05 compton scattering

8/3/2019 05 Compton Scattering

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FISIKA KUANTUMBab I. Wave as Particles

3. Compton Scattering


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I. Waves as Particles:

1. Blackbody Radiation

2. Photoelectric Effect

3.Compton Scattering


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3. Compton Scattering

X-rays


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The Compton Effect

The Compton effect is thescattering of a photon off ofan electron thats initially atrest.

if the photon has enoughenergy (X-ray energies orhigher), the scatteringbehaves like an elastic

collision between particles the energy and momentum ofthe system is conserved


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Classical Elastic Scattering

To calculating what happens, we use the sameprinciples as before for an elastic collision, however,the fact that one particle is mass less (the photon) hassome strange consequences. If the two particles were massive, wed had the situation we

studied before. Note in particular:

no deflection angle since particle 2 is at rest, the problem reducesto a one dimensional collision

the velocity of particle 1 will change after the impact, and if m2>m1,

particle 1 will get scattered backwards


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Compton Scattering

Since the photon is mass less, it always moves at thespeed of light. the photon does loose momentum and energy during the

collision (giving it to the electron), consequently itswavelength decreases.

the reason there is a deflection angle, is that otherwise itwould be impossible for the system to conserve bothenergy and linear momentum.


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The Collision of Particle

Classical Elastic

Scattering

Compton

Scattering


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The Compton Effect

In 1924, A. H. Comptonperformed an experiment

where X-rays impinged

on matter, and he

measured the scattered

radiation.

X-rays


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The Compton Experiment

M

A

T

T

E

R

Incident X-raywavelength

l1 l2 >l1Scattered X-ray

wavelength

l2e

Electron comes flying out


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Problem

According to the wave picture of light, the incident X-

ray should give up some of its energy to the electron,

and emerge with a lower energy (i.e., the amplitude islower), but should have l2 = l1.

It was found that the scattered X-ray did not have the

same wavelength?


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Quantum Picture to the Rescue

Incident X-ray

E1 = hc /l1e

Electron

initially atrest (almost)

l2 >l1

Scattered X-ray

E2 = hc / l2

e

Ee

e

Compton found that if you treat the photons as if they were particles

of zero mass, with energy E=hc/l and momentum p=h/l The collision behavesjust as if it were 2 billiard balls colliding !

Photon behaves like a particle with energy & momentum as given above!


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Compton

ScatteringData


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Compton Scattering Data


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Calculating The Compton Effect

Calculating the Compton effect:

The incident photon has frequency f,

hence wavelength l=c/f

The photon is scattered into an angle

q, and in the process its frequency

changes to f (and correspondingly

l=c/f)

The electron is initially at rest, and

afterwards gains a velocity v.

The angle at which the electron is

scattered is q

'q

v

l

'l


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Conservation of Energy

2

2

1 vmfhhf e


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Conservation of Momentum

)'cos()cos('

qqll

vmhh

e

Conservation of momentum in the x direction


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Conservation of Momentum

Conservation of momentum in the y direction

)'sin()sin('

0 qql

vmh

e


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The Compton Effect

The preceding is a rather messy set of equations tosolve here is the key result:

qll cos1' cm

h

e


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The Compton Wavelength of

the Electron

qll cos1' cm

h

e

The quantity h/mec is called the Compton wavelengthof the electron, and has a value of 2.43x10-12m.


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Energies of a photon at 500 keV and

an electron after Compton scattering
http://upload.wikimedia.org/wikipedia/commons/9/99/ComptonEnergy.jpg


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Example

In a Compton scattering experiment, the incident x-

rays have a wavelength of 0.2685 nm, while the

scattered x-rays have a wavelength of 0.2703 nm. Through what angle are the x-rays scattered?


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The Scatter of Lights from

Charged Particles

From the wave theory, we can understand that

charged particles would interact with the light sincethe light is an electromagnetic wave.

But the actual predictions of how the light scatters

from the charged particles does not fit our simple

wave model.


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Compton Scattering

If we consider the photon idea of light, some of the

photons would hit the charged particles and bounce

off. The laws of conservation of energy and momentum

should then predict the scattering.


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Energy and Momentum After

Collision

As we know Photons have momentum as well as

energy.

The scattered photons will have less energy and lessmomentum after collision with electrons, and so

should have a larger wavelength according to the

formula:

l = lscattered - lincident = (h/mc)[1-cos(q)]


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Plancks Constant & Compton

Scattering

Note that Plancks constant is in this relation as well,

and gives a further experimental way of getting thisvalue.

Again, the photon theory provides a nice explanation

of a phenomenon involving light.


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Maximum Change of Wavelength

Note that the maximum change in wavelength is (for

scattering from an electron):

2h/mc = 2(6.63 x 10-34 J-s) / (9.1 x 10-31 kg * 3 x 108

m/s) = 4.86 x 10-12 m

Which would be insignificant for visible light but NOT

for x-ray and -ray light.


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Scattering Problem

Incident X-ray

wavelengthli1.5 [nm]ee

Electron

initially atrest

KE=0.2 [keV]

lf

Before After

Compute the energy of the 1.5 [nm] X-ray photon.

E = hc/l = (6.6x10-34 [J s])(3x108 [m/s]) / (1.5x10-9 [m])

= 1.3x10-16 [J]


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Scattering Example (cont)

Express this energy in [keV].

1.3x10-16 [J] * (1 [eV] / 1.6 x10-19 [J]) = 825 [eV] = 0.825 [keV]

What is the magnitude of the momentum of this photon?

p = E /c = 0.825 [keV] /c = 0.825 [keV/c]

After the collision the electrons energy was found to be 0.2 [keV].

What is the energy of the scattered photon?

A) 0.2 [keV] B) 0.625 [keV] C) 1.025 [keV] D) 0.825 [keV]

Since energy must be conserved, the photon must have E=0.825-0.2 = 0.625 [keV]

What would be the wavelength of the scattered photon?

HW exercise !


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Compton Scattering

Application


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Radiobiology

Compton scattering is of prime importance to

radiobiology, as it happens to be the most probable

interaction of high energy X rays with atomic nuclei in

living beings and is applied in radiation therapy.
http://en.wikipedia.org/wiki/Radiation_therapyhttp://en.wikipedia.org/wiki/Radiation_therapy


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Material Physics

In material physics, Compton scattering can be used to

probe the wave function of the electrons in matter in

the momentum representation.
http://en.wikipedia.org/wiki/Wave_functionhttp://en.wikipedia.org/wiki/Wave_function


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Gamma Spectroscopy

Compton scattering is an important effect in gamma

spectroscopy which gives rise to the Compton edge, as

it is possible for the gamma rays to scatter out of the

detectors used. Compton suppression is used to detect stray scatter

gamma rays to counteract this effect.
http://en.wikipedia.org/wiki/Compton_edgehttp://en.wikipedia.org/wiki/Compton_suppressionhttp://en.wikipedia.org/wiki/Compton_suppressionhttp://en.wikipedia.org/wiki/Compton_edge


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Inverse Compton

Scattering


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Inverse Compton scattering

Inverse Compton scattering is important inastrophysics.

In X-ray astronomy, the accretion disk surrounding ablack hole is believed to produce a thermal spectrum.

The lower energy photons produced from thisspectrum are scattered to higher energies by relativisticelectrons in the surrounding corona.

This is believed to cause the power law component in

the X-ray spectra (0.2-10 keV) of accreting blackholes.


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Inverse Compton scattering

The effect is also observed when photons from thecosmic microwave background move through the hotgas surrounding a galaxy cluster.

The CMB photons are scattered to higher energies by

the electrons in this gas, resulting in the Sunyaev-Zel'dovich effect.

Observations of the Sunyaev-Zel'dovich effect providea nearly redshift-independent means of detectinggalaxy clusters.


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05 compton scattering

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