Natural units are based on universal physical constants. The units are defined so that the relevant physical constant becomes one (unity, that is). The downside is that potentially there is loss of understanding when the constants become irrelevant to statement of physical laws. Some claim this whole topic is a mind game; others say it is the only sensible way to define units. I don’t think the jury has even been chosen.

This page may be a bit extreme for those in Y11 and below, though the exercises are doable.

An obvious example is c, the speed of light. Picking units so that c=1 turns E=Mc² into E=m. It means that walking speed is around 5x10^{-11} and the speed of sound is around 10^{-8}. Fans or supporters of such an idea are enthused by the loss of the need for units; the opposite view says that, if we don’t have the unit, we still need to supply terminology to put the number into context. So 5x10^{-11} = 50x10^{-12} and so a walking speed might be 50 pico-cee, while sound would be 10 nano-cee.

There are several sets of these natural units and they have particular uses as explained below. However, before you lose touch with reality entirely, it is more appropriate to refer to the radian, which struck me as a special case at the time of first reading about these. I’ll put that differently; radians are the only natural unit we use at school. calling it the natural unit for angle (which it is). If you have academic access, read this.

The whole point about the radian is that it is dimensionless. Besides making loads of trig-type maths work nicely, the radian is that angle that is subtended by the arc of a circle whose length is the same as the radius of the arc itself.

We generally define our SI units by counting something:

• a __ second__ is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom”.

• a __ metre__ is then defined so that lightspeed is 299 792 458 m/s exactly, though I far preferred the 1960-1983 definition, counting 1 650 763.73 wavelengths of the orange-red emission line in the electromagnetic spectrum of the krypton-86 atom in a vacuum I note from wikipedia that a realisation of the metre is usually delineated (not defined) today in labs as 1579800.762042(33) wavelengths of helium-neon laser light in a vacuum, the error (33) stated being only that of frequency determination. 13 sf is pretty good, better than the nine of the other two systems.

• a kilogram is still measured against a prototype and since 2005 there has been intention to change this to a definition in terms of fundamentals, perhaps the Planck constant. See future definition on the wiki page, or here. I think it is likely to be slightly circular; the kilogram will be such that the Planck constant is 6.62607015x10^{-34} Joule seconds. [Js = kg m² s^{-1}]. So the kilogram would be dependent upon the other two definitions, of second and metre.

A fair article on this topic is here. I found this wikipedia article excellent. What we have need of is unts we can define with sufficient precision, preferably so that the % error in the measurement is of the order of 10^{-8}, but we would like better still. If you find this interesting, follow the BIPM, where you will find the current discussion of proposed changes. Including the problems of cleaning the prototype kilogram, which might make a good short story (challenge). You might find it interesting to read up on the Watt balance and the Avogadro Project (to count the number of atoms of Carbon_{12} in a kilogram, thus providing a definition to better than 2x10^{-8}. By comparison, the second is measured accurate to somethiing of the order of 10^{-16}. Many systems are working to within 10^{-12} second error across a year. Because the time measurement is so good, that work tends to define the precision work on other units. The sense in which the metre is next best is what causes the trend to be to define the other needed fundamental units to be defined in terms of the second and the metre, as above for the kilogram.

**Natural unit systems.**

There are several sets of units. In particle Physics, wikipedia tells me, the phrase "natural units" generally means: h=c=kB=1where h is the reduced Planck constant, c is the speed of light, and kB is the Boltzmann constant.

__ Planck units__ are similar but we add the gravitational constant G=1. Length, L

_{p}is then such that L²=hG/c³, so one planck length is 1.616x10

^{-35}m, or a metre is around 6.168x10

^{34}. A planck mass, m

_{p}, is m

_{p}²=hc/G, so one is 2.176x10

^{-8}kg and 1kg is 4.596x10⁷. Planck time, t

_{p}²=hG/c⁵, so 1=5.39x10

^{-44}, which means I’m around 3.5x10

^{52}in age. For temperature, T

_{p}²=hc⁵/GK², so 1 (Planck temperature unit) is 1.417x10

^{32}K, which makes room temperature a bit low, at around 2x10

^{-30}.

I have a regular problem with units of charge, since I don’t understand what the fine-structure constant, α, might be. Besides being 0.007297.

__ Stoney units__ set G=c=kB=e=1, where e is the elementary charge and h=1/α. Stoney (wikipedia again) didn’t propose the Planck constant in 1874, since the Planck constant was not discovered (is that the right word?) until 1900.

__ Atomic (Hartree) units __say e=me=h=KB=1 and c=1/α. In these units, the speed of light is 137, G is 10-45. I’m 7.7x10

^{34}mass units and 7.825x10

^{25}time units.

So these things look like they have sensible application in large scale and small scale physics, but not a lot of use in ‘real life’.

*Planck constant (reduced)*: the quantum of angular momentum in quantum mechanics. E=hv connects the energy of a photon and the frequency of its elctromagnetic wave. Using λv=c, we could write Eλ=hc. In traditional units, h=6.626 x10^{-34} Joule seconds and the reduced one is that value divided by 2π, or 1.054x10^{-34} Js. Planck’s constant is also called the __ quantum of action__, which (action) is something I want to chase down and attempt to understand.

Boltzmann constant (not understood, at any deep level, by DJS) Wikipedia says: Boltzmann's constant, k, is a bridge between macroscopic and microscopic physics. Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure p and volume V is proportional to the product of amount of substance n (in moles) and absolute temperature T: p V = n R T

[which is GCSE chemistry] where R is the gas constant (8.3144621(75) J⋅K−1⋅mol^{−1}[1]). Introducing the Boltzmann constant (k) transforms the ideal gas law into an alternative form:

p V = N k T ,

where N is the number of molecules of gas. For n = 1 mol, N is equal to the number of particles in one mole (Avogadro's number).

Boltzmann constant = 1.38064852 × 10^{-23} m² kg s^{-2} K^{-1} [which is J/K. I assume K refers to Kelvin and I note a proposal to change the definition so that Boltzmann’s constant becomes exactly 1.38065x10^{-23}.

*F*__ ine-structure constan__t, α. Straight from wikipedia, though I re-typed the formulae:

Three equivalent definitions of α in terms of other fundamental physical constants are:

where:

• e is the elementary charge;

• ħ = h/2π is the reduced Planck constant;

• c is the speed of light in vacuum;

• ε_{0} is the electric constant or permittivity of free space;

• or µ_{0} is the magnetic constant or permeability of free space;

• or k_{e} is the Coulomb constant.

In electrostatic cgs units, the unit of electric charge, the statcoulomb, is defined so that the Coulomb constant, k_{e}, or the permittivity factor, 4πε_{0}, is 1 and dimensionless. Then the expression of the fine-structure constant becomes the abbreviated α = e²/ħc which is an expression commonly appearing in physics literature.

In natural units, commonly used in high energy physics, where ε_{0} = c = ħ = 1, the value of the fine-structure constant is α=e²/4π. As such, the fine-structure constant is just an alternative expression of the elementary charge; e=√4πα ≈0.30282212 in terms of the natural unit of charge.

α = 7.2973525698x10^{-3}. Often used instead is α^{-1} = 137.035999074.

Exercise:

1. Light speed is 3x10^{10}m/s. Walking at 5kph is what in m/s? And as a decimal part of c? Now check the speed of sound 330m/s in terms of c.

2. Give your own definitions for Boltzmann and light. And then look them up properly.

3. Using Planck units, convert

(i) g=9.807 m/s/s

(ii) the speed of sound at 330m/s

(iii) atmospheric pressure of 1000 millibars (that’s one bar, surely?)

(iv) you pick something different from these three.