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By Charles Rhodes, P.Eng., Ph.D.

This web page presents both historical and up-to-date definitions of common physical units used on this web site.

Historically a second was defined by the relationship:
(60 seconds / minute) X (60 minutes / hour) X (24 hours / day) X (365 days / year)
Where a year was measured by counting exposures of the sun. Note that due to revolution of Earth around the sun there are 366 exposures to a fixed star per year.

Historically a metre was defined by the distance between two scratches on a metal bar located in a temperature controlled vault in Paris. A metre was intended to be approximately equal to (1 / 10,000,000) of the distance along Earth's surface between the equator and the geometric North Pole.

Historically a kilogram was defined as the mass of a chunk of metal stored in a vault in Paris.

During the later part of the 20th century it became increasingly apparent that these fundamental unit definitions were not sufficiently stable. Moreover it was shown that the perception of time depends on the path of an observer. For example, clocks on mountain tops run faster than clocks at sea level. Also the period of rotation of planet Earth changes over time.

Scratches on a metal bar lack the position resolution required for really accurate linear dimension measurements.

A standard mass in a storage vault potentially loses mass through corrosion and handling every time it is used as a mass comparison reference.

What is needed is a new set of metric unit standards which approximately equal to the old set of standards but which would be stable over time and universally accessible. The proposed new standards assign values to natural physical constants rather than to physical objects that must be stored in a vault and which are subject to deterioration with usage.

a) Speed of light in a vacuum = 299,792,458 m / s;
b) Ce-133 ground state hyperfine transition frequency Fc = 9,192,631,770 Hz;
c) Luminous efficacy of monochromatic 540 X 10^12 Hz radiation = 683 lumen / watt (lm / W);
d) Planck constant = h = 6.62607015 X 10^-34 joule-second (J-s);
e) Quantum charge Q = 1.602176634 X 10^-19 coulomb (C);
f) Boltzmann constant k = 1.380649 X 10^-23 J / deg K;
g) Avogadro number Na = 6.02214076 X 10^23 / mole.

A second object is to minimize uncertainty issues in data reporting relating to the fine structure constant. The solution to this problem is to express energy and mass units in terms of a fixed Planck Constant h so that the effect of the fine structure constant Alpha is clear.

Certain atoms absorb and emit electromagnetic radiation at specific frequencies. In a solar spectrum these absorption/emission frequencies appear as dark lines in the spectrum. One spectral line is very narrow and is due to a ground state hyperfine electron energy transition in the isotope cesium-133. For several practical reasons this particular energy transition of cesium-133 is convenient for use as a unit standard.

The speed of light C in combination with the cesium ground state hyperfine energy transition frequency defines a metre.

Incorporated in the Planck Constant h is the fine structure constant Alpha. The new system of units chooses a values for h which make new kg approximately equal to old kg. Due to the change in h to accomplish this goal there is a slight change in the size of a quantum charge Q and hence in the definition of a coulomb. There are further changes to vacuum permiability Muo and and vacuum permittivity Epsilono.

In a field free vacuum the relationship between electromagnetic wavelength Lamda and frequency F is given by:
(Lamda F) = C
C = speed of light.

For the hyperfine transition of cesium-133:
(Lamdac Fc) = C

The wavelength Lamdac of cesium as a fraction of a metre could in principle be measured using a geometrically precise diffusion grating.

Thus a metre can be redefined as (Na Lamdac), where Na is a number chosen to make the redefined metre closely match the historic metre. Thus:
Lamdac = (1 m / Na)

A second can be redefined as a number Nb of cycles of frequency Fc where Nb is a number chosen to make the redefined second closely match the historic second. Thus:
Fc = (Nb / 1 second)
Today the value of Nb is set at exactly:
Nb = 9,192,631,770
which value defines a second. If the cesium is at 0 degrees K in a field free enclosure the precision of this frequency is better than 1 second in 300 years.
[1 s / (3600 s / hour X 8766 hour / year X 300 years)]
= 1.05 X 10^-10

The the speed of light C is given by:
C = [Lamdac Fc]
= [(1 m / Na) (Nb / 1 second)]
= (Nb / Na) m / s

C = (Nb / Na)
is set at exactly:
C = (Nb / Na) = 299,792,458 m / s

A basic hypothesis of special relativity is that the value of:
C = (Nb / Na)
is the same for all inertial observers.

Thus 1 m is defined by:
1 m = Na Lamdac
= Na (C / Fc)
= Na (Nb / Na Fc)
= Nb / Fc

Note that from a practical perspective to optimize the match of new units to historic units it has been most convenient to precisely measure C and Nb using historic units and then determine Na using the formula:
Na = (Nb / second) / C
= (9,192,631,770 / s) / (299,792,458 m / s)
= 30.66331899 / m

Hence in the new units:
Lamdac = (1 m / Na)
= 1 / [(Nb / second) / C] = (299,792,458 m / s) / (9,192,631,770 / s)
= 0.0326122557 m
= 3.26122557 cm
which is in the microwave region.

Planck and Einstein found follow the relationship:
Ep = h Fp
where h is a constant which is actually a composite of othe constants. Thus in circumstances where Fp can be measured Ep can be determined.

The constant h can be experimentally measured or can be calculated from first principles as set out elsewhere on this web site.

The importance of this relationship is that:
E = M C^2
M = E / C^2
dM = dE / C^2
= Ep / C^2
= h dF / C^2

In the case of a cesium-133 hyperfine energy transition:
Ep = h Fc

Thus a change in mass dM can be expressed in terms of the corresponding frequency change dF. A kilogram can be redefined as:
kilograms = h F / C^2
where h is chosen as 6.636070150 X 10^-34 J-s to make the redefined kilogram closely match the historic kilogram. However, changing h in this manner modifies the definitions of quantum charge Q, vacuum permiability Muo and vacuum permittivity Epsilono.

The fine structure constant Alpha is defined by:
Muo C Q^2 = 2 h Alpha

In the proposed new units h is fixed at:
h = 6.636070150 X 10^-34 J-s
in order to match existing kg.

Alpha^-1 is a geometric ratio established on the web page titled: PLANCK CONSTANT AND FINE STRUCTURE CONSTANT to be:
Alphas^-1 = 137.0333724

Q AND Muo:
The quantum charge Q is chosen to be exactly:
Q = 1.602176634 X 10^-19 A s

The vacuum permeability Muo is chosen to be:
Muo = (2 h Alpha) / (C Q^2)
= (2 h) / (C Q^2 Alpha^-1)
= [2 (6.636070150 X 10^-34 J-s)]
/ [(299,792,458 m / s) (1.602176634 X 10^-19 A s)^2 (137.0333724)]
= 1.25855769 X 10^-6 J-s^2 / m coul^2
= 12.5855769 X 10^-7 J-s^2 / m coul^2

For comparison the previous value of Muo was:
Muo = 4 Pi X 10^-7 J-s^2 / m coul^2
= 4 (3.1415926535) X 10^-7 J-s^2 / m coul^2
= 12.56637061 X 10^-7 J-s^2 / m Coul^2

Maxwell found that in a vacuum:
C^2 = 1 / (Muo Epsilono)

The new value of Epsilono is given by:
Epsilono = 1 / (Muo C^2)
= (C Q^2) / (2 h Alpha C^2)
= (Q^2 Alpha^-1) / (2 h C)
= [(1.602176634 X 10^-19 A s)^2 (137.0333724)]
/ [2 (6.636070150 X 10^-34 J-s) (299,792,458 m / s)]
= 8.840675836 X 10^-12 Coul^2 / J-m

This web page last updated July 28, 2018.

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