Physics and Engineering of Radiation Detection

1.3. Radioactivity and Radioactive Decay 23

Activity

AP

AD

T1/2P>>T1/2D

Time

Figure 1.3.6: Activities of par-

ent and daughter nuclides as a

function of time for a material

that eventually reaches the state

of secular equilibrium. The half

life of parent in such a material

is so large that it can be consid-

ered stable.

Solution:

Since the half life of the parent is much larger than that of the daughter,

therefore we can safely assume that at the state of secular equilibrium the

activity of the daughter will be nearly equal to the initial activity of the

parent, that is.

ADA 0 P.

Substitution of this equality in equation 1.3.32 gives

λdD

λdD−λdP

e−λdPt  1

⇒t 

1

λdP

ln

(

λdD

λdD−λdP

)

.

The decay constants of the two materials are

λdP =

ln(2)

T 1 P/ 2

=

ln(2)

2. 14 × 106 × 365

=8. 87 × 10 −^10 day−^1

λdD =

ln(2)

T 1 D/ 2

=

ln(2)

27

=2. 57 × 10 −^2 day−^1

Hence the required time is given by

t 

1

8. 87 × 10 −^10

ln

(

2. 57 × 10 −^2

2. 57 × 10 −^2 − 8. 87 × 10 −^10

)

.

=38.9days