The term radioactive decay is used to
describe both the process of individual nuclei changing as they give
out radiation and the process of a sample of material gradually
losing its radioactivity. 
Radioactive decay is a
random process. 
This means that we cannot predict when any
specific nucleus will decay. 
However, remember that in even a small
sample of material we are going to have huge (like, really
astoundingly enormous) numbers of atoms. 
When we have large numbers of randomly
occurring events we can make some surprisingly accurate
predictions about what to expect. 

Before saying any more about radioactive
nuclei, consider another random process; throwing dice. 
Suppose we have 600 dice (why 600, I hear
you ask… well, each die has 6 sides, right?) 
Imagine throwing these 600 dice onto a flat
surface. 
Assuming that the dice are all
perfectly uniform, how many would we expect to find with, say,
the “3” face upwards? 
There’s an equal chance (of 1 in 6) for
each die to have any given face upwards, so you would expect to find
about 100 with the “3” face upwards. 
You probably wouldn’t bet much on any
given die finishing with a "3" upwards but I assume that, like
me, you would be prepared to risk a couple of dollars/euros etc on
the bet that there would be between, say, 90 and 110 dice with a 3
visible. 

Now, imagine having 6000 dice to throw,
then all this can be scaled up and we would expect to find around
1000 dice with any given face visible. 
In other words, the number of dice found
with any given face upwards upwards is directly proportional to the total number
of dice thrown. 

Returning to the radioactive nuclei: 
as stated above, although we cannot predict
when any specific nucleus will decay, we can say (comparing
with the dice) that the
number of nuclei which decay per unit time will be directly
proportional to the total number of unstable nuclei present in
the sample, N. 

The intensity of radiation emitted by a
source (the activity of the source) is often stated in
counts per unit time (imagine counting the "clicks" of a Geiger
counter or similar detector of radiation) 
The intensity of radiation from a source
decreases with time. 
A graph of activity against time always has
the same form but the time scale can vary from a small
fraction of a second to millions of years. 


Notice that the time taken for the activity to fall to half* its
initial value is a constant. 

*or any fraction but half is commonly considered. 

The Half Life of a Radioactive
Substance (symbol t_{˝}) 
The half life of a radioactive substance is the
time taken for the activity of a sample to decrease to half its
initial value. 

Note that, if the activity has fallen to half its initial value,
this is because the number of unstable nuclei has fallen to half its
initial value, so an alternative definition of the half life is: 

The half life of a radioactive substance is the time taken for
the number of unstable nuclei in a sample fall to half its initial
value. 

Mathematical Description of the
Decay Curve 
If ΔN is the number of nuclei
which decay in time Δt, then the
activity of the source is defined to be
ΔN/Δt 
Therefore, as explained above, we can write 

ΔN/Δt
represents the slope of the graph of N against t, which is always
negative. 
Therefore, we can write an equation to describe the curve by
including a (positive) constant of proportionality: 

the constant, λ is called the
decay constant for the nuclide (the given radioactive
substance). 

To get an idea of what the decay constant
means practically, we rearrange the equation to give (ignoring the
minus sign) 

So we can now see that
λ is the fraction of the
existing unstable nuclei which decay per unit time. 

Note that, as ΔN/Δt
is proportional to N, a graph of N against time will have the same
form as a graph of activity against time. 

It can be shown that a curve which has a slope which is
proportional to the value of the variable on the vertical axis
(as in this case) is called an exponential. 
In this case the curve can be represented
by an equation of the form 

where 
 N_{o} is the number of unstable nuclei at time t = 0
(or the activity at time t = 0) 
 N is the number of unstable nuclei remaining at time t (or the
activity at time t) 
 e is a constant (2.71828... the base of natural
logarithms) 

Relation between the Decay
Constant and the Half Life of a Nuclide 
From above we have 

and from the definition of half life we can
say, when t = t_{˝} then N = N_{o}/2, therefore 

and, by taking the natural logarithm of
both sides gives 

→ 


so the relation between half life and
the decay constant is given by 
