Basics
Detector & statistics in a nutshell
- Statistical data analysis in a nutshell
- Basic detector concepts
Detectors used in elementary particle and nuclear physics are based on the
principle to transfer radiation energy to detector mass. Charged particles
are transfering their energy through collisions to atomic electrons leading
to excitation and ionisation.
In most cases, neutral particles have to produce charged particles first
inside the detector volume which in turn are transfering their energy by
excitation or ionisation to the detector.
All these interaction processes are random processes.
There is no detector which is perfectly suited for all possible applications.
Depending on the application at hand detectors and complex detector systems
are designed according to several criteria.
There are several characteristics specifiying the features of a detector:
SENSITIVITY
The first question which arises is whether the detector is sensitive to
the type of radiation and energy range under consideration. The detector
sensitivity depends on the following factors:
* the interaction cross section with the detector material
* the detector mass
* the detector noise
* the protection material around the detector
Ex: If the interaction cross section is small (as e.g. for neutrinos
scattering on a detector material) one needs a large detector mass
in order to reach reasonable sensitivity if the incoming particle
flux can not be significantly increased.
DETECTOR RESPONSE (FUNCTION)
Related to sensitivity is the detector response (function) to the
radiation under study. Usually, the output signal of an electrical
detector is a current pulse where the time integral of the signal
corresponds to the amount of ionization produced by the
particle-detector interaction. If the shape of the signal does not
depend on the amount of ionization the amplitude of the signal is
a measure for the radiation energy deposited in the detector.
If this relation is linear the response of the detector is called
to be linear.
It is possible that a particle of defined energy leads to a spectrum
of signal amplitudes. This is called the detector response function.
E.g. photons with definite energy may interact with the detector
material by Compton scattering resulting into a broad spectrum of
deposited energies due to the subsequent interaction of the recoil
electrons.
In contrast, charged particles with definite energies which are
stopped within the detector material will rather lead to a Gaussian
distribution of signal amplitudes.
ENERGY RESOLUTION
Suppose the detector is designed to measure the energy of a particle.
Usually, due to fluctuations in the number of excitations and ionizations
in the detector material, one observes a Gaussian-like peak for a
monoenergetic particle beam instead of an ideal delta-function peak.
The width of this peak determines the capability to distinguish particles
with different energies.
The energy resolution
is given by the full-width-at-half-maximum (FWHM) of the signal peak.
For a Gaussian distribution with standard deviation σ we have
Energies closer than this resolution can not be separated.
The relative energy resolution is given by
In general, the average energy needed to produce a ionization is a constant
and only depends on the material.
As a consequence, the average number of ionizations increases with the
deposited energy. As the number of ionizations are fluctuating according
to a Poissonian distribution the relative energy resolution scales like
If the full energy of a particle is absorbed in the detector, Poissonian
statistics does not apply any more since the number of ionizations is
constrained by this energy value. As a consequence, the resolution may be
reduced by the so-called Fano factor F<1 resulting in improved resolution.
Example: We make use of the program
identification which
simulates the smearing of the measured energy in a electromagnetic calorimeter.
DETECTOR EFFICIENCY
The TOTAL EFFICIENCY of a detector is defined by the fraction of events
registered at the detector with respect to the number of events emitted
by a radiation source:
If the mean free path for an interaction with the detector material is
much smaller than the actual detector length then the total efficiency
can be written as the product
between the INTRINSIC EFFICIENCY and the GEOMETRICAL ACCEPTANCE of the
detector. The intrinsic efficiency is given by the fraction of events
registered by the detector with respect to the number of events hitting
the detector.
The geometrical acceptance depends in general on the polar and azimuthal
angle of the emitted particle. It may depend also on the momentum of the
particle (charged particles below a certain theshold momentum might be
stopped due to energy loss before reaching the detector). The acceptance
can also differ between different particle types: the detector device used
to reconstruct charged particles may cover an angular range different from
the detector device used to detect neutral particles.
In many cases, the intrinsic efficiency as well as the geometrical acceptance
are determined by a Monte Carlo simulation.
RESPONSE TIME
This is the time between the arrival of the radiation and the formation
of an output signal. If the signal is formed on a very short time scale
with a fast rising flank a precise moment in time can be marked by the
signal. This characteristics is of importance if timing information is
crucial e.g. in Time-Of-Flight measurements with scintillators or in
space determinations using a drift chamber through drift time measurements.
The duration of the signal is also important if dead time or a pile up
of subsequent signals may become important as discussed in the next
subsection.
DEAD TIME
Related to efficiency is the dead time of a detector.
The process of energy (charge) deposition and the 'readout' of the information
takes a finite time in which the detector and its associated electronics is not
able to register a subsequent signal. Depending on the type of detector and the
rate of particle interactions with the detector material the issue of dead time
can be rather important.
If the detector is insensitive to other events during the 'readout' time
these events are lost. If the detector is sensitive to additional events
during the 'readout' time these events may pile-up resulting in a distortion
of the signal.
In experiments with large particle fluxes dead time can be reduced by designing
the detector with a high granularity such that the occupancy per detector cell
is reasonably small. The occupancy measures the number of particles traversing
a detector cell per event. This will however increase the costs of building
the detector and its readout electronics.
Pile-up can be reduced by shaping the detector signal such that a very short
and as a consequence a small signal is formed. This however is limited by the
inherent noise of the detector.
- Problems