23.2 Compton's Equation

As noted on page 2, the change in wavelength of the incident photon varies depending on the scattered angle.  Using algebra and Einstein's theory of relativity, the Compton scattering can be analyzed mathematically with the following equation:

 

The change in wavelength of the X-ray before and after the collision is related to the angle at which it scatters.

Note: change in wavelength ∆ λ = λ f - λ i Quantity Symbol SI Unit wavelength λ m Planck's constant h 6.63 × 10 -34 J*s mass of an electron m 9.11 × 10 31 kg scattering angle-the angle between the incident ray and the scattered ray θ degrees

 

 

 

 Compton Scattering Video

Energy of a Photon

 

So far you've been working with two forms of the equation to find a photon's energy: and .  By rearranging the momentum formula above, you now have a third equation for a photon's energy: , where p is the momentum of the photon in kg$·$m/s and c is the speed of light in a vacuum.

 

 

Conservation of Energy

 

The collision observed in Compton's experiment is perfectly elastic, so the kinetic energy is conserved.

 

Recall : Most collisions observed in real life are not perfectly elastic.  In these collisions some of the initial kinetic energy is transformed into other forms, such as heat and sound.  In collisions that are not perfectly elastic, E _{k final} E _{k initial} .

 

The energy of the incident X-ray photon is equal to the energy of the scattered X-ray photon plus the kinetic energy of the recoil electron.

Quantity Symbol SI Unit momentum p kg$·$m/s speed of light in a vacuum c 3.00 × 10 8 m/s mass of an electron m 9.11 × 10 -31 kg velocity of the recoil electron v m/s