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Lecture 2
Radioactive Decay Kinetics
Basic Decay Equations • Radioactive decay is a first order process, ie, the number of decays/s is
proportional to the number of nuclei present • In eqn form, -dN/dt =λN where the constant λ is the decay constant • (A=λN) • Rearranging
!
"dNdt
= #N
dNN
= "#t
N = N0e"#t
where N0 is the number of nuclei present at t=0.
If we remember the basic equation relating activity to number of nuclei in a sample, A=λN, then we can write
!
A = A0e"#t
Thus we have two equations that look the same, but have very different meanings
!
N = N0e"#t
A = A0e"#t
Graphically
Decay constants, t1/2
!
AA0
=12
= e"#t1/2
t1/ 2 =ln2#
=0.693#
Note that if t1/2 has units of time, λ has units of time-1
λ is probability per unit time of getting a decay
Use of basic decay equation
!
A = "N
N =A"
N =massradionuclide
At.Wt6.02x1023
What do these equations really mean?
An easy decay rate to measure is 10 dpm of a nuclide with a t1/2 of 20 min. Then
!
N =A"
=10ln2
20 = 288
Mass =N
6.02x1023At.Wt.= 5x10#20g
Mean Life
!
" =1#
(" =1.443t1/ 2)
" =
ti1
N0
$N0
= %1N0
tdN =1N0
t#Ndt = # te%#t0
&
'0
&
't= 0
t=&
'
=%#t +1#
e%#t(
) * +
, - 0
t
=1#
Significance of mean life
!
"E •"t # !$ % "t
"E =!$
=0.658x10&15eV
$(s)= '
Units
1 Bq=1 Becqueral=1 d/s 1Curie=1Ci=3.7x1010 Bq
Mixture of independently decaying activities
!
Atot = A1 + A2 = A10e"#1t + A2
0e"#2t
Radioactive Decay Equilibria Consider 1è2è3è
rate of change of 2=rate of production of 2 by decay of 1 - rate of decay of 2
! !
dN2
dt= !1N1 !!2N2
dN2 +!2N2dt = !1N1dtN1 = N1
0e!!1t
dN2 +!2N2dt = !1N10e!!1tdt
e!2tdN2 +!2N2e!2tdt = !1N1
0e(!2!!1 )tdtd(N2e
!2t ) = !1N10e(!2!!1 )tdt
N2e!2t
0
t=!1N1
0e(!2!!1 )t
!2 !!1 0
t
N2e!2t ! N2
0 =!1N1
0 (e(!2!!1 )t !1)!2 !!1
N2 (t) =!1N1
0
!2 !!1(e!!1t ! e!!2t )+ N2
0e!!2t
A2 (t) =!1!2N1
0
!2 !!1(e!!1t ! e!!2t )+ A2
0e!!2t
N2 (t) =!1N1
0
!2 !!1(e!!1t ! e!!2t )+ N2
0e!!2t
A2 (t) =!1!2N1
0
!2 !!1(e!!1t ! e!!2t )+ A2
0e!!2t
if "!1 = !2 = !(see "homework)N2 = N1
0!te!!t
Special Cases • No equilibrium, product is stable (2=0)
!
dN2
dt= "1N1
dN2 = "1N1dt = "1N10e#"1tdt
N2 ="1N1
0
#"1(e#"1t )
0
t
= N10(1# e#"1t )
Special Cases
• Transient Equilibrium (λ2 ~ 10x λ1)
!
"2 > "1e#"2t << e#"1t
N2e#"2t $ 0
N2(t) ="1N1
0
"2 # "1(e#"1t # e#"2t ) + N2
0e#"2t
N2(t) %"1N1
0
"2 # "1e#"1t
N1 = N10e#"1t
N1N2
="2 # "1"1
Note that the daughter activity is maximum at tmax
!
tmax =1
"2 # "1ln"2"1
Special Cases • Secular Equilibrium (λ2 >> λ1)
!
N1N2
=("2 # "1)
"1N1N2
=("2)"1
"1N1 = "2N2
A1 = A2
Importance of Secular Equilibrium
• Naturally occurring decay series
Importance of Secular Equilibrium
• Production of radionuclides in a nuclear reaction
• Nuclear reaction è 2 è
!
N20 = 0
Rate = R " #1N1
N2(t) =#1N1
0
#2 " #1(e"#1t " e"#2t )
#1 << #2
N2(t) =#1N1
0
#2(1" e"#2t )
A2(t) = #2N2 = R(1" e"#2t )
The “economics” of irradiating samples
At t=∞, A=Asaturation
Bateman equations Consider the general case, 1è 2 è 3 è 4...n
Assume
!
N20 = N3
0 = N40 = ...= Nn
0 = 0
!
Nn = C1e"#1t + C2e
"#2t + C3e"#3t + ...+ Cne
"#nt
C1 =#1#2...#n"1
(#2 " #1)(#3 " #1)...(#n " #1)N10
C2 =#1#2...#n"1
(#1 " #2)(#3 " #2)...(#n " #2)N10
Cn =#1#2...#n"1
(#1 " #n )(#2 " #n )...(#n"1 " #n )N10
Branching Decay Suppose a nucleus decays by several different modes, such as α-decay, SF-decay and EC decay. Then the total decay probability, λtot, is the sum of the probabilities of decaying by each mode, i.e.,λtot = λα+λSF+λEC Foe each mode of decay there is an associated partial halflife, i.e., t1/2(α)= ln 2 /λα
Naturally occurring radionuclides
• Primordial • Cosmogenic
• Anthropogenic
Environmentally interesting radionuclides
• 222Rn • 40K • 3H
14C
Simple Radionuclide dating
!
A(t) = A0e"#t
t =ln(A0 /A)
#=ln(N0 /N)
#
Tricks
• AMS • Variations in A0 or N0
Parent->Daughter Dating
!
D(t) + P(t) = P(t0) = P0
P(t) = P0e"#t
t =1#ln 1+
D(t)P(t)
$
% &
'
( )
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