Download Poisson Stats in Particle Counting: Distributions & Waiting Times and more Study notes Physics in PDF only on Docsity!
Junior Physics Laboratory
Particle Counting Statistics
The following paragraphs are intended to supplement the text by Bevington and Robinson with a specific focus on counting experiments. You will need to read the text to obtain further details and background information.
A. Counts in a fixed interval Poisson statistics are the appropriate model for counting experiments, so it will be useful to express the Poisson distribution in a form that can be readily applied. If the average number of events in a time interval t is μ, it is convenient to define a counting rate r by
r =μ / t (1)
The probability of observing exactly events in t is then given by the Poisson distribution function with mean μ = r t
P μ ( ) =μ^
!
e^ μ^ = ( r^ t )
! e^ r t^ (2)
Two special cases are of interest: The probability of exactly zero events in t is
P μ (0) = e^ r t^ (3)
and the probability of exactly one event in t is
P μ (1) = r t e^ r ^ t (^) t 0 r dt (4)
These results will be directly applicable when interpreting experimental counting data, and are very helpful in the examples that follow. For large values of the mean the factorial makes it difficult to evaluate the Poisson distribution function numerically. Fortunately, it can be shown that P μ( ) approaches the
Gaussian form,
G μ , ( ) d = 1
exp 1 2
μ
^2
^
d (5)
in the limit of large μ, provided that we set ^2 = μ. You will need to use this mathematical result when you analyze Poisson-distributed data for large μ. To use the distribution functions with experimental data we must estimate μ and from the data. The mean is found from the usual formula
μ = 1 N i f (^) i i
where i is the value of the random variable (the number of counts/time interval in this case), f (^) i is the number of measurements yielding a result of i , and N is the total number of measurements. Similarly, is found from
2 s^2 = 1 N
f^ i (^ i )^2 (7)
where the symbols have the same meanings. It will also be useful to know the standard deviations of our estimates. For this is just the standard deviation of the mean,
(^) = / N s / N (8)
The standard deviation of s^2 , s 2 is found from the general formula
(^ s^2 )
2 =^ ( s
i
i
2 ^2 i^ (9)
which reduces to
( s 2 )
2 = 4 N 2 s^2 4 N s^4 (10)
when all the (^) i are equal, as is the case here.
to get
q 2 ( t ) dt = r^2 e rt^ dt 1 dt 0
t =^ tr
(^2) e rt (^) dt (14)
The general result for any j is
q (^) j ( t ) = r ( rt^ )
j (^1) e rt ( j 1)!
which is plotted in Fig. 2. Note that after j = 1 the function peaks near rt = j -1, as one might expect. This function also arises in other contexts, and is called the gamma distribution.
C. Accidental coincidence Suppose we have connected two counters to a coincidence module, with the intention of picking out simultaneous events. We might do this to discriminate against some background process which produces non-coincident counts, for example. Unfortunately some of the coincidences we observe will be due to unrelated events which happen to occur within the time resolution of the coincidence circuit. It is important to know how frequent such accidental coincidences are. As a simple approximation, we assume that detector pulses that arrive at the coincidence circuit within a resolving time r are counted as coincident, while pulses with a greater separation interval are not counted. The probability of an accidental coincidence is then just the
q (t)j
rt 0 2 4 6 8 10
Fig. 2 Plots of q (^) j(t) for j = 1 (solid line), j = 2 (dashed line) and j = 4 (dotted line).
probability of an event in counter 1 and an event in counter 2 within a short interval r. If the individual counting rates are r 1 and r 2 , this is just the product of the two probabilities
pc = ( r 1 r )( r 2 r ) (16)
where we assumed r 1 r « 1 and r 2 r « 1 because this is the only useful situation. The rate of accidental counts is just the probability per unit time, or
ra = r 1 r 2 r (17)
Actually this is a slight overestimate since some of the measured single-counter rate is due to real coincident events. One should then subtract the true coincidences before estimating accidentals, but when coincidence counting is useful this is usually a small correction.
D. Dead time correction All counter systems require a finite amount of time to process an event. During this time, the counter is unable to respond to another particle that may arrive. To model this phenomenon, we assume that the system is totally unresponsive for a "dead time" d , after which it will again accept an event. If another event arrives during the dead time it will extend the interval, so that very high event rates will paralyze this type of counter. To grasp the seriousness of the problem, we can ask for the probability of losing one or more counts following a particular event. This can be approximated as the Poisson probability of having one or more arrivals during d , which is
P μ ( 1) = 1 P μ (0) = 1 e^ r^ ^ d^ (18)
Numerically, the probability of losing counts is about 10% when r d is about 0.1. For a Geiger tube with a dead time of 250 μs, as might be used in a survey meter, this corresponds to only 400 counts per second. Clearly we need to correct for the dead time when highly accurate measurements are needed. Suppose we have events arriving at a true rate r for a period T. The counter registers only N counts, fewer than the true number of arrivals, rT. In fact, only those events which arrive at time intervals greater than d are recorded. The probability of a time interval between events greater than d is just an integral over the time interval distribution q 1 ( t ) derived above, namely
