5.1 Equivalence of various interactions
There is a number of electron-photon interaction that involve two incoming particles and two outgoing particles, two of which are electrons (or positrons) and the two are photons. They all have essentially the same Feynman diagrams and therefore they are closely related and basically have the same cross sections.
If the two incoming particles are a photon and an electron, so the outgoing particles are also a photon and an electron, we call the process Compton scattering. In the Feynman sense we can turn a particle from incoming to outgoing and then must replace it with its antiparticle, a positron for an electron while the photon is its own antiparticle. So two incoming photons can be turned into an electron-positron pair in a process called pair production, if the two photons carry enough energy in the center-of-momentum frame to account for the restmass energy of the pair. Likewise, an electron-positron pair can undergo pair annihilation and turn into two photons, for which no energy threshold must be satisfied.
A common characteristic of all these processes is the existence of two kinematic regimes. If in the center-of-momentum frame the particle energy is below m e c 2 , so the electrons would be non-relativistic, then the total cross section is given by the Thomson cross section σ
T and we speak of the Thomson limit. If the particle energy is above m e c 2 , the Klein-Nishina limit applies and the total cross section falls off with increasing energy.
Pair production on ambient photons is an important proces that limits the probability with which gamma rays from astronomical sources can reach our detectors near earth. On account of the high energy threshold of photon-photon pair production the process immediately turns into the Klein-Nishina regime, where the cross section drops, so a gamma ray most likely interacts with photons at a specific frequency. At about 400 TeV the gamma-ray interact with the entire cosmological microwave background, that has about 400 photons per cm 3 , so the mean free path of the gamma rays is only about l =
1
σ
T n ph
'
4
·
10 21 cm
'
4000 Lyr (5 .
1 .
1)
At lower energy the range of the gamma ray in intergalactic space can be billions of light-years, but then pair creation inside the source may be important.
A special case of these interactions is given if an incoming photon is replaced with a virtual photon representing the Coulomb field of an ion. Then the total cross section is given by the finestructure constant α
'
1 / 137 times the Thomson cross section, so
∼
σ
T
/ 137. The bremsstrahlung process that we discussed before is an example. Another example is pair production by one photon on the Coulomb field of an ion, for which the photon has to carry at least
2 m e c 2 in energy. This process is responsible for the shielding of cosmogenic gamma-rays by the
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earth atmosphere where a cascade of bremsstrahlung, Compton scattering, and pair creation and annihilation turns the energy of a single incoming gamma-ray into a shower of low-energy photons and electron-positron pairs that for not too high energy dies out before reaching the ground.
5.2 Compton scattering
In textbooks on usually considers Compton scattering of photons on electrons at rest. For photon energies far below m e c 2 the Thomson limit applies and the differential cross section for unpolarized radiation is dσ !
d Ω unpol
= r 2 e
2
(1 + cos 2 θ ) (5 .
2 .
1) where r e is the classical electron radius and θ is the scattering angle between incoming and outgoing photon. The total cross section is derived by integration over solid angle and is called the Thomson cross section.
σ =
I d Ω dσ !
d Ω unpol
=
8 π
3 r 2 e
= σ
T
(5 .
2 .
2)
The conservation of energy and momentum imply that the energy of the scattered photon, and the energy of the incoming photon, , satisfy s
=
1 + (1
− cos θ ) with = hν mc 2
(5 .
2 .
s
3)
,
The energy loss suffered by the photon
∆ = s
2 (1
− cos θ )
−
=
−
1 + (1
− cos θ )
(5 .
2 .
4) is therefore indeed minimal in the Thomson regime, when 1.
The correct differential cross section for all energies, that can be derived using QED, is given by the Klein-Nishina formula dσ !
d Ω unpol
= r e
2
2
2 s
2 s
+ s
− sin 2 θ (5 .
2 .
5)
For 1 the scattering angle lends to be small as shown in the figure. The total cross section then follows in the asymptotic limits as
σ =


σ
T

3
8
(1
σ
T
−
−
1
2 . . .
)
(1 + 2 ln 2 )
1 Thomson regime
1 Klein-Nishina-limit
(5 .
2 .
6)
What would happen if the electron moved? The total cross section is invariant to our choice of reference frame, of course, and the relation between the energies of the incoming and outgoing photons can be derived using the Lorentz transformation of relation (5.2.3).
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The differential cross section for Compton scattering, dσ/d Ω , as function of the scattering angle
θ in a polar diagram. The dotted lines correspond to r e
2 times a factor 0.25, 0.5, 0.75 and 1.0.
The thick solid lines mark the differential cross section for three different photon energies from the Thomson regime to the Klein-Nishina-limit with = 0 .
03 , = 0 .
3 and = 3 .
0 .
The electron restframe moves with Lorentz factor γ in the laboratory frame, which we denote with primed quantities. Chose polar coordinates, so the relative velocity of two frames, βc , is oriented along the polar axis. Then
= 0 γ (1
−
β cos θ 0 ) = 0 γ −
1 (1 + β cos θ ) −
1
(5 .
2 .
7)
0 s
= s
γ (1 + β cos θ s
) =
We see that the Thomson limit applies for 0 γ s
γ −
1 (1
−
β cos θ s
0 ) −
1
−
1 . We further note that in the case γ 1 for nearly all cos θ s
0 in the laboratory frame cos θ 0 s
= cos θ s
+ β
1 + β cos θ s
≈
β (5 .
2 .
8)
Thus the scattered photon essentially has the same direction as the scattering electron. For the photon energy we find
0 s
≈


γ 2
0

1
2
γ 0
0 γ −
1
γ −
1
Thomson- regime
Klein-Nishina-limit
(5 .
2 .
9)
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Compton scattering is just an energy exchange between the photon and the electron. If the incoming electron is much more energetic than the photon, the process is called inverse Compton scattering .
A good approximation to the differential cross section for inverse Compton scattering in the
Thomson regime is given by the Head-on approximation. Here we set the direction of the scattered photon identical to that of the electron and assume in the energy relations (5.2.7) cos θ 0 s
= β . With all quantities in the laboratory frame and dropping the primes we find dσ d s d Ω s
'
σ
T
δ ( s
−
γ 2 (1
−
β cos ψ )) δ (Ω s
−
Ω e
) (5 .
2 .
10) where cos ψ = cos θ ph cos θ e
+ sin θ ph sin θ e cos( φ ph
−
φ e
) (5 .
2 .
11) is the scattering angle (cos θ 0 then is in 5.2.7). The differential production rate of scattered photons s
( s
, Ω s
) = dn s dt d Ω s d s
= c
Z ∞
0 d
I d Ω ph
Z
(2 ) −
1
1 dγ
I d Ω e
(1
−
β cos ψ ) n e dσ n ph d s d Ω s
(5 .
2 .
12) where we integrate over all quantities in which the electron and soft-photon distributions are differential. We will now calculate the electron energy los rate and the emission spectrum.
For that purpose we assume an isotropic monoenergetic distribution of low-frequency photons, n ph
= n ph, 0
δ (
−
0
) / (4 π ). For the energy loss rate we will consider one electron, that is n e
= δ ( γ
−
γ
0
) δ ( φ e
) δ (cos θ e
−
1), and calculate
−
˙ =
I d Ω s
Z
∞ d s s
0 s
×
δ ( φ e
= c σ
T
4 n ph, 0
π
Z
∞ d
I
0
) δ (cos θ e
−
1) δ (
− d Ω ph
Z
(2 ) −
1 dγ
1
I d Ω e
0
) δ ( γ
−
γ
0
) γ 2 (1
−
β cos ψ ) 2
= c σ
T n ph, 0 0
γ
0
2
2
Z
1
−
1 d cos θ ph
(1
−
β cos θ ph
) 2
'
4 c σ
T
U ph
β 2 γ
0
2
3
(5 .
2 .
13)
In the Thomson limit the electron energy losses thus scale quadratically with the electron energy. If we insert an isotropic power-law spectrum of electrons, for the emission spectrum s
∝
−
( s +1) / 2 s n e
∝
γ − s / 4 π , then we find
(5 .
2 .
14)
4