About Axisymmetric Flow
Essay by Rusky Slava • January 25, 2017 • Research Paper • 1,632 Words (7 Pages) • 1,136 Views
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)
...
...