Question

A stationary free electron in a gas is struck by a 6.37 -nm X-ray, which experiences an increase in wavelength of \(1.13 \mathrm{pm}\). How fast is the electron moving after the interaction with the X-ray?

Short answer

Answer: To find the final speed of the electron, follow these steps: 1. Calculate the final wavelength of the X-ray: \(\lambda_f = \lambda_i + \Delta\lambda\). 2. Use the Compton formula to relate the initial and final wavelengths and the scattering angle: \(\lambda_f - \lambda_i = \dfrac{h}{m_ec}(1-\cos{\theta})\). 3. Solve for the scattering angle, \(\theta\), in the above equation. 4. Calculate the conservation of momentum using the equation \(p_e = \dfrac{h}{\lambda_i} - \dfrac{h}{\lambda_f}\). 5. Use the energy-momentum relation to find the final speed of the electron: \(v = \dfrac{p_e}{m_e}\). Substitute the given initial wavelength, increase in wavelength, and proper constants into these equations to find the final speed of the electron after the interaction with the X-ray.

Step-by-step solution

Identify the known variables

In this problem, we are given: - Initial wavelength of X-ray: \(\lambda_i = 6.37 \ \text{nm}\) - Increase in wavelength: \(\Delta\lambda = 1.13 \mathrm{pm}= 1.13 \times 10^{-3} \ \text{nm}\)

Calculate the final wavelength of the X-ray

Since we know the initial wavelength and the increase in wavelength, we can calculate the final wavelength by adding the initial wavelength and the increase in wavelength: $$ \lambda_f = \lambda_i + \Delta\lambda $$

Use the Compton formula

In the Compton scattering phenomenon, the photon scatters from an electron, causing a shift in the photon's wavelength, given by the Compton formula: $$ \Delta\lambda = \dfrac{h}{m_ec}(1-\cos{\theta}) $$ Here, \(h\) is the Planck's constant, \(m_e\) is the mass of an electron, \(c\) is the speed of light, and \(\theta\) is the scattering angle. We are not given the value of \(\theta\), but we can rewrite this formula in terms of the initial and final wavelengths: $$ \lambda_f - \lambda_i = \dfrac{h}{m_ec}(1-\cos{\theta}) $$

Solve for the scattering angle, \(\theta\)

Rearrange the equation to isolate \(\cos{\theta}\) and solve for \(\theta\): $$ \cos{\theta} = 1 - (\lambda_f - \lambda_i)\dfrac{m_ec}{h} $$

Calculate the conservation of momentum

We know that the total momentum should be conserved in this interaction. The momentum conservation equation for Compton scattering is: $$ p_e = \dfrac{h}{\lambda_i} - \dfrac{h}{\lambda_f} $$ Here, \(p_e\) is the momentum of the electron after the interaction.

Calculate the final speed of the electron

Using the energy-momentum relation, we can find the final speed of the electron: $$ p_e = m_ev $$ Solve for \(v\): $$ v = \dfrac{p_e}{m_e} $$ Now, insert the values for the initial and final wavelengths and constants to calculate the final speed of the electron: $$ v = \dfrac{h}{m_e} \times \dfrac{(1/\lambda_i) - (1/\lambda_f)}{m_e-\lambda_i-\lambda_f} $$ Remember to use the appropriate units for the constants and wavelengths. The final answer will give you the speed of the electron after the interaction with the X-ray.