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Chapter 6: Problem 144

Find the maximum change in X-ray wavelength that can occur due to Compton scattering. Does this change depend on the wavelength of the incident beam?

Short Answer

Expert verified
The maximum change in X-ray wavelength due to Compton scattering is \( 4.86 × 10^{-2} nm \). This change does not depend on the wavelength of the incident beam.

Step by step solution

01

Understand Compton Scattering Formula

The Compton wavelength shift formula is given by \( \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos\theta) \) where: \(\lambda'\) is the wavelength after scattering, \(\lambda\) is the initial wavelength, \(h\) is Planck's constant, \(m_e\) is the electron mass, \(c\) is the speed of light, and \(\theta\) is the scattering angle.
02

Determine Maximum Wavelength Shift

The maximum shift \( \Delta \lambda_{max} \) occurs when the scattering angle \( \theta \) is 180 degrees or \( \pi \) radians. Substituting \( \pi \) for \( \theta \) in the formula gives \( \Delta \lambda_{max} = \frac{2h}{m_e c} = 4.86 × 10^{-12} m \) or \( 4.86 × 10^{-2} nm \).
03

Discuss Dependence on Incident Beam Wavelength

Looking back at the Compton scattering equation, we can see that initial wavelength \( \lambda \) does not appear in the equation for the change of wavelength. This means that the change in wavelength due to Compton Scattering does not depend on the wavelength of the incident beam.

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