Посотјат различни процеси каде можат да се добијат пар електрон - позитрон. Во воздухот ( на пр. при електрични празнења ), но најпознато е расејувањето на фотоните од јадрата на атомите или молекулите.
Со помош на квантната механика процесот на создавање на парови може да се опише со квадруполниот диференцијален напречен пресек:[ 2]
d
4
σ
=
Z
2
α
f
i
n
e
3
c
2
(
2
π
)
2
ℏ
|
p
+
|
|
p
−
|
d
E
+
ω
3
d
Ω
+
d
Ω
−
d
Φ
|
q
|
4
×
×
[
−
p
−
2
sin
2
Θ
−
(
E
−
−
c
|
p
−
|
cos
Θ
−
)
2
(
4
E
+
2
−
c
2
q
2
)
−
p
+
2
sin
2
Θ
+
(
E
+
−
c
|
p
+
|
cos
Θ
+
)
2
(
4
E
−
2
−
c
2
q
2
)
+
2
ℏ
2
ω
2
p
+
2
sin
2
Θ
+
+
p
−
2
sin
2
Θ
−
(
E
+
−
c
|
p
+
|
cos
Θ
+
)
(
E
−
−
c
|
p
−
|
cos
Θ
−
)
+
2
|
p
+
|
|
p
−
|
sin
Θ
+
sin
Θ
−
cos
Φ
(
E
+
−
c
|
p
+
|
cos
Θ
+
)
(
E
−
−
c
|
p
−
|
cos
Θ
−
)
(
2
E
+
2
+
2
E
−
2
−
c
2
q
2
)
]
.
{\displaystyle {\begin{aligned}d^{4}\sigma &={\frac {Z^{2}\alpha _{fine}^{3}c^{2}}{(2\pi )^{2}\hbar }}|\mathbf {p} _{+}||\mathbf {p} _{-}|{\frac {dE_{+}}{\omega ^{3}}}{\frac {d\Omega _{+}d\Omega _{-}d\Phi }{|\mathbf {q} |^{4}}}\times \\&\times \left[-{\frac {\mathbf {p} _{-}^{2}\sin ^{2}\Theta _{-}}{(E_{-}-c|\mathbf {p} _{-}|\cos \Theta _{-})^{2}}}\left(4E_{+}^{2}-c^{2}\mathbf {q} ^{2}\right)\right.\\&-{\frac {\mathbf {p} _{+}^{2}\sin ^{2}\Theta _{+}}{(E_{+}-c|\mathbf {p} _{+}|\cos \Theta _{+})^{2}}}\left(4E_{-}^{2}-c^{2}\mathbf {q} ^{2}\right)\\&+2\hbar ^{2}\omega ^{2}{\frac {\mathbf {p} _{+}^{2}\sin ^{2}\Theta _{+}+\mathbf {p} _{-}^{2}\sin ^{2}\Theta _{-}}{(E_{+}-c|\mathbf {p} _{+}|\cos \Theta _{+})(E_{-}-c|\mathbf {p} _{-}|\cos \Theta _{-})}}\\&+2\left.{\frac {|\mathbf {p} _{+}||\mathbf {p} _{-}|\sin \Theta _{+}\sin \Theta _{-}\cos \Phi }{(E_{+}-c|\mathbf {p} _{+}|\cos \Theta _{+})(E_{-}-c|\mathbf {p} _{-}|\cos \Theta _{-})}}\left(2E_{+}^{2}+2E_{-}^{2}-c^{2}\mathbf {q} ^{2}\right)\right].\\\end{aligned}}}
каде
d
Ω
+
=
sin
Θ
+
d
Θ
+
,
d
Ω
−
=
sin
Θ
−
d
Θ
−
.
{\displaystyle {\begin{aligned}d\Omega _{+}&=\sin \Theta _{+}\ d\Theta _{+},\\d\Omega _{-}&=\sin \Theta _{-}\ d\Theta _{-}.\end{aligned}}}
Овој израз се добива со користење на квантно механичката симетрија меѓу создавањето на парови и запирното зрачење .
Z
{\displaystyle Z}
атомскиот број ,
α
f
i
n
e
≈
1
/
137
{\displaystyle \alpha _{fine}\approx 1/137}
константата на фината структура ,
ℏ
{\displaystyle \hbar }
Планкова константа и
c
{\displaystyle c}
брзината на светлината . Кинетичките енергии
E
k
i
n
,
+
/
−
{\displaystyle E_{kin,+/-}}
E
+
,
−
{\displaystyle E_{+,-}}
импулсите
p
+
,
−
{\displaystyle \mathbf {p} _{+,-}}
E
+
,
−
=
E
k
i
n
,
+
/
−
+
m
e
c
2
=
m
e
2
c
4
+
p
+
,
−
2
c
2
.
{\displaystyle E_{+,-}=E_{kin,+/-}+m_{e}c^{2}={\sqrt {m_{e}^{2}c^{4}+\mathbf {p} _{+,-}^{2}c^{2}}}.}
Зачувувањето на енергијата дава:
ℏ
ω
=
E
+
+
E
−
.
{\displaystyle \hbar \omega =E_{+}+E_{-}.}
Импулсот
q
{\displaystyle \mathbf {q} }
виртуелниот фотон меѓу упадниот фотон и јадрото е:
−
q
2
=
−
|
p
+
|
2
−
|
p
−
|
2
−
(
ℏ
c
ω
)
2
+
2
|
p
+
|
ℏ
c
ω
cos
Θ
+
+
2
|
p
−
|
ℏ
c
ω
cos
Θ
−
−
2
|
p
+
|
|
p
−
|
(
cos
Θ
+
cos
Θ
−
+
sin
Θ
+
sin
Θ
−
cos
Φ
)
,
{\displaystyle {\begin{aligned}-\mathbf {q} ^{2}&=-|\mathbf {p} _{+}|^{2}-|\mathbf {p} _{-}|^{2}-\left({\frac {\hbar }{c}}\omega \right)^{2}+2|\mathbf {p} _{+}|{\frac {\hbar }{c}}\omega \cos \Theta _{+}+2|\mathbf {p} _{-}|{\frac {\hbar }{c}}\omega \cos \Theta _{-}\\&-2|\mathbf {p} _{+}||\mathbf {p} _{-}|(\cos \Theta _{+}\cos \Theta _{-}+\sin \Theta _{+}\sin \Theta _{-}\cos \Phi ),\end{aligned}}}
каде насоките се дадени преку:
Θ
+
=
∢
(
p
+
,
k
)
,
Θ
−
=
∢
(
p
−
,
k
)
,
Φ
=
Agol megju ramninite
(
p
+
,
k
)
i
(
p
−
,
k
)
,
{\displaystyle {\begin{aligned}\Theta _{+}&=\sphericalangle (\mathbf {p} _{+},\mathbf {k} ),\\\Theta _{-}&=\sphericalangle (\mathbf {p} _{-},\mathbf {k} ),\\\Phi &={\text{Agol megju ramninite }}(\mathbf {p} _{+},\mathbf {k} ){\text{ i }}(\mathbf {p} _{-},\mathbf {k} ),\end{aligned}}}
каде
k
{\displaystyle \mathbf {k} }
За да се разгледа односот меѓу енергијата на фотонот
E
+
{\displaystyle E_{+}}
Θ
+
{\displaystyle \Theta _{+}}
Θ
−
{\displaystyle \Theta _{-}}
Φ
{\displaystyle \Phi }
[ 3]
d
2
σ
(
E
+
,
ω
,
Θ
+
)
d
E
+
d
Ω
+
=
∑
j
=
1
6
I
j
{\displaystyle {\begin{aligned}{\frac {d^{2}\sigma (E_{+},\omega ,\Theta _{+})}{dE_{+}d\Omega _{+}}}=\sum \limits _{j=1}^{6}I_{j}\end{aligned}}}
со
I
1
=
2
π
A
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
×
ln
(
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
−
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
(
Δ
1
(
p
)
+
Δ
2
(
p
)
)
+
Δ
1
(
p
)
Δ
2
(
p
)
−
(
Δ
2
(
p
)
)
2
−
4
p
+
2
p
−
2
sin
2
Θ
+
−
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
(
Δ
1
(
p
)
−
Δ
2
(
p
)
)
+
Δ
1
(
p
)
Δ
2
(
p
)
)
×
[
−
1
−
c
Δ
2
(
p
)
p
−
(
E
+
−
c
p
+
cos
Θ
+
)
+
p
+
2
c
2
sin
2
Θ
+
(
E
+
−
c
p
+
cos
Θ
+
)
2
−
2
ℏ
2
ω
2
p
−
Δ
2
(
p
)
c
(
E
+
−
c
p
+
cos
Θ
+
)
(
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
)
]
,
I
2
=
2
π
A
c
p
−
(
E
+
−
c
p
+
cos
Θ
+
)
ln
(
E
−
+
p
−
c
E
−
−
p
−
c
)
,
I
3
=
2
π
A
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
×
ln
(
(
(
E
−
+
p
−
c
)
(
4
p
+
2
p
−
2
sin
2
Θ
+
(
E
−
−
p
−
c
)
+
(
Δ
1
(
p
)
+
Δ
2
(
p
)
)
(
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
−
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
)
)
(
(
E
−
−
p
−
c
)
(
4
p
+
2
p
−
2
sin
2
Θ
+
(
−
E
−
−
p
−
c
)
+
(
Δ
1
(
p
)
−
Δ
2
(
p
)
)
(
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
−
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
)
)
−
1
)
×
[
c
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
p
−
(
E
+
−
c
p
+
cos
Θ
+
)
+
[
(
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
)
(
E
−
3
+
E
−
p
−
c
)
+
p
−
c
(
2
(
(
Δ
1
(
p
)
)
2
−
4
p
+
2
p
−
2
sin
2
Θ
+
)
E
−
p
−
c
+
Δ
1
(
p
)
Δ
2
(
p
)
(
3
E
−
2
+
p
−
2
c
2
)
)
]
[
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
]
−
1
+
[
−
8
p
+
2
p
−
2
m
2
c
4
sin
2
Θ
+
(
E
+
2
+
E
−
2
)
−
2
ℏ
2
ω
2
p
+
2
sin
2
Θ
+
p
−
c
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
+
2
ℏ
2
ω
2
p
−
m
2
c
3
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
]
[
(
E
+
−
c
p
+
cos
Θ
+
)
(
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
]
−
1
+
4
E
+
2
p
−
2
(
2
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
−
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
(
Δ
1
(
p
)
E
−
+
Δ
2
(
p
)
p
−
c
)
(
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
2
]
,
I
4
=
4
π
A
p
−
c
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
+
16
π
E
+
2
p
−
2
A
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
(
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
2
,
I
5
=
4
π
A
(
−
(
Δ
2
(
p
)
)
2
+
(
Δ
1
(
p
)
)
2
−
4
p
+
2
p
−
2
sin
2
Θ
+
)
(
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
×
[
ℏ
2
ω
2
p
−
2
E
+
c
p
+
cos
Θ
+
[
E
−
[
2
(
Δ
2
(
p
)
)
2
(
(
Δ
2
(
p
)
)
2
−
(
Δ
1
(
p
)
)
2
)
+
8
p
+
2
p
−
2
sin
2
Θ
+
(
(
Δ
2
(
p
)
)
2
+
(
Δ
1
(
p
)
)
2
)
]
+
p
−
c
[
2
Δ
1
(
p
)
Δ
2
(
p
)
(
(
Δ
2
(
p
)
)
2
−
(
Δ
1
(
p
)
)
2
)
+
16
Δ
1
(
p
)
Δ
2
(
p
)
p
+
2
p
−
2
sin
2
Θ
+
]
]
[
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
]
−
1
+
2
ℏ
2
ω
2
p
+
2
sin
2
Θ
+
(
2
Δ
1
(
p
)
Δ
2
(
p
)
p
−
c
+
2
(
Δ
2
(
p
)
)
2
E
−
+
8
p
+
2
p
−
2
sin
2
Θ
+
E
−
)
E
+
−
c
p
+
cos
Θ
+
−
[
2
E
+
2
p
−
2
{
2
(
(
Δ
2
(
p
)
)
2
−
(
Δ
1
(
p
)
)
2
)
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
8
p
+
2
p
−
2
sin
2
Θ
+
[
(
(
Δ
1
(
p
)
)
2
+
(
Δ
2
(
p
)
)
2
)
(
E
−
2
+
p
−
2
c
2
)
+
4
Δ
1
(
p
)
Δ
2
(
p
)
E
−
p
−
c
]
}
]
[
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
]
−
1
−
8
p
+
2
p
−
2
sin
2
Θ
+
(
E
+
2
+
E
−
2
)
(
Δ
2
(
p
)
p
−
c
+
Δ
1
(
p
)
E
−
)
E
+
−
c
p
+
cos
Θ
+
]
,
I
6
=
−
16
π
E
−
2
p
+
2
sin
2
Θ
+
A
(
E
+
−
c
p
+
cos
Θ
+
)
2
(
−
(
Δ
2
(
p
)
)
2
+
(
Δ
1
(
p
)
)
2
−
4
p
+
2
p
−
2
sin
2
Θ
+
)
{\displaystyle {\begin{aligned}I_{1}&={\frac {2\pi A}{\sqrt {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}}\\&\times \ln \left({\frac {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}-{\sqrt {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}(\Delta _{1}^{(p)}+\Delta _{2}^{(p)})+\Delta _{1}^{(p)}\Delta _{2}^{(p)}}{-(\Delta _{2}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}-{\sqrt {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}(\Delta _{1}^{(p)}-\Delta _{2}^{(p)})+\Delta _{1}^{(p)}\Delta _{2}^{(p)}}}\right)\\&\times \left[-1-{\frac {c\Delta _{2}^{(p)}}{p_{-}(E_{+}-cp_{+}\cos \Theta _{+})}}+{\frac {p_{+}^{2}c^{2}\sin ^{2}\Theta _{+}}{(E_{+}-cp_{+}\cos \Theta _{+})^{2}}}-{\frac {2\hbar ^{2}\omega ^{2}p_{-}\Delta _{2}^{(p)}}{c(E_{+}-cp_{+}\cos \Theta _{+})((\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})}}\right],\\I_{2}&={\frac {2\pi Ac}{p_{-}(E_{+}-cp_{+}\cos \Theta _{+})}}\ln \left({\frac {E_{-}+p_{-}c}{E_{-}-p_{-}c}}\right),\\I_{3}&={\frac {2\pi A}{\sqrt {(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}}\\&\times \ln {\Bigg (}{\Big (}(E_{-}+p_{-}c)(4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}(E_{-}-p_{-}c)+(\Delta _{1}^{(p)}+\Delta _{2}^{(p)})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)\\&-{\sqrt {(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}})){\Big )}{\Big (}(E_{-}-p_{-}c)(4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}(-E_{-}-p_{-}c)\\&+(\Delta _{1}^{(p)}-\Delta _{2}^{(p)})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)-{\sqrt {(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}})){\Big )}^{-1}{\Bigg )}\\&\times \left[{\frac {c(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)}{p_{-}(E_{+}-cp_{+}\cos \Theta _{+})}}\right.\\&+{\Big [}((\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})(E_{-}^{3}+E_{-}p_{-}c)+p_{-}c(2((\Delta _{1}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})E_{-}p_{-}c\\&+\Delta _{1}^{(p)}\Delta _{2}^{(p)}(3E_{-}^{2}+p_{-}^{2}c^{2})){\Big ]}{\Big [}(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}{\Big ]}^{-1}\\&+{\Big [}-8p_{+}^{2}p_{-}^{2}m^{2}c^{4}\sin ^{2}\Theta _{+}(E_{+}^{2}+E_{-}^{2})-2\hbar ^{2}\omega ^{2}p_{+}^{2}\sin ^{2}\Theta _{+}p_{-}c(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)\\&+2\hbar ^{2}\omega ^{2}p_{-}m^{2}c^{3}(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c){\Big ]}{\Big [}(E_{+}-cp_{+}\cos \Theta _{+})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}){\Big ]}^{-1}\\&+\left.{\frac {4E_{+}^{2}p_{-}^{2}(2(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}-4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})(\Delta _{1}^{(p)}E_{-}+\Delta _{2}^{(p)}p_{-}c)}{((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})^{2}}}\right],\\I_{4}&={\frac {4\pi Ap_{-}c(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)}{(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}+{\frac {16\pi E_{+}^{2}p_{-}^{2}A(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}}{((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})^{2}}},\\I_{5}&={\frac {4\pi A}{(-(\Delta _{2}^{(p)})^{2}+(\Delta _{1}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})}}\\&\times \left[{\frac {\hbar ^{2}\omega ^{2}p_{-}^{2}}{E_{+}cp_{+}\cos \Theta _{+}}}{\Big [}E_{-}[2(\Delta _{2}^{(p)})^{2}((\Delta _{2}^{(p)})^{2}-(\Delta _{1}^{(p)})^{2})+8p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}((\Delta _{2}^{(p)})^{2}+(\Delta _{1}^{(p)})^{2})]\right.\\&+p_{-}c[2\Delta _{1}^{(p)}\Delta _{2}^{(p)}((\Delta _{2}^{(p)})^{2}-(\Delta _{1}^{(p)})^{2})+16\Delta _{1}^{(p)}\Delta _{2}^{(p)}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}]{\Big ]}{\Big [}(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}{\Big ]}^{-1}\\&+{\frac {2\hbar ^{2}\omega ^{2}p_{+}^{2}\sin ^{2}\Theta _{+}(2\Delta _{1}^{(p)}\Delta _{2}^{(p)}p_{-}c+2(\Delta _{2}^{(p)})^{2}E_{-}+8p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}E_{-})}{E_{+}-cp_{+}\cos \Theta _{+}}}\\&-{\Big [}2E_{+}^{2}p_{-}^{2}\{2((\Delta _{2}^{(p)})^{2}-(\Delta _{1}^{(p)})^{2})(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+8p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}[((\Delta _{1}^{(p)})^{2}+(\Delta _{2}^{(p)})^{2})(E_{-}^{2}+p_{-}^{2}c^{2})\\&+4\Delta _{1}^{(p)}\Delta _{2}^{(p)}E_{-}p_{-}c]\}{\Big ]}{\Big [}(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}{\Big ]}^{-1}\\&-\left.{\frac {8p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}(E_{+}^{2}+E_{-}^{2})(\Delta _{2}^{(p)}p_{-}c+\Delta _{1}^{(p)}E_{-})}{E_{+}-cp_{+}\cos \Theta _{+}}}\right],\\I_{6}&=-{\frac {16\pi E_{-}^{2}p_{+}^{2}\sin ^{2}\Theta _{+}A}{(E_{+}-cp_{+}\cos \Theta _{+})^{2}(-(\Delta _{2}^{(p)})^{2}+(\Delta _{1}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})}}\end{aligned}}}
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{\displaystyle {\begin{aligned}A&={\frac {Z^{2}\alpha _{fine}^{3}c^{2}}{(2\pi )^{2}\hbar }}{\frac {|\mathbf {p} _{+}||\mathbf {p} _{-}|}{\omega ^{3}}},\\\Delta _{1}^{(p)}&:=-|\mathbf {p} _{+}|^{2}-|\mathbf {p} _{-}|^{2}-\left({\frac {\hbar }{c}}\omega \right)+2{\frac {\hbar }{c}}\omega |\mathbf {p} _{+}|\cos \Theta _{+},\\\Delta _{2}^{(p)}&:=2{\frac {\hbar }{c}}\omega |\mathbf {p} _{i}|-2|\mathbf {p} _{+}||\mathbf {p} _{-}|\cos \Theta _{+}+2.\end{aligned}}}
Овој напречен пресек може да се искорити во Монте Карло симулации. Анализата на овој израз покажува дека позитроните се главно оддадени во насока на упадниот фотон.
![{\displaystyle {\begin{aligned}d^{4}\sigma &={\frac {Z^{2}\alpha _{fine}^{3}c^{2}}{(2\pi )^{2}\hbar }}|\mathbf {p} _{+}||\mathbf {p} _{-}|{\frac {dE_{+}}{\omega ^{3}}}{\frac {d\Omega _{+}d\Omega _{-}d\Phi }{|\mathbf {q} |^{4}}}\times \\&\times \left[-{\frac {\mathbf {p} _{-}^{2}\sin ^{2}\Theta _{-}}{(E_{-}-c|\mathbf {p} _{-}|\cos \Theta _{-})^{2}}}\left(4E_{+}^{2}-c^{2}\mathbf {q} ^{2}\right)\right.\\&-{\frac {\mathbf {p} _{+}^{2}\sin ^{2}\Theta _{+}}{(E_{+}-c|\mathbf {p} _{+}|\cos \Theta _{+})^{2}}}\left(4E_{-}^{2}-c^{2}\mathbf {q} ^{2}\right)\\&+2\hbar ^{2}\omega ^{2}{\frac {\mathbf {p} _{+}^{2}\sin ^{2}\Theta _{+}+\mathbf {p} _{-}^{2}\sin ^{2}\Theta _{-}}{(E_{+}-c|\mathbf {p} _{+}|\cos \Theta _{+})(E_{-}-c|\mathbf {p} _{-}|\cos \Theta _{-})}}\\&+2\left.{\frac {|\mathbf {p} _{+}||\mathbf {p} _{-}|\sin \Theta _{+}\sin \Theta _{-}\cos \Phi }{(E_{+}-c|\mathbf {p} _{+}|\cos \Theta _{+})(E_{-}-c|\mathbf {p} _{-}|\cos \Theta _{-})}}\left(2E_{+}^{2}+2E_{-}^{2}-c^{2}\mathbf {q} ^{2}\right)\right].\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff9871e8a58506dabaab65e0a880ab92510f6f37)
![{\displaystyle {\begin{aligned}I_{1}&={\frac {2\pi A}{\sqrt {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}}\\&\times \ln \left({\frac {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}-{\sqrt {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}(\Delta _{1}^{(p)}+\Delta _{2}^{(p)})+\Delta _{1}^{(p)}\Delta _{2}^{(p)}}{-(\Delta _{2}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}-{\sqrt {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}(\Delta _{1}^{(p)}-\Delta _{2}^{(p)})+\Delta _{1}^{(p)}\Delta _{2}^{(p)}}}\right)\\&\times \left[-1-{\frac {c\Delta _{2}^{(p)}}{p_{-}(E_{+}-cp_{+}\cos \Theta _{+})}}+{\frac {p_{+}^{2}c^{2}\sin ^{2}\Theta _{+}}{(E_{+}-cp_{+}\cos \Theta _{+})^{2}}}-{\frac {2\hbar ^{2}\omega ^{2}p_{-}\Delta _{2}^{(p)}}{c(E_{+}-cp_{+}\cos \Theta _{+})((\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})}}\right],\\I_{2}&={\frac {2\pi Ac}{p_{-}(E_{+}-cp_{+}\cos \Theta _{+})}}\ln \left({\frac {E_{-}+p_{-}c}{E_{-}-p_{-}c}}\right),\\I_{3}&={\frac {2\pi A}{\sqrt {(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}}\\&\times \ln {\Bigg (}{\Big (}(E_{-}+p_{-}c)(4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}(E_{-}-p_{-}c)+(\Delta _{1}^{(p)}+\Delta _{2}^{(p)})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)\\&-{\sqrt {(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}})){\Big )}{\Big (}(E_{-}-p_{-}c)(4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}(-E_{-}-p_{-}c)\\&+(\Delta _{1}^{(p)}-\Delta _{2}^{(p)})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)-{\sqrt {(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}})){\Big )}^{-1}{\Bigg )}\\&\times \left[{\frac {c(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)}{p_{-}(E_{+}-cp_{+}\cos \Theta _{+})}}\right.\\&+{\Big [}((\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})(E_{-}^{3}+E_{-}p_{-}c)+p_{-}c(2((\Delta _{1}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})E_{-}p_{-}c\\&+\Delta _{1}^{(p)}\Delta _{2}^{(p)}(3E_{-}^{2}+p_{-}^{2}c^{2})){\Big ]}{\Big [}(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}{\Big ]}^{-1}\\&+{\Big [}-8p_{+}^{2}p_{-}^{2}m^{2}c^{4}\sin ^{2}\Theta _{+}(E_{+}^{2}+E_{-}^{2})-2\hbar ^{2}\omega ^{2}p_{+}^{2}\sin ^{2}\Theta _{+}p_{-}c(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)\\&+2\hbar ^{2}\omega ^{2}p_{-}m^{2}c^{3}(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c){\Big ]}{\Big [}(E_{+}-cp_{+}\cos \Theta _{+})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}){\Big ]}^{-1}\\&+\left.{\frac {4E_{+}^{2}p_{-}^{2}(2(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}-4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})(\Delta _{1}^{(p)}E_{-}+\Delta _{2}^{(p)}p_{-}c)}{((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})^{2}}}\right],\\I_{4}&={\frac {4\pi Ap_{-}c(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)}{(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}+{\frac {16\pi E_{+}^{2}p_{-}^{2}A(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}}{((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})^{2}}},\\I_{5}&={\frac {4\pi A}{(-(\Delta _{2}^{(p)})^{2}+(\Delta _{1}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})}}\\&\times \left[{\frac {\hbar ^{2}\omega ^{2}p_{-}^{2}}{E_{+}cp_{+}\cos \Theta _{+}}}{\Big [}E_{-}[2(\Delta _{2}^{(p)})^{2}((\Delta _{2}^{(p)})^{2}-(\Delta _{1}^{(p)})^{2})+8p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}((\Delta _{2}^{(p)})^{2}+(\Delta _{1}^{(p)})^{2})]\right.\\&+p_{-}c[2\Delta _{1}^{(p)}\Delta _{2}^{(p)}((\Delta _{2}^{(p)})^{2}-(\Delta _{1}^{(p)})^{2})+16\Delta _{1}^{(p)}\Delta _{2}^{(p)}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}]{\Big ]}{\Big [}(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}{\Big ]}^{-1}\\&+{\frac {2\hbar ^{2}\omega ^{2}p_{+}^{2}\sin ^{2}\Theta _{+}(2\Delta _{1}^{(p)}\Delta _{2}^{(p)}p_{-}c+2(\Delta _{2}^{(p)})^{2}E_{-}+8p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}E_{-})}{E_{+}-cp_{+}\cos \Theta _{+}}}\\&-{\Big [}2E_{+}^{2}p_{-}^{2}\{2((\Delta _{2}^{(p)})^{2}-(\Delta _{1}^{(p)})^{2})(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+8p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}[((\Delta _{1}^{(p)})^{2}+(\Delta _{2}^{(p)})^{2})(E_{-}^{2}+p_{-}^{2}c^{2})\\&+4\Delta _{1}^{(p)}\Delta _{2}^{(p)}E_{-}p_{-}c]\}{\Big ]}{\Big [}(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}{\Big ]}^{-1}\\&-\left.{\frac {8p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}(E_{+}^{2}+E_{-}^{2})(\Delta _{2}^{(p)}p_{-}c+\Delta _{1}^{(p)}E_{-})}{E_{+}-cp_{+}\cos \Theta _{+}}}\right],\\I_{6}&=-{\frac {16\pi E_{-}^{2}p_{+}^{2}\sin ^{2}\Theta _{+}A}{(E_{+}-cp_{+}\cos \Theta _{+})^{2}(-(\Delta _{2}^{(p)})^{2}+(\Delta _{1}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89082d84daae3b60c3793714c522e974deaa7739)
