Question

A free electron in a gas is struck by an \(8.5-\mathrm{nm} \mathrm{X}\) -ray, which experiences an increase in wavelength of \(1.5 \mathrm{pm} .\) How fast is the electron moving after the interaction with the X-ray?

Short answer

Final speed of the electron: approximately 587 m/s

Step-by-step solution

Identify the information given in the problem

The initial wavelength of the X-ray, \(\lambda_i\), is 8.5 nm and the change in wavelength, \(\Delta\lambda\), is 1.5 pm. We need to find the final speed of the electron, \(v_f\), after it interacts with the X-ray.

Convert to standard units

We'll convert the given wavelength measurements to standard SI units, i.e., meters. \(\lambda_i = 8.5 \ \text{nm} = 8.5 \times 10^{-9} \ \text{m}\), and \(\Delta\lambda = 1.5 \ \text{pm} = 1.5 \times 10^{-12} \ \text{m}\).

Calculate the final wavelength of the X-ray

To calculate the final wavelength of the X-ray, \(\lambda_f\), we add the change in wavelength to the initial wavelength. \(\lambda_f = \lambda_i + \Delta\lambda = 8.5 \times 10^{-9} \ \text{m} + 1.5 \times 10^{-12} \ \text{m} = (8.5 + 0.0015) \times 10^{-9} \ \text{m} = 8.5015 \times 10^{-9} \ \text{m}\).

Apply the Compton effect formula

The Compton effect formula is \(\Delta\lambda = \dfrac{h}{m_e c}(1 - \cos{\theta})\), where \(h\) is the Planck's constant, \(m_e\) is the mass of the electron, \(c\) is the speed of light, and \(\theta\) is the angle between the initial and final X-ray directions. Since we are interested in finding the speed of the electron after the interaction, we can use the conservation of momentum to relate the change in wavelength with the final speed of the electron: \(\Delta\lambda = \dfrac{h}{m_e c}(1 - \cos{\theta})\) and \(p_f = m_e v_f = h(\dfrac{1}{\lambda_i} - \dfrac{1}{\lambda_f})\).

Solve for the final speed of the electron

We can rewrite the momentum conservation equation in terms of the final speed of the electron: \(v_f = \dfrac{h}{m_e}(\dfrac{1}{\lambda_i} - \dfrac{1}{\lambda_f})\). Now, we can substitute the values of \(h\), \(m_e\), \(\lambda_i\), and \(\lambda_f\) to solve for the final speed of the electron: \(v_f = \dfrac{(6.626 \times 10^{-34} \ \text{J s})}{(9.109 \times 10^{-31} \ \text{kg})}(\dfrac{1}{(8.5 \times 10^{-9} \ \text{m})} - \dfrac{1}{(8.5015 \times 10^{-9} \ \text{m})})\). Calculating the final speed of the electron, we obtain: \(v_f \approx 587 \ \text{m/s}\). The electron is moving at approximately 587 m/s after interacting with the X-ray.

Key concepts

X-ray Scattering

X-ray scattering, a process fundamental to understanding the Compton effect, involves the interaction of X-rays with free electrons or atoms. As X-rays pass through a material, they may collide with electrons and scatter in different directions. This scattering changes the X-rays' energy and direction, which is closely linked to the behavior of the electrons in the material.

During this scattering event, physical properties such as wavelength and momentum can be altered. The change in wavelength, known as the Compton shift, is a key attribute of this effect and is observed as a difference between the incoming and outgoing wavelengths of the X-rays. The calculations often involve understanding how this shift relates to the change in energy and speed of the electrons, as seen in the worked out exercise.

Momentum Conservation

Momentum conservation is a principle in physics stating that within an isolated system, the total momentum remains constant if there are no external forces acting on it. In the context of the Compton effect, we consider the collision between an X-ray and a free electron to be such an isolated system.

When an X-ray photon collides with an electron, their interaction must adhere to the conservation of momentum. This means the combined momentum of the photon and the electron before the collision must equal their combined momentum afterward. Using this principle allows us to create equations that relate the change in wavelength of the X-ray to the momentum—and consequently the speed—of the electron after the collision, helping us solve for unknown variables in problems like the one we're examining. The steps outlined in the solution thoroughly apply momentum conservation to deduce the electron's final speed after being hit by an X-ray.

Electron Speed

Electron speed after an X-ray interaction is an integral part of analyzing the Compton effect. It refers to the velocity an electron gains as a result of the energy transferred to it by the X-ray photon during the scattering process. Knowing the electron's speed can give insights into the amount of energy exchanged during this interaction.

By computing the change in X-ray wavelength and utilizing the laws of momentum conservation, we can establish a direct relationship between the speed of the electron and the change in the X-ray's characteristics. This enables the solution to the original exercise, where the requirement was to find the final speed of the electron after collision with an X-ray, given the change in the X-ray wavelength. Using the Compton effect formula, in conjunction with momentum conservation principles, you can determine how fast an electron is moving after the event, vital for understanding the dynamics of X-ray scattering and the behavior of electrons under high-energy conditions.