112 RADIOACTIVITY; BIOLOGICAL TRACERS 



DISINTEGRATION (DECAY) 



Rate of Decay; Half-Life 



We have no control over the disintegration of individual nuclei: if a 

 nucleus is unstable, it will decay at a time which is completely unpre- 

 dictable. However it is possible to describe and predict the fraction of a 

 large number of unstable nuclei which will decay within a given period; that is, 

 AN/ At is easily measured. In fact the number of nuclei (J\ ) which do decay 

 within a given time is proportional to the number present which are able 

 to decay. 



Thus 



AN/ At oc N 



or the instantaneous rate 



dN/dt ex TV- 

 Insertion of the proportionately constant — A (called the "decay con- 

 stant") gives 



-dN/dt = \N 



After the summation in the fashion indicated in Chapter 1, 



N = N e~ Xl 



where N is the number present at any arbitrarily chosen zero of time. 



This expression says simply that the number, N, of nuclei which are 

 present at any time, t, is only a fraction of the number, N , which were 

 present at zero time — the fraction being e~ Xt . Now, it is useful and instruc- 

 tive to expand the fraction into the series it is, and write 



e~ Xi = l + + + 



l 2x1 3x2x1 



A 2 / 2 A 3 ; 3 

 1 _ \ t + + 



The value of A differs for different radioactive elements. For Sr 90 the value 

 has been measured to be 0.028 yr '. After five years, for example, 



.-* = 1 - (0.028 x 5) + (0 -° 28 X 5)2 - (°-° 28 X 5)3 + ^0.87 



2 6 



Therefore N = 0.87 N , or the fraction of N () left after five years is 87 per cent. 

 Calculations for 10, 15, 25, 50 yr would span a time at which N is just 

 50 per cent of N . For Sr 90 this time is about 25 yr, and it is called the "half- 

 life"— the time it takes active material to decay to 50 per cent of the original 

 concentration, N . Half-life, r - In 2/A = 0.693/A. 



