Sec. 7.5] 



RADIOACTIVITY 



167 



An estimate of the statistical error in the measurement of an activity N is 

 expressed usually in terms of the absolute probable error given by 



r = 0.6745 VN 

 or in terms of the per cent probable error by 



1 



t% = 67.45 



Vn 



It is assumed, of course, that the period of observation is very small compared 

 to 1/X. 



7.4. Simple Decay. The observed decay of a given quantity of radio- 

 active isotope follows a simple exponential law as is readily apparent on 

 integrating the expression dN/dt = —\N in Sec. 7.2. This assumes, of 

 course, that the isotope is not at the same time being produced by some 

 external agent. If N is the initial number of atoms of the radioactive 

 isotope, the number of atoms remaining after a time / is 



N = N e~ M 



where X = decay constant in units of reciprocal time 



However, the decay of an isotope is usually expressed in terms of half-life 

 T, which is the time required for an initial number of radioactive nuclei to 

 be reduced by one-half. The number of atoms present at time / is then 



N = N e-»-™ WT 



Also frequently used is the mean life r = T/0.693 = 1/X, which is 

 the time required for an initial quantity of radioisotope to be reduced by 1/e. 

 The activity of any substance as a function of the number of half-lives is 

 plotted in Fig. 47 for convenient reference. 



7.5. Growth of Radioactivity. When a radioisotope is produced at a 

 constant rate, for example, in a cyclotron or as a daughter substance of 



N 



N(t) 



12 3 4 5 12 3 



HALF -LIVES 



Fig. 48. Growth of a radioactive isotope that is produced at a constant rate, e.g., in a 

 cyclotron, and decay after production is stopped. 



