PHYS 352
Radiation Detectors: Resolution and Dead Time –
General Considerations
Statistics of Charge Collection
• interaction of radiation with matter is a statistical process; there is a quantum
mechanical probability for any given collision, excitation or absorption
• even after all the energy is deposited, charge collection is also statistical
• hence Poisson statistics are involved in the amount of charge collected
σ= N
• for large values of N, distribution becomes Gaussian
∴
ΔE σ
N
1
= =
=
E
N
N
N
resolution can be given as σ
or full width half maximum
frequency of occurrence
• voltage signal is proportional to N which is supposed to be
∝E
σ
FWHM = 2.354σ
N
number of collected charges
Energy Resolution
• energy resolution ideally is as good as statistics allows but it can be worse
• what makes energy resolution worse than statistics?
• e.g. random electrical noise adds/subtracts from the size of the signal; this
adds statistical spread that is not at all related to sqrt(N) charges collected
• e.g. N = k E (charges collected proportional to energy deposited);
fluctuations in the proportionality constant k add additional fluctuations to
the uncertainty in N charges collected
• but the scaling of the energy resolution with energy still has a major portion
that goes as 1/sqrt(E) because of Poisson statistics
• detector resolution is often quoted like this:
ΔEFWHM 3%
=
+ 0.2%
E
E at 1MeV
Fano Factor
• surprisingly, when you look at the energy resolution of radiation detectors, like
gas ionization detectors, you find that the resolution is better than sqrt(N)?!
• this was studied by Fano and the explanation goes like this...
• the energy deposited by the particle equals # of ionizations times the average
energy required to ionize plus # of excitations times the average energy per
excitation
E = N ion I + N exc Eexc = N ion w
• simplified as E = Nion w, w is the average energy loss per electron-ion pair
produced
• remember: w is around 30 eV per e-i pair in a gas (e.g. we calculated
1
for 300 keV that’s about 10,000 electron-ion pairs)
or 1%
N
• the energy deposited is a fixed quantity though; it’s the total energy in the
gamma ray photopeak, for example
• thus, the uncertainty ΔNion is correlated with ΔNexc
E = N ion I + N exc Eexc = N ion w
Fano Factor cont’d
• Poisson statistics applies for independent variables but these are constrained
in that their weighted sum must equal the total energy deposited
• e.g. so you might happen to have a few more ionization events; that
means you must have a few less excitation events, and vice versa
• make up some more numbers:
• 2 excitations for every ionization
ΔN ion I = −ΔN exc Eexc
I = 20 eV
Eexc = 5 eV
• 300 keV deposited is 10,000 ionizations and 20,000 excitations
• consider it as 1 independent variable and 1 correlated one
ΔN ion =
N exc Eexc 5 20000
=
 35.4
I
20
• smaller than sqrt(10,000) = 100!
Fano factor
definition:
2
F≡
σ actual
; F ≤1
2
σ Poisson
σ actual = FN
Dead Time
• some detectors need time to recover before they are sensitive to another
radiation interaction (e.g. Geiger counter) – they are dead during this period
• other detectors are forming an electrical pulse, maybe with a long tail; when
another radiation interaction takes place – this is called pulse pileup and
distorts the pulse shape and possibly the energy measurement (based upon
pulse amplitude)
• the ADC used for data acquisition
also might have dead time during
the digitization of a pulse that
renders it dead to any other pulses
that might arrive
any of these can result in
inaccurate measurement
of radiation intensity
Aside: How Do You Digitize a Random Pulse?
• the signal amplitude from a radiation detector is related to the energy
deposited
• what ADC do you use?
• the pulse comes at a random time
• the pulses tend to be of short duration
• a discriminator triggers the data acquisition when the pulse exceeds a (small)
threshold value
• this pulse is then integrated
• typically with dual-slope ADC (Wilkinson)
• ADC is dead during discharge/counting period
Correction for Dead Time
• we know the properties of the detector (or ADC); can know that it is dead for
τ [s] after it responds to a single pulse
• observed event rate: m [counts/s]
• we say counts or events or decays or interactions (per second)
• true event rate: n [counts/s]
• fraction of the time the detector is dead: m τ (in other words m times a
second the detector is dead for τ seconds, where τ is <1; so we know what
fraction of the second is unobservable)
• thus n =
m
1 − mτ