Problem 35 - University Physics with Modern Physics

Question

An incident x-ray photon of wavelength 0.0900 nm is scattered in the backward direction from a free electron that is initially at rest. (a) What is the magnitude of the momentum of the scattered photon? (b) What is the kinetic energy of the electron after the photon is scattered?

Step-by-Step Solution

Answer

(a) The magnitude of the momentum of the scattered photon is \(6.98 \times 10^{-24} \text{ kg} \cdot \text{m/s}\). (b) The kinetic energy of the electron is \(2.21 \times 10^{-15} \text{ J}\).

Step 1: Calculate the wavelength shift

The Compton wavelength shift formula is given by \( \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta) \), where \( \lambda \) is the initial wavelength, \( \lambda' \) the scattered wavelength, \( h \) the Planck constant (\(6.626 \times 10^{-34} \text{ m}^2 \text{ kg} / \text{s}\)), \( m_e \) the electron rest mass (\(9.11 \times 10^{-31} \text{ kg}\)), \( c \) the speed of light (\(3 \times 10^8 \text{ m/s}\)), and \( \theta \) the scattering angle, which is \( 180^\circ \) for backward scattering. Compute \( \Delta \lambda = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 3 \times 10^8} (1 - \cos 180^\circ) = 4.85 \times 10^{-12} \text{ m} \).

Step 2: Determine the scattered wavelength

The scattered wavelength \( \lambda' \) is calculated using \( \lambda' = \lambda + \Delta \lambda = 0.0900 \times 10^{-9} + 4.85 \times 10^{-12} = 0.09485 \times 10^{-9} \text{ m} \).

Step 3: Calculate momentum of the scattered photon

The momentum \( p \) of the photon is given by \( p = \frac{h}{\lambda} \). For the scattered photon, **\( p' = \frac{h}{\lambda'} = \frac{6.626 \times 10^{-34}}{0.09485 \times 10^{-9}} = 6.98 \times 10^{-24} \text{ kg} \cdot \text{m/s} \)**.

Step 4: Calculate the energy of the incident and scattered photons

The energy of a photon is given by \( E = \frac{hc}{\lambda} \). Calculate the initial and scattered energies: \( E_i = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{0.0900 \times 10^{-9}} \) and \( E_f = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{0.09485 \times 10^{-9}} \). The energy difference \( \Delta E = E_i - E_f = 2.21 \times 10^{-15} \text{ J} \).

Step 5: Calculate the kinetic energy of the electron

The kinetic energy \( K \) of the electron, which equals the energy lost by the photon, is \( K = \Delta E = 2.21 \times 10^{-15} \text{ J} \).

Key Concepts

Photon Momentum

Photon momentum is an essential concept when discussing Compton scattering. While light is often thought of as a wave, photons, which are particles of light, also exhibit particle-like properties, including momentum. Even without mass, a photon has momentum that can be calculated using the formula:

• \( p = \frac{h}{\lambda} \)

where \( p \) is the momentum, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \text{m}^2 \text{kg/s}) \), and \( \lambda \) is the photon's wavelength.
The momentum is inversely proportional to the wavelength, meaning the shorter the wavelength, the greater the momentum. This principle comes into play during scattering events, helping to determine how a photon's energy and momentum are distributed between the photon and other particles like electrons.

Kinetic Energy

In the context of Compton scattering, an electron gains kinetic energy as it absorbs part of the energy from the incident photon. Initially at rest, the electron receives energy transferred from the photon when scattered.
Post-collision, the electron's kinetic energy is determined by the energy difference between the incoming and outgoing photon. The equation to calculate the energy of a photon is:

• \( E = \frac{hc}{\lambda} \)

By calculating the difference in the energy of the initial photon and the scattered photon, you can find the kinetic energy \( K \) gained by the electron:

• \( K = E_i - E_f \)

Understanding this transfer of energy emphasizes the particle-like behavior of photons and how they can impart energy to other particles, causing motion in otherwise stationary electrons.

Wavelength Shift

Wavelength shift is a cornerstone of understanding Compton scattering. When a photon collides with an electron, its wavelength changes as a result of the scatter. This change is quantified using the Compton wavelength shift formula:

• \( \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c}(1 - \cos \theta) \)

Here, \( \lambda \) is the initial wavelength, \( \lambda' \) is the final scattered wavelength, \( m_e \) is the electron's rest mass, and \( \theta \) is the scatter angle.
For backward scattering where the scattering angle \( \theta \) is \( 180^\circ \), the formula simplifies to calculate the wavelength change due to the interaction, illustrating how energy is redistributed in the system.
This shift illustrates the change in energy states between different phases of photon-electron interactions, altering the wavelength as energy is passed or taken by the electron.

Scattered Wavelength

The scattered wavelength of a photon post-collision is an important result of Compton scattering. Once the wavelength shift \( \Delta \lambda \) has been calculated, the scattered wavelength \( \lambda' \) can easily be determined:

• \( \lambda' = \lambda + \Delta \lambda \)

Compton scattering causes an increase in wavelength, known as a redshift, due to energy loss by the photon to the electron.
This formula not only quantifies changes in wavelength due to scattering but also aids in understanding the redistribution of energy. The increased wavelength reveals crucial insights about how photons and electrons interact.
Knowing the scattered wavelength helps in calculations of new photon energies and momentums, proving crucial for resolving particle dynamics in high-energy physics and various technological applications.