lbj-wade / natural_units

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.

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

Data Source and Convention

Values are taken from PDG Review of Particle Physics 2014.

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.

Author

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