PulsedMuonBeam

The function defining one pulse of the ISIS Extracted Proton Beam (EPB) versus time is approximated by the positive part of an inverted parabola

{$p(t)=-at^2+c$}

centered around time {$t=0$}, If we assume the full width of this beam to be {$\Delta\approx 80$} ns and require an area normalised to {$N$} particles per pulse, we get {$a=6N/\Delta^3$} and {$c=3N/2\Delta$}.

By making use of the simple integrals

{$ \array{{1\over\tau}\int\,_a^b e^{x/\tau} dx &=& e^{x/\tau}|_a^b \\ {1\over\tau}\int\,_a^b x e^{x/\tau} dx &=& (x - \tau) e^{x/\tau}|_a^b \\ {1\over\tau}\int\,_a^b x^2 e^{x/\tau} dx &=& (x^2-2\tau x+2\tau^2)e^{x/\tau}|_a^b}$}

one can compute the pion beam shape {$\pi(t)$} as the convolution of the EPB with the pion decay probability, characterised by a mean lifetime {$\tau_\pi$}=26 ns

The pion beam shape resulting from the convolution

{$ \pi(t)={1\over\tau_\pi}\int\,_{-\infty}^t p(x)\,e^{-{{t-x}\over{\tau_\pi}}}\, dx $}

is

{$\pi(t)=\left{ {\array{e^{-{t\over{\tau_\pi}}}\,\left{\left[c+a({{\Delta^2}\over4}-2\tau_\pi^2)\right]2\sinh{\Delta\over{2\tau_\pi}}+2a\tau_\pi\Delta\cosh{\Delta\over{2\tau_\pi}} \right}\qquad\qquad(t>{\Delta\over2})\\ c-2a\tau_\pi^2+\left[a({{\Delta^2}\over4}+\tau_\pi\Delta+2\tau_\pi^2)-c\right]e^{-{\Delta-2t}\over{2\tau_\pi}}\, +\, 2a\tau_\pi t\,-\,at^2 \qquad\qquad(-{\Delta\over2}\le t \le {\Delta\over2})} \right.$}

Finally, the unpolarised muon beam shape {$\mu(t)$} is obtained by a further convolution of the pion beam with the muon decay probability, characterised by a mean lifetime {$\tau_\mu$}=2196 ns

{$ \mu(t)={1\over \tau_\mu} \int\,_{\!\!\!\!\!-{\Delta\over2}}^t\,\, \pi(x)\, e^{-{{t-x}\over{\tau_\mu}}}\, dx $}

The result is

{$ \mu(t)= \left{ \array{ \mu_1(t) \qquad (t>{\Delta\over2})\\ \mu_2(t)\qquad (-{\Delta\over2}\le t\le {\Delta\over2}) }\right.$}

where

{$ \array{\mu_1(t) & = & {{\tau_\pi}\over{\tau_\mu-\tau_\pi}} \left{ \left[ c+a({{\Delta^2}\over4}-2\tau_\pi^2) \right] 2\sinh{\Delta\over{2\tau_\pi}}+2a\tau_\pi\Delta\cosh{\Delta\over{2\tau_\pi}} \right} (e^{{\Delta\over{2\tau_\mu}}-{\Delta\over{2\tau_\pi}}-{t\over{\tau_\mu}}}\,-\,e^{-{t\over{\tau_\mu}}})\\ & & + \mu_2({\Delta\over 2})\, e^{\Delta\over{2\tau_\mu}}\, e^{-{t\over{\tau_\mu}}} }$}

and

{$ \array{\mu_2(t) & = & [c-2a\tau_\pi^2](1\, - \, e^{{{\Delta-2t}\over{2\tau_\mu}}})\, + \, \left[a({{\Delta^2}\over4}+\tau_\pi\Delta+2\tau_\pi^2)-c\right] {{\tau_\pi}\over{\tau_\mu-\tau_\pi}} (e^{-{{\Delta+2t}\over{2\tau_\mu}}}\,- \,e^{-{{\Delta+2t}\over{2\tau_\pi}}}) \\ & & + 2a\tau_\pi\left[t-\tau_\mu+({\Delta\over2}+\tau_\mu)e^{-{{\Delta+2t}\over{2\tau_\mu}}}\right]\,-\,a \left[t^2-2\tau_\mu t +2\tau_\mu^2 - ({{\Delta^2}\over4}+\tau_\mu\Delta+2\tau_\mu^2)e^{-{{\Delta+2t}\over{2\tau_\mu}}} \right] }$}