ASSAY OF RADIOACTIVITY 



where N is the number of atoms present at time t, 1 is the decay constant for 

 the particular isotope, and dNjdt is the rate of disintegration of the nuclei. 

 This expression embodies the observation that the chances of any nucleus 

 disintegrating within a given period are constant, no matter how long it has 

 already existed. It leads, of course, to the familiar exponential expression 



N = N^e-^ 



where Nq is the number of atoms at zero time, and it will be seen that A is 

 related to the half-life of the isotope (see Table 1) by 



t^ = 0-693/2 



Now since the individual disintegrations are not causally related to one 

 another, they occur not at regular intervals but at random. This has the 

 important consequence that if a sample is counted under fixed conditions 

 several times within a period which is short compared with its half-life, the 

 results will not be the same on each occasion but will vary according to the 

 Poisson distribution law for random events. There is not space here to 

 discuss the statistics of counting in detail, but some of the practical repercus- 

 sions of the random fluctuations in counting rate that thus inevitably occur 

 in all radioactivity measurements must next be considered. 



In the first place, in order to have reasonable confidence that an experimen- 

 tally determined counting rate can be rehed upon to be close to the true 

 counting rate for a sample, it is necessary to record a fairly large number of 

 counts from the sample. The more counts are recorded, the greater wiU the 

 reliability of the result be. For a Poisson distribution, the standard error (S.E.) 

 of a single determination of a counting rate is the square root of the total 

 number of counts recorded (this assumes, as is normally the case, that any 

 error in the timing is so small that it can be neglected). The S.E. is a standard 

 way of expressing the probable scatter of the results ; rather roughly, there 

 is a 1 in 20 chance that the value to which it is attached differs from the true 

 value by more than twice the S.E. To take a specific example: if only 100 

 counts were recorded from a sample in x minutes, the counting rate would be 

 100/x and its S.E. would be 10/x— that is to say the S.E. would be dzlO per 

 cent of the recorded result, which could not be regarded as a very reliable 

 estimate of the true counting rate. If 10,000 counts were recorded in y 

 minutes, then the counting rate would be 10,000/^ and its S.E. would be 

 lOO/j, which is only ±1 per cent of the recorded result. In most biological 

 experim.ents this would be an acceptable answer, since it is generally difficult 

 to reduce other sampling errors much below 1 per cent; but the counting 

 rate could be determined to within still closer limits by prolonging the count- 

 ing period so as to record an even larger number of counts. Conversely, it 

 may often be unnecessary to achieve an S.E. as low as ±1 per cent, and 

 shorter counting periods will then yield sufficiently accurate results. 



What has just been said is strictly true only for rather active samples, 

 since for weak samples a further complication is introduced by the back- 

 ground count of the Geiger tube. Most arrangements for recording ionizing 

 radiation will give a slow counting rate even when there is no radioactive 

 source close to them because of cosmic radiation and traces of radioactivity 



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