For Pb glass with radiation length 1.2 cm and density 5.4 g/cm^3: If you were to

Question

For Pb glass with radiation length 1.2 cm and density 5.4 g/cm^3:
If you were to use time-of-flight to distinguish between electrons and pions, calculate the separation between two blocks of pb glass that would make it possible to distinguish between 1-5 GeV electrons and pions if you need at least 0.25 ns difference between two signals For Pb glass with radiation length 1.2 cm and density 5.4 g/cm^3:
If you were to use time-of-flight to distinguish between electrons and pions, calculate the separation between two blocks of pb glass that would make it possible to distinguish between 1-5 GeV electrons and pions if you need at least 0.25 ns difference between two signals
If you were to use time-of-flight to distinguish between electrons and pions, calculate the separation between two blocks of pb glass that would make it possible to distinguish between 1-5 GeV electrons and pions if you need at least 0.25 ns difference between two signals

Explanation / Answer

we consider both electron and pion is moving with 2Gev/c(initial momentum between1-5Gev/c)

now we need time of fligt difference =.25ns

using pc=Lm0c^2/sqrt(t^2c^2-L^2)

we get time of flight t=sqrt(L^2m0^2+p^2L^2)/pc [where L is the seperation between blocks and c is the velocity in each case]

for electron velocity =c/3 and for neutral pion velocity =.2c

considering c is the velocity of light

then t for electron =sqrt(L^2*(.000511)^2+2^2L^2)/2*c/3 [m0=rest mass of electron=9.1*10^-31 kg=.000511Gev/c^2 energy equivalent]

again t for pion=sqrt(L^2*(.013497)^2+2^2L^2)/2*.2c [m0= rest mass of pion=.013497Gev/c^2]

we can write

sqrt(L^2*(.013497)^2+2^2L^2)/2*.2c -sqrt(L^2*(.000511)^2+2^2L^2)/2*c/3 =.25ns=.25*10^-9s

or L=1.6m

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