Electromagnetic cascade in a calorimeter

[Kara386]

Homework Statement

A calorimeter is made from layers of lead (1.75mm thick) alternated with layers of scintillator. The radiation length $X_0$ of lead is $0.64cm$.

In an EM shower the number of particles doubles and the energy of each particle halves per radiation length travelled. The shower stops when critical energy $E_c$ is reached. For lead $E_c$ is 9.6MeV. Estimate the calorimeter thickness required to completely contain a shower caused by a 10GeV electron. Neglect interactions in the scintillator.

Homework Equations

The Attempt at a Solution

I know a calorimeter has to have scintillator as the first and last layers. So if there are n layers of scintillator, there will be n-1 layers of lead.

Based on the information given. I'm thinking the equation should be something like
$E = \frac{E_0}{2^{t/X_0}}$
Where t is the thickness of lead the shower travels through. Then thickness would be $(0.175t) \times 0.4(t+1)$. Is that ok? Or do I need to somehow include the doubling in particle number in there?

Thanks for any help!

 

[mfb]

I know a calorimeter has to have scintillator as the first and last layers.

It does not have to. Is there a scintillator thickness given? Otherwise you can just calculate the required length of lead.

Your second expression grows quadratically with t, that cannot be right. Did you mean "+"? Where does the 0.4 come from?