Problem 24 - University Physics with Modern Physics

Question

A photon with wavelength \(\lambda=0.1385 \mathrm{nm}\) scatters from an electron that is initially at rest. What must be the angle between the direction of propagation of the incident and scattered photons if the speed of the electron immediately after the collision is \(8.90 \times 10^{6} \mathrm{m} / \mathrm{s} ?\)

Step-by-Step Solution

Answer

The scattering angle between the incident and scattered photon is approximately 150.2 degrees.

Step 1: Find Initial Photon Energy

To determine the initial energy of the photon, use the formula for energy of a photon, which is \[ E_i = \frac{hc}{\lambda_i} \]where \( h = 6.63 \times 10^{-34} \, \text{J s} \) (Planck's constant), \( c = 3 \times 10^8 \, \text{m/s} \), and \( \lambda_i = 0.1385 \, \text{nm} = 0.1385 \times 10^{-9} \, \text{m}. \) Substitute these values to find \( E_i. \)

Step 2: Calculate Electron's Kinetic Energy

The electron's kinetic energy \( K \) immediately after collision is given by \[ K = \frac{1}{2}mv^2 \]where \( m = 9.11 \times 10^{-31} \text{ kg} \) is the mass of an electron, and \( v = 8.90 \times 10^{6} \text{ m/s}. \) Calculate \( K. \)

Step 3: Determine Scattered Photon Energy

The kinetic energy of the electron \( K \) is equal to the energy lost by the incident photon. Therefore, the energy of the scattered photon \( E_s = E_i - K \). Calculate \( E_s. \)

Step 4: Find Scattered Photon Wavelength

Using the energy of the scattered photon, determine its wavelength \( \lambda_s \) with the equation \[ \lambda_s = \frac{hc}{E_s} \]Substitute the values of \( h \), \( c \), and \( E_s \) to find \( \lambda_s. \)

Step 5: Apply Compton's Wavelength Shift Equation

Use Compton's equation which relates the change in wavelength to the angle \( \theta \): \[ \lambda_s - \lambda_i = \frac{h}{mc}(1 - \cos \theta) \]Substitute \( \lambda_s \), \( \lambda_i \), \( h \), \( m \), and \( c \) into the equation and solve for \( \cos \theta \).

Step 6: Solve for Scattering Angle

Finally, use the inverse cosine function to determine \( \theta \): \[ \theta = \cos^{-1}(\text{value obtained}) \]Calculate \( \theta \) which is the angle between the incident and scattered photon directions.

Key Concepts

Photon Energy

Photon energy is the energy carried by a photon, which is a particle of light. Unlike classical particles, photons are unique because their energy is directly linked to their wavelength. The relationship between energy and wavelength is described by the formula:
\[E_i = \frac{hc}{\lambda_i}\]where:

• \(E_i\) represents the initial photon energy.
• \(h = 6.63 \times 10^{-34} \, \text{J s}\) is Planck's constant, a fundamental constant in physics that relates the energy of a photon to its frequency.
• \(c = 3 \times 10^8 \, \text{m/s}\) is the speed of light, illustrating how quickly light travels in a vacuum.
• \(\lambda_i\) is the initial wavelength of the photon, converted into meters for calculation.

This equation shows that as the wavelength of a photon decreases, its energy increases. Thus, short-wavelength photons like X-rays and gamma rays possess high energy. Understanding the photon's energy is crucial when analyzing photon interactions such as scattering, as it determines how photons behave upon interacting with particles.

Kinetic Energy

Kinetic energy is the energy that a body possesses due to its motion. It's an essential concept in physics, particularly when understanding particle collisions. When photons scatter from electrons, they transfer some of their energy to the electrons, increasing the electrons' kinetic energy. This is described by the formula:
\[K = \frac{1}{2}mv^2\]where:

• \(K\) is the kinetic energy of the electron.
• \(m = 9.11 \times 10^{-31} \, \text{kg}\) is the mass of the electron.
• \(v\) is the velocity of the electron after the collision, given as \(8.90 \times 10^{6} \, \text{m/s}\).

By calculating the kinetic energy using this formula, we find out how much energy was transferred to the electron. The law of conservation of energy dictates that this energy is equivalent to the energy lost by the photon during scattering. Therefore, the change in the electron's kinetic energy helps us track the photon's energy dynamics in the scattering event.

Wavelength Shift

Wavelength shift is a phenomenon resulting from the interaction of photons with matter, specifically during Compton scattering. During this process, the wavelength of the scattered photon changes compared to its original wavelength.
This shift is quantitatively expressed by Compton's equation:
\[\lambda_s - \lambda_i = \frac{h}{mc}(1 - \cos \theta)\]where:

• \(\lambda_s\) is the wavelength of the scattered photon.
• \(\lambda_i\) is the initial wavelength of the incident photon.
• \(h\) is Planck's constant, symbolizing the photon's quantum nature.
• \(m\) is the mass of the electron.
• \(c\) is the speed of light.
• \(\theta\) is the scattering angle, or the angle between the incident and scattered directions.

The equation reveals that the wavelength shift depends on the angle \(\theta\) at which scattering occurs. Maximum shift is observed when the photon scatters backward at an angle of \(180^\circ\). By analyzing the wavelength shift, we can gain insights into the specifics of the photon's interaction with an electron and further understand photon behavior in the quantum realm.