Problem 35 - College Physics

Question

An incident X-ray photon is scattered from a free electron that is initially at rest. The photon is scattered straight back at an angle of \(180^{\circ}\) from its initial direction. The wavelength of the scattered photon is \(0.0830 \mathrm{nm}\). (a) What is the wavelength of the incident photon? (b) What is the magnitude of the momentum of the electron after the collision? (c) What is the kinetic energy of the electron after the collision?

Step-by-Step Solution

Answer

(a) 0.0802 nm; (b) 1.67 × 10^{-24} kg·m/s; (c) 1.53 × 10^{-17} J.

Step 1: Understanding the Scattering Formula

For part (a), we need to find the wavelength of the incident photon using the Compton scattering formula. The formula is given by \( \lambda' - \lambda = \frac{h}{m_ec}(1 - \cos \theta) \), where \( \lambda' \) is the wavelength of the scattered photon, \( \lambda \) is the wavelength of the incident photon, \( \theta \) is the scattering angle, \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{ J}\cdot\text{s}\), \( m_e \) is the rest mass of the electron \(9.11 \times 10^{-31} \text{ kg}\), and \( c \) is the speed of light \(3 \times 10^8 \text{ m/s}\).

Step 2: Calculating Wavelength of the Incident Photon

Plugging the values into the scattering formula with \( \theta = 180^\circ \) (so \( \cos \theta = -1 \)), we have \( \lambda' - \lambda = \frac{h}{m_ec}(1 - (-1)) = \frac{2h}{m_ec} \). Thus, \( \lambda = \lambda' - \frac{2h}{m_ec} \). Substituting in the values, \( \lambda = 0.0830 \text{ nm} - \frac{2(6.626 \times 10^{-34})}{(9.11 \times 10^{-31})(3 \times 10^8)} \approx 0.0802 \text{ nm} \).

Step 3: Finding Momentum of the Electron

For part (b), we use the relation \( p' = \frac{h}{\lambda} - \frac{h}{\lambda'} \), where \( p' \) is the momentum of the photon after collision. This also equals the momentum of the electron due to conservation of momentum. Calculating, \( p' = \frac{6.626 \times 10^{-34}}{0.0802 \times 10^{-9}} - \frac{6.626 \times 10^{-34}}{0.0830 \times 10^{-9}} \approx 1.67 \times 10^{-24} \text{ kg}\cdot\text{m/s} \).

Step 4: Calculating Kinetic Energy of the Electron

For part (c), we calculate kinetic energy using \( KE = \frac{p'^2}{2m_e} \). Substituting \( p' = 1.67 \times 10^{-24} \text{ kg}\cdot\text{m/s} \) and \( m_e = 9.11 \times 10^{-31} \text{ kg} \), we find \( KE = \frac{(1.67 \times 10^{-24})^2}{2 \times 9.11 \times 10^{-31}} \approx 1.53 \times 10^{-17} \text{ J} \).

Key Concepts

X-ray Photon Scattering

In Compton scattering, an X-ray photon hits a stationary electron, causing the photon to scatter and change direction while transferring energy to the electron. This is a crucial phenomenon in quantum mechanics as it demonstrates the particle-like properties of light. The scattering angle, often denoted as \( \theta \), plays an essential role in determining the amount of energy transferred.

For this exercise, the scattering happens at an angle of \(180^{\circ}\), meaning the photon is scattered straight back. Using the Compton scattering formula,

• \(\lambda' - \lambda = \frac{h}{m_ec}(1 - \cos \theta)\),

where \(\lambda'\) is the scattered photon's wavelength, \(\lambda\) is the incident photon's wavelength, \(h\) is Planck's constant, \(m_e\) is the electron's rest mass, and \(c\) is the speed of light.

This relationship helps us calculate the initial wavelength of the photon once the scattered wavelength is known.

Photon-Electron Collision

The collision involved in Compton scattering is between a photon and an electron. The photon, a particle of light, transfers its energy to the electron, knocking it into motion. Initially, the electron is at rest, but after the collision, it moves with newfound momentum and energy. This exchange exemplifies the conservation of energy and momentum.

During the collision:

• The photon loses some energy and increases its wavelength.
• The electron gains kinetic energy and momentum.

These changes are directly calculated and play a vital role in understanding the photon's behavior and the effects on the electron. The changes in wavelength and energy help illustrate the particle nature inherent in quantum phenomena.

Momentum Conservation

In physics, momentum conservation is a fundamental principle where the total momentum before a collision equals the total momentum after a collision. In the context of Compton scattering, momentum conservation notably applies when a photon hits an electron.

Here, the principle states that:

• The initial momentum of the photon equals the sum of the final momentums of the photon and the electron after the collision.

Using the relation,

• \( p' = \frac{h}{\lambda} - \frac{h}{\lambda'} \),

where \( p' \) matches the electron's momentum, allows us to perform the calculations needed. This highlights the essence of these interactions and contributes to understanding the effects of momentum in quantum mechanics.

Kinetic Energy Calculation

The kinetic energy of a particle describes its motion energy. Following a photon-electron collision, calculating the kinetic energy of the electron is crucial to evaluate the energy distribution post-collision.

We use the kinetic energy formula:

• \( KE = \frac{p'^2}{2m_e} \),

where \( p' \) is the electron's momentum after the collision, and \( m_e \) is its rest mass. This calculation provides us the electron's energy gained during the interaction.

Understanding this helps us grasp how energy is transferred between particles, which is a critical aspect of quantum interactions. By calculating the kinetic energy, we better understand the dynamics at play during Compton scattering.