Decay Constant and Half-Life

Spec mapping: OCR H556 Module 6.4 — Nuclear and Particle Physics (the decay constant $\lambda$λ as the probability per unit time of decay; activity $A = \lambda N$A=λN in becquerels; exponential decay law $N = N_{0} e^{-\lambda t}$N=N0​e−λt for population, activity and mass; half-life $t_{1/2} = \ln 2 / \lambda$t1/2​=ln2/λ; randomness of decay and the role of background subtraction). Refer to the official OCR H556 specification document for exact wording.

In the last lesson we met the four modes of radioactive decay: alpha, beta-minus, beta-plus and gamma. We said that decay is spontaneous and random — any individual nucleus in a sample may or may not decay in the next second, and there is no way of predicting which. What saves us from complete ignorance is that, in a sufficiently large sample, the statistics are extraordinarily reliable. This lesson is about those statistics: the decay constant $\lambda$λ, the activity $A$A, the exponential decay law, and the half-life $t_{1/2}$t1/2​.

This material is the core of Module 6.4 of the OCR A-Level Physics A specification (H556). You will need it for every subsequent topic in the nuclear module — and for all of the medical physics at the end of the course, where half-life considerations determine which radioisotopes are suitable for clinical use.

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The Decay Constant

Imagine a sample containing $N$N identical radioactive nuclei. We cannot predict which ones will decay in the next second, but we can ask: what is the probability that any given nucleus will decay in a short time interval $\Delta t$Δt? Experiment shows that this probability is proportional to $\Delta t$Δt (for small enough $\Delta t$Δt), and that the constant of proportionality is the same for every nucleus of that species. We call this constant the decay constant, denoted $\lambda$λ.

Definition. The decay constant $\lambda$λ of a radioactive species is the probability per unit time that a single nucleus will decay. Its units are $\text{s}^{-1}$s−1 (or equivalently $\text{min}^{-1}$min−1, $\text{h}^{-1}$h−1, $\text{yr}^{-1}$yr−1).

A large $\lambda$λ corresponds to a very unstable isotope that decays quickly; a small $\lambda$λ corresponds to a long-lived one. For example:

• Polonium-214: $\lambda \approx 4.2 \times 10^{3}\,\text{s}^{-1}$λ≈4.2×103s−1 (very unstable, half-life $\approx 164\,\mu\text{s}$≈164μs).
• Technetium-99m: $\lambda \approx 3.2 \times 10^{-5}\,\text{s}^{-1}$λ≈3.2×10−5s−1 (half-life $\approx 6$≈6 hours).
• Carbon-14: $\lambda \approx 3.8 \times 10^{-12}\,\text{s}^{-1}$λ≈3.8×10−12s−1 (half-life $\approx 5730$≈5730 years).
• Uranium-238: $\lambda \approx 4.9 \times 10^{-18}\,\text{s}^{-1}$λ≈4.9×10−18s−1 (half-life $\approx 4.5$≈4.5 billion years).

The decay constant is a fundamental property of the isotope, determined entirely by the quantum-mechanical structure of its nucleus. It cannot be changed by heating, cooling, compressing, dissolving, or applying electric or magnetic fields. The decay constant of carbon-14 is the same whether the atom is in a tree trunk, a fossil, or a laboratory bench.

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Activity

The activity $A$A of a radioactive sample is the number of decays per unit time. If the sample contains $N$N nuclei, each with decay probability $\lambda$λ per unit time, then on average:

$A = \lambda N$A=λN

The activity is measured in becquerels (Bq), where $1\,\text{Bq} = 1$1Bq=1 decay per second. A sample with activity 1 MBq is experiencing one million nuclear decays every second. (An older unit, the curie, is still occasionally used: $1\,\text{Ci} = 3.7 \times 10^{10}\,\text{Bq}$1Ci=3.7×1010Bq, which was roughly the activity of 1 g of pure radium-226.)

Note that the activity depends on both the decay constant and the number of nuclei present. A small sample of a short-lived isotope can have the same activity as a large sample of a long-lived one. For example:

• $10^{10}$1010 atoms of I-131 ($\lambda \approx 10^{-6}\,\text{s}^{-1}$λ≈10−6s−1) gives $A \approx 10^{4}\,\text{Bq}$A≈104Bq.
• $10^{16}$1016 atoms of U-238 ($\lambda \approx 5 \times 10^{-18}\,\text{s}^{-1}$λ≈5×10−18s−1) gives $A \approx 5 \times 10^{-2}\,\text{Bq}$A≈5×10−2Bq.

To get significant activity from a long-lived isotope, you need enormous quantities of material; a short-lived isotope gives high activity from tiny samples.

Worked Example 1 — Activity of a small sample

A sample of technetium-99m contains $5.0 \times 10^{14}$5.0×1014 nuclei. The decay constant is $\lambda = 3.2 \times 10^{-5}\,\text{s}^{-1}$λ=3.2×10−5s−1. What is the activity?

$A = \lambda N = (3.2 \times 10^{-5})(5.0 \times 10^{14}) = 1.6 \times 10^{10}\,\text{Bq} = 16\,\text{GBq}.$A=λN=(3.2×10−5)(5.0×1014)=1.6×1010Bq=16GBq.

This is a substantial activity — 16 billion decays per second — but a typical clinical dose of Tc-99m delivered to a patient is much smaller, of the order of a few hundred MBq.

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The Exponential Decay Law

Because each nucleus has a fixed decay probability $\lambda$λ per unit time, the rate at which the number of nuclei decreases is proportional to the number present:

$\frac{dN}{dt} = -\lambda N.$dtdN​=−λN.

The minus sign indicates that $N$N decreases with time. This is the defining equation of exponential decay, familiar from any A-Level maths course. Its solution is:

$N = N_{0}\, e^{-\lambda t}$N=N0​e−λt

where $N_{0}$N0​ is the number of nuclei at $t = 0$t=0. Multiplying both sides by $\lambda$λ gives the same exponential law for the activity:

$A = A_{0}\, e^{-\lambda t}$A=A0​e−λt

Both $N$N and $A$A decrease exponentially with time, with the same exponent $-\lambda t$−λt. Because $A = \lambda N$A=λN at every instant, the ratio $A/N = \lambda$A/N=λ is constant — the activity, the population and the mass of the radioactive species all fall together at the same fractional rate.

A plot of $N$N or $A$A against $t$t is a decaying exponential curve. A plot of $\ln N$lnN or $\ln A$lnA against $t$t is a straight line with gradient $-\lambda$−λ. This is the standard experimental technique for measuring $\lambda$λ: record activity over time, take the natural log, and fit a straight line to the log-plot.

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The Half-Life

The half-life $t_{1/2}$t1/2​ is the time taken for the number of radioactive nuclei in a sample (or equivalently, the activity, or the mass of the radioisotope) to fall to half its initial value. It is a more intuitive measure of decay speed than $\lambda$λ.

Setting $N = N_{0}/2$N=N0​/2 in the exponential decay law:

\tfrac{1}{2} N_{0} &= N_{0}\, e^{-\lambda t_{1/2}} \\ \tfrac{1}{2} &= e^{-\lambda t_{1/2}} \\ \ln \tfrac{1}{2} &= -\lambda t_{1/2} \\ -\ln 2 &= -\lambda t_{1/2}. \end{aligned}$$ Hence: $$\boxed{\; t_{1/2} = \frac{\ln 2}{\lambda} \quad\Longleftrightarrow\quad \lambda = \frac{\ln 2}{t_{1/2}} \;}$$ with $\ln 2 \approx 0.693$. This is one of the most important relations in the nuclear module and you should know it by heart. Note in particular that $t_{1/2}$ is **not** $1/\lambda$ — that quantity is the mean lifetime $\tau$, which is longer than the half-life by a factor $1/\ln 2 \approx 1.443$. After one half-life, $N = N_{0}/2$. After two half-lives, $N = N_{0}/4$. After three, $N = N_{0}/8$. After $n$ half-lives, $N = N_{0}/2^{n}$. For example, after 10 half-lives the sample has fallen to about $N_{0}/1024 \approx 0.1\%$ of its original activity — a useful rule of thumb for estimating when a sample has "effectively decayed away" relative to background. ### Worked Example 2 — From half-life to decay constant Iodine-131, used in thyroid therapy, has a half-life of $8.02$ days. What is its decay constant in $\text{s}^{-1}$? First convert the half-life to seconds: $$t_{1/2} = 8.02 \times 24 \times 3600 = 6.93 \times 10^{5}\,\text{s}.$$ Then: $$\lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.693}{6.93 \times 10^{5}} \approx 1.00 \times 10^{-6}\,\text{s}^{-1}.$$ ### Worked Example 3 — From activity to mass A sample of carbon-14 has an activity of $1.0 \times 10^{3}\,\text{Bq}$. Given that the half-life is 5730 years and the molar mass is $14\,\text{g mol}^{-1}$, find the mass of C-14 in the sample. Decay constant first: $$t_{1/2} = 5730 \times 365.25 \times 24 \times 3600 \approx 1.81 \times 10^{11}\,\text{s},$$ $$\lambda = \frac{\ln 2}{t_{1/2}} \approx 3.83 \times 10^{-12}\,\text{s}^{-1}.$$ Number of C-14 nuclei from $A = \lambda N$: $$N = \frac{A}{\lambda} = \frac{1.0 \times 10^{3}}{3.83 \times 10^{-12}} \approx 2.6 \times 10^{14}\,\text{nuclei}.$$ Mass: $$m = \frac{N}{N_{A}} \times M = \frac{2.6 \times 10^{14}}{6.02 \times 10^{23}} \times 14\,\text{g} \approx 6.1 \times 10^{-9}\,\text{g}.$$ So just 6 nanograms of C-14 produces an activity of 1 kBq. This is why C-14 dating is so sensitive: tiny traces of the isotope are easily detected against the natural background. ### Worked Example 4 — Fraction remaining after a given time A radioactive source has a half-life of 10 minutes. What fraction of the initial nuclei remain after 45 minutes? The number of half-lives elapsed is $n = 45/10 = 4.5$, so: $$\frac{N}{N_{0}} = \left(\tfrac{1}{2}\right)^{4.5} = 2^{-4.5} \approx 0.0442.$$ About 4.4 % of the original nuclei remain. Alternatively, using the exponential law: $$\lambda = \frac{\ln 2}{10\,\text{min}} = 0.0693\,\text{min}^{-1},$$ $$\frac{N}{N_{0}} = e^{-\lambda t} = e^{-0.0693 \times 45} = e^{-3.12} \approx 0.0442\ \checkmark$$ The two approaches give the same answer, as they must. --- ## Randomness and Statistical Fluctuations The exponential law $N = N_{0} e^{-\lambda t}$ is a statistical statement about the *average* behaviour of a large number of nuclei. Individual decays are random events, and the observed count rate fluctuates about the mean. The standard deviation of a count of $N$ decays is $\sqrt{N}$ (Poisson statistics). So if you observe 100 decays in a given time interval, the expected fluctuation is about $\pm 10$. If you observe 10 000, the fluctuation is about $\pm 100$ — *fractionally* smaller as $N$ grows. For the OCR specification you need to know that decay is random and that successive counts taken under identical conditions will not be identical — they fluctuate statistically around the mean value predicted by the exponential law. This randomness is why backgrounds must be subtracted and why several readings are usually averaged. --- ## Background Radiation In any real experiment the detector also registers counts from **background radiation** — natural sources such as cosmic rays, terrestrial radioactivity (radon, thoron, uranium and thorium in rocks), and traces of potassium-40 and carbon-14 in your own body. A typical background rate is around 20–30 counts per minute in a Geiger counter. When measuring the activity of a sample, you must measure the background separately and subtract it from your count: $$A_{\text{sample}} = A_{\text{measured}} - A_{\text{background}}.$$ Failing to do this can give misleading half-life measurements, because the background is (essentially) constant while the sample activity decays exponentially. As the sample ages, the background becomes a larger fraction of the total count, and a careless analysis will appear to show the decay "levelling off" at the background rate. --- ## Half-Lives in Medicine The choice of radioisotope for a medical application involves a compromise. A *short* half-life is good because the patient is exposed to less total radiation, but bad because the isotope decays before it can be used. A *long* half-life is good for shelf life but bad for patient dose. Commonly used half-lives: | Isotope | Half-life | Use | |---|---|---| | F-18 | $\approx 110$ min | PET scans (FDG) | | Tc-99m | $\approx 6$ h | Gamma-camera imaging | | I-123 | $\approx 13$ h | Thyroid imaging | | I-131 | $\approx 8$ d | Thyroid therapy | | Co-60 | $\approx 5.3$ y | External beam radiotherapy | | Cs-137 | $\approx 30$ y | Brachytherapy sources | Diagnostic isotopes tend to have half-lives of hours (long enough for delivery, short enough to clear the body), while therapeutic and calibration sources have half-lives of days to decades. --- ## Synoptic Links