Spontaneous emission
From Academic Kids

Spontaneous emission is the process by which an atom or molecule in an excited state drops to the ground state, resulting in the creation of a photon.
If the atom is in the excited state with energy <math>E_2<math>, it may spontaneously decay into the ground state, with energy <math>E_1<math>, releasing the difference in energies between the two states as a photon. The photon will have frequency <math>\nu<math> and energy <math>h \nu<math>, given by:
 <math>E_2  E_1 = h \nu<math>,
where h is Planck's constant. The phase of the photon in spontaneous emission is random as is the direction the photon propagates in. This is not true for stimulated emission.
An energy level diagram illustrating the process is shown below:
Before emission After emission O  E2  Atom in  excited state  ~~~>  Photon hν  V  O E1 Atom in ground state
In a group of such atoms, if the number of atoms in the excited state is given by N, the rate at which spontaneous emission occurs is given by:
 <math>\frac{\partial N}{\partial t} = A_{21} N<math>,
where A_{21} is a proportionality constant for this particular transition in this particular atom. (The constant is referred to as an Einstein A coefficient.) The rate of emission is thus proportional to the number of atoms in the excited state, N.
The above equation can be solved to give:
 <math>N(t) = N(0) e^{  \frac{t}{\tau_{21} } } <math>,
where N(0) is the initial number of atoms in the excited state, and τ_{21} is the lifetime of the transition, τ_{21} = (A_{21})^{1}.
It can be seen that spontaneous emission occurs in a way rather similar to the decay of radioactive particles, in particular that the lifetime is analogous to a halflife.
There are two different ways in which decay or relaxation can occur: radiative and nonradiative. In nonradiative relaxation, the energy is absorbed as phonons, more commonly known as heat. Nonradiative relaxation is nearly impossible to measure and cannot be inferred except in very small particles because the difference in the temperature before and after a relaxation is so small that it is in the noise of any measurement for practical systems.
Nonradiative relaxations occur when the energy difference between the levels is very small, and these typically occur on a much faster time scale than radiative transitions. For many materials (for instance, semiconductors), electrons move quickly from a high energy level to a metastable level via small nonradiative transitions and then make the final move down to the bottom level via an optical or radiative transition (This final transition is the transition over the bandgap in semiconductors.). Large nonradiative transitions do not occur frequently because the crystal structure generally can not support large vibrations without destroying bonds (which generally doesn't happen for relaxation). Metastable states form a very important feature that is exploited in the construction of lasers. Specifically, since electrons decay slowly from them, they can be piled up in this state without too much loss and then stimulated emission can be used to boost an optical signal.
See also absorption, stimulated emission, laser science.