Integrated track length in electromagnetic shower

[vbrasic]

Homework Statement

Show that integrated track length in EM-shower is proportional to $E_0$.

Homework Equations

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

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?

 

[mfb]

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.