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About Axisymmetric Flow

Autor:   •  May 24, 2018  •  1,599 Words (7 Pages)  •  588 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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