natural_units

The Natural Units convention file for GNU units.

TL;DR

Debian/Ubuntu

$ apt-get install units

$ units -V   # check that you are running **GNU Units**, not BSD units.

GNU Units version 2.1x
with readline, with utf8, locale en_US
Units data file is '/usr/share/units/definitions.units'

$ units -f natural.units

113 units, 42 prefixes, 0 nonlinear units
You have: 1m
You want: s
    1m = 3.335641e-09 s
    1m = (1 / 2.9979246e+08) s

macOS + Homebrew

$ brew install gnu-units

$ gunits -V   # double-check that you are running **GNU Units**, not BSD units.

GNU Units version 2.1x

$ gunits -vf natural.units

113 units, 42 prefixes, 0 nonlinear units

You have: 1 / fm
You want: GeV
    1 / fm = 0.19732696 GeV
    1 / fm = (1 / 5.0677312) GeV

Overview

natural.units is a definition file of the natural units for GNU units.

This is only for GNU units, not for BSD units pre-installed in macOS. You can check version by units -v (for BSD units) or units -V (for GNU units).

Definition

In the natural units, all the units are given in terms of energy (giga electron volt, GeV) with defining c = ℏ = k_B = 1, where c, ℏ and k_B are respectively the speed of light, the Planck constant and the Boltzmann constant.

For electromagnetism, we use the Lorentz-Heaviside convention, where the vacuum permittivity, ε₀, and the vacuum permeability, μ₀, are defined by μ₀ = c²/ε₀ = 1. The fine structure constant is given by α = e^2/4π, where the elementary charge is given by e = 1.60×10^-19 C = 0.303.

Data Source

• CODATA recommended values of the fundamental physical constants: 2018, Rev. Mod. Phys. 93 (2021) 025010, https://physics.nist.gov/cuu/Constants/
• Particle Data Group, Review of Particle Physics 2022, to be published in Prog. Theor. Exp. Phys. 2022 (2022) 083C01, https://pdg.lbl.gov/

Notes

Planck mass is symbolized by M_pl, while reduced Planck mass is by M. Light year (ly) is defined as c × 365.25day following NIST publication 811, while yr denotes tropicalyear as is in PDG table.

How to use

1. Install GNU units.
2. Prepare the file natural.units.
3. Run and convert!

Installation of GNU units

You can install GNU Units with apt-get (Debian/Ubuntu) or via Homebrew (macOS) by

apt-get install units   # Debian / Ubuntu
brew install gnu-units  # macOS

You should check that you are running GNU units, not BSD units (pre-installed in macOS), by

$ units -V   # Debian / Ubuntu + apt-get
$ gunits -V  # macOS + Homebrew

GNU Units version 2.1x

Run with natural.units

Run GNU Units with the file natural.units, downloaded from this GitHub repository:

units -f natural.units    # Debian / Ubuntu + apt-get
gunits -f natural.units   # macOS + Homebrew

Troubleshooting

You will encounter error message below if you are running BSD Units:

units: redefinition of unit '#' on line 2 ignored
units: redefinition of unit '#' on line 3 ignored
units: redefinition of unit '#' on line 3 ignored
units: unexpected end of unit on line 3
units: redefinition of unit '#' on line 4 ignored
...

In this case, you have to recheck that you have installed GNU units and that you are using the correct command.

Examples and some validations

Basic examples

$ units -f natural.units

You have: 1s
You want: m
    * 2.9979246e+08
    / 3.335641e-09
You have: 100 fb
You want: /GeV^2
    * 2.5681899e-10
    / 3.893793e+09
You have: 3 pc
You want: m
    * 9.2570327e+16
    / 1.0802598e-17
You have: 4.2K
You want:
    Definition: 3.6192796e-13 GeV
You have: 200 fm
You want: MeV
    * 1.0135462
    / 0.98663482
You have: 200 fm
You want: MeV
    reciprocal conversion
    * 0.98663482
    / 1.0135462
You have: 1 Mpl
You want: m
    reciprocal conversion
    * 1.6161943e-35
You have: sqrt(4 pi epsilon0)
You want: C
    * 1.8755459e-18
    / 5.331781e+17

Basic validations

$ units -f natural.units

You have: 4pi hbar c / e^2
You want:
    Definition: 137.036

You have: 4pi/e^2
You want:
    Definition: 137.03599

You have: c
You want:
    Definition: 2.99792458e8 m/s = 1

You have: hbar
You want:
    Definition: 1.054571726e-34 J s = 1

You have: e
You want:
    Definition: 1.602176565e-19 C = 0.30282213

You have: kB
You want:
    Definition: 1.3806488e-23 J/K = 1

You have: mu0
You want:
    Definition: 4 pi 1e-7 H/m = 1

You have: epsilon0
You want:
    Definition: 1/mu0 c^2 = 1

More validations

See the Appendix of Peskin's QFT book.

$ units -f natural.units

You have: 1 GeV
You want: g
    * 1.7826618e-24

You have: 1/GeV
You want: fm
    * 0.19732696

You have: 1/GeV^2
You want: mb
    * 0.3893793

You have: 1 e V / m
You want:
    Definition: 1.9732696e-25 GeV^2

You have: 1 e tesla
You want:
    Definition: 5.9157135e-17 GeV^2

Author

Sho Iwamoto @ https://www.misho-web.com/