Appendix E: Radiometric Dating

Radioactive Decay

The different ages of these rocks is determined by a process known as radiometric dating. First described in 1907 by Bertram Boltwood, this method is now widely used for dating specimens throughout geology and uses known properties of atomic physics. All the baryonic matter that we interact with every day is made up of protons, neutrons and electrons. Protons and neutron are made up of quarks and have three “valence” quarks (Figure 1 below). Quarks are elementary particles and, as such, cannot be broken down any further. They possess intrinsic properties (some of which include charge and mass) and transfer these properties to the hadrons they make up. Hadrons is simply the term for something composed of quarks. Hadrons come in two types: mesons, which consist of a quark and an anti-quark, and our familiar baryons, which consist of three quarks. Quarks are studied by analyzing the way hadrons interact. As far as we have been able to tell, the electron is an elementary particle itself that cannot be broken down any further.

 

Figure 1 – The quarks that make up protons and neutrons. True to their baryon form, protons and neutrons are made of quarks. The outer valence quarks are draw here as blue, red, and green spheres. The u and d stand for up and down respectively, which describe the spin of each quark. Spin is one of the intrinsic properties that describe quarks. What would have to change for a neutron (left) to change into a proton (right)? How about for a neutron to change into a proton?

Recall that atoms can exist as several different isotopes, which contain different numbers of neutrons in their nucleus. Not all nuclei are stable. Generally heavier isotopes with an unbalanced number of neutrons relative to protons will undergo radioactive decay. For example, all carbon atoms have 6 protons, but additional neutrons are possible: carbon-12 and carbon-13 are stable isotopes, but carbon-14 is an unstable isotope. The unstable isotopes (here, carbon-14) are the parent isotope and they spontaneously decay into a different element or isotope, known as the daughter isotope.

There are two different types of statistically predictable spontaneous decay. The first is known as alpha decay (Figure 2 below), so named because the process emits an alpha particle (two protons and two neutrons). Alpha decay can only occur with very large nuclei. The parent isotope is left with a reduction of four in atomic mass. The loss of two protons means that the parent isotope has been converted to a lighter element in the Periodic Table.

Figure 2 – A Uranium-238 nucleus undergoing alpha decay. The uranium nucleus loses an alpha particle, or two protons and two neutrons. This is often referred to as a helium nucleus, which typically has those exact proportions. Because both protons and neutrons were loss, the uranium atom changes into a thorium atom after the decay event.

A second type of spontaneous decay is beta decay. The atomic mass (total number of protons + neutrons) remains the same, but the atomic number (number of protons) changes. A proton or neutron may change into the other by flipping the charge of one quark. These changes are possible because protons and neutrons are not elementary particles. With \beta ^- decay, a neutron decays into a proton plus an electron (to maintain charge balance) and an electron antineutrino to carry away energy. This changes the atom to a heavier element (plus one proton). An example of \beta ^- decay is the conversion of 14C (6 protons) to 14N (7 protons):

                    ^{14}_6C \rightarrow \ ^{14}_7 N + e^- + \nu_e

With \beta ^+ decay, the proton becomes a neutron, absorbing an electron, and the atom is changed to a lighter element (minus one proton). And example of \beta ^- decay is conversion of magnesium (12 protons) to sodium (11 protons):

                   ^{23}_{12} Mg \rightarrow ^{23}_{11}Na + e^+ + \nu_e

It is impossible to say for sure when a decay event will happen, but we can categorize the rate at which a sample of atoms will decay. If we know the amount of parent and daughter isotope, then knowing the rate of decay allows us to solve for how much time it must have taken for the parent isotope to decay into that much daughter isotope. This rate is characterized by the half-life, or the amount of time it takes for half of the parent isotope to decay into the daughter isotope. Different elements are useful for dating different age ranges. The Table below lists information for common isotopes.

 

Table 1 – Radioactive half lives of key isotopes. Which of these would be most useful for dating meteorites to learn about the age of the solar system? Which would be most useful for forensic evidence about a crime scene?

Half life

Imagine a bag of microwave popcorn kernels. We can think of the popcorn kernels as unstable parent isotopes. The process of popping will symbolically represent spontaneous radioactive decay and the resulting popped popcorn will be the daughter isotopes. Though it is impossible to say when a specific kernel of popcorn is going to pop, we know how long it takes for most of the kernels to pop. There exists a relationship between number of popped kernels and amount of time passed, known as the radioactive half life.  If we know the rate at which an isotope decays, we can calculate the age of a specimen given the fraction of parent and daughter isotopes in the sample.

 

Figure 3 – The exponential decay of radioactive isotopes. Time here is represented by number of half-lives. Note how with each half-life, the amount of parent isotope (red) is halved. How much of the parent isotope is left after two half-lives

 

Rates of radioactive decay can be determined in a laboratory setting. It is known that radioactive decay is an exponential process given by:

N(t) = N(t=0) e^{\frac{-y}{t_{mean}}}

  • where N(t) is the amount of the parent isotope that remains,
  • N(t=0), or sometime also written N_0, is the initial amount the parent isotope that a sample started out with,
  • t is the amount of time that has passed,
  • and t_{mean} = \sqrt{2} \times t_{half} is the half-life of the element in question.

Taking the natural logarithm of both sides of the equation above and solving for t, we get an equation for the amount of time that has passed:

t = - \ln \left( \frac{N}{N_0} \right) \times \sqrt{2} \times t_{half}

As an example, let us imagine a classroom of 150 students who exhibit some very radioactive behavior. At the beginning of class, all 150 students are awake and attentive. Though we certainly hope this is never the case, suppose the students are falling asleep at an exponential rate, similar to the way that radioactive isotopes decay. After thirty minutes, half of the students have already fallen asleep! How much time has passed when only 30 students, or 20%, remain awake?

Using the equation above for time, the number of “surviving” (i.e., awake – no students were harmed in this thought experiment) 30 students is N, the initial 150 awake students is N_0, and 30 minutes was the half-life, t_{half}. Because the half-life is in units of minutes, the answer will also be in minutes.

t = - \ln \left( \frac{30}{150} \right) \times \sqrt{2} \ times 30 \approx 68 minutes

So, an hour and 8 minutes into the class, only 30 students remain awake in this completely hypothetical classroom.

In the example above was a simple case of misbehaving students. When dating rocks, the use of many different radioactive isotopes gives even more information about the age of a specimen. The choice of isotopes depends largely on what is present in the rock sample and a sensible choice, given the relative half-lives of the different isotopes.  Half-lives can range from fractions of a second to billions of billions of years. Elements with longer half-lives are more useful for dating older rocks. Isotopes with half-lives comparable to the age of the substance being dated are ideal.

Atomic elements can also be changed by fission, which splits massive atomic elements into less massive elements. Spontaneous fission releases substantial amounts of energy.  Elements can also be changed by fusion of lighter elements to form heavier elements.  As discussed before, this process takes place in the cores of stars where hydrogen undergoes nuclear fusion to form helium. This process requires the input of a substantial amount of energy.

Statistical uncertainty

The accuracy of radiometric dating can be hard to ensure because the method depends on knowing both how much of the parent isotope was initially present, and how much of the daughter product is the result of decay. It is possible that the daughter isotope will preferentially escape from a sample, or a contaminating source will add more of either the parent or the daughter isotope. Returning back to the fictitious classroom example, this would happen if different students left and entered the room during the class. Then, someone observing the room an hour after the start of class would be uncertain about how many students were initially in the room.

There are ways to improve accuracy. For example multiple samples can be analyzed from different locations in the same rock in case one area suffered contamination. It is also helpful to calculate the age using several different isotopes to check for consistent results. This offers some insurance against potential loss of daughter isotopes since contamination or loss of daughter isotopes should behave differently. Counting accuracy is improved when there is a relatively high concentration of both the parent and daughter isotope.

Even with the best laboratory practices, radiometric dating depends inherently on the type of rock. Rocks are classified into three groups. Igneous rocks are made from molten magma or lava that solidifies into rock. Sedimentary rocks are layered rocks formed when sand and silt collect on the surface or in bodies of water and cement together to form new rock. Metamorphic rocks form when rocks undergoes intense temperature and/or pressure and transform into a different type of rock altogether. Through various processes, different types of rocks can transform into one another, as depicted in Figure 4 below.

 

Figure 4 – The rock cycle. Outlined here are the different ways one classification of rock can transform into another. The rocks on Earth are constantly changing and shifting through these different processes.

 

Sedimentary rocks and metamorphic rocks are not good for radiometric dating. Sedimentary rock is made up of a conglomeration of the particles eroded from different types and ages of rock. Metamorphic rock undergoes too much change. Radiometric dating is only secure for igneous rocks that remain stable. Even so, radiometric dating of igneous rocks gives only the time since they last melted. Radiometric data has helped date rocks that are billions of years old going back almost to almost 4 Gya.  The age of the Earth can be determined by radiometric dating of meteorites, the unprocessed specimens of planet formation.

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