## The photon loses energy but does not slow down

From the laws of conservation of momentum and energy it is possible to calculate the momentum of the scattered particle in terms of the scattering angle, the masses of the projectile and target particles, and the momentum of the incoming particle.

incident photon

electron

Figure 13.8 A photon scatters off an electron.

scattered photon incident photon

In an elastic collision the photon scattered at an angled undergoes a uniquely defined energy loss.

electron

### Figure 13.8 A photon scatters off an electron.

The situation here is somewhat different from that in classical mechanics, in that the photon has zero mass and always travels with the speed c. The scattered photon will lose energy, but its speed remains the same. The energy loss, appears as a change in frequency of the photon (hf ^ hf'), and is transferred to the target particle as kinetic energy.

Compton assumed that a photon has:

hc energy = hf = — l hf h momentum = — = — c X

Assuming that there is an elastic collision of a photon ('cue ball') of energy hc/X, and momentum h/X with a stationary electron of mass me, we can calculate the change in wavelength of the photon for a given scattering angle from the relation:

Compton effect: change in wavelength of X-rays due to scattering by an electron

The derivation of this formula is given in Appendix 13.1.

Note that the scattered photon wavelength X'

incident photon wavelength X

scattered photon wavelength X'

incident photon wavelength X

stationary electron 0

recoil electron stationary electron 0

recoil electron change in wavelength is independent of the wavelength of the incident radiation. It depends inversely on the mass of the target particle, confirming that energy transfer is negligible for X-rays scattered from an atomic nucleus. The quantity h/mec is a universal constant called the Compton wavelength of the electron and has a value X = 2.44 x 10-12 m = 2.44 x 10-3 nm.

The maximum wavelength change occurs for d = 180°, i.e. a (unlikely) head-on collision in which the photon bounces straight back. Even then AX = 0.004885 nm, which is difficult to detect unless the original wavelength is itself small and the wavelength change forms a significant fraction of the original wavelength (another reason for using X-rays).

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