Квантово-механическое рассеяние фотона и ядра

При образовании пар фотон создает пару электрон-позитрон. В процессе рассеяния фотонов в воздухе (например, при разрядах молний ) важнейшим взаимодействием является рассеяние фотонов на ядрах атомов или молекул . Полный квантовомеханический процесс образования пар можно описать приведенным здесь четырехкратным дифференциальным сечением:

${\begin{aligned}d^{4}\sigma &={\frac {Z^{2}\alpha _{\textrm {fine}}^{3}c^{2}}{(2\pi )^{2}\hbar }}|\mathbf {p} _{+}||\mathbf {p} _{-}|{\frac {dE_{+}}{\omega ^{3}}}{\frac {d\Omega _{+}d\Omega _{-}d\Phi }{|\mathbf {q} |^{4}}}\times \\&\times \left[-{\frac {\mathbf {p} _{-}^{2}\sin ^{2}\Theta _{-}}{(E_{-}-c|\mathbf {p} _{-}|\cos \Theta _{-})^{2}}}\left(4E_{+}^{2}-c^{2}\mathbf {q} ^{2}\right)\right.\\&-{\frac {\mathbf {p} _{+}^{2}\sin ^{2}\Theta _{+}}{(E_{+}-c|\mathbf {p} _{+}|\cos \Theta _{+})^{2}}}\left(4E_{-}^{2}-c^{2}\mathbf {q} ^{2}\right)\\&+2\hbar ^{2}\omega ^{2}{\frac {\mathbf {p} _{+}^{2}\sin ^{2}\Theta _{+}+\mathbf {p} _{-}^{2}\sin ^{2}\Theta _{-}}{(E_{+}-c|\mathbf {p} _{+}|\cos \Theta _{+})(E_{-}-c|\mathbf {p} _{-}|\cos \Theta _{-})}}\\&+2\left.{\frac {|\mathbf {p} _{+}||\mathbf {p} _{-}|\sin \Theta _{+}\sin \Theta _{-}\cos \Phi }{(E_{+}-c|\mathbf {p} _{+}|\cos \Theta _{+})(E_{-}-c|\mathbf {p} _{-}|\cos \Theta _{-})}}\left(2E_{+}^{2}+2E_{-}^{2}-c^{2}\mathbf {q} ^{2}\right)\right].\\\end{aligned}}$

с

${\begin{aligned}d\Omega _{+}&=\sin \Theta _{+}\ d\Theta _{+},\\d\Omega _{-}&=\sin \Theta _{-}\ d\Theta _{-}.\end{aligned}}$

Это выражение можно получить, используя квантовомеханическую симметрию между образованием пар и тормозным излучением .
$Z$ атомный номер , $\alpha _{fine}\approx 1/137$ постоянная тонкой структуры , $\hbar$ приведенная постоянная Планка и $c$ скорость света . Кинетические энергии $E_{kin,+/-}$ позитрона и электрона относятся к их полным энергиям $E_{+,-}$ и импульс $\mathbf {p} _{+,-}$ с помощью

$E_{+,-}=E_{kin,+/-}+m_{e}c^{2}={\sqrt {m_{e}^{2}c^{4}+\mathbf {p} _{+,-}^{2}c^{2}}}.$

Сохранение энергетических результатов

$\hbar \omega =E_{+}+E_{-}.$

Импульс $\mathbf {q}$ виртуального фотона между падающим фотоном и ядром:

${\begin{aligned}-\mathbf {q} ^{2}&=-|\mathbf {p} _{+}|^{2}-|\mathbf {p} _{-}|^{2}-\left({\frac {\hbar }{c}}\omega \right)^{2}+2|\mathbf {p} _{+}|{\frac {\hbar }{c}}\omega \cos \Theta _{+}+2|\mathbf {p} _{-}|{\frac {\hbar }{c}}\omega \cos \Theta _{-}\\&-2|\mathbf {p} _{+}||\mathbf {p} _{-}|(\cos \Theta _{+}\cos \Theta _{-}+\sin \Theta _{+}\sin \Theta _{-}\cos \Phi ),\end{aligned}}$

где направления даны через:

${\begin{aligned}\Theta _{+}&=\sphericalangle (\mathbf {p} _{+},\mathbf {k} ),\\\Theta _{-}&=\sphericalangle (\mathbf {p} _{-},\mathbf {k} ),\\\Phi &={\text{Angle between the planes }}(\mathbf {p} _{+},\mathbf {k} ){\text{ and }}(\mathbf {p} _{-},\mathbf {k} ),\end{aligned}}$

где $\mathbf {k}$ – импульс падающего фотона.

Чтобы проанализировать связь между энергией фотона $E_{+}$ и угол излучения $\Theta _{+}$ между фотоном и позитроном Кён и Эберт интегрировали четырехкратное дифференциальное сечение по $\Theta _{-}$ и $\Phi$. Двойное дифференциальное сечение равно:

${\begin{aligned}{\frac {d^{2}\sigma (E_{+},\omega ,\Theta _{+})}{dE_{+}d\Omega _{+}}}=\sum \limits _{j=1}^{6}I_{j}\end{aligned}}$

с

${\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}}$

и

${\begin{aligned}A&={\frac {Z^{2}\alpha _{fine}^{3}c^{2}}{(2\pi )^{2}\hbar }}{\frac {|\mathbf {p} _{+}||\mathbf {p} _{-}|}{\omega ^{3}}},\\\Delta _{1}^{(p)}&:=-|\mathbf {p} _{+}|^{2}-|\mathbf {p} _{-}|^{2}-\left({\frac {\hbar }{c}}\omega \right)+2{\frac {\hbar }{c}}\omega |\mathbf {p} _{+}|\cos \Theta _{+},\\\Delta _{2}^{(p)}&:=2{\frac {\hbar }{c}}\omega |\mathbf {p} _{i}|-2|\mathbf {p} _{+}||\mathbf {p} _{-}|\cos \Theta _{+}+2.\end{aligned}}$

Это сечение можно применять при моделировании Монте-Карло. Анализ этого выражения показывает, что позитроны испускаются преимущественно в направлении падающего фотона.