When a photon of the proper frequency interacts with this system, three types of optical processes occur
(i) Absorption
Let the atom initially be in the lower energy state Em. A photon
of proper energy hν = (En − Em) interacts with the atom. This photon is
absorbed, and the atom makes a transition from the lower state m to the higher
excited state n. This process is called absorption. Because it is caused by an
external photon, it is also called stimulated absorption.
The probability of such a transition depends on the energy density of
radiation.
Pmn ∝ u(ν)
Pmn = Bmn u(ν) (1)
where Bmn is a proportionality constant called the
Einstein absorption coefficient. This coefficient depends on the transition
probability between the two energy levels.
(ii) Spontaneous Emission
An atom has a short mean lifetime in the excited state (typically τ ≈
10⁻⁸ s). After that it spontaneously emits a photon and falls to the lower
state. The energy of the emitted photon is the energy difference of the two
levels:
hν
= En − Em
If the system is a collection of excited atoms, the phase and the
direction of the photons emitted by different atoms are irregular (random). As
a result, the radiation emitted is incoherent.
The transition probability per second for spontaneous emission is
Pnm
= Anm
(2)
(iii) Stimulated or
Induced Emission
If a photon of proper frequency interacts with an atom that is already
in the excited state, it need not be absorbed; it may induce the atom to emit a
new photon of the same frequency. This process is called induced or stimulated
emission.
The transition probability for stimulated emission per atom is also
proportional to the energy density of the incident radiation:
Pnm = Bnm u(ν) (3)
where Bnm is the Einstein coefficient for stimulated
emission.
Relations among Einstein’s Coefficients
Consider an assembly of atoms in thermal equilibrium at temperature T
with radiation of frequency ν and energy density u(ν). Let Nm
and Nn be the numbers of atoms in the mth and nth
states at any instant.
Thus the probability of transition from m to n transition
Pmn = Bmn u(ν)
Similarly, the probability of transition from n to m transition
Pnm = [
Anm +
Bnm u(ν) ]
Than rate of transition from m to n state
Rmn
= Nm Bmn
u(ν)
(4)
Similarly, rate of transition from m to n state
Rnm
= Nn [ Anm
+ Bnm u(ν) ] (5)
For thermal equilibrium, absorption and emission must balance:
Rmn = Rnm
Nm Bmn u(ν) = Nn
[ Anm + Bnm u(ν) ]
(Nm Bmn − Nn
Bnm) u(ν) = Nn Anm
(6)
According to Maxwell–Boltzmann statistics, at constant temperature in
thermal equilibrium the number of atoms in an energy level E is proportional to
e(−E/kT), where k is Boltzmann’s constant. Therefore
But En − Em = hν, the energy of the emitted photons, so
(7)
Substituting (7) into (6) gives
(8)
According to Planck’s law, the energy density of radiation at frequency
ν is
(9)
Comparing equations (8) and (9) we obtain the Einstein relations:
Anm / Bnm = 8π h
ν³ / c³ (10)
Bmn = Bnm
(11)
Conclusions
(i)
Bnm ≠ 0.
Stimulated emission is a physically allowed process: an external photon can
induce an excited atom to emit. This hypothesis was used by Townes (1954) in
the development of the maser and later the laser.
(ii)
Bmn = Bnm.
The Einstein coefficient of stimulated emission is equal to the coefficient of
absorption. Consequently the probability of stimulated emission by incident
light of a given energy density is the same as the probability of absorption.
(iii) Anm / Bnm = 8πhν³/c³. The ratio of the
spontaneous-emission coefficient to the stimulated-emission coefficient
increases as ν³. Spontaneous emission therefore becomes relatively more
important at optical frequencies than at microwave frequencies.