14 Chapter 1. Properties and Sources of Radiation
well. In fact, whenever a new radionuclide is discovered its half life is one of the first
quantities that are experimentally determined. The half life of a radionuclide can
range from a micro second to million of years. However this experimental method
to determine the half life does not work very well for nuclides having long half lives.
The reason is quite simple: for such a nuclide the disintegration rate is so low that
the counts difference between two points in time will be insignificantly small. As
we saw earlier in this section, for such radionuclides other techniques such as mass
spectroscopy are generally employed.
Example:
Derive the equations for mean and half lives of a radioactive sample.Solution:
To derive the equation for mean life we take the weighted mean of the decay
timet
τ=∫∞
∫^0 tdN
∞
0 dN
UsingN=N 0 e−λdt, the integral in the denominator becomes
∫∞0dN = −λdN 0∫∞
0e−λdtdt= N 0
∣
∣
∣e−λdt∣
∣
∣
∞
0
= −N 0.
The integral in the numerator can be solved through integration by parts as
follows.
∫∞0tdN = −λdN 0∫∞
0te−λdtdt= −λdN 0[∣∣
∣
∣−
teλdt
λd∣
∣
∣
∣
∞0+
1
λd∫∞
0e−λdtdt]
The first term on the right side vanishes fort=0andatt→∞(a function
vanishes at infinity if its derivative vanishes at infinity). Therefore the integral
becomes
∫∞0tdN = −N 0∫∞
0e−λdtdt=
N 0
λd∣
∣
∣e−λdt∣
∣
∣
∞
0= −N 0
λd.
Hence the mean life isτ =
−N 0 /λd
−N 0=1
λd