Energy spectrum of electromagnetic showers.

[Silversonic]

Homework Statement

Use the simple model for electromagnetic showers (explained below) to show that the energy spectrum of all secondary particles contained in an electromagnetic shower falls like $E^{-2}$ for $E_0 >> E >> E_c$

The Attempt at a Solution

An electron or a photon with energy $E_0$ goes through an electromagnetic calorimeter and undergoes bremmstrahlung or pair production respectively after a radiation length. Hence after 1 radiation length there are 2 particles (considering photons as particles for ease), after 2 lengths there are 4 etc. So after t radiation lengths there are $2^t$ particles and each particle has a mean energy of $\frac {E_0}{2^t}$.

When the particle energy goes below the critical energy $E_c$ the main source of energy loss is ionisation, the showering process then stops.

Now I'm confused by what the question actually wants and what is meant by "energy spectrum". Is it just asking for a given energy $E$, how many particles $N$ will exist? Because that would just be $\frac {E_0}{E}$. It's not though, because the answer says;

The number of particles with energy exceeding $E$ is

$N(>E) = \int^{t(E)}_{0} N(t)dt$

It then goes on to show $N(>E) = \frac {E_0}{ln(2) E}$ and then says

"implying $dN/dE \propto E^{-2}$".

I'm entirely confused here. So what exactly was the quantity that was wanted? The rate of change (with E) of the amount of particles with energy greater than E?

But the way I see it the number of particles with energy greater than E is the same as the area under a graph of E versus N from $E$ to $E_0$. i.e.

$N(>E) = \int^{E_0}_{E} N(E)dE$.

But $N(E) = \frac {E_0}{E}$

So the derivative of $\int^{E_0}_{E} N(E)dE$ with respect to E is just $N(E) = \frac {E_0}{E}$

Why are there two different results for the same thing?

 

[member 553083]

Am I missing something here?EDIT:The simple model for electromagnetic showers is that each particle undergoes bremmstrahlung or pair production after a radiation length and each daughter particle has half the energy of its parent.