Revise Radioactive decay for IB Physics (E.3)

Unstable nuclei emit radiation in three classic forms. Alpha ($\alpha$) is a helium nucleus, $^{4}_{2}\alpha$ — two protons and two neutrons bound together, so it carries charge $+2e$ and a relatively large mass. Beta-minus ($\beta^-$) is a fast-moving electron, $^{\ 0}_{-1}e$, produced when a neutron inside the nucleus turns into a proton; it carries charge $-e$ and has a tiny mass, and it is always emitted alongside an (anti)neutrino. A rarer partner, beta-plus ($\beta^+$), is a positron $^{0}_{+1}e$ — the positive antiparticle of the electron — produced when a proton turns into a neutron. Gamma ($\gamma$) is not a particle of matter at all but a high-energy photon, $^{0}_{0}\gamma$, with no charge and no mass, usually emitted just after an alpha or beta decay as the leftover nucleus sheds excess energy.

There is a neat inverse pattern worth remembering: the most strongly ionising radiation is the least penetrating. Alpha dumps all its energy into ionising nearby atoms and so is stopped almost at once; gamma interacts weakly, ionises little, and therefore travels far. In a magnetic field the deflection is a direct read-out of charge and mass — alpha curving one way, beta the other, and gamma sailing straight on.

A decay equation is a balance sheet for the nucleus. Every nuclide is written $^{A}_{Z}\text{X}$, where $A$ is the nucleon number (protons + neutrons, the top number) and $Z$ is the proton number (the bottom number). In any decay, two quantities are conserved: the nucleon numbers on the left must equal those on the right, and likewise the proton numbers. That single rule fixes the daughter nuclide completely.

Radioactive decay has two defining features. It is spontaneous: no external condition triggers or changes it. Heating, cooling, squeezing, or chemically bonding the atom leaves its decay rate untouched — a nucleus decays at the same rate in the heart of a star or frozen near absolute zero. It is also random: you cannot predict which nucleus will decay next, nor when a given nucleus will go. Every undecayed nucleus carries the same fixed probability of decaying in the next interval, regardless of its age or what its neighbours are doing.

These two properties are exactly why half-life is constant. Because each nucleus has the same fixed chance of decaying per unit time, any sample loses the same FRACTION of itself in a fixed interval — and that fraction reaches one half after one half-life, whether the sample is huge or tiny, old or fresh. Randomness at the level of one nucleus produces iron-clad regularity across billions of them.

The activity $A$ of a sample is the number of decays per second, measured in becquerel (Bq), where $1\ \text{Bq}$ is one decay per second. Because activity is proportional to the number of undecayed nuclei present, it falls off in exactly the same way that the number of nuclei does — following a smooth decay curve. Plot activity (or number of nuclei) against time and you get a curve that drops steeply at first and then flattens, halving over each successive half-life: from full to half in one $T_{1/2}$, to a quarter after two, to an eighth after three, and so on, approaching zero without ever reaching it.

The half-life $T_{1/2}$ is the time for the activity (or the number of undecayed nuclei) to fall to half its value. Its constancy is the signature of exponential decay: the curve takes the same time to drop from 800 Bq to 400 Bq as it does to drop from 400 Bq to 200 Bq. That property gives you a reliable way to read a half-life straight off a graph or a table — and a way to check your reading, since every halving interval should come out the same.

When the elapsed time is a whole number of half-lives, you never need a fancy formula — just halve repeatedly. Count the number of half-lives $n = t / T_{1/2}$, then multiply the starting value by $(1/2)^n$. This works equally for the number of nuclei, the mass of the isotope, or the activity, because all three follow the same decay curve.

At Higher Level we describe decay continuously rather than in whole-half-life steps. The rate of decay — the activity — is proportional to the number of undecayed nuclei present, and the constant of proportionality is the decay constant $\lambda$: the probability that any one nucleus decays per unit time. A larger $\lambda$ means a faster decay and a shorter half-life.

A = \lambda N

This proportionality leads to an exponential fall in the number of undecayed nuclei with time, and — since activity is proportional to $N$ — activity follows the identical law:

$N = N_0\, e^{-\lambda t} \qquad\text{and}\qquad A = A_0\, e^{-\lambda t}$

Here $N_0$ and $A_0$ are the initial number of nuclei and initial activity. The decay constant and the half-life are two ways of describing the same rate. Setting $N = N_0/2$ at $t = T_{1/2}$ in the exponential law gives their relationship:

$T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$

In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an answer mark (A) — and error-carried-forward (ECF) means a wrong number early on does not have to cost you the marks that follow. But that protection only exists if your method is written down. Study how each mark below is earned by a specific line in a whole-half-life activity calculation.

Radioactive decay is the gateway to the rest of nuclear physics. The energy carried off by alpha, beta and gamma emissions connects directly to mass–energy equivalence and nuclear binding energy; the same exponential mathematics reappears in the attenuation of radiation through matter and in any process governed by a fixed probability per unit time. The balancing skill — conserving nucleon and proton number — is exactly the bookkeeping you will reuse for nuclear reactions and, at HL, for fission and fusion. Master the habit here — identify the radiation, balance $A$ and $Z$, count the half-lives, and show every line — and the nuclear topics that follow become variations on a method you already own.