Conservation laws, mass-energy and antimatterEdexcel International A Level Physics: Revision notes

In every interaction between particles three quantities are conserved:

• charge: the total charge before equals the total charge after
• energy: including the rest energy (mass) of the particles, kinetic energy and photon energy
• momentum: a vector, so direction matters

These rules decide which interactions are possible. For example, an electron and a positron at rest have almost zero total momentum, so they must produce two photons moving in opposite directions: a single photon would carry momentum.

Mass and energy are linked by $E = mc^2$. The rest energy of a particle of mass $m$ is $mc^2$.

Particle physicists use MeV and GeV for energy, and MeV/c² and GeV/c² for mass:

• 1 eV = 1.60 × 10⁻¹⁹ J, so 1 MeV = 1.60 × 10⁻¹³ J and 1 GeV = 1.60 × 10⁻¹⁰ J
• 1 MeV/c² = $\frac{1.60 \times 10^{-13}}{(3.00 \times 10^{8})^{2}} = 1.78 \times 10^{-30}$ kg
• electron rest energy = 0.511 MeV; proton rest energy = 938 MeV = 0.938 GeV

Example. The muon, 105.7 MeV/c²: $m = \frac{105.7 \times 10^{6} \times 1.60 \times 10^{-19}}{(3.00 \times 10^{8})^{2}} = 1.88 \times 10^{-28}$ kg.

Every particle has an antiparticle with the same mass but opposite charge (for example the positron, the antiparticle of the electron).

• Pair production: a photon of enough energy turns into a particle and its antiparticle, e.g. $\gamma \to e^{-} + e^{+}$. The photon energy must be at least $2mc^2$ (1.02 MeV for an electron-positron pair); any excess becomes kinetic energy.
• Annihilation: a particle and its antiparticle meet and their mass is converted to energy, usually two photons, each of at least 0.511 MeV for $e^{-} + e^{+}$ at rest.

Charge, energy and momentum are conserved in both.

In a detector with a magnetic field:

• tracks of opposite charge curve in opposite directions
• a neutral particle leaves no track; its path can be deduced from the tracks it produces when it decays or pair-produces
• the radius is $r = \frac{p}{BQ}$, so higher momentum gives a larger radius
• tracks that spiral inwards belong to particles losing energy
• conservation of charge, energy and momentum tell you what the tracks mean: two oppositely charged tracks starting at one point (a V shape) imply a neutral particle or photon that created them

To investigate the structure of nucleons (radius about $10^{-15}$ m), the probe's de Broglie wavelength $\lambda = \frac{h}{p}$ must be about this size or smaller. This needs a momentum of order $\frac{6.63 \times 10^{-34}}{10^{-15}} = 6.6 \times 10^{-19}$ kg m s⁻¹, so very high energies, of the order of GeV. High energy also lets collisions create new massive particles.

For particles moving at speeds close to the speed of light, the lifetime measured in the laboratory is longer than the lifetime at rest (time dilation). This is significant only at a large fraction of $c$. It explains why unstable particles such as muons travel much further in an accelerator or through the atmosphere than $v \times$ (rest lifetime). You do not need the equations.