Natural Units (Units: A Unit Conversion Program and Scientific Calculator)

### 8.2 Natural Units ¶

Like the CGS units, “natural” units are an alternative to the SI system used primarily physicists in different fields, with different systems tailored to different fields of study. These systems are “natural” because the base measurements are defined using physical constants instead of arbitrary values such as the meter or second. In different branches of physics, different physical constants are more fundamental, which has given rise to a variety of incompatible natural unit systems.

The supported systems are the “natural” units (which seem to have no better name) used in high energy physics and cosmology, the Planck units, often used by scientists working with gravity, and the Hartree atomic units are favored by those working in physical chemistry and condensed matter physics.

You can select the various natural units using the `--units` option in the same way that you select the CGS units. The “natural” units come in two types, a rationalized system derived from the Heaviside–Lorentz units and an unrationalized system derived from the Gaussian system. You can select these using ‘`natural`’ and ‘`natural-gauss`’ respectively. For conversions in SI mode, several unit names starting with ‘`natural`’ are available. This “natural” system is defined by setting \({\hbar}\) , \({c}\) and the Boltzman constant to 1. Only a single base unit remains: the electron volt.

The Planck units exist in a variety of forms, and `units` supports two. Both supported forms are rationalized, in that factors of \({4\pi}\) do not appear in Maxwell’s equations. However, Planck units can also differ based on how the gravitational constant is treated. This system is similar to the natural units in that \({c}\) , \({\hbar}\) , and Boltzman’s constant are set to 1, but in this system, Newton’s gravitational constant, \({G}\) , is also fixed. In the “reduced” Planck system, \({8 \pi G=1}\) whereas in the unreduced system \({G = 1}\) . The reduced system eliminates factors of \({8 \pi}\) from the Einstein field equations for gravitation, so this is similar to the process of forming rationalized units to simplify Maxwell’s equations. To obtain the unreduced system use the name ‘`planck`’ and for the reduced Planck units, ‘`planck-red`’. Units such as ‘`planckenergy`’ and ‘`planckenergy_red`’ enable you to convert the unreduced and reduced Planck energy unit in SI mode between the various systems. In Planck units, all measurements are dimensionless.

The final natural unit system is the Hartree atomic units. Like the Planck units, all measurements in the Hartree units are dimensionless, but this system is defined by defined from completely different physical constants: the electron mass, Planck’s constant, the electron charge, and the Coulomb constant are the defining physical quantities, which are all set to unity. To invoke this system with the `--units` option use the name ‘`hartree`’.