Corrections for e-/e+ differences

The radiative energy loss for electrons or positrons is:

$-\{1\rho \}(\; \{dEdx\})$rad±	$=$	$\{N$Avαre2A}(T+m) Z2Φrad±(Z,T)
$\Phi$rad±(Z,T)	$=$	$\{1\alpha r$e2Z2(T+m)}∫0Tk{dσ±dk}dk

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 $T/Z2$ ''

In other words:

$\eta$

$=$

$\{\Phi$rad+(Z,T)Φrad-(Z,T)}= η({TZ2})

The authors have calculated this function in the range $10-7\le \{TZ2\}\le 0.5$ (here the kinetic energy T is expressed in MeV). Their data can be fairly accurately reproduced using a parametrisation:

where:

$x$	$=$	$log(\; C\; \{TZ2\})\; (T\; in\; GeV)$
$C$	$=$	$7.5221\; x106$
$a$1	$=$	$0.415$
$a$3	$=$	$0.0021$
$a$5	$=$	$0.00054$

This $e-/e+$ energy loss difference is not a pure low-energy phenomenon (at least for high Z), as it can be seen from Tables , and .

 

1c$\{TZ2\}(GeV)$	1c|T	1c|$\eta$	1c$(\; \{rad.\; losstotal\; loss\})$e-
$10-9$	$\sim 7\; keV$	$\sim 0.1$	$\sim 0$
$10-8$	$67\; keV$	$\sim 0.2$	$\sim 1$
$2\; x10-7$	$1.35\; MeV$	$\sim 0.5$	$\sim 15$
$2\; x10-6$	$13.5\; MeV$	$\sim 0.8$	$\sim 60$
$2\; x10-5$	$135.\; MeV$	$\sim 0.95$	$>\; 90$

Table: ratio of the $e-/e+$ radiative energy loss in lead (Z=82).

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:

$\{\Phi +\Phi -\}=\; \eta (\; \{TZ2\})$

$\{\{d\sigma +dk\}\{d\sigma -dk\}\}=\; does\; not\; scale$

If we consider the photon spectrum from bremsstrahlung reported in [] we see that:

 

1cT(MeV)	3|c|C	3cPb
&sp;	$\Delta E$l0	$\Delta E$l	$\Delta \sigma$l	$\Delta E$l0	$\Delta E$l	$\Delta \sigma$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
$102$	0.00	0.00	0.00	0.00	0.00	+9.00
$103$	0.00	0.00	0.00	0.00	0.00	+2.51
$104$	0.00	0.00	0.00	0.00	0.00	+0.00
7c&sp;
7c$\Delta E$l=100 {El--El+El-} &sp;and&sp;$\Delta \sigma$l=100 {σl--σl+σl-}

Table: Difference in the energy loss and bremsstrahlung cross-section for $e-/e+$ in Carbon and Lead with a cut for $\gamma$ and $e\pm$ of 10keV. $\Delta E$_l⁰ is the value without the correction for the difference $e-/e+$ .

We further assume that:

$\{d\sigma +dk\}=\; f(\epsilon )\; \{d\sigma -dk\}$

$\epsilon =\; \{kT\}$

In order to satisfy approximately the scaling law for the ratio of the total radiative energy loss, we require for $f(\epsilon )$ :

$\int$01f(ε)dε

$=$

$\eta$

From the photon spectra we require:

We have chosen a simple function f:

$f(\epsilon )$

$=$

 

1cT(MeV)	3|c|C	3cPb
&sp;	$\Delta E$l0	$\Delta E$l	$\Delta \sigma$l	$\Delta E$l0	$\Delta E$l	$\Delta \sigma$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
$102$	+0.08	+0.08	0.00	+0.06	+0.06	+9.10
$103$	0.00	0.00	0.00	0.00	0.00	+2.42
$104$	0.00	0.00	0.00	0.00	0.00	+0.00
7c&sp;
7c$\Delta E$l=100 {El--El+El-} &sp;and&sp;$\Delta \sigma$l=100 {σl--σl+σl-}

Table: Difference in the energy loss and bremsstrahlung cross-section for $e-/e+$ in Carbon and Lead with a cut for $\gamma$ and $e\pm$ of 1MeV. $\Delta E$_l⁰ is the value without the correction for the difference $e-/e+$ .

 

5c$100\; \{E$dep+-Edep-σ+2+σ-2}()
1cDepth	2cC	2|cPb
1c|($X$0 units)	No $e\pm$ diff	$e\pm$ diff	No $e\pm$ diff	$e\pm$ 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 $e-/e+$ in Carbon and Lead. No diff refers to the value without the correction for the difference $e-/e+$ .

from the conditions (), () we get:

We have defined weight factors $F$_l and $F$_σ for the positron continuous energy loss and discrete bremsstrahlung cross section:

$F$l= {1ε0}∫0ε0f(ε)dε

$F$σ= {11-ε0}∫ε01f(ε)dε

where $\epsilon$₀= {k_cT} and $k$_c is the photon cut BCUTE. In this scheme the positron energy loss and discrete bremsstrahlung can be calculated as:

$(\; -\; \{dEdx\})+=\; F$l( - {dEdx})-

$\sigma$brems+= Fσσbrems-

As in this approximation the photon spectra are identical, the same SUBROUTINE is used for generating $e-/e+$ bremsstrahlung. The following relations hold:

which is consistent with the spectra.

The effect of this $e-/e+$ bremsstrahlung difference can be also seen in e.m. shower development, when the primary energy is not too high. An example can be found in table .

PHYS341