Rami Arieli: "The Laser Adventure" Chapter 2.5 page 1
2.5 Thermodynamic (Thermal) Equilibrium

From thermodynamics we know that a collection of atoms, at a temperature T [0K], in thermodynamic equilibrium with its surrounding, is distributed so that at each energy level there is on the average a certain number of atoms.
The number of atoms (Ni) at specific energy level (Ei) is called Population Number.

The Boltzmann equation determines the relation between the population number of a specific energy level and the temperature:

Ni = const * exp (-Ei/kT)
Ni = Population Number = number of atoms per unit volume at certain energy level Ei.

k  = Boltzmann constant:  k = 1.38*1023 [Joule/0K].

Ei = Energy of level i. We assume that Ei> Ei-1.

Const = proportionality constant. It is not important when we consider population of one level compared to the population of another level as we shall see shortly.

T = Temperature in degrees Kelvin [0K] (Absolute Temperature).

The Boltzmann equation shows the dependence of the population number (Ni) on the energy level (Ei) at a temperature T.
From this equation we see that:

1. The higher the temperature, the higher the population number.

2. The higher the energy level, the lower the population number.