X-rays of wavelength 0.20 nm are scattered by a free electron.The change in the wavelength of the x-rays is observed to be 2.0 × 10^-12 m at a certain scattering angle measured relative to the incoming x-ray direction.What is the scattering angle of the x-rays? (m_electron = 9.11 × 10^-31 kg,c = 3.00 × 10⁸ m/s,h = 6.626 × 10^-34 J ∙ s)

Question

Quizplus · Accepted Answer

The change in wavelength ($\Delta \lambda$) due to the scattering of X-rays by a free electron can be calculated using the Compton scattering formula: $\Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta)$, where $h$ is Planck's constant, $m_e$ is the mass of the electron, $c$ is the speed of light, and $\theta$ is the scattering angle. Given $\Delta \lambda = 2.0 \times 10^{-12} m$, $h = 6.626 \times 10^{-34} J \cdot s$, $m_e = 9.11 \times 10^{-31} kg$, and $c = 3.00 \times 10^8 m/s$, we can solve for $\theta$.Rearranging the formula to solve for $\theta$, we get $\theta = \cos^{-1} \left(1 - \frac{\Delta \lambda \cdot m_e \cdot c}{h}\right)$.Substituting the given values, $\theta = \cos^{-1} \left(1 - \frac{2.0 \times 10^{-12} \cdot 9.11 \times 10^{-31} \cdot 3.00 \times 10^8}{6.626 \times 10^{-34}}\right)$.This calculation yields a value close to 80°, making option D the correct answer.

X-Rays of Wavelength 0