Medical and Archaeological Uses: Radiocarbon Dating
Radioactive isotopes are used in medicine to diagnose and treat disease and in archaeology and geology to measure age; in radiocarbon dating, the age of organic material is found from the decay of carbon-14, which has a half-life of 5730 years: t = (5730/ln 2) × ln(A0/A). In brief: a sample that has 25 % of the carbon-14 activity of living material is two half-lives, about 11,460 years, old, and technetium-99m is used in tens of millions of imaging procedures every year.
Radioisotopes behave chemically like the stable isotopes of the same element, but they can be found by the radiation they emit. This makes them excellent tracers in the body, and clocks in the study of the past. This article explains the main medical uses of radioisotopes in a table, shows how radiocarbon dating works and how to calculate ages, describes the limits and calibration of the method and mentions other dating methods for much older materials.
How are radioisotopes used in medicine?
- Diagnosis (nuclear medicine): a small amount of a radioactive tracer is attached to a molecule that goes to a certain organ. A gamma camera or a PET scanner shows where the radiation comes from. The tracer is chosen to have a short half-life (hours), and to emit gamma rays that can leave the body, but with little beta or alpha radiation, so that the dose to the patient is low.
- Therapy: radioisotopes that emit beta or alpha particles can destroy diseased tissue from within: iodine-131 collects in the thyroid, where it treats hyperthyroidism and thyroid cancer; lutetium-177 and radium-223 are used for tumors that have spread.
- External radiotherapy: beams of gamma rays from cobalt-60 (or X-rays from linear accelerators) are directed at tumors from several angles.
- Brachytherapy: sealed sources such as iodine-125 seeds are placed in or near the tumor.
- Sterilization: of syringes, dressings and implants with gamma rays.
Important medical radioisotopes
| Isotope | Half-life | Radiation | Use |
|---|---|---|---|
| 99mTc | 6.01 h | γ (140 keV) | Imaging of bone, heart, kidneys, lungs, thyroid |
| 18F | 110 min | β+ | PET scans (fluorodeoxyglucose) of tumors and the brain |
| 123I | 13.2 h | γ | Thyroid imaging |
| 131I | 8.02 d | β−, γ | Thyroid therapy |
| 82Rb | 75 s | β+ | PET imaging of heart blood flow |
| 60Co | 5.27 y | γ (1.17 and 1.33 MeV) | External radiotherapy, sterilization |
| 177Lu | 6.65 d | β−, γ | Targeted therapy of neuroendocrine and prostate tumors |
Technetium-99m is obtained from a “generator” containing molybdenum-99 (half-life 66 h), which decays to technetium-99m, and the hospital can “milk” the generator for fresh technetium each day.
How does radiocarbon dating work?
- Production: cosmic rays produce neutrons in the upper atmosphere, and the neutrons convert nitrogen-14 into carbon-14: 147N + 10n → 146C + 11H.
- Distribution: the carbon-14 is oxidized to 14CO2, which mixes with the atmospheric carbon dioxide, and enters plants by photosynthesis and animals through food. In living organisms, the ratio of 14C to 12C is in equilibrium with the atmosphere, about 1.2 × 10−12, which corresponds to an activity of about 0.226 Bq per gram of carbon (13.6 decays per minute per gram).
- Decay: when the organism dies, it stops taking up carbon, and the 14C decays by β− emission, with a half-life of 5730 years: 14C → 14N + e− + ν̄.
- Measurement: the remaining 14C is determined, either by counting the beta decays or, now mostly, by accelerator mass spectrometry (AMS), which counts the carbon-14 atoms directly and needs only milligrams of material.
- Calculation: the age comes from the decay law.
How do you calculate an age?
Since the decay is first order, A = A0e−λt, so:
t = (1/λ) ln(A0/A) = (t½/ln 2) ln(A0/A) = 8267 y × ln(A0/A)
Here A0 is the activity (or the 14C/12C ratio) of modern material and A that of the sample. Because the same carbon is analyzed in both, the amount of sample does not matter.
Worked examples
Example 1: the fraction method
A piece of charcoal has 25 % of the 14C activity of living wood. This is two half-lives, so the age is 2 × 5730 = 11,460 years. A sample with 12.5 % has gone through three half-lives: 17,190 years.
Example 2: using activities
A wooden artifact has an activity of 0.100 Bq per gram of carbon, compared with 0.226 Bq/g for living wood. t = 8267 × ln(0.226 / 0.100) = 8267 × ln 2.26 = 8267 × 0.815 = 6740 years.
Example 3: counting rate
A bone sample of 5.00 g of carbon gives 36.0 decays per minute, whereas 5.00 g of carbon from a living organism gives 5.00 × 13.6 = 68.0 decays per minute. A/A0 = 36.0 / 68.0 = 0.529. t = 8267 × ln(1/0.529) = 8267 × 0.636 = 5260 years.
Example 4: what is the expected activity?
What fraction of the original carbon-14 remains in an object that is 2000 years old (the age of the Dead Sea Scrolls is about this)? Fraction = e−λt = e−0.693 × 2000 / 5730 = e−0.2419 = 0.785, so 78.5 % of the original activity is left.
Example 5: the age at the detection limit
With a detection limit of about 0.2 % of the modern 14C content, the maximum age is t = 8267 × ln(1/0.002) = 8267 × 6.21 = 51,400 years. This is why the practical limit of radiocarbon dating is about 50,000 years.
What are the limits of the method?
- Age range: up to about 50,000 years. After about 10 half-lives, there is too little 14C to be measured.
- Only for carbon-containing material that was once alive: wood, charcoal, bone, shell, cloth, seeds. Not for rocks or metals.
- Variable atmospheric 14C: the production varies with the cosmic ray intensity and the magnetic field of the Earth, so radiocarbon years are not exactly calendar years. The method is calibrated against tree rings (dendrochronology), corals, cave deposits and lake sediments. Curves such as IntCal20 convert radiocarbon ages to calendar ages. Libby’s original calculations used a half-life of 5568 years, and by convention results are still reported with it, as “radiocarbon years before present” (BP, where “present” is 1950).
- Contamination: modern or old carbon added to the sample gives wrong ages, so careful pretreatment is needed.
- The reservoir effect: marine organisms have apparent ages hundreds of years older, because of the old carbon in the deep ocean. The burning of fossil fuels (which contain no 14C) lowered the atmospheric ratio, and atmospheric nuclear tests in the 1950s–60s almost doubled it.
What other dating methods exist?
| Method | Parent → daughter | Half-life | Range and use |
|---|---|---|---|
| Radiocarbon | 14C → 14N | 5730 y | Up to about 50,000 years; organic remains |
| Potassium–argon | 40K → 40Ar | 1.25 × 109 y | Volcanic rocks, from about 100,000 years to the age of the Earth |
| Uranium–lead | 238U → 206Pb | 4.47 × 109 y | Zircon crystals, oldest rocks and meteorites |
| Rubidium–strontium | 87Rb → 87Sr | 4.9 × 1010 y | Ancient rocks |
Uranium–lead dating of meteorites gives an age of 4.54 billion years for the Earth and the Solar System.
Who developed radiocarbon dating?
Willard Libby, an American chemist at the University of Chicago, developed radiocarbon dating in 1946–1949 and showed that the method gives the correct ages of samples of known age, such as wood from Egyptian tombs, whose ages are known from historical records. He received the Nobel Prize in Chemistry in 1960. The method changed archaeology: it gave the first universal and objective way of dating prehistoric sites. Famous results include the age of Ötzi the Iceman (about 5300 years), and the Turin Shroud, which was dated in 1988 by three laboratories to the 13th–14th century (1260–1390 AD).
Which mistakes are common?
- Using radiocarbon dating on rocks or dinosaur bones: the method is for organic material under 50,000 years old.
- Using the half-life formula with the wrong time unit: keep years throughout.
- Mixing up ln and log: the equation uses the natural logarithm.
- Ignoring calibration: radiocarbon years differ from calendar years.
- Forgetting that A0 is not fixed over time: it changed in the past, and this is accounted for by calibration curves.