Radiocarbon Dating - Computation of Ages and Dates

Computation of Ages and Dates

The number of decays per time is proportional to the current number of radioactive atoms. This is expressed by the following differential equation, where N is the number of radioactive atoms and λ is a positive number called the decay constant:

As the solution to this equation, the number of radioactive atoms N can be written as a function of time:

,

which describes an exponential decay over a timespan t with an initial condition of N₀ radioactive atoms at t = 0. Canonically, t is 0 when the decay started. In this case, N₀ is the initial number of 14C atoms when the decay started.

For radiocarbon dating a once living organism, the initial ratio of 14C atoms to the sum of all other carbon atoms at the point of the organism's death and hence the point when the decay started, is approximately the ratio in the atmosphere.

Two characteristic times can be defined:

• mean- or average-life: mean or average time each radiocarbon atom spends in a given sample until it decays.
• half-life: time lapsed for half the number of radiocarbon atoms in a given sample, to decay,

It can be shown that:

= = radiocarbon mean- or average-life = 8033 years (Libby value)

= = radiocarbon half-life = 5568 years (Libby value)

Notice that dates are customarily given in years BP which implies t(BP) = –t because the time arrow for dates runs in reverse direction from the time arrow for the corresponding ages. From these considerations and the above equation, it results:

For a raw radiocarbon date:

and for a raw radiocarbon age:

After replacing values, the raw radiocarbon age becomes any of the following equivalent formulae:

using logs base e and the average life:

and

using logs base 2 and the half-life:

Wiggle matching uses the non-linear relationship between the 14C age and calendar age to match the shape of a series of closely sequentially spaced 14C dates with the 14C calibration curve.