Definition of half-life Nuclear decay occurs in a probabilistic way. Suppose I have a sample of nuclei of type A. There is a certain number, called "half-life", that characterizes all nuclei of this type. As an example, suppose the half-life of A is 6 minutes. This means that at any given instant each nucleus of type A has a 50% probability of decaying within the next 6 minutes.

Examples

The same probability applies to any nucleus of type A, but that does not mean that every A waits 6 minutes and then decays. Some take a shorter time; some take longer. But if I have a large number of nuclei A, say 1 million, then after 6 minutes 500,000 (or a number very close to this) will have decayed, and 500,000 will still be left.

Suppose my starting time is 1:00 PM. The chart below shows how many A's I have at 1:00 and 1:06. The interesting thing is what happens after 1:06. Now I start off with 500,000 nuclei A, each with a chance of 50% to decay during the next 6 minutes. That means that during this period about half of them will decay. Between 1:06 and 1:12 there will be 250,000 decays. At precisely 1:12 there will be about 250,000 A's left. Similarly, during the next 6 minutes, half of these (namely, 125,000) will decay.

 Time Number of A's 1:00 PM 1,000,000 1:06 PM 500,000 1:12 PM 250,000 1:18 PM 125,000 1:24 PM 62,500 etc.
A graph of these numbers shows that the decrease does not follow a straight line.

The "law of averages"

Note how the logic of probability theory comes into this argument. At the time 1:06 we have 500,000 nuclei which definitely did not decay during the previous 6 minutes. Should we say that since these nuclei did not decay earlier, they are now more likely to decay than before? People sometimes call this the "law of averages". If I flip a coin 5 times and get heads every time, am I now more likely to get tails in the 6th toss? The answer is No, there is no law of averages; the chance is still 50-50 to get heads on the 6th toss. Similarly, the nuclei still have a 50% probability of decaying in the second 6-minute period, in the third 6-minute period, etc.

(A good baseball player might have a batting average of 300, meaning that on any given time at bat he has a 0.3 or 30% chance of getting a hit. But suppose he has gone hitless in 40 at bats. Is he less likely than 30% to get a hit in the 41st at bat because he is in a slump, perhaps caused by an injury that is not improving? Or maybe he is more likely than 30% to get a hit because he is "due". There's clearly no good answer to these questions because batting is not based on chance and probability.)

Exponential decay

The kind of decrease shown in the table and in the graph is called exponential decay. It occurs in various kinds of processes, not only radioactivity. For example, suppose there is a leak in the stove and a certain amount of gas collects in the kitchen. Now the leak is corrected and you wait for the gas to clear out of the kitchen. There will be a certain half-life (perhaps 20 minutes) after which the amount of gas in the kitchen is reduced to half. After another 20 minutes the gas is reduced to one-fourth of the original amount; then to one-eighth the original amount, etc. The half-life is 20 minutes. If you open the window, you might change the half-life to 5 minutes.

In the example of the radioactive A's, half of these nuclei are gone after the first 6 minutes. It is not correct to say that all radioactivity is gone after the first 12 minutes. If I wait a few half-lives though, then the radioactivity is reduced substantially. For example, after 18 minutes I have only 1/8 the number of radioactive nuclei (the A's) as I had the beginning. After 30 minutes I have 1/32 of the original number.

Application to weight

Half-life applies to the weight of a substance, as well as to numbers of nuclei. The half-life of plutonium-239 is 24,000 years. If I have 3 grams of plutonium today, then 24,000 years from now there will be 1.5 grams; 48,000 years from now there will be 0.75 grams; etc.

Determining time intervals

We can also use half-lives to determine times. If I have 500 grams of plutonium today, how long do I have wait until this is reduced to 125 grams? In one half-life (24,000 years) it will be reduced to 250 grams; it will then take a second half-life (another 24,000 years) to reduce this 250 grams to 125 grams. So the answer is two half-lives, or 48,000 years. To have the original 500 grams come down to 62.5 grams will take three half-lives, or 72,000 years.

What if I wanted to find out how long it would take for 500 grams to be reduced to 100 grams (a quantity between the 125 and 62.5 grams in the examples above)? You can see from the previous paragraph that the time would be between 48,000 years and 72,000 years. To find the exact answer you would have to use a more mathematical treatment of this kind of decay; or you could use a graph to find an approximate answer.

The half-life of carbon-14 is about 6,000 years. Suppose I know that a certain sample had 10 grams of carbon-14 at an unknown time in the past. Today I measure it and I find there are 5 grams. I deduce that the time in the past was 6,000 years ago. If my measurement today found there were 2.5 grams, I could conclude that the time in the past was 12,000 years. (Two half-lives: one half-life reduces the isotope from 10 grams to 5; the second half-life reduces it from 5 to 2.5 grams.)

Wide variation of half-lives

Half-lives vary widely, as you can see from the Nuclear Table. The half-life of 238U is 4.5 billion years; that of hahnium is 34 seconds. The artificial 28Al that you use in the lab has a half-life of a few minutes. Half-lives of millionths of a second have been measured.

 KEY CONCEPTS Definition of half-life Characteristics of exponential decay Determining time intervals