Everything in Natural Unit (High-Energy Physics)

In high-energy physics, scholars frequently adopt a convention where certain fundamental constants are set equal to one: c = ℏ = k_B = ε₀ = μ₀ = 1 (hence the elementary charge e = (4πα)^1/2 ≈ 0.303) and express other quantities in terms of the units of energy, such as eV (~energy of the visible light), keV (~temperature of the Sun), MeV (~electron mass), and GeV (~nucleon mass). This system of units manifests the significance of relativity and quantum physics. Familiarizing oneself with the relationships between the International System of Units (SI) and these natural units can be beneficial for practitioners in the field.

Basic unit conversion

SI unit → Natural unit	Natural unit → SI unit
1 m = 5.07e15 GeV−1	1 GeV−1 of (reduced) length = 1.97e−16 m ≈ 0.2 fm [E=ℏc/𝜆]
	1 GeV of normal length = 1.24e−15 m [E=2𝛑ℏc/𝜆]
1 kg = 5.61e26 GeV	1 GeV of mass = 1.78e−27 kg
1 sec = 1.52e24 GeV−1	1 GeV−1 of time = 6.58e−25 sec ≈ (2/3)e−24 sec
1 Hz = 6.58e−16 eV	1 eV of angular frequency = 1.52e15 Hz [E=ℏω]
	1 eV of normal frequency = 2.42e14 Hz [E=2𝛑ℏf]
1 K = 8.62e−5 eV	1 eV of temperature = 1.16e4 K
1 J = 6.24e9 GeV	1 GeV of energy = 1.60e−10 J
1 erg = 6.24e2 GeV	1 GeV of energy = 1.60e−3 erg
1 Watt = 4.11e3 eV2	1 eV2 of luminosity = 2.43e−4 Watt
1 erg/s = 4.11e−4 eV2	1 eV2 of luminosity = 2.43e3 erg/s

Useful unit in astrophysics & particle physics

• Gravitational constant G = 6.67e−11 m³kg⁻¹s⁻² = 6.67e−8 cm³g⁻¹s⁻² = 4.30e−6 kpc M_⊙⁻¹(km/s)² = 4.30e−3 pc M_⊙⁻¹(km/s)² = 4.50e−3 pc³ M_⊙⁻¹ Myr^-2 = (1.22e19 GeV)⁻²

• Hubble parameter H₀ = 100 h km sec⁻¹ Mpc⁻¹ = 2.13e−42 h GeV = (2997.9)⁻¹ h Mpc⁻¹ = (3.0857e17)⁻¹ h sec⁻¹ = (9.7778)⁻¹ h Gyr⁻¹

• 1 year = 3.16e7 sec = 4.79e31 GeV⁻¹

• 1 M_⊙ = 1.99e30 kg = 1.99e33 g = 1.12e57 GeV

• 1 L_⊙ = 3.83e33 erg/s = 3.83e26 J/s = 1.57e12 GeV²

• 1 M_⊕ = 5.97e24 kg = 5.97e27 g = 3.35e51 GeV

• Earth radius = 6.37e6 m = 3.23e22 GeV⁻¹ = (3.1e-14 eV)⁻¹

• 1 AU = 1.50e11 m = (1.3e-18 eV)⁻¹

• 1 light year = 9.46e15 m ≈ (1/3) pc = (2.1e-23 eV)⁻¹

• 1 pc = 3.09e16 m = 1.56e32 GeV⁻¹ = (6.4e-24 eV)⁻¹

• 1 kpc = 3.09e19 m = (6.4e-27 eV)⁻¹

• 1 Mpc = 3.09e22 m = (6.4e-30 eV)⁻¹

• Distance to the Galactic center = 8.5 kpc = 1.33e36 GeV⁻¹ = (7.5e-28 eV)⁻¹

• 1 km/s = 1.02 pc/Myr

• GM_⊙ = 1.33e11 km³ s⁻² = (1.3e-10 eV)⁻¹

• GM_⊕ = 3.99e5 km³ s⁻² = (4.4e-5 eV)⁻¹

• Stefan–Boltzmann constant σ = π²kB⁴ /(60 hbar³c²) = 0.164

• Thomson cross section σ_T = 8πα²/3m_e² = 6.65e−25 cm² = 5.52e9 GeV⁻²

• 1 GeV⁻² = 0.389 mb = 3.89e8 pb = 3.89e−28 cm²

• 1 cm² = 1e24 barn = 1e27 mb = 1e36 pb = 1e39 fb

• 1 cm⁻²s⁻¹ = 1e-33 nb⁻¹s⁻¹ ≈ 1e-32 fb⁻¹yr^-1

• 1 GeV cm^-3 = 2.63e-2 M_⊙ pc⁻³ = 2.63e7 M_⊙kpc⁻³ = 7.68e−42 GeV⁴ = 7.68e−6 eV⁴

• 1 M_⊙ kpc⁻³ = 3.80e−8 GeV cm⁻³ = 2.92e−49 GeV⁴ = 2.92e−13 eV⁴

• 1 cm²g^-1 = 1.78e−24 cm²GeV^-1 = 4.58e3 GeV⁻³ = 2.09e−4 pc² M_⊙^-1

• 1 e·cm = 1.53e13 GeV⁻¹ = 1.53e4 eV⁻¹

• 1 ton·day = 7.36e58

• 1 ton·year = 2.69e61

• Atomic binding energy = α² m_e = 27 eV

• Critical density ρ_crit = 3H₀²/(8πG) = 1.88e−26 h² kg m⁻³ = 1.88e−29 h² g cm⁻³ = 8.10e47 h² GeV⁴ = 1.05e4 h² eV cm⁻³ = 2.78e-7 h² M_⊙ pc⁻³ ≈ (0.003 h^1/2 eV)⁴ ≈ (2.5 meV)⁴ = 1.3e11 M_⊙ Mpc⁻³ = 5.1 protons m⁻³

• Vacuum energy density ρ_vaccum = 5.96e−27 kg m⁻³ = 2.56e-11 eV⁴ = (2.25 meV)⁴

• Cosmological constant Λ = 1.11e−52 m⁻² = 4.33e−66 eV²

• The background dark matter density today ρ_bkg,DM = 1.3 keV cm⁻³ = 1e-11 eV⁴

• The local dark matter density ρ_local,DM = 0.4 GeV cm⁻³ = 0.01 M_⊙ pc⁻³ = (0.04 eV)⁴

• The photon number density today n_γ0 = 411 cm⁻³

• The relic neutrino number density today n_ν0 = 3·2·56 cm⁻³ = 3·(3/11)·n_γ0

Useful unit conversion in electromagnetism

SI unit → Natural unit	Natural unit → SI unit
1 Coulomb = 1.89e18	1e of charge = 1.60e−19 Coulomb
1 Ampere = 1.24 keV	1 keV of electric current = 0.804 Ampere
1 Volt = 3.30 eV	1 eV of electric potential = 0.303 Volt
1 Volt/m = 6.52e−7 eV2	1 eV2 of electric field strength = 1.54e6 Volt/m
1 Tesla = 1.95e2 eV2	1 eV2 of magnetic field strength = 5.12e−3 Tesla
1 Gauss = 1.95e−2 eV2	1 eV2 of magnetic field strength = 51.2 Tesla
1 Weber = 5.02e15	1 of magnetic flux = 5.62e-17
1 Ω = 2.65e−3	1 of resistance = 376.72
1 Farad = 5.72e17 eV−1	1 eV-1 of capacitance = 1.75e-18 Farad
1 Henry = 4.03e12 eV−1	1 eV-1 of inductance = 2.48e-13 Henry

Note that the charge, the magnetic flux, and the resistance are unitless under the natural unit.

If you have any questions or comments, please contact Yiming Zhong (yiming.zhong@cityu.edu.hk).