Environmental Engineering Reference
In-Depth Information
Sidebar 18.3: Chains of Radionuclides
The safety analysis for a repository of radioactive waste includes the
modeling of the fate of radionuclides. In such a work it is necessary to look
at several radionuclides simultaneously, as these are connected in chains of
radionuclides. Let's explain the general behavior here without going too
much into detail.
With a characteristic half-life the concentration of a mother nuclide
declines by radioactive decay. Instead of half-life it is convenient to work
with a decay coefficient, as introduced in Chap. 5. The differential equation
(5.3) with
n ¼
1 is the basis for mathematical modeling of the mother
nuclide. As a result of the radioactive decay a daughter nuclide is produced.
The daughter nuclide usually is unstable itself, i.e. it also decays with
a characteristic constant, which is expressed by the differential equation:
@
c
daughther
@t
¼
l
mother
c
mother
l
daughter
c
daughter
The daughter nuclide itself is a mother nuclide for a next daughter nuclide.
Thus a chain of radionuclides can be identified. Let's denote the concen-
trations within that chain by
c
i
, where the index
i
indicates the
i
th member in
the chain. The entire system of species may thus be described by a system of
linear differential equations:
0
@
1
A
¼
0
@
1
A
0
@
1
A
c
1
c
2
c
3
:::
l
1
:::
c
1
c
2
c
3
:::
0
0
@
@t
l
2
:::
l
1
0
0
l
2
l
3
:::
:::
:::
:::
:::
(continued)
