept on y-axis. One can 87Rb/86Sr o b measure 87Sr/86Sr and 87Rb/86Sr ratios of a rock and its t2 constituent minerals with an instrument called the mass Sr t1 spectrometer. Note that individual mineral phases in a 86 rock will have different 87Rb/86Sr ratios because some Sr/ 87 []87Sr/86Sr minerals can hold more Rb relative to Sr in their atomic o t=0 structure than others; for example, biotite can have 87Rb/86Sr more Rb substituting for K atoms in its atomic structure than any K-poor mineral, such as, say, pyroxene. So, Fig. A.1 (a) Rb–Sr isochron diagram. The dots on the isochron repre- 87Rb/86Sr will be higher in biotite than pyroxene in a sent minerals and whole rock analyses from the same rock. (b) This rock containing both phases. Over time, 87Sr will be diagram schematically illustrates how the initial ratio stays constant while the isochrons develop steeper slopes with age generated by breakdown of 87Rb, and 87Sr/86Sr in bio- tite will be higher than in pyroxene. Even though the constituent minerals of a rock will have different Before the separation of the continental crust from the 87Rb/86Sr they should all have the same initial ratio mantle, it is commonly accepted that the bulk earth had [87Sr/86Sr] . Armed with the values of three unknowns, o an 87Sr/86Sr initial ratio of 0.69897, which is the so- one can now calculate the age of the rock. The straight called “BABI” (Basaltic Achondrite Best Initial ratio). line that gives the age of the rock is called an isochron. For a constant Rb/Sr ratio of 0.027, the present day Figure A.1 illustrates the above statements. Let us 87Sr/86Sr ratio of the bulk earth should be 0.704 via imagine a rock that gives the following analyses of its production of 87Sr via breakdown of 87Rb over the past mineral components and whole rock: 4.55 Ga (Wilson 1989). Rb is preferentially transferred Rock/mineral 87Rb/86Sr 87Sr/86Sr via magmas from the mantle to the continental crust. Mineral 1 0.05 0.7040 So, when a certain amount of continental crust with a Mineral 2 0.10 0.7050 much higher Rb/Sr ratio is extracted out of the mantle, Whole rock 0.15 0.7060 the mantle is depleted by that much Rb/Sr. The Mineral 3 0.20 0.7070 extracted crust with a high 87Rb/86Sr will generate Plotting these data and fitting a line through the data much more 87Sr/86Sr via decay than the upper mantle points give us Fig. A.1a. The straight line intercepts (with significantly lower 87Rb/86Sr) over the same 87 86 y-axis at 0.7030, which is the initial ratio, [ Sr/ Sr]o. amount of time passed since the separation. 87 86 87 86 Applying the values of [ Sr/ Sr]o, Sr/ Sr, and If a certain mass of continental crust with 87Rb/86Sr and using the lambda value in Table A.1 in Rb/Sr ¼ 0.18 had been extracted some 2.5 billion Eq. (A.6) give an age of 1,394 million years. Fig- years ago out of the bulk earth reservoir (i.e., a portion ure A.1b schematically shows how three minerals of the mantle), then its 87Sr/86Sr would have evolved to with different initial Rb/Sr evolve with time while a present day ratio of 0.718 (can be extrapolated in 87 86 maintaining a constant initial ratio [ Sr/ Sr]o. Time Fig. A.2). The “depleted” (i.e., depleted of continental increases from t ¼ 0tot1 and t2 as the slope of the crustal components) portion of the mantle, having a isochron also increases. lower Rb/Sr (0.024), would then have a lower present Sr isotope system is extremely useful as a tracer of day 87Sr/86Sr ratio than the calculated present day bulk geological processes and earth evolution. Following is earth ratio. Because the timing of extraction of the crust an example of how it can be used to trace back the influences the degree of enrichment of the crust and evolution of continental crust–upper mantle system. depletion of the mantle relative to the bulk earth Appendix A: A Brief Introduction to Isotope Geochemistry 353 Table A.1 Isotope ratios, decay constants, and half-lives (from W. White’s lecture notes: http://www.geo.cornell.edu/geology/classes/ Geo656/656notes09/656_09Lecture03.pdf) Parent isotope Daughter isotope Isotope ratio λ Half-life (years) 87Rb 87Sr 87Sr/86Sr 1.42  10À11/year 48.8  109 147Sm 143Nd 143Nd/144Nd 6.54  10À12/year 1.06  1011 187Re 187Os 187Os/188Os 1.64  10À11/year 4.23  1010 190Pt 186Os 186Os/188Os 1.54  10À12/year 4.50  1011 232Th 208Pb, 4He 208Pb/204Pb, 3He/4He 4.948  10À11/year 1.4  1010 235U 208Pb, 4He 207Pb/204Pb, 3He/4He 9.849  10À10/year 7.07  108 238U 206Pb, 4He 206Pb/204Pb, 3He/4He 1.551  10À10/year 4.47  109 0.710 20 Oceanic Basalts 0.708 10 Bulk Earth 0.706 0 87Sr 0.704 86Sr ε -10 Upper Continental Crust continental crust evolution path Nd 0.702 depleted oceanic mantle -20 0.700 bulk earth evolution path -30 BABI Lower Continental 0.698 Crust 4.5 3.5 2.5 1.5 0.5 0 time (billion years before present) -200 0 200 400 600 800 87 86 Fig. A.2 This diagram shows how the Sr/ Sr of BABI (representing εSr the bulk earth) would evolve with time since the earth’s formation 4.5 billion years ago.