X rays with an initial wavelength of 0.900 * 10-10 m undergo Compton scattering. For what scattering angle is the wavelength of the scattered x rays greater by 1.0% than that of the incident x rays?

Verified step by step guidance

1. Identify the initial wavelength of the X-rays, \( \lambda_i = 0.900 \times 10^{-10} \, \text{m} \).
2. Calculate the target wavelength after scattering, which is 1.0% greater than the initial wavelength. Use the formula \( \lambda_f = \lambda_i \times (1 + \frac{1.0}{100}) \).
3. Use the Compton scattering formula to relate the change in wavelength to the scattering angle, \( \theta \). The formula is \( \Delta \lambda = \lambda_f - \lambda_i = \frac{h}{m_e c} (1 - \cos(\theta)) \), where \( h \) is Planck's constant, \( m_e \) is the electron mass, and \( c \) is the speed of light.
4. Isolate \( \theta \) in the Compton scattering formula to find the scattering angle. Solve for \( \theta \) using the equation \( 1 - \cos(\theta) = \frac{\Delta \lambda \cdot m_e c}{h} \).
5. Calculate \( \theta \) using the inverse cosine function, \( \theta = \cos^{-1}(1 - \frac{\Delta \lambda \cdot m_e c}{h}) \).

Key Concepts

Compton Scattering

Compton scattering is a phenomenon where X-rays or gamma rays collide with matter, resulting in a change in the wavelength of the radiation. This effect occurs due to the interaction between photons and electrons, leading to the transfer of energy and momentum. The change in wavelength is dependent on the scattering angle, which can be calculated using the Compton wavelength shift equation.

Wavelength Shift

The wavelength shift in Compton scattering refers to the increase in wavelength of the scattered X-rays compared to the incident X-rays. This shift can be quantified as a percentage change, which is crucial for determining the scattering angle. In this case, a 1.0% increase in wavelength indicates a specific relationship between the initial and final wavelengths, which can be used to solve for the angle.

Scattering Angle

The scattering angle is the angle at which a photon is deflected after colliding with an electron during Compton scattering. It is a critical parameter in determining the change in wavelength of the scattered radiation. By applying the Compton wavelength shift formula, one can relate the scattering angle to the observed change in wavelength, allowing for the calculation of the angle when the wavelength increases by a specified percentage.