Answer either part (a) or part (b) - Leaving Cert Physics - Question 11 - 2014

The equation representing pair annihilation is:

To calculate the frequency, we use the energy-mass equivalence:

$hf = m_e c^2$

Substituting the values:

• Mass of electron, $m_e = 9.11 imes 10^{-31} kg$,
• Speed of light, $c = 3 imes 10^8 m/s$,

The frequency can be found as follows:

f = rac{m_ec^2}{h} = rac{(9.11 imes 10^{-31})(3 imes 10^8)^2}{6.626 imes 10^{-34}}

This will yield the frequency value in Hz.

The photons produced in pair annihilation travel in opposite directions due to the conservation of momentum. When the positron and electron annihilate, the total momentum must remain zero; therefore, one photon travels in one direction while the other travels in the opposite direction, ensuring that their momenta cancel each other out.

The nuclear equation for the decay of carbon-11 is:

The decay constant ($ext{λ}$) can be calculated using the formula:

ext{λ} = rac{ ext{ln}(2)}{T_{1/2}}

Where $T_{1/2}$ is the half-life of carbon-11, which is approximately 20 minutes (or 1200 seconds). Thus:

ext{λ} = rac{ ext{ln}(2)}{1200} ext{ s}^{-1} \\ ext{λ} ext{ is approximately } 0.000578 s^{-1}.

Carbon-11 has a short half-life (around 20 minutes). Therefore, it must be produced in proximity to the PET scanner to minimize decay before it can be used for imaging. If produced far away, too many nuclei would decay, and the amount available for scanning would be insufficient.

The expression for the momentum (p) of a particle in a cyclotron is given by:

$p = qB r$

Where:

• $p$ is the momentum of the particle,
• $q$ is the charge on the particle,
• $B$ is the magnetic flux density,
• $r$ is the radius of the circular path of the particle at any instant.