Spontaneous Emission of Radiation

We can 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 initially formulated by Einstein [38].
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 [91]

where is the Boltzmann constant. This well-known result was first obtained by Max Planck in 1900 [82,83].

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. (See Section 8.8.) 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

(8.154) |

where and are the rates of spontaneous and stimulated emission, respectively, (for a single atom) between states and , and is the number of atoms in the ensemble in state . Likewise, the latter rate takes the form

(8.155) |

where is the rate of absorption (for a single atom) between states and , and is the number of atoms in the ensemble in state . The previous expressions describe how atoms in the ensemble make transitions from state to state via a combination of spontaneous and stimulated emission, and make the reverse transition as a consequence of absorption. In thermal equilibrium, we have , which gives

(8.156) |

Equations (8.149) and (8.150) imply that

where , and the large angle brackets denote an average over all possible polarization directions of the incident radiation (because, in equilibrium, the radiation inside the cavity is isotropic). In fact, it is easily demonstrated that

where

(See Exercise 7.) 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 [91]. This implies that

Thus, it follows from Equations (8.156), (8.160), (8.161), and (8.163) that the rate of spontaneous emission between states and takes the form

Note, that, although the previous result has been derived for an atom in a radiation-filled cavity, it remains correct even in the absence of radiation.

The direction of propagation of a photon spontaneously emitted by an atom is specified by its normalized wavevector,
.
Likewise, the electric polarization direction of the photon is specified by the unit vector
**
**
.
However,
**
**
, which implies that
**
**
,
where
. Here, the unit vectors
**
**
,
**
**
, and
are
mutually perpendicular, and form a right-handed set (i.e.,
**
**
).
The vectors
**
**
and
**
**
represent the two possible independent electric polarization directions of
the emitted photon. Equation (8.164) generalizes to give [77]

where is the rate of spontaneous emission of photons whose propagation directions lie in the range of solid angle . Of course,

(8.163) |

(See Exercise 8.)