Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Chapter 27, Problem 82P
To determine

The energy of the γ-rays scattered through θ=90° and θ=180°.

Expert Solution & Answer
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Answer to Problem 82P

The energy of the γ-rays scattered through θ=90° is 136keV, and that through θ=180° is 108keV_

Explanation of Solution

Given that the energy of the photons is 186keV.

In Compton scattering, photons scattered from a target have longer wavelengths than the incident photons. The Compton shift is given by the expression;

    Δλ=hmec(1cosθ)                                                                                                  (I)

Here, Δλ is the Compton shift, h is the Planck’s constant, c is the speed of light, me is the mass of the electron, θ is the scattering angle.

Write the expression for the wavelength of an incident photon.

    λ=hcE                                                                                                                    (II)

Here, E is the energy of the photon.

Write the expression for the wavelength of the scattered x-rays.

  λ=λ+Δλ                                                                                                            (III)

Here, λ is the wavelength of the photon scattered at angle θ, λ is the wavelength of the incident photon.

Write the expression for the energy of the scattered ray.

    E=hcλ                                                                                                                  (IV)

Here, E is the energy of the scattered photon.

Conclusion:

Substitute 90° for θ, and 2.426pm for hmec in equation (I) to find Δλ90°.

  Δλ90°=2.426pm(1cos90°)=2.426pm

Substitute 180° for θ, and 2.426pm for hmec in equation (I) to find Δλ180°.

  Δλ180°=2.426pm(1cos180°)=4.852pm

Substitute 1240eVnm for hc, 186keV for E in equation (II) to find λ.

  λ=1240eVnm186keV=(1240eVnm×1m1×109nm)(186keV×1000eV1keV)=6.667×1012m×1pm1×1012m=6.667pm

Substitute 6.667pm for λ, 2.426pm for Δλ in equation (III) to0 find λ90.

    λ90=6.667pm+2.426pm=9.09pm

Substitute 6.667pm for λ, 4.852pm for Δλ in equation (III) and solve for λ180°.

    λ180°=6.667pm+4.852pm=11.52pm

Substitute 1240eVnm for hc and 9.09pm for λ90° in equation (IV) to find E90°.

    E90°=1240eVnm9.09pm=1240eVnm9.09pm×1nm1000pm=1.36×105eV×1keV1000eV=136keV

Substitute 1240eVnm for hc and 11.52pm for λ180° in equation (IV) to find E180°.

    E180°=1240eVnm11.52pm=1240eVnm11.52pm×1nm1000pm=1.08×105eV×1keV1000eV=108keV

Therefore, the energy of the γ-rays scattered through θ=90° is 136keV, and that through θ=180° is 108keV_.

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Chapter 27 Solutions

Physics

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