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Chapter 27: Problem 22

An x-ray photon of wavelength 0.150 nm collides with an electron initially at rest. The scattered photon moves off at an angle of \(80.0^{\circ}\) from the direction of the incident photon. Find (a) the Compton shift in wavelength and (b) the wavelength of the scattered photon.

Short Answer

Expert verified
$$\Delta\lambda = 2.43 \times (1 - \cos{(80.0^\circ)})$$ $$\Delta\lambda \approx 2.43 \times (1 - 0.1736)$$ $$\Delta\lambda \approx 2.43 \times 0.8264$$ $$\Delta\lambda \approx 2.01\,\text{pm}$$ The Compton shift in wavelength is approximately \(2.01\,\text{pm}\).

Step by step solution

01

Understand the Compton scattering formula

The Compton scattering formula relates the initial photon wavelength \(\lambda\), the scattered photon wavelength \(\lambda'\), the scattering angle \(\theta\), and the Compton wavelength of an electron \(h/mc\) (\(h\): Planck's constant, \(m\): electron mass, \(c\): speed of light). The formula is given by: $$\lambda' - \lambda = \frac{h}{mc} \left(1 - \cos{\theta}\right)$$
02

Calculate the Compton shift in wavelength

Given the incident photon wavelength \(\lambda = 0.150\,\text{nm}\), the angle of the scattered photon \(\theta = 80.0^\circ\), and the Compton wavelength of the electron as \(\frac{h}{mc} \approx 2.43 \times 10^{-12}\,\text{m}\) or \(2.43\,\text{pm}\), we can find the wavelength shift, denoted as \(\Delta\lambda = \lambda' - \lambda\), by substituting the values into the Compton scattering formula: $$\Delta\lambda = \frac{2.43\,\text{pm}}{1\,\text{pm}} \left(1 - \cos{(80.0^\circ)}\right)$$ Calculate the Compton shift:

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