Physics:Energy resolution in calorimeters

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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:

Hepb img242.gif

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:

Hepb img243.gif

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:

Hepb img244.gif

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

    Hepb img245.gif

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

    Hepb img246.gif

    with Hepb img247.gif 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 Hepb img248.gif   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

    Hepb img249.gif

    have been reported. For more quantitative values, Hepb img34.gif 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 Hepb img64.gif 's and e+/e- from Hepb img77.gif 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:

    Hepb img250.gif

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

    Hepb img251.gif

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

    Hepb img252.gif

    with Hepb img247.gif again the energy lost by a single charged particle in one sampling layer (note that Hepb img253.gif is a very small number).
    Note that these numbers refer to single hadronic particles; the Hepb img254.gif 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.

More reading can be found e.g. in Gordon95, Fabjan91, Wigmans91a, Brau90.