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 (λ2=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 For each mode of decay there is an associated partial half-life, i.e., t1/2(α)= ln 2 /λα

Radiation Dosage

•  When radiation interacts with matter, the matter is altered by ionization or atom/nucleus displacement

•  Need to know the amount of energy absorbed by matter

•  Unit of dose= Gray (Gy) •  1 Gray -> absorption of 1 Joule/kg •  1Gray -> 6.24 e12 MeV/kg

Naturally Radioactivity

•  ~ 70 naturally occurring radionuclides •  70 kg man-> 4400 Bq 40K, 3600 Bq

14C •  8000 decays/sec in your body •  You ingest about 1 pCi/day •  The air you breathe is radioactive, the

water you drink is radioactive, etc.

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

Health effects of natural radiation

Parent->Daughter Dating

!

D(t) + P(t) = P(t0) = P0

P(t) = P0e"#t

t =1#ln 1+

D(t)P(t)

$

% &

'

( )