Physics:Energy resolution in calorimeters

The ultimate limit for the energy resolution of a calorimeter is determined by fluctuations inherent in the development of showers, and by instrumental and calibration limits. The basic phenomena in showers are statistical processes, hence the intrinsic limiting accuracy, expressed as a fraction of total energy, improves with increasing energy as:

Over much of the useful range of calorimeters, this term dominates energy resolution.

There are other contributions than statistics, though: a second component is due to instrumental effects, being rather energy-independent (noise, pedestal); its relative contribution decreaseswith E:

This component may limit the low-energy performance of calorimeters.

A third component is due to calibration errors, non-uniformities and non-linearities in photomultipliers, proportional counters, ADC's, etc. This contribution is energy-independent:

This component sets the limit for the performance at very high energies.

The two types of showers have markedly different characteristics:

a) Electromagnetic Showers: electromagnetic showers the intrinsic limitation in resolution results from variations in the net track length of charged particles in the cascade; for homogeneous shower counters

In sampling calorimeters, one has to add the sampling fluctuations:

with the energy loss of a single charged particle in one sampling layer. There are also fluctuations arising from the Landau distribution; a comparison can be found in Fabjan91.
In practice, total energy resolution below the percent level for GeV can be achieved routinely in homogeneous calorimeters; the same seems more like a very tough lower limit for sampling calorimeters. At low energies and for crystal calorimeters, total energy resolutions

have been reported. For more quantitative values, Fabjan95a, Gratta94.
b) Hadronic Showers: Hadronic Shower the intrinsic limitation is due to fluctuations in the fractional energy loss accounted for by the many interaction mechanisms leaving behind non-hadronic debris (including muons and the 's and e⁺/e^- from decays) and slow neutrons, along with fast hadrons. The fluctuations in these production processes, much larger than for electromagnetic processes, are the major ingredient of the final performance of a hadron calorimeter.
Intrinsic shower fluctuations are given by:

for uncompensated calorimeters, and with compensation for nuclear effects (see Hadronic Shower, Compensating Calorimeter).

Compared with the intrinsic fluctuations, sampling fluctuations are normally small:

with again the energy lost by a single charged particle in one sampling layer (note that is a very small number).
Note that these numbers refer to single hadronic particles; the for jets is typically higher by a factor 1.3 or more.
Hadronic showers can spread over a large volume; a major source of systematic errors, therefore, is the geometric limitation of a calorimeter. The resolution figures determined by intrinsic shower and sampling fluctuations will not be reached if showers are not fully contained within the calorimeter volume. In practice some average fraction of the shower energy escapes through the sides (lateral leakage) or back (longitudinal leakage). While the corrections for longitudinal leakage are understood, and can partly be accounted for, corrections for lateral leakage need a careful inspection of the shower development and an estimate of the particle impact point.