Question

A $\gamma$-ray photon emitted from a $^{137}$Cs source collides with an electron at rest. If the Compton shift of the photon is $3.25 \times 10^{-13}$ m, then the scattering angle is closest to (Planck’s constant $h = 6.626 \times 10^{-34}$ J s, electron mass $m_e = 9.109 \times 10^{-31}$ kg and velocity of light in free space $c = 3 \times 10^8$ m/s)

• A. 45°
• B. 60°
• C. 30°
• D. 90°

Hint:

In Compton scattering problems, use the Compton wavelength shift equation to calculate the scattering angle from the wavelength change.

Answer 1

The Correct Option is C

Step 1: Compton scattering formula.
The Compton scattering equation is given by: \[ \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta) \] where \(\Delta \lambda\) is the change in wavelength, \(h\) is Planck's constant, \(m_e\) is the electron mass, \(c\) is the speed of light, and \(\theta\) is the scattering angle.

Step 2: Calculating the scattering angle.
Rearranging the Compton equation to solve for \(\theta\): \[ \cos \theta = 1 - \frac{\Delta \lambda m_e c}{h} \] Substituting the given values: \[ \Delta \lambda = 3.25 \times 10^{-13} \text{ m}, \, h = 6.626 \times 10^{-34} \text{ J s}, \, m_e = 9.109 \times 10^{-31} \text{ kg}, \, c = 3 \times 10^8 \text{ m/s} \] Solving for \(\theta\) gives approximately 60°.

Step 3: Conclusion.
The correct scattering angle is (B) 60°. This is the angle closest to the one calculated using the Compton scattering equation.