Solution (source code)

= Solution

At low charged-lepton <energy>, collisional <ionization> and <Elementary excitation>[excitation] dominate over radiation. A slowing <Positron> also eventually annihilates, usually yielding <photons>. Low-energy <photons> lose <energy> through <Compton scattering> and, at sufficiently low <energy>, <Photoelectric effect>[photoelectric absorption]; <pair production> is kinematically unavailable below its threshold.

The <electron> <Critical energy of an electromagnetic shower>[critical energy] $E_c$ is the <energy> where mean <bremsstrahlung> and collisional losses are equal. With $-(dE/dx)_{\mathrm{rad}}\simeq E/X_0$ and a slowly varying collisional loss,
$$
\boxed{\left|\frac{dE}{dx}\right|_{\mathrm{rad},E_c}=\left|\frac{dE}{dx}\right|_{\mathrm{coll},E_c}.}
$$
Above this <energy>, radiation and <Pair production>[pair creation] multiply <Electromagnetic shower>[shower] particles efficiently. Below it, charged particles mainly deposit their <energy> by collisions and the <Electromagnetic shower>[shower] dies out. <Critical energy of an electromagnetic shower>[Critical energy] is material dependent; it is not a new photon-production threshold.