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. A free proton decaying by p → n + e⁺ + ν_e would conserve charge, baryon number and electron lepton number. Yet it does not happen. The products' rest energies, about 939.6 MeV plus 0.511 MeV, add up to more than the proton's 938.3 MeV. No energy is available to make up the difference, so the decay is forbidden, and a free proton is stable. Inside a nucleus the energy balance is different, so bound protons can undergo β⁺ decay where free ones cannot.

Describe the drawing

Two bars drawn to one scale that starts at 930 MeV, so that a small difference can be seen. The upper bar is the proton's rest energy, 938.3 MeV. The lower bar is what the proposed products need, a neutron at 939.6 MeV with a positron on the end of it, and it runs about 1.8 MeV past the dashed line the proton reaches. A free proton has no spare energy to cover that difference, which is why the decay never happens although every quantum number balances.

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. No arrangement of that pair balances energy and momentum at once. That is why pair production needs 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. 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. A pair of tracks appearing out of empty space, with a clear run behind them, means a neutral particle crossed that gap unseen. It is drawn dashed because nobody saw it. A neutral kaon or a Λ⁰ appears in this way, as a V with nothing leading into it.

Describe the drawing

Two panels. On the left a dashed grey line, a neutral particle leaving no track, ends at a vertex where two curved tracks spring apart: a wide cyan arc curling upward for charge plus one and a tighter amber arc curling downward for charge minus one. On the right a single cyan track spirals inward, its radius shrinking turn by turn as the particle loses energy to the chamber.

A magnetic field across the chamber bends every charged track. The magnetic fields unit equates the magnetic force to the centripetal force to get the radius. It writes the result as r = mv/BQ, and Edexcel writes it 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 means 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. 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, and 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 reverses the sign of every charge in it.

At the vertex, apply the same conservation laws to the photograph. Charge is counted by curl: 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. Read each daughter's momentum from its radius and point it along its own track; together they must add to the parent's momentum along the invisible line. That is why the two arms of a V straddle that line instead of leaning to one side. Energy is the last check. The parent has to cover the daughters' rest energies, with enough left over for the kinetic energy those momenta imply.