Medical Physics

• Radioactivity occurs when we have an instability in the nuclear structure of an atom. There’s several different ways in which this can happen:
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  • Change in the number of protons at the center (i.e. change in $Z$)
    • Not only would we add protons or lose protons, but we’re also changing the chemical nature.
    • 3 major processes that this can occur:
      • Alpha decay
        • Where a cluster of protons and neutrons (e.g. ${}_2^4\text{He}$) are expelled from the nucleus.
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        • Common in very large atoms, such as uranium and plutonium.
      • Beta decay
        • Where either a proton sheds a positron (or a neutron sheds an electron), i.e. shed a little bit of charge either in the positive sense or in the negative sense → Z increases or decreases by 1, respectively, but A remains unchanged.
        • Does NOT change the number of nucleons, but it does change the amount of charge in the center of the nucleus.
      • Electron Capture
        • Electron cloud passes close by a large nucleus → grabs an electron from the electron cloud + a proton is changed to a neutron
  • Isomeric transitions: No change in the number of protons, i.e. no change in $Z$
    • Gamma Decay
      • We have an atom that maybe starts to raise its energy state → goes to a low energy state with no change in the number of nucleons or the number of protons at the center of it, but gives off energy and conserves energy → produces a gamma ray that then detected by the system
• Calculating the number of radioactive atoms in a sample
  • Is proportional to the number of atoms present
  • The change in the number is proportional to a constant ($k$, sometimes written as $\lambda$) times the number that’s present. When we integrate that we end up with the familiar exponential decay.
  • $\frac{dN}{dt}=-kN$ → Integration of this is $\int \frac{dN}{N}=-k\int dt$
  • $N=N_0\cdot \exp (-kt)$
  • Example: We have a five millicary (mCi) dose of technitium that spilled on the floor carpet in the in a laboratory. And with a half life of 72 hours, how long must we wait to reach 1/1,000th of the activity remaining?
    • $\text{Dose}={\text{Dose}}_0\cdot \exp (\frac{-t\cdot \ln (2)}{t_{1/2}})$
    • $t=\frac{t_{1/2}}{\ln (2)}\cdot \ln (1,000)$ = 717 hours

Compton Scattering

• The way that Compton scattering takes place is an incident photon scatters off an electron in the electron cloud and the photon heads off in a new direction and kicks out an electron into the media.
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• The amount of energy change is determined by the Compton formula.
  • The new energy is largely determined by the incident energy and the angle.
  • Higher incident energy → bigger energy change energy change
  • Higher unit angle → bigger energy change
• Example: A technologist opens the 140 keV photon window to +/-14 keV (20%). What is the minimum single scattering angle that the energy window can reject (assume perfect energy resolution and $m{c}^2$ = 511 keV?
  • See 18 minutes mark in lecture 1