Protocol

Received: 4 May 2011 Revised: 7 June 2011 Accepted: 8 June 2011 Published online in Wiley Online Library

Rapid Commun. Mass Spectrom. 2011, 25, 25382560

2538Guidelines and recommended terms for expression of stable-isotope-ratio and gas-ratio measurement results, {

Tyler B. Coplen*U.S. Geological Survey, 431 National Center, Reston, VA 20192, USA

To minimize confusion in the expression of measurement results of stable isotope and gas-ratio measurements,recommendations based on publications of the Commission on Isotopic Abundances and Atomic Weights of theInternational Union of Pure and Applied Chemistry (IUPAC) are presented. Whenever feasible, entries are consistentwith the Systme International dUnits, the SI (known in English as the International System of Units), and thethird edition of the International Vocabulary of Basic and General Terms inMetrology (VIM, 3rd edition). The recom-mendations presented herein are approved by the Commission on Isotopic Abundances and Atomic Weights and aredesigned to clarify expression of quantities related to measurement of isotope and gas ratios to ensure that quantityequations instead of numerical value equations are used for quantity definitions. Examples of column headings consis-tent with quantity calculus (also called the algebra of quantities) and examples of various deprecated usages connectedwith the terms recommended are presented. Published in 2011 by John Wiley & Sons, Ltd.

(wileyonlinelibrary.com) DOI: 10.1002/rcm.5129During the 2003 meeting in Ottawa, Canada, of the Commis-sion on Isotopic Abundances and Atomic Weights of theInternational Union of Pure and Applied Chemistry, membersrecognized that there is confusion about the notation of deltavalues in isotope studies. The delta value (d) is used to expressthe relative difference of a ratio of the numbers (or theamounts) of two isotopes in a specimen compared with thatof a reference, commonly an international measurementstandard. Results of such isotopic measurements are usedin anthropology, atmospheric sciences, biology, chemistry,environmental sciences, food and drug authentication, forensicapplications, geochemistry, geology, oceanography, and paleo-climatology. The Commission on Isotopic Abundances andAtomicWeights (theCommission hereafter) recommended thatan article be prepared to clarify the terminology and conceptsbehind the terms involved in expressing relative differences instable isotope ratios. These guidelines also apply to radioactiveisotopes with half-lives sufficiently great to be treated as stableisotopes, such as 238U and 235U.[1] This article is a result of thisinitiative and provides information on the reporting of rela-tive differences in ratios of volumes, numbers of molecules,and amounts of gases. Many of the recommendations in thisarticle also apply to reporting and expressing radioactiveisotope ratios although this is not the topic of this document.* Correspondence to: T. B. Coplen, U.S. Geological Survey, 431National Center, Reston, VA 20192, USA.E-mail: tbcoplen@usgs.gov

This article is a U.S. Government work and is in the publicdomain in the U.S.A.

{ Dedicated to Prof. Etienne Roth, the consummate gentle-man and expert on isotopic terminology a long-timefriend of and contributor to the Commission on IsotopicAbundances and Atomic Weights of the InternationalUnion of Pure and Applied Chemistry.

Rapid Commun. Mass Spectrom. 2011, 25, 25382560Most chemical elements have more than one stable isotope.Molecules, atoms, and ions having different stable isotopes ofthe same chemical element possess slightly different physicaland chemical properties, and they commonly will be fraction-ated during physical, chemical, and biological processes giv-ing rise to variations in isotopic abundances and in atomicweights.[2,3] The purpose of this article is to improve theglobal exchange of scientific information in different disci-plines that measure or make use of variations in isotopicabundances. This document should aid the reader in whatmay be called good scientific language in fields utilizingisotopic abundance variations. In those cases where certaincommon usages are deprecated, there are strong reasonsfor this and the reader should follow these recommenda-tions for consistency with the Systme InternationaldUnits, the SI (known in English as the InternationalSystem of Units),[4] and the International Vocabulary ofMetrology Basic and General Concepts and Associated Terms(VIM, 3rd edn.),[5] both published by BIPM (Bureau Interna-tional des Poids et Mesures). In general, published results ofmeasurements should be provided with their measurementuncertainties, which can be calculated following the Guide tothe Expression of Uncertainty in Measurement.[6] For purposesof brevity, uncertainty values are omitted in most of theexamples in this article.CONCEPTS AND ASSOCIATED TERMS

The entries in this article are organized in the form of aglossary and listed in alphabetical order. They conform tospecific requirements that are consistent with the InternationalSystem of Quantities (ISQ),[5] upon which the SI is based. Anumber of recommendations apply to several glossary itemsand are listed here. For definitions of terms in recommenda-tions, see corresponding entry below.Published in 2011 by John Wiley & Sons, Ltd.

Guidelines and recommended terms for expressing stable isotope results Themass number of a nuclide of a chemical element may bespecified by attaching the mass number as a left superscriptto the symbol for the element, as in 15N, or by adding itafter the name of the element, as in nitrogen-15.

An isotope of a chemical element in a substance may bespecified by adding the substance as a right subscriptfollowing the right parenthesis, e.g., n(15N)(NH4)232SO4.

Sub-subscripts and super-subscripts may improve readabil-ity and are acceptable in the specification of the substance.

The ionic charge is shown as a right superscript and bysign alone when the charge number is equal to plus oneor minus one, for example, Na+, 2H+, and 34S2. The widelyused notation S2 and the old notation S= are both obsolete.[7]

In writing the formula for a complex ion, spacing for thecharge number (ionic charge) and parentheses can beadded; for example, 34SO4

2 or (34SO4)2. This staggered

arrangement is now recommended[7] as opposed to theformat 34SO24 in which 2 is directly written above 4.

Molecular masses should not be written as superscripts tomolecular formulae. For example, writing 30CO for 12C18Oto indicate that the mass of the molecule is 30 is incorrectand should be avoided.

It is recommended that the heavy isotopes of hydrogen bewritten as 2H and 3H rather than as D and T, and that theybe written out as deuterium and tritium, which is inagreement with Nomenclature of Inorganic Chemistry.[7]

In specifically and selectively isotopically labeled compounds,the nuclide symbol is placed in square brackets before thename of the part of the compound that is isotopicallymodified,[7] i.e., [15N]H2[

2H] and [2H1,15N]ammonia. In

isotopically substituted compounds, the appropriate nuclidesymbol(s) is placed in parentheses before the name ofthe part of the compound that is isotopically substituted,[7]

i.e., compare H2HO and (2H1)water. A space should be printed between the numerical value

and symbol of the unit,[4] for example:

m 6:4 kg d2HVSMOWSLAP 16:4 % 253t 14 C x 13C 1:11 %Even when the value of a quantity is used as an adjec-

tive, a space is left between the numerical value and theunit symbol.[4] Only when the name of the unit is spelledout would the ordinary rules of grammar apply, so that inEnglish a hyphen would be used to separate the number fromthe unit.[4] Compare 2-mg sample and two milligram sample.

In compliance with established use in the literature,parentheses following an isotope-quantity symbol regularlyare omitted, e.g. compare e13C versus e(13C) and d18O versusd(18O), which is usually pronounced delta oxygen eighteenor delta oh eighteen, rather than delta eighteen oxygen.

The solidus (oblique stroke or forward slash,/) regularly isused as a record separator for isotopes in a quantity, e.g.R(44Ca/42Ca) or R(44/42Ca), and for separation of theunknown and standard, e.g. d34SP/VCDT.

The notation 13C/12C, as in 13C/12C determinations, isacceptable. However, on first use it should be defined,e.g. determinations of N(13C)/N(12C), abbreviated hereinas (13C/12C), are or determinations of n(13C)/n(12C),abbreviated herein as (13C/12C), are.

All quantity symbols are printed in italic font and theirsuperscripts and subscripts are printed in Roman uprightPublished in 2011 bRapid Commun. Mass Spectrom. 2011, 25, 25382560font,[4,5] e.g. P/Q, a, xP, e13C, and d44/42Castandard, unless

superscripts or subscripts are symbols of quantities, inwhich case they are printed in italic font, e.g. iE.

If the absolute value of the number is less than one, a zeroshould precede the decimal sign. Thus, .113% should bewritten 0.113 %.[4]

For camera-ready copy, an en dash symbol (), rather thana hyphen (-), should be used to indicate negative values.

The symbols%, %, ppm, and permeg are used in the expres-sion of values of dimensionless quantities because diE valuesare small. Examples of reporting that include the symbols%, %, and ppm are provided throughout these recommen-dations. When any of the terms %, ppm, etc., are used, it isimportant to state the dimensionless quantity whose valueis being specified; see section 5.3.7 of the 8th SI brochure.[4]

Use of the symbols % and ppm is encouraged when theirmeaning is clearly understood and their use clarifies andsimplifies comprehension of text. For example, manyreaders find with the value d13C=5.45 % easier to readand comprehend than with the value d13C=0.00545.

The isotope should be specified when using words such asdepleted and enriched because depleted or enrichedcould refer to either an isotope with lower mass numberor with higher mass number. The text The sample isenriched by crystallization and . . . or the enrichedspecimen. . .written to specify a sample enriched in a heavyisotope should be avoided. Instead, using carbon isotopes asan example, authors should write The sample is enriched in13C by. . . or the specimen enriched in 13C is. . ..

Each term is presented with notes as appropriate. To assistthe reader, cross-references to other terms are denoted in italictypeface, and a figure (Fig. 1) is provided that relates selectedquantities.

absolute isotopic abundance, xIsotopic abundance value that is free from all known sourcesof bias within stated uncertainty.Note 1:y John Wiley &These values can be determined by calibrating amass spectrometer by means of gravimetricallyprepared synthetic mixtures of materials inwhich an isotope is enriched (or depleted) by aknown or measurable factor.Note 2: This name is a misnomer in that absolute indi-cates absence of uncertainty, but each abundancevalue of an isotope has an uncertainty associatedwith its determination.Example: The absolute isotopic abundance of 13C in theinternational measurement standard NBS 19 calciteis x(13C)NBS19=0.011 0780.000 028.[8,9]See isotope-amount fraction.

absolute isotope ratio, RIsotope ratio that has been determined by isotope-ratio calibra-tion using an international measurement standard, certified iso-topic reference materials, or gravimetric mixtures of highlyenriched (or depleted) isotopes.Note 1: These values can be determined by calibrating amass spectrometer by means of gravimetricallyprepared synthetic mixtures of materials inwileyonlinelibrary.com/journal/rcmSons, Ltd.

9

amount of isotopes ofmass number i of element Ein substance P: n(iE)P

isotope-amount ratio: r(iE/ jE)

and r(iE/ jE)P = n(iE)P / n(

jE)P

isotope delta, relative difference of isotope

ratios: iE = iEP/std = (

iE/ jE)P/std= [R (iE/ jE)P R (

iE/ jE)std] / R (iE/ jE)std

number of isotopes of mass number i of element Ein substance P: N(iE)P

N(iE)P / NA = n(iE)P

where NA is the Avogadro constant,approximately 6.022 14 1023 mol1

isotope ratio,isotope-number ratio: R(iE/ jE)P

and R (iE/ jE)P = N(iE)P / N(

jE)P

R = r

X = xisotope-number fraction: X(iE)P

and X(iE)P = N(iE)P / N(kE)P

k

isotope-amount fraction,isotopic abundance,atom fraction: x(iE)P

and x(iE)P = n(iE)P /n(kE)P

k

relative difference of isotope-amount ratios:

iE = iEP/std = (iE/ jE)P/std

= [r (iE/ jE)P r (iE/ jE)std] / r (

iE/ jE)std

Macroscopic QuantityMicroscopic (Atomic) Quantity(a)

amount of isotopes ofmass number 13 of carbonin substance P: n(13C)P

isotope-amount ratio: r (13C/12C)

and r(13C/12C)P = n(13C)P / n(12C)P

isotope delta, relative difference of isotope

ratios: 13C = 13CP/VPDB = (13C/12C)P/VPDB= [R(13C/12C)P R(13C/12C)std] / R(13C/12C)std

number of isotopes of mass number 13 of carbonin substance P: N(13C)P

N(13C)P / NA = n(13C)Pwhere NA is the Avogadro constant,approximately 6.02214 1023 mol1

isotope ratio,isotope-number ratio: R(13C/12C)P

and R(13C/12C)P = N(13C)P / N(12C)P

R = r

X = xisotope-number fraction: X(13C)Pand X (13C)P = N(13C)P / N(kC)P

k

isotope-amount fraction,isotopic abundance,atom fraction: x(13C)P

and x(13C)P = n(13C)P / n(kC)Pk

relative difference of isotope-amount ratios:

13C = 13CP/VPDB = (13C/12C)P/VPDB= [r (13C/12C)P r (13C/12C)VPDB] / r (13C/12C)VPDB

Macroscopic QuantityMicroscopic (Atomic) Quantity(b)

Figure 1. (a) Relations among selected quantities. The heavier (higher atomic mass) andlighter (lower atomic mass) isotopes of element E are, respectively, iE and jE; k is the massnumber of any isotope for a summation over all isotopes of element E. The isotope iE insubstance P is specified by iEP. For a given non-Greek symbol, lowercase font signifiesthe macroscopic quantity and uppercase font denotes the microscopic (atomic) quantity,e.g., r and R. Alternative names are listed for the quantities isotope ratio, isotope delta,and isotope-amount fraction. (b) Example of selected quantities using stable carbon iso-topes, 12C and 13C.

T. B. Coplen

wileyonlinel

2540which an isotope is enriched (or depleted) by aknown or measurable factor.Note 2: To determine absolute isotope ratios by isotope-ratio calibration, the true isotope ratio that thecalibration material carries has to lie within theuncertainty interval stated for the material usedat a given statistical probability.Note 3: This name is a misnomer in that absoluteindicates absence of uncertainty, but a measuredisotope ratio has an uncertainty associated withits determination.ibrary.com/journal/rcm Published in 2011 by John WiNote 4:ley & Sons, LtdThe forms using a comma as a record separator,R(i, jE)P or R(

iE, jE)P, instead of a forward slash,have also been used in the literature. They areequivalent to, but not preferred over, the defini-tions given above.Example: The stable carbon absolute isotope ratio ofNBS 19 calcite as reported by Chang and Li[9]

is R(13C)NBS19=R(13C/12C)NBS19=R(

13/12C)NBS19=0.011 202 0.000 028.See also measured isotope ratio, true isotope ratio, and isotope-number ratio.. Rapid Commun. Mass Spectrom. 2011, 25, 25382560

Guidelines and recommended terms for expressing stable isotope resultsamount of substance, namountchemical amountBase quantity in the system of quantities upon which the SI isbased.[4] The name of the SI base unit is mole with symbolmol. One mole is the amount of substance of a system thatcontains as many defined elementary entities as there areatoms in 0.012 kg of 12C in their nuclear and electronicground states (approximately 6.022 141023 entities).Note 1:Rapid CommAmount of substance reflects the particulate natureof matter and reflects the fact that these particleshave a property of matter known as numerosityand requires that the substance always be speci-fied; e.g., when counting, the counted particlesmust be identical and specified.[10]Note 2: This quantity had no name prior to 1969 and wassimply referred to as the number of moles.[11]See Avogadro constant and Fig. 1.APEAbbreviation for the deprecated term atom percent excess. Therecommended name for this quantity is excess atom fraction, xE.

atom fraction, xAmount of a specified atom (isotope) of a chemical elementdivided by the total amount of atoms of the element withinthe mixture.Note 1: Previously, this term was called atom %, atompercent, or AP; however, these are deprecatedbecause they are not valid symbols of quantitynames. Neither the name of a quantity, nor thesymbol used to denote it, should imply any par-ticular choice of unit; see section 5.3.1 of the 8thSI brochure.[4] Symbols of ISQ quantity namesshould be, in general, a single character.Note 2: Atom fraction is a dimensionless quantity, and theunits of mmol/mol and mmol/mol (and similarunits) are permissible.Note 3:E

The atom fraction (isotope-amount fraction), x(iE)P, isa valid ISQ quantity of use in tracer studies andisotope balance (or amount-of-substance balance)studies,[12] commonly with isotopes of H, C, N,and O. Use of atom fraction is preferred over use ofdelta values for tracer and mixing calculationsbecause x(iE)P is linearly correlated to mixing frac-tion, whereas d is not.[12] Delta values generally aremeasured and then converted into atom fractions.Note 4: In the literature, the symbols f and F have beenused for atom fraction in isotope-balance equa-tions, and their use should be avoided.Note 5:254For an element with two isotopes iE and jE, theatom fraction, x(iE)P, in specimen P is given bythe relation:

x iE

P n iE

P

n iE

P n jE

P

where n(iE)P is the amount of isotopeiE of element

E in substance P.Published in 2011 bun. Mass Spectrom. 2011, 25, 25382560Note 6:y John WileyThe relation between x(iE)P, the isotope-amountratio r(iE/ jE)P, and the ratio of the numbers of iso-topes, R(iE/ jE)P, of specimen P for two isotopes is:

x iE

P r iE=jE

P

1 r iE=jE P R iE=jE

P

1 R iE=jE PThe relation between x(iE)P and the quantity d

iEP/stdof specimen P for two isotopes is:

x iE

P 1

1 1diEP=std1 r iE=jE std

11 1

diEP=std1 R iE=jE stdAs an example, the x(13C)P of specimen P

expressed relative to the VPDB (Vienna PeedeeBelemnite) scale, for which R(13/12C)VPDB=0.011 179 60 (ignoring its uncertainty for thisexample),[9] is given by the relation:

x 13C

P 1

1 1d13CP=VPDB1 0:011 179 60

In the literature, one can find equations such as:Note 7:AP 13C 1001 1d

1000 1 RPDBwhere AP stands for atom %. As recommendedby Milton and Wielgosz,[13] their use should beavoided. This relation contains the extraneousfactors 100 and 1000, both of which should bedeleted from the equation so that it is a coher-ent quantity equation.[4,13]See also isotopic abundance and Fig. 1.

atom percentatom %APDeprecated term that includes both a quantity and a unit or asymbol for a quantity that is not a valid symbol in the SIbecause symbols of ISQ quantities names must be a singlecharacter, except in a few cases. Neither the name of aquantity, nor the symbol used to denote it, should implyany particular choice of unit; see section 5.3.1 of the 8thSI brochure.[4] Replace with atom fraction expressed as percent.xamples:& SDeprecated:wileyonons, Ltd.The abundance of 13C insample P is 1.5 atom %.Recommended: The 13C fraction, x(13C), ofsample P is x(13C)P=1.5 %.atom percent excessAPEDeprecated term that includes both a quantity and a unit, or asymbol for a quantity that is not a valid symbol in the SI, becausesymbols of ISQ quantities names must be a single character,except in a few cases. In addition, neither the name of a quantity,nor the symbol used to denote it, should imply any particularchoice of unit; see section 5.3.1 of the 8th SI brochure.[4]linelibrary.com/journal/rcm

1

T. B. Coplen

2542APE is defined by two different relations (excess atom fractionand relative exceedance) that are not equivalent, but may yieldsimilar numerical values for isotopic compositions near nat-ural terrestrial values, depending upon the chemical ele-ment.[12] For example, the two versions of APE equationsyield substantially different values for Cl, B, and Li isotopicmeasurements.See excess atom fraction and relative exceedance.Avogadro constant, NAA universal constant that relates the number of entities to theamount of substance for any sample. Its value is the num-ber of 12C atoms in their nuclear and electronic groundstates in 0.012 kg of 12C (designated as 1 mole) and is about6.022 141023; thus, the Avogadro constant has the coherentSI unit reciprocal mole; see section 2.1.1.6 of the 8th SIbrochure.[4]Note:wileyonThe Avogadro constant has long been known as thescaling factor between the macroscopic and themicroscopic or atomic worlds, as shown in Fig. 1.[14]See also number of entities.certified isotopic reference materialReference material accompanied by documentation issuedby an authoritative body, such as a national measurementinstitute, and providing one or more isotope-number or isotope-amount ratios or fractions with associated uncertainties andtraceabilities using valid procedures.

consensus delta valueSee reference delta value.delta, dSee relative difference of isotope ratios and relative difference of gasratios.delta over baseline, DOBDifference between the isotope delta (relative difference of iso-tope ratios) of a sample (usually called a tracer sample) andthat of a stated baseline. The delta over baseline, symbol(iE/ jE)P/reference, is a measurement of the enrichment of aspecimen in a heavier isotope relative to a specified baseline(reference) according to the relation:

iE= jE

P=reference d iE= jE

P d iE= jE

reference

where d(iE/ jE)P is the delta value of isotopesiE and jE of

element E of specimen P and likewise for a stated reference(both of which could be measured relative to a laboratoryworking standard that is not indicated).Note 1:linelThe quantity (iE/ jE)P/reference should bedefined by authors to avoid confusion becausethis symbol can also be used to express mass-independent isotopic variation.Note 2: When it is clear from the context, (iE/ jE)P/referencemay be shortened to (iE)P/reference or

iEreference,ibrary.com/journal/rcm Published in 2011 by John Wiley & Sons, Ltdas appropriate. When the element has morethan two stable isotopes, (iE/ jE)P/reference ispreferred.Note 3: Delta over baseline regularly is used in tracerstudies and studies of isotope balance. For tracerstudies and isotope-mass balances at isotopetracer levels greater than those of natural abun-dance, the use of the isotope-amount fraction,x(iE)P/std, yields accurate results and is preferredto delta over baseline, which yields increasinglyincorrect mass-balance results with increasingvalues of diEP.

[12,13]Example: The stable carbon delta over baseline valueof a specimen relative to the baseline RefQ,(13C)RefQ=

13CRefQ=+9.55103=+9.55%.Delta values are difference measurements andauthors may give a leading+symbol to valuesgreater than zero.See also isotopic difference and relative difference of isotope ratios.dimensionless quantityquantity of dimension one

(a) Ratio of two quantities of the same kind. The coherent SI unitfor all such ratios is the number one (symbol 1) because theunit must be the ratio of two identical SI units.[5] The valuesof such quantities are simply expressed as numbers, andthe unit one is not explicitly shown. Because SI prefixsymbols can neither be attached to the symbol 1 nor tothe name one, the symbol and terms %, %, ppm, andsuch, are used to express values of dimensionless quanti-ties. The powers of 10 are used to express the values ofparticularly large or small dimensionless quantities.[5]

(b) Numbers that represent a count; e.g., number of molecules.Note 1: Either definition implies a quantity for whichthe sum of all of the exponents of the factorscorresponding to the base quantities in its quan-tity dimension is zero.Note 2: While the measurement units and values ofdimensionless quantities are numbers, suchquantities convey more information than anumber; they are characteristic of particulartypes of quantities of the same kind.Examples: mole fraction, isotope ratio, and isotope-numberfraction.excess atom fraction, xE(iE)P/referenceexcess (stable) isotope-amount fractionDifference between the mole fraction of an isotope iE of ele-ment E in substance P and that of a reference. The superscriptE signifies an excess quantity.Note 1: Previously, this term was called atom percentexcess, atom % excess, or APE. However, noneof these should be used because this quantity doesnot have a unique definition and because APE isnot a valid symbol for an ISQ quantity nameas symbols of ISQ quantities names must be asingle character, except in a few cases. In addition,. Rapid Commun. Mass Spectrom. 2011, 25, 25382560

Guidelines and recommended terms for expressing stable isotope results

Rapid Commneither the name of a quantity, nor the symbol usedto denote it, should imply any particular choiceof unit; see section 5.3.1 of the 8th SI brochure.[4]Note 2: One definition in the literature is that this quantityis the excess isotope tracer in substance P relativeto that in a reference; [12,15] thus, APE can bereplaced by xE(iE)P/reference, which is the excessstable isotope-amount fraction or excess atom frac-tion defined by the relation:

xE iE

P=reference x iE

P x iE

reference

where superscript E is the usual way in thermo-dynamics to indicate an excess quantity. When itis clear from the context, xE(iE)P/reference may beshortened to xE(iE)P, x

E(iE)reference, or xEreference , asappropriate.Note 3: The macroscopic quantity excess stable isotope-amount fraction is equal to the microscopic(atomic) quantity excess stable isotope-numberfraction, XEP , which is defined similarly to isotoperatios (isotope-number ratios).Note 4: For a second definition of this quantity,[16] seerelative exceedance.The following are examples of data reporting of xEP and XEP:

xE 34S

VCDT 0:93 % XE 34S

VCDT 0:93 %xE 2H

P=reference 0:008 XE 2H

P 0:008xE 13C

P=reference 9:3 103 XE 13C

P=reference 9:3 103106xE 17O

reference 8:42 106XE 17O

reference 8:42

xE 17O

reference 8:42 ppm XE 17O

P 8:42 ppm

See also atom fraction and relative exceedance.gas delta, dSee relative difference of gas ratios.

international measurement standardMeasurement standard recognized by signatories to an inter-national agreement and intended to serve worldwide.[5]Note 1: VSMOW2 reference water, SLAP2 water, IAEA-S-1silver sulfide, and L-SVEC lithium carbonate,which are distributed by the International AtomicEnergy Agency (Vienna, Austria), are such stan-dards for measurement of relative difference of iso-tope ratios. These materials are assigned referencedelta values by agreement.Note 2: The assigned uncertainties of delta values of someinternational measurement standards (e.g., NBS 19calcite, SLAP water, and IAEA-S-1 silver sulfide)are zero even though their isotope ratios may not bewell known.Note 3: Information on values of internationally dis-tributed isotopic reference materials and theirsources is provided by IUPACs Commissionon Isotopic Abundances and Atomic Weights.[17]254See also isotopic reference material, reference delta value, andreference isotope-ratio value.Published in 2011 bun. Mass Spectrom. 2011, 25, 25382560isotope balanceDetermination of the amount or isotope-amount ratio of asubstance in a mixture by making use of the fact that thesum of the amounts of the isotopes of each constituent ofthe mixture must equal the total in the system. Balance ofamount of material is performed with isotope-number fractionsor isotope-amount fractions to yield a relation such as:

nPx iE

P nQx iE

Q nP nQ x iE

F

where P and Q are two samples that are combined to formmixture F with isotope-amount fractions x(iE)P, x(

iE)Q, andx(iE)F, respectively, of isotope

iE of element E.Note 1:y John WileyIsotope-delta values commonly are used for massbalance or amount-of-substance balance calcula-tions in tracer and natural isotopic variation stu-dies. An example is the relation:

nP d13CP nQ d13CQ nP nQ d13CF

The quantity d is not proportional to atom fraction;for example, the relation above will give increasinglyincorrect results with larger differences betweend13CP and d

13CQ. For accurate isotope-balancecalculations, deltas can be replaced by atom frac-tions; Brenna et al.[12] show the disparity betweenbalances calculated with d values and atomfractions.Note 2: For tracer studies, Corso and Brenna[18] have pro-posed the use of the quantity relative differencein isotope-amount fraction between a specimen Pand a standard, which they write as iE. Writing(iE) is preferred and is defined by the relation:

iE

standard x iE

P x iE

standard

x iE

standard

where x(iE)P is the isotope-amount fraction of isotopeiE in specimen P and likewise for a measurementstandard or an international measurement standard.An analytical correct amount-of-carbon-balanceequation for 13C is written:

13C

Pn C P 13C

Qn C Q 13C F n C P n C Q

where n(C)P and n(C)Q are the amounts of carbonin substances P and Q of mixture F. For tracerstudies and isotope balances at isotope tracer levelsgreater than those of natural abundance, either ofthe quantities atom fraction, isotope-number fraction,or iE give accurate results as opposed to thequantities R(iE) or diE, which yield increasinglyincorrect mass or amount-of-substance balancewith increasing values of x(iE).[12,13,18] The symbol also is used for the quantity volume fraction.[20]

If there is any possibility of confusion, the format(iE)P/standard should be used.wileyonlinelibrary.com/journal/rcm& Sons, Ltd.

3

T. B. Coplen

2544isotope delta, dSee relative difference of isotope ratios.

isotope effectAlteration of either the equilibrium constant or the rate constantof a reaction if an atom in a reactantmolecule is replaced by oneof its isotopes.[11]

(a) Kinetic isotope effect.The effect of isotopic substitution ona rate constant is referred toas a kinetic isotope effect.[19] For example, in the reaction:

Q B ! P

where B is a reactant in which isotopic substitution does notappear and P is a product, the effect of isotopic substitutionin reactant Q is expressed as the ratio of rate constants jk/ ik,where the superscripts j and i represent reactions in whichthemolecules contain the light and heavy isotopes, respectively.A kinetic isotope effect is referred to as normal if thereactant with the lighter isotope reacts more rapidly, sothat jk/ ik is greater than 1. It is termed inverse if the reactantwith the heavier isotope reacts more quickly and jk/ ik is lessthan 1.

(b) Equilibrium isotope effect.The effect of isotopic substitution on an equilibrium constantis referred to as a thermodynamic (or equilibrium) isotopeeffect. The equilibrium constant for isotopic exchangereactions can be written in either of two ways, dependingupon the scientific discipline, and the reader must payattention to which form is being used. For example, the effectof isotopic substitution in reactant Q and product P thatparticipates in the equilibrium:

Q BP

is the ratio jK/ iK of the equilibrium constant for the reactionin which Q contains the light isotope compared with thereaction in which Q contains the heavy isotope, where Bis a reactant in which isotopic substitution does notappear.[19] The ratio can be expressed equivalently as theequilibrium constant for the isotopici exchange reaction:

jQ iPiQ jP

in which reactants that are not isotopically substituted, suchas B, do not appear. The inversely formulated reaction:

jP iQiP jQ

is widely used and is essentially identical from a chemicalpoint of view; however, the value of the equilibrium constantalso is the inverse of that of the previous equation.

(c) Primary isotope effectKinetic isotope effect attributable to isotopic substitution ofan atom to which a bond is made or broken in a specifiedreaction.[19] The corresponding isotope effect on the equili-brium constant of a reaction in which one or more bonds tothe isotopic atoms are broken is called a primary equilibriumisotope effect.wileyonlinelibrary.com/journal/rcm Published in 2011 by John Wi(d) Secondary isotope effectKinetic isotope effect that is attributable to isotopicsubstitution of an atom to which bonds are neither madenor broken in a specified reaction. One speaks of , (etc.)secondary isotope effects, where , (etc.) denote theposition of isotopic substitution relative to the reactioncenter. The corresponding isotope effect on the equilibriumconstant of such a reaction is called a secondary equilibriumisotope effect.Note 1:ley & Sons,An isotope effect is related to, and may be predictedfrom, molecular structures of reactants, transitionstates, and products of an elementary chemical reac-tion. It is position- and reaction-specific (i.e., for thesame reaction, each molecular position has a differ-ent isotope effect, and the same position shows dif-ferent isotope effects in different reactions).Note 2: If isotope effect is used to describe a weightedaverage over all molecular positions of a com-pound, this should be specified explicitly. To avoidmisunderstandings, such use is discouraged, and itis recommended that the quantity isotopic fraction-ation factor be used instead.[21]Note 3: If kinetic isotope effect is used to describe a cas-cade of reaction steps instead of an elementarychemical reaction (e.g., in enzyme reactions orenvironmental processes), the value will beaffected by all processes leading up to andincluding the first irreversible step.[22] In suchcases, it is recommended that the term apparentkinetic isotope effect be used.[21,22]See also isotopic fractionation factor and isotopic fractionation.

isotope ratio, Risotope-number ratiostable isotope ratioRatio of the number of atoms of one isotope to the numberof atoms of another isotope of the same chemical elementin the same system. The number ratio, R, is the numberobtained by counting a specified entity (usually molecules,atoms, or ions) divided by the number of another specifiedentity of the same kind in the same system.[23] For isotopes,the ratio of the number of atoms of one isotope to anotherisotope of a chemical element in a specimen is the isotope-number ratio. Because of historical use, isotope-number ratiois shortened to isotope ratio, and it is defined by therelation:

R iE=jE

P N iE

P

N jE

P

where N(iE)P and N(jE)P are the number of each isotope,

iEand jE, of chemical element E in substance P.Note 1:LtdThe mass number of a nuclide of chemicalelement E may be specified by attaching themass number as a left superscript to the symbolfor the element, as in 15N, and should not beconfused with isotope-number as in isotope-number fraction X or isotope-number ratio R.. Rapid Commun. Mass Spectrom. 2011, 25, 25382560

Guidelines and recommended terms for expressing stable isotope resultsNote 2:45RP

R iE=

Rapid Commun.Superscripts i and j should denote mass numbersof a heavier (higher atomic mass) and a lighter(lower atomic mass) isotope, respectively. In thismanner, a heavier sample will have a higheratomic weight.Note 3: Isotope pairs found in the literature to defineN(iE)/N(jE) or n(iE)/n(jE) include, but are notlimited to:

2H=1H 18O=16O 36S=32S 56Fe=54Fe 97Mo=95Mo7Li=6Li 26Mg=24Mg 37Cl=35Cl 57Fe=54Fe 98Mo=95Mo11B=10B 29Si=28Si 42Ca=40Ca 65Cu=63Cu 110Pd=104Pd13C=12C 30Si=28Si 44Ca=40Ca 66Zn=64Zn 130Te=122Te15N=14N 33S=32S 53Cr=52Cr 82Se=76Se 205Tl=203Tl17O=16O 34S=32S 54Cr=52Cr 88Sr=86Sr

238U and 235U compose a pair when U isotopesare treated as stable isotopes.[1]Note 4: In the mass spectrometric analysis of a gas, thequantity R commonly is used for the ratio ofthe number of ions collected by Faraday-cupcollectors. For example, for analysis of CO2, theratio of the number of ions on them/z-45 collectorto the number on the m/z-44 collector is:

45=44RP N 13C16O16OP N 12C16O17O P

N 12C16O16O

P

where N(13C16O16O), N(12C16C17O), andN(12C16O16O) are, respectively, the number of13C16O16O+, 12C16C17O+, and 12C16O16O+ ionscollected.Note 5: The atomic quantity isotope ratio, R(iE/ jE)P, isequal to the corresponding macroscopic quantityisotope-amount ratio, r(iE/ jE)P, as shown by therelation:

jEP

N iE

P

N jE

P

N iE PNA

N jE PNA

niE

P

n jE

P

r iE=jE Pwhere NA is the Avogadro constant and wheren(iE)P and n(

jE)P are the amounts of the two iso-topes iE and jE of chemical element E in sub-stance P.Note 6: When there can be no confusion about iE and jE,such as in the case for elements with twoisotopes, the quantity R commonly is used withthe convention that the isotope referredto is the heavier stable isotope. In a mannersimilar to that of r(13C)P, R(

13C)P stands forN(13C)P/N(

12C)P. R(iE)P is sometimes shortened

to iR.

Note 7: The unabbreviated format, R(iE/ jE)P, is preferred

for elements with more than two stable isotopesto prevent confusion, as with R(57Fe)P, whichcould be either R(57Fe/56Fe)P or R(

57Fe/54Fe)P.

Note 8: Although isotope ratio is a dimensionless quantity,

the units mmol/mol and mmol/mol should notbe used for R.Note 9:254The forms using a comma as a record separator,R(i,jE)P or R(

iE, jE)P, instead of a forward slash,also have been used in the literature. They arePublished in 2011 by John Wiley &Mass Spectrom. 2011, 25, 25382560equivalent to, but not preferred over, the notationgiven above. The form Ri,j also has been used inthe literature.Examples: (a) R(33S/32S)P=0.008 03;R(33S/32S)P=0.803 %;

R(46Ca/40Ca)P=43.2 ppm.(b) R(13C)Li2CO3=9.510

3.isotope scaleAn ordered set of isotope-delta values or isotopic abundancevalues used in ranking, according to magnitude.[5]Note 1: A scale of isotope-delta values must be normalized(anchored) by a substance that can bemeasured byother laboratories, and an international measurementstandard is preferred.Note 2: It is preferable that isotope scales be normalizedwith two international measurement standards, e.g.,consider the VSMOWSLAP scale and the VPDBLSVEC scale.Example: The hydrogen isotopic composition of the brineexpressed as a delta value on the VSMOWSLAPscale is d2HVSMOWSLAP=5 %.isotope-amount fraction, xatom fractionisotopic abundancestable isotope-amount fractionAmount of a specified isotope in a mixture divided by theamount of all atoms of all isotopes of an element in themixture.The isotope-amount fraction, x(iE)P, is specified for isotope

iEof element E in specimen P by the relation:

x iE

P n iE

PPn E P

N iE PNAPN E PNA

where the summation includes all isotopes of element E,NP is the number of entities, and NA is the Avogadroconstant.Note 1: The sum of the isotope-amount fractions of allisotopes of an element is 1.Note 2: The isotope-amount fraction is a macroscopicquantity, and it is equal to the atomic quantityisotope-number fraction, XP.Note 3: There is no preference between isotope-amountfraction, isotopic abundance, and atom frac-tion. The choice commonly is based uponhistorical use and depends upon the scientificdiscipline.Examples: (a) The 13C amount fraction in specimen P isx(13C)P=0.0095=9.5103=0.95 %.

(b) x(13C)aragonite=0.95 %=9.5 mmol/mol

See atom fraction, Note 2 of mole fraction, and Fig. 1.See also absolute isotopic abundance.

isotope-amount ratio, rstable isotope-amount ratioAmount of an isotope divided by the amount of another iso-tope of the element in the same system. The isotope-amountratio, r(iE/ jE)P , is defined by the relation:wileyonlinelibrary.com/journal/rcmSons, Ltd.

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2546r iE=jE

P n iE

P

n jE

P

where n(iE)P and n(jE)P are the amounts of the two isotopes,

iEand jE, of chemical element E in substance P. Superscripts iand j should denote mass numbers of a heavier (higher atomicmass) and a lighter (lower atomic mass) isotope, respectively.The primary exceptions are reporting of He isotope ratios andisotope ratios of extra-terrestrial materials in cosmochemistry.Note 1:wileyonlinelibWhen there can be no confusion about iE and jE,such as in the case for elements with twoisotopes, the quantity r commonly is used withthe convention that the isotope referred to is theheavier stable isotope. Thus, r(13C)P stands forn(13C)P/n(

12C)P. The quantity r(iE)P is some-

times shortened to ir .

Note 2: The unabbreviated format, r(iE/ jE)P, is preferred

for elements with more than two stable isotopeswhen there can be confusion about iE and jE.Note 3: The macroscopic quantity isotope-amount ratio,r(iE/ jE)P, is equal to the corresponding atomicquantity isotope ratio, R(iE/ jE)P, as shown bythe relation:

r iE=jE

P n iE

P

n jE

P

N iE PNA

N jE PNA

NiE

P

N jE

P

R iE=jE Pwhere NA is the Avogadro constant and whereN(iE)P and N(

jE)P are the numbers of the two iso-topes, iE and jE, of chemical element E in sub-stance P.Note 4: Isotope-amount ratio is a dimensionless quantity,and the units of mmol/mol and mmol/mol (andsimilar units) are permissible.Note 5: The forms using a comma as a record separator,r(i, jE)P or r(

iE, jE)P , instead of a forward slash,have also been used in the literature. Theyare equivalent to, but not preferred over, thenotation given above.Examples: r(33S/32S)P=0.00803, r(2H)methane=6.04103,

r(46Ca/40Ca)P=43.2106, and r(46/40Ca)calcite=r(46Ca/40Ca)calcite=43.2 mmol/mol. The form4.32105 is preferred to 43.2106.See also Fig. 1.isotope-number fraction, Xstable isotope-number fractionNumber of atoms of a specified isotope in a mixture dividedby the total number of atoms of all isotopes of an elementin the mixture. The isotope-number fraction, X, is definedfor isotope iE in specimen P as

X iE

P N iE

PPN E P

where the summation includes all isotopes of element E. Thechoice of uppercase X for the number fraction is in agreementwith the German standard.[23,24]rary.com/journal/rcm Published in 2011 by John WiNote 1: The mass number of a nuclide of chemicalley & Sons, Ltdelement E may be specified by attachingthe mass number as a left superscript tothe symbol for the element, as in 15N, andshould not be confused with isotope-numberas in isotope-number fraction, X, or isotope-number ratio, R, which is also known as the iso-tope ratio.Note 2: The sum of the isotope-number fractions of allisotopes of an element is 1.Note 3: The units mol/mol, mmol/mol, g/g, or similarunits should not be used for isotope-numberfraction although isotope-number fraction isa dimensionless quantity because neither di-mensions of amount of substance nor massappear in N. Adding extraneous units ofmol/mol or g/g to quantities whose dimensionnever included amount of substance or massis confusing and therefore should not bedone.Note 4: The atomic quantity isotope-number fraction,X(iE)P, is equal to the corresponding macro-scopic quantity isotope-amount fraction, x(iE)P.Examples: (a) X(13C)methane=0.993 %, X(15N)P=0.00485,

and X(17O)P=8.42 ppm.(b) X(2H)methane=2.5103.See also Note 3 of isotope-amount fraction, number fraction, andFig. 1.

isotope-number ratio, RSee isotope ratio. See also Fig. 1.isotope-ratio calibrationA two-step operation in which the first step is to establisha relation between an isotope-ratio value of an internationalmeasurement standard (or a certified isotopic reference material orgravimetric mixtures of isotopes) and values of ion currentsor counts of ions provided from an isotope-ratio massspectrometer (or other measuring device or system).[5] Thesecond step is to use this information to establish the isotoperatio of an unknown sample, using ion-current measurementsor counts of ions provided by the mass spectrometer. Mea-surement results should be provided with uncertaintiesthroughout this two-step process.Note 1: A calibration may be expressed by a statement,calibration function, calibration diagram, calibrationcurve, or calibration table. In some cases, it mayconsist of an additive or multiplicative correc-tion of the indication with associated measure-ment uncertainty.Note 2: Calibration should not be confused with adjust-ment of a measuring system, often mistakenlycalled self-calibration, nor with verification ofcalibration. The first step alone in the above defi-nition often is mistakenly perceived as beingcalibration.Note 3: Factors that enter into an assessment of calibratedisotope-ratio measurements by IUPACs Commis-sion on Isotopic Abundances and Atomic Weightsare discussed by Wieser and Berglund.[25]. Rapid Commun. Mass Spectrom. 2011, 25, 25382560

iEP

Guidelines and recommended terms for expressing stable isotope resultsisotopic abundance, xSee isotope-amount fraction and atom fraction. See also absoluteisotopic abundance.isotopic compositionTerm indicating that qualitative or generalized, non-numericalisotopic information has been gathered.Note 1:Rapid CommunAlthough this is not an SI quantity, this termoften is followed by SI-quantity values.Note 2: Use such as Sample Q is not homogeneous; itscopper-65 isotopic composition varies by 0.5 %should be avoided because the reader does notknow if this is a variation of isotope-amountfraction or of isotope-amount ratio, each ofwhich would have a different value.Note 3: The term absolute isotopic composition shouldbe avoided; instead use absolute isotopic abun-dance. IUPACs Commission on Isotopic Abun-dances and Atomic Weights regularly publishesa Table of Isotopic Compositions of the Elementsthat provides evaluated isotopic abundance data.[8]Example: The isotopic composition of total dissolved sulfurin Lake Audubon is highly variable, d34SVCDT=12.2 to +6.55 %.isotopic difference, The difference between the isotope deltas of two substances.The isotopic difference between the two substances P and Qis given by the relation:

iEP=Q iEP;Q;reference i=jEP;Q;reference di=jEP=reference di=jEQ=reference

where di/ jEP/reference is the delta value of substance P relativeto a reference for heavier isotope iE and lighter isotope jE ofelement E. The quantity di/jEQ/reference is defined similarly forsubstance Q. The reference should be clearly specified; usingthe form iEP,Q,reference clearly identifies the reference.Note 1: The quantity iEP/Q should be defined byauthors to avoid confusion because this symbolcan also be used to express mass-independentisotopic variation and this quantity has beendefined in different ways in the literature.[26]Note 2: The value of iEP/Q is dependent upon thereference and can vary greatly depending uponthe isotopic composition of the reference (e.g.,for hydrogen isotopes).Note 3: This quantity is not proportional to isotope-amount fraction.Note 4:254Closely related to iEP/Q are the isotopic fraction-ation eiEP/Q and the isotopic fractionation factoraiEP/Q, which are related by the expressions:

iEP;Q;reference

ei EP=Q

a i EP=Q

1

di EP=reference 1

diEQ=reference 1 1

andPublished in 2011 by John Wiley &. Mass Spectrom. 2011, 25, 25382560=Q iEP;Q; reference diEP diEQ ln aiEP=Q

The quantities aiEP/Q and eiEP/Q are preferred to

iEP,Q,reference because the former two quantitiesare independent of the reference. Note theapproximation symbols in these two equations.In some cases, the approximation can be poor.Note 5: The forms using a comma as a record separator ofisotopes, (i, jE)P,Q,reference and (

iE, jE)P,Q,reference,have been used in the literature. They areequivalent to, but not preferred over, the nota-tion given above.Example: The stable carbon isotopic difference betweenthe two specimens P and Q is 13CP/Q=+8.55103=+8.55 %.See also delta over baseline and relative difference of isotope ratios.

isotopic fractionation, eThe quantity e=a 1, where a is the isotopic fractionation fac-tor. The isotopic fractionation, eP/Q, is useful in discussing dis-tributions of isotopes between substances, and it is definedfor substances P and Q by the relations:

e iE=jE

P=Q e iE;jE

P;Q aP=Q 1 a iE=jE

P=Q 1

eiEP=Q ei=jEP=Q e iE= jE

P=Q aiEP=Q 1 ai=jEP=Q 1

where a is the isotopic fractionation factor of heavier isotope iEand lighter isotope jE of element E between substances Pand Q.Note 1: In the literature, the quantity e (or an equivalentquantity defined as a 1) has been called theisotopic enrichment factor, discrimination,isotopic discrimination, and isotopic fraction-ation constant. The IUPAC Gold Book[11] hasanother definition for isotopic enrichment fac-tor. Of these names, isotopic fractionation ispreferred. The name isotopic fractionationshould not be shortened to fractionationbecause it is important to distinguish betweenchemical fractionation and isotopic fractionation.Note 2: When it is clear from the context, e(iE/ jE)P/Qmay be shortened to eiEP/Q or eP/Q, as appropri-ate. When the element has more than two stableisotopes, e(iE/ jE)P/Q or e

i/jEP/Q is preferred.

Note 3: The isotopic fractionation is a dimensionless

quantity (SI unit 1) and because values of ecommonly are small, they regularly are expressedwith the symbol%, which is 103.Note 4: The quantity eP/Q has been defined in the litera-ture as:

eP=Q 1000 aP=Q 1

This equation is incorrect. The factor 1000 is anextraneous numerical factor and should bedeleted to give a coherent equation.[4]Note 5: The quantities isotopic fractionation (e), isotopicfractionation factor (a), and isotope delta (d), ofsample P and a reference, and a sample Q andthe reference are related by:wileyonlinelibrary.com/journal/rcmSons, Ltd.

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2548eiE P=Q a iE P=Q 1 diE P=reference 1diE Q=reference 1

1

If the reference is an international measurementstandard, reference can be updated asappropriate.Note 6: The quantity e, like the quantity a from which itis derived, can refer to (1) an apparent or netdistribution of isotopes between phases in com-plex biogeochemical systems having unknownreactions or pathways (thus, unknown whetherit is a kinetic or an equilibrium process), (2) athermodynamic (equilibrium) process, or (3) akinetic process. If aiEP/Q is the isotopic fraction-ation factor of an equilibrium process, eiEP/Q isthe equilibrium isotopic fractionation, mostcommonly with the isotopic exchange reactionwritten as:

jP iQiP jQ

However, the reader should be alert because theexchange reaction written as:

jQ iPiQ jP

also is found in the literature. Readers need topay attention to the format used to write theisotopic exchange reaction.Note 7: If the isotopic fractionation factor is that of akinetic process, the kinetic isotopic fractiona-tion can thus be designated by adding subscriptkin, as ekin(

iE)P/Q. If akin(iE)P/Q is the kinetic

isotopic fractionation factor of an irreversiblereaction, ekin(

iE)P/Q reflects the difference in iso-topic composition of the substrate and theweighted average of all instantaneously formedproducts. A negative value of ekin(

iE)P/Q indi-cates that product P is enriched in light isotopejE and the underlying isotope effect is normal. Apositive value of ekin(

iE)P/Q indicates that theproduct P is enriched in the heavy isotope iEand the isotope effect is inverse. Readers shouldbe aware that e sometimes was defined withthe heavy isotope in the denominator, such thata positive e value corresponds to a normal iso-tope effect.Examples: (a) The equilibrium oxygen isotopic fraction-ation factor between calcite and water, writ-ten for the exchange of one atom of oxygenas:

13CaC16O3 H218O1 3 CaC18O3 H216O

is acalcite/water. Its value with combined standarduncertainty at 33.7 C is 1.028 490.000 13,[27]which can be written 1.028 49(13). The valueof ecalcite/water, which equals 1.028 49(13) 1,rary.com/journal/rcm Published in 2011 by John Wiley & Sons, Ltdis 0.028 49(13). Thus, the value of ecalcite/waterat 33.7 C is 28.49(13) %. The following arethree examples of data reporting:

e18OCaCO3=H2O e 18O=16O

calcite=water

ecalcite=water 28:49 13 % at 33:7 C

ecalcite=water 28:49 13 103 at 33:7 C

103e 18O=16O

calcite=H2O 103e 18=16O

calcite=H2O

28:49 13 at 33:7 C

The first form with the symbol% (or % or ppm)is suggested because e is a quantity of dimen-sion 1. The value 28.49(13)103 usually iswritten 2.849(13)102.

(b) The kinetic isotopic fractionation for treatmentof aragonite at 25 C with 100 % H3PO4 isakin(

18O/16O)CO2/aragonite=1.010 63,[28] and

it can be expressed by ekin(18O/16O, 25 C,

H3PO4)CO2/aragonite=10.63 %.(c) In a biological controlled diet experiment, the

net carbon isotopic fractionation betweenconsumer food and consumer blood plasmais expressed by enet(

13C)food/plasma=9.1 %.See also isotopic difference, isotopic fractionation factor, and iso-tope effect.

isotopic fractionation factor, aRatio of R(iE/ jE)P and R(

iE/ jE)Q, where R is the isotope ratioof heavier isotope iE and lighter isotope jE of element E insubstances P and Q. In many biogeochemical systems, thechanges in isotopic composition between reactant and pro-duct can be a complex function of multiple reaction path-ways, branching pathways, fractional yields of variousproducts, and quite frequently of unknown reactions orpathways. It frequently is unknown whether an isotopicfractionation between two substances is an equilibrium ora kinetic isotopic fractionation. One cannot know whichit is without a full understanding of the reaction networkthat connects the two, and often this is impossible.Advances in the understanding of such systems are madeby measuring the isotopic compositions of two or moresubstances and reporting a net or apparent distribution(or fractionation) of isotopes between them.[29] The distri-bution of isotopes between substances P and Q is given bythe isotopic fractionation factor, aiEP/Q, which is defined bythe relation:

aiEP=Q ai=jEP=Q a iE=jE

P;Q N iE

P =NjE

P

N iE

Q =NjE

Q

RiE=jE

P

R iE=jE

Q

where N is the number of isotopes and R is the isotope ratioof heavier isotope iE and lighter isotope jE of element E.. Rapid Commun. Mass Spectrom. 2011, 25, 25382560

Guidelines and recommended terms for expressing stable isotope resultsNote 1:Rapid Commun.The name isotopic fractionation factor shouldnot be shortened to fractionation factorbecause it is important to distinguish betweenchemical fractionation factor and isotopicfractionation factor.Note 2: One also can define aiEP/Q using isotope-amount ratios or isotope amounts.Note 3: When it is clear from the context, ai/ jEP/Q maybe shortened to aiEP/Q or aP/Q, as appropriate.Note 4: The isotopic fractionation factor is a dimension-less quantity (SI unit 1).Note 5: The isotopic fractionation factor is related to thequantity d by the relation:

a iE P=Q diEP=reference 1diEQ=reference 1

where diEP/reference is the relative difference of isotoperatios of sample P and a reference and thesame for Q. If the reference is an international mea-surement standard, reference can be updatedaccordingly.Note 6: In well-constrained reactions, isotopic fraction-ation factors may be linked to the causative iso-tope effects and can be classified as equilibrium orkinetic. For a system with two substances orphases, P and Q, in chemical and isotopic equili-brium and for the exchange of one atom betweenQ and P with heavier isotope iE and lighter iso-tope jE of chemical element E according to:

Q iE P jE Q jE P iE

the distribution of isotopes between Q and Pis given by the isotopic fractionation factor,ai/jEP/Q. The equilibrium isotopic fractionationfactor is exactly the inverse of the equilibriumisotope effect when the reaction is written in thesame direction.Note 7:254The kinetic isotopic fractionation factor of thereaction of product P formed irreversibly fromreactant Q is akin(

iE/ jE)P/Q, where the subscriptkin indicates a kinetic process; it is expressed bythe relation:

akin iE=jE

P=Q Rinstantaneous iE=jE

P

R iE=jE

Q

Rreacting iE=jE

Q

R iE=jE

Q

where Rinstantaneous(iE/ jE)P is the corresponding

isotope ratio of the weighted average of allreaction products that are being formed inan infinitesimally short time. Rreacting(

iE/ jE)Qis the isotope ratio of reactant molecules that areconsumed through reaction in this short time,and R(iE/ iE)Q is the isotope ratio of the remainingreactant. This kinetic isotopic fractionationfactor differs from a kinetic isotope effect in thatkinetic isotope effects are position-specific,Published in 2011 by John Wiley &Mass Spectrom. 2011, 25, 25382560whereas isotopic fractionation factors arederived from isotope ratios of reacting moleculesand express average effects. Taking into accountnon-reacting positions and intramolecularcompetition, it is possible to estimate kineticisotope effects from kinetic isotopic fractionationfactors.[21] When it is clear from the context,akin(

iE/ jE)P/Q may be shortened to akin(iE)P/Q

or akin(iE), as appropriate.Note 8: The isotopic fractionation factor (and the relatedisotopic fractionation, e) is therefore a term thatcan be used to describe several different con-cepts: two or more phases in chemical and iso-topic equilibrium, reactants and products ofnon-reversible reactions, or the measured dis-tributions of isotopes in poorly understoodcomplex systems. Despite the same name,these quantities have fundamentally differentmeanings in particular, the first two quantitiesare assumed to be invariant while the third isnot. Thus, a clear definition by authors isrequired as to whether a is descriptive of athermodynamic (equilibrium) isotopic fraction-ation factor, a kinetic isotopic fractionation factor,or a measured distribution.Note 9: Position specificity may be indicated by lettingP and Q be specific isotopic molecular species.For example, suppose one is investigating theisotopic fractionation between carbonate isoto-pic species, (CO3)

2, having specified isotopesin a carbonate mineral. For the reaction:

Q iE P jE Q jE P iE

if P is (12C16O2)O2 and Q is (12C16O18O)O2,

the reaction for the exchange of one 16O atomand one 17O atom in the carbonate mineral(slightly modified from Eqn.(2) of Schaubleet al.[30]) is:

12C16O217O 2 12C16O218O 2 12C16O3 2

12C16O18O17O 2for which the extended form of alpha at 25 C is:

a17=16O; 25 C12C16O18OO2=12C16O2O2

or alternatively as

a 17=16O; 12C16O18OO2; 12C16O2O2; 25Ch i

:

(a) The oxygen isotopic fractionation factorExamples:

between calcite and water written for theexchange reaction

13CaC16O3 H218O1 3CaC18O3 H216O

has been determined as a18Ocalcite/water=a18/16Ocalcite/water=1.028 49(13),

[27] with the un-certainty in parentheses, following the lastwileyonlinelibrary.com/journal/rcmSons, Ltd.

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2550significant figure to which it is attributed.The value lna18Ocalcite/water=0.028 09(13). Thesevalues generally are expressed as 103ln a values,and 103ln a18Ocalcite/water=28.09(13).

(b) The kinetic isotopic fractionation for treatmentof aragonite at 25 Cwith 100%H3PO4, whichhas a reported akin(

18O/16O)CO2/aragonitevalue of 1.010 63,[28] can be expressed byany of the following:akin 18O=16O

CO2=aragonite

1:010 63 at 25 C

n18=16O; 25 C; H3PO4

CO2=aragonite

1:010 63

akin 18=16O

CO2=aragonite 1:010 63 at 25 C

otopic difference, isotope effect, and isotopicSee also isfractionation.isotopic reference materialSubstance that is sufficiently stable and homogeneous inisotopic composition to serve as a measurement standardin the measurement of isotope ratios.[5]Note 1: An isotopic reference material can be locallyprepared in an individual laboratory or can bean internationally distributed isotopic referencematerial, e.g., USGS40 L-glutamic acid, whichis used for the measurement of d13C and d15Nvalues (see relative difference of isotope ratios).Note 2: Reference materials with or without assignedisotope ratios or delta values can be used for mea-surement precision control, whereas only referencematerials with assigned isotope ratios or deltavalues can be used for calibration or measurementtrueness control.Note 3: A reference materials delta value can be assigneda reference delta value that will have no uncertainty;however, its isotope ratios will have uncertaintyvalues.See also certified isotopic reference material and international mea-surement standard.isotopoculesMolecular species that only differ in either the number orpositions of isotopic substitutions. The term is a contrac-tion of isotopically substituted molecules.[31]Note: The molecular species can be an isotopologue, anisotopomer, or neither. For example, the three molecu-lar species 15N2

16O, 14N15N16O, and 15N14N16O areisotopocules, but they are neither isotopologues(because the latter two do not differ in isotopic com-position) nor isotopomers (only the latter two areisotopomers).rary.com/journal/rcm Published in 2011 by John WiisotopologueisotopologMolecular species that differ only in isotopic composition(number of isotopic substitutions) and relative molecularmass, e.g., C35Cl4, C

35Cl337Cl, C35Cl2

37Cl2.[11] The term

isotopologue is a contraction of isotopic analogue.isotopomersMolecular species having the same number of each isotopicatom (thus, the same relative molecular mass) but differingin their positions.[11] The term is a contraction of isotopicisomer.Note:ley & SonIsotopomers can be either constitutional isomers (e.g.,CH2

2HCH=O and CH3C2H=O or 14N15N16O and

15N14N16O) or isotopic stereoisomers [e.g., (R)- and(S)-CH3CH

2HOH or (Z)- and (E)-CH3CH=CH2H].[11]K-factor, KA correction factor for mass discrimination effects in a massspectrometer that relates an isotope ratio and a measured isotoperatio.Note 1:s,In an isotope-ratio mass spectrometer, the cor-rection factor for mass discrimination, K, is equalto the true isotope ratio, R, divided by the observedisotope ratio. The true isotope ratio can, in practice,be an isotope ratio of a certified isotopic referencematerial or an IUPAC-recommended value. The fac-tor K can then be used to convert the measured iso-tope ratio of a sample P into a true isotope ratio withthe relation RP=KrPm, where RP is the true iso-tope ratio and rPm is the measured isotope-amountratio.Note 2: In evaluating isotope ratios, IUPACs Commissionon Isotopic Abundances and Atomic Weightsincludes system linearity, baseline correction, iso-baric interferences, instrumental fractionation, andsample preparation in its determination ofK-factors,where r is the published isotope-amount ratio.[25]kind of quantityAspect common to mutually comparable quantities.[5]Note: Examples of kind of quantity include length,energy, electrical charge, and amount of substance.Quantities of the same kind within a given systemof quantities have the same quantity dimension.However, quantities of the same dimension arenot necessarily of the same kind. For example, theisotope ratio, R, and the isotope-amount ratio, r, areboth quantities of dimension 1, but they are notthe same kind of quantity.mass-independent isotopic variation, mass-independent (isotopic) fractionationnon-mass-dependent (isotopic) fractionationLtd. Rapid Commun. Mass Spectrom. 2011, 25, 25382560

Guidelines and recommended terms for expressing stable isotope resultsMeasure of the deviation of the isotopic composition of a sam-ple from a specifiedmass-dependent isotopic fractionation rela-tion for an element with three or more stable isotopes.Note 1:Rapid CommThe majority of the isotopic variations observed instable isotope amounts of elements with more thantwo stable isotopes in terrestrial materials areapproximately linearly correlated with differencesin isotope masses. For example, classical isotopeeffects generally will produce d17/16O values thatare about one half those of d18/16O values, d34/32Svalues that are about twice those of d33/32S values,and d36/32S values that are approximately fourtimes those of d33/32S values. Such isotopic varia-tions are termed mass-dependent isotopic fraction-ation because the physical and chemical processesthat produce these isotope effects have a strongand direct dependence on isotope mass. Some iso-tope effects are produced by physical and chemicalprocesses that have strong dependences on para-meters other than isotope mass and are termedmass-independent isotope effects. Mass-independentisotopic fractionation has been observed for the ele-ments oxygen,[32,33] sulfur, and mercury. All ele-ments of even atomic number greater than that ofoxygen have three or more stable isotopes andhave the potential to exhibit non-mass-dependentisotopic fractionation. Mass-independent isotopicfractionation (MIF) has also been referred to asnon-mass-dependent fractionation and anomalousfractionation of isotopes of an element. Mass-independent isotope effects can yield large valuesof . For this reason, has been used to describesituations where mass-independent isotopic frac-tionation has occurred. How mass-independenteffects are defined and expressed for elements is amatter of active discussion and investigation.[33]Note 2: To avoid confusion, should be defined byauthors. The symbol also is used for the quantitydelta over baseline.Note 3: The factors of 1000, 10000, and 1000000 are extra-neous numerical factors and none of them shouldappear in any equation used to define iE to ensurethat the equation is coherent.[4]measured isotope ratiomeasured isotope-number ratioThe isotope ratio of an element in a sample obtained as a mea-surement result by mass spectrometric or chemical measure-ment procedures or from standard reference data.[5]Note: A measurement procedure is sometimes called astandard operating procedure, abbreviated SOP.255See also true isotope ratio.

mole fraction, xamount-of-substance fractionamount fractionAmount of a constituent divided by the total amount of allconstituents in a mixture.[4,11,20]Published in 2011 bun. Mass Spectrom. 2011, 25, 25382560Note 1:y John WileyWhen the defined constituents are isotopes of an ele-ment, the mole fraction is the amount of a definedatom (isotope) in a mixture divided by the amountof all isotopes of the same element in the mixture.This macroscopic quantity is equal to the atomicquantity isotope fraction, X.Note 2: Mole fraction is a dimensionless quantity, and theunits of mmol/mol and mmol/mol (and similarunits) are permissible.See also atom fraction, isotope-amount fraction, and Fig. 1.

mole ratio, ramount-of-substance ratioamount ratioAmount of a specified constituent (usually molecules, atoms,or ions) divided by the amount of another constituent in thesame system.[4,23,34]Note 1: When the defined constituents are isotopes of anelement, themole ratio is the amount of the specifiedatom (isotope) in a mixture divided by the amountof another specified isotope of the same element inthe mixture. This macroscopic quantity is equal tothe atomic quantity isotope ratio, R.Note 2: Mole ratio is a dimensionless quantity, and the unitsof mmol/mol and mmol/mol (and similar units)are permissible.See isotope-amount ratio.non-mass-dependent (isotopic) fractionationSee mass-independent isotopic variation.

number fraction, XNumber of specified entities of a mixture divided by the totalnumber of entities in the mixture. The number fraction is numeri-cally equal to the mole fraction: the amount of a constituentdivided by the total amount of all constituents in the mixture.Note 1: When the specified entity is an isotope of an element,the number fraction is the isotope-number fraction orthe stable isotope-number fraction, and this micro-scopic (atomic) quantity is equal to the macroscopicquantities atom fraction and isotopic abundance.Note 2: The sum of the isotope-number fractions of all iso-topes of an element is 1.Note 3: Although number fraction is a dimensionless quan-tity, the units mmol/mol or mmol/mol (or any suchunits) should not be used for X.Note 4: The symbol X for number fraction is in accord withthe German standard.[24]See also isotope-number fraction.

number of entities, NInteger number obtained by counting of entities, which areusually molecules, atoms, ions, or formula units.[4,5]Note 1: The number of entities, N, is a fundamental basequantity in any system of units, such as the SI,because it can be regarded as a base quantity inall systems of quantities.[4,5]wileyonlinelibrary.com/journal/rcm& Sons, Ltd.

1

T. B. Coplen

2552Note 2:wileyonlinThe microscopic (atomic) quantity N is related tothe macroscopic quantity amount of substance, n,through the scaling factor NA, the Avogadroconstant,[14] as shown in Fig. 1 and discussed inthe section on isotope-amount fraction.Note 3: Number of entities is a dimensionless quantity andthe kind of quantity is not the same as mass frac-tion or isotope-amount fraction even though theyhave the same dimension.[5]Note 4: Measurement of the numbers of isotopes, e.g. of6Li and 7Li, or of isotopologues, e.g. 28SiF4 and

30SiF4,or of the ratio of the numbers of isotopes orisotopologues, is the basis of isotope-ratio massspectrometry.See also Fig. 1 and amount of substance.

number ratio, RNumber obtained by counting a specified entity (usually mole-cules, atoms, or ions) divided by the number of another spe-cified entity of the same kind in the same system.[23]Note 1:elWhen the specified entities are isotopes of an ele-ment, the microscopic (atomic) quantity numberratio is the isotope-number ratio, also termed thestable isotope ratio and isotope ratio, and it isnumerically equal to the macroscopic quantityisotope-amount ratio.Note 2: Although number ratio is a dimensionless quan-tity, the units mmol/mol or mmol/mol (or anysuch units) should not be used for R.Example: The atomic ratio of carbon to nitrogen in amolecule of acetanilide is R(C/N)acetanilide=N(C)acetanilide/N(N)acetanilide=8, where N is thenumber of entities.See also isotope ratio.

part per million, ppmOne part in one million parts, with value 106.Note 1:ibThe unit part per million is not an SI unit, but itcommonly is used.[4]Note 2: This unit is dimensionless. It does not implyany quantity, and the value associated with thesymbol may be negative or positive.Note 3: Units and their symbols should not be modifiedto provide further information about the quan-tity, as for instance ppmv, where v indicatesvolume; see section 5.3.2 of the 8th SI brochure[4]

and Cvita.[35] All ppm are equal in the samesense as all meters are equal irrespective of wheth-er we express heights, depths, lengths, widths,diameters, etc. A p.p.m.v. or ppmv is not anSI symbol or even a symbol to be usedwith the SI.Note 4: The symbol ppm may be replaced by mmol/molfor expressing the ratio of two quantities ofthe same kind, the kind of quantity being amountof substance; for example, it may be used withmole fraction, isotope-amount fraction, isotope-amount ratio, and isotopic abundance. The unitsmmol/mol or similar should not be used withrary.com/journal/rcm Published in 2011 by John Wiley & Sons, Ltdthe quantities isotopic fractionation factor, isotoperatio, isotopic fractionation, nor with relative differ-ence of isotope ratios.Note 5: The symbol ppm may be replaced by mg/g forexpressing the ratio of two quantities of thesame kind, where the kind of quantity is mass.Examples: (a) The relative difference of gas ratios of sample Prelative to standard AIR04 isd(O2/N2)P/AIR04=50 ppm=50 mmol/mol.

(b) The mass-independent 17O enrichment ofspecimen P expressed by its 17O value=88106=88 ppm.

(c) For a specimen of chromium(III) nitrate,d54/52CrCr(NO3)3/SRM979=9.810

6=9.8 ppm.

(d) The molybdenum isotopic fractionationbetween substances P and Q ise98/95MoP/Q=75106=75 ppm.See also Note 4 of per mil.

part per ten thousand, ppttOne part in ten thousand parts, with value 104.Note 1: The unit pptt is not an SI unit, but is used insome specialized scientific fields. When used,pptt should be explained in the text or a footnote.Note 2: This unit is dimensionless. It does not implyany quantity, and the value associated with thesymbol may be negative or positive.Example: For a zinc-bearing specimen and a referenceZnREF, d66/64ZnZnREF=5104=5 pptt.See also Note 4 of per mil.

percent, %per centOne part in one hundred parts, with value 102.Note 1: Units and their symbols should not be modified toprovide further information about the quantity, asfor instance %%,%v, etc; see section 5.3.2 of the 8thSI brochure[4] and Cvita.[35] All percents areequal in the same sense as all meters are equal,irrespective of whether we express heights,depths, lengths, widths, diameters, etc.Note 2: The symbol % may be replaced by cg/g, e.g., forexpressing the ratio of two quantities of thesame kind, where the kind of quantity is mass.Note 3: The symbol % may be replaced by cmol/mol,e.g., for expressing the ratio of two quantities ofthe same kind, the kind of quantity being amountof substance; for example, it may be used formole fraction, isotope-amount fraction, isotope-amount ratio, and isotopic abundance. The unitscmol/mol or similar units should not be usedwith the quantities isotopic fractionation factor,isotope ratio, isotopic fractionation, nor with relativedifference of isotope ratios.Example: The 13C fraction of sample P is x(13C)=1.5102=1.5 %.See also Notes 2 and 4 of per mil.. Rapid Commun. Mass Spectrom. 2011, 25, 25382560

Guidelines and recommended terms for expressing stable isotope resultsper megOne part in one million parts, with value 106.Note 1:Rapid Commun.The unit permeg is not an SI unit, but it is used insome specialized scientific fields.When used, permeg should be explained in the text or a footnote.Note 2: This unit is dimensionless. It does not implyany quantity, and the value associated with thesymbol may be negative or positive.Note 3: The value of per meg and ppm are both 106;therefore, ppm can be substituted for per meg.Note 4: The symbol per meg may be replaced by mmol/mol for expressing the ratio of two quantities ofthe same kind, the kind of quantity being amountof substance; for example, it may be used formolefraction, isotope-amount fraction, isotope-amountratio, and isotopic abundance. The units mmol/molor similar units should not be used with thequantities relative difference of isotope ratios, rela-tive difference of gas ratios, isotopic fractionationfactor, isotope ratio, nor with isotopic fractionation.Examples: (a) The relative difference of gas ratios of asample relative to standard AIR04 isd(O2/N2)AIR04=50 permeg=50 ppm.

(b) The approximate preindustrial molefraction of CO2 in the Earths atmosphereis x(CO2)AIR=275 permeg=275 ppm=275mmol/mol=275106=0.275 %=0.0275 %.

(c) The mass-independent 17O enrichmentof specimen P expressed by the value of17O=+88 per meg=+88106=88 ppm.

(d) For a specimen of chromium(III) boride,d5 4 / 5 2 CrCrB/SRM979 =9.8 per meg =9.8106=9.8 ppm.

(e) The molybdenum isotopic fractionationbetween substances P and Q is e98/95MoP/Q=75 per meg=75106=75 ppm=0.075%.per mil, %per millper millepermilpermilleOne part in one thousand parts, with value 103.Note 1: The unit per mil is not an SI unit, but it com-monly is used.Note 2: The unit is dimensionless. It does not implyany quantity, and the value associated with thesymbol may be negative or positive.Note 3: Units and their symbols should not be modified,as for instance to provide further informationabout the quantity %%, %v, etc; see section 5.3.2of the 8th SI brochure[4] and Cvita.[35] All permils are equal in the same sense as all metersare equal irrespective of whether we expressheights, depths, lengths, widths, diameters, etc.Note 4:255The symbol may be used in expressing the ratioof two quantities where the kind of quantity isnumber of entities; for example, it may be usedPublished in 2011 by John Wiley &Mass Spectrom. 2011, 25, 25382560for number fraction, number ratio, isotope-numberfraction, isotope ratio, and relative difference ofisotope ratios.Note 5: The symbol%may be replaced bymg/g, for ex-ample, for expressing the ratio of two quantities ofthe same kind, where the kind of quantity is mass.Note 6: The symbol % may be replaced by mmol/mol,for example, for expressing the ratio of two quan-tities of the same kind, the kind of quantity beingamount of substance; for example, itmay be usedwith mole fraction, isotope-amount fraction, isotope-amount ratio, and isotopic abundance. The unitsmmol/mol or similar should not be used withthe quantities isotopic fractionation factor, isotoperatio, isotopic fractionation, nor with isotope delta.Examples: (a) The hydrogen isotopic composition of the brineexpressed as a delta value on the VSMOWSLAP scale is d2HVSMOWSLAP=5 %.

(b) For a zinc sample and a reference ZnREF2,d66/64ZnZnREF2=1.18103=1.18 %.

(c) The value of the oxygen isotopicfractionation between calcite and water ise18/16Ocalcite/water=2.849102=28.49 %.quantityProperty of a phenomenon, body, or substance, where the prop-erty has a magnitude that can be expressed as a number and areference.[4,5] The reference can be a measurement unit, a mea-surement procedure, a referencematerial, or a combination of such.Note 1: Quantities are variables and their symbols arewritten in italic font, even if their symbols areGreek characters.Note 2: A given symbol can indicate different quantities.

Example: (a) The deuterium mole fraction of the

methane is x(2H)CH4=15 %; the atomfraction of deuterium in the methane isx(2H)CH4=15 %; the isotopic abundance of2H in the methane is x(2H)CH4=15 %.

(b) For a sample, d18OVSMOWSLAP=9.1 %.quantity of dimension oneSee dimensionless quantity.

reference delta value, dconsensus delta valueValue of the relative difference of isotope ratios (d) carried byan isotopic reference material that is used as a basis for com-parison of a specified d(iE/ jE) value between a sample anda reference material, where iE and jE are heavier and lighterisotopes, respectively, of element E.Note: The uncertainty value may be zero, such as inthe case of international agreement of a quantityvalue of an international measurement standard.For example, the d18/16O of NBS 19 calcite rela-tive to VPDB is d18/16ONBS19/VPDB=2.2 %.Example: The hydrogen isotopic composition of SLAP2reference water on the VSMOWSLAP scale isd2HSLAP2/VSMOW-SLAP=427.50.3 %.[36]wileyonlinelibrary.com/journal/rcmSons, Ltd.

3

T. B. Coplen

2554reference isotope-ratio valueValue of an isotope ratio carried by an isotopic reference materialthat is used as a basis for comparison of a specified isotoperatio between a sample and a reference material.Note:wileyonlinelThe value can be either an isotope (number) ratioor an isotope-amount ratio.Example: The reference isotope-ratio value of SRM 975asodiumchlorideisR(35Cl/37Cl)=3.12790.0047.[37]relative difference of gas ratios, dgas deltarelative gas-ratio differenceRelative differences in ratios of numbers, amounts, orvolumes occupied by two gaseous components in two sys-tems. The components usually are gases, such as O2, N2,and Ar. The synonym gas delta may be preferred to relativedifference of gas ratios because the former is shorter.The relative difference in the amounts of components B1

and B2 in a sample P and a standard (std), d(B1/ B2)P/std, isexpressed by the relation:

d B1=B2 P=std

n B1 Pn B2 P

n B1 stdn B2 std

n B1 stdn B2 std

n B1 Pn B2 Pn B1 stdn B2 std

1

Alternatively, d(B1/ B2)P/std can equally well be defined withnumber of entities, N, of B1 and B2.For measurement of the relative difference in the amounts of

O2 and N2 in a sample P and a standard (std), one can write:

d O2=N2 P=std n O2 Pn N2 P

n O2 stdn N2 std

n O2 stdn N2 std

n O2 Pn N2 Pn O2 stdn N2 std

1Note 1:ibWhen it is clear from the context, either or bothof the subscripts P and std may be deleted, e.g.,d(O2/N2)=6. 1106.Note 2: It is important for authors to state the quantityon which d(B1/B2) is based, e.g., mole ratio,volume ratio, number ratio, or mass ratiobecause the numerical value will be differentwhen the quantity is defined with mass ratios,mole ratios, or volume ratios.Note 3: The gas delta is a dimensionless quantity (SIunit 1). Because variations in this quantity typi-cally are small, values commonly are reportedin part per million (ppm) or per meg (one partin one million parts).Note 4: The relative difference of gas ratios of O2 andN2 between a sample P and a standard can bedefined with volumes of gas by the relation:

dV O2=N2 P=std V O2 PV N2 PV O2 stdV N2 std

1

where the subscript V of dV indicates a deltavalue determined from volume measurements.This symbol and equation make it clear thatrary.com/journal/rcm Published in 2011 by John Wiley & Sons, Ltdvolume ratios are measured. Because gasesusually are not ideal, d(O2/N2) and dV(O2/N2)are not numerically identical. Compressionfactors or fugacity coefficients can be usedto convert from d(O2/N2) into dV(O2/N2),and vice versa.Note 5: In the literature, one finds equations such as:

d O2=N2 O2=N2 sampleO2=N2 ref 1

1000 000

The factor 1000 000 is an extraneous numericalfactor that should be deleted to make therelation coherent.[4]Examples: (a) For an air sample, d(O2/N2)std=7.1 ppm=7.1 per meg=7.1106.

(b) For a sample P, 106 d(O2/N2)P/std=106 d(O2/N2)std=7.1.See also per meg and part per million.

relative difference of isotope ratios, disotope deltarelative isotope-ratio differenceRatio (R(iE/ jE)P R(

iE/ jE)Q) / R(iE/ jE)Q equals d

i/jEP,Q,where R is the isotope ratio of heavier (higher atomic mass)isotope iE and lighter (lower atomicmass) isotope jE of elementE in substances P and Q. The same relative differences can beused to calculate delta values from frequencies,[38] ion currents,and numbers or amounts of isotopomers or isotopologues.When delta values are expressed relative to an internationalmeasurement standard, the symbol of the standard replaces Q.Equal weights of different isotopes correspond to differentamounts of isotopes. The synonym isotope delta may bepreferred to relative difference of isotopic ratios because theformer is shorter.

Although the delta value, di/jE, can be determined betweenany two substances P and Q, most published delta values aremeasured using an international measurement standard (std),and the delta value is defined by:

diEstd di=jEstd R iE=jE

P R iE=jE

std

R iE=jE

std

N iE PN jE P

N iE stdN jE std

N iE stdN jE std

or the mathematically equivalent version:

diEstd R iE=jE

P

R iE=jE

std

1 NiE

P=NjE

P

N iE

std=N jE

std

1

where N(iE)P and N(jE)P are the numbers of the two isotopes

iE and jE of chemical element E in specimen P, R(iE/ jE)P isN(iE)P / N(

jE)P in specimen P, and equivalent parametersfollow for the international measurement standard, std.Superscripts i and j should denote mass numbers of a heavierand a lighter isotope, respectively, so that the quantity of the. Rapid Commun. Mass Spectrom. 2011, 25, 25382560

Guidelines and recommended terms for expressing stable isotope resultsheavier isotope is in the numerator and that of the lighterisotope is in the denominator. In this manner, when onesays colloquially that sample P is heavier than the standard,the delta value of P is more positive than that of the standard,and the atomic weight of the element in question is higher inP than in the standard. The latter would not be the case if theisotope with the lower atomic mass were in the numerator.A positive di/jE value indicates that the number fraction (and

mole fraction) of the heavier isotope is greater in substanceP than in the international measurement standard.A negative di/jE value indicates that the number fraction(and mole fraction) of the heavier isotope is lower in sub-stance P than in the standard. Numerically identical deltavalues can be determined by substituting equivalent isotope-amount ratios, r(iE/ jE)P and r(

iE/ jE)std, or isotope amounts,n(iE)P and n(

jE)P, for isotope ratios or number of isotopes inequations above.Note 1:Rapid Commun.Commonly, di/jE is shortened to diE, but for ele-ments having three or more isotopes, di/jE ispreferred. With the fully expanded format, itis easy to differentiate between d44/42Ca andd44/40Ca, which could otherwise be indicatedas d44Ca.Note 2: The forms using a comma instead of a forwardslash as a record separator and parentheses,e.g. d(i, jE)P or d(

iE, jE)P , have also been usedin the literature. They are equivalent to, butnot preferred over, the definitions given above.Note 3: The name of the quantity derives from the defi-nition equation. N(iE)P/N(

jE)P is the isotoperatio. The numerator of the delta definitionequation, N(iE)P/N(

jE)P N(iE)std/N(

jE)std, isa difference of isotope ratios. Dividing the nu-merator by its denominator, N(iE)std/N(

jE)std,is indicated by adding the word relative, andd i/ jE is a relative difference of isotope ratios.However, this term is too long for convenientuse and it is suggested that diE or d i/ jE be usedin text. When the isotope is clear from the con-text, authors may also write text such as Thedelta value of specimen P relative to VCDT is44 %. In the literature, the quantity diEstdhas been called the isotopic abundance, isoto-pic composition, and isotope ratio. Isotopedelta or its synonym, relative difference ofisotopic ratios, is preferred.Note 4: The symbol d is pronounced delta; del isincorrect because it is the name for the deloperator, symbol r.[20]Note 5: The standard should be a substance that can bedistributed or a scale defined in terms of such asubstance. Constructs, such as average compo-sition of the Earth or Bulk Earth, areunacceptable.Note 6: Delta values are difference measurements andauthors are encouraged to give a leading+symbol to values greater than zero.Note 7:255The isotope delta is a dimensionless quantity (SIunit 1). Because variations in isotopic abun-dances typically are small, the range in diEvalues for many commonly studied chemicalPublished in 2011 by John Wiley &Mass Spectrom. 2011, 25, 25382560elements[3] is less than 0.2 and values arereported in part per hundred (% or percent),part per thousand (% or per mil), part per million(ppm), and per meg. For expressing an isotoped value, the units mg/g, mg/g, mmol/mol, orany such units, should not be substituted forthe symbols %, pptt, per meg, or ppm.Note 8: To express delta values in part per ten thou-sand, authors can use the abbreviation pptt,which should be explained in the text or in afootnote.Note 9: Equations used to define the quantity d i/ jEstdshould be coherent quantity equations.Because of their universality, they hold forany units as they do not contain extraneousnumerical factors or units.[4,5] Equations, suchas:

d18O 18O=16O

P18O=16O

std

1" #

1000

are found in the literature. A shortened versionof this equation is:

d18O 18O=16O

P18O=16O

std

1

where the author should define (18O/16O)P intheir text, and this parameter typically will beeither N(18O)P/N(

16O)P or R(18O)P and the

parameter (18O/16O)std is defined in a similarmanner. The problem with the former equa-tion is that it can be confusing. For example,suppose that N(18O)P/N(

16O)P=0.001 986 andN(18O)std/N(

16O)std=0.002 005. One calculates:d(18O)=(0.001986/0.002 005) 1=0.0095.Withuse of the first equation in this note, one cal-culates with these same data: d18O=(0.001986/0.002 005 1)1000=9.5, not9.5 %. But, con-vention requires users to place the % symbolfo