Boltwood’s uranium-lead ages (1907)

Boltwood noticed that uranium minerals always contain lead, and more of it in older rocks. Taking lead as the end of the uranium decay series, he turned the lead-to-uranium ratio of minerals into ages of hundreds of millions to billions of years, the first radiometric ages. With a known decay constant the age is log(1 + D/P) / lambda; Boltwood’s simpler ratio-over-rate is the first term of that logarithm and overestimates old ages.

import matplotlib.pyplot as plt
import numpy as np

from earthsciencekit.constants import LAMBDA_U238
from earthsciencekit.units import convert
from earthsciencekit.volcanology import radiometric_age

ratio = np.linspace(0, 0.6, 121)  # radiogenic 206Pb atoms per 238U atom
age = convert(radiometric_age(ratio, LAMBDA_U238), "s", "Ma")
linear = convert(ratio / LAMBDA_U238, "s", "Ma")
assert np.all(age <= linear) and np.isclose(age[1], linear[1], rtol=3e-3)
assert np.isclose(convert(radiometric_age(np.e - 1, LAMBDA_U238), "s", "Ma"), 1e-6 / 1.55125e-10)  # one mean life

minerals = np.array([0.05, 0.12, 0.2, 0.3])  # synthetic Pb/U ratios
fig, ax = plt.subplots()
ax.plot(ratio, age, label="log(1 + D/P) / lambda")
ax.plot(ratio, linear, "--", label="D / (P lambda), Boltwood's form")
ax.plot(minerals, convert(radiometric_age(minerals, LAMBDA_U238), "s", "Ma"), "ko", label="synthetic minerals")
ax.set(xlabel="206Pb / 238U", ylabel="age [Ma]", title="Lead accumulates in uranium minerals")
ax.legend()

plt.show()