The Physics
Opus in profectus

Half Life

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Activity (A) is the number of nuclear decays (n) occurring in a radioactive source per time (t). The SI unit of activity is the becquerel [Bq], which is equal to an inverse second [1/s or s−1]. The becquerel is named in honor of the French scientist Henri Becquerel (1852–1908) who was the first person to detect radioactive decay. The becquerel replaces an earlier unit called the curie [Ci], named in honor of the French scientists Marie Curie (1867–1934) and Pierre Curie (1859–1906) who were pioneers in nuclear physics and chemistry. One curie is equal to 3.7 × 1010 Bq, or 37 gigabecquerels, by definition.

A = n

half life

N = N02t/T½

radiocarbon dating

Der Tod startet die Stoppuhr. Immer wenn ein Lebewesen stirbt, beginnt eine Stoppuhr zu laufen. Die Wissenschaft kann diese Uhr ablesen und so das Alter eines Fundes ermitteln.

Every time a living being dies a stopwatch starts ticking. Death starts the stopwatch. Science can read it.

Source unknown — possibly das Museum für Vor‑ und Frühgeschichte (the Museum for pre‑ and early history) in Berlin.

Radiocarbon dating is used to determine the age of previously living things based on the abundance of an unstable isotope of carbon. The isotopic distribution of carbon on the Earth is roughly 99% carbon 12 (with 6 protons and 6 neutrons) and 1% carbon 13 (with 6 protons and 7 neutrons). These isotopes are stable, which is why they are with us today, but unstable isotopes are also present in minute amounts. About one carbon atom in a trillion (1012) contains a radioactive nucleus with 6 protons and 8 neutrons — carbon 14. This rare, unstable isotope is produced from ordinary nitrogen 14.

In Earth's upper atmosphere, on the edge of what is commonly called outer space, light atomic nuclei from unknown sources outside of our solar system traveling at speeds approaching the speed of light called cosmic rays rain down continuously. These highly energetic nuclear bullets wreak havoc on the atoms in the upper atmosphere: tearing electrons from their orbitals and setting them free, knocking neutrons and protons from the tight confines of the nucleus and setting them free, generating x-rays and gamma rays as they decelerate, and creating exotic particles like muons and pions directly from their excessive kinetic energy. These secondary cosmic rays are also highly energetic and will ionize atoms, transmute nuclei, and generate x-rays themselves. A secondary cosmic ray neutron of sufficient energy striking a common nitrogen 14 nucleus can force it to eject a proton.

147N + 10n → 146C + 11p

This is the process by which all of the carbon 14 on the Earth is produced. (Produced naturally to be more precise. More on that later.)

All organic material contains carbon. (By definition, organic chemistry is the chemistry of carbon compounds.) Plants absorb 14C like they absorb other isotopes of carbon — through the respiration of carbon dioxide — and then use this carbon to produce sugars, fats, proteins, and vitamins. Bacteria, fungi, and animals eat these plants and each other. In this way, atmospheric carbon is distributed throughout the web of life until every living thing has the same ratio of 14C:12C as the atmosphere. (Well, nearly the same ratio. Plants and animals tend to favor lighter nuclei just a bit. Serious technicians know how to compensate for this preference when dating samples.)

With a half life of 5730 years, 14C decays by beta emission back into the 14N from which it originated.

146C → 147N + 0−1e + 00ν

Since living creatures are constantly swapping atoms with their environment, the abundance of 14C within them remains fixed. After death, however, no new radioactive carbon comes along to replenish that which has decayed and the abundance of 14C decreases. The ratio of 14C:12C in a piece of living organic matter will be the same as it is in the atmosphere but larger than in a piece of dead organic material. A timber found in a home built 5730 years ago (one half life) would have half the 14C:12C ratio that a person living today would. A discarded oyster shell from someone's dinner eaten 11,460 years ago (two half lives) would have one-quarter the 14C:12C ratio that a cotton shirt worn today would. A tusk from a mammoth that died 17,190 years ago (three half lives) would have one eighth the 14C:12C ratio that a cardboard box manufactured today would. And so on. Death starts the stopwatch. Science can read it.

Radiocarbon dating of a hypothetical organic sample * In the United States and a few other countries 1012 is called a trillion. The unit ppt means "parts per trillion".
age (half-lives) age (years) 14C (atoms) 12C (atoms) 14C:12C (ppt)*
0 0 128 128 × 1012 1
1 5,730 64 128 × 1012 0.5
2 11,460 32 128 × 1012 0.25
3 17,190 16 128 × 1012 0.125
4 22,920 8 128 × 1012 0.0625
5 28,650 4 128 × 1012 0.03125
6 34,380 2 128 × 1012 0.015625
7 40,110 1 128 × 1012 0.0078125

Calculations of this sort are based on the assumption that the ratio of 14C:12C in the Earth's atmosphere has remained constant. As a first approximation one can assume this, but more accurate results must take into account fluctuations in the intensity of the cosmic rays entering the Earth's atmosphere. These deviations were determined from the comparative dating of ancient tree rings (a field called dendrochronology) and the results were then compiled into a calibration curve. Current calibration curves go back as far as 26,000 years ago. The range of radiocarbon dating extends back to about 50,000 years. For items older than this, there isn't enough undecayed 14C left to measure the ratio reliably.

Radiocarbon dating in the future will have to include adjustments for human activities. Beginning in the late 1950s, considerable amounts of anthropogenic (human-produced) 14C have been added to the atmosphere, mostly as a result of nuclear weapons testing. This activity reached its peak in the early 1960s when an atmospheric blast occurred somewhere on Earth every two to three days. Nuclear bombs generate large numbers of high energy neutrons, which can in turn transmute nitrogen 14 into carbon 14 in exactly the same way as naturally occurring secondary cosmic rays. By 1965, atmospheric 14C concentrations were double their pre "atomic age" values.

Coal and petroleum are the fuels that powered the Industrial Revolution. Coal is nearly pure carbon and petroleum is a mixture of hydrocarbons. By the end of the 20th century, humans were adding carbon from fossil fuels to the atmosphere at a rate of about 10 billion tonnes per year (1013 kg/year). The total human contribution to the atmosphere passed one trillion tonnes of carbon (1015 kg) a few years into the 21st century. Fossil fuels are the remains of long dead plants that were buried in sediment tens to hundreds of millions of years ago (coal being made primarily from land plants and petroleum from plankton and algae). The fraction of the original 14C left in a plant that has been dead for 5,730,000 years (one thousand half lives) is…

1  ≈ 10−301

For all intents and purposes, this number is zero. Coal and petroleum have been dead for so long they no longer contain any 14C. Burning these fossil fuels is diluting the 14C content of the atmosphere. This effect this has on radiocarbon dating was first measured by the Austrian chemist Hans Suess (1909–1993) in the 1950s and is now known as the Suess effect.

Stacked bar graph

Drinking ethanol (which is usually just called alcohol) is made from the fermented sugars of plants (grains like barley, wheat, rye corn, or rice; fruits like grapes or apples; vegetables like sugarcane or agave; or the nectar of plants collected by bees called honey). Industrial ethanol is made from petroleum. The carbon in the ethanol that came from plants will be relatively rich in 14C, since the plants it came from were relatively recently alive. The carbon found in the ethanol that came from petroleum will not contain a single atom of 14C, since the plants it came from died hundreds of millions of years ago and every last atom of 14C in it will have decayed long ago. 14C is radioactive. The other common isotopes 12C and 13C are not. This means that the ethanol you might put inside yourself (the ethanol you might drink) will be slightly radioactive, while the ethanol you'd never consume (the ethanol you might wipe your skin with) will not be.

potassium-argon dating

Potassium-argon dating is used to determine the age of igneous rocks based on the ratio of an unstable isotope of potassium to that of argon. Potassium is a common element found in many minerals. The isotopic distribution of potassium on the Earth is approximately 93% 39K and 7% 41K. Since these values are only approximate, the total percent abundance of these two isotopes is not 100%, but 99.9883%. The remaining 0.0117% is 40K — an unstable isotope with a half life of 1.26 × 109 years (1.26 billion years). Potassium 40 has three decay modes: beta decay, positron emission, and electron capture.

4019K → 4020Ca + 0−1e + 00ν   (beta decay)
4019K → 4018Ar + 0+1e + 00ν   (positron emission)
0−1e + 4019K → 4018Ar + 00ν   (electron capture)

When 40K undergoes beta decay it transmutes into 40Ca — the most abundant isotope of calcium. Since calcium is also common in minerals, it is not possible to distinguish the 40Ca produced from the decay of 40K from the 40Ca present when the rock was formed. However, when 40K undergoes positron emission or electron capture it transmutes into 40Ar. Argon is an inert substance, which means that it basically will not combine chemically with other elements. It is also a gas over a wide range of temperatures, which means that any 40Ar would escape while the rock was molten like carbon dioxide escaping from a glass of soda. After solidification, those 40Ar nuclei that appeared as a result of radioactive decay would be trapped by the crystal structure and accumulate as the mineral aged.

In a hypothetical mineral sample with an initial population of 64 40K atoms, the ratio of 40K:40Ar would evolve as follows.

Potassium-argon dating of a hypothetical mineral sample
age (half-lives) age (109 years) 40K (atoms) 40Ar (atoms) 40K:40Ar
0 0 64 0 -
1 1.26 32 32 1:1
2 2.52 16 48 1:3
3 3.78 8 56 1:7
4 5.04 4 60 1:15
5 6.30 2 62 1:31
6 7.56 1 63 1:63

Potassium-Argon dating techniques have been used to date minerals covering the entire span of geologic history from 10 thousand to 3 billion years old.

other radioisotopic dating techniques

There are several other dating techniques that rely on the principle of exponential decay and half-life. Each has its own range of validity.

Radioisotopic dating techniques
technique range (years) dateable items
lead 210 1 150 lake and ocean sediments, glacial ice
carbon 14 1 40,000 previously living things
uranium series 1 400,000 bone, teeth, coral, shells, eggs
potassium-argon 10,000 3 billion minerals, igneous rocks
uranium-lead 1 million 4.5 billion minerals, igneous rocks
rubidium-strontium 60 million 4.5 billion minerals, igneous rocks

Horzontal bar graph of radiodating ranges

odds 'n' ends