Nuclear Spectroscopy

A P P E N D I X C
Mathematics of Radioactive Decay
Consider a chain of nuclides, one related to another
by radioactive decay. Three so-related nuclides will
suffice for this discussion. The initial nuclide, #1, is
called the parent, the decay product of the parent is
the daughter, #2, and the decay product of the daughter
is the granddaughter, #3. A typical radioactive
decay chain is shown symbolically in Figure C.1.
where N(t) is the number of parent nuclei at time t,
and dN is the number of parent nuclei that decayed
in the time between t and t+dt. Integration of equation
C.1 for the parent atoms, assuming the number
of parent ions at time t=0, N 1
(0), is zero, gives the
number of parent nuclei remaining after time, t, as
C.2
N () t = N ( ) e
−λ1t
1 1
0
where λ 1 is related to the parent’s half-life, T 1/2
, by
C.3
T 1 2
= 2 λ 1
= 0 693 λ 1
/
ln / . /
Grandaughter
#3
Parent
#1
β
–
α
Daughter
#2
N 1
(t) is a mathematically smooth function that decreases
exponentially with time, but N 1
(t) is not a
completely correct mathematical model for radioactive
decay of a parent nucleus. It mathematically
represents the number of parent nuclei that most
probably survive after time, t. There can and will be
statistical fluctuations in N 1
(t). In fact it can be shown
(see experiment 9 on counting statistics) that if N is
the ‘most probable’ number of disintegrations in a
period of time ∆t, there is about a 68% chance that the
observed number is between
Figure C1. A schematic representation of nuclear decay. The
horizontal axis is the atomic number and the vertical axis is
the mass/energy axis.
The fundamental experimental foundation of radioactive
decay is that the probability, P(t)dt =λdt, for
any one nucleus to decay in a small time interval, dt,
is independent of any external influence (in very
special situations this is not true), including the decay
of other nuclei. All nuclei of a particular nuclide have
the same decay probability (rate, λ) such that
C.1
Z-1 Z Z+1
− dN = N() t P() t dt = λN()
t dt
C.4
N − N and N + N
The rate at which atoms, say the parent of Figure C.1,
decay, is called the activity, A(t), and is a function of
time. Units for activity are the Curie (1 Ci = 3.7 x 10 10
decays/sec) and the SI unit, the Becquerel (Bq = 1
decay/sec). For the decay symbolized in Figure C.1,
let [λ 1
, N 1
(t)], [λ 2
, N 2
(t)], and [λ 3
=0, N 3
(t)] represent
the decay constants and the number of atoms of the
parent, daughter, and granddaughter atoms, respectively.
Then the rate of population change, dN, for
each generation of atom is
C.5
C.6
C.7
dN
=−λ N () t dt
1 1 1
dN = λ N () t dt −λ
N () t dt
2 1 1 2 2
dN3 = λ
2N2()
t dt
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