Radioactive dating is one of the primary tools of science. Geologists, biologists, paleontologists and archaelogists, among others, depend on radiometric analysis to place their discoveries chronologically. It provides the evidence for statements that the earth is 4.6 billion years old, or that dinosaurs were wiped out 65 million years ago.

Since it is so fundamental a tool, we should take the time to understand how it works. Physics to the rescue!

Let’s start with the basics. There are 90 naturally occurring elements, each with a unique atomic structure. We order elements according to the number of protons in the nucleus — the atomic number. For example, carbon is element 12 and uranium element 92, because they contain that many protons in their nuclei.

Additionally, each element (atom) varies in the number of neutrons in the nucleus. These varieties of atoms are called isotopes. Each isotope has a different atomic mass (or weight), conveniently measured by counting the total of the protons and neutrons in the nucleus. Isotopes all have the same chemical behavior.

Hydrogen, the simplest atom, has one proton. It has three isotopes: H-1 (one proton), H-2 (one proton, one neutron) and H-3 (one proton, two neutrons). The last two isotopes are also known as deuterium and tritium, but they are still hydrogen, and can still combine with other atoms to make water, coal or tomatoes, for example.

While isotopes have the same chemical behavior, they do not share the same nuclear behavior. Some isotopes, such as H-1 or Carbon-12, have very stable nuclei. Others, like Carbon-14, do not. Many “heavy” nuclei, such as radium, radon, uranium and thorium, share this instability.

Each nucleus is in constant turmoil, as protons and neutrons dance around each other, subject to forces beyond their control. Protons try to repel each other, since they share the same positive electical charge. The strong nuclear force holds the protons and neutrons together. Meanwhile, the weak nuclear force is trying to split the family up.

This instability results in what we conveniently call radioactivity. In the process of reducing their inner turmoil, some nuclei spit out matter and energy to calm things down. It’s a little like bouncers throwing trouble-makers out of a dance club.

The process happens with some mathematical regularity. Each nucleus has its own internal “clock,” so we can use that clock to determine how long a sample containing that nucleus has been around. This clock-like property of an isotope is its “half-life.”

Radioactive emission typically changes the atomic number or atomic mass of a nucleus. For example, alpha emission reduces the atomic number by two and the atomic mass by four. Beta emission increases the atomic number by one, while decreasing the atomic mass by one.

So alpha emitters, over time, transmute into lighter elements at a very predictable rate. A convenient way to measure this rate is the half-life – how long it takes half the available nuclei to transmute into lighter ones.

Mathematics allows to predict half-lives, some of which are millions of years. To demonstrate, let us suppose we have a sample of 1000 nickels. At the end of every minute, we replace one-fourth the number of nickels with pennies — the rate of radioactive decay. At the end of the first minute, we would have 1000-250, or 750 nickels and 250 pennies. If we pretend we can have fractions of coins, after two minutes, we would have 750-187.5, or 562.5 nickels and 437.5 pennies. We can tabulate the results to get a sense of the “history” of the decay.

Elapsed time in minutes | nickels | pennies |

0 | 1000 | 0 |

1 | 750 | 250 |

2 | 562.5 | 437.5 |

3 | 421.88 | 578.13 |

4 | 316.41 | 683.59 |

5 | 237.3 | 762.7 |

6 | 177.98 | 822.02 |

7 | 133.48 | 866.52 |

8 | 100.11 | 899.89 |

9 | 75.08 | 924.92 |

10 | 56.31 | 943.69 |

11 | 42.24 | 957.76 |

12 | 31.68 | 968.32 |

13 | 23.76 | 976.24 |

14 | 17.82 | 982.18 |

15 | 13.36 | 986.64 |

16 | 10.02 | 989.98 |

17 | 7.52 | 992.48 |

18 | 5.64 | 994.36 |

19 | 4.23 | 995.77 |

20 | 3.17 | 996.83 |

21 | 2.38 | 997.62 |

22 | 1.78 | 998.22 |

23 | 1.34 | 998.66 |

24 | 1 | 999 |

After 24 minutes we would have one nickel and 999 pennies. But the half-life, the time it takes for half the nickels to turn into pennies, is somewhere around 2-3 minutes. After another half-life, or 4-5 minutes, we have one-quarter of the original 1000 nickels. In two half-lives, we’re down to 125 nickels, and so.

This kind of decrease is called exponential decay, and can be modeled in math with the exponential function. For our imaginary sample of “nickelium,” the half-life is more precisely 2.4094 minutes.

Larger samples of nickelium would take longer to transmute completely into pennies, even with the same half-life. If we had started with 2000 nickels, after 24 minutes we would have two nickels and after 26.4094 minutes we have have just one nickel. For a sample containing 1,000,000 nickels, we would have to wait about 49 minutes to end up with just one nickel.

Since all nickelium acts the same way, we do not have to wait around for the last nickel to turn into a penny. We can calculate how long the nickelium sample has been around by counting how many nickels and pennies there are. For isotopes with half-lives longer than several days, this principle is especially handy.

Let’s see how can apply this concept to a sample of something dug up from the ground. We estimate, based on its chemical composition and mass, that there are 1,000,000 nuclei of isotope Z and 1,000,000 nuclei of isotope X in it. We know from previous experiments that isotope Z has a half-life of 5,000 years and that on average it is about as common as isotope X.

So, over 5,000 years, half of Z turned into X. Since we have 1,000,000 of each now, this means that 5,000 years ago, there were 2,000,000 Z and zero X. Our sample then is approximately 5,000 years old.

The previous examples are over-simplified to demonstrate the basics of radioactive dating. In reality, we cannot know precisely the number of radioactive nuclei in a sample, since the number is much, much larger than 1,000,000. We can estimate that number by finding the mass of the sample and the relative amount of the radioactive isotope in it.

Also, we assume that the different isotopes that exist today also existed in the past. In the case of the isotope Z sample, for simplicity we pretended that at one time there was no X in the sample. In reality, some isotopes exist indefinitely. If X has always existed and Z is quite rare compared to X, then our estimate of 5,000 years is too low.

Further, the rate of decay is not as regular as clockwork. External factors may affect the decay rate. A sample might have been surrounded by more radioactive material, accelerating its own decay rate. In that case, we might think it’s younger than it really is.

Scientists typically compare their radioactive dating results with estimates obtained through other means to verify their conclusions are reliable. Rarely do they rely on just one method of dating a sample.

Carbon-14 is a rare, radioactive isotope of carbon, the most prevalent element in organic matter. The proportion of C-12 to C-14 in living organisms is relatively constant, since the carbon is constantly replenished. Once the organism dies, it no longer takes in new carbon. C-12 is stable, but C-14 decays with a half-life of 5730 years. The longer a sample has been dead, the less C-14 it will contain, compared to a living organism. So C-14 is especially handy in determining the age of items that are younger than 60,000 years. Older samples usually have too few C-14 nuclei for results to be reliable.

Potassium, another element found in organic matter, has a radioactive isotope K-40 (K is the chemical symbol for potassium; it was once known as kalium). As with carbon, living organisms replenish their potassium. Dead tissue will have decreasing proportions of K-40 in them. K-40 has a half-life of 1.3 billion years, so age results from C-14 analysis can be compared to results from K-40 analysis.

If our organic sample were embedded in a layer of rock or soil, we could use the radioactive minerals in the rock or soil to further refine our age estimates. Geologists have a pretty firm grasp on the age of the various layers of rock and soil, based on their own radioactive dating methods and estimates on how long it would take geologic processes to “lay down” a new layer of rock.

With these methods of dating rocks, fossils and organic materials, scientists can confidently say the earth and moon are 4.6 billion years old, dinosaurs died off 65 million years ago and mummies were buried 4,000 years ago. It’s all in the physics of the nucleus.

Handy link: An animated model of radioactive decay

More technical link: Radiometric Dating, explained by a Christian scientist