Recalling that the geometric cross section of a nucleus goes like <math>a_N^2</math> and that the nuclear radius goes like aN ~ A1/3, one can write the nuclear mean free path as:
λI ~ [35 g/cm²]A1/3
ν = x / λI
Hadronic showers develop through hadronic collisions. We expected most of the emitted particles to be pions, the lightest hadron, in equal fractions of π+, π-, and π0, which means 2/3 of the products will be charged. Because neutral pions decay to two photons almost immediately, they convert to an electromagnetic cascade. As in electromagnetic showers, there is a critical energy below which hadrons no longer have the energy to produce more pions. Empirically, this value, ETH, is equal to twice the pion mass. Below this energy, charged hadrons ionize. There is also a minimum transverse momentum limit of 0.4 GeV required for showers to continue. Approximately 9 particles are emitted in every hadronic interaction.
A shower maximum can be defined when the average energy of charged particles in the shower is equal to the threshold energy. For an incident pion with 250 GeV energy, the maximum occurs after three generations. Though we assume that neutral pions take 1/3 of the energy from each interaction, overall a much higher percentage is carried away by neutral pions because of their decay. In the infinite incident energy limit, the neutral fraction goes to 1.
In electromagnetic interactions, the nucleus basically allows balancing energy and momentum, but there are no contributions to Bremsstrahlung or pair production (for the most part). In hadronic showers, the interactions are with the nuclei, so a large amount of energy can be absorbed through binding energy effects, nuclear excitation, or pion ionization. The exact contributions of these effects and electromagnetic processes are not well understood and model dependent. This energy is only seen when the nuclei de-excites. These processes limit both the speed and resolution of hadronic calorimeters.
Because there are so many fluctuations in nuclear interactions, it is not possible to define a well reproduced hadronic shower shape. They can be thought of as a group of electromagnetic clusters.
When a calorimeter has a different response to electrons and hadrons, which is entirely expected, the ratio e/h defines the difference in this response. When this ratio is not 1, there is a constant term contributing to the energy resolution, which is proportional to the fractional fluctuation in the neutral energy fraction.
The stochastic contribution to the energy resolution is the coefficient of the term a/√E. The ratio of ah/ae goes like √(ETH/Ec) which is roughly 6. This term comes from the sampling fluctuations of the detector.
The transverse shower shape for hadronic showers is caused by the transverse momentum of secondary particles. The radius of the shower is defined as the transverse distance traveled during the last interaction length. The radius and the interaction length are roughly the same in hadronic interactions.
Because of finite detector depth, it's necessary to consider energy leakage. For a hadron of 1 TeV in incident energy, 99% of the shower energy will be contained with a depth of 11 λI. Since the depth increases as ln(E), the added expense for higher energy experiments is modest. It is possible to correct for leakage by weighting the last sampling portion of the calorimeter. There is also leakage caused by pion decays to muons, which only ionize in the calorimeter.