The Standard Definition of the Kilogram: A Grave Matter

The kilogram is the sole base unit in the International System of Units (SI) system defined by a physical artifact, and recent popular science news articles (with varying degrees of accuracy) describe changes in the relative masses of the standard prototype and its replicas around the world. These articles imply a potential problem exists, which is untrue, because the changes are documented and tracked. However, proposed redefinitions of the kilogram that do not involve the physical prototype are now being considered by the metrology community, and these involve defining the kilogram in terms of a natural constant.

In 1793, the term "grave" was proposed by an early committee looking at metrological standards for the mass standard we know today as the kilogram. The term was not adopted and "kilogram" became the approved term just a few years later. But how to define the kilogram and provide it with the authority of a standard has been a topic of continuing interest for more than 200 years now. Eventually, a physical prototype was constructed and became the consensus kilogram. Davis (1) provides a brief history of the kilogram standard unit and the creation of the prototypes and its copies, and discusses how they are compared in verification protocols.

Of course, the definition of mass is central to mass spectrometry (MS), and redefinitions are not only theoretical but may have a practical consequence. Here, we describe the two distinct redefinitions of mass. First, we review the transition in the 1960s from the ¹⁶O mass standard to the ¹²C mass standard; the former is known as the physical atomic mass scale, and the latter as the chemical atomic mass scale. This transition had immediate effects on MS, and defined the unified atomic mass unit, u. Lingering variations in nomenclature such as amu, u, and daltons (Da) have been peppered throughout the research literature and persist to this day. Secondly, we review some current research and proposals for redefining the kilogram in terms of natural constants, and not the current physical artifact. The anticipated consequences on practical analyses in MS may seem distant, but any redefinition percolates through the SI hierarchy, and we best be prepared. A previous column discussed the transition between physical and chemical atomic mass scales (2); the reiteration here will be brief and will amplify different points. The development of mass standards for commerce and industry (within the range of milligrams to kilograms) proceeded independently and is not covered here.

A Brief History

The term Da (3) honors the early contributions of John Dalton, who studied chemical reactivity of elements and formulated the atomic theory that attempted to explain how they combined. As the theory was developed (with a few notable missteps derived from mistaken assumptions of formula), Dalton and others observed that reacting elements react in simple and consistent integer ratios. Assigning a value of 1 unit to the mass of hydrogen allowed many of the chemical reactions to be conveniently expressed as ratios of mass. As more and more relative masses were determined through the study of chemical reactions, it was suggested by The International Union of Pure and Applied Chemistry (IUPAC) in 1920 (through the creation of an International Commission on Atomic Weights) that the mass of oxygen be set at exactly 16, and that this standard be used as a scale definition. However, the oxygen isotopes ¹⁷O and ¹⁸O were discovered in 1929, and this discovery resulted in the bifurcation of mass scales and a variance in the values of constants associated with them. Physicists used solely ¹⁶O as mass 16 and chemists used the average mass of oxygen in its natural state (which included all the isotopes). Therefore, the physical atomic mass scale and the chemical atomic mass scale differed by a nominal factor of 1.000275. Actually, that factor varied somewhat; oxygen drawn from different sources varies in its isotopic abundances.

The role of A.O.C. Nier in his service on the Atomic Weights Commission from 1947 to 1961 is emblematic of the extended effort required to resolve the dispute (4,5). Nier and A. Ölander each proposed to use ¹²C as the basis for the atomic weights table, and that suggestion was adopted in 1961. Mattauch was an early champion (6) and succinctly discussed the reasons for adopting the ¹²C standard. In the blur of a 50-year historical retrospective, it is not often noted that ¹⁹ F was also proposed as a mass standard (7). The agreement to create the unified atomic mass unit u was indeed a unification of the disparate research communities at the time and accommodated the need for a standard that reflected the precision with which measurements could be made. Duckworth's memoir (8) provides a great deal of personal insight into those communities, and the development of high precision MS in isotopic research.

Where Does Mass Spectrometry Fit In?

For mass spectrometrists (who could be affiliated with any of several different research communities), the unification of the mass scales required heightened attention to correct reporting of research results. Compilations were updated, and constants recalculated as necessary. Through it all, MS continued to be viewed as what is delightfully termed a "rogue community" by SI notation purists (9). Proper nomenclature eluded us — we published mass spectra with the x-axis denoted by amu, AMU, u, m/z, u/z, Da, Th (10), and sometimes simply mass. Petley (11) places the atomic mass unit in its proper, current perspective. The relevant committees that are part of the SI system do not approve of any of these units. The sole current standard of mass is the artifact kilogram, and it represents a mass far removed from MS as it is practiced. Our accurate and precise determinations of ¹H, ¹²C, and ¹⁶O (and all of the isotopes) continue in one part of the field, but as modern organic MS became the method of choice for the analysis of organic compounds, the scale shifted to molecular masses of several hundred (choose your favorite term for mass unit and insert here), and then several thousands, and now we create mass spectra from biological molecules of several orders of magnitude greater mass. We are proud of the high mass reach of our molecular analysis, but a molecule with a mass of 10⁶ Da is still only about 10^-21 kg!

Redefining the Kilogram

For this reason, it might seem that modern MS is not involved with research and discussions in the metrology community about a possible redefinition of the kilogram. Whatever direction the redefinition takes, the effect on our community will likely be smaller in the short term than it was with the consolidation of the mass scales in 1961. The remainder of this column provides an overview of some proposals for the kilogram redefinition. The driving force is to define the kilogram in terms of a fundamental natural physical constant, with the expectation that such a definition will lead to increased accuracy. Other standards are defined with an accuracy that exceeds the accuracy variations already documented for the kilogram artifact, and there is a certain intrinsic satisfaction in delineating an underlying connection between natural constants. A few selected publications that describe aspects of proposed redefinitions are here (12–15). It is noteworthy that the proposal of Mills and colleagues (15) for the redefinition of the kilogram offers two alternatives. The first defines the kilogram in terms of the mass of a photon of a specified frequency. The second alternative defines the kilogram in terms of Avogadro's number, or equivalently, the unified mass unit u familiar to mass spectrometrists. The equivalency derives from the definition that states an Avogadro's number of ¹²C atoms has a mass of exactly 12 g.

Avogadro's number is currently known to an uncertainty of approximately 0.1 ppm. The goal of the international program known as the Avogadro Project is to reduce the uncertainty, and then to redefine the kilogram in terms of Avogadro's number (N_A). As noted above, this is the equivalent of defining the kilogram in terms of the unified atomic mass unit. The project involves counting the number of atoms in a highly purified crystalline sphere consisting of ²⁸Si. Silicon can be produced with very high isotopic purity, and the isotopic composition of the silicon in the spheres is characterized by isotopic dilution MS. It has been claimed that the spheres are the roundest objects known. Let's consider the numbers, recognizing that they describe an artifact as tangible as the cylinder that currently defines the kilogram. The final spheres used for experimental measurements are 1 kg in mass. The silicon began at an enriched 99.995% ²⁸Si, and was carried through extensive refining afterwards. The final composition of the silicon artifact is determined by a specialized measurement in isotopic dilution MS that can accommodate the large dynamic range. Each of the two spheres created is unique, but as an example one of the spheres was determined to have only about 40 ppm ²⁹Si and 2 ppm ³⁰Si. The sphere has a diameter of 93.6 mm, and the diameter is known to an uncertainty of 0.6 nm. This represents about a one-atom spacing at the surface of the sphere, which is actually not silicon but a 3–4 nm thick layer of silicon oxide. Additionally, if the sphere is not within the vacuum water is adsorbed on the surface. The atoms in the sphere are counted by using the ordered arrangement of a known crystal structure and its associated unit volume. The description here is a tremendous simplification of an extensive collaborative project for which many more detailed publications are available (16,17). The most recent measured value of N_A is 6.02214082(18) × 10²³. An alternative experimental approach using a watt balance method (linking the kilogram to Planck's constant) yields a slightly different value, but the "slight differences" are in the 10^-8 N_A range.

Hill and colleagues (14) describe the two alternative approaches to redefining the kilogram as an "electronic kilogram" (linking the kilogram to Planck's constant, currently pursued through the watt balance experiments) and the "atomic kilogram" (linking the kilogram to Avogadro's number). Clearly, the Avogadro Project is an approach of the latter sort. However, another approach to the "atomic kilogram" links directly to ¹²C, and therefore to u, and may be especially attractive to mass spectrometrists. Hill (14) suggests that "a kilogram is the mass of 84,446,88983 × 1000/12 unbound atoms of carbon-12 at rest and in their ground state." It is suggested that this different definition lends itself to visualization and perhaps creation of an artifact, or a prototype. In this case, the prototype would be a cube approximately 8.11 cm in each of its dimensions, created so that it would contain 368,855,762 atoms of ¹²C. Do the calculation to show that this is 1 kg. There would be corrections for the form of carbon in the cube, and the binding energies, but the process for such correction has already been established. It is also suggested that an atomic kilogram has an educational advantage in that once the concept of an atom is understood by a student, then a kilogram is defined simply by counting atoms. The fact that there are a lot of atoms to be counted helps to emphasize the magnitude of Avogadro's number.

Fraundorf (18) approaches proposed redefinitions of the kilogram from the standpoint of ¹²C, further developing the atomic kilogram and again using a counting process. The mass of ¹²C is a constant, and then the kilogram would be defined as the mass of an aggregate number of ¹²C atoms, and that number should be a value divisible by 12. A kilogram prototype can be constructed both virtually and physically. Fraundorf suggests that graphene hexagons can be layered to form such a prototype, and derives the geometry that would contain Avogadro's number of carbon atoms. A 1 mole amount of carbon is contained in a hexagonal stacked structure about 1.71 cm high and about the width of a dime. From the current value of N_A, that crystal would contain 50,184,513,209,183,067,081,900 atoms of ¹²C. If one instead specifies the number of carbon atoms in the structure (by definition), then a value of N_A follows. If the mass of ¹²C (u) is the starting point, the mass of the kilogram follows, and then the kilogram is therefore defined in terms of a fundamental natural constant.

Conclusion

Figure 1 compares the various kilogram standards as prototypes; the Pt/Ir standard exists, the ²⁸Si boule exists, and the ¹²C hexagonal stacked structure is currently a suggestion. The ultimate goal in a definition of the kilogram is to define it in terms of a natural physical constant. The mass of ¹²C is a natural constant, and the mass of an electron is a natural constant, and so the mass of ¹²C⁺ is a natural constant. It's also straightforward to create carbon ions in a variety of sizes. The Penning ion-trap mass spectrometer has been used for absolute mass measurements on carbon ion clusters (19). A systematic study of the clusters containing integer numbers of carbon atoms provides a definitive measure of the errors in the experimental measurement. The residual systematic uncertainty is about 8 × 10^-9 . The system is used to perform absolute mass measurements of radioactive nuclides based on a direct comparison of the measured masses with the masses of the carbon cluster ions as references. From one perspective, the redefinition of the kilogram simplifies to deciding on N_A and dividing by 12.

Figure 1: Summary of actual (Pt/Ir prototype), proposed and constructed (28Si boule), and proposed virtual (12C hexagonal stacked graphene structure) standards for the kilogram.

References

(1) R. Davis, Metrologia 40, 299–305 (2003).

(2) K.L. Busch, Spectroscopy 16(11), 28–32 (2001).

(3) H.F. Dixon, Trends Biochem. Sci. 8, 49–53 (1983).

(4) J.R. De Laeter, Int. J. Mass Spectrom. 178, 1–7 (1998).

(5) H.E. Duckworth and A.O. Nier, Int. J. Mass Spectrom. Ion Proc. 86, 1–19 (1988).

(6) T.P. Kohman, J.H.E. Mattauch, and A.H. Wapstra, Phys. Today 12, 30–32 (1959).

(7) A.F. Scott and W.R. Ware, J. Amer. Chem. Soc. 79, 4253–4257 (1957).

(8) H.E. Duckworth, One Version of the Facts: My Life in Academe (University of Manitoba Press, Winnipeg, Canada 2000).

(9) O.D. Sparkman made this observation.

(10) R.G. Cooks and A.L. Rockwood, Rapid Commun. Mass Spectrom. 5, 93 (1991).

(11) B.W. Petley, IEEE Trans. Instrum. Measurement 38, 175–179 (1989).

(12) J.L. Flowers and B.W. Petley, Metrologia 42, L31–L34 (2005).

(13) B.W. Petley, Metrologia 44, 68–72 (2007).

(14) T.P. Hill, J. Miller, and A.C. Censullo, Metrologia 48, 83–86 (2011).

(15) I.M. Mills, P.J. Mohr, T.J. Quinn, B.N. Taylor, and E.R. Williams, Metrologia 43, 227–246 (2006).

(16) A. Picard, P. Barat, M. Borys, M. Firius, and S. Mizushima, Metrologia 48, S112–S119(2011).

(17) P. Becker, Contemp. Phys. 53, 461–479 (2012)

(18) P. Fraundorf, arXiv:1201.5537v1 (2012).

(19) A. Kellerbauer, K. Blaum, G. Bollen, F. Herfurth, J.-J. Kluge, M. Kuckein, E. Sauvan. C. Scheidenberger, and L. Schweikhard, Eur. Phys. J. D. 22, 53–64 (2003).

Kenneth L. Busch discovered the missing mass of the universe in the form of a fruitcake he received as a holiday gift. Late at night, he considers whether a ¹²C nanotube of specified length could be considered as a mass standard, but that's just linear thinking. This column is the sole responsibility of the author, who can be reached at wyvernassoc@yahoo.com

Kenneth L. Busch