Differential cross-section

The differential cross-section of the $\delta$ -ray production can be written as in equations (, ) [], [], []. For the electron/electron (Möller) scattering we have: $\{d\sigma d\; \epsilon \}=\{2\; \pi Z\; r$₀²m β²(E-m)}[ {(γ-1 )²γ²}+{1ε}({1ε}-{2 γ-1 γ²}) +{11- ε}({11- ε}{2 γ- 1γ²}) ]

and for the positron-electron (Bhabha) scattering: $\{d\; \sigma d\; \epsilon \}=\{2\; \pi Z\; r$₀²m (E-m)}[{1β²ε²}-{B₁ε}+B₂- B₃ε+ B₄ε²]

where

$Z$	$=$	$atomic\; number\; of\; the\; medium$	$E$	$=$	$energy\; of\; the\; incident\; particle$
$M$	$=$	$rest\; mass\; of\; the\; incident\; particle$	$\gamma$	$=$	$\{EM\}$
$\beta 2$	$=$	$1-\; \{1\gamma 2\}$	$y$	$=$	$\{1\gamma +\; 1\}$
$B$1	$=$	$2-y2$	$B$2	$=$	$(1-2y)(3+y2)$
$B$3	$=$	$(1-2y)2+(1-2y)3$	$B$4	$=$	$(1-2y)3$
$\epsilon$	$=$	$\{TE-m\}$

with T the kinematic energy of the scattered electron (of the lower energy in the case of $e-e+$ scattering).

The kinematical limits for the variable $\epsilon$ are:

$\epsilon$₀= {TCUTE-m}≤ε≤{12}&sp;for e^-e^-&sp;ε₀= {TCUTE-m}≤ε≤1&sp;for e⁺e^-

For the other charged particles the differential cross-section can be written:

$\{d\sigma dT\}$	$=$	$2\; \pi Z\; r$02m{1β2}{1T2}[ 1- β2{TTMAX }]	$for\; spin\; 0\; particle$
$\{d\; \sigma dT\}$	$=$	$2\; \pi Z\; r$02m {1β2}{1T2}[1- β2{T TMAX}+ {T22E2}]	$for\; spin\; 1/2\; particle$

where TMAX is the maximum energy transferable to the free electron:

$TMAX=\; \{2m(\gamma 2-1)1+2\gamma (m/M)\; +(m/M)2\}$

and TCUT is the energy threshold for the $\delta$ -ray emission: $TCUT\le T\; \le TMAX$