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

**PURPOSE:**

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

**HISTORICAL METRIC UNIT STANDARDS:**

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 Earth's equator and its 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 and relative motion of observers. For example, clocks on mountain tops run faster than clocks at sea level. A further problem is that 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.

Today we have adopted a new set of metric unit standards which approximately equal to the old set of standards but which will be stable over time and will remain universally accessible. The contemplated 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.

**NEW METRIC UNIT STANDARDS:**

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 Alpha. 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 time unit standard.

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

Incorporated in the Planck Constant h is the Fine Structure constant Alpha. The new system of units chooses a value for h which makes new kg approximately equal to old kg. Due to the change in h required 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 slight changes to the vacuum permiability Muo and and vacuum permittivity Epsilono. There is a complication that Alpha is not unique. Alpha takes a series of close values that are only unique if the measured system is in the same quantum state as the system used to measure Alpha. This quantum state is dependent upon the environmental conditions. For example Alpha for a free electron in a vacuum may differ slightly from Alpha for an electron in a solid. Hence if the Fine Structure constant Alpha is used as a standard it is essential that the quantum state of the Alpha measurement system be specified.

**DISTANCE AND TIME:**

In a field free vacuum the relationship between electromagnetic wavelength Lamda and frequency F is given by:

(Lamda F) = C

where:

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**

Today:

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.

**ENERGY:**

Planck and Einstein found follow the relationship:

Ep = h Fp

where h is a constant which is actually a composite of other 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

or

M = E / C^2

or

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.

**CHOICE OF h AND Alpha:**

The fine structure constant Alpha is defined by:

Muo C Q^2 = 2 h Alpha

In the 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 = 137.035999**

Note that the measured value of Alpha is slightly dependent on the system quantum state that can vary depending on the system environment. For example, the quantum state of a free electron in a vacuum may differ from the quantum state of a free electron in a metal which may differ again from the quantum state of an electron in a superconductor.

As set out on the web page titled: PLANCK CONSTANT AND FINE STRUCTURE CONSTANT the quantum state is specified by two numbers, Np and Nt. The most common quantum states are: Np/Nt = 223/303 and 222/305. However, the theoretically possible Np/Nt states include but are not limited to:

226/297, 225/299, 224/301, 223/303, 222/305, 221/307, 220/309, 219/ 311, 218/313, 217/315

Note that in a particular quantum state the integers Np and Nt cannot have common factors other than one. For the purpose of unit definition the appropriate quantum states are 222/305 and 223/303 which for a spheromak shape parameter of:

So = 2.026 give:

(1 / Alpha) = 137.035999

**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 (137.035999)]

= 1.25855769 X 10^-6 J-s^2 / m coul^2

= **12.5853357 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

**Epsilono:**

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 (137.035999)]

/ [2 (6.636070150 X 10^-34 J-s) (299,792,458 m / s)]

= **8.84084529 X 10^-12 Coul^2 / J-m**

This web page last updated September 16, 2018.

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