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About Axisymmetric Flow

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Radioactive Decay

Radioactive Decay

v Dynamic , or time-varying, properties of nuclei: Ø Discovery of the radioactivity by Henri Becquerel in 1896.

§ radioactive decay and § nuclear reaction. o Discovered the radioactivity of Uranium while

investigating the fluorescence properties of Uranium salts.

Ø Both characterized by:

· Transition from some initial system to some final system, occurring either:

o spontaneously (radioactive Decay) or Ø Discovery of two more radioactive elements: Polonium

o artificially (nuclear reaction) and Radium by Pierre & Marie Curie, 1898.

v Radioactive decay: the nucleus of an atom Ø Discovery of more radioactive elements: Thorium,

· emits: Actinium, Radiothorium, ..., in a few years.

o an alpha particle, o a beta particle, o a gamma ray Ø Today hundreds of radioactive isotopes of different

· Or capture: elements are known.

o an electron from an extra-nuclear shell

Ø Man-made radioactive element (30P) by Irene Curie &

v parent: the initial radioactive nuclide in any decay mode Pierre Joliot, 1934

v daughter: the product nuclide

v radioactive decay chain: several successive radioactive

generation.

1

2


The Radioactive Decay law:

Assume:

Ø N radioactive nuclei

Ø At time t

The number dN decaying in a time dt is proportional to N :

N

half-life,

t 21

: the time necessary for half of the nuclei to

decay

§ put NN =

0

2

t

21

== 693.02ln

l l mean lifetime, t : the average time before a nucleus decays

l

-= ( dtdN ) where: l o disintegration or decay constant.

l

Ø rhs of the equation is the probability per unit time for

the decay of an atom.

Ø The Probability of decay is CONSTANT, regardless

of the age.

Integrating the decay constant equation gives the exponential law of radioactive decay:

l dNdtN = ò l dtN = ò dN eNtN = - l t 0

t

=

ò ò 0 0 ¥

¥

dtdtdNt

dtdtdN

= 1

t = 2ln t

21 =Þ t

21

693.02ln t = t The number of decays between t and ( tΔ ) :

=D+-=D

eNttNtNN )()(

0 - l t 1( - e - l t ) If

tt<

, then we can say:

=D teNN l

0 - l t D )(

tNN

o

=o )0( dN dt

=

l eN - l t 3

4


dn

=

)

eAtNA o

l = 0 Then we can define the activity A: (since

dt

lN

)( - l t where:

tAA 0

l=== )0( N 0 Ø Measuring the number of counts NΔ in a time

interval tD gives the activity of the sample only if <

tt 21

The number of decay in the time interval from

1

t :

ò

t to

2

Note:

Ø The activity tells us ONLY the number of disintegration.

=D

N ttt

12

D+=

t

1

dtA Ø It says NOTHING about the kind or energies of

radiations emeited. which equals tAD only if

<

tt 21

If radioactive decay of isotope is simple:

nuclide-1 (initial unstable) ® nuclide-2 (stable, end product)

5

6


Then there is a simple relation between

N 1

and

N 2

:

eNN

1

0 1

t where

lll t

+= a b , is the total decay constant.

= - l dN

2

λ=-=

Nd 1 11 dtN A fraction

ll ta

of the decay will go to mode a, and Nd

2 td

=

l 11

N

= l 1 eN o

- l t

likewise a fraction

ll tb

eNN

2

-=

0 1( l-

1

t ) to mode b :

eNN 1

=

l- t

t 0 NNN 0

+=

1 2 N 2

a

=

(

ll ta 1() eN 0 - - l t

t ) But if it is not simple ( nuclide-2 be radioactive, or nuclide-1

2

1()( 0 ) be produced) then these equation do not apply.

If a single nuclide can decay by more than one process (mode), for example:

nb

N b

=

ll tb eN - - l t

We can not “turn off ” one or the other, so they both must appear in the exponential at all times.

or

branching ratio t

t

l

l

paX +®

a l t

l

b

o

+® Then, there are more than one decay constant,

l a

and

l b

, (here two for two different modes aand b ), called “partial decay constant”:

l

a

-= (

dtdN

N

) a

l

b

-= (

dtdN

N

) b

The total decay rate

)( dN dt

t is:

- )()()(

dN dt

t

-= dN dt

a - dN dt

b = N

( ll a =+ b ) N l t 7

8


Production and decay of radioactivity:

Assume the nuclide

N 1

is produced at a constant rate R and decay with the constant decay rate

l 1

:

dtRdN

1

-= l 11 dtN tN 1

1()(

=

l R

1

- e

- l

1 t ) tA 1

)(

=

l 11 eRtN 1()( -= - l 1 t ) Special cases:

tRtA 1

)( » l 1 tt << 21 (const. production) RtA 1

)( = tt >> 21 (Secular equilibrium)

...

...

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