PHY 102: Waves & Quanta

Topic 10The Compton Effect

John Cockburn (j.cockburn@... Room E15)

Photons: summary so far

•Einstein postulated the existence of a particle called a photon, to explain detailed results of photoelectric experiment.

hc

hfEp

•Photon has zero rest mass, travels at speed of light

•Explains “instantaneous” emission of electrons in photoelectric effect, frequency dependence.

•Further confirmation of the photon picture provided by the COMPTON EFFECT (1922-23)…………………

A little bit about relativity……

•Einstein’s Special Theory of Relativity, 1905:

•The laws of physics are the same in every inertial frame of reference (in which Newton’s first law is valid)

•The speed of light in a vacuum is the same in all inertial frames of reference, and is independent of the motion of the source.

(corollary: the velocity of light can’t be exceeded)

Many important consequences: length contraction, time dilation effects at high speeds, mass/energy equivalence……

A little bit about relativity……

Momentum and kinetic energy at high speeds…………….

Relativistic expressions for energy and momentum

2

2

2

1cv

mcE

(energy of stationaryparticle = mc2)

2

2

1cv

mvp

Relativistic expressions for energy and momentum

2

2

2

1cv

mcE

2

2

1cv

mvp

2222 pcmcE

2222 pcmcE

consequently, particle with zero rest mass (eg photon) has momentum pgiven by:

h

c

hf

c

Ep

The Compton Effect

•x-rays scattered from target containing very loosely bound electrons

•Wavelength of scattered x-rays found to be different from that of incident X-rays AND to depend on detection angle :

cos1cm

h

e

The Compton Effect

Compton Effect explained by photon model:

Treat Compton scattering as a 2-particle collision between photon and initially stationary electron, obeying conservation laws for energy and momentum:

pi

photon electronBefore

photon

pf

pe

After

Compton Scattering: Conservation of momentum

Vector triangle:

pi

pf pe

Consider magnitudes of vectors pi, pf and pe:

cos2222ifife ppppp

Compton Scattering: Conservation of energy

We are dealing with velocities at, or close to, speed of light so need to use relativistic expressions:

Initial energy:

Final energy

2cmcp ei

ef Ecp

222222 cmcpcpcpcmE efieee

2cmcpcpE efie

general expression

Compton Scattering: Conservation of energy

22222 cmcpcpcpcm efiee

Divide both sides by c2 (be careful!………)

222 cmpppcm efiee

cmppcmpppcm efiefiee 22222

cmppppp efifie 222

Compton Scattering: Conservation of energy & momentum

cmppppp efifie 222

cmppppppp efififie 22222

Energy

cos2222ifife ppppp momentum

Compton Scattering: Conservation of energy & momentum

cmpppppp efififi 2222

cos222ifif pppp =

cos222 fiefifi ppcmpppp

cos1

fi

efi

pp

cmpp

Compton Scattering: Conservation of energy & momentum

cos1

fi

efi

pp

cmpp

cos1if p

mc

p

mc

)cos1(111 mcpp if

)cos1( mc

h

Compton Scattering: Summary

The observed experimental result:

)cos1( mc

h

Is entirely explained by the photon-electron scattering model. Further proof of the validity of the photon concept.

•maximum wavelength shift for = 180°, Δλ=2h/mc

•h/mc is known as the COMPTON WAVELENGTH of the electron.

•very small (work it out!) so Compton effect only observed for short wavelength radiation (x-rays, gamma rays)

A final word….

We’ve seen that light can be described as a stream of particles called photons, but we don’t have to take this picture too literally.

Instead can think of the situation as the electromagnetic wave exchanging energy and momentum with a charged particle in quantised amounts………….