24.2.3 Detecting Gamma-Rays from PET Scanning

• The patient lays stationary in a tube surrounded by a ring of detectors
• Images of slices of the body can be taken to show the position of the radioactive tracers
• The detector consists of two parts:
• Crystal Scintillator – when the gamma-ray (γ-ray) photon is incident on a crystal, an electron in the crystal is excited to a higher energy state
  • As the excited electron travels through the crystal, it excites more electrons
  • When the excited electrons move back down to their original state, the lost energy is transmitted as visible light photons

• Photomultiplier -The photons produced by the scintillator are very faint, so they need to be amplified and converted to an electrical signal by a photomultiplier tube

Detecting gamma rays with a PET scanner

Creating an Image from PET Scanning

• The γ rays travel in straight lines in opposite directions when formed from a positron-electron annihilation
  • This happens in order to conserve momentum

• They hit the detectors in a line – known as the line of response
• The tracers will emit lots of γ rays simultaneously, and the computers will use this information to create an image
• The more photons from a particular point, the more tracer that is present in the tissue being studied, and this will appear as a bright point on the image
• An image of the tracer concentration in the tissue can be created by processing the arrival times of the gamma-ray photons

• In the annihilation process, both mass-energy and momentum are conserved
• The gamma-ray photons produced have an energy and frequency that is determined solely by the mass-energy of the positron-electron pair
• The energy E of the photon is given by

E = hf = m_ec²

• The momentum p of the photon is given by

• Where:
  • m_e = mass of the electron or positron (kg)
  • h = Planck's constant (J s)
  • f = frequency of the photon (Hz)
  • c = the speed of light in a vacuum (m s^–1)

Part (a)

Step 1: Write down the known quantities

• • Mass of an electron = mass of a positron, m_e = 9.11 × 10^–31 kg
  • Total mass is equal to the mass of the electron and positron = 2m_e

Step 2: Write out the equation for mass-energy equivalence

E = m_ec²

Step 3: Substitute in values and calculate energy E

E = 2 × (9.11 × 10^-31) × (3.0 × 10⁸)² = 1.6 × 10^–13 J

Part (b)

Step 1: Determine the energy of one photon

• • Planck's constant, h = 6.63 × 10⁻³⁴ J s
  • Two photons are produced, so, the energy of one photon is equal to half of the total energy from part (a):

Step 2: Write out the equation for the energy of a photon

E = hf

Step 3: Rearrange for frequency f, and calculate

Part (c)

Step 1: Write out the equation for the momentum of a photon

Step 2: Substitute in values and calculate momentum, p