# Planck particle

A Planck particle, named after physicist Max Planck, is a hypothetical particle defined as a tiny black hole whose Compton wavelength is equal to its Schwarzschild radius.[1] Its mass is thus approximately the Planck mass, and its Compton wavelength and Schwarzschild radius are about the Planck length.[2] Planck particles are sometimes used as an exercise to define the Planck mass and Planck length.[3] They play a role in some models of the evolution of the universe during the Planck epoch.[4]

Compared to a proton, for example, the Planck particle would be extremely small (its radius being equal to the Planck length, which is about 10−20 times the proton's radius) and massive (the Planck mass being 1019 times the proton's mass).[5] The Planck particle would also have a very fleeting existence, evaporating due to Hawking radiation after approximately 5×10−39 seconds.

## Derivation

While opinions vary as to its proper definition, the most common definition of a Planck particle is a particle whose Compton wavelength is equal to its Schwarzschild radius. This sets the relationship:

${\displaystyle \lambda ={\frac {h}{mc}}={\frac {2Gm}{c^{2}}}}$

Thus making the mass of such a particle:

${\displaystyle m={\sqrt {\frac {hc}{2G}}}}$

This mass will be ${\displaystyle {\sqrt {\pi }}}$ times larger than the Planck mass, making a Planck particle 1.772 times more massive than the Planck unit mass.

Its radius will be the Compton wavelength:

${\displaystyle r={\frac {h}{mc}}={\sqrt {\frac {2Gh}{c^{3}}}}}$

The Planck length P is defined as

${\displaystyle \ell _{\mathrm {P} }={\frac {r}{2{\sqrt {\pi }}}}={\sqrt {\frac {\hbar G}{c^{3}}}}}$

## Dimensions

Using the above derivations we can substitute the universal constants h, G, and c, and determine physical values for the particle's mass and radius. Assuming this radius represents a sphere of uniform density, we can further determine the particle's volume and density.

Table 1: Physical dimensions of a Planck particle
Parameter Dimension Value in SI units
Mass M 3.85763×10−8 kg