Question

An X-ray photon with an energy of \(50.0 \mathrm{keV}\) strikes an electron that is initially at rest inside a metal. The photon is scattered at an angle of \(45^{\circ} .\) What is the kinetic energy and momentum (magnitude and direction) of the electron after the collision? You may use the nonrelativistic relationship connecting the kinetic energy and momentum of the electron.

Short answer

Question: Calculate the kinetic energy and momentum (magnitude and direction) of an electron inside a metal after being struck by a 50.0 keV X-ray photon that scatters at a 45° angle with respect to the initial direction of the photon. Answer: To find the kinetic energy and momentum of the electron after the photon collision, follow these steps: 1. Calculate the initial momentum of the photon: \(p_{photon} \approx 5.33 \times 10^{-22} \mathrm{kg \cdot m/s}\). 2. Use the conservation of momentum to find the momentum components of the electron after the collision: \(p_{e_x} = p_{photon} - p_{x}\), and \(p_{e_y} = -p_{y}\). 3. Calculate the magnitude of the electron's momentum and use the nonrelativistic relationship to find the kinetic energy: \(K_e = \frac{p_e^2}{2m_e}\). 4. Determine the direction of the electron's momentum after the collision using the arctangent function: \(\theta = \arctan{\frac{p_{e_y}}{p_{e_x}}}\).

Step-by-step solution

Calculate the initial momentum of the photon

First, we need to find the initial momentum of the photon. The formula to find the momentum of a photon can be given as: \(p_{photon} = \frac{E_{photon}}{c}\), where \(E_{photon}\) is the energy of the photon and \(c\) is the speed of light. Plug in the given energy and the speed of light: \(p_{photon} = \frac{50.0 \times 10^3 \mathrm{ eV} \times 1.6 \times 10^{-19} J\,/\mathrm{eV}}{3 \times 10^8 \mathrm{m/s}}\approx 5.33 \times 10^{-22} \mathrm{kg \cdot m/s}\).

Use the law of conservation of momentum to find the momentum of the electron after the collision

According to the conservation of momentum, the total momentum before and after the collision should be equal. Let's denote the components of the momentum vectors of the scattered photon as \(p_{x}\) and \(p_{y}\) respectively. Using the conservation of momentum: Initial momentum = Final momentum \(p_{photon}\) = \(p_{x} + p_{y}\). Now, using the given angle of \(45°\), we can find the components of the final momentum: \(p_{x} = p_{photon} \cos{45^\circ}\) \(p_{y} = p_{photon} \sin{45^\circ}\) The electron's change in momentum in the horizontal direction must be equal to the photon's change in momentum in the horizontal direction; similar condition holds for the vertical direction as well. Therefore, the final momentum of the electron would be equal to the difference between the photon's initial and scattered momentum: \(p_{e_x} = p_{photon} - p_{x}\), and \(p_{e_y} = -p_{y}\) (Note: negative sign indicates the momentum direction due to the scattered photon).

Calculate the kinetic energy of the electron from its momentum

Now that we have the components of the electron's momentum, we can calculate the magnitude of its momentum and then find the kinetic energy using the nonrelativistic relationship. First, calculate the magnitude of the momentum vector: \(p_e = \sqrt{p_{e_x}^2 + p_{e_y}^2}\) Next, we use the relationship between kinetic energy and momentum for an electron, given by: \(K_e = \frac{p_e^2}{2m_e}\), where \(m_e\) is the mass of the electron (\(9.11 \times 10^{-31} \mathrm{kg}\)). Plug in the magnitude of the momentum, and the mass of the electron to find the kinetic energy.

Determine the direction of the electron's momentum after the collision

To find the direction of the electron's momentum after the collision, we can calculate the angle formed by the momentum vector with respect to the horizontal axis. We can use the arctangent function to find this angle: \(\theta = \arctan{\frac{p_{e_y}}{p_{e_x}}}\) Now, we have the direction of the electron's momentum after the collision. In summary, follow these steps to find the kinetic energy and momentum of the electron after the photon collision: 1. Calculate the initial momentum of the photon using its energy. 2. Use the conservation of momentum to find the momentum components of the electron after the collision. 3. Calculate the magnitude of the electron's momentum and use the nonrelativistic relationship to find the kinetic energy. 4. Determine the direction of the electron's momentum after the collision using the arctangent function.

Key concepts

Momentum Conservation

Momentum conservation is a foundational concept in physics, particularly when dealing with collisions. It states that the total momentum of a closed system remains constant if no external forces are acting on it. During a photon-electron collision, the momentum of the incoming photon is transferred to the electron, and because there is no outside force, the total momentum before and after the collision must be the same.

In practical terms, for the given exercise where an X-ray photon hits an electron, this principle allows us to equate the momentum of the photon before the collision to the combined momentum of the photon and electron after the collision. This lays the groundwork for solving problems by enabling us to predict the final momentums of the particles involved post-collision, provided we have the initial conditions.

Kinetic Energy Calculation

The kinetic energy calculation for an electron post-collision involves translating its momentum into a value which represents the energy of motion. Unlike momentum conservation, which must hold true in all scenarios, the kinetic energy of the system may not be conserved in inelastic collisions, such as when the X-ray photon is absorbed by the electron.

However, in this exercise scenario, we can assume an elastic collision where kinetic energy is conserved. To find the kinetic energy of the electron after collision with a photon, we can use the nonrelativistic formula:
\(K_e = \frac{p_e^2}{2m_e}\),
where \(p_e\) is the magnitude of the electron's momentum, and \(m_e\) is its mass. This formula provides a direct link between an electron's momentum and its kinetic energy, allowing for calculations of one if the other is known.

X-ray Photon Interactions

X-ray photon interactions with matter, including electrons, are essential for understanding phenomena such as X-ray imaging and radiation scattering. When an X-ray photon strikes an electron, several processes can occur: the photon can be absorbed, causing the electron to be ejected from its atom (photoelectric effect), or it can be scattered, changing direction and potentially transferring some of its energy to the electron (Compton scattering).

In the given exercise, the interaction is of the Compton scattering type. The energy and momentum transfer from the X-ray photon to the electron are crucial for determining the electron's kinetic energy and its new trajectory post-collision.

Nonrelativistic Energy-Momentum Relationship

The nonrelativistic energy-momentum relationship is applied when dealing with velocities much lower than the speed of light, which is suitable for most everyday scenarios, including our exercise. This relationship is especially handy for quickly determining kinetic energy from momentum without dealing with the individual velocities of particles.

Given by the formula:\[K = \frac{p^2}{2m}\], where \(K\) represents the kinetic energy, \(p\) the linear momentum, and \(m\) the mass, it provides a simplistic, yet accurate, calculation of kinetic energy for an electron that has been struck by a photon. Using this relation, students can solve for kinetic energy directly, bypassing the need to calculate the velocity of the electron, thus simplifying the problem-solving process significantly.