Einstein coefficients

In Einstein's picture rates the Einstein coefficients are used to calculate the spontaneous and stimulated ( induced ) emission and absorption. You can find statistical physics, among others in spectroscopy and laser physics application. They were introduced in 1916 by Albert Einstein.

Einstein distinguishes three processes in radiative equilibrium:

• By absorption of a photon of an electromagnetic field is created, an excited state, eg an atom.
• An n -fold -studded fashion an electromagnetic field stimulates the emission of another photon in this mode, where the atom from the excited transitions to the ground state. In the same fashion means same direction, frequency and phase.
• The atom emits spontaneously - without external influence - a photon in an unoccupied mode ( in free space means in particular: in any direction ).

We refer hereinafter to the ground state as the state 1 and the excited state as a state 2, the probability of the three processes obviously depend on the number of atoms at the end of state (). In addition, the stimulated processes depend on the occupation of the modes of the electromagnetic field. Einstein introduced the coefficients B12, B21, and A21 as initially indeterminate proportionality, so that the probability of stimulated by absorption, the probability of stimulated emission of the spontaneous emission and is given by, with a spectral radiation density.

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

At the thermodynamic equilibrium this sum is zero, so that:

From the Boltzmann distribution, however, we know that the occupation of the states associated with their energies as follows:

Which represent the weights of the degeneracy. Solving this by the energy density, we obtain:

From the coefficient comparison with the Planck 's radiation law or the Rayleigh-Jeans law - in the latter case using the boundary conditions and a series expansion of the exponential function - one obtains the following relations between the three Einstein coefficients:

If the states are not degenerate, so is so.

The lifetime of the excited state, ie, the average time to an atom without external action initiated through the spontaneous decay to the ground state is,

The Einstein coefficient A21 is a property of transition and substance-specific.

The Einstein coefficients are independent of temperature. The temperature dependence of the energy distribution of the thermal radiation is a result of the different occupation probabilities N1 and N2, depending on the temperature, which is usually described by the Boltzmann distribution.