Question

X- rays of wavelength 15 pm are scattered from a target. The wavelength of the X-rays scattered through 60° is (Given compton wavelength = 2.426 pm)

• A. 13.787 pm (pm is pico metre)
• B. 16.213 pm (pm is pico metre)
• C. 15.5 pm (pm is pico metre)
• D. 1.6213 pm (pm is pico metre)

Hint:

Remember that in Compton scattering, the scattered photon always has a longer wavelength (and therefore lower energy) than the incident photon, except for forward scattering (\(\theta=0\)), where the wavelength does not change. This helps you eliminate any options that suggest a decrease in wavelength.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem describes Compton scattering, an effect where X-rays or gamma rays scatter off charged particles (usually electrons), resulting in an increase in the wavelength of the scattered radiation. The change in wavelength depends on the scattering angle.
Step 2: Key Formula or Approach:
The Compton scattering formula gives the change in wavelength \( \Delta\lambda \):
\[ \Delta\lambda = \lambda' - \lambda = \lambda_c (1 - \cos\theta) \] where \( \lambda' \) is the scattered wavelength, \( \lambda \) is the incident wavelength, \( \lambda_c \) is the Compton wavelength of the electron (\(h/m_e c\)), and \( \theta \) is the scattering angle.
Step 3: Detailed Explanation:
1. Identify the given values:
Incident wavelength, \( \lambda = 15 \) pm.
Compton wavelength, \( \lambda_c = 2.426 \) pm.
Scattering angle, \( \theta = 60^\circ \).
2. Calculate the change in wavelength (\( \Delta\lambda \)):
First, find the cosine of the scattering angle: \( \cos(60^\circ) = 0.5 \).
Now, substitute the values into the Compton formula:
\[ \Delta\lambda = 2.426 \, \text{pm} \times (1 - 0.5) \] \[ \Delta\lambda = 2.426 \, \text{pm} \times 0.5 = 1.213 \, \text{pm} \] 3. Calculate the scattered wavelength (\( \lambda' \)):
The new wavelength is the original wavelength plus the change.
\[ \lambda' = \lambda + \Delta\lambda \] \[ \lambda' = 15 \, \text{pm} + 1.213 \, \text{pm} = 16.213 \, \text{pm} \] Step 4: Final Answer:
The wavelength of the X-rays scattered through 60° is 16.213 pm.