# Kilogram

Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]

## Overview

Shown above is a computer-generated image of the International Prototype Kilogram (“IPK”). The IPK is the kilogram. It sits next to an inch-based ruler for scale. The IPK is made of a platinum-iridium alloy and is stored in a vault at the BIPM in Sèvres, France. For other kilogram-related images, see Links to photographs, below.

The kilogram or kilogramme (symbol: kg) is the base unit of mass in the International System of Units (known also by its French-language initials “SI”). The kilogram is defined as being equal to the mass of the International Prototype Kilogram (IPK; known also by its French-language name Le Grand K), which is almost exactly equal to the mass of one liter of water.[1] It is the only SI base unit with an SI prefix as part of its name. It is also the only SI unit that is still defined in relation to an artifact rather than to a fundamental physical property that can be reproduced in different laboratories.

In everyday usage, the mass of an object in kilograms is often referred to as its weight, although strictly speaking the weight of an object is the gravitational force on it, measured in newtons (see also Kilogram-force). Similarly, the avoirdupois pound, used in both the Imperial system and U.S. customary units, is a unit of mass and its related unit of force is the pound-force. The avoirdupois pound is defined as exactly 0.45359237 kg, making one kilogram approximately equal to 2.205 avoirdupois pounds.

Many units in the SI system are defined relative to the kilogram so its stability is important. After the International Prototype Kilogram had been found to vary in mass over time, the International Committee for Weights and Measures (known also by its French-language initials CIPM) recommended in 2005 that the kilogram be redefined in terms of fundamental constants of nature.[2]

## The nature of mass

The kilogram is a unit of mass, the measurement of which corresponds to the general, everyday notion of how “heavy” something is. However, mass is actually an inertial property; that is, the tendency of an object to remain at constant velocity unless acted upon by an outside force. An object with a mass of one kilogram will accelerate at one meter per second squared (about one-tenth the acceleration due to Earth’s gravity) when acted upon by a force of one newton (symbol: N).

While the weight of matter is entirely dependent upon the strength of local gravity, the mass of matter is constant (assuming it is not traveling at a relativistic speed with respect to an observer). Accordingly, for astronauts in microgravity, no effort is required to hold objects off the cabin floor; they are “weightless.” However, since objects in microgravity still retain their mass, an astronaut must exert ten times as much force to accelerate a 10-kilogram object at the same rate as a 1-kilogram object.

## History

#### Early definitions

See also Grave (mass) for more on the history of the kilogram.

On 7 April 1795, the gram was decreed in France to be equal to “the absolute weight of a volume of water equal to the cube of the hundredth part of the meter, at the temperature of melting ice.”[3] Since trade and commerce typically involve items significantly more massive than one gram, and since a mass standard made of water would be inconvenient and unstable, the regulation of commerce necessitated the manufacture of a practical realization of the water-based definition of mass. Accordingly, a provisional mass standard was made as a single-piece, metallic artifact one thousand times more massive than the gram—the kilogram.

At the same time, work was commissioned to precisely determine the mass of a cubic decimeter (one liter[4]) of water. Although the decreed definition of the kilogram specified water at 0 °C—its highly stable temperature point—the scientists in 1799 chose to redefine the standard to water’s most stable density point: the temperature at which water reaches maximum density, which was measured at the time as 4 °C.[5] They concluded that one cubic decimeter of water at its maximum density was equal to 99.92072% of the mass of the provisional kilogram made four years earlier.[6] That same year, 1799, an all-platinum kilogram prototype was fabricated with the objective that it would equal, as close as was scientifically feasible for the day, the mass of a cubic decimeter of water at 4 °C. The prototype was presented to the Archives of the Republic in June and on 10 December 1799, the prototype was formally ratified as the Kilogramme des Archive (Kilogram of the Archives) and the kilogram was defined as being equal to its mass. This standard stood for the next ninety years.

#### International Prototype Kilogram

Since 1889, the SI system defines the magnitude of the kilogram to be equal to the mass of the International Prototype Kilogram, often referred to in the professional metrology world as the “IPK”. The IPK is made of an alloy of 90% platinum and 10% iridium (by weight) and is machined into a right-circular cylinder (height = diameter) of 39.17 mm to minimize its surface area.[7] The addition of 10% iridium improved upon the all-platinum Kilogram of the Archives by greatly increasing hardness while still retaining platinum’s many virtues: extreme resistance to oxidation, extremely high density, satisfactory electrical and thermal conductivities, and low magnetic susceptibility. The IPK and its six sister copies are stored in an environmentally monitored safe in the lower vault located in the basement of the BIPM’s House of Breteuil in Sèvres on the outskirts of Paris (see Links to photographs, below for images). Three independently controlled keys are required to open the vault. Official copies of the IPK were made available to other nations to serve as their national standards. These are compared to the IPK roughly every 50 years.

The IPK is one of three cylinders made in 1879. In 1883, it was found to be indistinguishable from the mass of the Kilogram of the Archives made eighty-four years prior, and was formally ratified as the kilogram by the 1st CGPM in 1889.[7] Modern measurements of the density of Vienna Standard Mean Ocean Water—purified water that has a carefully controlled isotopic composition—show that a cubic decimeter of water at its point of maximum density, 3.984 °C, has a mass that is 25.05 parts per million less than the kilogram.[8] This small, 25 ppm difference, and the fact that the mass of the IPK was indistinguishable from the mass of the Kilogram of the Archives, speak volumes of the scientists’ skills over 221 years ago when making their measurements of water’s properties and in manufacturing the Kilogram of the Archives.

## Stability of the International Prototype Kilogram

Mass drift over time of national prototypes K21–K40, plus two of the IPK’s sister copies: K8(41) and K32. All mass changes are relative to the IPK. The initial 1889 starting-value offsets relative to the IPK have been nulled.[9] The above are all relative measurements; no historical mass-measurement data is available to determine which of the prototypes has been most stable relative to an invariant of nature. There is the distinct possibility that all the prototypes gained mass over 100 years and that K21, K35, K40, and the IPK simply gained less than the others.

By definition, the error in the measured value of the IPK’s mass is exactly zero; the IPK is the kilogram. However, any changes in the IPK’s mass over time can be deduced by comparing its mass to that of its official copies stored throughout the world, a process called “periodic verification.” For instance, the U.S. owns four platinum-iridium (“Pt-10Ir”) kilogram standards, two of which, K4 and K20, are from the original batch of 40 replicas delivered in 1884.[10] The K20 prototype was designated as the primary national standard of mass for the U.S. Both of these, as well as those from other nations, are periodically returned to the BIPM for verification.[11]

Note that the masses of the replicas are not precisely equal to that of the IPK; their masses are calibrated and documented as offset values. For instance, K20, the U.S.’s primary standard, originally had an official mass of 1 kg39 µg in 1889; that is to say, K20 was 39 µg less than the IPK. A verification performed in 1948 showed a mass of 1 kg19 µg. The latest verification performed in 1999 shows a mass identical to its original 1889 value. The mass of K4, the U.S.’s check standard, as of 1999 was officially calibrated as 1 kg116 µg. However, it was 41 µg more massive (in comparison to the IPK) in 1889.

Since the IPK and its replicas are stored in air (albeit under two or more nested bell jars), they adsorb atmospheric contamination onto their surfaces and gain mass. Accordingly, they are cleaned in a process the BIPM developed between 1939 and 1946 known as “the BIPM cleaning method” that comprises lightly rubbing with a chamois soaked in equal parts ether and ethanol, steam-cleaning with bi-distilled water, and allowing the prototypes to settle for 7–10 days before period verification.[12] Cleaning the prototypes removes between 5 and 60 µg of contamination depending largely on the time elapsed since the last cleaning. Further, a second cleaning can remove up to 10 µg more. After cleaning—even when they are stored under their bell jars—the IPK and its replicas immediately begin gaining mass again. The BIPM even developed a model of this gain and concluded that it averaged 1.11 µg per month for the first 3 months after cleaning and then decreased to an average of about 1 µg per year thereafter. Since check standards like K4 are not cleaned for routine calibrations of other mass standards—a precaution to minimize the potential for wear and handling damage—the BIPM’s model has been used as an “after cleaning” correction factor.

K48, above, came from the second batch of kilogram replicas to be produced. It was delivered to Denmark in 1949 with an official mass of 1 kg+81 µg. Like all other replicas, it is stored under two nested bell jars virtually all the time. Still, its mass and that of the IPK diverged markedly in only 40 years; the mass of K48 was certified as 1 kg+112 µg during the 1988–1992 periodic verification.[9]

Because the first forty official copies are made of the same alloy as the IPK and are stored under similar conditions, periodic verifications using a large number of replicas—especially the national primary standards, which are rarely used—can convincingly demonstrate the stability of the IPK. What has become clear after the third periodic verification performed between 1988 and 1992 is that masses of the entire worldwide ensemble of prototypes have been slowly but inexorably diverging from each other. It is also clear that the mass of the IPK lost perhaps 50 µg over the last century, and possibly significantly more, in comparison to its official copies.[9][13] The answer as to why this might be the case has proved elusive for physicists who have dedicated their careers to the SI unit of mass. No plausible mechanism has been proposed to explain either a steady decrease in the mass of the IPK, or an increase in that of its replicas dispersed throughout the world.[14] This relative nature of the changes amongst the world’s kilogram prototypes is often misreported in the popular press, and even some notable scientific magazines, which often state that the IPK simply “lost 50 µg” and omit the very important caveat of “in comparison to its official copies.”[15] Further, there is no technical means available to determine whether or not the entire worldwide ensemble of prototypes suffer from even greater long-term trends upwards or downwards because their mass “relative to an invariant of nature is unknown at a level below 1000 µg over a period of 100 or even 50 years.”[13] Given the lack of data identifying which of the world’s kilogram prototypes has been most stable in absolute terms, it is equally as valid to state that the first batch of replicas has, as a group, gained an average of about 25 µg over one hundred years in comparison to the IPK.[16]

What is known specifically about the IPK is that it exhibits a short-term instability of about 30 µg over a period of about a month in its after-cleaned mass.[17] The precise reason for this short-term instability is not understood but is thought to entail surface effects: microscopic differences between the prototypes’ polished surfaces, possibly aggravated by hydrogen absorption due to catalysis of the volatile organic compounds that slowly deposit onto the prototypes as well as the hydrocarbon-based solvents used to clean them.[18]

Scientists are seeing far greater variability in the prototypes than previously believed. The increasing divergence in the masses of the world’s prototypes and the short-term instability in the IPK has prompted research into improved methods to obtain a smooth surface finish using diamond-turning on newly manufactured replicas and has intensified the search for a new definition of the kilogram. See Proposed future definitions, below.[19]

## Importance of the kilogram

The magnitude of many of the units comprising the SI system of measurement, including most of those used in the measurement of electricity and light, are defined in part by the mass of the IPK.

The stability of the IPK is crucial because the kilogram underpins much of the SI system of measurement as it is currently defined and structured. For instance, the newton is defined as the force necessary to accelerate the kilogram by one meter per second squared. If the mass of the IPK were to change slightly, so too must the newton by a proportional degree. In turn, the pascal, the SI unit of pressure, is defined in terms of the newton. This chain of dependency follows to many other SI units of measure. For instance, the joule, the SI unit of energy, is defined as that expended when a force of one newton acts through one meter. Next to be affected is the SI unit of power, the watt, which is one joule per second. The ampere too is defined relative to the newton, and ultimately, the kilogram. With the magnitude of the primary units of electricity thus determined by the kilogram, so too follow many others; namely, the coulomb, volt, tesla, and weber. Even units used in the measure of light would be affected. The candela—following the change in the watt—would in turn affect the lumen and lux.

Because the magnitude of many of the units comprising the SI system of measurement is ultimately defined by the mass of a 141-year-old, golf ball-size piece of metal, the quality of the IPK must be diligently protected in order to preserve the integrity of the SI system. Yet, in spite of the best stewardship, the IPK has likely already lost 4.4 µg relative to the average mass of the worldwide ensemble of prototypes since the third periodic verification 31 years ago.[20] Further, the world’s national metrology labs must wait for the fourth periodic verification to confirm whether the historical trends persisted.

Fortunately, definitions of the SI units are quite different from their practical realizations. For instance, the meter is defined as the distance light travels in a vacuum during a time interval of 1⁄299,792,458 of a second. However, the meter’s practical realization typically takes the form of a helium-neon laser, and the meter’s length is delineated—not defined—as 1,579,800.298728 wavelengths of light from this laser. Now suppose that the official measurement of the second was found to have drifted by a few parts per billion (it is actually exquisitely stable). There would be no automatic effect on the meter because the second—and thus the meter’s length—is abstracted via the laser comprising the meter’s practical realization. Scientists performing meter calibrations would simply continue to measure out the same number of laser wavelengths until an agreement was reached to do otherwise. The same is true with regard to the real-world dependency on the kilogram: if the mass of the IPK was found to have changed slightly, there would be no automatic effect upon the other units of measure because their practical realizations provide an insulating layer of abstraction. Any discrepancy would eventually have to be reconciled though because the virtue of the SI system is its precise mathematical and logical harmony amongst its units. If the IPK’s value were definitively proven to have changed, one solution would be to simply redefine the kilogram as being equal to the mass of the IPK plus an offset value, similarly to what is currently done with its replicas; e.g., “the kilogram is equal to the mass of the IPK + 42 ppb.”

The long-term solution to this problem, however, is to liberate the SI system’s dependency on the IPK by developing a practical realization of the kilogram that can be reproduced in different laboratories by following a written specification. The units of measure in such a practical realization would have their magnitudes precisely defined and expressed in terms of fundamental physical constants. While major portions of the SI system would still be based on the kilogram, the kilogram would in turn be based on invariant, universal constants of nature. While this is a worthwhile objective and much work towards that end is ongoing, no alternative has achieved the uncertainty of a couple parts in 108 (~20 µg) required to improve upon the IPK. However, as of April 2007, the NIST’s implementation of the watt balance was approaching this goal, with a demonstrated uncertainty of 36 µg.[21] See Watt balance, below.

## Proposed future definitions

In the following section, wherever numeric equalities are shown in ‘concise form’—such as 1.85487(14)×1043—the two digits between the parentheses denotes the uncertainty at standard deviation (68% confidence level) in the two least significant digits of the significand.

The kilogram is the only SI unit that is still defined in relation to an artifact. Note that the meter was also once defined as an artifact (a single platinum-iridium bar with two marks on it). However, it was eventually redefined in terms of invariant, fundamental constants of nature that are delineated via practical realizations (apparatus) that can be reproduced in different laboratories by following a written specification. Today, physicists are investigating various approaches to do the same with the kilogram. Some of the approaches are fundamentally very different from each other. Some are based on equipment and procedures that enable the reproducible production of new, kilogram-mass prototypes on demand (albeit with extraordinary effort) using measurement techniques and material properties that are ultimately based on, or traceable to, fundamental constants. Others are devices that measure either the acceleration or weight of hand-tuned, kilogram test masses and which express their magnitudes in electrical terms via special components that permit traceability to fundamental constants. Measuring the weight of test masses requires the precise measurement of the strength of gravity in laboratories. All approaches would precisely fix one or more constants of nature at a defined value. These different approaches are as follows:

#### Atom-counting approaches

##### Carbon-12

Though not offering a practical realization, this definition would precisely define the magnitude of the kilogram in terms of a certain number of carbon-12 atoms. Carbon-12 is a certain isotope of carbon. The mole is currently defined as “the quantity of entities (elementary particles like atoms or molecules) equal to the number of atoms in 12 grams of carbon-12.” Thus, the current definition of the mole requires that 1000/12 (83⅓) moles of C-12 has a mass of precisely one kilogram. The number of atoms in a mole, a quantity known as the Avogadro constant, is an experimentally determined value that is currently measured as being 6.02214179(30)×1023 atoms (2006 CODATA value). This new definition of the kilogram proposes to fix the Avogadro constant at precisely 6.02214179×1023 and the kilogram would be defined as “the mass equal to that of 83⅓ · 6.02214179×1023 atoms of carbon-12.”

Currently, the uncertainty in the Avogadro constant is determined by the uncertainty in the measured mass of carbon-12 atoms (a relative standard uncertainty of 50 parts per billion at this time). By fixing the Avogadro constant, the practical effect of this proposal would be that the precise magnitude of the kilogram would be subject to future refinement as improved measurements of the mass of carbon-12 atoms become available; electronic realizations of the kilogram would be recalibrated as required. In an electronic definition of the kilogram, 83⅓ moles of carbon-12 would—by definition—continue to have a mass of precisely one kilogram and the Avogadro constant would continue to have uncertainty in its precise value.

A variation on a carbon-12-based definition proposes to define the Avogadro constant as being precisely 84,446,8863 (≈6.02214098×1023 atoms. An imaginary realization of a 12-gram mass prototype would be a cube of carbon-12 atoms measuring precisely 84,446,886 atoms across on a side. With this proposal, the kilogram would be defined as “the mass equal to 84,446,8863 × 83⅓ atoms of carbon-12.” The value 84,446,886 was chosen because it has a special property; its cube (the proposed new value for the Avogadro constant) is evenly divisible by twelve. Thus with this definition of the kilogram, there would be an integer number of atoms in one gram of carbon-12: 50,184,508,190,229,061,679,538 atoms.[22]

Shown above, one of the master opticians at the Australian Centre for Precision Optics (ACPO) is holding a 1 kg, single-crystal silicon sphere for the Avogadro project. These spheres are among the roundest man-made objects in the world. If the best of these spheres were scaled to the size of Earth, its high point—a continent-size area—would gently rise to a maximum elevation of only 2.4 meters above “sea level”.[23]

Another Avogadro constant-based approach, known as the Avogadro project, would define and delineate the kilogram as a softball-size sphere of silicon atoms. Silicon was chosen because a commercial infrastructure with mature processes for creating defect-free, ultra-pure monocrystalline silicon already exists to service the semiconductor industry. To make a practical realization of the kilogram, a silicon boule (a rod-like, single-crystal ingot) would be produced. Its isotopic composition would be measured with a mass spectrometer to determine its average atomic mass. The rod would be cut, ground, and polished into spheres. The size of a select sphere would be measured using optical interferometry to an uncertainty of about 0.3 nm on the radius—roughly a single atomic layer. The precise lattice spacing between the atoms in its crystal structure (≈192 pm) would be measured using a scanning X-ray interferometer. This permits its atomic spacing to be determined with an uncertainty of only three parts per billion. With the size of the sphere, its average atomic mass, and its atomic spacing known, the required sphere diameter can be calculated with sufficient precision and uncertainty to enable it to be finish-polished to a target mass of one kilogram.

Experiments are being performed on the Avogadro Project’s silicon spheres to determine whether their masses are most stable when stored in a vacuum, a partial vacuum, or ambient pressure. However, no technical means currently exist to prove a long-term stability any better than that of the IPK’s because the most sensitive and accurate measurements of mass are made with dual-pan balances like the BIPM’s FB-2 flexure-strip balance (see Links to photographs, below). Balances can only compare the mass of a silicon sphere to that of a reference mass. Given the latest understanding of the lack of long-term mass stability with the IPK and its replicas, there is no known, perfectly stable mass artifact to compare against. Single-pan scales capable of measuring weight relative to an invariant of nature with a long-term uncertainty of only 10–20 parts per billion do not yet exist. Another issue to be overcome is that silicon oxidizes and forms a thin layer (equivalent to 5–20 silicon atoms) of silicon dioxide (common glass) and silicon monoxide. This layer slightly increases the mass of the sphere, an effect which must be accounted for when polishing the sphere to its finish dimension. Oxidation is not an issue with platinum and iridium, both of which are noble metals that are roughly as cathodic as oxygen and therefore don’t oxidize unless coaxed to do so in the laboratory. The presence of the thin oxide layer on a silicon-sphere mass prototype places additional restrictions on the procedures that might be suitable to clean it to avoid changing the layer’s thickness or oxide stoichiometry.

All silicon-based approaches would fix the Avogadro constant but vary in the details of the definition of the kilogram. One approach would use silicon with all three of its natural isotopes present. About 7.77% of silicon comprises the two heavier isotopes: silicon-29 and silicon-30. As described in Carbon–12 above, this method would define the magnitude of the kilogram in terms of a certain number of carbon-12 atoms by fixing the Avogadro constant; the silicon sphere would be the practical realization. This approach could accurately delineate the magnitude of the kilogram because the mass of the three silicon isotopes relative to carbon-12 are known with great precision. An alternative method for creating a silicon sphere-based kilogram proposes to use isotopic separation techniques to enrich the silicon until it is nearly pure silicon-28, which has an atomic mass of 27.9769271(7)u=g/mol. With this approach, the Avogadro constant would not only be fixed, but so too would the atomic mass of silicon-28. As such, the definition of the kilogram would be decoupled from carbon-12 and the kilogram would instead be defined as 1000/27.9769271 · 6.022141793×1023 atoms of silicon-28 (≈35.7437397 fixed moles of silicon-28 atoms). Physicists could elect to define the kilogram in terms of silicon-28 even when kilogram prototypes are made of natural silicon (all three isotopes present). Even with a kilogram definition based on silicon-28, a silicon-sphere prototype made of nearly pure silicon-28 would necessarily deviate slightly from the defined number of moles of silicon in order to compensate for various chemical and isotopic impurities as well as the effect of surface oxides.[24]

##### Ion accumulation

Another Avogadro-based approach, ion accumulation, would define and delineate the kilogram by precisely creating new metal prototypes on demand. It would do so by accumulating gold or bismuth ions (atoms stripped of an electron) and counting them by measuring the electrical current required to neutralize the ions. Gold and bismuth were chosen because they can be safely handled and have the two highest atomic masses among the mononuclidic elements that can be treated as if they are non-radioactive (bismuth) or perfectly stable (gold). See also Table of nuclides.[25]

With a gold-based definition of the kilogram for instance, the atomic mass of gold would be fixed as precisely 196.966569g/mol, from the current value of 196.966569(4)g/mol. As with a definition based upon carbon-12, the Avogadro constant would also be fixed. The kilogram would then be defined as “the mass equal to that of precisely 1000/196.966569 · 6.02214179×1023 atoms of gold” (3,057,443,616,231,138,188,734,962 atoms of gold ≈5.0770037021 fixed moles).

Ion-accumulation techniques, while a relatively new field of study, have advanced rapidly. In 2003, experiments with gold at a current of only 10 µA demonstrated a relative uncertainty of 1.5%. Yet, follow-on experiments using bismuth ions and a current of 30 mA were expected to accumulate a mass of 30 g in six days and to have a relative uncertainty of better than 1 part in 106.[26]

Among the many technical challenges of the ion-deposition apparatus is in obtaining sufficiently high ion currents while simultaneously decelerating the ions so they can all deposit onto the target electrode imbedded in a balance pan. Experiments with gold showed that the ions must be decelerated to very low energies to avoid sputtering effects, which essentially gold-plates the inside of the deposition chamber and diminishes the mass retained on the electrode. The deposited mass fraction approaches very close to 100% at ion energies of less than around 1 eV (<1 km/s).[27]

If the kilogram is one day defined as a precise quantity of gold or bismuth atoms deposited with an electric current, not only would the Avogadro constant and the atomic mass of gold or bismuth be precisely fixed, but so too would the value of the ampere. This would be brought about by precisely fixing the value of the elementary charge (e), likely to 1.602176487×10-19 coulomb (from the present 2006 CODATA value of 1.602176487(40)×10-19 . Doing so would effectively define the ampere as a flow of 1/1.602176487×10-19 (6,241,509,647,120,417,390) electrons per second past a fixed point in an electric circuit. The SI unit of mass would thus be fully defined by having precisely fixed the values of the Avogadro constant and elementary charge, and by exploiting the fact that the atomic masses of bismuth and gold atoms are invariant, universal constants of nature.

One challenge in maturing an ion-accumulation-based kilogram into a practical realization is the development of a deposition chamber/balance system that enables the convenient calibration of a reasonable quantity of transfer standards relative to any single internal ion-deposited prototype. The mass prototypes produced by ion deposition techniques are nothing like the freestanding platinum-iridium prototypes currently in use; they are deposited onto—and become part of—an electrode imbedded into one pan of a special balance that is part of the device. Further, the ion-deposited material doesn’t have a hard, highly polished surface that can be vigorously cleaned like those of current prototypes. Gold, while dense and a noble metal (resistant to oxidation and the formation of other compounds), is extremely soft so an internal gold prototype would have to be kept well isolated and scrupulously clean to avoid contamination and the potential of wear from having to remove the contamination. Bismuth, which is an inexpensive metal used in low-temperature solders, slowly oxidizes when exposed to room-temperature air and forms other chemical compounds and so, would not produce a stable reference mass unless it was continually maintained in a vacuum or inert atmosphere.

#### Electronic approaches

##### Watt balance
The NIST’s watt balance is a project of the U.S. Government to develop an “electronic kilogram.” The vacuum chamber dome, which lowers over the entire apparatus, is visible at top.

The watt balance is essentially a single-pan weighing scale that measures the electric power necessary to oppose the weight of a kilogram test mass as it is accelerated by gravity. It is a variation of an ampere balance in that it employs an extra calibration step that nulls the effect of geometry. The electric potential in the watt balance is delineated by a Josephson voltage standard, which allows voltage to be linked to an invariant constant of nature with extremely high precision and stability. Its circuit resistance is calibrated against a quantum Hall resistance standard. The watt balance requires exquisitely precise measurement of gravity in a laboratory (see “FG-5 absolute gravimeter” in Links to photographs, below). For instance, the gravity gradient of 3.1 µGal/cm (≈3 ppb/cm) is accounted for when the elevation of the center of the gravimeter differs from that of the nearby test mass. As of April 2007, the NIST’s implementation of the watt balance was demonstrating a combined relative standard uncertainty (CRSU) at 68% probability of 36 µg and a short-term resolution of 10–15 µg.[21] The UK’s National Physical Laboratory’s watt balance as of 2007, was demonstrating a CRSU of 70.3 µg.[28]

Ultimately, the watt balance would define the kilogram in terms of the Planck constant, which is a measure that relates the energy of photons to their frequency. The Planck constant would be fixed, where h = 6.62606896×10-34J·s (from the 2006 CODATA value of 6.62606896(33)×10-34J·s) and the kilogram would be defined as “the mass of a body at rest whose equivalent energy equals the energy of photons whose frequencies sum to 1.356392733×1050 Hz.”[29]

Gravity is measured with exceptional precision with the help of a laser interferometer. The laser’s pattern of interference fringes—the dark and light bands above—blooms at an ever faster rate as a free-falling corner reflector drops inside an absolute gravimeter. The pattern’s frequency sweep is timed by an atomic clock.

The virtue of electronic realizations like the watt balance is that the definition and dissemination of the kilogram would no longer be dependent upon the stability of kilogram prototypes, which must be very carefully handled and stored. It would free physicists from the need to rely on assumptions about the stability of those prototypes, including those that would be manufactured under atom-counting schemes. Instead, hand-tuned, close-approximation mass standards would simply be weighed and documented as being equal to one kilogram plus an offset value. With scales, the kilogram would not only be defined in electrical terms, it would also be delineated in electrical terms. Mass artifacts calibrated in a watt balance would effectively become transfer standards. Further, one additional term in all scale-based realizations—acceleration due to gravity—is currently measured using dropping-mass absolute gravimeters that contain an iodine-stabilized HeNe laser interferometer. The fringe-signal, frequency-sweep output from the interferometer is measured with a rubidium atomic clock. Thus, the ‘gravity’ term in the delineation of an all-electronic kilogram would also be measured relative to invariants of nature.

Scales like the watt balance also permit more flexibility in choosing materials with especially desirable properties for mass standards. For instance, Pt-10Ir could continue to be used so that the specific gravity of newly produced mass standards would be the same as existing national primary and check standards (≈21.55 g/ml). This would reduce the relative uncertainty when making mass comparisons in air. Alternately, entirely different materials and constructions could be explored with the objective of producing mass standards with greater stability. For instance, osmium-iridium alloys could be investigated if platinum’s propensity to absorb hydrogen (due to catalysis of VOCs and hydrocarbon-based cleaning solvents) and atmospheric mercury proved to be sources of instability. Also, vapor-deposited, protective ceramic coatings like nitrides could be investigated for their suitability to isolate these new alloys.

The challenge with watt balances is not only in reducing their uncertainty, but also in making them truly practical realizations of the kilogram. Nearly every aspect of watt balances and their support equipment requires such extraordinarily precise and accurate, state-of-the-art technology that—unlike a device like an atomic clock—few countries would currently choose to fund their operation. For instance, the NIST’s watt balance uses four resistance standards, each of which is rotated through the watt balance every two to six weeks after being calibrated in a different part of the facility. Simply moving the resistance standards down the hall to the watt balance alters their values 10 ppb (equivalent to 10 µg) or more. The plan is to eventually collocate the watt balance and the calibration equipment so the resistance standards can be calibrated in place. Still, present-day technology is insufficient to permit stable operation of watt balances between even biannual calibrations. If the kilogram is defined in terms of the Planck constant, it is likely there may only be a few—at most—watt balances initially operating in the world.

##### Ampere-based force
A magnet floating above a superconductor bathed in liquid nitrogen demonstrates perfect diamagnetic levitation via the Meissner effect. Experiments with an ampere-based definition of the kilogram flipped this arrangement upside-down: an electric field accelerated a superconducting test mass supported by fixed magnets.

This approach would define the kilogram as “the mass which would be accelerated at precisely 2 × 10–7 m/s² when subjected to the per-meter force between two straight parallel conductors of infinite length, of negligible circular cross section, placed 1 meter apart in vacuum, through which flow a constant current of 1/1.602176487×10−19 (6,241,509,647,120,417,390) elementary charges per second”.

Effectively, this would define the kilogram as a derivative of the ampere, rather than present relationship, which defines the ampere as a derivative of the kilogram. This redefinition of the kilogram would result from fixing the elementary charge (e) to be precisely 1.602176487×10−19 coulomb (from the current 2006 CODATA value of 1.602176487(40)×10−19), which effectively defines the coulomb as being the sum of 6,241,509,647,120,417,390 elementary charges. It would necessarily follow that the ampere then becomes an electrical current of this same quantity of elementary charges per second.

The virtue of a practical realization based upon this definition is that unlike the watt balance and other scale-based methods, all of which require the careful characterization of gravity in the laboratory, this method delineates the magnitude of the kilogram directly in the very terms that define the nature of mass: acceleration due to an applied force. Unfortunately, it is extremely difficult to develop a practical realization based upon accelerating masses. Experiments over a period of years in Japan with a superconducting, 30-gram mass supported by diamagnetic levitation never achieved an uncertainty better than ten parts per million. Magnetic hysteresis was one of the limiting issues. Other groups are continuing this line of research using different techniques to levitate the mass.[30]

## SI multiples

Because SI prefixes may not be concatenated (serially linked) within the name or symbol for a unit of measure, SI prefixes are used with the gram, not the kilogram, which already has a prefix as part of its name.[31] For instance, one-millionth of a kilogram is 1 mg (one milligram), not 1 µkg (one microkilogram).

SI multiples for gram (g)

Submultiples Multiples
Value Symbol Name Value Symbol Name
10–1 g dg decigram 101 g dag decagram
10–2 g cg centigram 102 g hg hectogram
10–3 g mg milligram 103 g kg kilogram
10–6 g µg microgram (mcg) 106 g Mg megagram (tonne)
10–9 g ng nanogram 109 g Gg gigagram
10–12 g pg picogram 1012 g Tg teragram
10–15 g fg femtogram 1015 g Pg petagram
10–18 g ag attogram 1018 g Eg exagram
10–21 g zg zeptogram 1021 g Zg zettagram
10–24 g yg yoctogram 1024 g Yg yottagram
Common prefixes are in bold face.[32]

• When the Greek lowercase “µ” (mu) in the symbol of microgram is typographically unavailable, it is occasionally—although not properly—replaced by Latin lowercase “u”.
• The microgram is often abbreviated “mcg”, particularly in pharmaceutical and nutritional supplement labeling, to avoid confusion since the “µ” prefix is not well recognized outside of technical disciplines.[33] Note however, that the abbreviation “mcg”, is also the symbol for an obsolete CGS unit of measure known as the “millicentigram”, which is equal to 10 µg.
• The unit name “megagram” is rarely used, and even then, typically only in technical fields in contexts where especially rigorous consistency with the units of measure is desired. For most purposes, the unit “tonne” is instead used. The tonne and its symbol, t, were adopted by the CIPM in 1879. It is a non-SI unit accepted by the BIPM for use with the SI. In English speaking countries it is usually called “metric ton”.[34] Note also that the unit name “megatonne” or “megaton” (Mt) is often used in current literature on global climate change and its equivalent value in scientific literature on the subject is the “teragram” (Tg).

## Glossary

• Abstracted: Isolated and its effect changed in form, often simplified or made more accessible in the process.
• Artifact: A human-made object used as a comparative standard in the measurement of a physical quantity.
• Check standard:
1. A standard body’s backup replica of the International Prototype Kilogram (IPK).
2. A secondary kilogram mass standard used as a stand-in for the primary standard during routine calibrations.
• Definition: A formal, specific, and exact specification.
• Delineation: The physical means used to mark a boundary or express the magnitude of an entity.
• Disseminate: To widely distribute the magnitude of a unit of measure, typically via replicas and transfer standards.
• IPK: Abbreviation of “International Prototype Kilogram” (CG image), the artifact which has a mass defined as precisely one kilogram.
• Magnitude: The extent or numeric value of a property
• National prototype: A replica of the IPK possessed by a nation.
• Practical realization: An artifact or readily reproducible apparatus to conveniently delineate the magnitude of a unit of measure.
• Primary national standard:
1. A replica of the IPK possessed by a nation
2. The least used replica of the IPK when a nation possesses more than one.
• Prototype:
1. A human-made object that serves as the defining comparative standard in the measurement of a physical quantity.
2. A human-made object that serves as the comparative standard in the measurement of a physical quantity.
3. The IPK and any of its replicas
• Replica: An official copy of the IPK.
• Sister copy: One of six official copies of the IPK that are stored in the same safe as the IPK and are used as check standards by the BIPM.
• Transfer standard: An artifact or apparatus that reproduces the magnitude of a unit of measure in a different, usually more practical, form.

## References

1. Proceedings of the 94th meeting (October 2005) of the International Committee for Weights and Measures, (1.1 MB zip file, here)
2. Citation: Decree on weights and measures, 7 April 1795 Original text: “Gramme , le poids absolu d'un volume d'eau pure égal au cube de la centième partie du mètre , et à la température de la glace fondante.”
3. The same decree (Decree on weights and measures, 7 April 1795) also defined the liter as follows: “Liter, the measure of volume, both for liquid and solids, for which the displacement would be that of a cube [with sides measuring] one-tenth of a meter.” Original text: “Litre, la mesure de capacité, tant pour les liquides que pour les matières sèches, dont la contenance sera celle du cube de la dixièrne partie du mètre.”
4. Modern measurements show the temperature at which water reaches maximum density is 3.984 °C. However, the French scientists at the close of the 18th century concluded that the temperature was 4 °C. Citation: L'Histoire Du Mètre, La Détermination De L'Unité De Poids, link to Web site here.
5. Citation: History of the kilogram
6. New Techniques in the Manufacture of Platinum-Iridium Mass Standards, T. J. Quinn, Platinum Metals Rev., 1986, 30, (2), pp. 74–79
7. Water Structure and Science, Water Properties, Density maximum (and molar volume) at temperature of maximum density, a (by London South Bank University). Link to Web site.
8. The Third Periodic Verification of National Prototypes of the Kilogram (1988–1992), G. Girard, Metrologia 31 (1994) 317–336
9. The other two Pt-10Ir standards owned by the U.S. are K85, which is used for watt balance experiments (see Watt balance, above), and K650, which was an early attempt with a new series of prototypes (K64–K80) that were diamond-turned directly to a finish mass. K650 has an unremarkable density of 21.53537 g/ml. However, as its finished mass was roughly 2000 µg less than one kilogram, it is unsuitable for use as a national prototype and is instead known as a mass “standard”—not a “prototype.”. However, it serves well for special duties, such as a stability reference when K4 and K20 are transported to the BIPM and back. There are three other diamond-turned, Pt-10Ir replicas that are not formally considered to be “prototypes”: K651, K690, and K691.
10. Extraordinary care is exercised when transporting prototypes. In 1984, the K4 and K20 prototypes were hand-carried in the passenger section of a commercial airliner.
11. Before the BIPM’s published report in 1994 detailing the relative change in mass of the prototypes, different standard bodies used different techniques to clean their prototypes. The NIST’s practice before then was to soak and rinse its two prototypes first in benzene, then in ethanol, and to then clean them with a jet of bi-distilled water steam.
12. Redefinition of the kilogram: a decision whose time has come, Ian M. Mills et al., Metrologia 42 (2005), 71–80
13. Citations: The SI unit of mass, Richard Davis, Metrologia 40 (2003), 299–305; and Recalibration of the U.S. National Prototype Kilogram, R. S. Davis, Journal of Research of the National Bureau of Standards, 90, No. 4, July–August 1985 (5.5 MB PDF, here). Note that if the ∆ 50 µg between the IPK and its replicas was entirely due to wear, the IPK would have to have lost 150 million billion more platinum and iridium atoms over the last century than its replicas. That there would be this much wear, much less a difference of this magnitude, is thought unlikely; 50 µg is roughly the mass of a fingerprint. Specialists at the BIPM in 1946 carefully conducted cleaning experiments and concluded that even vigorous rubbing with a chamois—if done carefully—did not alter the prototypes’ mass. More recent cleaning experiments at the BIPM, which were conducted on one particular prototype (K63), and which benefited from the then-new NBS-2 balance, demonstrated 2 µg stability. Many theories have been advanced to explain the data, including one that begins with the observation that the IPK is uniquely stored under three bell jars whereas its six sister copies stored in the vault with it and the other replicas dispersed throughout the world are stored under only two. This theory is founded on two other facts: that platinum has a strong affinity for mercury, and that atmospheric mercury is significantly more abundant in the atmosphere today than at the time the IPK and its replicas were manufactured. This theory posits that the relative change in mass between the IPK and its replicas is not one of loss at all, and is instead a simple matter that the IPK has gained less than the replicas. This theory is just one of many advanced by the specialists to account for the relative change in mass. To date, each theory has either proven implausible, or there is insufficient data or technical means to either prove or disprove it. Citation: Conjecture why the IPK drifts, R. Steiner, NIST, 11 Sept. 2007.
14. Even well respected organizations incorrectly represent the relative nature of the mass divergence as being one of mass loss, as exemplified by this site at Science Daily, and this site at PhysOrg.com, and this site at Sandia National Laboratories. The root of the problem is often the reporters’ failure to correctly interpret or paraphrase nuanced scientific concepts, as exemplified by this 12 September 2007 story by the Associated Press published on PhysOrg.com. In that AP story, Richard Davis—who used to be the NIST’s kilogram specialist and now works for the BIPM in France—was correctly quoted by the AP when he stated that the mass change is a relative issue. Then the AP summarized the nature of issue with this lead-in to the story: “A kilogram just isn't what it used to be. The 118-year-old cylinder that is the international prototype for the metric mass, kept tightly under lock and key outside Paris, is mysteriously losing weight - if ever so slightly.” Like many of the above-linked sites, the AP also misreported the age of the IPK, using the date of its adoption as the mass prototype, not the date of the cylinder’s manufacture. This is a mistake even Scientific American fell victim to in a print edition.
15. The mean change in mass of the first batch of replicas relative to the IPK over one hundred years is +23.5 µg with a standard deviation of 30 µg. Per The Third Periodic Verification of National Prototypes of the Kilogram (1988–1992), G. Girard, Metrologia 31 (1994) Pg. 323, Table 3. Data is for prototypes K1, K5, K6, K7, K8(41), K12, K16, K18, K20, K21, K24, K32, K34, K35, K36, K37, K38, and K40; and excludes K2, K23, and K39, which are treated as outliers. This is a larger data set than is shown in the chart at the top of this section, which corresponds to Figure 7 of G. Girard’s paper.
16. Report to the CGPM, 14th meeting of the Consultative Committee for Units (CCU), April 2001, 2. (ii); General Conference on Weights and Measures, 22nd Meeting, October 2003, which stated “The kilogram is in need of a new definition because the mass of the prototype is known to vary by several parts in 108 over periods of time of the order of a month…” (3.2 MB ZIP file, here).
17. Citations: Conjecture why the IPK drifts, R. Steiner, NIST, 11 Sept. 2007; and the BBC’s Getting the measure of a kilogram.
18. General section citations: Recalibration of the U.S. National Prototype Kilogram, R. S. Davis, Journal of Research of the National Bureau of Standards, 90, No. 4, July–August 1985 (5.5 MB PDF, here); and The Kilogram and Measurements of Mass and Force, Z. J. Jabbour et al., J. Res. Natl. Inst. Stand. Technol. 106, 2001, 25–46 (3.5 MB PDF, here)
19. Assuming the past trend continues, whereby the mean change in mass of the first batch of replicas relative to the IPK over one hundred years was +23.5 σ30 µg.
20. Uncertainty Improvements of the NIST Electronic Kilogram, R. L. Steiner et al., Instrumentation and Measurement, IEEE Transactions on, 56 Issue 2, April 2007, 592–596
21. Georgia Tech, 21 Sept. 2007 press release: A Better Definition for the Kilogram? Note that the uncertainty in the Avogadro constant narrowed since this proposal was first submitted to American Scientist for publication. The 2006 CODATA value for the Avogadro constant has a relative standard uncertainty of 50 parts per billion and the only cube root values within this uncertainty must fall within the range of 84,446,889.8 ±1.4; that is, there are only three integer cube roots (…89, …90, and …91) in this range and the value 84,446,886 falls outside of it. Unfortunately, none of the three integer values within the range possess the property of their cubes being divisible by twelve; one gram of C-12 could not comprise an integer number of atoms. If the value 84,446,886 was adopted to define the kilogram, many other constants of nature and electrical units would have to be revised an average of about 0.13 part per million. The straightforward adjustment to this approach advanced by the group would instead define the kilogram as “the mass equal to 84,446,8903 × 83⅓ atoms of carbon-12.” This proposed value for the Avogadro constant falls neatly within the measured value (≈6.02214184×1023 vs. the 2006 CODATA value of 6.02214179(30)×1023 and the proposed definition of the kilogram produces an integer number of atoms in 12 grams of carbon-12, but not for 1 gram or 1 kilogram.
22. The sphere shown in the photograph has an out-of-roundness value (peak to valley on the radius) of 50 nm. According to ACPO, they improved on that with an out-of-roundness of 35 nm. On the Ø 93.6 mm sphere, an out-of-roundness of 35 nm (undulations of ±17.5 nm) is a fractional roundness (∆r/r) = 3.7 × 10–7. Scaled to the size of Earth, this is equivalent to a maximum deviation from sea level of only 2.4 meters. The roundness of that ACPO sphere is exceeded only by two of the four fused-quartz gyroscope rotors flown on Gravity Probe B, which were manufactured in the late 1990s and given their final figure at the W.W. Hansen Experimental Physics Lab at Stanford University. Particularly, “Gyro 4” is recorded in the Guinness Database of World Records (their database, not in their book) as the world’s roundest man-made object. According to a published report (221 kB PDF, here) and the GP-B public affairs coordinator at Stanford University, of the four gyroscopes onboard the probe, Gyro 4 has a maximum surface undulation from a perfect sphere of 3.4 ±0.4 nm on the Ø 38.1 mm sphere, which is a ∆r/r = 1.8 × 10–7. Scaled to the size of Earth, this is equivalent to an undulation the size of North America rising slowly up out of the sea (in molecular-layer terraces 11.9 cm high), reaching a maximum elevation of 1.14 ±0.13 meters in Nebraska, and then gradually sloping back down to sea level on the other side of the continent. The only known objects in the universe that are rounder than this are neutron stars.
23. Citations: NPL: Avogadro Project; Australian National Measurement Institute: Redefining the kilogram through the Avogadro constant; and Australian Centre for Precision Optics: The Avogadro Project
24. In 2003, the same year the first gold-deposition experiments were conducted, physicists found that the only naturally occurring isotope of bismuth, 209Bi, is actually very slightly radioactive, with the longest known radioactive half-life of any naturally occurring element that decays via alpha radiation—a half-life of (19±2)×1018 years. As this is 1.4 billion times the age of the universe, 209Bi is considered a stable isotope for most practical applications (those unrelated to such disciplines as nucleocosmochronology and geochronology). In other terms, 99.999999983% of the bismuth that existed on Earth 4.567 billion years ago still exists today. Only two mononuclidic elements are heavier than bismuth and only one approaches its stability: thorium. Long considered a possible replacement for uranium in nuclear reactors, thorium can cause cancer when inhaled because it is over 1.2 billion times more radioactive than bismuth. It also has such a strong tendency to oxidize that its powders are pyrophoric. These characteristics make thorium unsuitable in ion-deposition experiments. See also Isotopes of bismuth, Isotopes of gold and Isotopes of thorium.
25. General Conference on Weights and Measures, 22nd Meeting, October 2003 (3.2 MB ZIP file, here).
26. The German national metrology institute, known as the Physikalisch-Technische Bundesanstalt (PTB): Working group 1.24, Ion Accumulation
27. Citations: An initial measurement of Planck's constant using the NPL Mark II watt balance, I. A. Robinson et al., Metrologia 44 (2007), 427–440; and NPL: NPL Watt Balance
28. Hysteresis and Related Error Mechanisms in the NIST Watt Balance Experiment, Joshua P. Schwarz et al., Journal of Research of the National Bureau of Standards and Technology, 106, No. 4, July–August 2001 (888 KB PDF, here); and On the redefinition of the kilogram, B. N. Taylor et al., Metrologia 36 (1999), 63–64; and the NIST’s Fundamental Physical Constants: “Energy Equivalents” calculator.
29. NIST, Beyond the Kilogram: Redefining the International System of Units; and A Watt Balance On Its Side, R. Steiner, NIST, 24 Sept. 2007.
30. BIPM: SI Brochure: Section 3.2, The kilogram
31. Criterion: A combined total of at least 250,000 Google hits on both the U.S. spelling (-gram) and the U.K./International spelling (-gramme).
32. The practice of using the abbreviation “mcg” rather than the SI symbol “µg” was formally mandated for medical practitioners in 2004 by the Joint Commission on Accreditation of Healthcare Organizations (JCAHO) in their “Do Not Use” List: Abbreviations, Acronyms, and Symbols because hand-written expressions of “µg” can be confused with “mg”, resulting in a thousand-fold overdosing. The mandate was also adopted by the Institute for Safe Medication Practices.
33. BIPM: SI Brochure: Section 4.1, Non-SI units accepted for use with the SI, and units based on fundamental constants: Table 6