Special Counting Situations

In general, we assume that the statistics of counting can be adequately described by the Poisson distribution. When we calculated the various decision limits, we effectively assumed, for simplicity, the Normal distribution for the counts. We know, however, that Poisson statistics are only applicable when the probability of detection of the decay of any particular radioactive atom within the count period is small and when the statistical sample size is large. There are a number of circumstances when these conditions may not be met and we should consider whether the statistical treatment above is still valid. 

If a sample is counted for a long time compared to the half life, the probability of decay within the count period is high. This condition is seldom met in most routine gamma spectrometry situations where half-lives are long, but may 

However, if the count period were long and the detection efficiency high (perhaps a low-energy gamma-ray emitted with high probability and measured close to the detector or inside a well detector), then the assumptions underpinning our use of the Poisson distribution are no longer valid. It is then necessary to return to the binomial distribution. There is no place here for the mathematics involved but it can be shown that if we observe a count of n, then the expected true count is  

If is very small, then these relationships approximate to Equations (5.25) and (5.24) discussed above. If p is large, then var(n) tends towards zero. This is not unreasonable if we could detect every disintegration, then there would be no uncertainty associated with the number of counts detected. 

however, that even if the counts due to the measured species cannot be assumed to be Poisson distributed, it is most likely that the background count, be it a single count or a peak, will be. If this is so, then the gross count will also be Poisson distributed. 

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In this chapter, I will examine the statistical nature of radioactivity counting. Statistics is unavoidably mathematical in nature and many equations will emerge from the discussion. However, only as much general statistical mathematics wiU be introduced as is necessary to understand the relevant matters. I wiU go on to discuss the statistical aspects of peak area measurement, background subtraction, choosing optimum counting parameters and the often superficially understood critical limits and minimum detectable activity. I end with an examination of some special counting situations. 

To see that this phase has no relation to the number of ci s encircled (if this statement is not already obvious), we note that this last result is true no matter what the values of the coefficients k, X, and so on are provided only that the latter is nonzero. In contrast, the number of ci s depends on their values for example, for some values of the parameters the vanishing of the off-diagonal matrix elements occurs for complex values of q, and these do not represent physical ci s. The model used in [270] represents a special case, in which it was possible to derive a relation between the number of ci s and the Berry phase acquired upon circling about them. We are concerned with more general situations. For these it is not warranted, for example, to count up the total number of ci s by circling with a large radius. 

In general, the step count method of estimation can be apphed to any special situation to derive a model equation for that particular industry or group of processes. 

This is the classical argument introduced by van Heerden in 1953 for the adiabatic stirred tank. It is a most important one to grasp firmly for it can be used in more complicated situations to get some insight into the stability of a system. However, its limitations must be also thoroughly understood. In particular, it can be used to establish instability, but it does not count conclusively for stability because of several reasons. First, we should be suspicious of a single condition for a system in which there are two variables. Second, the diagram for the heat generation was drawn in a rather special way, for the steady state-mass balance equation, f 7), was first... 

MPG President Adolf Butenandt, likewise to be counted amongst the older generation, made oracular reference to the special situation of the FHI in a congratulatory piece on the occasion of Rolf Hosemann s 60 birthday, writing ... 

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