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Radioactive Decay
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
- No 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 = No/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
 
 
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