The Compton effect was studied in 1922 and was additional evidence of the wave particle duality of photons. It was performed using x-rays, high-frequency electromagnetic radiation, scattered from electrons that are bound in atoms. For this work Arthur Holly Compton was awarded the Nobel Prize in Physics in 1927 the citation for which read “for his discovery of the effect named after him.” Because the Compton effect is of considerable importance we will derive the result.

Figure 1.4 shows a schematic diagram of the scattering process. A photon of frequency ν is incident on an electron at rest. The electron is not actually at rest, but its kinetic energy is small compared with the energy of the x-rays. The initial momentum of the photon is pp = hν/c where c is the speed of light. The photon is assumed scattered at an angle θ with momentum p'p = hν'/c where ν' is the frequency of the scattered photon. The momentum of the scattered electron is p ' , a vector. We wish to find the wavelength λ' = c/ν' of the

scattered photon.

Fig. 1.4 The kinematics of Compton scattering |

where me is the mass of the electron. We (necessarily) used the relativistic formula for the energy of the electron.We can isolate in Equation 1.11 by squaring:

To eliminate we note from Fig. 1.4 that

Substituting Equation 1.13 into Equation 1.12 and writing the resulting equation in terms of the difference in wavelengths between the incident and scattered photons we have

where λc is known as the Compton wavelength of the electron:

Equation 1.15 can be put in another form by multiplying the numerator and denominator by c, the speed of light. The denominator is thus the rest mass of the electron, 0.51 × 106 MeV, while the numerator is 1240 (see Equation 1.10). Equation 1.14 is known as the Compton equation. One of the remarkable features of it is that the change in wavelength of the photon does not depend upon its incident wavelength. The maximum difference in wavelength that can be detected is twice (when θ = π) the Compton wavelength, ∼ 5 × 10−3nm. For this reason it is very difficult to perform Compton scattering experiments using visible light (λ ≈ 400−700 nm) because the λ would be only a tiny fraction of the wavelength of the incident photon wavelength. For much shorter wavelengths, as short as ∼ λc, however, λ/λ can be large enough to measure. Thus, an incident photon of energy comparable with the rest energy of the electron, ∼ 500 keV, is required. Compton

used x-rays having wavelength 0.071nm, roughly 17 keV in his experiments.While 17 keV is more than an order of magnitude lower than the rest mass of the electron, the effect was indeed detectable.

The Compton wavelength is often seen written as

The reason for this is that the actual value of the Compton wavelength is not really important. It is the order of magnitude of it that is significant. This will be discussed later in this chapter.

Equation 1.14 shows that the wavelength of the scattered photon is always longer than the wavelength of the incident photon because cos θ is always less than unity. Thus, λ > 0. The process can thus be envisioned as one in which the photon is elastically scattered by the electron, imparting momentum and kinetic energy to the electron. Because conservation of energy dictates that the photon loses energy, it must, in accord with Equations 1.2, have lower frequency and longer wavelength.

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reference : C.E. Burkhardt, J.J. Leventhal, Foundations of Quantum Physics, 1

DOI: 10.1007/978-0-387-77652-1 1, C Springer Science+Business Media, LLC 2008

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