Problem 61 - University Physics with Modern Physics

Question

An incident x-ray photon of wavelength 0.0900 \(\mathrm{nm}\) 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 momentum of the scattered photon is approximately \( 6.99 \times 10^{-24} \ \mathrm{kg \cdot m/s} \). (b) The kinetic energy of the electron is \( 1.1 \times 10^{-16} \ \mathrm{J} \).

Step 1: Calculate the Change in Wavelength

The Compton wavelength shift formula is given by \[ \Delta \lambda = \lambda' - \lambda = \dfrac{h}{m_e c} (1 - \cos \theta) \]where \( \lambda \) is the initial wavelength, \( \lambda' \) is the wavelength after scattering, \( h = 6.63 \times 10^{-34} \ \mathrm{m^2 \ kg/s} \) is Planck's constant, \( m_e = 9.11 \times 10^{-31} \ \mathrm{kg} \) is the electron mass, \( c = 3 \times 10^8 \ \mathrm{m/s} \) is the speed of light, and \( \theta = 180^\circ \) for backward scattering. Solving for \( \Delta \lambda \):\[ \Delta \lambda = \dfrac{6.63 \times 10^{-34}}{9.11 \times 10^{-31} \times 3 \times 10^8} (1 - (-1)) = 4.86 \times 10^{-12} \ \mathrm{m} \].

Step 2: Find the New Wavelength of the Photon

To find the new wavelength, add the change in wavelength to the initial wavelength:\[ \lambda' = \lambda + \Delta \lambda = 0.0900 \ \mathrm{nm} + 4.86 \times 10^{-3} \ \mathrm{nm} = 0.09486 \ \mathrm{nm} \].

Step 3: Compute the Momentum of the Scattered Photon

The momentum \( p \) of a photon is given by:\[ p = \dfrac{h}{\lambda'} \].Substitute \( \lambda' = 0.09486 \ \mathrm{nm} = 0.09486 \times 10^{-9} \ \mathrm{m} \): \[ p = \dfrac{6.63 \times 10^{-34}}{0.09486 \times 10^{-9}} = 6.99 \times 10^{-24} \ \mathrm{kg \cdot m/s} \].

Step 4: Calculate the Initial Energy of the X-ray Photon

The energy \( E \) of a photon can be calculated with:\[ E = \dfrac{hc}{\lambda} \].Substitute \( \lambda = 0.0900 \times 10^{-9} \ \mathrm{m} \): \[ E = \dfrac{6.63 \times 10^{-34} \times 3 \times 10^8}{0.0900 \times 10^{-9}} = 2.21 \times 10^{-15} \ \mathrm{J} \].

Step 5: Calculate the Energy of the Scattered Photon

Using the new wavelength \( \lambda' = 0.09486 \times 10^{-9} \ \mathrm{m} \), find the energy of the scattered photon:\[ E' = \dfrac{hc}{\lambda'} = \dfrac{6.63 \times 10^{-34} \times 3 \times 10^8}{0.09486 \times 10^{-9}} = 2.10 \times 10^{-15} \ \mathrm{J} \].

Step 6: Find the Kinetic Energy of the Recoil Electron

The kinetic energy \( KE \) of the electron is the energy lost by the photon, or:\[ KE = E - E' = 2.21 \times 10^{-15} - 2.10 \times 10^{-15} = 1.1 \times 10^{-16} \ \mathrm{J} \].

Key Concepts

Photon Momentum

Photons, unlike objects with mass, carry momentum without having mass. This is a unique property that distinguishes them in the realm of physics. The momentum of a photon is calculated using the equation

• \( p = \frac{h}{\lambda} \),

where h is Planck's constant, and \(\lambda\) is the wavelength of the photon.
The momentum is directly related to how the photon behaves during scattering events, such as Compton Scattering, where its wavelength, and therefore its momentum, changes. Understanding photon momentum is essential to describing how light interacts with matter.
In Compton scattering scenarios, a photon's momentum before and after scattering helps determine the kinetic energy transferred to electrons, influencing their motion and energy. This interaction is core to demonstrating both the particle and wave nature of light.

Kinetic Energy of Electron

When a photon scatters, it transfers some of its energy to an electron, giving the electron kinetic energy. Initially, an electron may be at rest, so any energy it gains leads to motion. The kinetic energy after scattering can be calculated by finding the difference between the initial and final photon energies:

• \( KE = E - E' \)

where \( E \) is the initial energy and \( E' \) the energy after scattering.
This equation is essential in determining the energy budget in photon-electron interactions.
The kinetic energy provides insights into the internal processes occurring at an atomic scale, illustrating how electrons behave when exposed to energetic radiation.
When solving problems involving Compton Scattering, this energy transfer concept is pivotal in comprehending how high-energy photons interact with matter. Hence, accurately calculating the initial and final energies can shed light on electron behavior post-collision.

Wavelength Shift

Wavelength shift is a phenomenon that occurs when photons collide with particles such as electrons. This process is quantified by the Compton wavelength shift formula:

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

Where \( \lambda \) is the original wavelength, m_e is the electron mass, \(c\) is the speed of light, and \( \theta \) is the scatter angle.
This formula calculates how much the wavelength changes after the scattering event. In backward scattering, \( \theta \) is \(180^\circ\), maximizing the wavelength change.
The shift in wavelength is a crucial indicator of the scattering interaction's specifics, affecting photon energy and momentum. Recognizing the direction and magnitude of this change can reveal underlying physical dynamics and corroborate the particle nature of light, as photons react conspicuously different than waves in such interactions.

X-ray Scattering

X-ray scattering is a key mechanism for probing the structural details of materials. In Compton scattering, which is a subset of X-ray scattering, photons are deflected by electrons. This process is governed by the principles of energy and momentum conservation, allowing us to study the microscopic behavior of materials.
X-ray photons differ from visible light due to their extremely short wavelengths and high energy, making them ideal for exploring arrangements within atomic structures. When an X-ray photon undergoes scattering, the change in both angle and energy offers detailed insights into the material it interacts with.
This kind of scattering is not only vital for fundamental research but also has practical applications in fields such as medical imaging and crystallography, helping to map structures at nanoscales.