Problem 33 - College Physics

Question

If a photon of wavelength \(0.04250 \mathrm{nm}\) strikes a free electron and is scattered at an angle of \(35.0^{\circ}\) from its original direction, find (a) the change in the wavelength of this photon, (b) the wavelength of the scattered light, (c) the change in energy of the photon (is it a loss or a gain?), and (d) the energy gained by the electron.

Step-by-Step Solution

Answer

(a) \( \Delta \lambda = 0.000115 \text{ nm}\), (b) \( \lambda' = 0.042615 \text{ nm}\), (c) Photon loses energy \( 4.35 \times 10^{-22} \text{ J}\), (d) Electron gains energy \( 4.35 \times 10^{-22} \text{ J}\).

Step 1: Understand Compton Scattering Formula

Compton scattering describes how a photon changes direction and loses energy when it collides with a free electron. The change in wavelength \( \Delta \lambda \) can be determined using the formula:\[ \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta) \]where \( \lambda' \) is the wavelength after scattering, \( \lambda \) is the initial wavelength, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ J s}) \), \( m_e \) is the mass of the electron \( (9.11 \times 10^{-31} \text{ kg}) \), \( c \) is the speed of light \( (3 \times 10^8 \text{ m/s}) \), and \( \theta \) is the scattering angle.

Step 2: Calculate Change in Wavelength \( \Delta \lambda \)

Firstly, calculate the Compton wavelength shift for the given angle: \( \theta = 35.0^\circ \). Convert the angle to radians: \( 35.0^\circ = 0.6109 \text{ radians} \). Then using the formula:\[ \Delta \lambda = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 3 \times 10^8} (1 - \cos 0.6109) \]Perform the calculation to find \( \Delta \lambda \approx 0.000115 \text{ nm} \).

Step 3: Determine Wavelength of Scattered Light

The wavelength of the scattered light \( \lambda' \) is the initial wavelength \( \lambda \) plus the change in wavelength \( \Delta \lambda \): \[ \lambda' = \lambda + \Delta \lambda = 0.04250 \text{ nm} + 0.000115 \text{ nm} = 0.042615 \text{ nm} \]

Step 4: Find Change in Photon's Energy

The energy of a photon is related to its wavelength by the equation \( E = \frac{hc}{\lambda} \). The change in energy \( \Delta E \) can be expressed as:\[ \Delta E = \frac{hc}{\lambda'} - \frac{hc}{\lambda} \]Substitute \( \lambda = 0.04250 \text{ nm} \) and \( \lambda' = 0.042615 \text{ nm} \) and solve to find \( \Delta E \approx -4.35 \times 10^{-22} \text{ J} \), indicating a loss.

Step 5: Calculate Energy Gained by Electron

The energy lost by the photon is gained by the electron. Thus, the energy gained by the electron is \( 4.35 \times 10^{-22} \text{ J} \). Since energy is conserved, the magnitude of energy gain for the electron is equal to the magnitude of energy loss for the photon.

Key Concepts

Photon Wavelength

Photon wavelength is a key feature that describes the distance between consecutive peaks of a photon's wave. In Compton scattering, understanding changes in the photon's wavelength helps us grasp how energy transfers during the electron-photon interaction. Photons can often be described by the wavelength of their electromagnetic wave. When a photon collides with a free electron, its wavelength generally increases meaning it moves towards the red end of the spectrum.
By conservation principles, the formula for the change in wavelength \[ \Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta) \]comes into play, where \(h\) is Planck's constant, \(m_e\) the electron mass, \(c\) the speed of light, and \(\theta\) the scattering angle. This increase is known as the Compton shift. Understanding this concept is essential to comprehend the broader part of quantum mechanics where photon behaviors are analyzed and quantified.

Energy Conservation

Energy conservation is a fundamental principle in physics, stating that energy cannot be created or destroyed; it can only change form. Within Compton scattering, energy conservation plays a significant role.
During the process, when a photon hits an electron, it loses a portion of its energy, and this loss is equivalent to the gain of energy by the electron. The energy of photons before and after a collision can be calculated by \[ E = \frac{hc}{\lambda} \]where \(E\) is energy, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength. When \[ \Delta E = \frac{hc}{\lambda'} - \frac{hc}{\lambda} \]is calculated, the difference reveals whether energy is gained or lost. In this case, the photon's energy decreases, showing a transfer of energy to the electron. This outcome aligns perfectly with the principle of energy conservation.

Scattering Angle

In Compton scattering, the scattering angle \(\theta\) is the angle at which the photon deviates from its original path due to the collision with an electron. The scattering angle is vital in determining how much the wavelength changes. It is within this scattered movement that crucial calculations take root to establish the degree of change in wavelength.

• At \(0\,^{\circ}\), no scattering and thus no wavelength change occurs.
• At \(180\,^{\circ}\), maximum scattering occurs, resulting in a more substantial wavelength increase.

The Compton formula uses the scattering angle to predict changes in the photon's wavelength by including \( \cos \theta \). Understanding this angle helps scientists determine not just the trajectory alteration, but also the energy shifts during such interactions.

Electron-Photon Interaction

Electron-photon interaction is the dynamic between a light particle (photon) and an electron, typically a fast-moving process that leads to energy transformation. When a photon collides with an electron, such interactions fall under quantum mechanics, specifically embodied in Compton scattering.
During the interaction:

• The photon hits a stationary or slow-moving electron.
• Some of the photon's energy transfers to the electron.
• As a result, the photon changes its direction and increases in wavelength.

These interactions are valuable for understanding phenomena in fields like medical imaging, where X-rays are involved, and in the exploration of cosmic events, allowing physicists to grasp deeper insights into subatomic particles' behaviors. Understanding this interaction alone can significantly enhance appreciation of quantum fields and particle studies.