Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Question
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Chapter 27, Problem 85P
To determine

The wavelengths of incident and scattered x-rays.

Expert Solution & Answer
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Answer to Problem 85P

The wavelengths of incident and scattered x-rays are 171pm and 176pm.

Explanation of Solution

Write the relation for the Compton shift in wavelength of x-ray.

  λλ=hmec(1cosθ)                                                                                            (1)

Here, λ is wavelength of the scattered x-ray, λ is wavelength of the incident x-ray, h is Planck’s constant, me is mass of the electron, c is speed of the x-ray photon, and θ is scattering angle.

The term h/mec in equation (1) is called Compton wavelength has the value of 2.426pm.

Rearrange equation (1) for the wavelength of the scattered x-ray.

  λ=λhmec(1cosθ)                                                                                            (2)

Write the relation for the kinetic energy of the electron.

  Ke=EE                                                                                                              (3)

Here, Ke is kinetic energy of the electron, E is kinetic energy of incident x-ray photon, and E is kinetic energy of scattered x-ray photon.

The change in kinetic energy of the x-ray photon is the kinetic energy of the electron.

Substitute hc/λ for E and hc/λ for E in equation (3) and simplify.

  Ke=hcλhcλKehc=1λ1λ                                                                                                           (4)

Substitute equation (2) in (4) to find λ.

  Kehc=1[λhmec(1cosθ)]1λ

Substitute 2.426pm for h/mec and 180° for θ, 0.20keV for Ke, 1240eVnm for hc to find λ.

  (0.20keV)(103eV1keV)(1240eVnm)(103pmnm)=1[λ(2.426pm)(1cos180°)]1λλ2(4.852pm)λ=3.008×104pm2λ2(4.852pm)λ3.008×104pm2=0                            (5)

Solve the quadratic equation in (5) to find λ.

  λ=(4.852pm)±(4.852pm)2(4)(1)(3.008×104pm2)(2)(1)=176pm,171pm

The value of wavelength of scattered x-ray photon is 176pm and the negative solution is not taken into account due to the wavelength never be negative.

Rewrite equation (2) for the wavelength of the incident x-ray photon.

  λ=λhmec(1cosθ)

Substitute 2.426pm for h/mec and 180° for θ, and 176pm for λ to find λ.

  λ=(176p)m(2.426pm)(1cos180°)=171pm

Thus, the wavelength of the incident x-ray photon is 171pm.

Conclusion:

Therefore, the wavelengths of incident and scattered x-rays are 171pm and 176pm.

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Chapter 27 Solutions

Physics

Ch. 27.7 - Prob. 27.7PPCh. 27.7 - Prob. 27.8PPCh. 27.8 - Prob. 27.9PPCh. 27 - Prob. 1CQCh. 27 - Prob. 2CQCh. 27 - Prob. 3CQCh. 27 - Prob. 4CQCh. 27 - Prob. 5CQCh. 27 - Prob. 6CQCh. 27 - Prob. 7CQCh. 27 - Prob. 8CQCh. 27 - Prob. 9CQCh. 27 - Prob. 10CQCh. 27 - Prob. 11CQCh. 27 - Prob. 12CQCh. 27 - Prob. 13CQCh. 27 - Prob. 14CQCh. 27 - Prob. 15CQCh. 27 - Prob. 16CQCh. 27 - Prob. 17CQCh. 27 - Prob. 18CQCh. 27 - Prob. 19CQCh. 27 - Prob. 20CQCh. 27 - Prob. 21CQCh. 27 - Prob. 22CQCh. 27 - Prob. 23CQCh. 27 - Prob. 1MCQCh. 27 - Prob. 2MCQCh. 27 - Prob. 3MCQCh. 27 - Prob. 4MCQCh. 27 - Prob. 5MCQCh. 27 - Prob. 6MCQCh. 27 - Prob. 7MCQCh. 27 - Prob. 8MCQCh. 27 - Prob. 9MCQCh. 27 - Prob. 10MCQCh. 27 - Prob. 1PCh. 27 - Prob. 2PCh. 27 - Prob. 3PCh. 27 - Prob. 4PCh. 27 - Prob. 5PCh. 27 - Prob. 6PCh. 27 - Prob. 7PCh. 27 - Prob. 8PCh. 27 - Prob. 9PCh. 27 - Prob. 10PCh. 27 - Prob. 11PCh. 27 - Prob. 12PCh. 27 - Prob. 13PCh. 27 - Prob. 14PCh. 27 - Prob. 15PCh. 27 - Prob. 16PCh. 27 - Prob. 17PCh. 27 - Prob. 18PCh. 27 - Prob. 19PCh. 27 - Prob. 20PCh. 27 - Prob. 21PCh. 27 - Prob. 22PCh. 27 - Prob. 23PCh. 27 - Prob. 24PCh. 27 - Prob. 25PCh. 27 - Prob. 26PCh. 27 - Prob. 27PCh. 27 - Prob. 28PCh. 27 - Prob. 29PCh. 27 - Prob. 30PCh. 27 - Prob. 31PCh. 27 - Prob. 32PCh. 27 - Prob. 33PCh. 27 - Prob. 34PCh. 27 - Prob. 35PCh. 27 - Prob. 36PCh. 27 - Prob. 37PCh. 27 - Prob. 38PCh. 27 - Prob. 39PCh. 27 - Prob. 40PCh. 27 - Prob. 41PCh. 27 - Prob. 42PCh. 27 - Prob. 43PCh. 27 - Prob. 44PCh. 27 - Prob. 45PCh. 27 - Prob. 46PCh. 27 - Prob. 47PCh. 27 - Prob. 48PCh. 27 - Prob. 49PCh. 27 - Prob. 50PCh. 27 - Prob. 51PCh. 27 - Prob. 52PCh. 27 - Prob. 53PCh. 27 - Prob. 54PCh. 27 - Prob. 55PCh. 27 - Prob. 56PCh. 27 - Prob. 57PCh. 27 - Prob. 58PCh. 27 - Prob. 59PCh. 27 - Prob. 61PCh. 27 - Prob. 60PCh. 27 - Prob. 62PCh. 27 - Prob. 64PCh. 27 - Prob. 63PCh. 27 - Prob. 66PCh. 27 - Prob. 65PCh. 27 - Prob. 68PCh. 27 - Prob. 67PCh. 27 - Prob. 70PCh. 27 - Prob. 69PCh. 27 - Prob. 72PCh. 27 - Prob. 71PCh. 27 - Prob. 74PCh. 27 - Prob. 73PCh. 27 - Prob. 75PCh. 27 - Prob. 76PCh. 27 - Prob. 77PCh. 27 - Prob. 78PCh. 27 - Prob. 79PCh. 27 - Prob. 80PCh. 27 - Prob. 82PCh. 27 - Prob. 81PCh. 27 - Prob. 84PCh. 27 - Prob. 83PCh. 27 - Prob. 86PCh. 27 - Prob. 85PCh. 27 - Prob. 88PCh. 27 - Prob. 87PCh. 27 - Prob. 89PCh. 27 - Prob. 90PCh. 27 - Prob. 91PCh. 27 - Prob. 92PCh. 27 - Prob. 93PCh. 27 - Prob. 94PCh. 27 - Prob. 95PCh. 27 - Prob. 96P
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