Why use a term like half-life rather than lifetime? The answer can be found by examining Figure 1, 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, t1/2t1/2 size 12{t rSub { size 8{1/2} } } {}. 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 NN size 12{N} {} to N/2N/2 size 12{N/2} {} in one half-life, then to N/4N/4 size 12{N/4} {} in the next, and to N/8N/8 size 12{N/8} {} in the next, and so on. If NN size 12{N} {} 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 t1/2t1/2 size 12{t rSub { size 8{1/2} } } {}. Thus, if NN size 12{N} {} 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.

Figure 1: Radioactive decay reduces the number of radioactive nuclei over time. In one half-life t1/2t1/2 size 12{t rSub { size 8{1/2} } } {}, the number decreases to half of its original value. Half of what remains decay in the next half-life, and half of those in the next, and so on. This is an exponential decay, as seen in the graph of the number of nuclei present as a function of time.

There is a tremendous range in the half-lives of various nuclides, from as short as 10−2310−23 size 12{"10" rSup { size 8{ - "23"} } } {} s for the most unstable, to more than 10161016 size 12{"10" rSup { size 8{"16"} } } {} 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 (N0N0 size 12{N rSub { size 8{0} } } {}) and the number (NN size 12{N} {}) at a later time tt size 12{t} {}:

where e=2.71828...e=2.71828... size 12{e=2 "." "71828" "." "." "." } {} is the base of the natural logarithm, and λλ size 12{λ} {} is the decay constant for the nuclide. The shorter the half-life, the larger is the value of λλ size 12{λ} {}, and the faster the exponential e−λte−λt size 12{e rSup { size 8{ - λt} } } {} decreases with time. The relationship between the decay constant λλ size 12{λ} {} and the half-life t1/2t1/2 size 12{t rSub { size 8{1/2} } } {} is

To see how the number of nuclei declines to half its original value in one half-life, let t=t1/2t=t1/2 size 12{t=t rSub { size 8{1/2} } } {} in the exponential in the equation N=N0e−λtN=N0e−λt . This gives N=N0 e−λt =N0e−0.693 =0.500N0N=N0 e−λt =N0e−0.693 =0.500N0. 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 NN size 12{N} {} by 2 ten times. This reduces it to N/1024N/1024 size 12{ {N} slash {"1024"} } {}. 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 14N14N size 12{"" lSup { size 8{"14"} } N} {} 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 1.3×10−121.3×10−12 to find the number of 14C14C nuclei in the object. When an organism dies, carbon exchange with the environment ceases, and 14C14C is not replenished as it decays. By comparing the abundance of 14C14C 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 14C14C nuclei in them is greater. Very old biological materials contain no 14C14C 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 Figure 2). 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 14C14C size 12{"" lSup { size 8{"14"} } C} {} found in living tissues, allowing the shroud to be dated (see Example 1).

Figure 2: Part of the Shroud of Turin, which shows a remarkable negative imprint likeness of Jesus complete with evidence of crucifixion wounds. The shroud first surfaced in the 14th century and was only recently carbon-14 dated. It has not been determined how the image was placed on the material. (credit: Butko, Wikimedia Commons)

There are other forms of radioactive dating. Rocks, for example, can sometimes be dated based on the decay of 238U238U. The decay series for 238U238U ends with 206Pb206Pb, 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 238U238U has a half-life of 4.5×1094.5×109 y, it is useful for dating only very old materials, showing, for example, that the oldest rocks on Earth solidified about 3.5×1093.5×109 size 12{3 "." 5 times "10" rSup { size 8{9} } } {} years ago.

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 RR size 12{R} {} to be the rate of decay expressed in decays per unit time. In equation form, this is

where ΔNΔN size 12{ΔN} {} is the number of decays that occur in time ΔtΔt size 12{Δt} {}. 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 RR size 12{R} {} 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 226Ra226Ra, in honor of Marie Curie’s work with radium. The definition of curie is

or 3.70×10103.70×1010 size 12{3 "." "70" times "10" rSup { size 8{"10"} } } {} decays per second. A curie is a large unit of activity, while a becquerel is a relatively small unit. 1 MBq=100 microcuries (μCi)1 MBq=100 microcuries (μCi) size 12{"1 MBq"="100 microcuries " \( μ"Ci" \) } {}. 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 RR size 12{R} {} should be proportional to the number of radioactive nuclei, NN size 12{N} {}, and inversely proportional to their half-life, t1/2t1/2 size 12{t rSub { size 8{1/2} } } {}. In fact, your intuition is correct. It can be shown that the activity of a source is

where NN size 12{N} {} is the number of radioactive nuclei present, having half-life t1/2t1/2 size 12{t rSub { size 8{1/2} } } {}. 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 Figure 3). 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 131I131I, 90Sr90Sr, 137Cs137Cs, 239Pu239Pu, 238U238U, and 235U235U. Estimates are that the total amount of radiation released was about 100 million curies.

Figure 3: The Chernobyl reactor. More than 100 people died soon after its meltdown, and there will be thousands of deaths from radiation-induced cancer in the future. While the accident was due to a series of human errors, the cleanup efforts were heroic. Most of the immediate fatalities were firefighters and reactor personnel. (credit: Elena Filatova)

Activity RR size 12{R} {} 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 R=0.693Nt1/2R=0.693Nt1/2 size 12{R= { {0 "." "693"N} over {t rSub { size 8{1/2} } } } } {}, the activity decreases as the number of radioactive nuclei decreases. The equation for RR size 12{R} {} as a function of time is found by combining the equations N=N0e−λtN=N0e−λt size 12{N=N rSub { size 8{0} } e rSup { size 8{ - λt} } } {} and R=0.693Nt1/2R=0.693Nt1/2 size 12{R= { {0 "." "693"N} over {t rSub { size 8{1/2} } } } } {}, yielding

where R0R0 size 12{R rSub { size 8{0} } } {} is the activity at t=0t=0 size 12{t=0} {}. 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 R=R0e−λtR=R0e−λt size 12{R=R rSub { size 8{0} } e rSup { size 8{ - λt} } } {} must be used to find RR size 12{R} {}.

Data from the appendices and the periodic table may be needed for these problems.