# Einstein coefficient

The two energy levels and , the spontaneous emission (A) as well as the absorption (B 12 ) and the induced emission (B 21 ) are shown.${\ displaystyle E_ {1}}$${\ displaystyle E_ {2}}$

In Einstein's rate picture , the Einstein coefficients B 12 , B 21 and A 21 are used to calculate the spontaneous and stimulated (induced) emission and absorption. In addition to statistical physics u. a. used in spectroscopy and laser physics and was introduced by Albert Einstein in 1916 .

Einstein distinguishes three processes in the radiation equilibrium:

In the following we refer to the ground state as state 1 and the excited state as state 2. The probability of the three processes obviously depends on the number of atoms in the outgoing state. In addition, the stimulated processes depend on the occupation of the modes of the electromagnetic field ( spectral radiance ). Einstein introduced the coefficients B 12 , B 21 and A 21 as initially indeterminate proportionality constants , so that ${\ displaystyle N_ {i}}$ ${\ displaystyle u}$

• the likelihood of absorption through ${\ displaystyle B_ {12} \ cdot N_ {1} \ cdot u}$
• the probability of stimulated emission by and${\ displaystyle B_ {21} \ cdot N_ {2} \ cdot u}$
• the probability of spontaneous emission through ${\ displaystyle A_ {21} \ cdot N_ {2}}$

given is.

The increase in the number of particles in the ground state and the decrease in the number of particles in the excited state is then given by:

${\ displaystyle {\ frac {dN_ {1}} {dt}} = - {\ frac {dN_ {2}} {dt}} = - N_ {1} \ cdot B_ {12} \ cdot u + N_ {2 } \ cdot B_ {21} \ cdot u + N_ {2} \ cdot A_ {21}}$

In thermodynamic equilibrium this sum is zero:

${\ displaystyle {\ frac {dN_ {1}} {dt}} = {\ frac {dN_ {2}} {dt}} = 0 \ Rightarrow {\ frac {N_ {2}} {N_ {1}}} = {\ frac {B_ {12} \ cdot u} {A_ {21} + B_ {21} \ cdot u}}}$

From the Boltzmann distribution we know that the occupation of the states is related to their energies as follows:

${\ displaystyle {\ frac {N_ {2}} {N_ {1}}} = {\ frac {g_ {2}} {g_ {1}}} \ cdot {\ frac {e ^ {- E_ {2} / (k _ {\ mathrm {B}} \ cdot T)}} {e ^ {- E_ {1} / (k _ {\ mathrm {B}} \ cdot T)}}} = {\ frac {g_ {2 }} {g_ {1}}} \ cdot e ^ {- \ Delta E / (k _ {\ mathrm {B}} \ cdot T)} \ ,,}$

where they represent the weights of degeneracy . ${\ displaystyle g_ {i}}$

Equating and resolving the spectral radiance yields:

${\ displaystyle u = {\ frac {A_ {21}} {B_ {21}}} \ cdot {\ frac {1} {{\ frac {B_ {12}} {B_ {21}}} {\ frac { g_ {1}} {g_ {2}}} \ cdot e ^ {\ Delta E / (k _ {\ mathrm {B}} \ cdot T)} - 1}}}$

By comparing coefficients with Planck's law of radiation or Rayleigh-Jeans law - in the latter case using the boundary conditions and a series expansion of the exponential function - the following relationships are obtained between the three Einstein coefficients:

${\ displaystyle g_ {1} \ cdot B_ {12} = g_ {2} \ cdot B_ {21}}$
${\ displaystyle B_ {21} = A_ {21} \ cdot {\ frac {\ lambda ^ {3}} {8 \ pi h}}}$

With

• the wavelength ${\ displaystyle \ lambda}$
• the Planck's constant .${\ displaystyle h}$

If the states are not degenerate, then so is . ${\ displaystyle g_ {1} = g_ {2} = 1}$${\ displaystyle B_ {12} = B_ {21} =: B}$

The lifetime of the excited state, i.e. the average time until an atom changes into the ground state through spontaneous decay without any external influence , is

${\ displaystyle \ tau = {\ frac {1} {A_ {21}}}.}$

The Einstein coefficient  A 21 is a substance-specific property of the transition and can be determined quantum mechanically with the help of the transition dipole moment. ${\ displaystyle {\ vec {M}} _ {ik}}$

The Einstein coefficients depend not on the temperature from. The temperature dependence of the energy distribution of the thermal radiation is instead a consequence of the temperature dependence of the occupation probabilities  N 1 and  N 2 , which is usually described by the Boltzmann distribution .