Electromagnetic Calorimeter

What is modelled

A high-energy electron or photon entering dense material starts a cascade: electrons radiate photons and photons make electron pairs, doubling the number of particles every radiation length until the particles have too little energy to continue (the critical energy). The calorimeter absorbs all of this and its signal measures the energy, provided the shower is deep enough to stay inside. Longer showers appear only logarithmically with energy.

Equations

• Shower maximum tmax = ln(E0/Ec) ± 0.5 (minus for electrons, plus for photons), in radiation lengths.
• Longitudinal profile (Longo-Sestili) dE/dt = E0 b (bt)a−1 e−bt / Γ(a) with b = 0.5 and a = 1 + b tmax.
• Containment to depth T is the regularised incomplete Gamma function P(a, bT).
• Resolution σ/E = √( s²/E + c² + (0.7 · leakage)² ).

Checks run in the page

• The Gamma function and the integral of the profile against one.
• The numerical peak of the profile against (a − 1)/b.
• Numerical integration to the back face against the incomplete Gamma function.
• A Monte Carlo of 100,000 measurements against the resolution formula.

Honest simplifications

Only the longitudinal profile is modelled, the lateral spread is shown through the Moliere radius only as a picture, the parameterisation is for electromagnetic showers (hadronic showers are longer and fluctuate more), the stochastic and constant terms are free parameters and not computed from the material, and the leakage term is a crude fixed fraction.

• Press Too shallow, 1 TeV and watch the leaking fraction and the resolution worsen.
• Compare Lead-glass block with Lead tungstate crystal: the longer radiation length means the same energy needs a deeper block.
• Press Low energy photon and see how the shower maximum moves toward the front.