The radiative energy loss for electrons or positrons is:
rad± | Avαre2
| |
rad±(Z,T) | e2Z2(T+m)}∫0Tk{dσ±
|
Reference [] says that:
``The differences between the radiative loss of positrons
and electrons are considerable and cannot be disregarded.
[...] The ratio of the radiative energy loss for positrons to that for electrons obeys a simple scaling law, [...] is a function only of the quantity 2 ''
In other words:
rad+(Z,T)
|
The authors have calculated this function in the range -7≤{T
where:
2}) (T in GeV)
| ||
6
| ||
1 |
| |
3 |
| |
5 |
|
This -/,
and
.
1c2}(GeV) | 1c|T | 1c| | 1ce-
|
-9 |
| ||
-8 |
| ||
-7 |
| ||
-6 |
| ||
-5 |
| ||
Table: ratio of the -/
The scaling holds for the ratio of the total radiative energy losses, but it is significantly broken for the photon spectrum in the screened case. In case of a point Coulomb charge the scaling would hold also for the spectrum. The scaling can be expressed by:
+ | +
|
If we consider the photon spectrum from bremsstrahlung reported in [] we see that:
± | +(1) = 0 &sp;S-(1) > 0
|
1cT(MeV) | 3|c|C | 3cPb | ||||
&sp; | l0 | l | l | l0 | l | l
|
0.02 | -2.86 | -2.86 | +52.00 | -4.89 | -4.69 | +99.80 |
0.1 | -0.33 | -0.33 | +21.10 | -0.52 | -0.47 | +81.08 |
1 | +0.07 | +0.07 | +6.49 | +0.11 | +0.11 | +48.99 |
10 | 0.00 | 0.00 | +1.75 | 0.00 | +0.01 | +23.89 |
2 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | +9.00 |
3 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | +2.51 |
4 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | +0.00 |
7c&sp; | ||||||
7cl=100 {El--El+
|
Table: Difference in the energy loss and bremsstrahlung
cross-section for -/
We further assume that:
+ |
|
In order to satisfy approximately the scaling law for the ratio of the total radiative energy loss, we require for :
01f(ε)dε |
|
From the photon spectra we require:
We have chosen a simple function f:
α&sp;C,α> 0
|
1cT(MeV) | 3|c|C | 3cPb | ||||
&sp; | l0 | l | l | l0 | l | l
|
2 | +4.19 | +4.21 | +7.29 | +4.47 | +6.88 | +61.78 |
10 | +0.87 | +0.87 | +1.93 | +0.87 | +1.14 | +26.29 |
2 | +0.08 | +0.08 | 0.00 | +0.06 | +0.06 | +9.10 |
3 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | +2.42 |
4 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | +0.00 |
7c&sp; | ||||||
7cl=100 {El--El+
|
Table: Difference in the energy loss and bremsstrahlung
cross-section for -/
5cdep+-Edep-
| ||||
1cDepth | 2cC | 2|cPb | ||
1c|(0 units) | No ± diff | ± diff | No ± diff | ± diff |
0.5 | -11.7 | -13.0 | -0.8 | -3.9 |
1.0 | -5.3 | -4.9 | -1.0 | -4.1 |
1.5 | +7.3 | +8.0 | -1.4 | -3.5 |
2.0 | +7.1 | +5.3 | -0.7 | -0.0 |
2.5 | +4.9 | +4.3 | +1.7 | +3.6 |
3.0 | +4.8 | +4.1 | +1.1 | +4.3 |
3.5 | +3.3 | +2.7 | +2.7 | +3.1 |
4.0 | +3.6 | +5.3 | +2.9 | +3.0 |
4.5 | +1.7 | +2.8 | +0.5 | +2.3 |
5.0 | +3.4 | +3.5 | -1.9 | +1.8 |
Table: Difference in the shower development
for -/
from the conditions (), (
) we get:
We have defined weight factors l and σ for the positron continuous energy loss and discrete bremsstrahlung cross section:
l= {1 | σ= {1
|
where 0= {kc
+= Fl( - {dE | brems+= Fσσbrems-
|
As in this approximation the photon spectra are identical, the
same SUBROUTINE is used for generating -/
which is consistent with the spectra.
The effect of this -/.