Population Inversion - Boltzmann Distributions and Thermal Equilibrium

Boltzmann Distributions and Thermal Equilibrium

To understand the concept of a population inversion, it is necessary to understand some thermodynamics and the way that light interacts with matter. To do so, it is useful to consider a very simple assembly of atoms forming a laser medium.

Assume there are a group of N atoms, each of which is capable of being in one of two energy states, either

1. The ground state, with energy E₁; or
2. The excited state, with energy E₂, with E₂ > E₁.

The number of these atoms which are in the ground state is given by N₁, and the number in the excited state N₂. Since there are N atoms in total,

The energy difference between the two states, given by

determines the characteristic frequency ν₁₂ of light which will interact with the atoms; This is given by the relation

h being Planck's constant.

If the group of atoms is in thermal equilibrium, it can be shown from thermodynamics that the ratio of the number of atoms in each state is given by a Boltzmann distribution:

where T is the thermodynamic temperature of the group of atoms, and k is Boltzmann's constant.

We may calculate the ratio of the populations of the two states at room temperature (T ≈ 300 K) for an energy difference ΔE that corresponds to light of a frequency corresponding to visible light (ν ≈ 5×1014 Hz). In this case ΔE = E₂ - E₁ ≈ 2.07 eV, and kT ≈ 0.026 eV. Since E₂ - E₁ ≫ kT, it follows that the argument of the exponential in the equation above is a large negative number, and as such N₂/N₁ is vanishingly small; i.e., there are almost no atoms in the excited state. When in thermal equilibrium, then, it is seen that the lower energy state is more populated than the higher energy state, and this is the normal state of the system. As T increases, the number of electrons in the high-energy state (N₂) increases, but N₂ never exceeds N₁ for a system at thermal equilibrium; rather, at infinite temperature, the populations N₂ and N₁ become equal. In other words, a population inversion (N₂/N₁ > 1) can never exist for a system at thermal equilibrium. To achieve population inversion therefore requires pushing the system into a non-equilibrated state.