Physics and Engineering of Radiation Detection

12 Chapter 1. Properties and Sources of Radiation

decay constant. In this book the subscriptdinλdwill be used to distinguish it from
the wavelength symbolλthat was introduced earlier in the chapter. Conventionally
both of these quantities are represented by the same symbolλ. The negative sign
signifies the fact that the number of nuclei in the sample decrease with time. This
equation, when solved for the numberN of the radioactive atoms present in the
sample at timet,gives
N=N 0 e−λdt, (1.3.13)

whereN 0 represents the number of radioactive atoms in the sample att=0.
Equation 1.3.12 can be used to determine the decay constant of a radionuclide,
provided we can somehow measure the amount ofdecayedradionuclide in the sam-
ple. This can be fairly accurately accomplished by a technique known as mass
spectroscopy (the details can be found in the chapter on spectroscopy). If the mass
of the isotope in the sample is known, the number of atoms can be estimated from

N=

NA

wn

mn, (1.3.14)

whereNA=6. 02 × 1023 is the Avagadro number,wnis the atomic weight of the
radionuclide andmnis its mass as determined by mass spectroscopy.
Although this technique gives quite accurate results but it requires sophisticated
equipments that are not always available in general laboratories. Fortunately there
is a straightforward experimental method that works almost equally well for nuclides
that do not have very long half lives. In such a method the rate of decay of the sample
is measured through a particle detector capable of counting individual particles
emitted by the radionuclide. The rate of decayA, also called the activity, is defined
as

A=−

dN

dt

=λdN. (1.3.15)

Using this definition of activity, equation 1.3.13 can also be written as

A=A 0 e−λdt, (1.3.16)

whereA 0 =λdN 0 is the initial activity of the sample.
Since every detection system has some intrinsic efficiencywith which it can
detect particles therefore the measured activityCwould be lower than the actual
activity by the factor.

C = A

= 

[

−

dN

dt

]

= λdN (1.3.17)

The detection efficiency of a good detection system should not depend on the count
rate (as it would imply nonlinear detector response and consequent uncertainty
in determining the actual activity from the observed data). and hence the above
equation can be used to determine the count rate att=0.

C 0 =λdN 0 (1.3.18)

The above two equation can be substituted in equation 1.3.13 to give

C=C 0 e−λdt. (1.3.19)