Defining Constant Table 1 - The seven defining constants of the SI Exact numerical value, {Q}SI hyperfine transition frequency of Cs speed of light in vacuum Planck constant elementary charge Boltzmann constant Avogadro constant luminous efficacy Symbol, Q ΔvCs c h e k NA Kcd How Units Are Now Defined in the SI The above paragraphs are sufficient preparation for defining units in the present SI [2]. All SI units can still be defined in terms of the definitions of the seven base units [5], where essential information is given in Table 1, adapted from Table 1 of [2]. Note that the units of the seven defining constants are mostly blends of the base units. Only the shaded portion of the table is discussed in this article. This subgroup of defining constants applies to all units that can be accommodated by (1). The units in the last column of Table 1 are written as they are in [2], using special names whenever possible. To aid the reader, the parentheses give the equivalent in base units of the units with special names and symbols. Thus, the hertz (symbol: Hz) is a special name for the inverse second but must only be used for periodic phenomena; the joule (symbol: J) is a special name for the SI unit of energy; the coulomb (symbol: C) is a special name for the SI unit of electrical charge. The Q symbols are not found in [2] but they are discussed in the text of this article. Although not dealt with in this article, K, mol, lm, W, cd and sr are symbols for the units kelvin, mole, lumen, watt, candela, and steradian. The last of these has dimension 1. The Nature of the First Four Defining Constants (the updated MKSA system) The definition of SI units based on the information in Table 1 will again be illustrated with only four defining constants, which are sufficient to define all SI units dealt with in this article. The subset of four defining constants is shaded in Table 1. Of these, ΔvCs and c have had the exact numerical values shown in Table 1 since 1967 and 1983 respectively. The speed of light in vacuum needs no introduction; the hyperfine transition frequency of the cesium-133 atom is the frequency of the atomic clocks that directly realize the definition of the second. The elementary charge e is the charge of the proton, which is positive; the electron charge is exactly equal to −e. The Planck constant h, which has the dimension of angular momentum, is the fundamental constant of quantum physics. It appears, for instance, in quantum electrical metrology, which was a major February 2023 mv r e e0 22/4π (4) Bohr also postulated several rules that have no analog in Newtonian physics. One of these is that the angular momentum of the electron in its lowest-possible orbit is equal to h/(2π) which has the symbol (pronounced h-bar). This results in an additional mathematical relation that must be satisfied: m vr e (5) Both h and have the dimension of angular momentum but Bohr had his reasons for insisting on . Replacing me in (4) by immediately predicts that the velocity v of the vr IEEE Instrumentation & Measurement Magazine 7 9 192 631 770 299 792 458 6.626 070 15 × 10−34 1.602 176 634 × 10−19 1.380 649 × 10−23 6.022 140 76 × 1023 683 SI coherent unit, [Q]SI Hz (= s−1 ) m s−1 J s (= kg m2 s−1 C (= A s) J K−1 lm W−1 ) (= kg m2 s−2 K−1 mol−1 (= cd sr kg−1 ) m−2 s3 ) driver for including both h and e in the list of seven defining constants [6]. These constants replaced the mass of the international prototype (which is not a physical constant but was treated as one), and the permeability of vacuum, μ0 . Another example gives some idea of why the Planck constant is essential to model observed properties of the microscopic world. This example is Bohr's primitive model of the hydrogen atom. The Primitive Bohr Model and the Fine-Structure Constant: Niels Bohr (1885-1962) once suggested that the electron of the hydrogen atom orbits the relatively massive proton in analogy to a small planet of mass m in a circular orbit around a massive sun of mass M. In the planetary model, the attractive force is gravitational and given by F=GMm/r2 attractive force balances a centrifugal force given by F=mv2 . For a circular orbit, the /r, where r is the separation between the centers of the two masses and v is the velocity of m. In the hydrogen-atom analogy, the dominant attractive mv r e /(4 : e0 v2 22/4π r2 force between the proton and its orbiting electron is electrostatic; Coulomb's law applies. For the SI electrical analog, F=e2 ), and this force balances the centrifugal force F=me /r on the electron in a circular orbit. Equating the righthand sides of these two equations and multiplying both sides by r2

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