phys586-lec08-photons2

Transcript phys586-lec08-photons2

Compton Scattering

 There are three related processes  Thomson scattering (classical)  Photon-electron   Compton scattering (QED)  Photon-electron Rayleigh scattering (coherent)  Photon-atom  Thomson and Rayleigh scattering are elastic only the direction of the photon changes, not its energy  Plus Thomson and Rayleigh scattering are only important at low energies where the photoelectric effect dominates 1

Thomson Scattering

 In 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 wave  The electron absorbs energy from the EM wave and scatters it in a different direction  In particular, the wavelength of the scattered wave is the same as that of the incident wave 2

Thomson Scattering

F a S

 

0 





8   sin

0 

 sin



8  

 power/area 



 2 3 8  3

3   

2    

1 2 2

4 3

0 2  power emitted

2  this is the power subtracted from the incoming beam  T  8  3   

2    2  8  3

r e

2  0 .

655  10  24

2 3

Rayleigh Scattering

 Rayleigh scattering is scattering of light from a harmonically bound electron Assuming SHO with frequency  0 for an electron in an atom 

Rayleigh

 

hom

son

    2  4   0 2  2    You may recall the probability for Rayleigh scattering goes as 1/λ 4  Why is the sky blue?

Compton Scattering

 Compton scattering is the scattering of light (photons) from free electrons 5

Compton Scattering

 Calculations  The change in wavelength can be found by applying  Energy conservation

 

m e c

2 

  

E e



   

p e

2 

m e

4  1 / 2  Momentum conservation 

p p e

2   



 2  

p e p

 2  2 

 

 

2 

 2  2



 cos  6

Compton Scattering

 From energy conservation

m e

4  (

 

  ) 2  2

m e c

2 

 

   

m e

4 

p e

2 



2 

 

2  2



  

2  From momentum conservation 2

m e





p e

2 

2 

 2  2 

 

 

2  

  

 2  2



 cos 

p e

2    



   2    



    2  2



c h

  cos 

 Eliminating p e 2

m e c

2 

 

   



   1  cos   7

Compton Scattering

 Continuing on        

h m e c

2 ( 1  cos  )  And using v=c/λ effect      we arrive at the Compton

h m e c

 1  cos    And h/mc is called the Compton wavelength 



h m e c

 2 .

43  10  12

Compton Scattering

 Summarizing and adding a few other useful results are     

h m e c

 1  cos  



T e

  

1     

hv h m e c

2  

   1  cos   cot     1 

hv m e c

2   tan  2 9

Compton Scattering

 The differential and total cross sections are calculated in a straightforward manner using QED  Called the Klein-Nishina formula



  

Compton r e

2 2   1   2 

r e

2  1 1  cos    1   2     2   1  1  2     1    2   cos 2  1 ln   1   1   2  2   1   1  cos  cos   2              1 2  ln  1  2     1 1   3  2   2   10

Compton Scattering

 On the previous slide  

hv m e c

2  At low energies 

Compton

 

hom

son

 8  3  At high energies

r e

2 

Compton

 8 

r e

2 3  3 8  ln 2   1 2 11

Compton Scattering

 Thus at high energies, the Compton scattering cross section  C goes as 

Compton

Z hv

Compton Scattering

 Graphically, d  /d  13

Compton Scattering

 In polar form, assume a photon incident from the left 14

Compton Scattering

 At high energies, say > 10 MeV, most of the photons are scattered in the forward direction  Because of the high forward momentum of the incident photons, most of the electrons will also be scattered in the forward direction 15

Compton Scattering

 Concerning kerma and absorbed dose, we are particularly interested in the scattered electron because it is ionizing  We can split the Compton cross section into two parts: one giving the fraction of energy transferred to the electron and the other the fraction of energy contained in the scattered photon 16

Compton Scattering



 

C tr

 

C sc



tr C



C sc

 

C T h



h v



  

similarly





h v

 for the mass energy tra nsfer attenuatio n coefficien t  

C tr



T h

  



T h



N Av



C A

Compton Scattering

Here  en =  tr 18

Compton Scattering

 Another useful form of the differential cross section is d  /dT, which gives the energy distribution of the electron 19

Compton Scattering

 The maximum electron kinetic energy is given by

max 

 



max 2 1   2    and  

  1  2  1  2  

hv m e c

2    2

m e hvm e c

2 

and for

large



max 

m e c

2 2  0 .

2555

MeV

Compton Scattering

 In cases where the scattered photon leaves a detector without interaction one would observe 21

Compton Scattering

h v

 |   

h v

 |     1 

m e c

2  255

keV



m e c

2 2 23

Pair Production

 Pair production is the dominant photon interaction at high energies (> 10 MeV)  In order to create a pair, the photon must have > 2m e = 1.022 MeV  In order to conserve energy and momentum, pair production must take place in the Coulomb field of a nucleus or electron   For nuclear field, E threshold > 2 x m e For atomic electron field, E threshold > 4 x m e 24

Pair Production

 Energy and momentum conservation give Energy



 

 Momentum Momentum

(x) (y) 0

 



 sin cos      

 cos  

 sin    Energy conservation can be re-written



2 

2 

4 

2 

4  But momentum conservation (x) shows

max 





 Thus energy and momentum are not simultaneously conserved 26

Pair Production

 The processes of pair production and bremsstrahlung are related (crossed processes)  Thus we’d expect the cross section to depend on the screening of atomic electrons surrounding the nucleus  Does the photon see nuclear charge Ze or 0 or something in between?

 The relevant screening parameter is   100

m e c

hv E





1 / 3 27

Pair Production

 In the Born approximation (which is not very accurate for low energy or high Z) one finds No screening    1  and

m e c

2 

  137

m e c

 1 / 3 

pair

 4

Complete 2 

r e

2    7 9   screening 2

 ln  

m e c

2  

0  and

    109 54     137

m e c

 1 / 3 

pair

 4

2 

r e

2   7 9  ln 183

 1 / 3  

   1 54 28

Pair Production

 Notes     pair ~ Z 2 Above some photon energy (say > 1 GeV),  pair becomes a constant In order to account for pair production from the Coulomb field of atomic electrons, Z 2 is replaced by Z(Z+1) approximately since the cross section is smaller by a factor of Z  Usually we don’t distinguish between the source of the field 29

Pair Production

 Notes  In the case of the nuclear field and for large photon energies, the mean scattering angle of the electron and positron is 

 

m e c



 1 .

022 For

 2  5

MeV



 2

MeV

and   15  30

Pair Production

 The probability for pair production 31

Pair Production

 2m e (1.022 MeV) of the photon’s energy goes into creating the electron and positron  The electron will typically be absorbed in a detector  The positron will typically annihilate with an electron producing two annihilation photons of energy m e (0.511 MeV) each  If these photons are not absorbed in the detector than the pair production energy spectrum will look like 32

Pair Production

 Similar to the photoelectric effect and Compton scattering we define the mass attenuation and mass energy transfer coefficients as 

pair

 

tr pair

 

N Av



pair A

  

 2

m e c

  

pair

 34

Photonuclear Interactions

 Here a nucleus is excited by the absorption of a photon, subsequently emitting a neutron or proton  Most important when the energy of the photon is approximately the binding energy of nucleons (5-15 MeV)   Called giant nuclear dipole resonance Still a small fraction compared to pair production however 35

Photonuclear Interactions

 Giant dipole resonance 36

Photonuclear Interactions

 These interactions would be observed with higher energy x-ray machines   A 25 MV x-ray beam will contain neutron contamination from photonuclear interactions Small effect compared to the photon beam itself  Also important in designing shielding since ~MeV neutrons are difficult to contain 37

Photon Interactions

 Typical photon cross sections 38

Photon Interactions

 Typical photon cross sections 39

Photon Interactions

 Notes  Of course different interactions can occur at a given photon energy   





Compton

 

pair

    





Compton

  

pair

  A polyenergetic beam such as an x-ray beam is not attenuated exponentially    Lower energy x-rays have higher attenuation coefficients than higher energy x-rays Thus the attenuation coefficient changes as the beam proceeds through material An effective attenuation length  eff can be estimated as 

eff

 0 .

693

HVL

Beam Hardening

Photon Interactions

 Let’s return to our first slide  

x I



I e

0  As we’ve seen in the different photon interactions   Secondary charged particles are produced Photons can lose energy through Compton  We define  Narrow beam geometry and attenuation   Only primaries strike the detector or are recorded Broad beam geometry and attenuation   All or some of the secondary or scattered photons strike the detector or are recorded Effective attenuation coefficient  ’ <  42

Photon Interactions

 In ideal broad beam geometry all surviving primary, secondary, and scattered photons (from primaries aimed at the detector) is recorded  In this case  ’ =  en 44

Photon Interactions

 There are three relevant mass coefficients   

N Av



mass absorption coefficien t 

 mass energy tra nsfer coefficien t  

mass energy absorption coefficien  

  

 1 

 t where g is the average fraction of secondary electron energy lost to radiative interactio ns (bremsstra hlung and annhilatio n) 45

Photon Interactions

 Tables of photon cross sections, mass attenuation, and mass-energy absorption coefficients can be found in numerous places  http://physics.nist.gov/PhysRefData/contents.html   NIST also gives material constants and composition Useful since  

mixture

  

A f A

  

B f B

 ...

where

f i

are the weight fractions of separate elements 46

 =1/(  /  )

Photon Interactions

 Sometimes easy to loose sight of real thickness of material involved 48

Photon Interactions

 X-ray contrast depends on differing attenuation lengths 49

Photon Interactions

 What is a cross section?

 What is the relation of  for the physical process?

to the cross section   has units  



2 and  has units where

is the density 1 /

of atoms     in

N Av

  is the linear

A cm

absorption coefficien is more common t 50