Download Random Coincidence between two Independent Pulses

Random Coincidence between two Independent
Pulses
Sean O’Brien
June 16, 2006
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Random Coincidence Rates
In nuclear experimentation there are genuine coincident events, those that are
detected by two or more detectors that correspond to the same event. There
are in addition to these true coincidence events signals that arise for which the
detectors are responding to two separate events during the detectors resolving
time. These signals can be due to one of the detectors not seeing the event or
due to a event that has single a emission and no partner emission to be detected
by its partner detector. Due to their random distribution over time and high
rate of occurrence, there is a probability that some of these unrelated events will
occur simultaneously, one in each detector leading to an over estimate of the
true coincidence events. The time interval during which these random signals
overlap is the resolving time. The random interval’s size is dependent on the
two single rates. If the time duration of the signals is sufficiently small, then
the resolving time distribution will be linear throughout time [1] [2].
The magnitude of the random distribution is derived as follows:
Let r1 and r2 be the rates of two uncorrelated start and stop pulses and
T be the interval of time between the two pulses. After each start pulse the
probability that over the length of time T a stop pulse will not occur is:
e−T r2
(1)
. The differential probability of the arrival of a stop pulse during the following
differential time dT is r2 dT . Since both independent events must occur, the
total probability of creating an interval of dT and ∆T + T is:
r2 e−T r2 dT
(2)
The total differential rate of the interval is the rate of arrival multiplied by the
probability,
r1 r2 e−T r2 dT
(3)
The exponential portion, e−T r2 , can be approximated to ∆T if dT ∼
= ∆T , e−T r2
∼
= 1, T r2 1 and is not large in comparison with the inverse of the resolving
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time, then r2 T will also be small. Now we have that the random coincidence
rate, r12 , is [1]:
r12 = r1 r2 ∆T
(4)
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Circuit Construction
The above equation was experimentally verified in the following way. The
two single rates (r1 and r2 ) were obtained by two independent NIM Pocket
Pulsers(model 417). Each pulser generates −800mV into 50Ω, with a risetime
of 1.5ns , a falltime of 5ns, with a width of 6ns at a 10KHz rate. These two
pulers were input into the following circuit in order to measure their random
coincidence rate. Each pulser was connected to an individual channel in the
Scalar
ch2
gate
pulser 1
CH1
Logic
variable
width
width
5ns
width
5ns
AND
gate
Scalar
ch3
Linear
fan out
CH3
Logic
CH2
pulser 2
Scalar
ch2
gate
All cable lengths 4ns.
Figure 1: A Random Coincidence Circuit Constructed with NIM bins.
sixteen channel discriminator(module 706), which converts the pulses into logic
pulses. A copy of each pulse is sent to a scalar (module N114), so the that each
pulser’s rate can be determined and then to a quad majority logic unit(module
755). This module allows for adjustment of the width of each signal. In this
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Scalar
timer
Figure 2: Setup Coincidence Circuit
circuit one channel was set at 5ns, while the other was variable, ranging from
approximately 5ns to 950ns. Each pulse then was duplicated again, with one
running to a channel on the oscilloscope, so that the width adjustments of each
pulse could be measured and the other leading back into the majority logic
module set to a coincidence level of two. From here the outgoing signal, only
present when the two pulses overlapped and set to a width of 5ns, goes to the
quad scalar to determine the random coincidence rate and to the oscilloscope.
Before each scalar channel is a gate, this gate is used to tell each of the scalars
to stop counting. This portion of the circuit runs from the scalar timer to a
quad linear fan-in/out(module 740), this unit simply accepts the timers stop
pulse and copies it and sends it to the three scalar gates.
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Data and Analysis
The above circuit was then used to experimentally verify the derived expression,
r12 = r1 r2 ∆T . By adjusting the width (resolving time),∆T , of a single pulser
while the other remained fixed allowed a sample of resolving times ranging from
5ns to 950ns. The data was taken over 10s intervals on the scalar and forty
runs were made with each run corresponding to a different width.
Figure 3: Random Coincidence Plot: Resolving Time vs. Coincidence Rate
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The data corresponds well to the expected result. The expected slope calculated by averaging all r1 r2 is in good agreement with the observed slope.
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Summary and Conclusions
The construction and principles of random coincidence circuit are important
concepts and skills necessary to nuclear instrumentation. They are important
for particle measurements and help weed out false data, allowing greater statistical precession by understanding how to distinguish them from genuine coincidences [1] [2].
References
[1] Glen F. Knoll. Radiation Detection and Measurement: Third Edition. John
Wiley and Sons, Inc. New York, 2000.
[2] E. Fenyves and O. Haiman The Physical Principles of Nuclear Radiation
Measurements Academic Press, New York, 1969.
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