Delta Notation

By convention, the heavier isotope is written first to give a heavy/light isotope ratio.

For example, δ 2H or δ D = 2H/1H.  δ 18O = 18O/16O and δ 17O = 17O/16O. 

The δ of an isotope is always measured relative to a known standard. In the case of O and H, the standard is VSMOW (Vienna Standard Mean Ocean Water). Other standards are used for other isotopes.

To make these differences more apparent, the ratio is multiplied by 1000. The value of  δ 18O is stated as per mil (‰, parts per thousand, in the same way that % is per cent, for parts per hundred).  


In absolute values, δ 18O = 2.0052 * 10-3

Δ (Big delta, not to be confused with small delta) is the difference between two δs.


Stable Isotopes

Stable isotopes are those isotopes that do not undergo radioactive decay. Their nuclei are stable and their masses remain the same. However, they may themselves be the product of radioactive isotope decay. The isotopic composition of stable isotopes is, however, subject to natural variation due to mass-dependent fractionation. Differences in mass between isotopes result in isotopic fractionation during chemical processes. Stable isotopes of interest for the hydrologist are generally H, C, N, O, S, B, and Li.


Isotopic Fractionation

During isotopic fractionation, heavy and light isotopes partition differently between two compounds or phases. Isotope fractionation occurs because the bond energy of each isotope is slightly different, with heavier isotopes having stronger bonds and slower reaction rates. This is because the bond energy and the reaction rates are proportional to the mass difference between isotopes. Thus, light elements are more likely to exhibit isotopic fractionation than heavy isotopes.

For example, the relatively light 12C and 13C isotopes have an 8% mass difference and undergo stable isotope fractionation. In contrast, the heavy isotopes 87Sr and 86Sr have a 1.1% mass difference and do not exhibit detectable mass fractionation. Isotopes especially susceptible to fractionation are among the most abundant on earth: H, C, N, O, and S.


Equilibrium Isotope Effects

Equilibrium reactions explain how isotopes will be distributed when products and reactants remain in a chemical reaction that moves forwards and backwards. This is a reversible process, and reactants in contact long enough can equilibrate. The equilibrium constant is the concentration of reactants/ concentration of products.

Heavier isotopes have stronger bonds, and these bonds have higher activation energies. As a result, heavier isotopes have slower reaction rates. Heavy isotopes are more likely to move into a chemical compound where the element is most strongly bound. For example, 18O is preferentially held in CO2.

Because the bonds of lighter isotopes are broken more easily, lighter isotopes have faster reaction times. By taking hydrogen as an example, we see that the bonds between D-D are stronger than H-D which are still stronger than H-H. For every mole of hydrogen gas there is a 3.53 kJ difference required to break the bond of H-H versus H-D versus D-D.

Due to enthalpy and entropy mixing, isotope fractionation is temperature dependent. Temperature is a measure of energy in a system, and this is seen in the energy of the bond. Greater isotopic fractionation is expected at low temperatures, and no isotopic fractionation is expected at very high temperatures. This principle is used to reconstruct temperature and create proxies for temperatures in the past using oxygen as an isotopic tracer. δ 18O from ice cores, sediment cores, and corals can be analyzed to reconstruct past climates. And, assuming minerals grow in conditions allowing isotopic equilibrium, δ 18O can be used to calculate past temperatures from when minerals such as quart and magnetite crystalized.


Kinetic Fractionation (Non-Equilibrium Isotope Effects)

Unlike with equilibrium reactions where the reaction may proceed in both directions, many chemical and biochemical reactions only proceed in one direction and do not reverse. Photosynthesis is an example of such a uni-directional reaction. Normally, because lighter isotopes have weaker bonds and are utilized first, if a reaction doesn’t proceed to completion, the product will be enriched in lighter isotopes.

Kinetic fractionation can be seen in carbon isotope variability as it progresses through biochemical reactions. Biological reaction tend to favor the lighter 12C isotopes, so carbon reservoirs can be differentiated using carbon as an isotopic tracer. Carbon isotopes extracted from C4 plants can be differentiated from CAM and C3 plants. Freshwater carbonates are distinct from marine limestone. With some overlap, coal can be distinguished from petroleum, and sources of methane (CH4) can be identified as atmospheric, biogenic, or thermogenic.

Non-equilibrium isotope effects encompass kinetic (unidirectional, chemical reactions), biological or metabolic reactions, evaporation, and diffusion/effusion.


Evaporation and Condensation

Hypothetically, evaporation can take place under purely equilibrium conditions (when there is 100% humidity and no breeze), but in practice, the products become isolated from the reactants (if there is wind), and the isotopic composition is subject to kinetic fractionation. 

The bonds of liquid water are stronger than those of water vapour, and this means that heavier isotopes become concentrated in the liquid. Because lighter isotopes move faster they can break through the surface of the water into the vapour leading to greater fractionation.

Rayleigh fractionation explains how δ 18O in rain is removed as water vapour condenses and the storm moves inland away from the ocean.  This results in more negative 18O values further inland.

Temperature is the primary control on isotopic values in precipitation. The origins of rain storms can be identified by examining δ 18O.

* For a more detailed discussion with diagrams explaining temperature controls on precipitation and the Rayleigh rainout effect, please visit oxygen on the periodic table of this website.


Radioactive Isotopes

Radioactive isotopes respond differently than stable isotopes. Radioactive isotopes are nuclides (isotope-specific atoms) that have unstable nuclei that decay, emitting alpha, beta, and sometimes gamma rays. These isotopes eventually reach stability in the form of non-radioactive isotopes of other chemical elements called "radiogenic daughters." Decay of a radionuclide to a stable radiogenic daughter is a function of time measured in units of half-lives.

Types of radioactive decay:

  • alpha (a) decay results from an excess of mass. In this type of decay, alpha particles (consisting of two protons and two neutrons) are emitted from the nucleus. Both the atomic number and neutron number of the daughter are reduced by two, so the mass number decreases by four. An example is the decay of 238U:

  • ß+ - or "positron decay" results from an excess of protons. In this type of decay, a positively charged beta particle and a neutrino are emitted from the nucleus. The atomic number decreases by one and the neutron number is increased by one. An example is the decay of radioactive 18F to stable 18O:

where ß+ is the positron, v is the neutrino, and Q is the total energy given off.

  • ß- - or "negatron decay" results from an excess of neutrons. In this type of decay, a negatively charged beta particle and a neutrino are emitted from the nucleus. The atomic number increases by one and the neutron number is reduced by one. An example is the decay of radioactive 14C to stable 14N:

where ß- is the beta particle, v is the antineutrino, and Q is the end point energy (0.156 MeV).

  • electron capture also results from an excess of protons. In this type of decay, an electron is spontaneously incorporated into the nucleus and a neutrino is emitted from the nucleus. The atomic number decreases by one and the neutron number increases by one. Electron capture may be followed by the emission of a gamma ray. An example is the decay of 123I to 123Te:


Radioactive isotopes can be grouped according to their origin.

Long-lived radioactive nuclides. Some radioactive nuclides that have very long half-lives were created during the formation of the solar system (~4.6 billion years ago) and are still present in the earth. These include 40K (t½ = 1.28 billion years), 87Rb (t½ = 48.8 billion years), 238U (t½ = 447 billion years), and 186Os (t½ = 2 x 106 billion years, or 2 million billion years).

Cosmogenic. Cosmogenic isotopes are a result of cosmic ray activity in the atmosphere. Cosmic rays are atomic particles that are ejected from stars at a rate of speed sufficient to shatter other atoms when they collide. This process of transformation is called spallation. Some of the resulting fragments produced are unstable atoms having a different atomic structure (and atomic number), and so are isotopes of another element. The resulting atoms are considered to have cosmogenic radioactivity. Cosmogenic isotopes are also produced at the surface of the earth by direct cosmic ray irradiation of atoms in solid geologic materials.

Examples of cosmogenic nuclides include 14C, 36Cl, 3H, 32Si, and 10Be. Cosmogenic nuclides, since they are produced in the atmosphere or on the surface of the earth and have relatively short half-lives (10 to 30,000 years), are often used for age dating of waters.

Anthropogenic. Anthropogenic isotopes result from human activities, such as the processing of nuclear fuels, reactor accidents, and nuclear weapons testing. Such testing in the 1950s and 1960s greatly increased the amounts of tritium (3H) and 14C in the atmosphere; tracking these isotopes in the deep ocean, for instance, allows oceanographers to study ocean flow, currents, and rates of sedimentation. Likewise, in hydrology it allows for the tracking of recent groundwater recharge and flow rates in the vadose zone. Examples of hydrologically useful anthropogenic isotopes include many of the cosmogenic isotopes mentioned above: 3H, 14C, 36Cl, and 85Kr.

Radiogenic. Radiogenic isotopes are typically stable daughter isotopes produced from radioactive decay. In the geosciences, radiogenic isotopes help to determine the nature and timing of geological events and processes. Isotopic systems useful in this research are primarily K-Ar, Rb-Sr, Re-Os, Sm-Nd, U-Th-Pb, and the noble gases (4H, 3H-3He, 40Ar).

Because of their stable evolution in groundwater, such naturally occurring isotopes are useful hydrologic tracers, allowing evaluation of large geographic areas to determine flowpaths and flow rates. Consequently, they are helpful in building models that predict fracturing, aquifer thickness, and other subterranean features.


Using Time-Dependent Tracers

Isotopes can also be divided into their usefulness as tracers for modern water, submodern water, and palaeowater.

Current tracers of modern groundwater include SF6, 85Kr, and tritium used in conjunction with 3He. 36Cl and 14C can also be useful tracers of modern groundwater, but their atmospheric concentrations have recovered to natural levels. Radiocarbon accumulates in biomass, so analyzing the 14C in dissolved organic carbon (DOC), for example, may prove an effective strategy. CFCs, SF6, and 85Kr are analyzed from dissolved concentrations of atmospheric gasses and should be corrected for using a soluble gas with a known and stable concentration such as Ar, Ne, or N2. Testing is still done for CFCs, though this is best done in conjunction with SF6.

Some radioisotopes are very short-lived. 133Xe has a half-life of 5 days, 37Ar t1/2= 35 days, and 85Kr t1/2= 10.76 years.

Submodern water can be thought of as water recharged between 50 to 1000 years before present. Tritium-free groundwater is generally considered to be older than 50 years. Since 14C is best used in applications of Holocene and late Pleistocene recharge, it is not an ideal selection for submodern recharge and should be used for applications where water is greater than 1000 years old.

On the other hand, 39Ar has a half-life of 269 years and is not reactive in the subsurface. This makes 39Ar an ideal choice for dating submodern recharge, although the method of analysis is important in deciding if this technique is feasible. A new method for counting atoms rather than decay events (atom trap trace analysis—ATTA) greatly reduces the amount of sample required. Other tracers of submodern water include 32Si and 4He, although complications may arise with these tracers since silicon is highly reactive in soils and helium is highly diffuse.

Radiocarbon remains the most routine way of dating groundwater older than 1000 years. The challenge is in distinguishing the modern component of dissolved carbon acquired during the recharge process from that gained through geochemical reactions below the surface. Cosmogenic 14C is produced in the upper atmosphere, mixes in the troposphere, and cycles into the marine system through dissolved inorganic carbon (DIC) and into the terrestrial system through photosynthesis. The production of 14C has fluctuated over time. 14C dating remains a solid tool for dating organic compounds such as wood and bone to ages beyond 50,000 years ago. For groundwater applications, 14C dating can really only be calibrated to ages of 25,000 to 30,000 years ago due to geochemical reactions that limit precision.

Stable isotopes may be used to trace different origins of groundwater, and palaeowater may be distinguished from modern recharge based on climatic changes. Noble gases may also be used as indicators of recharge during past climates. For very old groundwater (> 50,000 years), qualitative age estimates can be made analyzing the ingrowth of radiogenic isotopes of noble gases including 4He, 21Ne, 40Ar, and several isotopes of Xe. These radiogenic isotopes include 81Kr, with a half-life of 229,000 years, and 36Cl, with a half-life of 301,000 years.