Population inversion

Population inversion ( Latin inversio , reverse ') is a term used in physics of systems (for example, atoms with) that only certain states discrete energies can accept, as determined by the quantum mechanics are described. Population inversion occurs when more particles are in an energetically higher state  E ₂ than in the energetically lower state  E ₁ . It cannot occur in thermal equilibrium .

Situation in thermal equilibrium

In thermal equilibrium according to the Boltzmann distribution , if a uniform temperature is assumed: ${\displaystyle T}$

${\displaystyle {\frac {N_{2}}{N_{1}}}={\frac {g_{2}}{g_{1}}}\cdot \exp \left(-{\frac {E_{2}-E_{1}}{k_{\mathrm {B} }\cdot T}}\right)}$

With

• ${\displaystyle N_{1}}$the density of the particles in the lower state
• ${\displaystyle N_{2}}$ the density of the particles in the upper state
• ${\displaystyle g_{i}}$the statistical weights of the states
• ${\displaystyle E_{i}}$ their energy and
• ${\displaystyle k_{\mathrm {B} }}$the Boltzmann constant .

Since the energy gap between two levels is always greater than 0:

${\displaystyle \Delta E=E_{2}-E_{1}>0,}$

the exponential function can never be greater than 1:

${\displaystyle \Rightarrow -{\frac {E_{2}-E_{1}}{k_{\mathrm {B} }\cdot T}}<0\qquad \Rightarrow \exp \left(-{\frac {E_{2}-E_{1}}{k_{\mathrm {B} }\cdot T}}\right)<1}$

Thus in the natural equilibrium there are fewer particles in a higher energy state than in the lower energy state:

${\displaystyle \Rightarrow {\frac {g_{2}}{g_{1}}}\cdot \exp \left(-{\frac {E_{2}-E_{1}}{k_{\mathrm {B} }\cdot T}}\right)<{\frac {g_{2}}{g_{1}}}\qquad \Rightarrow {\frac {N_{2}}{N_{1}}}<{\frac {g_{2}}{g_{1}}}}$

Generating an inversion by pumping

It follows that a population inversion

${\displaystyle {\frac {N_{2}}{N_{1}}}>{\frac {g_{2}}{g_{1}}}}$

can only exist when the system is not in thermal equilibrium.

Every system strives to maximize its entropy , i.e. to minimize its free energy . The population inversion represents a deviation from the local thermodynamic equilibrium and is therefore not stable. It can therefore only be artificially brought about and maintained in non-equilibrium systems with a constant supply of energy, known as pumping. Pumping must be selective; H. only certain particles may be supplied with energy. This means that selected levels are occupied more strongly than would be the case in natural equilibrium.

If the excitation source (e.g. optical pumping, gas discharge ) is switched off, the thermal overpopulation of the inverted electronic state is broken down by emission and collisions with other atoms or molecules . The local thermal equilibrium is reached when excited electronic states, degree of ionization , and the kinetic energy of the atoms / molecules of the Boltzmann statistics are correspondingly distributed again. Depending on the lifespan of the states and the particle density in the system, the process can take a few milliseconds .

Optical pumping

A common type of pumping is optical pumping , using flash lamps or the radiation from other lasers. The radiation from the pump source must be more energetic than the light that is later emitted by the laser pumped with it. Population inversion is achieved when the energy difference between the basic and a higher excited electronic state of the particle and the photon energy of the pump source match. The energy of a photon is the product of its frequency and Planck's constant h : ${\displaystyle \nu }$

${\displaystyle E=h\cdot \nu .}$

Shock pumps

Another form of selective excitation is the collision with another excited particle B, which can exchange the energy difference through de-excitation in order to bring the first particle A into the more highly excited state instead. In order to bring the particles of type B back into the excited state after the shock de-excitation, energy, e.g. B. by electron impacts supplied (see He-Ne laser ). The energy can be introduced into the medium in the form of an electrical discharge (e.g. glow discharge , hollow cathode , microwaves ).

laser

→ Main article : Laser

Excitation scheme of a 4-level laser with the excitation level that empties quickly and without radiation to the laser level . After being cleared by stimulated emission, the electrons quickly change from to the ground state , from which they are pumped into again. ${\displaystyle E_{P}}$ ${\displaystyle E_{M}}$ ${\displaystyle E_{L}}$ ${\displaystyle E_{0}}$ ${\displaystyle E_{P}}$

A laser represents an arrangement to generate a light beam whose photons are characterized by the same frequency , phase (together: coherence ) and polarization . The usable radiation is decoupled from the radiation field of the resonator , e.g. B. by partially transparent mirrors .

A necessary, but not sufficient, requirement for the operation of a laser is the amplification of a beam through stimulated emission . For this purpose, in the simplest case ( 3-level laser ) there must be population inversion between the ground state and the laser level . The picture on the right shows a 4-level laser that works in the same way, but has an additional level above (namely ), which in turn is quickly emptied into the basic state . In the 4-level laser, population inversion is therefore easier to produce because it is practically empty. ${\displaystyle E_{0}}$ ${\displaystyle E_{M}}$ ${\displaystyle E_{0}}$ ${\displaystyle E_{L}}$ ${\displaystyle E_{0}}$ ${\displaystyle E_{L}}$

The population inversion can only be achieved in a steady state if the state relaxes quickly (empties, happens in the µs range) and, if present, has a short lifespan , or if the excitation occurs quickly enough. The laser-active level , on the other hand, must have a long service life (ms), since otherwise it is quickly depopulated by spontaneous emission and a thermal equilibrium is established according to the Boltzmann distribution. ${\displaystyle E_{P}}$ ${\displaystyle E_{L}}$ ${\displaystyle E_{0}}$ ${\displaystyle E_{M}}$

The detailed list of the equilibria of individual radiation processes is as follows:

${\displaystyle A_{21}\,N_{2}+B_{21}\,N_{2}\,u_{\nu }=B_{12}\,N_{1}\,u_{\nu }}$

(spontaneous emission (low) + stimulated emission = absorption)

${\displaystyle A_{21}:}$ Einstein coefficient for spontaneous emission

${\displaystyle B_{12}:}$ Einstein coefficient for absorption

${\displaystyle B_{21}:}$ Einstein coefficient for stimulated emission

${\displaystyle u_{\nu }:}$ Energy density of the radiation field

The Einstein coefficients represent transition probabilities between levels is The. Coefficient for stimulated emission is related to the absorption in for context: . ${\displaystyle B_{21}={\tfrac {g_{1}}{g_{2}}}\cdot B_{12}}$

The detailed equilibrium is only microscopic in the non-equilibrium state ; the radiation density increases exponentially over the path length within the resonator. In a laser, radiation of the laser wavelength is optically amplified, while other wavelengths are suppressed for several reasons. This includes the amplification characteristics of the active laser medium (only amplification of certain wavelength ranges) and the laser condition (formation of sharp wavelengths due to the resonator dimensions).

See also

• Rydberg state