Random Coincidence between two Independent

Pulses

Sean O’Brien

June 16, 2006

1 Random Coincidence Rates

In nuclear experimentation there are genuine coincident events, those that aredetected by two or more detectors that correspond to the same event. Thereare in addition to these true coincidence events signals that arise for which thedetectors are responding to two separate events during the detectors resolvingtime. These signals can be due to one of the detectors not seeing the event ordue to a event that has single a emission and no partner emission to be detectedby its partner detector. Due to their random distribution over time and highrate of occurrence, there is a probability that some of these unrelated events willoccur simultaneously, one in each detector leading to an over estimate of thetrue coincidence events. The time interval during which these random signalsoverlap is the resolving time. The random interval’s size is dependent on thetwo single rates. If the time duration of the signals is sufficiently small, thenthe 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 theprobability that over the length of time T a stop pulse will not occur is:

e−Tr2 (1)

. The differential probability of the arrival of a stop pulse during the followingdifferential time dT is r2dT . Since both independent events must occur, thetotal probability of creating an interval of dT and ∆T + T is:

r2e−Tr2dT (2)

The total differential rate of the interval is the rate of arrival multiplied by theprobability,

r1r2e−Tr2dT (3)

The exponential portion, e−Tr2 , can be approximated to ∆T if dT ∼= ∆T , e−Tr2

∼= 1, Tr2 1 and is not large in comparison with the inverse of the resolving

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time, then r2T will also be small. Now we have that the random coincidencerate, r12, is [1]:

r12 = r1r2∆T (4)

2 Circuit Construction

The above equation was experimentally verified in the following way. Thetwo single rates (r1 and r2) were obtained by two independent NIM PocketPulsers(model 417). Each pulser generates −800mV into 50Ω, with a risetimeof 1.5ns , a falltime of 5ns, with a width of 6ns at a 10KHz rate. These twopulers were input into the following circuit in order to measure their randomcoincidence rate. Each pulser was connected to an individual channel in the

pulser 1

pulser 2

Logic

Logic

AND

CH1

CH2

CH3

Scalar ch2gate

gate Scalar ch2

gateScalar

ch3Scalar timer

Linear fan out

All cable lengths 4ns.

width 5ns

width 5nsvariable

width

Figure 1: A Random Coincidence Circuit Constructed with NIM bins.

sixteen channel discriminator(module 706), which converts the pulses into logicpulses. A copy of each pulse is sent to a scalar (module N114), so the that eachpulser’s rate can be determined and then to a quad majority logic unit(module755). This module allows for adjustment of the width of each signal. In this

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Figure 2: Setup Coincidence Circuit

circuit one channel was set at 5ns, while the other was variable, ranging fromapproximately 5ns to 950ns. Each pulse then was duplicated again, with onerunning to a channel on the oscilloscope, so that the width adjustments of eachpulse could be measured and the other leading back into the majority logicmodule set to a coincidence level of two. From here the outgoing signal, onlypresent when the two pulses overlapped and set to a width of 5ns, goes to thequad 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 scalarsto stop counting. This portion of the circuit runs from the scalar timer to aquad linear fan-in/out(module 740), this unit simply accepts the timers stoppulse and copies it and sends it to the three scalar gates.

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3 Data and Analysis

The above circuit was then used to experimentally verify the derived expression,r12 = r1r2∆T . By adjusting the width (resolving time),∆T , of a single pulserwhile the other remained fixed allowed a sample of resolving times ranging from5ns to 950ns. The data was taken over 10s intervals on the scalar and fortyruns 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 cal-culated by averaging all r1r2 is in good agreement with the observed slope.

4 Summary and Conclusions

The construction and principles of random coincidence circuit are importantconcepts and skills necessary to nuclear instrumentation. They are importantfor particle measurements and help weed out false data, allowing greater statis-tical precession by understanding how to distinguish them from genuine coinci-dences [1] [2].

References

[1] Glen F. Knoll. Radiation Detection and Measurement: Third Edition. JohnWiley and Sons, Inc. New York, 2000.

[2] E. Fenyves and O. Haiman The Physical Principles of Nuclear RadiationMeasurements Academic Press, New York, 1969.

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