LOG#222. The New SI.

By the time we will find new physics, we have already redefined the SI in terms of base units and fundamental constants.

The definition of the new SI is the next one: the SI is the system in which the following constants are taken to be exact

  • The unperturbed ground state hyperfine splitting (transition) frequency of the caesium-133 atom \Delta f(Cs-133) is exactly

    \[\Delta f=9192631770 Hz\]

Thus, frequency is fundamental, and the time is a base unit from frequency. One second is the time

    \[1s=\dfrac{9192631770}{\Delta f}\]

and 1 Hz is the reciprocal of the above quantity, exactly too.

  • The speed of light in vacuum c is exactly the quantity

    \[c=299792458m\cdot s^{-1}\]

Using the previous and this definition, you can define the meter to be exactly the amount of length

    \[1m=\dfrac{9192631770c}{299792458\Delta f}\]

  • The elementary charge e is exactly the quantity

    \[e=1.602176565\cdot 10^{-19}\]

Thus, the old electric current unit, base unit, the ampère, is the unit in which you can express charge into current, in corresponding units, with the next conversion constants:

    \[1C=\dfrac{e}{1.602176565\cdot 10^{-19}}=6.241509343\cdot 10^8e\]

    \[1A=\dfrac{1C}{1s}=\dfrac{6.241509343\cdot 10^8e\Delta f}{9192631770}=\dfrac{e}{(1.602176565\cdot 10^{-19})(9192631770)}\]

  • The Planck constant is exactly defined to be

    \[h=6.62607015\cdot 10^{-34}J\cdot s\]

and thus the kilogram is defined in terms of fundamental constants as

    \[1kg=\dfrac{(299792458)^2h\Delta f}{(6.62607015\cdot 10^{-34})(9192631770)c^2}\]

  • The Boltzmann constant is exactly

    \[k_B=1.3806488\cdot 10^{-23}J\cdot K^{-1}\]

so (with 1J=1kg\cdot 1m^2\cdot 1s^{-2} and then

    \[1K=\dfrac{1.3806488\cdot 10^{-23}}{k_B}=\dfrac{1.3806488\cdot 10^{-23}h\Delta f}{(6.62607015\cdot 10^{-34})(9192631770)k_B}\]

  • The Avogadro constant is defined exactly to be

    \[N_A=6.02214129\cdot 10^{23}\]

so the mole is

    \[1mole=\dfrac{6.02214129\cdot 10^{23}}{N_A}\]

  • The luminous efficacy K_{cd} of monochromatic radiation of frequency 540 THz is exactly defined to be 683 lumen/(W\cdot sr). Thus, as stereradian is dimensionless, lm=cd\cdot sr and the candela definition holds exactly to be as well

    \[1cd=1\dfrac{lm}{683W\cdot sr}=\dfrac{K_{cd}}{683}kg\cdot m^2\cdot s^{-3}\cdot sr^{-1}=\dfrac{1\cdot \Delta f^2 hK_{cd}}{(9192631770)^2(6.62607015\cdot 10^{-34})683}\]

See you in another blog post!

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