208
C. W. Fabjan and D. Fournier
dicular to the scattering plane and the polarization vector of the initial photon (in
case it is linearly polarized) reads (ε being the ratio between the scattered and the
incident photon energy ε = 1/(1 + k/m e c 2 (1 − cosθ )):
dσ/d = 0.5 r
2
e
ε + 1/ε − 2 sin
2 θ cos
2 η
.
(6.12)
At low energy (k not larger than a few MeV), the η-dependence can be exploited
for polarization measurements (Compton polarimetry). In the same energy range the
probability of backward scattering is also sizeable.
Photoelectric Effect
For sufficiently low photon energies the atomic electrons can no longer be considered as free. The cross section for photon absorption, followed by electron emission
(photoelectric effect) presents discontinuities whenever the photon energy crosses
the electron binding energy of a deeper shell.
Explicit calculations [4] show that above the K-shell the cross section decreases
like E −3.5 .
In the section devoted to shower formation, the relevance of the photoelectric
effect will be considered. The coherent scattering (or Rayleigh scattering) is
comparatively smaller than the photoelectric effect and its role negligible for shower
formation.
High Energy Effects (LPM)
In the collinear approximation of bremsstrahlung, the longitudinal momentum
difference q || between the initial electron (energy E) and the sum of the final electron
and photon (energy k) is equal to
q || = m e
2 c
3 k/2E (E − k) .
(6.13)
This quantity can be extremely small, being for example 0.002 eV/c for a 25 GeV
electron radiating a 10 MeV photon. Such a small longitudinal momentum transfer
implies a large formation length, L f (L f q || ≥ h/2π), about 100 μm in the above
example. Secondary interactions (like multiple scattering) taking place over this
distance will perturb the final state and will in general diminish the bremsstrahlung
cross section and the pair production cross section in case of photon interactions.
Coherent interaction of the produced photons with the medium (dielectric effect)
also affects, and reduces, the bremsstrahlung cross-section.
Such effects, already anticipated by Landau and Pomeranchuk [9] were considered in detail by several authors, and were measured by the experiment E146 at
SLAC. A recent overview is given in [10]. The high k/E part of the bremsstrahlung
spectrum is comparatively less affected (because of much larger q || values) while
the low k/E part is significantly influenced for E above ~100 GeV, see Fig. 6.7. Only
at much higher energies (>10 TeV) is the pair production cross-section affected.
In crystalline media the strong intercrystalline electrical fields may result in
coherent suppression or enhancement of bremsstrahlung. Net effects depend on the
C. W. Fabjan and D. Fournier
dicular to the scattering plane and the polarization vector of the initial photon (in
case it is linearly polarized) reads (ε being the ratio between the scattered and the
incident photon energy ε = 1/(1 + k/m e c 2 (1 − cosθ )):
dσ/d = 0.5 r
2
e
ε + 1/ε − 2 sin
2 θ cos
2 η
.
(6.12)
At low energy (k not larger than a few MeV), the η-dependence can be exploited
for polarization measurements (Compton polarimetry). In the same energy range the
probability of backward scattering is also sizeable.
Photoelectric Effect
For sufficiently low photon energies the atomic electrons can no longer be considered as free. The cross section for photon absorption, followed by electron emission
(photoelectric effect) presents discontinuities whenever the photon energy crosses
the electron binding energy of a deeper shell.
Explicit calculations [4] show that above the K-shell the cross section decreases
like E −3.5 .
In the section devoted to shower formation, the relevance of the photoelectric
effect will be considered. The coherent scattering (or Rayleigh scattering) is
comparatively smaller than the photoelectric effect and its role negligible for shower
formation.
High Energy Effects (LPM)
In the collinear approximation of bremsstrahlung, the longitudinal momentum
difference q || between the initial electron (energy E) and the sum of the final electron
and photon (energy k) is equal to
q || = m e
2 c
3 k/2E (E − k) .
(6.13)
This quantity can be extremely small, being for example 0.002 eV/c for a 25 GeV
electron radiating a 10 MeV photon. Such a small longitudinal momentum transfer
implies a large formation length, L f (L f q || ≥ h/2π), about 100 μm in the above
example. Secondary interactions (like multiple scattering) taking place over this
distance will perturb the final state and will in general diminish the bremsstrahlung
cross section and the pair production cross section in case of photon interactions.
Coherent interaction of the produced photons with the medium (dielectric effect)
also affects, and reduces, the bremsstrahlung cross-section.
Such effects, already anticipated by Landau and Pomeranchuk [9] were considered in detail by several authors, and were measured by the experiment E146 at
SLAC. A recent overview is given in [10]. The high k/E part of the bremsstrahlung
spectrum is comparatively less affected (because of much larger q || values) while
the low k/E part is significantly influenced for E above ~100 GeV, see Fig. 6.7. Only
at much higher energies (>10 TeV) is the pair production cross-section affected.
In crystalline media the strong intercrystalline electrical fields may result in
coherent suppression or enhancement of bremsstrahlung. Net effects depend on the