Spontaneous emission

A simple model of spontaneous emission consists of an atom which may be in two electronic energy states, the ground state (1) and the excited state (2), with energies E₁ and E₂ respectively.

If the atom is in the excited state, it may spontaneously decay into the ground state, releasing the difference in energies between the two states as a photon. The photon will have frequency ν and energy hν, given by:

E₂ - E₁ = hν ,

where h is Planck's constant.

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:

∂N / ∂t = - A₂₁N ,

where A₂₁ is a proportionality constant for this particular transistion in this particular atom. (The constant is referred to as an Einstein A co-efficient.) 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:

N(t) = N(0) exp( - t / τ₂₁ ),

where N(0) is the inital number of atoms in the excited state, and τ₂₁ is the lifetime of the transition, τ₂₁ = (A₂₁)^-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 half-life.

See also absorption, stimulated emission, laser science.