24.3.2 Annihilation

Annihilation

Definition: Annihilation

• Annihilation occurs when a particle interacts with its antiparticle.
• Mass-energy and momentum are conserved in the process.

Annihilation of a positron and electron

Positron and Electron Annihilation

• The positron emitted by the decay of a Fluorine-18 nuclide travels a very short distance before it collides with an electron.
• The positron and electron undergo pair annihilation.
• The particles disappear, releasing their combined mass-energy as photons.
• The mass of the electron and positron are converted into energy through the relation $E=\Delta m{c}^2$. -Assuming the electron and positron have negligible kinetic energy when they interact, the two photons appear with equal energies.
• The photons have frequencies in the gamma ray range.
• Assuming both particles were initially at rest before the collision, their initial momentum was negligible.
• To conserve momentum, two photons must be emitted in opposite directions.

Energy of Gamma Photons

Before Annihilation

• Positrons and Electrons have the same mass $9.11\times 10^{-31}\text{ kg}$
• Their total mass is $1.822\times 10^{-30}\text{ kg}$

After Annihilation

• The positron and electron no longer exist.
• Their combined mass is converted into photon energy.

Using the relation $\Delta E=\Delta m{c}^2$: $\Delta E=1.822\times 10^{-30}\times (3.0\times 10^8{)}^2$ $\Delta E=1.6398\times 10^{-13}\text{ J}=1024\text{ keV}$

• Two photons are emitted in opposite directions in order to conserve momentum.
• The photons have the same energy.

The energy of one photon is therefore: $\frac{1.6398\times 10^{-13}}{2}=8.199\times 10^{-14}\text{ J}=512\text{ keV}$

Frequency of Photon

Using $E=hf$: $f=\frac{E}{h}$ $f=\frac{8.199\times 10^{-14}}{6.63\times 10^{-34}}=1.24\times 10^{20}\text{ Hz}$

Momentum of Photon

Using $E=pc$ $p=\frac{E}{c}$ $f=\frac{8.199\times 10^{-14}}{3.0\times 10^8}=2.73\times 10^{-22}{\text{ kg ms}}^{-1}$