Remstraling

Remstraling ontstaat doordat een elektron afgebogen wordt in het elektrische veld van een atoomkern

Remstraling (uit het Duits, Bremsstrahlung) is een vorm van elektromagnetische straling die wordt uitgestraald als een geladen deeltje een acceleratie (of deceleratie) ondergaat, typisch bij het op een doel botsen van een versneld elektron in een röntgenbuis.

Het met hoge snelheid vliegende elektron wordt door botsing met de atomen in de anode sterk geremd en verliest hierbij veel van zijn snelheid; de verloren kinetische energie wordt als röntgenstraling uitgezonden.

Remstraling ontstaat wanneer elektronen aan een hoge snelheid een atoom binnenvliegen en onder de invloed van de kern van het atoom in hun baan afgebogen of zelfs volledig gestopt worden. Hierbij wordt de kinetische energie omgezet in röntgenstraling, in dit geval remstraling. Een andere manier om röntgenstraling op te wekken is aan de hand van karakteristieke straling.

Remstraling ontstaat ook in het heelal in emissienevels zoals H-II-gebieden en planetaire nevels die geïoniseerd gas bevatten met een temperatuur van ongeveer 10.000 K.

Kwantummechanische beschrijving

De volledige kwantummechanische beschrijving werd ten eerste door Bethe en Heitler uitgevoerd . Zij namen vlakke materiegolven voor elektronen aan die aan de kern van een atoom worden gebotst, en leidden daarmee een werkzame doorsnede af die de volledgie geometrie van dit process in verband met de frequentie van het geëmitteerde foton brengt. De differentielle werkzame doorsnede, die een kwantummechanische symmetrie voor paarproductie toont, is:

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

Daarbij zijn $Z$$Z$ het atoomnummer, $\alpha _{fine}\approx 1/137$$\alpha _{fine}\approx 1/137$ de fijnstructuurconstante, $\hbar$$\hbar$ de gereduceerde constante van Planck en $c$$c$ de lichtsnelheid. De kinetische energie $E_{kin,i/f}$$E_{kin,i/f}$ van het elektron in de begin- en eindtoestand hangen door

$E_{i,f}=E_{kin,i/f}+m_{e}c^{2}={\sqrt {m_{e}^{2}c^{4}+{\vec {p}}_{i,f}^{2}c^{2}}},$$E_{i,f}=E_{kin,i/f}+m_{e}c^{2}={\sqrt {m_{e}^{2}c^{4}+{\vec {p}}_{i,f}^{2}c^{2}}},$

waarbij $m_{e}$$m_{e}$ de elektronmassa is, met zijn totale energie $E_{i,f}$$E_{i,f}$ of zijn impuls ${\vec {p}}_{i,f}$${\vec {p}}_{i,f}$ samen. De wet van behoud van energie levert

$E_{f}=E_{i}-\hbar \omega ,$$E_{f}=E_{i}-\hbar \omega ,$

waarbij $\hbar \omega$$\hbar \omega$ de energie van het foton is. De richtingen van het geëmitteerde foton en het gestrooide elektron zijn gegeven door

${\begin{aligned}\Theta _{i}&=\sphericalangle ({\vec {p}}_{i},{\vec {k}}),\\\Theta _{f}&=\sphericalangle ({\vec {p}}_{f},{\vec {k}}),\\\Phi &={\text{Hoek tussen de vlakten }}({\vec {p}}_{i},{\vec {k}}){\text{ en }}({\vec {p}}_{f},{\vec {k}}).\end{aligned}}$${\begin{aligned}\Theta _{i}&=\sphericalangle ({\vec {p}}_{i},{\vec {k}}),\\\Theta _{f}&=\sphericalangle ({\vec {p}}_{f},{\vec {k}}),\\\Phi &={\text{Hoek tussen de vlakten }}({\vec {p}}_{i},{\vec {k}}){\text{ en }}({\vec {p}}_{f},{\vec {k}}).\end{aligned}}$

De differentialen zijn gegeven door

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

De absolute waarde van het virtuele foton tussen kern en elektron is

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

De geldigheid is door de Born-nadering

$v\gg {\frac {Zc}{137}}$$v\gg {\frac {Zc}{137}}$

gegeven, waar $v$$v$ zowel de snelheid van het elektron in de begin- als in de eindtoestand kenmerkt.

Voor praktische toepassingen (bvb. in Monte Carlo codes) kan het van belang zijn de prioriteit op de relatie tussen de frequentie $\omega$$\omega$ van het geëmitteerde foton en de hoek tussen dit foton en het inlopende elektron te leggen. Köhn und Ebert integreerden de bovenstaande term over $\Phi$$\Phi$ en $\Theta _{f}$$\Theta _{f}$ en kregen:

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

met

${\begin{aligned}I_{1}&={\frac {2\pi A}{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}}\ln \left({\frac {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}-{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}(\Delta _{1}+\Delta _{2})+\Delta _{1}\Delta _{2}}{-\Delta _{2}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}-{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}(\Delta _{1}-\Delta _{2})+\Delta _{1}\Delta _{2}}}\right)\\&\times \left[1+{\frac {c\Delta _{2}}{p_{f}(E_{i}-cp_{i}\cos \Theta _{i})}}-{\frac {p_{i}^{2}c^{2}\sin ^{2}\Theta _{i}}{(E_{i}-cp_{i}\cos \Theta _{i})^{2}}}-{\frac {2\hbar ^{2}\omega ^{2}p_{f}\Delta _{2}}{c(E_{i}-cp_{i}\cos \Theta _{i})(\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})}}\right],\\I_{2}&=-{\frac {2\pi Ac}{p_{f}(E_{i}-cp_{i}\cos \Theta _{i})}}\ln \left({\frac {E_{f}+p_{f}c}{E_{f}-p_{f}c}}\right),\\I_{3}&={\frac {2\pi A}{\sqrt {(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}}\\&\times \ln {\Bigg (}{\Big (}(E_{f}+p_{f}c)(4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}(E_{f}-p_{f}c)+(\Delta _{1}+\Delta _{2})((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)\\&-{\sqrt {(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}})){\Big )}{\Big (}(E_{f}-p_{f}c)(4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}(-E_{f}-p_{f}c)\\&+(\Delta _{1}-\Delta _{2})((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)-{\sqrt {(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}})){\Big )}^{-1}{\Bigg )}\\&\times \left[-{\frac {(\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})(E_{f}^{3}+E_{f}p_{f}^{2}c^{2})+p_{f}c(2(\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})E_{f}p_{f}c+\Delta _{1}\Delta _{2}(3E_{f}^{2}+p_{f}^{2}c^{2}))}{(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\right.\\&-{\frac {c(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)}{p_{f}(E_{i}-cp_{i}\cos \Theta _{i})}}\\&-{\frac {4E_{i}^{2}p_{f}^{2}(2(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}-4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})(\Delta _{1}E_{f}+\Delta _{2}p_{f}c)}{((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})^{2}}}\\&+\left.{\frac {8p_{i}^{2}p_{f}^{2}m^{2}c^{4}\sin ^{2}\Theta _{i}(E_{i}^{2}+E_{f}^{2})-2\hbar ^{2}\omega ^{2}p_{i}^{2}\sin ^{2}\Theta _{i}p_{f}c(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)+2\hbar ^{2}\omega ^{2}p_{f}m^{2}c^{3}(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)}{(E_{i}-cp_{i}\cos \Theta _{i})((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})}}\right],\\I_{4}&=-{\frac {4\pi Ap_{f}c(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)}{(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}-{\frac {16\pi E_{i}^{2}p_{f}^{2}A(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}}{((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})^{2}}},\\I_{5}&={\frac {4\pi A}{(-\Delta _{2}^{2}+\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})}}\\&\times \left[{\frac {\hbar ^{2}\omega ^{2}p_{f}^{2}}{E_{i}-cp_{i}\cos \Theta _{i}}}\right.\\&\times {\frac {E_{f}[2\Delta _{2}^{2}(\Delta _{2}^{2}-\Delta _{1}^{2})+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}(\Delta _{2}^{2}+\Delta _{1}^{2})]+p_{f}c[2\Delta _{1}\Delta _{2}(\Delta _{2}^{2}-\Delta _{1}^{2})+16\Delta _{1}\Delta _{2}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}]}{\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\\&+{\frac {2\hbar ^{2}\omega ^{2}p_{i}^{2}\sin ^{2}\Theta _{i}(2\Delta _{1}\Delta _{2}p_{f}c+2\Delta _{2}^{2}E_{f}+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}E_{f})}{E_{i}-cp_{i}\cos \Theta _{i}}}\\&+{\frac {2E_{i}^{2}p_{f}^{2}\{2(\Delta _{2}^{2}-\Delta _{1}^{2})(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}[(\Delta _{1}^{2}+\Delta _{2}^{2})(E_{f}^{2}+p_{f}^{2}c^{2})+4\Delta _{1}\Delta _{2}E_{f}p_{f}c]\}}{((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})}}\\&+\left.{\frac {8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}(E_{i}^{2}+E_{f}^{2})(\Delta _{2}p_{f}c+\Delta _{1}E_{f})}{E_{i}-cp_{i}\cos \Theta _{i}}}\right],\\I_{6}&={\frac {16\pi E_{f}^{2}p_{i}^{2}\sin ^{2}\Theta _{i}A}{(E_{i}-cp_{i}\cos \Theta _{i})^{2}(-\Delta _{2}^{2}+\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})}},\end{aligned}}$${\begin{aligned}I_{1}&={\frac {2\pi A}{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}}\ln \left({\frac {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}-{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}(\Delta _{1}+\Delta _{2})+\Delta _{1}\Delta _{2}}{-\Delta _{2}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}-{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}(\Delta _{1}-\Delta _{2})+\Delta _{1}\Delta _{2}}}\right)\\&\times \left[1+{\frac {c\Delta _{2}}{p_{f}(E_{i}-cp_{i}\cos \Theta _{i})}}-{\frac {p_{i}^{2}c^{2}\sin ^{2}\Theta _{i}}{(E_{i}-cp_{i}\cos \Theta _{i})^{2}}}-{\frac {2\hbar ^{2}\omega ^{2}p_{f}\Delta _{2}}{c(E_{i}-cp_{i}\cos \Theta _{i})(\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})}}\right],\\I_{2}&=-{\frac {2\pi Ac}{p_{f}(E_{i}-cp_{i}\cos \Theta _{i})}}\ln \left({\frac {E_{f}+p_{f}c}{E_{f}-p_{f}c}}\right),\\I_{3}&={\frac {2\pi A}{\sqrt {(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}}\\&\times \ln {\Bigg (}{\Big (}(E_{f}+p_{f}c)(4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}(E_{f}-p_{f}c)+(\Delta _{1}+\Delta _{2})((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)\\&-{\sqrt {(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}})){\Big )}{\Big (}(E_{f}-p_{f}c)(4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}(-E_{f}-p_{f}c)\\&+(\Delta _{1}-\Delta _{2})((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)-{\sqrt {(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}})){\Big )}^{-1}{\Bigg )}\\&\times \left[-{\frac {(\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})(E_{f}^{3}+E_{f}p_{f}^{2}c^{2})+p_{f}c(2(\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})E_{f}p_{f}c+\Delta _{1}\Delta _{2}(3E_{f}^{2}+p_{f}^{2}c^{2}))}{(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\right.\\&-{\frac {c(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)}{p_{f}(E_{i}-cp_{i}\cos \Theta _{i})}}\\&-{\frac {4E_{i}^{2}p_{f}^{2}(2(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}-4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})(\Delta _{1}E_{f}+\Delta _{2}p_{f}c)}{((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})^{2}}}\\&+\left.{\frac {8p_{i}^{2}p_{f}^{2}m^{2}c^{4}\sin ^{2}\Theta _{i}(E_{i}^{2}+E_{f}^{2})-2\hbar ^{2}\omega ^{2}p_{i}^{2}\sin ^{2}\Theta _{i}p_{f}c(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)+2\hbar ^{2}\omega ^{2}p_{f}m^{2}c^{3}(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)}{(E_{i}-cp_{i}\cos \Theta _{i})((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})}}\right],\\I_{4}&=-{\frac {4\pi Ap_{f}c(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)}{(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}-{\frac {16\pi E_{i}^{2}p_{f}^{2}A(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}}{((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})^{2}}},\\I_{5}&={\frac {4\pi A}{(-\Delta _{2}^{2}+\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})}}\\&\times \left[{\frac {\hbar ^{2}\omega ^{2}p_{f}^{2}}{E_{i}-cp_{i}\cos \Theta _{i}}}\right.\\&\times {\frac {E_{f}[2\Delta _{2}^{2}(\Delta _{2}^{2}-\Delta _{1}^{2})+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}(\Delta _{2}^{2}+\Delta _{1}^{2})]+p_{f}c[2\Delta _{1}\Delta _{2}(\Delta _{2}^{2}-\Delta _{1}^{2})+16\Delta _{1}\Delta _{2}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}]}{\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\\&+{\frac {2\hbar ^{2}\omega ^{2}p_{i}^{2}\sin ^{2}\Theta _{i}(2\Delta _{1}\Delta _{2}p_{f}c+2\Delta _{2}^{2}E_{f}+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}E_{f})}{E_{i}-cp_{i}\cos \Theta _{i}}}\\&+{\frac {2E_{i}^{2}p_{f}^{2}\{2(\Delta _{2}^{2}-\Delta _{1}^{2})(\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}[(\Delta _{1}^{2}+\Delta _{2}^{2})(E_{f}^{2}+p_{f}^{2}c^{2})+4\Delta _{1}\Delta _{2}E_{f}p_{f}c]\}}{((\Delta _{2}E_{f}+\Delta _{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})}}\\&+\left.{\frac {8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}(E_{i}^{2}+E_{f}^{2})(\Delta _{2}p_{f}c+\Delta _{1}E_{f})}{E_{i}-cp_{i}\cos \Theta _{i}}}\right],\\I_{6}&={\frac {16\pi E_{f}^{2}p_{i}^{2}\sin ^{2}\Theta _{i}A}{(E_{i}-cp_{i}\cos \Theta _{i})^{2}(-\Delta _{2}^{2}+\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i})}},\end{aligned}}$

en

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

Een analyse van de tweevoudige differentiële werkzame doorsnede toont bvb dat elektronen van wie de kinetische energie groter is dan de rustenergie (511 keV), fotonen overwegend naar voren uitzenden, waartegen elektronen met een lagere energie fotonen isotroop uitzenden.

Bronnen, noten en/of referenties