Geol. 656 Isotope Geochemistry Lecture 4 Spring 2007 BASICS OF RADIOACTIVE ISOTOPE GEOCHEMISTRY INTRODUCTION We can broadly define two principal applications of radiogenic isotope geochemistry. The first is geo- chronology. Geochronology makes use of the constancy of the rate of radioactive decay. Since a radioac- tive nuclide decays to its daughter at a rate independent of everything, we can determine a time simply by determining how much of the nuclide has decayed. We will discuss the significance of this time at a later point. The other application is tracer studies. These studies make use of the differences in the ratio of the radiogenic daughter isotope to other isotopes of the element. We can understand the utility of such studies without, for the moment, understanding why such differences in isotope ratios arise. These sorts of studies are analogous to tracer studies performed in biology. For example, biologists will inject an organism with a radioactive nuclide, such as one of carbon or phosphorus, to investigate the organism’s metabolism and physiology. Unfortunately, the time scales of geology are too long for us to inject the Earth with a radioactive tracer to study its physiology. However, we can make use of natural tracers, the radiogenic isotopes, to investigate fundamental Earth processes. For example, Nd isotope ratios have been used to distinguish different ocean water masses. Radiogenic isotope ratios are also used in a more sophisticated way, namely to understand evolutionary histories. Table 4.1 lists the prin- cipal decay systems used in geology; these are also illustrated in Figure 4.1. TABLE 4.1: Geologically Useful Long-Lived Radioactive Decay Schemes Parent Decay Mode λ Half-life Daughter Ratio 40K β+, e.c, β− 5.543 x 10-10y-1 1.28 x 109yr 40Ar, 40Ca 40Ar/36Ar 87Rb β− 1.42 x 10-11y-1* 48.8 x 109yr 87Sr 87Sr/86Sr 138La β− 2.67 x 10-12y-1 2.59 x 1011yr 138Ce, 138Ba 138Ce/142Ce, 138Ce/136Ce 147Sm α 6.54 x 10-12y-1 1.06 x 1011yr 143Nd 143Nd/144Nd 176Lu β− 1.867† x 10-11y-1 3.6 x 1010yr 176Hf 176Hf/177Hf 187Re β− 1.64 x 10-11y-1 4.23 x 1010yr 187Os 187Os/188Os, (187Os/186Os) 190Pt α 1.54 x 10-12y-1 4.50 x 1011yr 186Os 186Os/188Os 232Th α 4.948 x 10-11y-1 1.4 x 1010yr 208Pb, 4He 208Pb/204Pb, 3He/4He 235U α 9.849 x 10-10y-1 7.07 x 108yr 207Pb, 4He 207Pb/204Pb, 3He/4He 238U α 1.551 x 10-10y-1 4.47 x 109yr 206Pb, 4He 206Pb/204Pb, 3He/4He Note: the branching ratio, i.e. ratios of decays to 40Ar to total decays of 40K is 0.117. 147Sm and 190Pt also produce 4He, but a trivial amount compared to U and Th. *The officially accepted decay constant for 87Rb is that shown here. However, recent determinations of this con- stant range from 1.421 x 10-11y-1 by Rotenberg (2005) to 1.399 x 10-11y-1 by Nebel et al. (2006). †This is the value recommended by Söderlund et al. (2004) and is consistent with most, but not all, other recent re- sults. THE BASIC EQUATIONS The basic equation of radioactive decay is: dN = !"N 4.1 dt 30 January 26, 2007 Geol. 656 Isotope Geochemistry Lecture 4 Spring 2007 Sm Radioactive (Parent) H He Os Radiogenic (Daughter) Li Be B C N O F Ne Rd Radiogenic and Radioactive Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rd Fr Ra Ac La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Ac Th Pa U Figure 4.1. Periodic Table showing the elements having naturally occurring radioactive isotopes and the elements produced by their decay. λ is the decay constant, which we define as the probability that a given atom would decay in some time dt. It has units of time-1. Let's rearrange equation 4.1 and integrate: N dN t = "#dt 4.2 !N0 N !0 where N0 is the number of atoms of the radioactive, or parent, isotope present at time t=0. Integrating, we obtain: N ln = !"t 4.3 N0 N !"t !"t This can be expressed as: = e or N = N0e 4.4 N0 Suppose we want to know the amount of time for the number of parent atoms to decrease to half the original number, i.e., t when N/N0 = 1/2. Setting N/N0 to 1/2, we can rearrange 4.3 to get: 1 ln = -λt or ln 2 = λt 2 1/2 1/2 ln2 and finally: t = 4.5 1/ 2 ! This is the definition of the half-life, t1/2. Now the decay of the parent produces a daughter, or radiogenic, nuclide. The number of daughters produced, D*, is simply the difference between the initial number of parents and the number remaining after time t: D* = N0 – N 4.6 Rearranging 4.4 to isolate N0 and substituting that into 4.6, we obtain: 31 January 26, 2007 Geol. 656 Isotope Geochemistry Lecture 4 Spring 2007 D* = Neλt – N = N(eλt – 1) 4.7 This tells us that the number of daughters produced is a function of the number of parents present and time. Since in general there will be some atoms of the daughter nuclide around to begin with, i.e., when t = 0, a more general expression is: !t D = D0 + N(e "1) 4.8 where D is the total number of daughters and D0 is the number of daughters originally present. As an aside, we’ll note that there is a simple linear approximation of this function for times short compared to the decay constant. An exponential function can be expressed as a Taylor Series expan- sion: 2 (!t) (!t)3 e!t = 1+ !t + + +… 4.9 2! 3! Provided λt << 1, the higher order terms become very small and can be ignored; hence for times that are short compared to the decay constant (i.e., for t << 1/λ), equation 4.8 can be written as: D ≅ D0 + N λt 4.10 Let’s now write equation 4.8 using a concrete example, such as the decay of 87Rb to 87Sr: 87 87 87 t Sr = Sr0 + Rb(eλ – 1) 4.11 As it turns out, it is generally much easier, and usually more meaningful, to measure to ratio of two iso- topes than the absolute abundance of one. We therefore measure the ratio of 87Sr to a non-radiogenic isotope, which by convention is 86Sr. Thus the useful form of 4.11 is: 87Sr ! 87Sr $ 87 Rb e't 1 4.12 86 = # 86 & + 86 ( ( ) Sr " Sr % 0 Sr Similar expressions can be written for other decay systems. It must be emphasized that 87Rb/86Sr ratio in equation 4.12, which we will call this the “parent- daughter ratio”, is the ratio at time t, i.e., present ratio. If we need this ratio at some other time, we need to calculate it using equation 4.4. A SPECIAL CASE: THE U-TH-PB SYSTEM The U-Th-Pb system is somewhat of a special case as there are 3 decay schemes producing isotopes of Pb. In particular two U isotopes decay to two Pb isotopes, and since the two parents and two daugh- ters are chemically identical, combining the two provides a particularly powerful tool. Lets explore the mathematics of this. First some terminology. The 238U/204Pb ratio is called µ, the 232Th/238U is called κ. The ratio 238U/235U is constant at any given time in the Earth and today is 137.88 (except in nuclear reactors and Oklo!). Now, we can write two versions of equation 4.8: ! 207 Pb / 204 Pb = ( 207 Pb / 204 Pb) + (e !235t " 1) 4.13 0 137.88 206 204 206 204 ! t and Pb / Pb = ( Pb / Pb) + !(e 238 " 1) 4.14 0 These can be rearranged by subtracting the initial ratio from both sides and calling the difference be- tween the initial and the present ratio ∆. For example, equation 4.13 becomes ! ! 207 Pb / 204 Pb = (e !235t " 1) 4.15 137.88 Dividing the equivalent equation for 235U-207Pb by equation 4.15 yields: 32 January 26, 2007 Geol. 656 Isotope Geochemistry Lecture 4 Spring 2007 ! 207 Pb / 204 Pb (e !235t " 1) 206 204 = 4.16 ! Pb / Pb 137.88(e !238t 1) " Notice the absence of the µ term. The equation holds for any present-day ratio of 207Pb/204Pb and 206Pb/204Pb we measure and thus for all pairs of ratios. The left-hand side is simply the slope of a series of data points from rocks or minerals formed at the same time (and remaining closed systems since time t) on a plot of 207Pb/204Pb vs. 206Pb/204Pb. This means we can determine the age of a system without knowing the parent-daughter ratio. The bad news is that equation 4.16 cannot be solved for t. How- ever, we can guess a value of t, plug it into the equation, calculate the slope, compared the calculated slope with the observed one, revise our guess of t, calculated again, etc.