About Axisymmetric Flow
Autor: Rachel • May 24, 2018 • 1,599 Words (7 Pages) • 747 Views
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9
Production of radioisotope by a decaying parent:
Assume that the daughter nucleus in the decay is not stable:
1 ® 2 ® 3 ® ...
At time t 0= , there are only nuclide of type 1 present:
tN 1
)0( == N 0 tN
2
0...)0()0( == tN 3 === Let’s choose an easy case, and assume that the granddaughter is stable:
21 ® l
1
l
2
® ̄
3
dN
1
-= l 11 dtN eNtN
1
)( =
l- `1 t 0 Nuclide 2 :
o Increases as 1 decays
o Decreases as 2 decays
10
---------------------------------------------------------------
So:
tdN
2
)( = l 11 dtN - l 22 dtN thus
NtN 2
)(
=
l
1 0 ll
12
-
( e - l
1
t - e - l 2 t ) The activity caused by the decay of nuclide type 2 is:
tA 2
)(
=
l 22 NtN )( = ll
21 0 ll
12
-
( e - l
1
t - e - l 2 t ) And the stable granddaughter, nuclide type 3, will be:
tdN
3
)( = l 22 dtN NtN
3 )(
=
ll 21 0 lll
12
-
( 1
-
e
- l 1
t -
1 -
e - l 2 t l )
11
12
---------------------------------------------------------------
Special cases:
Ø If
ll 1
o Parent is so long-lived that for the time scale we
are interested in, we can treat: e - l
1
t @ 1
o number of granddaughter is:
NtN 2
)(
=
l
1 0 l
2
1( - e - l
2 t ) As t ¥® , nuclei 2 are decaying at the same rate that they are formed:
l 22
NN = l 11 (secular equilibrium)
o If
ll 1
13
ll Ø If
1
o Parent is long lived compared to the daughter o After a long time t (
t >> 1 l 2
)
NtN
2
)( @
l
1 0 ll
2
-
e -
l
1 t 1 \ The decay of 2 after a long time is determined by
the half-life of 1.
o Under these conditions:
N
2 N
1
@
l
1 ll
1
- 2 l
N
activity
of 2
l and
l
N
ll
1
- 2 Which is called: “transient equilibrium”.
Ø
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