## 22.7 Energy in Annihilation Reactions **Annihilation reactions** are processes in particle physics where a particle collides with its antiparticle, resulting in their mutual destruction and the complete conversion of their mass into energy. These reactions are fundamental to understanding subatomic phenomena and have practical applications in medical imaging such as . ### Definition of Annihilation Reactions A process in particle physics where a particle collides with its corresponding antiparticle. This collision leads to their mutual destruction, and their combined mass is converted entirely into energy. The most common example is an electron colliding with a positron. ### Fundamental Conservation Laws Annihilation reactions are governed by two fundamental conservation laws: 1. **Conservation of Mass-Energy**: The total energy (including the mass-energy equivalent, $E=mc^2$) before and after the annihilation reaction remains constant. 2. **Conservation of Momentum**: The total momentum of the particles before annihilation equals the total momentum of the resulting particles after the reaction. To conserve momentum, when a particle and antiparticle at rest annihilate, at least two photons must be produced traveling in opposite directions. ### Common Examples of Annihilation Reactions #### Electron-Positron Annihilation The most common example, where an electron and a positron annihilate to produce two gamma ray photons ($\gamma$): $$\mathrm{e}^{-} + \mathrm{e}^{+} \rightarrow \gamma + \gamma$$ #### Proton-Antiproton Annihilation A proton and an antiproton can annihilate to produce: * Two gamma ray photons: $$\mathrm{p}^{+} + \mathrm{p}^{-} \rightarrow \gamma + \gamma$$ * Or various other particles like pions ($\pi$) and kaons: $$\mathrm{p}^{+} + \mathrm{p}^{-} \rightarrow \pi^{+} + \pi^{-} + \pi^{0}$$ ### Calculating Energy in Electron-Positron Annihilation (Particles at Rest) The rest mass energy of a single electron or positron is found using Einstein's equation $E = m_0 c^2$. This is the reverse process of pair production. Rest mass of electron/positron ($m_0$) is $9.1 \times 10^{-31}$ kg. Energy for one particle: $$E = m_0 c^2 = 9.1 \times 10^{-31} \text{ kg} \times (3 \times 10^8 \text{ m/s})^2 = 8.19 \times 10^{-14} \text{ J}$$ Converting this energy into electron-volt units: $$E = \frac{8.19 \times 10^{-14} \text{ J}}{1.6 \times 10^{-19} \text{ J/eV}} = 511,875 \text{ eV} \approx 0.51 \text{ MeV}$$ Since a positron has the same mass as an electron, it also has the same rest mass energy. The **total minimum energy** released when an electron and a positron (both at rest) annihilate is: $$E_{\text{total}} = 2 \times 511,875 \text{ eV} = 1.02 \text{ MeV}$$ This total energy is shared equally between the two emitted gamma ray photons, so each photon carries 0.51 MeV. ### Annihilation with Kinetic Energy If the electron and positron are moving before annihilation, their kinetic energy is also converted into the energy of the emitted gamma ray photons. The total energy equation in this case is: $$K.E_{\text{electron}} + K.E_{\text{positron}} + 2m_0 c^2 = 2hf$$ where $hf$ represents the energy of a single gamma ray photon. ### Practical Application Annihilation reactions have significant practical applications in medicine, particularly in , which is used for diagnosing diseases by detecting the gamma rays produced from positron annihilation within the body.