Isotopes and Radioactivity Tutorial


Elements are defined by the number of protons, positively charged subatomic particles, in an atom's nucleus. The number of protons in an atom's nucleus is termed its atomic number. Isotopes of a given element carry different numbers of neutrons, or neutrally charged particles, in their nuclei. The sum of the number of neutrons and protons in an atom's nucleaus defines its approximate atomic weight. For example, all carbon atoms have six protons; isotopes of carbon can have 6, 7, or 8 neutrons (Table 1).


Radioactive isotopes (also called radioisotopes) have unstable nuclei. These isotopes disintegrate to form atoms with stable nuclei by the release of subatomic particles and gamma rays (akin to X-rays). The radioactive elements are referred to as parents atoms; the atoms they disintegrate to form are called daughter products.

Some isotopes release an alpha particle during nuclear disintegration; an alpha particle consists of two protons and two neutrons (equivalent to the nucleus of an atom of helium). Others release a beta particle, which is an electron, or negatively charged nuclear particle. Beta particles originate in the nucleus, presumably by breakdown of a neutron into its proton-electron components. Gamma rays are released during both types of radioactive decay.

When an isotope emits an alpha particle, the resultant daughter product has an atomic number two units less than its parent's atomic number, and an atomic weight four units less than its parent's atomic weight. When an isotope emits a beta particle, it decays to a daughter with an atomic number one unit greater and an essentially unchanged atomic weight. The Table of Radionuclides documents the naturally-occurring radioisotopes.

Some isotopes decay and immediately produce a stable daughter product. For example, one-step decays to stable daughters are completed by the radiogenic isotopes 14C (decaying to 14N by the beta process), and 87Rb (decaying to 87Sr by the beta process). Others decay and produce unstable daughters, which then become the parent products of their own daughters. Unstable isotopes producing unstable daughters form a radioactive decay chain. For example, the 235U decay chain eventually produces 207Pb, a stable daughter.

Using empirical data, it is possible to statistically forecast what percentage of a radioisotope's popoulation will decay over a given period of time. This has enabled workers to define a half-life for each radioisotope, the period required for one-half of the original parent population to decay to its stable daughter product. Each radioisotope has its own characteristic half-life.

Suppose that at its inception, a sample contains 100 units of a parent radioisotope. After one half-life has passed, there will remain 50 units of the parent isotope, and 50 units of the daughter product will have been produced. After another half-life, 25 units of the parent isotope will remain, and 75 units of the daughter product will have been produced. After another half-life, 12.5 units of the parent isotope will remain, and 87.5 units of the daughter product will have been produced. Through time, the number of parents constantly decreases while the number of daughters constantly increases. Theoretically, although the number of parents will become insignificantly small, there should never come a time when all of the parent population has decayed to daughters.

Knowing the value of a specific isotope's half-life, it is possible to determine the age of a geologic or archaeologic sample by evaluating the amount of parent and daughter isotopes in it. For example, given the half-life of U-235 is 7 x108 y, suppose you have a rock sample containing minerals having 1 unit of 235U and 3 units of 207Pb. The sample must have originally contained 4 units of parent material, and 25% of the parent material (U-235) remains. Examination of the curve above shows that time equivalent to two half-lives have passed, or approximately 1.4 billion years.

The decay of a radioactive substance follows an exponential relationship. This relationship can be written:

N = No e - ct

where N is the number of parent atoms at time t, and No is the number of parent atoms at t=0 (the original amount in the sample). The decay constant, c, with dimensions of reciprocal time, is related to the half-life (t1/2) by the following relationship:

c = 0.693 / t1/2


t1/2 = ln 2/c

These relatiohsips can be used to determine the age of a geologic or archaeologic sample. Results of such studies are most effeective if enough time has passed to let a substantial amount of the daughter product grow (perhaps 10%), and are of limited use if morethan six half-lives have passed (because not enough of the parent material remains to study). Dating of archaeological samples is commonly conducted using C-14, which has a half-life of 5730 y. Dating of geologic samples is most often accomplished using K-40 (with a K-Ar half-life of 1.3 x 109 y), Rb-87 (with a half- life of 4.9 x 1010 y), U-235 (with a half-life of 7.0 x 108 y), and U-238 (with a half-life of 4.5 x 109 y).

Interpretation of data must take into consideration several factors that can yield inaccurate results. For example, metamorphic processes can "reset" radiometric clocks. If daughter products are noble gases - for example, Ar or Rn - loss of the daughter product can occur as the gases diffuse from minerals.

In other instances, a mineral can be created with a substantial amount of a daughter product already incorporated. For example, 40K is commonly used to determine radiometric ages. About 90% of all 40K undergoes a beta decay to produce a daughter of 40Ca; 40K can also undergo a process called electron capture to produce a daughter of 40Ar. (In electron capture, a proton is transformed to a neutron.) Radiometric dating techniques focus on the 40K-40Ar system because Ca is a common constituent in many rock-forming minerals, and it is not possible to distinguish the Ca that was derived from decay of 40K from the Ca that was originally in the sample.


Three isotopes of uranium occur in nature: U-238 (99.3% of all U in natural systems), U-235 (0.7%), and U-234 (0.005%). All of them are naturally-occuring radiogenic isotopes, and (as we have seen above) begin decay chains with geologically long half-lives. Nuclear energy makes uses of the heat released during fission (splitting) of U-235, the only naturally-occurring fissionable material. Nuclear fuel is uranium that has been enriched through processing so that it contains about 3% U-235.

Nuclear fuel in a reactor is bombarded by neutrons. U-235 is split during neutron bombardment, producing two atoms with smaller nuclei, more free neutrons, and heat. Although U-238 is not naturally fissionable, it can capture neutrons and convert to Pu (plutonium)-239, which is fissionable. Neutrons released during fission of U-235 and Pu-239 bombard other atoms of U-235, sustaining a chain reaction. The rate of the reaction is controlled by the insertion or removal of rods made of boron or hafnium, elements which absorb the free neutrons, and the neutrons are slowed by a moderating material such as water or graphite. The heat released during fission is used to heat water to produce steam that turns turbines to produce electricity.

Because uranium has an atomic number of 92, and atoms are likely to split into two roughly equivalent pieces, the elements most frequently produced in a nuclear reaction have atomic numbers ranging from about 36-56, and include isotopes of Rb, Sr, Zr, Nb, I, Cs, Ba and La. Some of the isotopes produced are radioactive (Table 2).

Most of the radioactive wastes that are produced during fission of U-235 accumulate in a reactor's fuel rods. Every year about one-third of the fuel rods are replaced, and used rods are stored in large tanks of water at the reactor sites. The U.S. currently stores more than 10,000 tons of uranium in fuel rod waste. After storage for a year, radioactivity levels decrease by more than 90% because many of the radioisotopes produced have relatively short half-lives (Table 2).

Careful waste management is particularly important for isotopes of strontium, cesium, and iodine, which are readily absorbed by humans. In general, there are three approaches to dealing with high-level wastes related to nuclear energy:

1) For products with relatively short half-lives: accepted methods of dealing with these include waiting until the products decay ("delay and decay"), or diluting them in water or air and releasing them into the environment ("dilute and disperse").

2) For products with half-lives ranging from a year to several hundred years, radioactive wastes are concentrated and stored in stainless steel or ceramic containers.

3) For products that have half-lives that exceed a few hundred years, storage (containment) considerations must include an understanding of geologic processes. At the present time, these are typically kept on site until a suitable long-term storage facility is developed.

Nuclear power plants also generate a great deal of low-level radioactive waste. This does not inherently emit radiation, but has become contaminated through contact with radioactive materials. It may include tools, clothing, or sludges. These are buried (like low-level waste from hospitals and universities) at federal or commercial disposal sites.

The U.S. has yet to develop a high-level nuclear waste storage facility that the community can agree meets the government's criteria of geologic stability for the next 10,000 years. In the 1980s, nine sites in three rock types were selected as potentially acceptable ones for development of a high-level waste disposal site (Table 3). Three of these (Yucca Mountain, Deaf Smith County and Hanford) were selected for further study. Low-level commercial wastes are currently being stored at Hanford, Yucca Mountain and at a site in South Carolina.

Others promoted disposal of radioactive wastes by jettisoning them into space, but this proposal has not been seriously considered because of cost, the difficulty of recovery, and the potential for them to fall back to Earth. The U.S. abandoned discussion of seabed disposal of nuclear wastes in 1983, because of the difficulty of waste recovery and the possibility that waste containment could fail, contaminating seawater and the biosphere. Many countries in Europe, limited by the availability of geologically reasonable disposal areas and their relatively small areas, do currently dispose of nuclear wastes on the seabed.

Measurements of Radioactivity and Hazard Assessment

There are several ways in which measures of radiation are presented. Some of these rely on direct measurements of disintegrations over time, and others are used to determine radiation hazard to living tissue (Table 4). The Systeme International (SI) units of measurement for radioactivity are Becquerels, which are defined in disintegrations per second. (A Becquerel is equivalent to 2.7 x 10-11 Ci, where Ci is Curie, a unit that was formerly used.)

Energy emitted during radioactive processes can be measured in joules (J). The absorbed dose of radiation is considered in terms of grays (Gy), where 1 Gy is equivalent to absorption of 1 joule (J) of radiation by 1 kg of material (for example, a human body).. (While Gy is the SI unit for absorbed radiation, a commonly used unit is the rad where 1 Gy = 100 rads). However, the situation is complicated for living matter because certain types of radiation energy do more damage to living tissue than others. The radiation dose equivalent, which takes such differences into account, is the sievert (Sv), also with dimensions of joules per kilogram. For the most penetrating radiation (like X-rays, gamma rays or beta rays) 1 Sv = 1 Gy. For neutrons and alpha particles, however, a multiplication factor is required. For neutrons, 1 Gy is considered equivalent to 10 Sv; for alpha particles, 1 Gy is considered equivalent to 20 Sv.

For a substance with a known activity, the dose is calculated by taking into account the energy released during each decay. For example, consider that a mass Mt of radioactive waste contains some weight percent (w) of a radioisotope, the mass of the radioisotope can be found:

Mi = Mt * w/100 (kg)

The number of moles of the isotope (mi) can be determined by :

mi = Mi / wi

where wi is the atomic weight of the radioisotope. The number of atoms of the isotope can be determined by

ni = N * mi

where N is Avagadro's Number, 6.02 x 1023. Activity (in Becquerels, or disintegrations per second) is measured as

A = nc

where c is the decay rate. Assuming that all energy emitted is in heat, radioactive heat production, H (in cal/s), can be evaluated using

H = A * E * f

where E is the energy per disintegration in MeV, and f is a conversion factor:

f = 3.83 x 10-14 cal/MeV

so that by dimensional analysis:

H (cal/s) = A (disintegrations/s) * E (MeV/disintegration) * f (cal/Mev)

Radiation is emitted in all directions from a source. The total radiation emitted from a source at some distance, r, can be modelled for the area of a sphere (Sa) with radius r:

Sa = 4p * r2

The approximate dose, D (in cal), that would be received by a 100 kg person with an area of 0.5 m2 standing at distance r from the radioactive source for one minute can be modelled as:

D = H/(4p* r2) * 0.5 m2 * 60 seconds

The absorbed dose in Gy (J/kg) could then be estimated as:

[D (cal) * 4.18 J/cal] / 100 kg


To decrease doseage, precautions are taken to limit a worker's exposure in nuclear waste areas. Radiation in the form of alpha particles cannot pass through material merely as thick as a piece of paper. Beta particles cannot pass through metal. Gamma rays can be halted by lead shielding. So it is possible to wear protective clothing and to use respirators to limit exposure to radiation. The federal government sets a limit on the amount of radiation a worker can be exposed to. The permissible occupational dose is 5 rems a year. The exposure any workers receive is monitored carefully and evaluated at regual periods.


Estimates suggest the average person in the U.S. receives over 150 mrem of radiation exposure per annum. Half of the people who receive a dose of 400-450 rads (about 3000 times greater) over a 30-day period die from the exposure within a period of weeks. A dose of 100 rads generally makes people very sick; a dose of 5000 rads usually will kill someone within hours. (For comparison, adult cockroaches can withstand doses of 100,000 rads, and some viruses can survive doses of millions of rads.)