Number of electron-positron pairs/sec out of Al converte
LINAC parameters used in calculations
1) pulse width 50 ps
2) pulse current 50 A
3) repetition rate 300 Hz
4) energy 44 MeV
Number of electrons/sec on radiator
[math]\displaystyle{ 50\ \frac{C}{sec} \times \frac{1\cdot e^-}{1.6\cdot 10^{-19}\ C} \times 50\ \mbox{ps} \times 300\ \mbox{Hz} = 0.47 \cdot 10^{13}\ \frac{e^-}{sec} }[/math]
Number of photons/sec out of radiator
- [math]\displaystyle{ \sigma_{brems}=0.1\ \mbox{photons/electrons/MeV/r.l} }[/math]
- [math]\displaystyle{ \mbox{r.l.(Ti)} = 3.59\ \mbox{cm} }[/math]
- [math]\displaystyle{ \mbox{radiator}\ \mbox{thickness} = 12.5\ \mu m }[/math]
[math]\displaystyle{ \frac{12.5\ \mu m}{3.59\ cm} = 3.48 \cdot 10^{-4}\ r.l. }[/math]
[math]\displaystyle{ 0.47 \cdot 10^{13} \frac{e^-}{sec} \times 0.1\ \frac{\gamma 's}{(e^- \cdot MeV \cdot r.l.)} \times 3.48 \cdot 10^{-4}\ r.l. \times 10\ MeV =1.63 \cdot 10^{9} \frac{\gamma}{sec} }[/math]
Alex factor is 6.85 %
[math]\displaystyle{ 1.64 \cdot 10^{9} \frac{\gamma}{sec} \cdot 6.85\ % = 1.12 \cdot 10^{8} \frac{\gamma}{sec} }[/math]
Pair production rate
out of Al converter
[math]\displaystyle{ \sigma_{pairs} = 0.5\ \frac{barns}{atom} }[/math]
[math]\displaystyle{ l = 3.0\ \mu m }[/math] (by varying width we can vary the yield)
[math]\displaystyle{ N_{Al} = \frac{2.375 \frac{g}{cm^3} \times 6.02 \cdot 10^{23} \frac{atoms}{mol} \times l} {26.98 \frac{g}{mol}} = 1.59 \cdot 10^{23}\ \frac{atoms}{m^2} }[/math]
[math]\displaystyle{ \frac {1.12 \cdot 10^{8} \frac{\gamma}{sec} \times \sigma_{pairs} \times N_{Al}} {f} = 2.96\ \frac{pairs}{pulse} }[/math]
through 1 m of air
[math]\displaystyle{ \sigma_{pairs} = 0.5\ \frac{barns}{atom} }[/math]
[math]\displaystyle{ l = 3.0\ \mu m }[/math] (by varying width we can vary the yield)
[math]\displaystyle{ N_{Al} = \frac{2.375 \frac{g}{cm^3} \times 6.02 \cdot 10^{23} \frac{atoms}{mol} \times l} {26.98 \frac{g}{mol}} = 1.59 \cdot 10^{23}\ \frac{atoms}{m^2} }[/math]
[math]\displaystyle{ \frac {1.12 \cdot 10^{8} \frac{\gamma}{sec} \times \sigma_{pairs} \times N_{Al}} {f} = 2.96\ \frac{pairs}{pulse} }[/math]