Once an atom has decayed, it is no longer that type of atom. (The carbon atom that decays is no longer a carbon atom.) If we start with a certain number of atoms - say 100 million of them, then after a certain period we will notice that the number of atoms remaining is decreasing. The number remaining depends on how stable the atom is (or isn't)! We end up with a graph which curves down towards zero.

This remarkable graph halves in equal time intervals
called **HALF-LIVES (T _{1/2})**.

**The half-life of an
isotope is the time taken for half of the starting quantity to decay away.**

In one half-life the amount of original material reduces by half, from 100% to 50%. During the second the amount of original material reduces by half, from 50% to 25%. And so on.

The half -life varies from isotope to isotope, some have values in the order of seconds, others such as U-238 , thousands of millions of years.

Generally isotopes used for medical treatment
have half-lives of the order of hours so that they do not linger in the
body. A tiny amount is made, usually at Lucas Heights, prepared into
the correct chemical form, flown to the hospital and injected within a
matter of half a day from being made. It is a small amount so that the
radiation dose received is tiny but the activity is initially reasonably
high because of the short half-life. (Clearly a short half-life means the
sample must be fairly active compared with the same number of long lived
atoms.)

Longer lived atoms are used for dating rocks
or fossils or archaeological sites. The most famous technique is the **Carbon
- 14 dating**. This isotope has a half-life of 5760 years. Bombardment
of the Nitrogen - 14 by solar generated neutrons in the upper atmosphere
creates a continual supply of C-14. This becomes part of all organic material.
When the organic material dies, the uptake of C-14 ceases. Due to
decay the amount of C-14 drops in proportion to the stable isotope C-12.
Referring to the graph, 100% is the amount of C-14 in, say, a gram of organic
material right now. As time passes we measure the amount of C-14
in the sample, put it on the graph and measure the time.

(In reality corrections must be made for fluctuations in the solar output. This is measured by the amount of C-14 in tree rings from log samples found in bogs going back over 7000 years.)

Potassium -Argon dating and U-238 decay are used to measure the age of extremely old rock crystals. The latter has now measured the age of the Solar System with confidence at 4.7 billion years!

A span of sample half-lives can be seen in the
decay
of U-238 to lead-206, which is stable. Dating
rocks

**A Formula Approach; **( NOT ON SYLLABUS
)

The decay curve can be shown to have an exact mathematical equation.

The equation is a strange looking one and can be rewritten in several ways.

Start with **N _{0}** , this can represent
numbers of atoms, mass or activity.

Amount remaining Time elapsed ( in half-lives,
T_{1/2} )

N_{o} = N_{o}
/2^{0}^{
}0
T_{1/2}

N_{o} /2 = N_{o} /2^{1}
1 T_{1/2}

N_{o} /4 = N_{o} /2^{2}
2 T_{1/2}

N_{o} /8 = N_{o} /2^{3
}3 T_{1/2}

Clearly a pattern is emerging, it can be generalised
to any time, **t**, by measuring the number of T_{1/2}'s in
t.

The number of T_{1/2} 's
in t = t /T_{1/2}

So, at a time, t, the amount remaining, **N**,is;

__PROBLEM;__ Strontium - 90 has a half-life
of 28.1 years. A sample has an initial activity of 400 MBq . What will
be the activity after 92 years?

Solution;

From the graph, about 38 MBq.

OR (not on syllabus )

Using the formula, let N_{o} =
400MBq , T_{1/2} = 28.1 years , t = 92 years.

**Using **N
= __N___{0}

^{t/T}_{1/2}

**
2**

= __400__

**2**^{92/28.1}

= __400__

2^{3.274}

= __400 __
= 41.3 MBq

9.67

The formula is more precise than the graph and gives the activity at 41.3 MBq.

Greater precision could be gained using the graph
if, say, the graph was recalibrated for after two half lives, starting
at 100 MBq, then looking up a time of (93 - 56.2 ) years. One could do
this because the SHAPE of the graph does NOT change despite starting point
or calibration in half-life.

One of the standard simulations for getting the concept of half-life across involves collecting about a hundred dice in an ice cream tub. The dice are all tossed at the same time on to a large surface. All dice showing the face '6' uppermost are removed (decayed). The remaining dice are counted and returned to the tub, the number being recorded.

These steps are repeated for around twenty throws or until all the dice have 'decayed'.

* * * * * * *

Owing to the difficulty of gathering 100 dice (not to mention the time and opportunities for error) you may wish to refer to a computer simulation of the dice simulation of atomic half-life. Sorry, you don't get little graphics of tumbling dice, just the results of five trials over 100 dice, plus the average number of dice remaining after each roll.

Take a list of the averages because you will
want to draw a graph with number of rolls on the horizontal axis and dice
*remaining*
on the vertical. Don't forget the usual rules
for drawing graphs.

From your graph you want to determine the half-life for a dice. (I know the singular of 'dice' is supposed to be 'die', but really?) That is, 100 dice remain after 0 throws, after how many throws do 50 remain? How many throws are required to reduce the number of dice remaining from 80 to 40? from 60 to 30? Read these values from your most excellent graph and assume you will get a decimal result (at least one decimal place).

Turn in your graph, annotated, and your best
guess at the half-life of a six sided dice.