Nuclear Engineering

Basic Formulation on Radioactive Dating process

Determination of age of any geological or archeological object is an extremely important task to know the history of planetary object as well as human civilization. Among geological time scale Uranium-lead dating, Sr-Rb dating etc. are popular. Radiocarbon dating is important in archeological time scale and to understand the history of human civilization.

Here I am going to discuss the basic mathematical formulation of the above mentioned dating methods and a brief description of each of these.

Case 1:

First Let us discuss the basic mathematical formulation of U-Pb time scale.

Let us consider that at time t = t₀, the number of parent radioactive nuclide present in any specimen is N_P (t₀), and at a time t = t₁, the number of parent and daughter radioactive nuclides are N_P (t₁) and N_D (t₁) respectively. Therefore, we can write

Now,

Therefore,

So, given the decay constantλ, and the ratio

the age of the sample can be obtained from Eqn. (m4.32).

Let us now see the U-Pb time scale based on this formulation.

The determination of geological ages is done very often by uranium-lead technique. It involves the following nuclear process.

In order to estimate the time scale, let us consider ²³⁸₉₂U series, where stable end product is ²⁰⁶₈₂Pb having decay constant, λ_Pb = 0. The half-life of ²³⁸U is 4.5 x 10⁹ years. Hence after sufficient time, (say, a billion years), the only elements in any appreciable amount will be uranium and lead. Other elements will be in secular equilibrium with the parent ²³⁸U, whereas ²⁰⁶Pb will not be in equilibrium and will continuously increase. Let as take ²³⁸U as element P and ²⁰⁶Pb as element D, and considering transformation of X → Y, we can write,

Now λ_P = λ_U = 4.5 x 10⁹ years

and λ_D = λ_Pb = 0

Now from Eqn. (m4.32) we can write

The oldest surface rocks have been found to have an age of about 3 x 10⁹ years by this technique. Applying the same technique to determine the age of the meteorites, it is found that oldest of these are about 4.5 x 10⁹ years. This corresponding to the age of the earth and is different from the segregation of surface rocks.

Case 2:

Let us consider that daughter nucleus is also present at time t = t₀. We can, therefore, write

Let us also consider an isotope of a daughter nucleus D’ is also present in the sample, which is formed from the decay of a long-lived parent and non-radioactive. Therefore, we can consider that N_D’(t₀) = N_D’(t₁).From Eqn. (m4.35) we can write,

Equation m4.36 can be expressed by a straight line y = mx + c with a slope m =

and intercept c =

Let us now see Rb-Sr time scale based on this formulation.

For example, from the decay of ⁸⁷Rb → ⁸⁷Sr (T_1/2 = 4.8 x 10¹⁰ y) in which the comparison is made with the isotope ⁸⁶Sr of the daughter nucleus ⁸⁷Sr. Variation of ⁸⁷Sr/⁸⁶Sr and ⁸⁷Rb/⁸⁶Sr is plotted, which is a straight line as per equation m4.36. The time interval Δt calculated from this data is 4.53 x 10⁹ year.²

This method indicates
(i) no loss of parent nuclei and
(ii) no loss of daughter nuclei during the decay processes.

Case 3:

Now we are going to discuss about radiocarbon dating method to determine the age of archeological objects and to evaluate history of living objects. Radiocarbon dating is a radiometric dating method that uses the naturally occurring isotope carbon-14 (¹⁴C) to determine the age of carbonaceous materials up to about 60,000 years2. Carbon has two non-radioactive isotopes; carbon-12 (¹²C) and carbon-13 (¹³C) and a very small amount of radioactive isotope, carbon-14 (¹⁴C). Carbon-14 continuously forms in atmosphere due to nuclear reaction of cosmic ray neutrons and ¹⁴N. The nuclear reaction is

is a neutron rich isotope of carbon and it decays through the following reaction

Half-life of

is 5730 years. This decay can be used to get a measure of how long ago a piece of once-living material died. It can be considered that C-14 is produced at a constant rate, as the cosmic ray flux is constant over a long period of time. Since ¹⁴C has a half-life of 5730 years it is an ideal radioactive isotope for studying the age of civilization and is extensively used in archaeology and anthropology. Plants take up atmospheric carbon dioxide by photosynthesis, and are eaten by animals, so every living being is constantly exchanging carbon-14 with its environment as long as it lives. When an animal or plant dies, its intake of carbon stops and from that moment decay of ¹⁴C is the only process that continues. In order to estimate the age (t_age) of any archaeological object, Eqns. 6.1 and 6.2 are to be used and we can write,

It is important to count the radioactive decay of individual carbon atom, as the proportion of radioactive to non-radioactive carbon atom is ^~ 1 parts per billion (ppb). The counting is generally done using a gas proportional counter or a liquid scintillation counter. However, these are relatively insensitive and subject to relatively large statistical uncertainties for small samples (below about 1g carbon). In the recent days after discovery of mass spectrometry and later accelerator based mass spectrometry (AMS), the sensitivity is greatly increased and we are going to see it in the next lecture.

Example m4.3: In an Archaeological expedition, charcoal from an ancient fire-pit was excavated. The sample showed a ¹⁴C activity of 11.3 counts per gm per min. The absolute activity of ¹⁴C in a living tree is independent of species and it is ^~ 15.3 counts per gm per min. Estimate the age of the charcoal sample.

Solution:

We have from the above data

11.3 = 15.3 e _-λt

where, λ = 0.693/T_1/2 = 0.693/5730 y

∴ 1.354 = eλt = e^0.000121t where t is in years

Thus, t = 2504.65 years = age of the charcoal

————————————————————————————

There are many important examples of radiocarbon dating methods.
Some of them are (i) 6000 year old human footprints in volcanic mud near lake Managua, Nicaragua,
(ii) prehistoric man found on a bank of the Columbia River near Kennewick, Washington, dated approximately 7000 years,
(iii) Shroud of Turin³ etc.

Nuclear Engineering

Nuclear Engineering

Curriculum

Basic Formulation on Radioactive Dating process

Modal title