Caesium standard

Last updated May 04, 2024

The caesium standard is a primary frequency standard in which the photon absorption by transitions between the two hyperfine ground states of caesium-133 atoms is used to control the output frequency. The first caesium clock was built by Louis Essen in 1955 at the National Physical Laboratory in the UK.^[1] and promoted worldwide by Gernot M. R. Winkler of the United States Naval Observatory.

Caesium atomic clocks are one of the most accurate time and frequency standards, and serve as the primary standard for the definition of the second in the International System of Units (SI), the modern metric system. By definition, radiation produced by the transition between the two hyperfine ground states of caesium-133 (in the absence of external influences such as the Earth's magnetic field) has a frequency, $Δ ν Cs$ , of exactly 9192631770 Hz . That value was chosen so that the caesium second equaled, to the limit of measuring ability in 1960 when it was adopted, the existing standard ephemeris second based on the Earth's orbit around the Sun.^[2] Because no other measurement involving time had been as precise, the effect of the change was less than the experimental uncertainty of all existing measurements.

While the second is the only base unit to be explicitly defined in terms of the caesium standard, the majority of SI units have definitions that mention either the second, or other units defined using the second. Consequently, every base unit except the mole and every named derived unit except the coulomb, ohm, siemens, weber, gray, sievert, radian, and steradian have values that are implicitly defined by the properties of the caesium-133 hyperfine transition radiation. And of these, all but the mole, the coulomb, and the dimensionless radian and steradian are implicitly defined by the general properties of electromagnetic radiation.

Technical details

The official definition of the second was first given by the BIPM at the 13th General Conference on Weights and Measures in 1967 as: "The second is the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom." At its 1997 meeting the BIPM added to the previous definition the following specification: "This definition refers to a caesium atom at rest at a temperature of 0 K."^[3]

The BIPM restated this definition in its 26th conference (2018), "The second is defined by taking the fixed numerical value of the caesium frequency ∆Cs, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s^–1."^[4]

The meaning of the preceding definition is as follows. The caesium atom has a ground state electron state with configuration [Xe] 6s¹ and, consequently, atomic term symbol ²S_1/2. This means that there is one unpaired electron and the total electron spin of the atom is 1/2. Moreover, the nucleus of caesium-133 has a nuclear spin equal to 7/2. The simultaneous presence of electron spin and nuclear spin leads, by a mechanism called hyperfine interaction, to a (small) splitting of all energy levels into two sub-levels. One of the sub-levels corresponds to the electron and nuclear spin being parallel (i.e., pointing in the same direction), leading to a total spin F equal to F = 7/2 + 1/2 = 4; the other sub-level corresponds to anti-parallel electron and nuclear spin (i.e., pointing in opposite directions), leading to a total spin F = 7/2 − 1/2 = 3. In the caesium atom it so happens that the sub-level lowest in energy is the one with F = 3, while the F = 4 sub-level lies energetically slightly above. When the atom is irradiated with electromagnetic radiation having an energy corresponding to the energetic difference between the two sub-levels the radiation is absorbed and the atom is excited, going from the F = 3 sub-level to the F = 4 one. After a small fraction of a second the atom will re-emit the radiation and return to its F = 3 ground state. From the definition of the second it follows that the radiation in question has a frequency of exactly 9.19263177 GHz, corresponding to a wavelength of about 3.26 cm and therefore belonging to the microwave range.

This particular caesium resonance was agreed upon under la Convention du Mètre and remains to the present time as the official definition of the second for the world community.

Note that a common confusion involves the conversion from angular frequency ( $\omega$ ) to frequency ( $f$ ), or vice versa. Angular frequencies are conventionally given as s^–1 in scientific literature, but here the units implicitly mean radians per second. In contrast, the unit Hz should be interpreted as cycles per second. The conversion formula is $\omega =2\pi f$ , which implies that 1 Hz corresponds to an angular frequency of approximately 6.28 radians per second (or 6.28 s^–1 where radians is omitted for brevity by convention).

Parameters and significance in the second and other SI units

Suppose the caesium standard has the parameters:

Velocity: c

Energy/frequency: h

Time period: $Δ t Cs$

Frequency: $Δ ν Cs$

Wavelength: $Δ λ Cs$

Photon energy: $Δ E Cs$

Photon mass equivalent: $Δ M Cs$

Time and frequency

The first set of units defined using the caesium standard were those relating to time, with the second being defined in 1967 as "the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom" meaning that:

1 second, s, = 9,192,631,770 $Δ t Cs$

1 hertz, Hz, = 1/s = $Δ ν Cs$ /9,192,631,770

1 becquerel, Bq, = 1 nuclear decay/s = 1/9,192,631,770 nuclear decays/ $Δ t Cs$

This also linked the definitions of the derived units relating to force and energy (see below) and of the ampere, whose definition at the time made reference to the newton, to the caesium standard. Before 1967 the SI units of time and frequency were defined using the tropical year and before 1960 by the length of the mean solar day ^[5]

Length

In 1983, the meter was, indirectly, defined in terms of the caesium standard with the formal definition "The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. This implied:

1 metre, m, = c s/299,792,458 = 9,192,631,770/299,792,458c $Δ t Cs$ = 9,192,631,770/299,792,458 $Δ λ Cs$

1 radian, rad, = 1 m/m = $Δ λ Cs$ / $Δ λ Cs$ = 1 (dimensionless unit of angle)

1 steradian, sr, = 1 m²/m² = $Δ λ Cs$ ²/ $Δ λ Cs$ ² = 1 (dimensionless unit of solid angle)

Between 1960 and 1983, the metre had been defined by the wavelength of a different transition frequency associated with the krypton 86 atom. This had a much higher frequency and shorter wavelength than the caesium standard, falling inside the visible spectrum. The first definition, used between 1889 and 1960, was by the international prototype meter.^[6]

Mass, energy, and force

Following the 2019 redefinition of the SI base units, electromagnetic radiation, in general, was explicitly defined to have the exact parameters:

c = 299,792,458 m/s

h = 6.62607015×10⁻³⁴ J s

The caesium-133 hyperfine transition radiation was explicitly defined to have frequency:

$Δ ν Cs$ = 9,192,631,770 Hz^[7]

Though the above values for c and $Δ ν Cs$ were already obviously implicit in the definitions of the metre and second. Together they imply:

$Δ t Cs$ = 1/ $Δ ν Cs$ = s/9,192,631,770

$Δ λ Cs$ = c $Δ t Cs$ = 299,792,458/9,192,631,770 m

$Δ E Cs$ = h $Δ ν Cs$ = 9,192,631,770 Hz × 6.62607015×10⁻³⁴ J s = 6.09110229711386655×10⁻²⁴ J

$Δ M Cs$ = $Δ E Cs$ /c² = 6.09110229711386655×10⁻²⁴ J/89,875,517,873,681,764 m²/s² = 6.09110229711386655/8.9875517873681764×10⁴⁰ kg

Notably, the wavelength has a fairly human-sized value of about 3.26 centimetres and the photon energy is surprisingly close to the average molecular kinetic energy per degree of freedom per kelvin. From these it follows that:

1 kilogram, kg, = 8.9875517873681764×10⁴⁰/6.09110229711386655 $Δ M Cs$

1 joule, J, = 10²⁴/6.09110229711386655 $Δ E Cs$

1 watt, W, = 1 J/s = 10¹⁴/5.59932604907689089550702935 $Δ E Cs$ $Δ ν Cs$

1 newton, N, = 1 J/m = 2.99792458×10²²/5.59932604907689089550702935 $Δ E Cs$ / $Δ λ Cs$

1 pascal, Pa, = 1 N/m² = 2.6944002417373989539335912×10¹⁹/4.73168129737820913189287698892486811451620615 $Δ E Cs$ / $Δ λ Cs$ ³

1 gray, Gy, = 1 J/kg = 1/89,875,517,873,681,764 $Δ E Cs$ / $Δ M Cs$ = c²/89,875,517,873,681,764

1 sievert, Sv, = the ionizing radiation dose equivalent to 1 gray of gamma rays

Prior to the revision, between 1889 and 2019, the family of metric (and later SI) units relating to mass, force, and energy were somewhat notoriously defined by the mass of the International Prototype of the Kilogram (IPK), a specific object stored at the headquarters of the International Bureau of Weights and Measures in Paris, meaning that any change to the mass of that object would have resulted in a change to the size of the kilogram and of the many other units whose value at the time depended on that of the kilogram.^[8]

Temperature

From 1954 to 2019, the SI temperature scales were defined using the triple point of water and absolute zero.^[9] The 2019 revision replaced these with an assigned value for the Boltzmann constant, k, of 1.380649×10⁻²³ J/K, implying:

1 kelvin, K, = 1.380649×10⁻²³ J/2 per degree of freedom = 1.380649×10⁻²³ × 10²⁴/2/6.09110229711386655 $Δ E Cs$ per degree of freedom = 1.380649/1.21822045942277331 $Δ E Cs$ per degree of freedom

Temperature in degrees Celsius, °C, = temperature in kelvins - 273.15 = 1.21822045942277331 × kinetic energy per degree of freedom - 377.12427435 $Δ E Cs$ /1.380649 $Δ E Cs$

Amount of substance

The mole is an extremely large number of "elementary entities" (i.e. atoms, molecules, ions, etc). From 1969 to 2019, this number was 0.012 × the mass ratio between the IPK and a carbon 12 atom.^[10] The 2019 revision simplified this by assigning the Avogadro constant the exact value 6.02214076×10²³ elementary entities per mole, thus, uniquely among the base units, the mole maintained its independence from the caesium standard:

1 mole, mol, = 6.02214076×10²³ elementary entities

1 katal, kat, = 1 mol/s = 6.02214076×10¹⁴/9.19263177 elementary entities/ $Δ t Cs$

Electromagnetic units

Prior to the revision, the ampere was defined as the current needed to produce a force between 2 parallel wires 1 m apart of 0.2 μN per meter. The 2019 revision replaced this definition by giving the charge on the electron, e, the exact value 1.602176634×10⁻¹⁹ coulombs. Somewhat incongruously, the coulomb is still considered a derived unit and the ampere a base unit, rather than vice versa.^[11] In any case, this convention entailed the following exact relationships between the SI electromagnetic units, elementary charge, and the caesium-133 hyperfine transition radiation:

1 coulomb, C, = 10¹⁹/1.602176634e

1 ampere, or amp, A, = 1 C/s = 10⁹/1.472821982686006218e $Δ ν Cs$

1 volt, V, = 1 J/C = 1.602176634×10⁵/6.09110229711386655 $Δ E Cs$ /e

1 farad, F, = 1 C/V = 6.09110229711386655×10¹⁴/2.566969966535569956e²/ $Δ E Cs$

1 ohm, Ω, = 1 V/A = 2.359720966701071721258310212×10⁻⁴/6.09110229711386655 $Δ E Cs$ / $Δ ν Cs$ e² = 2.359720966701071721258310212×10⁻⁴/6.09110229711386655h/e²

1 siemens, S, = 1/Ω = 6.09110229711386655×10⁴/2.359720966701071721258310212e²/h

1 weber, Wb, = 1 V s = 1.602176634×10¹⁵/6.62607015 $Δ E Cs$ $Δ t Cs$ /e = 1.602176634×10¹⁵/6.62607015h/e

1 tesla, T, = 1 Wb/m² = 1.43996454705862285832702376×10¹²/5.59932604907689089550702935 $Δ E Cs$ $Δ t Cs$ /e $Δ λ Cs$ ² = 1.43996454705862285832702376×10¹²/5.59932604907689089550702935E/e c $Δ λ Cs$

1 henry, H, = Ω s = 2.359720966701071721258310212×10⁶/6.62607015h $Δ t Cs$ /e²

Optical units

From 1967 to 1979 the SI optical units, lumen, lux, and candela are defined using the Incandescent glow of platinum at its melting point. After 1979, the candela was defined as the luminous intensity of a monochromatic visible light source of frequency 540 THz (i.e 6000/1.02140353 that of the caesium standard) and radiant intensity 1/683 watts per steradian. This linked the definition of the candela to the caesium standard and, until 2019, to the IPK. Unlike the units relating to mass, energy, temperature, amount of substance, and electromagnetism, the optical units were not massively redefined in 2019, though they were indirectly affected since their values depend on that of the watt, and hence of the kilogram.^[12] The frequency used to define the optical units has the parameters:

Frequency: 540 THz

Time period: 50/27 fs

Wavelength: 14.9896229/27 μm

Photon energy: 5.4×10¹⁴ Hz × 6.62607015×10⁻³⁴ J s = 3.578077881×10⁻¹⁹ J

luminous efficacy, K_CD, = 683 lm/W

Luminous energy per photon, $Q_{\mathrm {v} }$ , = 3.578077881×10⁻¹⁹ J × 683 lm/W = 2.443827192723×10⁻¹⁶ lm s

This implies:

1 lumen, lm, = 10⁶/2.246520349221536260971 $Q_{\mathrm {v} }$ $Δ ν Cs$

1 candela, cd, = 1 lm/sr = 10⁶/2.246520349221536260971 $Q_{\mathrm {v} }$ $Δ ν Cs$ /sr

1 Lux, lx, = 1 lm/m² = 8.9875517873681764×10²/1.898410313566852566340456048807087002459 $Q_{\mathrm {v} }$ $Δ ν Cs$ / $Δ λ Cs$ ²

Summary

The parameters of the caesium 133 hyperfine transition radiation expressed exactly in SI units are:

Frequency = 9,192,631,770 Hz

Time period = s/9,192,631,770

Wavelength = 299,792,458/9,192,631,770 m

Photon energy = 6.09110229711386655×10⁻²⁴ J

Photon mass equivalent = 6.09110229711386655×10⁻⁴⁰/8.9875517873681764 kg

If the 7 base units of the SI are expressed explicitly in terms of the SI defining constants, they are:

1 second = 9,192,631,770/ $Δ ν Cs$

1 metre = 9,192,631,770/299,792,458c/ $Δ ν Cs$

1 kilogram = 8.9875517873681764×10⁴⁰/6.09110229711386655h $Δ ν Cs$ /c²

1 ampere = 10⁹/1.472821982686006218e $Δ ν Cs$

1 kelvin = 13.80649/6.09110229711386655h $Δ ν Cs$ /k

1 mole = 6.02214076×10²³ elementary entities

1 candela = 10¹¹/3.82433969151951648163130104605h $Δ ν Cs$ ²K_CD/sr

Ultimately, 6 of the 7 base units notably have values that depend on that of $Δ ν Cs$ , which appears far more often than any of the other defining constants.

Related Research Articles

The ampere, often shortened to amp, is the unit of electric current in the International System of Units (SI). One ampere is equal to 1 coulomb (C) moving past a point per second. It is named after French mathematician and physicist André-Marie Ampère (1775–1836), considered the father of electromagnetism along with Danish physicist Hans Christian Ørsted.

<span class="mw-page-title-main">Candela</span> SI unit of luminous intensity

The candela is the unit of luminous intensity in the International System of Units (SI). It measures luminous power per unit solid angle emitted by a light source in a particular direction. Luminous intensity is analogous to radiant intensity, but instead of simply adding up the contributions of every wavelength of light in the source's spectrum, the contribution of each wavelength is weighted by the luminous efficiency function, the model of the sensitivity of the human eye to different wavelengths, standardized by the CIE and ISO. A common wax candle emits light with a luminous intensity of roughly one candela. If emission in some directions is blocked by an opaque barrier, the emission would still be approximately one candela in the directions that are not obscured.

<span class="mw-page-title-main">Caesium</span> Chemical element, symbol Cs and atomic number 55

Caesium is a chemical element; it has symbol Cs and atomic number 55. It is a soft, silvery-golden alkali metal with a melting point of 28.5 °C, which makes it one of only five elemental metals that are liquid at or near room temperature. Caesium has physical and chemical properties similar to those of rubidium and potassium. It is pyrophoric and reacts with water even at −116 °C (−177 °F). It is the least electronegative element, with a value of 0.79 on the Pauling scale. It has only one stable isotope, caesium-133. Caesium is mined mostly from pollucite. Caesium-137, a fission product, is extracted from waste produced by nuclear reactors. It has the largest atomic radius of all elements whose radii have been measured or calculated, at about 260 picometers.

<span class="mw-page-title-main">Frequency</span> Number of occurrences or cycles per unit time

Frequency, most often measured in hertz, is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as temporal frequency for clarity and to distinguish it from spatial frequency. Ordinary frequency is related to angular frequency by a factor of 2 $π$ . The period is the interval of time between events, so the period is the reciprocal of the frequency: f = 1/T.

<span class="mw-page-title-main">Hertz</span> SI unit for frequency

The hertz is the unit of frequency in the International System of Units (SI), equivalent to one event per second. The hertz is an SI derived unit whose expression in terms of SI base units is s⁻¹, meaning that one hertz is the reciprocal of one second. It is named after Heinrich Rudolf Hertz (1857–1894), the first person to provide conclusive proof of the existence of electromagnetic waves. Hertz are commonly expressed in multiples: kilohertz (kHz), megahertz (MHz), gigahertz (GHz), terahertz (THz).

The kilogram is the base unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially. It means 'one thousand grams'.

The International System of Units, internationally known by the abbreviation SI, is the modern form of the metric system and the world's most widely used system of measurement. Coordinated by the International Bureau of Weights and Measures it is the only system of measurement with an official status in nearly every country in the world, employed in science, technology, industry, and everyday commerce.

<span class="mw-page-title-main">Second</span> SI unit of time

The second is the unit of time in the International System of Units (SI), historically defined as 1⁄86400 of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds each. "Minute" comes from the Latin pars minuta prima, meaning "first small part", and "second" comes from the pars minuta secunda, "second small part".

The mole (symbol mol) is a unit of measurement, the base unit in the International System of Units (SI) for amount of substance, a quantity proportional to the number of elementary entities of a substance. One mole contains exactly 6.02214076×10²³ elementary entities (approximately 602 sextillion or 602 billion times a trillion), which can be atoms, molecules, ions, or other particles. The number of particles in a mole is the Avogadro number (symbol N₀) and the numerical value of the Avogadro constant (symbol N_A) expressed in mol^-1. The value was chosen based on the historical definition of the mole as the amount of substance that corresponds to the number of atoms in 12 grams of ¹²C, which made the mass of a mole of a compound expressed in grams numerically equal to the average molecular mass of the compound expressed in daltons. With the 2019 redefinition of the SI base units, the numerical equivalence is now only approximate but may be assumed for all practical purposes.

The einstein (symbol E) is an obsolete unit with two conflicting definitions. It was originally defined as the energy in one mole of photons (6.022×10²³ photons). Because energy is inversely proportional to wavelength, the unit is frequency dependent. This unit is not part of the International System of Units (SI) and is redundant with the joule. If it were still in use, as of the 2019 redefinition of the SI base units, its value would be related to the frequency of the electromagnetic radiation by

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Cs with a half-life of 30.1671 years and ¹³⁴Cs with a half-life of 2.0652 years. All other isotopes have half-lives less than 2 weeks, most under an hour.

A unit of time is any particular time interval, used as a standard way of measuring or expressing duration. The base unit of time in the International System of Units (SI), and by extension most of the Western world, is the second, defined as about 9 billion oscillations of the caesium atom. The exact modern SI definition is "[The second] is defined by taking the fixed numerical value of the cesium frequency, $Δ ν Cs$ , the unperturbed ground-state hyperfine transition frequency of the cesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s⁻¹."

A mononuclidic element or monotopic element is one of the 21 chemical elements that is found naturally on Earth essentially as a single nuclide. This single nuclide will have a characteristic atomic mass. Thus, the element's natural isotopic abundance is dominated by one isotope that is either stable or very long-lived. There are 19 elements in the first category, and 2 in the second category. A list of the 21 mononuclidic elements is given at the end of this article.

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<span class="mw-page-title-main">Kelvin</span> SI unit of temperature

The kelvin, symbol K, is the base unit of measurement for temperature in the International System of Units (SI). The Kelvin scale is an absolute temperature scale that starts from the coldest possible temperature, absolute zero, 0 K or -273.15 °C, then rises by exactly 1 K for each 1 °C. Thus, a Celsius temperature is converted to kelvin by adding 273.15. For example, 0 °C, the approximate melting point of ice, is 273.15 K.

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An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions between such states they interact with a very specific frequency of electromagnetic radiation. This phenomenon serves as the basis for the International System of Units' (SI) definition of a second:

The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency, $, the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be 9192631770 when expressed in the unit Hz, which is equal to s -1 .$

In 2019, four of the seven SI base units specified in the International System of Quantities were redefined in terms of natural physical constants, rather than human artifacts such as the standard kilogram. Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are now defined by setting exact numerical values, when expressed in SI units, for the Planck constant, the elementary electric charge, the Boltzmann constant, and the Avogadro constant, respectively. The second, metre, and candela had previously been redefined using physical constants. The four new definitions aimed to improve the SI without changing the value of any units, ensuring continuity with existing measurements. In November 2018, the 26th General Conference on Weights and Measures (CGPM) unanimously approved these changes, which the International Committee for Weights and Measures (CIPM) had proposed earlier that year after determining that previously agreed conditions for the change had been met. These conditions were satisfied by a series of experiments that measured the constants to high accuracy relative to the old SI definitions, and were the culmination of decades of research.

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References

↑ L. Essen, J.V.L. Parry (1955). "An Atomic Standard of Frequency and Time Interval: A Caesium Resonator". Nature . 176 (4476): 280–282. Bibcode:1955Natur.176..280E. doi:10.1038/176280a0. S2CID 4191481.
↑ Markowitz, W.; Hall, R.; Essen, L.; Parry, J. (1958). "Frequency of Cesium in Terms of Ephemeris Time". Physical Review Letters. 1 (3): 105. Bibcode:1958PhRvL...1..105M. doi:10.1103/PhysRevLett.1.105.
↑ "Comité international des poids et mesures (CIPM): Proceedings of the Sessions of the 86th Meeting" (PDF) (in French and English). Paris: Bureau International des Poids et Mesures. 23–25 Sep 1997. p. 229. Archived from the original (PDF) on 4 December 2020. Retrieved 30 December 2019.
↑ "Resolution 1 of the 26th CGPM" (in French and English). Paris: Bureau International des Poids et Mesures. 2018. pp. 472 of the official French publication. Archived from the original on 2021-02-04. Retrieved 2019-12-29.
↑ "Second - BIPM".
↑ "Metre - BIPM".
↑ "Resolution 1 (2018) - BIPM".
↑ "Kilogram - BIPM".
↑ "Kelvin - BIPM".
↑ "Mole - BIPM".
↑ "Ampere - BIPM".
↑ "Candela - BIPM".

This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22. (in support of MIL-STD-188).

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[1] L. Essen, J.V.L. Parry (1955). "An Atomic Standard of Frequency and Time Interval: A Caesium Resonator". Nature . 176 (4476): 280–282. Bibcode:1955Natur.176..280E. doi:10.1038/176280a0. S2CID 4191481.

[2] Markowitz, W.; Hall, R.; Essen, L.; Parry, J. (1958). "Frequency of Cesium in Terms of Ephemeris Time". Physical Review Letters. 1 (3): 105. Bibcode:1958PhRvL...1..105M. doi:10.1103/PhysRevLett.1.105.

[3] "Comité international des poids et mesures (CIPM): Proceedings of the Sessions of the 86th Meeting" (PDF) (in French and English). Paris: Bureau International des Poids et Mesures. 23–25 Sep 1997. p. 229. Archived from the original (PDF) on 4 December 2020. Retrieved 30 December 2019.

[4] "Resolution 1 of the 26th CGPM" (in French and English). Paris: Bureau International des Poids et Mesures. 2018. pp. 472 of the official French publication. Archived from the original on 2021-02-04. Retrieved 2019-12-29.

[5] "Second - BIPM".

[6] "Metre - BIPM".

[7] "Resolution 1 (2018) - BIPM".

[8] "Kilogram - BIPM".

[9] "Kelvin - BIPM".

[10] "Mole - BIPM".

[11] "Ampere - BIPM".

[12] "Candela - BIPM".

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