QUESTION:

Consider Compton scattering of a photon by a moving electron. Before the collision the photon has wavelength \(\lambda\) and is moving in the \(+x\)-direction, and the electron is moving in the \(-x\)-direction with total energy E (including its rest energy \(m c^{2}\)). The photon and electron collide head-on. After the collision, both are moving in the -direction (that is, the photon has been scattered by \(180^{\circ}\)).  (a) Derive an expression for the wavelength \(\lambda^{\prime}\) of the scattered photon. Show that if \(E>>m c^{2}\), where  is the rest mass of the electron, your result reduces to

                    \(\lambda^{\prime}=\frac{h c}{E}\left(1+\frac{m^{2} c^{4} \lambda}{4 h c E}\right)\)

(b) A beam of infrared radiation from a \(\mathrm{CO}_{2}\) laser (\(\lambda=10.6\) \(\mu \mathrm{m}\)) collides head-on with a beam of electrons, each of total energy \(E=10.0\) GeV(1 GeV = \(10^{9}\) eV). Calculate the wavelength \(\lambda^{\prime}\) of the scattered photons, assuming a  \(180^{\circ}\) scattering angle. (c) What kind of scattered photons are these (infrared, microwave, ultraviolet, etc.)? Can you think of an application of this effect?

Equation Transcription:

Text Transcription:

lambda

+x

-x

mc^2

-x

180deg

lambda'

E>>mc2

lambda'=hc over E(1+m^2c^4lambda over 4hcE)

CO_2

lambda=10.6

mu m

E=10.0

10^9

lambda'

180deg