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IsoNumbers includes at the moment isotopic numbers for stable isotopes of Hydrogen, Carbon, Nitrogen, Oxygen and Sulfur, and to a (much) lesser extent of Boron, Lithium, Selenium, and Silicon. In this section we present some basic definitions and terminology. Please read it before submitting an isotopic number!

Stable isotope abundance (x), ratio (R), and composition (δ)

Abundance and ratio

Isotope abundance (x, dimensionless) is defined as the proportion of the rare stable isotope of an element (see Table 1 for some examples).

x = quantity of the element's rare stable isotope / total quantity of the element

example for carbon stable isotope 13C: x13C = n(13C) / (n(12C) + n(13C))
with n the number of atoms

Isotope ratio (R, dimensionless) is defined as:

R = quantity of the element's rare stable isotope / quantity of the element's abundant isotope

example for carbon stable isotope 13C: n(R13C) = n(13C) / n(12C)

x and R therefore are linked by: R = x rare stable isotope / x abundant isotope

The above definitions are generalized to compounds.

Isotope composition

In isotope geochemistry, it is usual practice to express isotope ratio in terms of isotope "composition" (or "signature") (δ, dimensionless), the normalized deviation of the ratio of a sample from that of an accepted standard.

δ = (Rsample - R standard ) / R standard = Rsample / R standard − 1

As this is normally a small number it is also multiplied by 1000 and therefore expressed in parts per thousands or "permil" (‰).

Isotope ratios of some primary standards used by the international stable isotope community are summarized at Table 2.
It is common practice for stable isotope laboratories to establish working standards calibrated against these International Reference Standards.

Some terminology: isotope fractionation factor (α), isotope fractionation (enrichment or depletion, ε), and isotope discrimination (Δ)

As they have slight different masses, isotopologues (compounds that differ only in their isotope composition, e.g. nitrates 14N16O3- and 15N16O3-) have distinct physical as well as chemical properties. Heavier isotopologues have lower mobilities, higher binding energies, lower collision probabilities. These small differences result in observed isotope "fractionation" effects that can be subcategorized into thermodynamic (equilibrium) and kinetic (non-equilibrium) effects. The thermodynamic effect is related to the differences of equilibrium constants between isotopologues during chemical reactions or phase equilibrium (A<=>B), where A and B refer to source and product (chemical reaction) or to different phases of a compound (e.g. liquid and vapour). The kinetic effect refers to differences of rate constant (chemical reaction) or transport rate between isotopologues under non equilibrium conditions (A=>B).

In the following section, we provide definitions for isotope fractionation factor (α), fractionation (ε), and discrimination (Δ) that are traditionally used in different disciplines. We however emphasize the fact that these definitions may vary or be approximate.

The isotope fractionation factor (α, dimensionless) is defined as the ratio of the isotope ratios of the element of interest in A and B during a process or reaction:

α B/A = R B / R A

where, again, A and B are traditionally source and product.
example for oxygen stable isotope 18O during liquid-vapour equilibrium: 18αvap/liq = R18H2O_vap / R18H2O_liq
example for sulfur stable isotope 34S during hydrolysis of native S: 34αH2SO4/S = R34H2SO4 / R34S

In geochemistry, it is common practice to use the deviation of α from 1:

ε = α − 1

ε (dimensionless) is the computed isotope fractionation (also called isotope enrichment if ε>0 or isotope depletion if ε<0). In hydrology, it is more convenient to define fractionation as:

ε* = 1 - α

Since it is a small number, it is usually multiplied by 1000 and therefore expressed in ‰: ε (‰) = (α − 1)*1000. Note also that the following approximation is sometimes made for ε (‰):

ε (‰) = 1000 * ε = 1000 * ln(1 + ε) = 1000 * ln(α)

In eco-physiological studies, it is common to use discrimination (Δ, dimensionless) values, defined by:

Δ B = R A / R B - 1

Note that in this case: Δ = 1/α − 1.

In delta (dimensionless) notation:

Δ B = (δ A - δ B) / (1 + δ B)

In delta (‰) notation, the latter becomes: Δ B = (δ A - δ B)/ (1000 + δ B).
Finally Δ B (‰) = 1000 * (δ A - δ B)/ (1000 + δ B)

Certain authors approximate the isotope discrimination by Δ B = δ A - δ B (considering δ B negligible compared to 1).

Alternatively, other authors use D (dimensionless) fractionation defined as: D = - ε. D is calculated from the "Rayleigh" equation which gives the isotope ratio of some substrate relative to its remaining fraction:

ln(R t / R 0) = -D * ln(f)

where R 0 and R t are respectively the initial isotope ratio and that of the remaining fraction f.

Relation between fractionation of several stable isotopes of an element

In some studies, it is useful to measure the isotope compositions of more than one isotope of the element of interest.
example for water as far as oxygen is concerned: H216O, H217O, and H218O

It is theoretically postulated that fractionation of two stable isotopes (of atomic masses mx and my) of the same element during chemical reaction or phase change under equilibrium on non-equilibrium conditions (A<=>B or A=>B) should be described as:

(mxR A / mxR standard) = (myR A / myR standard)λ

where λ is defined as:

λ = mxε / myε = (mxα B/A - 1) / (myα B/A - 1)

Note that some authors use θ instead of λ. Others use γ, especially in the case of kinetic isotope effects.

Another way of accessing to λ values is by using the "Rayleigh" equation defined above. By converting ratios into deltas in the equation and rearranging it for stable isotopes of masses mx and my, we have:

λ = mxε / myε = [ln(mxδt + 1) - ln(mxδ0 + 1)] / [ln(myδt + 1) - ln(myδ0 + 1)]

Isotope compositions that deviate from these relationships (i.e., have λ that differ from theoretically calculated or commonly observed values) are termed "mass-independent" or "mass-anomalous".
Mass-independent fractionation is often denoted by Δ. For the example of 17O and 18O stable isotopes in O2, 17Δ is defined by:

17Δ = ln(δ17O + 1) - λ * ln(δ18O + 1)

or alternatively as:

17Δ = δ17O - λ * δ18O

Useful Stable Isotopes Handbooks, research articles, and internet websites / links

Fry, B. Stable Isotope Ecology. 2006. Springer. 308 p.

Hoefs, J. 2008. Stable isotope geochemistry. 6th edition. Springer.

Lajtha, J., and Michener, R.H. (eds.)1997. Stable isotopes in ecology and environmental science - methods in ecology. Blackwell Scientific Publications, 316 p.

Sharp, Z. Principles of Stable Isotope Geochemistry. 2007. Pearson Prentice Hall. Printed in the United States of America. 344 p.

For guidelines when reporting stable isotope ratios, compositions, or fractionations:
Coplen, T. B., 2011, Guidelines and recommended terms for expression of stable-isotope-ratio and gas-ratio measurement results, Rapid Commun. Mass Spectrom., 25, 2538–2560.

See also IAEA documents here
W. G. Mook, Environmental Isotopes in the Hydrological Cycle Principles and Applications, Vol 1, Introduction: theory, methods, review. UNESCO/IAEA series.

For equilibrium fractionation computations, visit Alphadelta website here
Stable Isotope Fractionation Calculator AlphaDelta (© (1999-2011) G. Beaudoin & P. Therrien)

Visit also Isogeochem here
Isogeochem is a discussion list for promoting the exchange of news and information among those with an interest in stable isotope geochemistry, providing new contacts within the stable isotope community, and enhancing collaborative efforts among researchers from varying disciplines.