Speculative physics: carbon with a tenfold half-life

Problem

Suppose carbon-14 had a half-life $T_{1/2} = 57\,300$ years, i.e. ten times the real value ($\approx 5730$ years). How would radiocarbon dating change? Discuss in particular how the minimum and maximum datable-age limits and the precision of the measurement would change. Reason about the physical cause, not just the numbers.

Solution

Why the half-life sets the dating “window”. The radiocarbon method works by measuring the fraction $N/N_0$ of $^{14}\text{C}$ atoms still present relative to the initial value. The measurement is reliable only when this fraction is appreciably different from $1$ (find not too young) but still above the background noise (find not too old). In practice the useful window covers about ten half-lives, because after $10\,T_{1/2}$ only $2^{-10} \approx 1/1000$ of the radioactive material remains, by then indistinguishable from the background.

Maximum limit (oldest ages). With the real value $T_{1/2} \approx 5730$ years the window extends to $\sim 50\,000$–$60\,000$ years. Multiplying $T_{1/2}$ tenfold to $57\,300$ years, ten half-lives would cover $\sim 500\,000$ years: it would be possible to date much older finds, well beyond $100\,000$ years. Slower decay preserves a measurable signal for longer.

Minimum limit and precision for recent finds. Here is the flip side. With a $T_{1/2}$ ten times longer, in a few thousand years the decayed fraction would be minimal: the decay constant $\lambda = \ln 2 / T_{1/2}$ becomes ten times smaller, so $N/N_0 = e^{-\lambda t}$ stays very close to $1$. For a find of, say, $2000$ years, the change in $N/N_0$ would be tiny, hard to distinguish from measurement errors and background.

Conclusion (trade-off). We would gain on the maximum limit (much older finds datable) but lose on the minimum limit and on the precision for recent finds, because the decay signal becomes too weak over short times.

$\ev{\text{wider window but worse precision for recent finds}}$

Connections

Topics: Nuclear physics Concepts: Radioactive decay Skills: Analysis of limiting cases and speculative physics