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Integrated track length in electromagnetic shower

  1. Mar 11, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that integrated track length in EM-shower is proportional to ##E_0##.

    2. Relevant equations
    ##E(t)=\frac{E_0}{2^t}##, with radiation length, ##x_0##. Knowledge that shower terminates at ##E_c##.

    3. The attempt at a solution
    The total track length is naturally the total number of particles in the shower (until terminal ##E_c##) multiplied by the radiation length. Therefore, I have, $$T_{int}=x_0\int_0^{t_{max}}2^t\,dt,$$ where ##t_{max}=\log_2{\frac{E_0}{E_c}}##. Therefore, I have, $$T_{int}=x_0(\frac{2^{t_{max}}}{\ln(2)}-\frac{1}{\ln(2)})=\frac{x_0}{\ln(2)}(\frac{E_0}{E_c}-1).$$

    However, this is not exactly proportional to ##E_0##, evidently differing by a constant. Any ideas on where I may be going wrong?
     
  2. jcsd
  3. Mar 12, 2017 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    You can assume that ##\frac {E_0}{ E_c} \gg 1## - otherwise there would be no shower.

    The assumption that your track number is a continuous real number is an approximation anyway - you never have 1.1, 1.2462, ... tracks.
     
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