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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), which is almost exactly equal to the mass of one liter of water. 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 by the initials CIPM) recommended in 2005 that the kilogram be redefined in terms of fundamental constants of nature.
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 since such objects naturally hover; they are “weightless.” However, since objects in microgravity still retain their mass, an astronaut must exert one hundred times more force to accelerate a 100-kilogram object at the same rate as a 1-kilogram object.
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. For instance, one-millionth of a kilogram is 1 mg (one milligram), not 1 µkg (one microkilogram).
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, and the temperature of melting ice” Since trade and commerce at the time (and still today) typically involved commodities transactions that were 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.
Concurrent with fabrication of the provisional kilogram mass standard, work was also commissioned to precisely determine the mass of a cubic decimeter (one liter) 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. 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. 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 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. 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 replicas 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. 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. 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 213 years ago when making their measurements of water’s properties and in manufacturing the Kilogram of the Archives.
Stability of the International Prototype Kilogram
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 three kilogram standards, two of which, K4 and K20, are from the original batch of 40 replicas delivered in 1884. The K20 standard 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.
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 kg – 39 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 kg – 19 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 kg – 116 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 preparation for periodic verifications—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. 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 in 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.
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 the mass of the IPK lost perhaps 50 µg over the last century, and possibly significantly more, in comparison to its official copies. 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. 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.” 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.” Beyond the problem of long-term drift, the IPK further exhibits a short-term instability of about 30 µg over a period of about a month in its after-cleaned mass. 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 VOCs and the hydrocarbon-based solvents used to clean the prototypes. Scientists are seeing far greater variability in the prototypes than previously believed.
The relative change in mass and the 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.
Importance of the kilogram
The kilogram underpins the entire SI system of measurement as it is currently defined and structured so the stability of the IPK is crucial. For instance, the newton—the SI unit of force—is defined as the force necessary to accelerate the kilogram by one meter per second². Accordingly, if the mass of the IPK were to change slightly, so too would the newton by a proportional degree so that the acceleration would remain at precisely one meter/second². In turn, the pascal—the SI unit of pressure—is defined in terms of the newton. This chain of dependency follows to most of the electrical units. For instance, the joule, which is the electrical and mechanical unit of energy, is defined as the energy expended when a force of one newton acts through one meter. The ampere is also defined relative to the newton. With the magnitude of two of the primary units of electricity thus determined by the kilogram, so too follow most of the rest; namely, the watt, ohm, coulomb, farad, siemens, henry, and weber. From there, the measurement of light (the candela, lumen, and lux) is in turn affected.
Clearly, having the magnitude of many of the units comprising the SI system of measurement ultimately defined by the mass of a 133-year-old, golf ball-size piece piece of metal is a tenuous state of affairs. The quality of the IPK must be diligently protected in order to preserve the integrity of the SI system. 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. Note that the redefinition of the meter in terms of a duration of one second reduced the uncertainty in the wavelength of the laser light. Now suppose that the official measurement of the second were found to have drifted by a few parts per billion (it is actually exquisitely stable). There would be no automatic effect on many of the SI units of measurement because, as with the meter, the duration of the second is sometimes abstracted through other physical principles underlying their practical realizations. 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 physicists were to definitively prove that the IPK’s value had changed, one solution would be to simply redefine the kilogram as being equal to 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 µg.”
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 getting close, with a demonstrated uncertainty of 36 µg. See Watt balance, below.
Proposed future definitions
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 upon 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:
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) as there are 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.
Another Avogadro constant-based approach, known as the Avogadro project, attempts to define and delineate the kilogram as a quantity 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. The precise lattice spacing between the atoms in its crystal structure (≈192 pm) would be measured using a scanning X-ray interferometer. Amazingly, this permits its atomic spacing to be determined with an uncertainty of only three parts in a billion. With the size of the sphere, its average atomic mass, and the atomic spacing known, the required number of atoms in the sphere could be calculated with sufficient precision and uncertainty to enable it to be ground down to the desired quantity of atoms (mass).
Experiments are planned for the Avogadro Project’s silicon sphere to determine whether its mass is most stable when stored in a vacuum, a partial vacuum, or ambient pressure. However, no technical means currently exist to prove a 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 an 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 and its effect will be accounted for when grinding 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 is 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) 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 kilogram would be defined as 1000/27.9769271 × 6.02214179 × 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 definition in order to compensate for various chemical and isotopic impurities as well as the effect of surface oxides.
Another Avogadro-based approach, ion accumulation, would define and delineate the kilogram by creating metal mass artifacts. It would do so by accumulating of 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 are used because, unlike most other elements, they each have only one naturally occurring isotope.
With a gold-based definition of the kilogram for instance, the atomic mass of gold would be fixed as precisely 196.966569 g/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” (≅5.0770037021 fixed moles of gold atoms).
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.
The difficulty with ion-accumulation-based standards is in obtaining truly practical mass artifacts. Gold, while dense and a noble metal (resistant to oxidation and the formation of other compounds), is extremely soft so mass artifacts would require extraordinary care to avoid wear. Bismuth, while an inexpensive metal for experiments, would not produce stable artifacts because it readily oxidizes and forms other chemical compounds.
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 @ 68% probability) of 36 µg and a short-term resolution of 10–15 µg. The U.K.’s National Physical Laboratory’s watt balance as of 2007, was demonstrating a CRSU of 70.9 µg.
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.
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 10 parts in 106. Magnetic hysteresis was one of the limiting issues. Other groups are continuing this line of research using different techniques to levitate the mass.
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