Radiation and Half Lives

Radiation

By the early 1900's, it was known that there were three types of radiation. These are alpha (α), beta (β), and gamma (γ) rays. The α and β rays were discovered in the late 1800's by Ernest Rutherford and J. J. Thompson. γ radiation was discovered by P. Villard some time later.

Radioactivity (just to give you a pseudo-definition) is the property possessed by some natural (and artificial) elements of spontaneously undergoing nuclear decay by emitting one of three types of radiation, α, β, or γ.

α radiation was discovered to be composed of particles, later exposed as helium nuclei (He²⁺) that were being ejected at high speeds. A significant feature of this α radiation is that the particles, when put through a magnetic field, tend to be drawn towards the negative side of the field due to their positive charge. Also significant is the fact that α rays are not very penetrating, easily stopped by such flimsy barriers as paper.

β radiation was discovered in the same experiment as α radiation, and was noticably more penetrating, requiring a good half-centimeter of lead to stop. It, too, is composed of particles, but of an opposite polarity than those of α rays. In later work Henri Becquerel discovered that the particles that make up β radiation are equal in electric charge and mass to an electron, therefore discovering that β particles are fast-moving ejected electrons.

γ radiation was discovered some time later by a French scientist named P. Villard. Breaking the trend set by the previous two types, γ radiation is not composed of particles. Instead, it is electromagnetic radiation, similar to x-radiation but more energetic. Also different from the previous two types, γ radiation has no charge, and a good ten centimeters of lead are required to shield objects from γ ray penetration.

Neutron radiation is also common, as unstable nuclei often eject neutrons as they disintegrate. Neutron radiation is the most important in terms of nuclear reactions, because neutrons are the trigger for nuclear fission. Like the other types of radiation, these fast-moving neutrons are harmful to living tissue. Large amounts of shielding are required to stop neutron radiation, and radioactive materials may continue to emit neutrons for thousands of years.

Half Lives

The term "Half-life" is the rate of decay of a given radioactive element, or more simply, how long it takes for one half of the amount of that given element to change into a non-radioactive substance. The rate of decay is often described for in terms of activity (A), which is simply how much disintegration is occurring over time. This activity is proportional to the number of atoms of the element present (N). So in other words, the rate of decay (A) is equal to the number of atoms present (N) multiplied by a proportionality constant (k). This constant is also called the decay constant. Confused? Probably. Just keep this simple equation in mind, then:

A = kN
Basically, the idea of the above equation is to find the rate of decay. But a rate requires a time interval, so multiple readings must be taken to find how much the activity is changing over time t. A comparison must be made to see how many atoms exist at time t as opposed to how many existed in the beginning of the measurement process (at time t_o). Thus, another related equation is formed:
(A/A_o) = (kN/kN_o)
Again, simply put this is the decay at time t in proportion to the activity in the beginning (t_o). The proportionality constants can be cancelled out, leaving us with just the activity in relation to the number of atoms. This means that that activity is directly related to the number of radiactive atoms that still exist.

A third equation now comes to bear as we try and relate the time period in which the measurements are taken to the fraction of radioactive atoms that are still present after the time has passed. This equation is:

ln(N/N_o) = -kt
This equation can be used in three ways:

• If the activity (A/A_o or its equal, N/N_o) is measured over a period of time t, then the proportionality constant k can be measured and the half-life of the sample can be found using the now-defined value of k.
• If k is known, whatever fraction of the sample the remains (after t time has elapsed) can be calculated.
• If k is known for a known isotope, one could calculate how long it would take for that isotope to decayto a fraction of the original activity.

From the definition of a half-life, it is given that at t_1/2, there will be 1/2 of the sample being measured left. Knowing this, it follows that
ln (1/2N_o/N_o) = -kt_1/2

or

ln1/2 = -kt_1/2

which leads to

-0.693 = -kt_1/2

which gives us the nice, simple equation you see below:

A half-life can be found using this equation, provided that one first utilizes ln(N/N_o) = -kt to find k. Sounds complicated, but it really isn't that bad. And in the end, viola! You have found the half-life of your radioactive substance.

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