Instrumentation & Measurement Magazine 26-1 - 10

fundamentalsmeasurement continued
of
Fig. 3. Array V is shown in gray. The numbers are exponents of the units
shown to the left and are shown in (8).
1Ω follows from division by the numerical coefficient of Ω, and
agrees with the result obtained by matrix multiplication. In the
present SI, all units are exactly defined in terms of one or more
of the seven defining constants.
Natural Unit Systems
Returning to [1], one notes the long history of defining " natural "
unit systems. The original goal of these was to uncover
some deeper truths about the universe rather than to facilitate
precision measurements. In any case, natural unit systems
never have units that lend themselves for use in our daily lives.
The Planck Units
One venerable natural-unit system is comprised of the socalled
Planck units [1], [7]. The Planck units for the dimensions
of mass, length and time are constructed from three fundamental
constants that were introduced, above: the Newtonian
constant of gravitation, G, the speed of light in vacuum, c,
and the Planck constant divided by 2π, . The exact numerical
value of each of these is taken to be 1 when expressed in its
Planck unit. We can therefore construct the Planck time tP
, the
Planck length lP
, and the Planck mass mp
by dimensional analysis,
and these can then be converted to the corresponding
values in SI. (Planck also included the Boltzmann constant k to
define a natural temperature, TP
; k also appears in Table 1 and
is needed to define the SI unit of temperature, the kelvin.) The
first matrix, M, is shown in Fig. 4. Matrix M is then inverted.
The result, M−1
, is shown in Fig. 5.
Fig. 5. Matrix M−1
columns:
m cG 
P
P
 /
l Gc   /
3
P
1/2
1/2
for Planck units, in analogy to Fig. 2.
The Planck units are revealed by the exponents in the three
t Gc   /
5
1/2
To convert to SI units, simply substitute SI values for the
three constants. The most accurate results have been tabulated
in [7] and are approximately:
mp = 2.2 × 10−8
lp = 1.6 × 10−35
kg
m
tp = 5.4 × 10−44
s
Theoretical physicists have pondered the meaning of these
units. For instance, does the Planck length indicate the distance
over which space itself is " quantized " ? Or, why is the Planck
mass big enough to be measured in a laboratory while the Planck
length and time obviously are not? As an aside, the factor of 1040
in
the present definition of the kilogram (7) is not a topic of learned
debate among theoretical physicists, and for good reason: ΔvCs
" an atomic parameter " [2] rather than a fundamental constant.
is
A Natural Unit System with a Single
Measurement Unit
There is a natural system that physicists in many fields prefer
[1], [3]. The system has a single base unit, the electronvolt (eV),
which is the unit of energy, E (dimension E). Other dimensions
are powers of E and other units are powers of eV corresponding
to the dimensions. (The eV is not the coherent SI unit of energy,
but the exact conversion to joules is given in section 4 of [2]. The
numerical conversion factor is simply {e}SI
which is found in
Table 1.) Crucially, the natural constants  and c have been chosen
to have dimension 1 (E0
) and exact numerical value equal to
Fig. 4. Matrix M for Planck units, in analogy to Fig. 1.
10
1. At the same time,  and c remain natural constants of angular
momentum and velocity, respectively. Values of mass (meV
IEEE Instrumentation & Measurement Magazine
),
February 2023
Instrumentation & Measurement Magazine 26-1

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