3.2.1.3
Annihilation converts mass into photon energy
Particles, Antiparticles & Photons — AQA A-Level Physics
Key Definition
Annihilation — When a particle meets its corresponding antiparticle, both are destroyed and their mass is converted into energy in the form of two gamma-ray photons.
$$E = 2m_e c^2$$
- The two most common examples: electron-positron annihilationThe process in which a particle and its corresponding antiparticle collide and convert their combined rest mass into energyThe capacity to do work. Measured in joules (J)., typically producing two gamma photons. and proton-antiproton annihilationThe process in which a particle and its corresponding antiparticle collide and convert their combined rest mass into energyThe capacity to do work. Measured in joules (J)., typically producing two gamma photons..
- Two photons are produced, moving in opposite directions to conserve momentum.
- The minimum energyThe capacity to do work. Measured in joules (J). of each photonA quantum (discrete packet) of electromagnetic radiation. Its energy is proportional to its frequency. equals the rest mass-energy of one of the particles.
$$\begin{aligned}
E_{\min} &= hf_{\min} \\
&= E_0
\end{aligned}$$
- $E_min$: minimum energy of one photonA quantum (discrete packet) of electromagnetic radiation. Its energy is proportional to its frequency. produced (J)
- $h$: Planck's constant (J s)
- $f_min$: minimum frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz). of one photonA quantum (discrete packet) of electromagnetic radiation. Its energy is proportional to its frequency. (Hz)
- $E_0$: rest mass-energy of one of the particles (J)
Worked Example
Calculate the maximum wavelengthThe minimum distance between two points on a wave that are in phase (e.g. crest to crest). Measured in metres (m). of one of the photons produced when a proton and antiproton annihilate.
Show Solution
1
List known values
Rest mass-energy of a proton $= 938.3$ MeV
$1$ MeV $= 1.60 \times 10^{-13}$ J
2
Write minimum photon energy
$$E_{\min} = E_0 = 938.3 \times 1.60 \times 10^{-13} \text{ J}$$
3
Rearrange for wavelengthThe minimum distance between two points on a wave that are in phase (e.g. crest to crest). Measured in metres (m).
Maximum wavelengthThe minimum distance between two points on a wave that are in phase (e.g. crest to crest). Measured in metres (m). corresponds to minimum energy:
$$\lambda_{\max} = \frac{hc}{E_0}$$4
Substitute values
$$\lambda_{\max} = \frac{(6.63 \times 10^{-34}) \times (3.0 \times 10^{8})}{938.3 \times (1.60 \times 10^{-13})} = 1.32 \times 10^{-15} \text{ m}$$
Answer
$\lambda_{\max} = 1.32 \times 10^{-15}$ m
Common Mistake
MEDIUM
Students often: Using MeV directly in $E = hc/\lambda$ without converting to joules.
Instead: Planck's constant is in J s, so all energies must be in joules. Multiply MeV by 1.60 × 10⁻¹³ to convert.
Instead: Planck's constant is in J s, so all energies must be in joules. Multiply MeV by 1.60 × 10⁻¹³ to convert.