Kvantový mechanický rozptyl fotonu a jádra - Quantum mechanical scattering of photon and nucleus

tento článek může být pro většinu čtenářů příliš technická na to, aby tomu rozuměli. Prosím pomozte to vylepšit na aby to bylo srozumitelné pro neodborníky, aniž by byly odstraněny technické podrobnosti. (Dubna 2015) (Zjistěte, jak a kdy odstranit tuto zprávu šablony)

v výroba párů, foton vytváří pár elektronových pozitronů. V procesu rozptylu fotonů vzduch (např. v Blesk nejdůležitějšími interakcemi je rozptyl fotonů v jádrech atomy nebo molekuly. Plný kvantově mechanické proces výroby párů lze popsat zde uvedeným čtyřnásobným diferenciálním průřezem:^[1]

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

s

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

Tento výraz lze odvodit pomocí kvantově mechanické symetrie mezi produkcí páru a Bremsstrahlung.
${ displaystyle Z}$ je protonové číslo, ${ displaystyle alpha _ {v pořádku} přibližně 1/137}$ the konstanta jemné struktury, ${ displaystyle hbar}$ snížený Planckova konstanta a ${ displaystyle c}$ the rychlost světla. Kinetické energie ${ displaystyle E_ {kin, + / -}}$ pozitronu a elektronu souvisí s jejich celkovou energií ${ displaystyle E _ {+, -}}$ a momenta ${ displaystyle mathbf {p} _ {+, -}}$ přes

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

Uchování energie výnosy

${ displaystyle hbar omega = E _ {+} + E _ {-}.}$

Hybnost ${ displaystyle mathbf {q}}$ z virtuální foton mezi dopadajícím fotonem a jádrem je:

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

kde jsou pokyny uvedeny pomocí:

${ displaystyle { begin {aligned} Theta _ {+} & = sphericalangle ( mathbf {p} _ {+}, mathbf {k}), Theta _ {-} & = sphericalangle ( mathbf {p} _ {-}, mathbf {k}), Phi & = { text {Úhel mezi rovinami}} ( mathbf {p} _ {+}, mathbf {k}) { text {and}} ( mathbf {p} _ {-}, mathbf {k}), end {zarovnáno}}}$

kde ${ displaystyle mathbf {k}}$ je hybnost dopadajícího fotonu.

Aby bylo možné analyzovat vztah mezi energií fotonu ${ displaystyle E _ {+}}$ a emisní úhel ${ displaystyle Theta _ {+}}$ mezi fotonem a pozitronem byly integrovány Köhn a Ebert ^[2] čtyřnásobný diferenciální průřez ${ displaystyle Theta _ {-}}$ a ${ displaystyle Phi}$. Dvojitý diferenciální průřez je:

${ displaystyle { begin {aligned} { frac {d ^ {2} sigma (E _ {+}, omega, Theta _ {+})} {dE _ {+} d Omega _ {+}} } = sum limity _ {j = 1} ^ {6} I_ {j} end {zarovnáno}}}$

s

${ 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}} vpravo), I_ {3} & = { frac {2 pi A} { sqrt {( Delta _ {2} ^ {(p)} E _ {-} + Delta _ {1} ^ {(p)} p _ {-} c) ^ {2} + 4 m ^ {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} + 4 m ^ {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} + 4 m ^ {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 _ {+})}} vpravo. & + { 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} + 4 m ^ {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} + 4 m ^ {2} c ^ {4} p _ {+} ^ {2} p _ {-} ^ {2} sin ^ {2} Theta _ {+}) { Big]} ^ {- 1} & + vlevo. { 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} + 4 m ^ {2} c ^ {4} p _ {+} ^ {2} p _ {-} ^ {2} v ^ {2} Theta _ {+}) ^ {2}}} vpravo], 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} + 4 m ^ {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} + 4 m ^ {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} + 4 m ^ {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})] vpravo. & + 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} + 4 m ^ {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 _ {+}}} vpravo], 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 {zarovnáno}}}$

a

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

Tento průřez lze použít v simulacích Monte Carlo. Analýza tohoto výrazu ukazuje, že pozitrony jsou emitovány hlavně ve směru dopadajícího fotonu.

Reference