Why does N_max scale roughly as E₀/E_c?

In the equal-split toy model each generation doubles the particle count while halving energies. Multiplication stops when energies reach E_c, so the number of particles at maximum is of order the primary energy divided by the critical energy. Real showers are smoother and stochastic, but the same energy bookkeeping sets the scale.

What does the critical energy E_c represent?

E_c is the energy where radiative losses (bremsstrahlung) become comparable to ionization losses. Below it, electrons and positrons mainly deposit energy by ionization rather than creating new photons, so the cascade stops branching.

Is this the same as a Geant4 calorimeter simulation?

No. Geant4 tracks continuous energy loss, angular distributions, multiple scattering, and material-specific cross sections. This page is a didactic branching sketch that highlights N_max and t_max scalings, not a design-grade shower Monte Carlo.

EM Calorimeter Shower (Heitler / Rossi)

Toy electromagnetic cascade in a calorimeter: γ → e⁺e⁻, e⁻ → e⁻γ each X₀, with critical-energy cutoff E_c. Animate the branching tree, see N_max ≈ E₀/E_c, t_max ≈ log₂(E₀/E_c) live; presets for Pb, Cu, Si, air.

About this model

An electromagnetic cascade in matter is a branching tree of pair production and bremsstrahlung. This simulator implements a toy Heitler–Rossi model: each radiation length X₀ a high-energy photon converts to e⁺e⁻, and each charged lepton radiates a photon, until particle energies fall below a critical energy E_c where ionization dominates. With primary energy E₀ the approximate shower maximum has particle count N_max ≈ E₀/E_c at depth t_max ≈ log₂(E₀/E_c) radiation lengths. The animation shows the branching tree and these live estimates; material presets (Pb, Cu, Si, air) change X₀ and E_c. Assumptions are binary equal-energy splits, fixed generation per X₀, no lateral spread, no hadronic channel, and no full Monte Carlo cross sections. Vary E₀, E_c, and material to see how deeper, denser showers develop.

Who it's for: Advanced undergrad and graduate particle physics, cosmic-ray, and detector courses covering calorimetry and EM showers.

Key terms

Electromagnetic cascade
Heitler model
Radiation length
Critical energy
Pair production
Calorimeter shower

How it works

Heitler–Rossi electromagnetic shower — a toy model of how a high-energy γ or electron deposits energy in a calorimeter. Every radiation length X₀, an electron emits a bremsstrahlung γ and every γ pair-produces an e⁺e⁻ with the energy split 50:50, until each daughter falls below the critical energy E_c, at which point ionisation losses dominate and the cascade dies. The number of particles roughly doubles each X₀, giving the classic N_max ≈ E₀/E_c at depth t_max ≈ log₂(E₀/E_c) — the basis of all sampling-calorimeter design (LHC, CMS ECAL, ATLAS LAr). Animate the cascade growing into a lead absorber.

Key equations

e⁻ → e⁻ + γ (each X₀) γ → e⁺ e⁻ (each X₀)

N(t) ≈ 2ᵗ, ⟨E⟩(t) ≈ E₀ / 2ᵗ, stops when ⟨E⟩ = E_c

N_max ≈ E₀ / E_c, t_max ≈ log₂ (E₀ / E_c)

Frequently asked questions

Why does N_max scale roughly as E₀/E_c?: In the equal-split toy model each generation doubles the particle count while halving energies. Multiplication stops when energies reach E_c, so the number of particles at maximum is of order the primary energy divided by the critical energy. Real showers are smoother and stochastic, but the same energy bookkeeping sets the scale.
What does the critical energy E_c represent?: E_c is the energy where radiative losses (bremsstrahlung) become comparable to ionization losses. Below it, electrons and positrons mainly deposit energy by ionization rather than creating new photons, so the cascade stops branching.
Is this the same as a Geant4 calorimeter simulation?: No. Geant4 tracks continuous energy loss, angular distributions, multiple scattering, and material-specific cross sections. This page is a didactic branching sketch that highlights N_max and t_max scalings, not a design-grade shower Monte Carlo.

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Key equations

Frequently asked questions

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Frequently asked questions