Instrumentation & Measurement Magazine 26-1 - 7

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
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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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