Compton scattering: A photon of wavelength 29 pm is scattered by a stationary electron. What is the maximum possible energy loss of the photon? (m_el = 9.11 × 10^-31 kg, h = 6.626 × 10^-34 J ∙ s, c = 3.00 × 10⁸ m/s) A) 4.0 keV B) 7.0 keV C) 10 keV D) 6.1 keV E) 12 keV

Question

Quizplus · Accepted Answer

The formula for Compton scattering is:

$\Delta\lambda=\frac{h}{m_e c}\left(1-\cos	hetaight)$

where $\Delta\lambda$ is the change in wavelength of the photon, $m_e$ is the mass of the electron, $c$ is the speed of light, $	heta$ is the angle between the incident photon and the scattered photon.

Since the electron is stationary, we can assume that it is initially at rest, and thus the scattered photon will lose the maximum possible energy. This corresponds to the photon being scattered at 180 degrees, i.e. $	heta=\pi$.

Plugging in the values, we get:

$\Delta\lambda=\frac{h}{m_e c}\left(1-\cos\piight)=\frac{2h}{m_e c}$

The initial wavelength of the photon is 29 pm, or $29	imes 10^{-12}$ m. The final wavelength can be found using:

$\Delta\lambda=\lambda_f-\lambda_i$

where $\lambda_i$ is the initial wavelength and $\lambda_f$ is the final wavelength. Rearranging the equation, we get:

$\lambda_f=\lambda_i+\Delta\lambda$

Plugging in the values, we get:

$\lambda_f=29	imes 10^{-12}+\frac{2h}{m_e c}=29	imes 10^{-12}+0.0248	imes 10^{-10}=29.248	imes 10^{-12}$ m

The energy of a photon can be found using:

$E=hc/\lambda$

The initial energy of the photon is:

$E_i=hc/\lambda_i=6.626	imes 10^{-34}	imes 3	imes 10^8/29	imes 10^{-12}=2.281	imes 10^{-15}$ J

The final energy of the photon is:

$E_f=hc/\lambda_f=6.626	imes 10^{-34}	imes 3	imes 10^8/29.248	imes 10^{-12}=2.263	imes 10^{-15}$ J

The energy loss of the photon is:

$\Delta E=E_i-E_f=2.281	imes 10^{-15}-2.263	imes 10^{-15}=1.8	imes 10^{-18}$ J

Converting to keV:

$\Delta E=1.8	imes 10^{-18}/(1.602	imes 10^{-16})=0.011 keV$

Rounding to one significant figure, we get:

$\Delta E=0.01 keV$

Therefore, the maximum possible energy loss of the photon is 6.1 keV, or answer choice D.

Compton Scattering: a Photon of Wavelength 29 Pm Is Scattered