A banana is slightly radioactive. It holds potassium, and about one atom in 8,500 of that potassium is the isotope potassium-40, which decays on its own, now and then, with a half-life of about 1.25 billion years. Every day, somewhere in your body, a few thousand of those atoms do it. Nothing pushed them. Nothing about the atom changed. One of its neutrons simply turned into a proton, an electron and an almost undetectable neutrino, and flew apart.
Now ask a stranger question. Why does your banana not do other things? Why does a proton in it sit still for longer than any experiment has been able to measure, when the proton is heavy and could, as far as raw energy goes, fall apart into lighter pieces? A graduate in particle and nuclear physics sees the banana, the proton and the sun as one subject: a short list of ingredients, and a short list of bookkeeping rules about what can turn into what.
Einstein's best-known equation is usually read as a statement about bombs. Particle physicists read it as a price list. Mass and energy are the same currency, and the exchange rate is the speed of light squared. If you hit two things together hard enough, some of their motion can be spent on a brand-new particle, which then exists until it decays.
The catch is that not all your energy is spendable. Momentum must still balance after the collision, so a particle fired at a stationary target keeps most of its energy as leftover motion. Fire two protons at each other head-on and the whole lot is spendable, because the total momentum is zero. The usable energy is the centre-of-mass energy, written √s. For two beams of energy E head-on it is just 2E. For one beam of energy E on a stationary proton it is only √(2m² + 2mE), in units where c = 1.
A worked number. To make a proton–antiproton pair in a proton–proton collision you need √s of at least four proton masses, 3.75 GeV. With a stationary target that takes a beam kinetic energy of 5.63 GeV, which is why the Bevatron in Berkeley, built to about 6 GeV, found the antiproton in 1955. At a collider you need only about 1.9 GeV in each beam. Try heavier targets below. Producing a Higgs boson of mass about 125 GeV needs roughly 63.5 GeV per beam in a collider, but about 8,600 GeV (8.6 TeV) in a fixed-target beam. That gap is why the biggest accelerators are colliders.
Two protons in, the new stuff must have mass M, so √s ≥ 2mp + M (mp = 0.938272 GeV). Collider: each beam needs √s/2. Fixed target: E = (s − 2mp²)/(2mp). This is a threshold: real production at the LHC also depends on how the proton's quarks and gluons share the energy, so real machines run well above it.
For scale, the Large Hadron Collider has run at 13.6 TeV collision energy since July 2022. Reaching that with a stationary target would need a beam of roughly 98,600 TeV. No ring could hold it.
The Standard Model's list is shorter than the periodic table. Matter particles come in two families. Quarks (up, down, charm, strange, top, bottom) feel the strong force and are never seen alone. Leptons (the electron, muon and tau, each with its own neutrino) do not feel the strong force. Each family comes in three "generations" with the same charges and growing mass; ordinary matter uses only the first: up and down quarks, the electron and its neutrino. Every particle has an antiparticle with opposite charges.
Quark charges are odd: up-type quarks carry +2/3 and down-type −1/3 of a proton's charge. But you never see a third of a charge, because quarks are confined. They only appear glued into bundles with total strong "colour charge" zero: three quarks (baryons), three antiquarks, or a quark with an antiquark (mesons). A proton is two up and one down: 2/3 + 2/3 − 1/3 = +1. A neutron is up, down, down: 0. Build some yourself.
Charges: u, c +2/3; d, s, b −1/3; antiquarks opposite. Baryon number ±1/3 per quark. Strangeness −1 per s quark. The top quark is left out because it decays before it can bind. The list of named hadrons is a small sample.
The quarks' bound pairs and triples are by far the biggest source of everyday mass. The up and down quarks together are only about 1% of a proton's mass; the rest is the energy of the strong force binding them, via E = mc². Most of you is binding energy.
A reaction is allowed if three things hold. Add up the electric charge before and after: it must match. Add up baryon number (quarks count +1/3, antiquarks −1/3) and lepton number, counting each lepton family separately: they must match. And the total energy in the rest masses must be at least that of the products. If a rule is not forbidden by one of these, it happens, at some rate. If it is forbidden, it has never been seen.
Those rules are why the proton, the lightest baryon, is stable. It has nothing lighter with the same baryon number to decay into. Experiments such as Super-Kamiokande in Japan have searched for it to fall apart and found nothing, setting a lifetime limit above 1034 years, many orders of magnitude longer than the universe has existed. In the same spirit, the muon should be able to decay into an electron and a photon on energy grounds alone, but flavour-specific lepton number forbids it, and searches find none at the level of about 1 in a trillion.
Predict first, then check. Pick a reaction, say whether you think it happens, and the page runs the arithmetic.
Rest masses in MeV from the Particle Data Group (neutrinos treated as massless; their real masses are tiny). Charge, baryon number and electron/muon lepton numbers are summed on each side.
In quantum field theory, forces are carried by particles. Two electrons repel by swapping photons. The strong force binds quarks by swapping gluons. The weak force, behind the neutron's decay, is carried by the W and Z bosons. And here is the part a graduate remembers: the heavier the carrier, the shorter the force's reach. Borrowing a heavy particle from the vacuum is expensive in energy, so the loan must be repaid quickly, and it cannot travel far. The range is about ħc divided by the carrier's rest energy.
The photon is massless, so electric forces reach across the galaxy. The pion, the lightest meson at 139.6 MeV, gives 1.41 fm, which is why the nuclear force binds neighbours in a nucleus and fades beyond. The W boson (80.4 GeV) gives 0.0025 fm, about 300 times smaller than a proton. That is why the weak force is "weak" at ordinary distances: it hardly reaches.
Yukawa estimate, range = ħc/mc². The strong force between quarks (gluons) is a different case: gluons are massless but the force is still confined, so this is the classic picture for pions between nucleons and for the weak force, not for gluons.
A nucleus weighs a little less than its separate protons and neutrons. The missing mass, multiplied by c², is the binding energy, the work needed to pull it apart. Divide by the number of nucleons A and you get a curve with a shape every nuclear scientist knows: it rises steeply for light nuclei, peaks around iron and nickel (about 8.8 MeV per nucleon), then falls slowly to uranium.
Anything that moves a nucleus up the curve releases energy. Light nuclei going up by fusing; heavy ones going up by splitting. Worked numbers from measured masses: helium-4 has 7.07 MeV per nucleon, iron-56 8.79, uranium-235 7.59. Deuterium plus tritium fusing into helium-4 and a neutron frees 17.6 MeV, about 3.4 × 10¹⁴ joules per kilogram of fuel. Fission of uranium-235 frees roughly 200 MeV per atom, about 8 × 10¹³ J/kg. TNT is 4.2 × 10⁶ J/kg: fusion fuel carries about 80 million times as much.
Line: the semi-empirical mass formula (volume, surface, Coulomb and asymmetry terms; pairing left out for a smooth curve), with stable Z. Dots: measured values computed from atomic masses (He-4, C-12, O-16, Fe-56, Ni-62, U-235, U-238). The formula is poor below A ≈ 20 (it gives 5.7 for helium-4 where the real value is 7.07), a reminder that it is a model.
The same curve explains the sun, which fuses hydrogen into helium, and why stars much heavier than the sun make iron and stop: past iron, fusion costs energy instead of paying it. That is the hinge to Astrophysics & Cosmology (34/88).
An unstable nucleus has no clock. A carbon-14 atom that has survived 5,730 years is exactly as likely to decay in the next minute as one made a second ago. So nothing in any single atom tells you when it will go. Yet a sample of trillions behaves with high precision: after one half-life, half remain; after two, a quarter.
Carbon dating uses it: living things hold carbon-14 at the atmosphere's level and stop taking more when they die. A bone with a quarter of the level has lived two half-lives, about 11,460 years. After 20,000 years only about 8.9% remain, which is why the method runs out near 50,000 years. The half-lives span an enormous range: below, from microseconds to billions of years. Watch the randomness shrink as the sample grows.
Each atom draws its own random lifetime (exponential), so no atom's fate is known in advance. Half-lives: Po-214 164 µs, I-131 8.02 days, Cs-137 30.04 years, C-14 5,730 years, U-238 4.468 billion years (standard tables).
With 10 atoms, the survivors after one half-life can easily be 3 or 7. With 1,000, they are close to 500 every time. That is the whole content of "the law of large numbers", and the reason a particle detector needs many events before it says anything.
The Standard Model is the most precisely tested theory in science, and every physicist expects it to be incomplete. The decisive experiments: neutrinos oscillating between flavours, which means they have mass (Super-Kamiokande and SNO, Nobel Prize 2015); the Higgs boson, announced on 4 July 2012 by ATLAS and CMS at about 125 GeV; and the muon's magnetic moment, where Fermilab's final 2025 result is 127 parts per billion precise and where theory predictions were revised, shrinking a famous disagreement.
What is left? Neutrinos have mass but tiny: the KATRIN experiment's 2025 limit is under 0.45 eV, at 90% confidence, at least a million times lighter than the electron, and the Standard Model does not say why. Dark matter makes up about five times as much as ordinary matter, and no Standard Model particle fits. The universe has far more matter than antimatter, and the model's known effects are too small to explain it. Gravity is not in the model at all. These are open, and physicists disagree on which gap will give first.
Matter is a short list of quarks and leptons, held together by forces whose reach is set by the mass of their carriers. A small set of conservation laws says what may happen; energy conversion by E = mc² says what it costs. The binding-energy curve explains why both the sun and a reactor work, and random decay with an exact crowd behaviour explains why radioactivity is unpredictable and measurable at once.
Try this: next time you see a smoke detector, a banana or a bag of potassium-based salt substitute, say aloud that some atoms in it are decaying right now, each one at a random moment, and that the count per second is nevertheless steady. That is the exponential law at work.
Sources. Constants are CODATA values (ħc = 197.3269804 MeV·fm; atomic mass unit 931.49410242 MeV/c²). Masses and half-lives are standard tables (Particle Data Group; NNDC). Higgs: ATLAS and CMS announced a boson of about 125 GeV on 4 July 2012 at CERN. KATRIN: Science 388, 180 (2025), upper limit 0.45 eV (90% CL). Muon g-2: Fermilab final result, 3 June 2025, precision 127 ppb; the Standard Model prediction was revised by newer lattice calculations. Super-Kamiokande proton-decay searches give lifetimes above 10³⁴ years for the channels searched. The LHC began Run 3 at 13.6 TeV in July 2022. Antiproton found at the Bevatron in 1955 (Chamberlain, Segrè, Wiegand, Ypsilantis), threshold kinetic energy 6 mp = 5.63 GeV. Every number an interactive shows was computed in node: threshold beams for the Higgs 63.5 GeV (collider) and 8,604 GeV (fixed target); 13.6 TeV collider equals 9.86 × 10⁷ GeV fixed target; Yukawa ranges 1.41 fm (pion), 0.399 fm (kaon), 0.00245 fm (W); binding energies per nucleon He-4 7.074, C-12 7.680, O-16 7.976, Fe-56 8.790, Ni-62 8.795, U-235 7.591, U-238 7.570 MeV; D-T release 17.59 MeV, 3.37 × 10¹⁴ J/kg; C-14 after 20,000 years = 8.9%.