Physics and Engineering of Radiation Detection

1.3. Radioactivity and Radioactive Decay 19

Activity

max

Time

AP

t

AD

D

Figure 1.3.5: Typical parent

and daughter nuclide activities.

Example:

Derive the relation for the time behavior of buildup of a stable nuclide from

a radioactive element.

Solution:

Assuming the initial concentration of daughter to be zero (N 0 D= 0), equation

1.3.26 can be used to determine the concentration of daughter nuclide at time

t.

ND=

λdP

λdD−λdP

N 0 P

(

e−λdPt−e−λdDt

)

Since the daughter nuclide is stable therefore we can substituteλdD=0in

the above equation to get the required relation.

ND=N 0 P

(

1 −e−λdPt

)

In most cases the radioactive decay process does not stop at the decay of the
daughter nuclide as depicted by equation 1.3.26. Instead the nuclides continue to
decay into other unstable nuclides until a stable state is reached. Assuming that the
initial concentrations of all the nuclides except for the parent is zero, equation 1.3.28
can be generalized for a material that undergoes several decays. The generalization
was first done by Bateman in 1910 (8). The Bateman equation for the concentration
ofithradionuclide is

Ni(t)=λd 1 λd 2 .....λd(i−1)N 01

∑i

j=1

e−λdjt

∏

k=1,k=j(λdk−λdj)

, (1.3.30)

providedN 0 i=0fori>1. In terms of activity the Bateman equation can be
written as

Ai(t)=λd 2 .....λdiA 01

∑i

j=1

e−λdjt

∏

k=1,k=j(λdk−λdj)

. (1.3.31)