Editing Population inversion (section)

== 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 [[atom]]s forming a [[active laser medium|laser medium]].

Assume there is a group of ''N'' atoms, each of which is capable of being in one of two [[energy state]]s: either
# The ''ground state'', with energy ''E''<sub>1</sub>; or
# The ''excited state'', with energy ''E''<sub>2</sub>, with ''E''<sub>2</sub> &gt; ''E''<sub>1</sub>.
The number of these atoms which are in the [[ground state]] is given by ''N''<sub>1</sub>, and the number in the excited state ''N''<sub>2</sub>. Since there are ''N'' atoms in total,
: <math>N_1+N_2 = N</math>
The energy difference between the two states, given by
: <math>\Delta E_{12} = E_2-E_1,</math>
determines the characteristic [[frequency]] <math display="inline">\nu_{12}</math> of light which will interact with the atoms; This is given by the relation
: <math>E_2-E_1 = \Delta E_{12} = h\nu_{12},</math>
''h'' being the [[Planck constant]].

If the group of atoms is in [[thermal equilibrium]], it can be shown from [[Maxwell–Boltzmann statistics]] that the ratio of the number of atoms in each state is given by the ratio of two [[Boltzmann distribution]]s, the Boltzmann factor:
: <math>\frac{N_2}{N_1} = \frac{g_2}{g_1}\exp{\frac{-(E_2-E_1)}{kT}},</math>
where ''T'' is the [[thermodynamic temperature]] of the group of atoms, ''k'' is the [[Boltzmann constant]] and ''g''<sub>1</sub> and ''g''<sub>2</sub> are the [[Degenerate energy levels|degeneracies]] of each state.

Calculable is the ratio of the populations of the two states at [[room temperature]] (''T''&nbsp;≈&nbsp;300&nbsp;[[kelvin|K]]) for an energy difference Δ''E'' that corresponds to light of a frequency corresponding to visible light (''ν''&nbsp;≈&nbsp;{{val|5|e=14|u=Hz}}). In this case Δ''E'' = {{nowrap|''E''<sub>2</sub> − ''E''<sub>1</sub>}} ≈ 2.07&nbsp;eV, and ''kT'' ≈ 0.026&nbsp;eV. Since {{nowrap|''E''<sub>2</sub> − ''E''<sub>1</sub> ≫ ''kT''}}, it follows that the argument of the exponential in the equation above is a large negative number, and as such ''N''<sub>2</sub>/''N''<sub>1</sub> 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 (''N''<sub>2</sub>) increases, but ''N''<sub>2</sub> never exceeds ''N''<sub>1</sub> for a system at thermal equilibrium; rather, at infinite temperature, the populations ''N''<sub>2</sub> and ''N''<sub>1</sub> become equal. In other words, a population inversion ({{nowrap|''N''<sub>2</sub>/''N''<sub>1</sub> &gt; 1}}) can never exist for a system at thermal equilibrium. To achieve population inversion therefore requires pushing the system into a non-equilibrated state.