EENS 211 
Earth Materials 
Tulane University 
Prof. Stephen A. Nelson 
Radiometric Dating 
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:
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 l, the decay constant. So, we can write
Rearranging, and integrating, we get
or ln(N/N_{o}) = l(t  t_{o})

If we let t_{o} = 0, i.e. the time the process started, then
We next define the halflife, t_{1/2}, the time necessary for 1/2 of the atoms present to decay. This is where N = N_{o}/2. Thus,
or ln 2 = lt, so that

The halflife 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 halflife there will be 0.5 gram of the parent isotope
left.
After the passage of two halflives only 0.25 gram will remain, and after 3 half lives only 0.125 will remain etc. 

Knowledge of t_{1/2} or l 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 ^{14}C, since we know N_{0} from the atmospheric
ratio, assumed to be constant through time. For other systems we have to
proceed further.

Some examples of isotope systems used to date geologic materials.

Parent 
Daughter 
t_{1/2} 
Useful Range 
Type of Material 
^{238U}  ^{206Pb}  4.47 b.y 
>10 million years 
Igneous & sometimes metamorphic rocks and minerals 
^{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 

^{147}Sm  ^{143}Nd  106 b.y.  
^{14C}  ^{14N}  5,730 y 
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* = N_{0}  N from equation (1) So, D* = Ne^{lt}N = N(e^{lt}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. 
The Rb/Sr System by b decay. The neutron emits an electron to become a proton. For this decay reaction, l = 1.42 x 10^{11} /yr, t_{1/2} = 4.8 x 10^{10} yr at present, 27.85% of natural Rb is ^{87}Rb. If we use this system to plug into equation (2), then ^{87}Sr* = ^{87}Rb (e^{lt}1) (3) but, ^{87}Sr_{t} = ^{87}Sr_{0} + ^{87}Sr* or ^{87}Sr* = ^{87}Sr_{t}  ^{87}Sr_{0} Plugging this into equation (3) ^{87}Sr_{t} = ^{87}Sr_{0} + ^{87}Rb (4) We still don't know ^{87}Sr_{0} , the amount of ^{87}Sr daughter element initially present. To account for this, we first note that there is an isotope of Sr, ^{86}Sr, that is:
Thus, ^{86}Sr is a stable isotope, and the amount of ^{86}Sr does not change through time 
If we divide equation (4) through by the amount
of ^{86}Sr, then we get: 


(5) 
This is known as the isochron equation. 
We can measure the present ratios of (^{87}Sr/^{86}Sr)_{t }and_{ }(^{87}Rb/^{86}Sr)_{t} with a mass spectrometer, thus these quantities are known. The only unknowns are thus (^{87}Sr/^{86}Sr)_{0} and t.Note also that equation (5) has the form of a linear equation, i.e. y = mx +b where b, the y intercept, is(^{87}Sr/^{86}Sr)_{0} and m, the slope is (e^{lt}  1). How can we use this?

First note that the time t=0 is the time when Sr was isotopically homogeneous, i.e. ^{87}Sr/^{86}Sr was the same in every mineral in the rock (such as at the time of crystallization of an igneous rock). In nature, however, each mineral in the rock is likely to have a different amount of ^{87}Rb. So that each mineral will also have a different ^{87}Rb/^{86}Sr ratio at the time of crystallization. Thus, once the rock has cooled to the point where diffusion of elements does not occur, the ^{87}Rb in each mineral will decay to ^{87}Sr, and each mineral will have a different ^{87}Rb and ^{87}Sr after passage of time. 
We can simplify our isochron equation somewhat by noting that if
x is
small,
so that (e^{lt}  1) = lt, when lt is small. 
So, applying this simplification,  

(6) 
and solving for t
The initial ratio, (^{87}Sr/^{86}Sr)_{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 ^{87}Rb/^{86}Sr ratio and would not change its ^{87}Sr/^{86}Sr 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 ^{87}Rb/^{86}Sr ratio. Over time, this results in crustal rocks having a much higher ^{87}Sr/^{86}Sr ratio than mantle rocks. Thus if the mantle has a ^{87}Sr/^{86}Sr of say 0.7025, melting of the mantle would produce a magma with a ^{87}Sr/^{86}Sr ratio of 0.7025, and all rocks derived from that mantle would have an initial ^{87}Sr/^{86}Sr ratio of 0.7025. On the other hand, if the crust with a ^{87}Sr/^{86}Sr of 0.710 melts, then the resulting magma would have a ^{87}Sr/^{86}Sr of 0.710 and rocks derived from that magma would have an initial ^{87}Sr/^{86}Sr ratio of 0.710. Thus we could tell whether the rock was derived from the mantle or crust be determining its initial Sr isotopic ratio as we discussed previously in the section on igneous rocks.

The U, Th, Pb System Two isotopes of Uranium and one isotope of Th are radioactive and decay to produce various isotopes of Pb. The decay schemes are as follows
^{232}Th has such along half life that it is generally not used in dating. ^{204}Pb is a stable nonradiogenic isotope of Pb, so we can write two isochron equations and get two independent dates from the U  Pb system. 


(7) and 

(8) 
If these two independent dates are the same, we say they are concordant.
We can also construct a Concordia diagram, which shows the values of Pb isotopes that would give concordant dates. The Concordia curve can be calculated by defining the following: (9) and (10) We can plug in t and solve for the ratios ^{206}Pb^{*}/^{238}U and ^{207}Pb^{*}/^{235}U to define a curve called the Concordia.

The Concordia is particularly useful in dating of the mineral Zircon
(ZrSiO_{4}). Zircon has a high hardness (7.5) which makes it
resistant to mechanical weathering, and it is also very resistant to
chemical weathering. Zircon can also survive metamorphism.
Chemically, zircon usually contains high amounts of U and low amounts of
Pb, so that large amounts of radiogenic Pb are produced. Other
minerals that also show these properties, but are less commonly used in
radiometric dating are Apatite and sphene.
If a zircon crystal originally crystallizes from a magma and remains a closed system (no loss or gain of U or Pb) from the time of crystallization to the present, then the ^{206}Pb^{*}/^{238}U and ^{207}Pb^{*}/^{235}U ratios in the zircon will plot on the Concordia and the age of the zircon can be determined from its position on the plot. 
Discordant dates will not fall on the Concordia curve. Sometimes, however, numerous discordant dates from the same rock will plot along a line representing a chord on the Concordia diagram. Such a chord is called a discordia.

The discordia is often interpreted by extrapolating both ends to intersect the Concordia. The older date, t_{0} is then interpreted to be the date that the system became closed, and the younger date, t*, the age of an event (such as metamorphism) that was responsible for Pb leakage. Pb leakage is the most likely cause of discordant dates, since Pb will be occupying a site in the crystal that has suffered radiation damage as a result of U decay. U would have been stable in the crystallographic site, but the site is now occupied by by Pb. An event like metamorphism could heat the crystal to the point where Pb will become mobile. 
Another possible scenario involves U leakage, again possibly as a result of a metamorphic event. U leakage would cause discordant points to plot above the cocordia. But, again, exptrapolation of the discordia back to the two points where it intersects the Concordia, would give two ages  t* representing the possible metamorphic event and t_{0} representing the initial crystallization age of the zircon. 
We can also define what are called PbPb Isochrons by combining the two isochron equations (7) and (8). (11) Since we know that the , and assuming that the ^{206}Pb and ^{207}Pb dates are the same, then equation (11) is the equation for a family of lines that have a slope
that passes through the point ,

The Age of the Earth



Other Dating Methods Sm  Nd Dating
^{147}Sm
®
^{143}Nd l = 6.54 x 10^{12} /yr, t_{1/2} = 1.06 x 10^{11} yr ^{144}Nd is stable and nonradiogenic, so we can write the isochron equation as:
The isochron equation is applied just like that for the RbSr system, by determining the ^{143}Nd/^{144}Nd and ^{147}Sm/^{144}Nd ratios on several minerals with a mass spectrometer and then from the slope determine the age of the rock. The initial ratio has particular importance for studying the chemical evolution of the Earth's mantle and crust, as we discussed in the section on igneous rocks. 
KAr Dating ^{40}K is the radioactive isotope of K, and makes up 0.119% of natural K. Since K is one of the 10 most abundant elements in the Earth's crust, the decay of ^{40}K is important in dating rocks. ^{40}K decays in two ways: by b decay. 89% of follows this branch. But this scheme is not used because ^{40}Ca can be present as both radiogenic and nonradiogenic Ca. ^{40}K ® ^{40}Arby electron capture For the combined process, l = 5.305 x 10^{10}/ yr, t_{1/2} = 1.31 x 10^{9} yr and for the Ar branch of the decay scheme l_{e} = 0.585 x 10^{10}/ yr Since Ar is a noble gas, it can escape from a magma or liquid easily, and it is thus assumed that no ^{40}Ar is present initially. Note that this is not always true. If a magma cools quickly on the surface of the Earth, some of the Ar may be trapped. If this happens, then the date obtained will be older than the date at which the magma erupted. For example lavas dated by KAr that are historic in age, usually show 1 to 2 my old ages due to trapped Ar. Such trapped Ar is not problematical when the age of the rock is in hundreds of millions of years. The dating equation used for KAr is: where = 0.11 and refers to fraction of ^{40}K that decays to ^{40}Ar. 
Some of the problems associated with KAr dating are

^{14}Carbon Dating
Radiocarbon dating is different than the other methods of dating because it cannot be used to directly date rocks, but can only be used to date organic material produced by once living organisms.

Other Uses of Isotopes

