An x-ray diffraction experiment using 16 -keV x-rays is repeated using electrons instead of \(x\) -rays. What should the kinetic energy of the electrons be in order to produce the same diffraction pattern as the x-rays (using the same crystal)?

Answer: To find the kinetic energy of electrons required to produce the same diffraction pattern as the x-rays, follow these steps: 1. Calculate the x-ray wavelength using the energy-wavelength relationship: \(\lambda_x = \frac{hc}{E_x}\) 2. Find the de Broglie wavelength of electrons with the same wavelength as the x-rays (\(\lambda_e = \lambda_x\)). 3. Determine the electron momentum using the de Broglie wavelength formula: \(p_e = \frac{h}{\lambda_e}\) 4. Relate electron momentum to its kinetic energy: \(K_e = \frac{p_e^2}{2m_e}\) 5. Calculate the kinetic energy of electrons in Joules and convert it to electron volts (eV): \(K_e (eV) = \frac{K_e (J)}{1.6 \times 10^{-19}\text{J/eV}}\) By following these steps and plugging in the given values and known constants, you can find the kinetic energy of electrons required to produce the same diffraction pattern as 16-keV x-rays using the same crystal.