# Elementary Particle Populations

When the temperature is well above a threshold temperature defined by

kTthreshold = mc2

then particles may be treated statistically. Here k is the Boltzmann constant, and this definition of threshold temperature involves setting the average thermal energy equal to the rest mass energy. When the temperature is much higher than the threshold temperature, then one can presume that particle-antiparticle pairs can be freely created from the thermal energy, and will therefore exist in thermal equilibrium with their surroundings. The contribution of a given type of particle will then be proportional to the "effective number of species" of that particle.

 Particle Symbol Rest energy (MeV) Threshold temperature (109 K) Effective number of species Mean lifetime (s) Photon g 0 0 1 x 2 x 1 = 2 Stable Neutrinos ne,ne 0 0 2 x 1 x 7/8 = 7/4 Stable Neutrinos nm,nm 0 0 2 x 1 x 7/8 = 7/4 Stable Electron e-,e+ 0.5110 5.930 2 x 2 x 7/8 = 7/2 Stable Muon m-,m+ 105.66 1226.2 2 x 2 x 7/8 = 7/2 2.197 x 10-6 Pion p0 134.96 1566.2 1 x 1 x 1 = 1 0.8 x 10-16 Pion p+,p- 139.57 1619.7 2 x 1 x 1 = 2 2.60 x 10-8 Proton p,p 938.26 10,888 2 x 2 x 7/8 = 7/2 Stable Neutron n,n 939.55 10,903 2 x 2 x 7/8 = 7/2 920
After Weinberg, First Three Minutes.

The effective number of species is the product of three factors:

1. 2 if the particle has a distinct antiparticle, 1 if not.
2. The number of possible orientations of the particles spin.
3. 7/8 if the particle must obey the Pauli exclusion principle, 1 if not.

The effective number of species is important for the calculation of the energy when a collection of particle is at a high enough temperature so that it can be considered to act like radiation. Under those condition, such as in modeling the early processes in the big bang, the Stefan Boltzmann law can be used to calculate the energy associated with the particles.

 Balance of energy and matter in the universe
Index

Particle concepts

Reference
Weinberg
First Three Minutes
HyperPhysics***** Quantum Physics Go Back