Half-Life

Why use a term like half-life rather than lifetime? The answer can be found by examining [link], which shows how the number of radioactive nuclei in a sample decreases with time. The time in which half of the original number of nuclei decay is defined as the half-life, [latex]{t}_{1/2}[/latex]. Half of the remaining nuclei decay in the next half-life. Further, half of that amount decays in the following half-life. Therefore, the number of radioactive nuclei decreases from [latex]N[/latex] to [latex]N/2[/latex] in one half-life, then to [latex]N/4[/latex] in the next, and to [latex]N/8[/latex] in the next, and so on. If [latex]N[/latex] is a large number, then many half-lives (not just two) pass before all of the nuclei decay. Nuclear decay is an example of a purely statistical process. A more precise definition of half-life is that each nucleus has a 50% chance of living for a time equal to one half-life [latex]{t}_{1/2}[/latex]. Thus, if [latex]N[/latex] is reasonably large, half of the original nuclei decay in a time of one half-life. If an individual nucleus makes it through that time, it still has a 50% chance of surviving through another half-life. Even if it happens to make it through hundreds of half-lives, it still has a 50% chance of surviving through one more. The probability of decay is the same no matter when you start counting. This is like random coin flipping. The chance of heads is 50%, no matter what has happened before.

There is a tremendous range in the half-lives of various nuclides, from as short as [latex]{\text{10}}^{-\text{23}}[/latex] s for the most unstable, to more than [latex]{\text{10}}^{\text{16}}[/latex] y for the least unstable, or about 46 orders of magnitude. Nuclides with the shortest half-lives are those for which the nuclear forces are least attractive, an indication of the extent to which the nuclear force can depend on the particular combination of neutrons and protons. The concept of half-life is applicable to other subatomic particles, as will be discussed in Particle Physics. It is also applicable to the decay of excited states in atoms and nuclei. The following equation gives the quantitative relationship between the original number of nuclei present at time zero ([latex]{N}_{0}[/latex]) and the number ([latex]N[/latex]) at a later time [latex]t[/latex]:

where [latex]e=\text{2.71828}\text{...}[/latex] is the base of the natural logarithm, and [latex]\lambda[/latex] is the decay constant for the nuclide. The shorter the half-life, the larger is the value of [latex]\lambda[/latex], and the faster the exponential [latex]{e}^{-\mathrm{\lambda t}}[/latex] decreases with time. The relationship between the decay constant [latex]\lambda[/latex] and the half-life [latex]{t}_{1/2}[/latex] is

To see how the number of nuclei declines to half its original value in one half-life, let [latex]t={t}_{1/2}[/latex] in the exponential in the equation [latex]N={N}_{0}{e}^{-\mathrm{\lambda t}}[/latex]. This gives [latex]N={N}_{0}{e}^{-\mathrm{\lambda t}}={N}_{0}{e}^{-0.693}=0.500{N}_{0}[/latex]. For integral numbers of half-lives, you can just divide the original number by 2 over and over again, rather than using the exponential relationship. For example, if ten half-lives have passed, we divide [latex]N[/latex] by 2 ten times. This reduces it to [latex]N/\text{1024}[/latex]. For an arbitrary time, not just a multiple of the half-life, the exponential relationship must be used.

Radioactive dating is a clever use of naturally occurring radioactivity. Its most famous application is carbon-14 dating. Carbon-14 has a half-life of 5730 years and is produced in a nuclear reaction induced when solar neutrinos strike [latex]{}^{\text{14}}N[/latex] in the atmosphere. Radioactive carbon has the same chemistry as stable carbon, and so it mixes into the ecosphere, where it is consumed and becomes part of every living organism. Carbon-14 has an abundance of 1.3 parts per trillion of normal carbon. Thus, if you know the number of carbon nuclei in an object (perhaps determined by mass and Avogadro’s number), you multiply that number by [latex]1\text{.}3×{\text{10}}^{-\text{12}}[/latex] to find the number of
[latex]{}^{\text{14}}\text{C}[/latex] nuclei in the object. When an organism dies, carbon exchange with the environment ceases, and
[latex]{}^{\text{14}}\text{C}[/latex] is not replenished as it decays. By comparing the abundance of
[latex]{}^{\text{14}}\text{C}[/latex] in an artifact, such as mummy wrappings, with the normal abundance in living tissue, it is possible to determine the artifact’s age (or time since death). Carbon-14 dating can be used for biological tissues as old as 50 or 60 thousand years, but is most accurate for younger samples, since the abundance of
[latex]{}^{\text{14}}\text{C}[/latex] nuclei in them is greater. Very old biological materials contain no
[latex]{}^{\text{14}}\text{C}[/latex] at all. There are instances in which the date of an artifact can be determined by other means, such as historical knowledge or tree-ring counting. These cross-references have confirmed the validity of carbon-14 dating and permitted us to calibrate the technique as well. Carbon-14 dating revolutionized parts of archaeology and is of such importance that it earned the 1960 Nobel Prize in chemistry for its developer, the American chemist Willard Libby (1908–1980).

One of the most famous cases of carbon-14 dating involves the Shroud of Turin, a long piece of fabric purported to be the burial shroud of Jesus (see [link]). This relic was first displayed in Turin in 1354 and was denounced as a fraud at that time by a French bishop. Its remarkable negative imprint of an apparently crucified body resembles the then-accepted image of Jesus, and so the shroud was never disregarded completely and remained controversial over the centuries. Carbon-14 dating was not performed on the shroud until 1988, when the process had been refined to the point where only a small amount of material needed to be destroyed. Samples were tested at three independent laboratories, each being given four pieces of cloth, with only one unidentified piece from the shroud, to avoid prejudice. All three laboratories found samples of the shroud contain 92% of the [latex]{}^{\text{14}}\text{C}[/latex] found in living tissues, allowing the shroud to be dated (see [link]).

There are other forms of radioactive dating. Rocks, for example, can sometimes be dated based on the decay of [latex]{}^{\text{238}}\text{U}[/latex]. The decay series for [latex]{}^{\text{238}}\text{U}[/latex] ends with [latex]{}^{\text{206}}\text{Pb}[/latex], so that the ratio of these nuclides in a rock is an indication of how long it has been since the rock solidified. The original composition of the rock, such as the absence of lead, must be known with some confidence. However, as with carbon-14 dating, the technique can be verified by a consistent body of knowledge. Since [latex]{}^{\text{238}}\text{U}[/latex] has a half-life of [latex]4\text{.}5×{\text{10}}^{9}[/latex] y, it is useful for dating only very old materials, showing, for example, that the oldest rocks on Earth solidified about [latex]3\text{.}5×{\text{10}}^{9}[/latex] years ago.

Activity, the Rate of Decay

What do we mean when we say a source is highly radioactive? Generally, this means the number of decays per unit time is very high. We define activity [latex]R[/latex] to be the rate of decay expressed in decays per unit time. In equation form, this is

where [latex]\text{Δ}N[/latex] is the number of decays that occur in time [latex]\text{Δ}t[/latex]. The SI unit for activity is one decay per second and is given the name becquerel (Bq) in honor of the discoverer of radioactivity. That is,

Activity [latex]R[/latex] is often expressed in other units, such as decays per minute or decays per year. One of the most common units for activity is the curie (Ci), defined to be the activity of 1 g of [latex]{}^{\text{226}}\text{Ra}[/latex], in honor of Marie Curie’s work with radium. The definition of curie is

or [latex]3\text{.}\text{70}×{\text{10}}^{\text{10}}[/latex] decays per second. A curie is a large unit of activity, while a becquerel is a relatively small unit. [latex]\text{1 MBq}=\text{100 microcuries}\phantom{\rule{0.25em}{0ex}}\left(\mu \text{Ci}\right)[/latex]. In countries like Australia and New Zealand that adhere more to SI units, most radioactive sources, such as those used in medical diagnostics or in physics laboratories, are labeled in Bq or megabecquerel (MBq).

Intuitively, you would expect the activity of a source to depend on two things: the amount of the radioactive substance present, and its half-life. The greater the number of radioactive nuclei present in the sample, the more will decay per unit of time. The shorter the half-life, the more decays per unit time, for a given number of nuclei. So activity [latex]R[/latex] should be proportional to the number of radioactive nuclei, [latex]N[/latex], and inversely proportional to their half-life, [latex]{t}_{1/2}[/latex]. In fact, your intuition is correct. It can be shown that the activity of a source is

where [latex]N[/latex] is the number of radioactive nuclei present, having half-life [latex]{t}_{1/2}[/latex]. This relationship is useful in a variety of calculations, as the next two examples illustrate.

Human-made (or artificial) radioactivity has been produced for decades and has many uses. Some of these include medical therapy for cancer, medical imaging and diagnostics, and food preservation by irradiation. Many applications as well as the biological effects of radiation are explored in Medical Applications of Nuclear Physics, but it is clear that radiation is hazardous. A number of tragic examples of this exist, one of the most disastrous being the meltdown and fire at the Chernobyl reactor complex in the Ukraine (see [link]). Several radioactive isotopes were released in huge quantities, contaminating many thousands of square kilometers and directly affecting hundreds of thousands of people. The most significant releases were of [latex]{}^{\text{131}}\text{I}[/latex], [latex]{}^{\text{90}}\text{Sr}[/latex], [latex]{}^{\text{137}}\text{Cs}[/latex], [latex]{}^{\text{239}}\text{Pu}[/latex], [latex]{}^{\text{238}}\text{U}[/latex], and [latex]{}^{\text{235}}\text{U}[/latex]. Estimates are that the total amount of radiation released was about 100 million curies.

Human and Medical Applications

Activity [latex]R[/latex] decreases in time, going to half its original value in one half-life, then to one-fourth its original value in the next half-life, and so on. Since [latex]R=\frac{0\text{.}\text{693}N}{{t}_{1/2}}[/latex], the activity decreases as the number of radioactive nuclei decreases. The equation for [latex]R[/latex] as a function of time is found by combining the equations [latex]N={N}_{0}{e}^{-\mathrm{\lambda t}}[/latex] and [latex]R=\frac{0\text{.}\text{693}N}{{t}_{1/2}}[/latex], yielding

where [latex]{R}_{0}[/latex] is the activity at [latex]t=0[/latex]. This equation shows exponential decay of radioactive nuclei. For example, if a source originally has a 1.00-mCi activity, it declines to 0.500 mCi in one half-life, to 0.250 mCi in two half-lives, to 0.125 mCi in three half-lives, and so on. For times other than whole half-lives, the equation [latex]R={R}_{0}{e}^{-\mathrm{\lambda t}}[/latex] must be used to find [latex]R[/latex].