$\mathrm { X }$-rays of wavelength $0.20 \mathrm {~nm}$ are scattered by a free electron. The change in the wavelength of the x-rays is observed to be $2.0 \times 10 ^ { - 12 } \mathrm {~m}$ at a certain scattering angle measured relative to the incoming x-ray direction. What is the scattering angle of the $x$-rays? $\left( m _ { \text {electron } } = 9.11 \times 10 ^ { - 31 } \mathrm {~kg} \right.$, $\left. c = 3.00 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } , h = 6.626 \times 10 ^ { - 34 } \mathrm {~J} \cdot \mathrm { s } \right)$
A) $60 ^ { \circ }$
B) $40 ^ { \circ }$
C) $20 ^ { \circ }$
D) $80 ^ { \circ }$

Question

$\mathrm { X }$-rays of wavelength $0.20 \mathrm {~nm}$ are scattered by a free electron. The change in the wavelength of the x-rays is observed to be $2.0 \times 10 ^ { - 12 } \mathrm {~m}$ at a certain scattering angle measured relative to the incoming x-ray direction. What is the scattering angle of the $x$-rays? $\left( m _ { \text {electron } } = 9.11 \times 10 ^ { - 31 } \mathrm {~kg} \right.$, $\left. c = 3.00 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } , h = 6.626 \times 10 ^ { - 34 } \mathrm {~J} \cdot \mathrm { s } \right)$ 
A) $60 ^ { \circ }$
B) $40 ^ { \circ }$
C) $20 ^ { \circ }$
D) $80 ^ { \circ }$

Quizplus · Accepted Answer

The change in wavelength ($\Delta \lambda$) due to the scattering of X-rays by an 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 $\Delta \lambda = 2.0 \times 10^{-12} \, \text{m}$, $h = 6.626 \times 10^{-34} \, \text{J} \cdot \text{s}$, $m_e = 9.11 \times 10^{-31} \, \text{kg}$, and $c = 3.00 \times 10^8 \, \text{m/s}$, solving for $\theta$ yields a value close to $80^\circ$.

X\mathrm { X }X -Rays of Wavelength 0.20 nm0.20 \mathrm {~nm}0.20 nm Are Scattered by a Free Electron

$\mathrm { X }$ -Rays of Wavelength $0.20 \mathrm {~nm}$ Are Scattered by a Free Electron