Spontaneous Emission

It is possible to derive the rate of spontaneous emission between two atomic states
from a knowledge of the corresponding absorption and stimulated
emission rates using a famous thermodynamic argument due to Einstein.
Consider a very large ensemble of similar atoms placed inside a closed cavity whose walls (which are assumed to be perfect emitters and absorbers of radiation) are held at
the constant temperature . Let the system have attained thermal equilibrium.
According to statistical thermodynamics, the cavity is filled with so-called ``black-body'' electromagnetic
radiation whose energy spectrum is

Consider two atomic states, labeled and , with . One
of the tenants of statistical thermodynamics is that in thermal equilibrium
we have so-called *detailed balance*. This means that, irrespective
of any other atomic states, the rate at which atoms in the ensemble leave
state due to transitions to state is exactly balanced by the
rate at which atoms enter state due to transitions from state .
The former rate (*i.e.*, number of transitions per unit time in the ensemble) is written

(1126) |

(1127) |

(1128) |

Now, another famous result in statistical thermodynamics is that in thermal equilibrium the number of atoms in an ensemble occupying a state of energy is proportional to . This implies that

Thus, it follows from Eq. (1125), (1129), and (1130) that the rate of spontaneous emission between states and takes the form

Note, that, although the above result has been derived for an atom in a radiation-filled cavity, it remains correct even in the absence of radiation. Finally, the corresponding absorption and stimulated emission rates for an atom in a radiation-filled cavity are

(1132) | |||

(1133) |

respectively.

Let us estimate the typical value of the spontaneous emission rate for a
hydrogen atom. We expect the dipole moment to be
of order , where is the Bohr radius [see Eq. (679)].
We also expect to be of order , where
is the energy of the ground-state [see Eq. (678)]. It thus
follows from Eq. (1131) that

(1134) |