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Table 3.3 lists several radioactive parent to stable daughter transformations that can be used to determine the formation ages of Earth materials. All these are based on the principle that after the radioactive isotope is incorporated into Earth materials, the ratio of radioactive parent isotopes to stable daughter isotopes decreases through time by radioactive decay. The rate at which such parent: daughter ratios decrease depends on the rate of decay, which is given by the decay constant (λ) , the proportion of the remaining radioactive atoms that will decay per unit of time. One useful formula that governs decay series states that the number of radioactive atoms remaining at any given time (N) is equal to the number of radioactive atoms originally present in the sample (N₀) multiplied by a negative exponential factor (e^–λt) that increases with the rate of decay ( λ ) and the time since the sample formed (t), that is, its age. These relationships are given by:

Table 3.3 Systematics of radioactive isotopes important in age determinations in Earth materials.

Decay series	Decay process	Decay constant (λ)	Half‐life	Applicable dating range
14C → 14N	Beta decay	1.29 × 10−4/year	5.37 Ka	<60 Ka
40K → 40Ar	Electron capture	4.69 × 10−10/year	1.25 Ga	25 Ka to >4.5 Ga
87Rb → 87Sr	Beta decay	1.42 × 10−11/year	48.8 Ga	10 Ma to >4.5 Ga
147Sm → 143Nd	Alpha decay	6.54 × 10−12/year	106 Ga	200 Ma to >4.15 Ga
232Th → 208Pb	Beta and alpha decays	4.95 × 10−11/year	14.0 Ga	10 Ma to >4.5 Ga
235U → 207Pb	Beta and alpha decays	9.85 × 10−10/year	704 Ma	10 Ma to >4.5 Ga
238U → 206Pb	Beta and alpha decays	1.55 × 10−10/year	4.47 Ga	10 Ma to >4.5 Ga

It should be clear from the formula that when t = 0, N = N₀, and that N becomes smaller through time as a function of the rate of decay given by the decay constant; rapidly for a large decay constant, more slowly for a small one. Figure 3.13 illustrates a typical decay curve, showing how the abundance of the radioactive parent isotope decreases exponentially over time while the abundance of the daughter isotope increases in a reciprocal manner. The two curves cross where the number of radioactive parent and stable daughter atoms is equal. The time required for this to occur is called the half‐life of the decay series and is the time required for one half of the radioactive isotopes to decay into stable daughter isotopes.

Figure 3.13 Progressive change in the proportions of radioactive parent (N) and daughter (D) isotopes over time, in terms of number of half‐lives.

More generally, the age of any sample may be calculated from the following equation:

where t = age, λ = the decay constant, d = number of stable daughter atoms and p = number of radioactive parent atoms. Where stable daughter atoms were present in the original sample, a correction must be made to account for them, as explained below.

Many radioactive to stable isotope decay series, each with a unique decay constant, can be used, often together, to determine robust formation ages for Earth materials, especially for older materials. Table 3.3 summarizes some common examples. Three of these are discussed in more detail in the sections that follow.