X rays with initial wavelength 0.0665 nm undergo Compton scattering. What is the longest wavelength found in the scattered x rays? At which scattering angle is this wavelength observed?

Verified step by step guidance

1. Identify the initial wavelength of the X-rays, denoted as \( \lambda_0 \), which is given as 0.0665 nm.
2. Recall the Compton scattering formula which relates the change in wavelength (\( \Delta \lambda \)) to the scattering angle (\( \theta \)): \( \Delta \lambda = \frac{h}{m_e c} (1 - \cos(\theta)) \), where \( h \) is Planck's constant, \( m_e \) is the electron mass, and \( c \) is the speed of light.
3. Understand that the maximum change in wavelength, \( \Delta \lambda_{max} \), occurs when \( \theta = 180^\circ \) because \( 1 - \cos(180^\circ) \) reaches its maximum value of 2.
4. Calculate \( \Delta \lambda_{max} \) using the Compton formula with \( \theta = 180^\circ \).
5. Add \( \Delta \lambda_{max} \) to the initial wavelength \( \lambda_0 \) to find the longest wavelength \( \lambda_{max} \) in the scattered X-rays: \( \lambda_{max} = \lambda_0 + \Delta \lambda_{max} \).

Key Concepts

Compton Scattering

Compton scattering is a phenomenon where X-rays or gamma rays collide with matter, typically electrons, resulting in a change in the wavelength of the incident radiation. This effect demonstrates the particle-like behavior of light, as photons transfer energy and momentum to electrons, causing the photons to scatter at different angles with longer wavelengths.

Wavelength Shift

The wavelength shift in Compton scattering is determined by the Compton formula, which relates the change in wavelength to the scattering angle. The formula is given by Δλ = λ' - λ = (h/m_ec)(1 - cos(θ)), where h is Planck's constant, m_e is the electron mass, c is the speed of light, and θ is the scattering angle. This shift indicates how much the wavelength increases as a result of the scattering process.

Scattering Angle

The scattering angle in Compton scattering is the angle at which the X-ray photon is deflected after colliding with an electron. It plays a crucial role in determining the final wavelength of the scattered X-rays. The longest wavelength occurs at a scattering angle of 180 degrees, where the photon is scattered directly backward, maximizing the energy transfer to the electron.