2 Compton Effect
• In the photoelectric effect, the energy of a photon is transferred to the electron by first
supplying enough energy to release it from its bound state to a nucleus and then transferring
the rest of its energy into kinetic energy of the freed electron. Typically, the electron that
is ejected is a K shell electron whose binding energy is a few keV. This is relatively a small
amount compared to the energy of the gamma ray so that most of the incident energy is then
transferred to kinetic energy of the electron. Therefore, if the energy of the gamma ray is
$E_γ$, its energy will be first transferred to the binding energy $E_b$ of the electron to free it from
its nucleus and then the remainder of its energy will be transferred to the electron's kinetic
energy $E_e$- or
$E = E_e + E_b \Rightarrow E_e = E_γ - E_b$.
The photoelectric absorption process then converts electromagnetic energy of a gamma ray
photon into kinetic energy of a charged particle, an electron.
• Just as a gamma ray can transfer its energy to an electron in a scintillator material by the
photoelectric effect, it can also transfer a portion of its energy to an electron by the Compton
Effect. Whereas the transfer of gamma ray energy to an electron via the photoelectric effect
is always nearly 100%, the transfer of energy via the Compton Effect can range from 0% to
nearly 100%, depending on the energy of the gamma ray and the angle that it is scattered.
• Applying the law of conservation of energy, the energy given to an electron by Compton
Scattering is
$E_e = E - E' = E - \frac{E_γmc^2}{mc^2 + E(1 - cos θ)}$
$= E \left[ 1 - \frac{1}{1 + \frac{E}{mc^2}(1 - cos θ)} \right]$.
• The maximum energy (where θ = 180°) transferred to the electron is
$E_{max} = E_γ \left( 1 - \frac{1}{1 + \frac{2E_γ}{mc^2}} \right) = E_γ \left( 1 - \frac{mc^2}{mc^2 + 2E_γ} \right) = \frac{2E_γ^2}{mc^2 + 2E_γ}$.
• If $E >> mc^2 \Rightarrow E_{max} \approx E$. If $E << mc^2 \Rightarrow E_{max} \approx 0$.
• Find the electron rest energy $mc^2$ for $^{137}Cs$ and $^{22}Na$ radioactive sources.
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