The Table of Standard Atomic Weights—An exercise in consensus

The present Table of Standard Atomic Weights (TSAW) of the elements is perhaps one of the most familiar data sets in science. Unlike most parameters in physical science whose values and uncertainties are evaluated using the “Guide to the Expression of Uncertainty in Measurement” (GUM), the majority of standard atomic‐weight values and their uncertainties are consensus values, not GUM‐evaluated values. The Commission on Isotopic Abundances and Atomic Weights of the International Union of Pure and Applied Chemistry (IUPAC) regularly evaluates the literature for new isotopic‐abundance measurements that can lead to revised standard atomic‐weight values, Ar °(E) for element E. The Commission strives to provide utmost clarity in products it disseminates, namely the TSAW and the Table of Isotopic Compositions of the Elements (TICE). In 2016, the Commission recognized that a guideline recommending the expression of uncertainty listed in parentheses following the standard atomic‐weight value, for example, Ar °(Se) = 78.971(8), did not agree with the GUM, which suggests that this parenthetic notation be reserved to express standard uncertainty, not the expanded uncertainty used in the TSAW and TICE. In 2017, to eliminate this noncompliance with the GUM, a new format was adopted in which the uncertainty value is specified by the “±” symbol, for example, Ar°(Se) = 78.971 ± 0.008. To clarify the definition of uncertainty, a new footnote has been added to the TSAW. This footnote emphasizes that an atomic‐weight uncertainty is a consensus (decisional) uncertainty. Not only has the Commission shielded users of the TSAW and TICE from unreliable measurements that appear in the literature as a result of unduly small uncertainties, but the aim of IUPAC has been fulfilled by which any scientist, taking any natural sample from commerce or research, can expect the sample atomic weight to lie within Ar °(E) ± its uncertainty almost all of the time.

The present Table of  The Commission strives to provide utmost clarity in products it disseminates, namely the TSAW and the Table of Isotopic Compositions of the Elements (TICE). In 2016, the Commission recognized that a guideline recommending the expression of uncertainty listed in parentheses following the standard atomic-weight value, for example, A r (Se) = 78.971 (8), did not agree with the GUM, which suggests that this parenthetic notation be reserved to express standard uncertainty, not the expanded uncertainty used in the TSAW and TICE. In 2017, to eliminate this noncompliance with the GUM, a new format was adopted in which the uncertainty value is specified by the "±" symbol, for example, A r (Se) = 78.971 ± 0.008. To clarify the definition of uncertainty, a new footnote has been added to the TSAW. This footnote emphasizes that an atomic-weight uncertainty is a consensus (decisional) uncertainty.
Not only has the Commission shielded users of the TSAW and TICE from unreliable measurements that appear in the literature as a result of unduly small uncertainties, but the aim of IUPAC has been fulfilled by which any scientist, taking any natural sample from commerce or research, can expect the sample atomic weight to lie within A r (E) ± its uncertainty almost all of the time. Many in the scientific public do not have a full understanding of the amount of effort and the processes that occur regularly to update standard atomic-weight values and continuously improve the Table of Standard Atomic Weights (TSAW). For example, the following questions are pertinent: • How are new standard atomic-weight values determined, and what are the guidelines for updating them?
• What are the uncertainties on standard atomic-weight valuesstandard uncertainties or expanded uncertainties?
• What is the new format for expressing standard atomic-weight values and their uncertainties to make it clear that they are expanded uncertainties?
• Why do 14 elements now have standard atomic-weight values expressed as intervals?
• When an element has a standard atomic-weight value expressed as an interval, what single value is provided for use in education, commerce, and trade?
• Why do 34 elements have no standard atomic weight?
This article was prepared to provide answers to these and other questions about standard atomic weights and the TSAW. Commission adopted the following definition of atomic weight (mean relative atomic mass) of an element from a specific source:

| Standard atomic weight
"the ratio of the average mass per atom of the element to 1/12 of the mass of an atom of 12 C." The novelty of this definition was emphasized by the following clarifying remarks: 2 • Atomic weight can be defined for any sample.
• Atomic weights are evaluated for atoms in their electronic and nuclear ground states.
• The "average mass per atom" in a specified source is the total mass of the element divided by the total number of atoms of that element.
• , the result is a combined standard uncertainty, u c . 9 The combined standard uncertainty may be thought of as equivalent to "±1 SD." 9 The Commission wishes to express atomic-weight values with a high level of confidence. This rescaling is performed by multiplying the combined standard uncertainty by a coverage factor, k, which gives a quantity that is called the expanded uncertainty, U. 9 Therefore, U = ku c . The confidence levels (for normal distributions) for coverage factors of 1, 2, and 3, respectively, are 68%, 95%, and 99.7%. 9 The Commission aims at disseminating value pairs of standard atomic weight and uncertainty, A r (E) and U[A r (E)], such that it can claim at a high level of confidence that any element in question in all known normal sources will have an atomic weight that will not differ from the relevant A r (E) by more than U[A r (E)]. At an even higher level of confidence, bordering on complete certainty, any chemist sampling any given "normal" material, 10 be it any ore in trade, any product at a chemical plant, or any substance at any chemical laboratory, shall be justified in expecting all elements in that material to possess atomic weights within the implied tabulated ranges of the standard atomic-weight values. 7 To achieve its aim of providing highly reliable and precise standard atomic-weight values, at the 1983 Commission meeting in Lyngby, Denmark, 11 a working party was formed to examine procedures that had been used to assign uncertainties to standard atomic-weight values. The working party reported to the Commission at the next meeting in Lyon, France. Its recommendations and unpublished "Technical Guidelines" were adopted by the Commission. 12 Subsequently, they were modified at the 1995 Commission meeting at Guildford, UK (see Supporting Information). 13 One of the issues addressed by the working party was a decision adopted in the 1981 TSAW 14 17 The TICE has been published at 6-or 8-year intervals and was last published in 2013. 18 The Commission has several subcommittees. Working through its Subcommittee on Isotopic Abundance Measurements (SIAM), the peer-reviewed literature is evaluated biennially for new isotopic-abundance measurements that might lead to revised standard atomic-weight values. SIAM searches for new "best measurements" using mass spectrometry of isotopic abundances of an element from a single terrestrial source, preferably an isotopic reference material. The best measurement is defined "as a set of analyses of the isotope-amount ratio or isotope-number ratio of an element in a well-characterized, representative material with small combined uncertainty." 16 Table of the Elements and Isotopes is presented with isotopic abundances shown as pie diagrams in cells of the periodic table. Figure 1 shows zirconium. An electronic interactive version of the IPTEI, which has been designed to be used both as a stand-alone digital learning object and as an object to be embedded in a set of electronic learning resources, can be found at www. isotopesmatter.com.

| Normal materials
The standard atomic weights of elements apply to normal materials, and this definition was redefined in 2017 as follows: 10 Normal materials include all substances, except (1) those subjected to substantial deliberate, undisclosed, or inadvertent artificial isotopic modification, (2) extraterrestrial materials, and (3) isotopically anomalous specimens, such as natural nuclear reactor products from Oklo (Gabon) or other unique occurrences.
This definition makes it clear that standard atomic-weight values do not apply to extraterrestrial materials or to the decay product 87 Sr in a natural rubidium source, but they do apply to reagents on the benchtops of chemists.  18 In the IPTEI, the background color of these element cells is white. 21 Prior to the 1983 TSAW, a longest half-life isotope, a most-abundant isotope, another radioactive isotope, or an atomic mass number was tabulated for these 34 elements. Twenty-one elements in Table 1 have one isotope that determines their standard atomic weight.

| A TSAW OF THE ELEMENTS
These elements have an isotope-amount fraction of 1 in the TICE, 18 and the uncertainty arises entirely from the measurement of the atomic mass of the isotope. In the IPTEI, the background color of these element cells is blue. 21 Forty-nine elements have two or more isotopes that are used to determine their standard atomic weights.
F I G U R E 1 Example cell (zirconium) from the IUPAC Periodic weight, a value of A r (E) with a lower uncertainty might be obtained by measurement of an individual specimen. g Geological and biological materials are known in which the element has an isotopic composition outside the limits for normal materials. The difference between the atomic weight of the element in such materials and that given in the table may exceed the stated uncertainty. m Modified isotopic compositions may be found in commercially available material because the material has been subjected to an undisclosed or inadvertent isotopic fractionation. Substantial deviations in atomic weight of the element from that given in the table can occur. r The range in isotopic composition of the normal terrestrial material prevents a more-precise standard atomic weight from being given; the tabulated value and uncertainty should be applicable to normal materials.

| Annotations and footnotes
The importance of annotations and footnotes in the TSAW is highlighted by Peiser et al: 7,8 For quite extraordinary occurrences and other abnormal sources with abnormal atomic weights outside an otherwise acceptable range, the Commission uses annotations given in footnotes that are an integral part of the Tables of Standard Atomic Weights. Describing such abnormalities merely in the text of biennial reports would surely cause the warnings to be overlooked by more of the affected users.
Footnote "g" (Table 1) identifies elements for which geological or biological materials are known in which the isotopic composition and atomic weight are outside the limits for normal materials. Footnote "m" (Table 1) identifies elements for which modified isotopic compositions may be found in commercially available material because of undisclosed or inadvertent isotopic fractionation. Footnote "r" (Table 1) identifies elements (currently six) whose range in the isotopic composition in normal materials prevents a more-precise standard atomic weight from being given. Footnote "*" is attached to element names and identifies the 38 elements having no stable isotopes, and 34 of these elements have no standard atomic weight because they do not have a characteristic terrestrial isotopic composition. In natural terrestrial substances, a radioactive isotope with a sufficiently long half-life is said to have a characteristic terrestrial isotopic composition, for example, xenon-136, potassium-40, and protactinium-231 (half-life = 3.25 × 10 4 a), and these isotopes are included in the TICE. A standard atomic weight is tabulated for bismuth, thorium, protactinium, and uranium, which have characteristic terrestrial isotopic compositions. Alternative formats were considered (see section 2 of Supporting Information), and a format to delineate the uncertainty value with the symbol "±", for example, A r (Se) = 78.971 ± 0.008, was adopted.

| A new column and footnote for standard atomic-weight uncertainty
To clarify that the tabulated uncertainties in this new column are consensus values, not GUM-evaluated values, the column heading is identified with the symbol double dagger ( ‡) for a new footnote which was slightly modified by the Commission in 2021. 17,25 This new footnote in the TSAW is the first in more than two decades: 32

| Intervals and lower-and upper atomicweight bounds
The span of atomic-weight values in normal materials is termed the respectively. 44 Writing the standard atomic weight of lithium as [6.938, 6.997] indicates that its atomic weight in any normal material will be greater than or equal to 6.938 and will be less than or equal to 6.997. The atomic-weight interval is said to encompass atomic-weight values of all normal materials. The range of an interval is the difference between b and a, that is, ba; 44 thus, the range of the atomic-weight interval of lithium is 6.997-6.938 = 0.059.
The lower bound of an atomic-weight interval is determined from the lowest atomic weight, and it considers the uncertainty of the measurement, which is commonly an isotope-delta measurement on a sample and a reference material, whose isotopic abundance has been measured and serves as a best measurement of a material in the TICE. 31,38 A delta measurement is a differential isotope-ratio measurement commonly performed using an isotope-ratio mass spectrometer. The isotope ratio of heavier and lighter isotopes i E and F I G U R E 2 Variation in atomic weight with isotopic composition of selected lithium-bearing materials (modified from Wieser and Coplen 30 ). LSVEC is the lithium carbonate isotopic reference material for the lithium isotope-delta scale, 38 which is assigned an isotope-delta value of zero. 39 The δ 7 Li LSVEC isotopedelta scale and the 7 Li-mole-fraction scale were matched using the data of Qi et al. 40 The expanded uncertainty in matching the atomic-weight and 7 Li-mole-fraction scales with the δ 7 Li LSVEC scale is equivalent to 3 ‰. The lower bound of the lithium standard atomic-weight interval is 6.938, and the upper bound is 6.997. The relatively high mole fraction of 7 38 The upper bound is determined in an equivalent manner. In addition to the uncertainty of the isotope-delta value, the uncertainty of the isotopic reference material anchoring the isotope-delta scale must be considered, as discussed in detail by Wieser et al. 31 An example of the symbol for a delta value is δ 7 Li LSVEC , shown in Figure 2, where LSVEC is the lithium isotopic reference material. 39 The Commission's rules and comments on determining atomicweight intervals are listed in section 3 of the Supporting Information.
Of pertinence to this discussion is rule 2: The Exemplifying why rule 2 was enacted, Figure 2 shows a graphical plot of lithium standard atomic weight and atomic-weight intervals for selected lithium-bearing substances. The standard atomic weight of lithium, [6.938, 6.997], is shown at the top of Figure 2. Were one to assume that the probability distribution function of the standard atomic weight interval were rectangular (or uniform), one might express the standard atomic-weight value as the average of the lower and upper bounds, (6.938 + 6.997)/2 = 6.9675. Figure 2 shows that this value is a poor estimate of the atomic weight of lithium. Only a laboratory reagent substantially depleted in 6 Li could have this atomic-weight value.
For the 14 elements whose standard atomic-weight values are expressed as intervals, more precise atomic-weight values of materials might be obtained by referring to published graphical plots of atomic weight for selected materials and compounds of each of these elements. 17,[35][36][37] For example, the atomic weight of lithium in sea water is well defined as a small interval (6.942 28 ± 0.000 07). 37 Possolo et al. 55 provide information on determining values and uncertainties of standard atomic weight intervals of elements from specified sources.

| THE PROCESS OF REVISING A STANDARD ATOMIC WEIGHT
3.1 | Elements having one isotope to determine its standard atomic weight Twenty-one elements have one isotope each to determine their standard atomic weight, which is determined from the latest Atomic Mass Evaluation. 45 The uncertainty of each element arises entirely from the measurement of the atomic mass of this single isotope.
These elements have symmetric uncertainties (expressed with the symbol ±). Since 1969, 1 the Commission has applied a coverage factor of 6 to the GUM-evaluated Gaussian combined standard uncertainties of atomic masses to improve the reliability of their A r (E) values to minimize the number of changes at each revision of a TSAW (rule 10 of the Technical Guidelines in the Supporting Information). For example, the standard atomic weight of rhodium is determined from the relative atomic mass of 103 Rh, which is 102.905 4941 ± 0.000 0025. 45 Expanding the coverage factor by 6 yields the Commission's A r (Rh) = 102.905 49 ± 0.000 02 value in Table 1. 3.2 | Elements having two or more isotopes to determine their standard atomic weight Forty-nine elements have two or more isotopes to determine their standard atomic weights. These elements have symmetric uncertainties (expressed by the symbol ±). An example is hafnium with A r (Hf) = 178.486 ± 0.006. 17,26 The standard atomic weight is given as a single value with an uncertainty that includes measurement uncertainty and may include uncertainty due to isotope-abundance variations. The variations in isotopic abundances may be very small and not exceed the measurement uncertainty and affect the atomicweight value; for example, high-purity gallium reagents vary in isotopic composition, but the variation is too small to affect its standard atomic weight. 19,46 The following steps comprise the process to revise a standard atomic weight of these elements.

A researcher publishes isotopic abundances of an element with
low uncertainties, commonly measured using thermal ionization mass spectrometry or multi-collector inductively coupled plasmamass spectrometry.
2. Beginning several months before a Commission meeting, SIAM reviews the literature, seeking peer-reviewed publications of isotopicabundance measurements of samples that might achieve bestmeasurement status. Preference is given to analyses of chemically stable materials that are distributed internationally as isotopic reference materials. 47 Guidance was provided in the 2013 TSAW: 16 The Commission seeks evidence that mass-   49. This situation is exemplified in Figure 4B. The maximum of the probability density function (123.456) differs from the A r (E) value of 123.46 ( Figure 4B). 3.3 | Elements whose standard atomic weights are expressed as an interval For an element having two or more isotopes to determine its standard atomic weight to move from having a standard atomic weight with symmetric uncertainty to one having a standard atomic weight expressed as an interval, two components are required: (a) a best measurement of isotopic abundances of the element with sufficiently low uncertainties and an isotope-delta value must both exist, and (b) a detailed investigation of peer-reviewed, published variations in isotopic abundances in normal materials must have been completed. The literature survey is conducted by one or more experts participating in an IUPAC project. 30 Figure 2 shows the result of a literature survey of lithium isotopic variations. Commonly, isotope-delta measurements 38 of various materials are published, but the missing link is a best measurement of isotopic abundances with low uncertainties on the isotopic reference materials used for the isotope-delta measurements of the element-this measurement enables the isotope-delta scale to be linked to the mole-fraction scale. This link is exemplified for lithium by the text "The expanded uncertainty in matching the atomic-weight and 7 Li-mole-fraction scales with the δ 7 Li LSVEC scale is equivalent to 3 ‰" in the caption of Figure 2. Thirteen Commission rules and comments on determining atomic-weight intervals can be found in section 3 of the Supporting Information.
Footnote "r" in Table 1 identifies elements whose range in the isotopic composition of normal materials prevents a more-precise standard atomic weight being assigned, and they are good candidates for elements that may move to the category of having an atomic weight that might be expressed as an interval. These elements include helium, nickel, copper, zinc, selenium, and strontium (Table 1), and there is an ongoing project for strontium. 48 Although probability density functions of the 14 elements having N. Greenwood's argument decades earlier that the standard atomic weights are consensus values enunciated by qualified experts. The qualifier, "decisional," appears neither in the GUM 29 nor in the International Vocabulary in Metrology. 44 But in the Commission's publication TICE, the best measurement of isotopic abundances of an element is a Commission outcome that has GUM-evaluated measurement uncertainties. 16,47 For each of the 49 elements whose A r (E) and U[A r (E)] values are determined from two or more isotopes of an element ( Figure 4B), the quantities A r (E) and U[A r (E)] are the Commission's decisional standard atomic-weight value and decisional uncertainty ( Figure 4B).

| Reliability of atomic-weight values
The question arises, "how reliable are the standard atomic weights of the elements?" Evidence suggests that standard atomic weights are highly reliable. In past reports, the Commission has referred to relative uncertainty, the uncertainty divided by an element's standard atomic weight, U[A r (E)]/A r (E). Between 1969 and 1997, there was no change in the relative uncertainty of 14 elements, the relative uncertainty of 69 elements improved, and only one element (xenon) had an "improvement factor" of less than 1, indicating a loss in the estimate of relative uncertainty. 50,51 A second method to measure the reliability of decisional standard atomic-weight values and their associated decisional uncertainty values is to estimate their coverage factor. The decisional uncertainty of the standard atomic weight that the Commission determines (e.g., 0.008 for selenium) is an expanded uncertainty and is the product of a combined standard uncertainty and the coverage factor k or K. 9,29 The coverage factor of most elements is intentionally not specified as indicated by Coplen and Peiser, 52 who state: Although the Commission has declined to specify the degree of expansion, i.e. the recommended K value, we believe it is expected to correspond to at least two standard deviations.
Nevertheless, the lower and upper bounds are assigned so that standard atomic weights are highly reliable and have great certitude. 7,8 For elements having two or more isotopes that contribute to the standard atomic weight, the coverage factor is sufficiently high that De Bièvre et al state: 53 The aim of IUPAC has been fulfilled by which any chemisttaking any natural sample from research, industry, or commerce can confidently expect his or her true sample atomic weight to lie within the tabulated range with a probability far in excess of 95%.