Step 1: Understanding the Concept:
Compton scattering refers to the inelastic scattering of a photon by a charged particle (usually an electron), which results in a decrease in the photon's energy and an increase in its wavelength.
The change in wavelength is known as the Compton shift.
Step 2: Key Formula or Approach:
The formula for the Compton shift (\(\Delta \lambda\)) is given by:
\[ \Delta \lambda = \lambda' - \lambda = \frac{h}{mc} (1 - \cos \theta) \]
where \(h\) is Planck's constant, \(m\) is the rest mass of the electron, \(c\) is the speed of light, and \(\theta\) is the scattering angle.
Step 3: Detailed Explanation:
The question states the photon is backscattered at \(180^\circ\).
Substituting \(\theta = 180^\circ\) into the formula:
\[ \cos(180^\circ) = -1 \]
\[ \Delta \lambda = \frac{h}{mc} (1 - (-1)) \]
\[ \Delta \lambda = \frac{h}{mc} (1 + 1) \]
\[ \Delta \lambda = \frac{2h}{mc} \]
Step 4: Final Answer:
The Compton shift for a photon backscattered at \(180^\circ\) is \(\frac{2h}{mc}\), which is the maximum possible Compton shift.