Energy, momentum and particle tracks

The two laws that always apply

Beyond the quantum numbers, energy and momentum are conserved in every interaction, and either can forbid a process that the quantum numbers allow. The standard example follows. A free proton decaying by p → n + e⁺ + ν_e balances charge, baryon number and electron lepton number perfectly, and it never happens, because the products' rest energies, about 939.6 MeV plus 0.511 MeV, exceed the proton's 938.3 MeV. No energy is available to make up the difference, so the decay is forbidden, and that is why a free proton is stable. Inside a nucleus the energy balance is different, and bound protons can undergo β⁺ decay where free ones cannot.

Momentum forbids processes just as firmly, and the antimatter lesson has already shown it doing so. A lone photon in empty space cannot become an electron-positron pair however much energy it carries, because no arrangement of that pair balances energy and momentum at once, which is why pair production requires a nearby nucleus to take the recoil. Keep the two checks separate. Energy sets whether there is enough available, and momentum sets whether the products can be arranged so that it is conserved.

Reading particle tracks, an Edexcel skill

Interpreting a detector photograph is Edexcel content; the other three boards do not set it. The physics is the same set of conservation checks, applied to a picture instead of an equation.

A detector records a charged particle because the particle ionises whatever it flies through, and that trail of ions is made visible, as a string of bubbles in a bubble chamber or as a run of hits in a modern tracker. A neutral particle ionises nothing and so leaves no track at all. Treat the absence as evidence rather than a hole in the picture. A pair of tracks springing out of empty space, with a clear run behind them, says a neutral particle crossed that gap unseen, and it is drawn dashed because nobody saw it. A neutral kaon or a Λ⁰ appears in exactly this way, as a V with nothing leading into it.

A magnetic field across the chamber bends every charged track, and the magnetic fields unit sets that magnetic force against the centripetal one to get the radius, which it writes as r = mv/BQ and Edexcel writes as r = p/(BQ)$r = p/(BQ)$r = p/(BQ). They are one relation, since p = mv.

Two readings come straight off the photograph. Which way a track curls gives the sign of the charge, once the field direction and the direction of travel are known; without those, the curl still separates the charges, since opposite signs bend opposite ways in one field. How tightly a track curls gives the momentum, a wide gentle arc marking high momentum and a tight one low; the photograph is read in the plane perpendicular to the field, so strictly the radius measures the momentum component in that plane, and speed follows only once the particle is identified. A straight track is simply the wide end of that scale, too much momentum to bend measurably.

A track that spirals is the same rule applied twice. The particle is ionising as it goes, every ion it makes takes energy from it, so its momentum falls, and by r = p/(BQ) the radius falls with it until the curve winds itself in. So a spiral reads two things at once. This particle is losing energy, and the tight end is the late end. Direction of travel follows from it, which matters more than it looks, because reading the same picture backwards turns every charge in it upside down.

At the vertex, apply the same conservation laws to the photograph. Charge is counted by curl, so two daughters curling opposite ways carry opposite signs, and if nothing led into the vertex those charges have to cancel exactly. Momentum is a vector sum, so the daughters' momenta, each read from a radius and pointed along its own track, must add to the parent's momentum along the invisible line, which is why the two arms of a V straddle that line instead of leaning to one side. Energy is the last check, the parent having to cover the daughters' rest energies with enough left over for the kinetic energy those momenta imply.