D EPARTMENT OF P HYSICS , C OLORADO S CHOOL OF M INES
PHGN 422: Nuclear Physics
Introduction to Radioactive Decay
Kyle G. Leach
September 27, 2015
Various atomic nuclei undergo spontaneous decay processes in order to decrease their total energy. There are four main decay modes; spontaneous fission, α decay, β decay, and γ decay. The
first two decay modes occur via the strong interaction, β decay is an electroweak process, and
γ decay is an electromagnetic process. Spontaneous fission and α decay involve the emission
of nucleons, β decay involves the conversion of a proton to a neutron (or vice versa) in order
for the nucleus to become more stable, whereas γ decay decreases the energy of the nucleus
through transitions from excited nuclear states to (eventually) the ground state.
1 D ECAY S TATISTICS
Radioactive nuclei are characterized by the rate at which they decay. This rate is given by the decay constant λ, which has dimensions of inverse time. Due to the statistical nature of radioactive decay, no specific prediction can be given for an individual nucleus. The characterization
of these decays is reliant instead on decay probabilities. The probability that a nucleus decays
within the time interval d t is λd t . Therefore, in a collection of identical radioactive nuclei, the
number of decays per unit time is proportional to the number of nuclei that are present:
−d N (t ) = λN (t )d t .
(1.1)
This result can be integrated to give an expression for the number of nuclei remaining as a function of time:
N (t ) = N0 e −λt ,
(1.2)
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where N0 is the number of parent nuclei present at t = 0. Equation 1.2 can be used to determine
the time required for half of the nuclei present to decay. This is known as the half life, denoted
by t 1/2 . By substituting N (t ) = N (0)/2 into Eqn. 1.2 and rearranging we obtain:
t 1/2 =
`n(2)
.
λ
(1.3)
The mean lifetime, τ, of a certain nuclear state is related to the half life via:
τ=
1
t 1/2
=
.
λ `n(2)
(1.4)
There are many cases in which there is more than one decay mode of the parent nuclear state.
These are known as multi-modal decays. For example, if the parent nucleus decays by way of two
branches, there would be two distinct decay constants λ1 and λ2 . These are known as partial
decay constants. For the total decay of the parent, λ = λ1 + λ2 , where λ is the total decay constant. These decay constants represent decay rates for each branch of the decay. The branching
fraction (or branching ratio) is a measure of the fraction of the total decays that proceed via a
given branch:
B1 =
λ1
,
λ
B2 =
λ2
,
λ
and in general B n =
λn
,
λ
(1.5)
where B n is the branching ratio for the n th branch. Other cases where the daughter nucleus is
also unstable and decays to a granddaughter occur as well. These are referred to as sequential
decays. Sequential decays often lead to large decay chains which may contain many generations
of α and β decays, as well as cascades of γ rays from excited nuclear states.
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