In physics, specifically statistical mechanics, a population inversion occurs when a system (such as a group of atoms or molecules) exists in a state in which more members of the system are in higher, excited states than in lower, unexcited energy states. It is called an "inversion" because in many familiar and commonly encountered physical systems in thermal equilibrium, this is not possible. This concept is of fundamental importance in laser science because the production of a population inversion is a necessary step in the workings of a standard laser.
Condition 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 is a group of N atoms, each of which is capable of being in one of two energy states: either
The ground state, with energy E1; or 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,
N 1 + N 2 = N {\displaystyle N_{1}+N_{2}=N}
The energy difference between the two states, given by
Δ E 12 = E 2 − E 1 , {\displaystyle \Delta E_{12}=E_{2}-E_{1},}
determines the characteristic frequency ν12 of light which will interact with the atoms; This is given by the relation
E 2 − E 1 = Δ E 12 = h ν 12 , {\displaystyle E_{2}-E_{1}=\Delta E_{12}=h\nu _{12},}
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 distributions, the Boltzmann factor:
N 2 N 1 = g 2 g 1 exp − ( E 2 − E 1 ) k T , {\displaystyle {\frac {N_{2}}{N_{1}}}={\frac {g_{2}}{g_{1}}}\exp {\frac {-(E_{2}-E_{1})}{kT}},}
where T is the thermodynamic temperature of the group of atoms, k is the Boltzmann constant and g1 and g2 are the degeneracies of each state. Calculable is 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.
Interaction of light with matter
There are three types of possible interactions between a system of atoms and light that are of interest:
Absorption
If light (photons) of frequency ν12 passes through the group of atoms, there is a possibility of the light being absorbed by electrons which are in the ground state, which will cause them to be excited to the higher energy state. The rate of absorption is proportional to the radiation density of the light, and also to the number of atoms currently in the ground state, N1.
Spontaneous emission
If atoms are in the excited state, spontaneous decay events to the ground state will occur at a rate proportional to N2, the number of atoms in the excited state. The energy difference between the two states ΔE21 is emitted from the atom as a photon of frequency ν21 as given by the frequency-energy relation above. The photons are emitted stochastically, and there is no fixed phase relationship between photons emitted from a group of excited atoms; in other words, spontaneous emission is incoherent. In the absence of other processes, the number of atoms in the excited state at time t, is given by
N 2 ( t ) = N 2 ( 0 ) exp − t τ 21 , {\displaystyle N_{2}(t)=N_{2}(0)\exp {\frac {-t}{\tau _{21}}},}
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