X-rays of frequency 1.210 ×10^18 Hz are scattered from electrons in an aluminum foil. The frequ

step 1 In this question, it is about content scattering. So the x -ray is shining on aluminum foil, and

X-rays of frequency 1.210 ×10^18 Hz are scattered from...

X-rays of frequency $1.210 \times 10^{18} \mathrm{~Hz}$ are scattered from electrons in an aluminum foil. The frequency of the scattered X-rays is $1.203 \times 10^{18} \mathrm{~Hz}$. (a) What is the scattering angle?
(b) What is the recoiling speed of the electrons?

Key Concepts

Compton Scattering

This concept explains that when high?energy photons (such as X?rays) collide with electrons, the photons lose energy and change wavelength depending on the scattering angle. The phenomenon is governed by the Compton formula, which relates the change in wavelength (or equivalently, frequency shift) to the scattering angle via the Compton wavelength of the electron.

Wavelength–Frequency Relationship

Since the speed of light is constant, the wavelength and frequency of electromagnetic radiation are inversely related through the equation c = ??. This relationship allows the conversion between frequency and wavelength, which is essential when applying the Compton scattering equation that is naturally expressed in terms of wavelength.

Photon Energy and Momentum

Photons carry energy and momentum that are given by the relations E = h? and p = E/c, respectively. These expressions are crucial in analyzing the energy and momentum transfer during the scattering process, as well as in determining the recoil speed of the electrons.

Conservation of Energy and Momentum

In any scattering event, the total energy and momentum before and after the collision must be conserved. This principle is used to analyze the collision between the photon and the electron: the energy lost by the photon appears as an increase in the electron’s kinetic energy. This conservation principle is key to determining outcomes such as the electron’s recoil speed.

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