thomson scattering - university of...
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Thomson Scattering
How does a photon (light) scatter from an electron?Classically, the answer is Thomson scattering
Assume The wavelength of light is small compared to an atomThe energy of the light is large compared to the binding energy of atomic electronsThe energy of the light is smaller than mec2
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Thomson Scattering
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Thomson ScatteringIn Thomson scattering an electromagnetic (EM) wave of frequency f is incident on an electron
What happens to the electron?
Thus the electron will emit EM waves of the same frequency and in phase with the incident waveThe electron absorbs energy from the EM wave and scatters it in a different directionIn particular, the wavelength of the scattered wave is the same as that of the incident wave
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Rayleigh Scattering
An aside, Rayleigh scattering is scattering of light from a harmonically bound electronYou may recall the probability for Rayleigh scattering goes as 1/λ4
Why is the sky blue?Why are sunsets red?What color is the moon’s sky?
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Compton ScatteringX-ray spectrum produced by bombarding a metal with electrons
Line spectra correspond to atomic electron transitions in an excited atomContinuum corresponds to the emission of radiation from accelerated electrons (scattered by the Coulomb force of atomic nuclei)
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22
32
caqS =
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Cross Section
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Compton EffectThe change in wavelength can be found by applying
Energy conservation
Momentum conservation
( ) 2/142222 cmcpfhEfhcmhf eeee ++′=+′=+
θcos22 22222 ppppppppp
ppp
e
e
′⋅−′+=′⋅−′+=
+′=rr
rrr
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Compton EffectFrom energy conservation
From momentum conservation
Eliminating pe2
( )
( )fhhfmc
fhfhcfh
chfp
cpcmfhhfcmfhhfcm
ee
eeee
′−+′
−⎟⎠⎞
⎜⎝⎛ ′
+⎟⎠⎞
⎜⎝⎛=
+=′−+′−+
22
2)(
2
222
22422242
θ
θ
cos2
cos2222
2
22222
cfh
chf
cfh
chfp
ppppppppp
e
e
′−⎟
⎠⎞
⎜⎝⎛ ′
+⎟⎠⎞
⎜⎝⎛=
′⋅−′−=′⋅−′+=rr
( ) ( )θcos12 −′=′− fhfhfhhfcme
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Compton EffectContinuing on
And using f=c/λ we arrive at the Compton effect
And h/mc is called the Compton wavelength
)cos1(2 θ−=′′−
cmh
ffff
e
( )θλλ cos1−=−′cm
h
e
mcm
h
eC
121043.2 −×==λ
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Compton Effect
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Pair ProductionAnother process by which photons can interact with matter is electron-positron pair productionγ → e++e-
The process as shown cannot take place because energy and momentum are not simultaneously conserved
Consider the center-of-momentum frame for the e+ and e-. What is the momentum of the photon?
However energy and momentum are both conserved in the presence of a Coulomb field from an atomic nucleus or atomic electron
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Pair ProductionEnergy and momentum conservation give
Energy conservation can be re-written
But momentum conservation (x) shows
Thus energy and momentum are not simultaneously conserved
++−−
++−−
+−
+=
+=
+=
θθ
θθ
sinsin0 (y) Momentum
coscos (x) Momentum
Energy
pp
ppchf
EEhf
42224222 cmcpcmcphf +++= +−
cpcphf +− +=max
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Pair Production
Conservation of energy and momentum does hold in the presence of an atomic nucleus (or electron) where the recoil of the nucleus ensures momentum conservationIn order for pair production to occur, the energy of the photon must be at least twice the electron rest mass
MeV022.12 2 => cmhf e
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Pair Production
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Pair ProductionA related process to electron-positron pair production is pair annihilation
e++e- → γ γ A positron passing through matter will lose energy through collisions with atomic electronsIt eventually slows down and annihilates with an electron (possibly first forming a bound system called positronium)
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PET
Pair annihilation is the basis of PET (Positron Emission Tomography)scanning
PET scans are most often used to detect cancer and to examine the effects of cancer therapy by characterizing biochemical changes in the cancerPET scans are also used to study heart function and brain disorders
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PETPET scanning
Positron emitting radioactive nuclei such as 11C, 13N, 15O, 18F are produced (at accelerators)The nuclei are incorporated into compounds used by the body such as sugar or ammoniaOnce taken into the body, positrons are emitted, lose energy in a few mm, and annihilate with electrons producing two 0.511 MeV photons that produced back-to-backSoftware reconstructs the point of origin of the annihilation producing a map showing tissues where the radiotracers have become concentrated
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PET
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PET
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Photon Interactions
Interactions of photons with matterThe primary processes by which photons interact with matter are From low energy to high energy
Photoelectric effectCompton scatteringPair production
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Photon Interactions