Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay

## An Ideal Decay

Imagine we have an undiscovered element, Parentium, that has a radioactive isotope, Parentium-123, which decays to stable Daughterium-123. This is the only way Parentium-123 decays, and there is no other source of Daughterium-123. Furthermore, Parentium and Daughterium are so different in chemical properties that they don't otherwise occur together.

If there were such a pair of isotopes, radiometric dating would be very simple. We could be sure that a mineral containing parentium originally had no daughterium. If the mineral contained 1 part per million Parentium-123 and 3 parts per million Daughterium-123, we could be sure all the Daughterium-123 was originally Parentium-123. In other words there was originally 4 parts per million Parentium-123 and 0 parts per million Daughterium-123. Since there is now only 1/4 of the original amount of Parentium-123, we know that two half-lives of  Parentium-123 have elapsed.

But there are some questions that come to mind:

• If the half-life is very long, how can we know how long it is?
• How can we be sure the mineral doesn't lose one or the other isotope?
• Can we be sure the mineral doesn't gain one or the other isotope?
• Can we really be sure the daughter isotope wasn't present to begin with?
• Could nuclear reactions have affected the results?

## How Do We Know the Half Life?

Calculus students typically meet this problem somewhere in the second semester. It is one of the simplest examples of a differential equation.

What radioactive materials actually do is decay according to a law:

Decays/Time = K * Number of atoms

K is a constant called the decay constant. Let t stand for time and N(t) stand for the number of atoms at time t . In calculus terms, we write:

dN(t)/dt = -K * N(t)

or

dN(t)/n(t) = -K dt

The minus sign means that each decay decreases the total number of atoms. Integrating both sides, we get:

ln N(t) = -Kt + C

C is the constant of integration that we can often ignore, but not here. When t = 0,

ln N(0) = C

Taking exponentials of both sides, we get

N(t) = N(0)exp(-Kt)

If t = one half life, then N(t)/n(0) = 1/2 = exp(-Kt), and:

ln(1/2) = -ln2 = -Kt, so

t = ln2 / K

So what we do in practice is determine the decay constant and calculate half life from it. If the decay constant is very small, even tiny amounts of contamination by other radioactive materials can be very significant. So accurate determinations require very pure samples, very accurate and selective detectors, or both.

## True, Maximum, and Minimum Ages

The true age of a sample is self-explanatory, but unless the material dates from historic times, the true age is rarely known. A minimum age is the youngest the object can possibly be. A maximum age is the oldest the object can possibly be.

Suppose, in repaving your driveway, you find a stash of old coins buried in the ground. The driveway was poured in 1950, and the coins are all dated 1920. When was the stash buried?

The coins were obviously buried before 1950. That's the minimum age of the burial. But they obviously have to have been made first, so 1920 is the maximum age of the burial. Of course there are more outlandish explanations, like somebody counterfeiting 1920 coins in 1900 (and successfully anticipating any changes in design in the meantime), or secretly tearing up part of the driveway after 1950, but unless someone comes up with really persuasive evidence, we're justified in ignoring these hypotheses.

## Open and Closed Systems

Radiometric dating generally requires that a system be closed- in other words, has not had material added or removed. Crystallization of a mineral is a good way to close a system. In a closed system, daughter products are trapped. Any disturbance of the system effectively resets the clock to zero by allowing decay products to escape or reshuffling the abundances of elements.

Weathering and metamorphism are the two most common ways to disturb a system. Potassium-argon dating is very susceptible to resetting because the argon decay products are merely held in place mechanically by surrounding atoms. Argon, an inert gas, is not chemically bonded to neighboring atoms at all, and even minor thermal disturbance allows them to escape. Rubidium-strontium dating is more robust, and uranium-lead dating can survive fairly significant metamorphism without resetting.

If a system gains or loses isotopes in a predictable way, it maybe possible to estimate the loss and correct the age. Uranium-lead dating methods often use this approach because some of the minerals used in dating lose the lead decay products over time.

## The Right Stuff

It's amazing how often people fail to realize that you can't date materials if they don't have the necessary ingredients.

Carbon-14 dating is often used for historical objects and young prehistoric objects, but it's based on the fact that all living things start out with a known amount of carbon-14. No carbon, no carbon-14 dating. It's thatsimple. You can't use carbon-14 to date an arrowhead with no carbon in it. If the arrowhead is stuck in a bone, you can date the bone.

The most common dating methods for rocks are based on radioactive isotopes of potassium, rubidium, uranium, and thorium. If you don't have minerals with those elements, you can't date the rock. In particular, quartzites and carbonate rocks almost always don't have enough to permit dating. Sedimentary rocks are generally hard to date because common cements like silica don't have datable radioisotopes, and minerals like glauconite that are common in sedimentary rocks are very prone to resetting. If only there were long-lived isotopes of silicon, calcium, and magnesium! If the rocks have an interbedded lava flow or volcanic ash bed, it's gold.

## Gain or Loss

The older our sample is, the more daughter isotope it will contain relative to the parent. So:

• Gaining parent or losing daughter gives an age that is too young. The sample is older than its calculated age.

• Losing parent or gaining daughter gives an age that is too old. The sample is younger than its calculated age.

The general approach to assessing gain or loss is to look at the isotope abundances in different minerals and see if there's a pattern. If the ratio is constant, we can be pretty sure there's been no gain or loss.

Most people who reject radiometric dating do so because they believe the dated objects are much younger than the ages indicate. So it's worth noting here that almost everything that can perturb radiometric ages in nature makes the ages too young; in other words, the objects are older than the radiometric ages indicate. Radiometric ages are almost always minimum ages.

## Original Daughter Isotopes

If there is daughter isotope in the sample to begin with, then obviously the sample will give an age that is too old. This can happen through contamination of the sample or because the daughter isotope is present in nature and is naturally incorporated into the sample.

This is actually a very common situation, and the solution is to graph the parent and daughter isotopes on an isochron plot. Isochron plots are used for a number of dating methods, especially rubidium-strontium.

## Nuclear Bombardment?

Got a nickel? U.S. Nickels are 70% copper and only 30% nickel. Natural nickel is 68% Ni-58, 26% Ni-60, and smaller amounts of Ni-61, 62, and64. In between Ni-58 and Ni-60 is Ni-59, with a half life of 80,000 years. If the rocks had been bombarded in the recent past to change the amounts of radioactive isotopes, it's hard to envision any way of doing it without converting a lot of the world's nickel to Ni-59. Your pocket change should be distinctly radioactive.

Also, why is there so little technetium in the world? Technetium has three isotopes with half lives of 150,000-260,000 years. The preceding element, molybdenum, is fairly abundant. Surely any bombardment ought to have converted a lot of molybdenum to technetium. Where is it? (It's present in very miniscule amounts, mostly from spontaneous fission of uranium.)

So, it didn't happen. End of story.

## Real-World Dating

Cosmogenic isotopes are those created by bombardment of atoms by extraterrestrial particles. Carbon-14 is the best known. In recent years, beryllium-10 (2.5 million years) has been used for estimating erosion rates. We know it's cosmogenic because it's absent from very freshly exposed rocks and decreases with depth on a scale of a meter or so in older rocks. In other words, it forms on the surface, not within the earth.

Primordial isotopes are those left over from the formation of the earth. Decay-chain isotopes, like radon and radium, form by decay of primordial isotopes like uranium and thorium.

There are several dozen isotopes, like Ni-59, Fe-60 (300,000years), Ca-41 (80,000 years) Mn-53 (2 million years) and Se-79 (65,000 years)with long half lives that are absent in nature. The element technetium has three long-lived isotopes, all absent in nature. The only radioactive isotopes of any significance in the earth are either cosmogenic, primordial, or decay chain elements. (If you look hard enough, you can find a few atoms of almost anything- once in a while a solar particle will create an atom of Ca-41 or Fe-60 - but the amounts are as miniscule as you'd expect.) The absence of isotopes with half lives of thousands or millions of years indicates pretty clearly:

• Any original amounts have decayed away

• No new isotopes have been created by nuclear reactions

• Radiometric dating works. None of the evasions of radiometric dating work.

None of the real world dating methods are as clean as our hypothetical Parentium - Daughterium pair above. They all have some complication. Dating methods based on cosmogenic isotopes can only date young materials. Dating methods based on primordial isotopes can only date old materials. In the range of a few hundred thousand years is a gap where precise dates are very hard to get. Add to that the fact that a lot of materials we'd like to date, like glacial deposits, are unconsolidated and are not closed systems.

Cosmogenic
Isotopes produced by bombardment of particles from space. Carbon-14 was the first type of cosmogenic dating, but as detectors become more accurate, others like Be-10 have come into use.
Carbon-14
Cosmogenic, and present in a fixed amount in all living things, but ceases accumulating after death. Complicated by non-uniform solar activity.
Potassium-Argon
Perhaps the closest approach to our simple system above, but only 1/9 of all K-40 atoms decay to Ar-40. The others decay to Ca-40, which is the only natural isotope of calcium. The correction is simple: multiply the number of argon atoms by 9 to find the total number of potassium atoms that decayed.
Rubidium-Strontium
Rubidium-87 decays to strontium-87, but most Sr-87 is primordial. This is the principal method that uses the isochron technique.
Uranium and Thorium
Since there are numerous isotopes of uranium, each with its own distinct end product (a lead isotope), and thorium also decays to lead, there are numerous plots that can be used to sort out daughter from primordial isotopes. Some minerals used for uranium-lead dating, like zircon, are prone to lead loss, but the plots can reveal this as well.

Created 21 February 2002, Last Update