Basics of Radiogenic Isotope Geochemistry

Geol. 656 Isotope Geochemistry Lecture 4

Spring 2003

BASICS OF RADIOACTIVE ISOTOPE GEOCHEMISTRY INTRODUCTION We can broadly define two principal applications of radiogenic isotope geochemistry. The first is geochronology. Geochronology makes use of the constancy of the rate of radioactive decay. Since a radioactive 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 principal decay systems used in geology; these are also illustrated in Figure 4.1.

TABLE 4.1: Geologically Useful Long-Lived Radioactive Decay Schemes Parent 40

K Rb 138 La 147 Sm 176 Lu 187 Re 190 Pt 232 Th 235 U 238 U 87

Decay Mode b+, e.c, bbba bba a a a

Half-life

l -10 -1

9

5.543 x 10 y 1.42 x 10-11y -1 2.67 x 10-12y -1 6.54 x 10-12y -1 1.94 x 10-11y -1 1.64 x 10-11y -1 1.54 x 10-12y -1 4.948 x 10-11y -1 9.849 x 10-10y -1 1.551 x 10-10y -1

Note: the branching ratio, i.e. ratios of decays to 4 He, but a trivial amount compared to U and Th.

1.28 x 10 yr 4.8 x 1010yr 2.59 x 1011yr 1.06 x 1011yr 3.6 x 1010yr 4.23 x 1010yr 4.50 x 1011yr 1.4 x 1010yr 7.07 x 108yr 4.47 x 109yr

4 0Ar

to total decays of

Daughter 40

40

Ar, Ca 87 Sr 138 Ce, 138B a 143 Nd 176 Hf 187 Os 186 Os 208 Pb, 4He 207 Pb, 4He 206 Pb, 4He 4 0K

is 0.117.

Ratio 40

Ar/36Ar Sr/86Sr 138 Ce/142Ce, 138Ce/136Ce 143 Nd/144Nd 176 Hf/177H f 187 Os/188Os, ( 187Os/186Os) 186 Os/188Os 208 Pb/204Pb, 3He/4He 207 Pb/204Pb, 3He/4He 206 Pb/204Pb, 3He/4He 87

1 4 7Sm

and

190

Pt also produce

THE BASIC EQUATIONS The basic equation of radioactive decay is:

dN = – lN dt

4.1

l 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

N0

t

dN = N

– l dt

4.2

0

where N 0 is the number of atoms of the radioactive, or parent, isotope present at time t=0. Integrating, we obtain:

28

January 27, 2003


Spring 2003

H Li Be Na Mg

Sm

Radioactive (Parent)

Os

Radiogenic (Daughter)

Rd

Radiogenic and Radioactive Cr Mn Fe Co

B

C

N

O

F

Ne

Al

Si

P

S

Cl

Ar

Ni Cu Zn Ga Ge As

Se

Br Kr I

K Ca

Sc

Ti

Rb Sr

Y

Zr Nb Mo Tc Ru Rh Pd Ag Cd

In Sn

Sb

Te

Hf

Tl Pb

Bi

Po At Rd

Cs Ba La

V

He

Ta W

Re Os

Ir

Pt

Au Hg

Xe

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.

This can be expressed as:

N = e–l t N0

ln N = – l t N0 or

N = N0 e–l t

4.3 4.4

Suppose we want to know the amount of time for the number of parent atoms to decrease to half t h e original number, i.e., t when N/N0 = 1/2. Setting N/N0 to 1/2, we can rearrange 4.3 to get: ln 12 = -lt1/2 or ln 2 = lt1/2

t1/2 = ln2 l

4.5

D* = N0 – N

4.6

and finally:

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: Rearranging 4.4 to isolate N0 and substituting that into 4.6, we obtain: D* = Nelt – N = N(elt – 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:

D = D0 + N(elt – 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 expansion:

29

January 27, 2003


Spring 2003 el t = 1 + l t +

(l t)2 (l t)3 + + 2! 3!

4.9

Provided lt