9.11. Counting Statistics 565
9.11CountingStatistics............................
Detection of single events is perhaps the most widely used investigation method in
particle physics, nuclear chemistry, nuclear medicine, radiology, and other related
disciplines. In such applications one is interested in determining whether a particle
interacted in the detector or not. This is not an enormously difficult task since here
one is not really interested in the shape of the pulse, rather its presence or absence. In
other words, one simply counts theinterestingpulses. Which pulses are interesting,
depends on a whole lot of factors including type of particles to be detected, signal to
noise ratio of the system, and precision required. Theuninterestingevents or pulses
are then either blocked from reaching the analysis software or are eliminated by the
analysis software. For example, with a neutron detector one might want to eliminate
theα-particle events. This can be accomplished easily since the energy deposited
by anα-particle is generally much higher than deposited by a neutron and all one
needs to do is to set an upper limit on the pulse amplitude.
In counting experiments, statistics plays a crucial role. The reason is that most
of the underlying phenomena are random in nature. Take the example of measuring
activity of a radioactive sample. The emission of particles from the sample is a
random phenomenon. To detect these particles one must allow them to interact
with some detection medium. If the medium is a scintillator, it will produce light
as a result of particle interaction. This process is governed by quantum mechanics,
which is essentially a statistical theory. The light thus produced can be detected by
a photomultiplier tube by first converting it into electrons and then by multiplying
the electron population. All these steps are characterized by statistics.
Hence in essence, we can say that statistics is of primary importance in counting
experiments. The physical and detection limits should therefore be deduced by
statistical methods. This is the subject of our next section.
9.11.A Measurement Precision and Detection Limits
Most of the counting experiments follow Poisson statistics. Accordingly, the stan-
dard deviation of a measurement is equal to the square root of its magnitude, that
is
σ=
√
N, (9.11.1)
whereN represents the observed counts. Note that here we have not yet taken
into account any detector related issues. The standard error in the measurement
obtained in this way reflects only the physics of the underlying processes. In other
words, it gives us thephysical limitof the measurement error. One can not have a
dataset that is spread out less than what is suggested byσ. The physical limit of
the error in measurement is then given by
δN =σ
N(9.11.2)
=
√
N
N
=
1
√
N
. (9.11.3)
This quantity after multiplying by 100 is sometimes referred to as the physical limit
of the measurement precision (multiplying by 100 simply evaluates the precision
as a percentage). That is, one can not achieve a precision better than this value.