Energy, mass and antimatterEdexcel A-Level Physics: Revision notes

In every interaction between particles, charge, energy (including the rest energy of the particles) and momentum are conserved. Momentum is a vector, so the momenta must cancel in direction as well as size.

Tracks in bubble chambers and cloud chambers are used to test this. Only charged particles leave tracks, and a photon or neutron leaves none. In a magnetic field the magnetic force is perpendicular to the velocity, so tracks curve: oppositely charged particles curve in opposite senses, and a faster, higher-momentum particle has a track of larger radius (r = p/BQ). A particle that loses energy to the chamber spirals inwards.

A pair of tracks that start at one point and curve in opposite senses shows that a neutral particle has produced two oppositely charged particles.

Mass and energy are equivalent: ΔE = c²Δm. A particle of mass m has a rest energy E = mc².

Particle energies are measured in electronvolts: 1 eV = 1.60 × 10⁻¹⁹ J, so 1 MeV = 1.60 × 10⁻¹³ J and 1 GeV = 1.60 × 10⁻¹⁰ J.

Masses are quoted in MeV/c² or GeV/c², which is just rest energy divided by c². The electron has mass 0.511 MeV/c² and the proton about 938 MeV/c².

Worked example: proton mass in kg = (938 × 10⁶ × 1.60 × 10⁻¹⁹) ÷ (3.00 × 10⁸)² = 1.50 × 10⁻¹⁰ ÷ 9.00 × 10¹⁶ = 1.67 × 10⁻²⁷ kg.

Every particle has an antiparticle with the same rest mass and opposite charge (and opposite baryon and lepton number). The photon is its own antiparticle.

Annihilation: a particle and its antiparticle meet and their mass is converted entirely to energy, usually as two photons, e.g. e⁻ + e⁺ → 2γ. For an electron and positron at rest the total energy is 2 × 0.511 = 1.022 MeV, so each photon has 0.511 MeV. Two photons are needed because the total momentum is zero, so they leave in opposite directions with equal energies. PET scanners detect these photon pairs.

Pair production: a photon of energy at least 2mc² (1.022 MeV for an electron–positron pair) creates a particle and its antiparticle, e.g. γ → e⁻ + e⁺. It happens close to a nucleus, which takes up momentum. Any extra photon energy becomes the kinetic energy of the pair.

Particles behave as waves with de Broglie wavelength λ = h/p. To resolve detail the wavelength must be comparable to or smaller than the object, so probing a nucleon (about 10⁻¹⁵ m) needs a very large momentum.

For a high-energy electron, E ≫ rest energy, so p ≈ E/c and λ ≈ hc/E.

Worked example: for 1.0 GeV electrons, E = 1.60 × 10⁻¹⁰ J and λ = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) ÷ 1.60 × 10⁻¹⁰ = 1.2 × 10⁻¹⁵ m.

Electrons of 10 GeV or more, with λ well below 10⁻¹⁵ m, scatter from point-like quarks inside protons. High energies are also needed to create massive new particles, since the collision energy must supply their rest energy.

An unstable particle has a mean lifetime that is shortest when measured in its own rest frame. For an observer who sees the particle moving, the lifetime is longer. This is the relativistic increase in lifetime (time dilation). You do not need the relativistic equations.

The effect is only significant when the speed is a large fraction of the speed of light. At everyday speeds it is negligible.

Worked example: a muon (rest lifetime 2.2 μs) at 0.99c travels about 0.99 × 3.00 × 10⁸ × 2.2 × 10⁻⁶ = 650 m before decaying if the effect is ignored. Muons formed many kilometres up reach the ground, so their lifetime in the Earth frame must be much longer. Fast particles in accelerators also survive for longer, which is what makes beam experiments possible.