Two lumps of matter sit on a bench. One is a speck of polonium-214; the other a grain of uranium-238. They are close cousins — heavy nuclei, each straining under the weight of its own protons. Yet the polonium is gone in about 160\ \mu\text{s}, a fraction of a heartbeat, while the uranium takes 4.5\times 10^{9} years — the age of the Earth — to get halfway through decaying. Same kind of decay, both spitting out an alpha particle, and yet their lifetimes differ by a factor of 10^{24}. What on Earth could stretch a process across twenty-four orders of magnitude?

We already know the bookkeeping of decay — the number surviving falls exponentially, radioactive decay with its constant \lambda and the N = N_0 e^{-\lambda t} law. This page asks the deeper question that the law leaves open: why does a given nucleus decay at all, and what fixes its \lambda? The answer is three quite different pieces of physics — quantum tunnelling for alpha decay, the weak force for beta decay, and simple photon emission for gamma decay — and understanding them turns that baffling 10^{24} from a mystery into a one-line estimate.

Three ways for a nucleus to change

Before the mechanisms, fix the bookkeeping. A nuclide is labelled {}^{A}_{Z}\text{X}: Z protons (the atomic number, which names the element) and A nucleons in total (the mass number). Every decay must conserve charge and nucleon number, and that alone tells you exactly what the daughter is.

So \alpha and \beta transmute the element; \gamma only sheds energy. Keep those two columns — change in A, change in Z — in your head, and you can balance any decay equation on sight. The rest of the page is about the machinery behind each line.

Alpha decay: a particle that climbs a wall it cannot climb

Deep inside a heavy nucleus, two protons and two neutrons occasionally find each other and cluster into a pre-formed alpha particle — a ready-made {}^{4}_{2}\text{He} nucleus, tightly bound and rattling around inside. It wants to leave: for a big nucleus like uranium, releasing an alpha lowers the total energy. But between it and freedom stands the Coulomb barrier. Once the alpha pokes past the short-range nuclear force (radius R), the +2e alpha and the +(Z-2)e daughter repel fiercely, a potential V(r) = \dfrac{2(Z-2)e^{2}}{4\pi\varepsilon_0\,r} that rises to a peak B far above the alpha's actual energy E.

Classically, this is the end of the story: a ball with energy E below a hill of height B simply rolls back down. The alpha should be trapped forever. Quantum mechanically, its wavefunction does not stop dead at the barrier — it leaks through, decaying exponentially across the forbidden zone from r = R out to the radius r_{\text{exit}} where the Coulomb slope has fallen back to E. This is quantum tunnelling: on each of its \sim 10^{21} collisions per second with the wall, the alpha has a tiny probability of appearing on the far side.

Why a tiny energy change means an enormous lifetime change

The probability of getting through the barrier on one attempt is the Gamow factor. For a barrier of "area" G,

The decay constant is then roughly \lambda \approx f\,e^{-2G}, where f\sim 10^{21}\ \text{s}^{-1} is the rate at which the alpha bangs on the wall. The whole drama lives in that exponent. Because G sits inside an exponential, and because G itself grows as the alpha's energy E falls (a lower E means a wider, taller effective barrier), a small drop in E produces a gigantic rise in lifetime. Doing the integral for a point charge gives the celebrated result:

The graph below plots the (base-10) log of half-life against the alpha energy for real alpha emitters. A modest slide along the energy axis sends the half-life plunging over dozens of decades — the visual signature of an exponential hiding inside the physics.

Worked example: the alpha Q-value from a mass table

Consider {}^{238}_{92}\text{U} \to {}^{234}_{90}\text{Th} + {}^{4}_{2}\text{He}. The energy released — the Q-value — is the mass that goes missing, times c^2. It is cleanest to use mass excesses \Delta = (m - A\,u)c^2, because the A-terms cancel automatically (238 = 234 + 4):

Plugging in the tabulated values \Delta(^{238}\text{U}) = 47.309\ \text{MeV}, \Delta(^{234}\text{Th}) = 40.612\ \text{MeV}, and \Delta(^{4}\text{He}) = 2.425\ \text{MeV}:

Because Q > 0 the decay is energetically allowed — and indeed uranium-238 is an alpha emitter. Almost all of this Q becomes the alpha's kinetic energy (the heavy thorium recoils only gently, sharing the momentum), so the emitted alpha carries about \tfrac{234}{238}\times 4.27 \approx 4.20\ \text{MeV}. That 4.2\ \text{MeV} — near the very bottom of the alpha-energy range — is exactly why uranium-238's half-life is billions of years rather than microseconds.

Beta decay: the weak force reshuffles a quark

Alpha decay ejects a whole chunk of nucleus. Beta decay does something stranger: it converts one nucleon into another, changing the proton–neutron balance without touching A. In \beta^{-} decay a neutron becomes a proton:

There is no electron pre-sitting inside the neutron waiting to be flung out — the electron is created at the instant of decay, together with an electron antineutrino \bar{\nu}_e. Dig one level deeper and it is a single quark that flips: a neutron is (udd) and a proton is (uud), so at heart

A down quark turns into an up quark by emitting a W^{-} boson — the heavy carrier of the weak interaction — which immediately splits into the electron and antineutrino we detect. Because the W is so massive, this coupling is feeble, which is exactly why weak-force lifetimes are long compared with the electromagnetic gamma transitions. The mirror processes are \beta^{+} decay, p \to n + e^{+} + \nu_e (an u \to d flip), and electron capture, p + e^{-} \to n + \nu_e, where the nucleus swallows an inner atomic electron. All three drive a nuclide toward the valley of beta stability: too many neutrons and it uses \beta^{-}; too many protons and it uses \beta^{+} or capture.

Gamma decay: shedding energy without changing identity

After an alpha or beta decay, the daughter nucleus is very often left in an excited state — its nucleons are in a higher energy arrangement, just as an atom can have an excited electron. The nucleus relaxes to a lower state by emitting the energy difference as a single high-energy photon, a gamma ray:

Notice that A and Z are both unchanged — gamma decay does not transmute the element. It is the same nucleus, the same isotope, simply at lower energy. Nuclear energy levels are spaced by keV to MeV (versus eV for atomic electrons), which is why nuclear de-excitation produces gamma rays rather than visible light. A rival route to the same end is internal conversion: instead of emitting a photon, the excited nucleus hands its energy directly to one of the atom's own inner electrons, which is then ejected. Same relaxation, no gamma ray — a reminder that gamma emission is one channel, not the only one.

Decay chains and secular equilibrium

A single decay rarely lands a heavy nucleus on stable ground. Uranium-238, for instance, is still far too heavy after shedding one alpha, so its daughter decays too, and its daughter, and on down a decay chain — a cascade of alternating \alpha and \beta^{-} steps that only halts at a stable lead isotope. The uranium series (^{238}\text{U} \to \cdots \to {}^{206}\text{Pb}) runs through 8 alpha and 6 beta decays; the thorium series (^{232}\text{Th} \to \cdots \to {}^{208}\text{Pb}) through 6 alpha and 4 beta. You can count the alphas straight from the mass number: each alpha removes 4, so n_\alpha = (A_{\text{parent}} - A_{\text{stable}})/4. The alphas alone would overshoot on charge (each removes 2 protons), and the \beta^{-} steps (each adds one proton back) make up the difference: n_\beta = 2\,n_\alpha - (Z_{\text{parent}} - Z_{\text{stable}}).

Here is the beautiful consequence. The parent (uranium-238, half-life 4.5\times 10^{9} yr) decays far more slowly than any of its daughters. So after a long time the chain reaches secular equilibrium: every intermediate nuclide decays at exactly the rate it is produced, and the activity of each member of the chain becomes equal,

A short-lived daughter is present in tiny amounts (small N) but decays rapidly (large \lambda); a long-lived one is abundant but sluggish — and the products \lambda N match. This is why an old lump of uranium ore holds a fixed, self-regulating cocktail of radium, radon, polonium and the rest, all ticking over in lock-step, governed by the slow patriarch at the top of the chain.

Worked example: reading a decay chain

The thorium series starts at {}^{232}_{90}\text{Th} and ends at stable {}^{208}_{82}\text{Pb}, using only \alpha and \beta^{-} decays. How many of each?

Alphas from the mass number. Only alpha decay changes A, by -4 each:

Betas from the charge. Those 6 alphas remove 6\times 2 = 12 protons, taking Z from 90 down to 78. But the stable lead has Z = 82, four more than that — so four \beta^{-} decays must each have nudged Z back up by one:

So the thorium series is 6 alpha and 4 beta decays — 10 transmutations to turn thorium into lead, no matter which order the wandering path actually takes.