The Einstein
coefficients for a two-level system: $A_{\rm UL}$ for
spontaneous
emission, $B_{\rm LU}$ for absorption, and $B_{\rm UL}$ for stimulated
emission.

Consider any two energy levels
$E_{\rm
U}$ and $E_{\rm L}$ of a
quantum system such as a single atom or molecule. The photon
emitted or absorbed during a transition between the upper and lower
states will have energy

$$E = E_{\rm U} - E_{\rm L}$$ and
contribute to a spectral line with frequency $\nu_0 = E / h$. The
energy
levels actually have small but finite widths, so the spectral line has
some narrow line profile $\phi(\nu)$ centered on $\nu = \nu_0$ and
conventionally normalized such that $\int_0^\infty \phi(\nu) d \nu = 1$.
If this system is in its lower energy
state, it may absorb a photon
of frequency $\nu \approx \nu_0$ and go to the upper state. The rate
for this
process is proportional to the profile-weighted mean energy density

$$\bar{U} \equiv \int_0^\infty U_\nu \,\phi(\nu) d\nu$$ of the
radiation field,
so the Einstein absorption coefficient $B_{\rm LU}$ is defined to make
the
product

$$\bbox[border:3px blue solid,7pt]{B_{\rm
LU} \bar{U}}\rlap{\quad \rm {(7B1)}}$$ equal the
rate (s$^{-1}$) of photon absorption from the radiation field by a
single atomic or molecular system in its lower energy state.

Einstein's critical insight was that
there must be a third process
in addition to spontaneous emission and absorption. It is
stimulated emission, in which a photon of energy $h \nu_0$
stimulates
the system in the upper energy state to emit a second photon with the
same energy and direction. The rate for this process is also
proportional to
$\bar{U}$, so by analogy with Equation 7B1 the Einstein
stimulated-emission
coefficient $B_{\rm UL}$ is defined to make the
product

$$\bbox[border:3px blue solid,7pt]{B_{\rm UL} \bar{U}}\rlap{\quad
\rm {(7B2)}}$$ equal the rate (s$^{-1}$) of
stimulated
photon emission from a single quantum system in its upper energy state.
Stimulated emission is sometimes called negative
absorption. It is
not intuitively familiar because negative absorption is much weaker
than ordinary absorption in room-temperature
objects at visible
wavelengths, but it competes effectively with ordinary absorption at
radio wavelengths where $h \nu
/ ( k T) \ll 1$.

Suppose we have a macroscopic physical
system containing a large
number of atoms or molecules in full thermodynamic equilibrium
(TE) with the surrounding radiation field. TE is a
stationary
state, so the
average rates
of photon creation and destruction must be equal. If the macroscopic
system
contains $(N_{\rm U},\,N_{\rm L})$ atoms or molecules per unit volume
in the (upper,
lower) energy states, then the balance of photon
creation by
spontaneous emission or stimulated emission and photon destruction by
absorption implies

$$\bbox[border:3px blue solid,7pt]{N_{\rm U} A_{\rm UL} + N_{\rm U}
B_{\rm UL} \bar{U} = N_{\rm
L} B_{\rm LU} \bar{U}}\rlap{\quad \rm {(7B3)}}$$

In TE, the ratio of $N_{\rm U}$ to
$N_{\rm L}$ is fixed by the
Boltzmann equation:

$${ N_{\rm U} \over N_{\rm L}} = {g_{\rm U} \over g_{\rm L}} \, \exp
\biggl[ - {(E_{\rm U} - E_{\rm L}) \over k T} \biggr] = {g_{\rm U}
\over g_{\rm L}} \, \exp \biggl( - {h \nu_0 \over k T} \biggr)~,$$
where $g_{\rm U}$ and $g_{\rm L}$ are the numbers of states with
energies $E_{\rm U}$ and $E_{\rm L}$. The quantities $g_{\rm U}$ and
$g_{\rm L}$ are called the
statistical weights of those energy
states. Examples of statistical weights include:

(1) Hydrogen atoms have $g_{\rm n}
=
2 n^2$, where ${n} =
1,~2,~3,\, ...$ is the electronic energy level. The number $2n^2$ is
the product of the 2 electron spin states and $n^2$ orbital angular
momentum states in the $n$th energy level.

(2) Rotating linear molecules
(e.g.,
carbon monoxide, CO) have $g =
2J +1$, where $J = 0,~1,~2,\, ...$ is the angular momentum quantum
number. For each $J$, there are $2J + 1$ possible values of the
$z$-component of the angular momentum: $J_z = -J,~-(J -
1),~...,~-1,~0,~1,~...,~(J-1),~J$.

(3) Hydrogen atoms have two
hyperfine
energy levels whose
difference yields the $\lambda = 21$ cm ($\nu_0 = 1420.406\dots$ MHz)
HI line; $g_{\rm U} = 3$ and $g_{\rm L} = 1$.

The balance of photon creation
and destruction in TE connects the mean energy density of blackbody
radiation to
properties of the quantum system (atom or molecule):

$$\bar{U} = { N_{\rm U} A_{\rm UL} \over N_{\rm L} B_{\rm LU} -
N_{\rm U} B_{\rm UL} } = {A_{\rm UL} \over (N_{\rm L} / N_{\rm
U})B_{\rm LU} - B_{\rm UL} }$$ Full TE at temperature $T$ implies both

$$ \bar{U} = A_{\rm UL} \biggl[ {g_{\rm L}
\over g_{\rm U}} \exp\biggl( {h \nu_0 \over k T} \biggr) B_{\rm LU} -
B_{\rm UL} \biggr]^{-1}$$ and

$$\bar{U} = {4 \pi \over c}\int_0^\infty B_\nu(T)
\,\phi(\nu) d \nu~.$$ Inserting the Planck radiation law for $B_\nu(T)$
near $\nu = \nu_0$ gives $$\bar{U} \approx {4 \pi \over c} { 2 h
\nu_0^3 \over c^2} \biggl[
\exp \biggl( { h \nu_0 \over k T} \biggr) - 1 \biggr]^{-1}~.$$

Next we
equate these two expressions for $\bar{U}$ at the line center
frequency:

$$A_{\rm UL} \biggl[ {g_{\rm L} \over g_{\rm U}} \exp\biggl(
{h \nu_0 \over k T} \biggr) B_{\rm LU} - B_{\rm UL} \biggr]^{-1} = {4
\pi \over c} { 2 h \nu_0^3 \over c^2} \biggl[ \exp \biggl( { h \nu_0
\over k T} \biggr) - 1 \biggr]^{-1}~.$$ This equality holds for * all*
temperatures
$T$, so

$$ {A_{\rm UL} \over B_{\rm UL}} \biggl[ {g_{\rm L} \over
g_{\rm U}} {B_{\rm LU} \over B_{\rm UL}} \exp\biggl({ h \nu_0 \over k
T}\biggr) - 1 \biggr]^{-1} = {8 \pi h \nu_0^3 \over c^3} \biggl[ \exp
\biggl( { h \nu_0 \over k T} \biggr) - 1 \biggr]^{-1}$$ implies both

$$\bbox[border:3px blue solid,7pt]{{ g_{\rm L} \over g_{\rm U} }
{B_{\rm LU} \over
B_{\rm UL} } = 1}\rlap{\quad \rm {(7B4)}}$$
and

$$\bbox[border:3px blue solid,7pt]{{A_{\rm UL} \over B_{\rm UL}}
= {8 \pi h \nu_0^3 \over c^3}}\rlap{\quad \rm {(7B5)}}$$
These two equations are called the equations
of detailed balance. They relate $A_{\rm UL}$,
$B_{\rm LU}$, and $B_{\rm UL}$, so all three quantities can be computed
if only
one (e.g., the spontaneous emission coefficient $A_{\rm UL}$) is
known. Equations 7B4 and 7B5 also prove that $B_{\rm UL}$ is not
zero; that is,
stimulated emission must occur.

Note that these equations are valid
for * any*
microscopic physical
system because they relate constants characteristic of
individual atoms
or molecules for which the * macroscopic statistical* concepts of
TE or LTE are meaningless. Even though TE was used to motivate the
derivation, the dependences on temperature $T$ and frequency $\nu$
dropped out for a line at a single frequency $\nu_0$. Thus these
equations are also valid for macroscopic systems whether or not they
are in TE or
LTE. [Recall the derivation of Kirchoff's law, which also made use of
full TE but which yielded

$$ {\epsilon_\nu (T) \over \kappa_\nu (T)} =
B_\nu(T)$$ relating the emission and absorption coefficients of any
matter in LTE, independent of the actual radiation field.]

**Quantum Radiative Transfer**

We can use the two equations (7B4 and
7B5) relating the
three Einstein coefficients to
solve the spectral-line radiative transfer problem in terms of the
spontaneous emission coefficient $A_{\rm UL}$ alone. The
radiative transfer equation (Eq. 2B4) is:

$$ {d I_\nu \over ds} = -\kappa_\nu I_\nu + \epsilon_\nu~,$$ where
$I_\nu$ is the specific intensity, $\kappa_\nu$ is the net fraction of
photons absorbed (the difference between ordinary absorption and
negative absorption) per unit length, and $\epsilon_\nu$ is the volume
emission coefficient.

Three processes must be considered:
(1) absorption from the lower to upper level, (2)
stimulated emission, which we treat as negative absorption from the
upper to lower level, and (3) spontaneous emission. They
contribute the three terms in the equation below:

$$ { d I_\nu \over ds} = -{h \nu_0 \over c} [N_{\rm L} B_{\rm LU}I_\nu
\,\phi(\nu)]- {h \nu_0 \over c} [-N_{\rm U} B_{\rm UL} I_\nu
\,\phi(\nu)] + {h \nu_0 \over 4 \pi} N_{\rm
U} A_{\rm UL} \phi(\nu) = -\kappa_\nu I_\nu + \epsilon_\nu$$ The
net absorption coefficient is

$$
\kappa_\nu = {h \nu_0 \over c} (N_{\rm L} B_{\rm LU} - N_{\rm U} B_{\rm
UL} ) \phi(\nu)$$ Equation 7B4 can be used to eliminate the stimulated
emission coefficient $B_{\rm UL}$ to yield

$$\kappa_\nu = {h \nu_0 \over c} N_{\rm L}
B_{\rm LU} \biggl( 1 - {N_{\rm U} \over N_{\rm L}}{g_{\rm L} \over
g_{\rm U}} \biggr) \phi(\nu)$$ The emission coefficient is
$$\epsilon_\nu = { h \nu_0 \over 4
\pi} N_{\rm U} A_{\rm UL} \phi(\nu)~.$$

The ratio of these emission and
(net) absorption coefficients is

$${\epsilon_\nu \over \kappa_\nu} = {c N_{\rm U} A_{\rm UL} \over 4
\pi N_{\rm L} B_{\rm LU}} \biggl( 1 - {N_{\rm U} \over N_{\rm
L}}{g_{\rm L} \over g_{\rm U}} \biggr)^{-1}$$ Equation 7B5 can be used
to eliminate $A_{\rm UL}$: $${\epsilon_\nu \over
\kappa_\nu} = {N_{\rm U} (8 \pi h \nu_0^3 / c^2) B_{\rm UL} \over 4 \pi
N_{\rm L} B_{\rm LU}} \biggl( 1 - {N_{\rm U} \over N_{\rm L}}{g_{\rm L}
\over g_{\rm U}} \biggr)^{-1}$$ $${\epsilon_\nu \over \kappa_\nu} = {2
h \nu_0^3 \over c^2} {B_{\rm UL} \over B_{\rm LU}} \biggl( {N_{\rm L}
\over N_{\rm U}} - {g_{\rm L} \over g_{\rm U}} \biggr)^{-1}$$
Finally, Equation 7B4 can be used to eliminate both $B_{\rm UL}$ and
$B_{\rm LU}$: $${\epsilon_\nu \over \kappa_\nu} = {2 h \nu_0^3 \over
c^2} \biggl(
{g_{\rm U} \over g_{\rm L}} {N_{\rm L} \over N_{\rm U}} - 1
\biggr)^{-1}$$

In LTE, Kirchoff's law independently
implies

$${\epsilon_\nu \over
\kappa_\nu} = B_\nu(T) = {2 h \nu^3 \over c^2} \biggl[ \exp \biggl(
{h \nu \over k T} \biggr) - 1 \biggr]^{-1}$$ so $${g_{\rm U} \over
g_{\rm L}} {N_{\rm L} \over N_{\rm U}} = \exp \biggl( { h \nu_0 \over k
T} \biggr)$$ and we recover the Boltzmann
distribution for LTE (not
just for full TE): $$\bbox[border:3px blue solid,7pt]{{N_{\rm U} \over
N_{\rm L}} =
{g_{\rm U} \over
g_{\rm L}} \exp \biggl( - { h \nu_0 \over k T} \biggr)}\rlap{\quad \rm
{(7B6)}}$$

Using

$$\kappa_\nu = {h \nu_0 \over c} N_{\rm L} B_{\rm LU} \biggl(
1 - {N_{\rm U} \over N_{\rm L}} {g_{\rm L} \over g_{\rm U}} \biggr)
\phi(\nu)$$ and the assumption of LTE, we can substitute

$$B_{\rm LU} =
{g_{\rm U} \over g_{\rm L}} B_{\rm UL} = {g_{\rm U} \over g_{\rm L}}
{A_{\rm UL} c^3 \over 8 \pi h \nu_0^3}$$ and $${N_{\rm U} \over N_{\rm
L}} {g_{\rm L} \over g_{\rm U}} = \exp \biggl( - {h \nu_0 \over k T}
\biggr)$$ to get the line
opacity coefficient

$$\bbox[border:3px blue solid,7pt]{\kappa_\nu = {c^2 \over 8 \pi
\nu_0^2} {g_{\rm U} \over g_{\rm L}}
N_{\rm L} A_{\rm UL} \biggl[ 1 - \exp \biggl( - {h \nu_0 \over k T}
\biggr) \biggr] \phi(\nu) {\rm \qquad (in~LTE)}}\rlap{\quad \rm
{(7B7)}}$$ in terms of the spontaneous
emission rate $A_{\rm UL}$ only; the
stimulated emission coefficient $B_{\rm UL}$ and absorption coefficient
$B_{\rm LU}$ have been eliminated.

The quantity $$ \biggl[ 1 - \exp
\biggl( - {h \nu_0 \over k T}
\biggr) \biggr]$$ in the line opacity equation above has two terms. The
positive term (1) comes from absorption and the negative exponential
term represents the negative opacity of stimulated emission. In the
Rayleigh-Jeans limit $h \nu_0 \ll k T$,

$$ \biggl[ 1 - \exp \biggl( - {h \nu_0
\over k T} \biggr) \biggr] \approx {h \nu_0 \over k T} \ll 1 $$

Thus
stimulated emission nearly cancels absorption and significantly reduces
the net line opacity. Since $\kappa_\nu \propto
T^{-1}$, $\kappa_\nu B_\nu \propto T^0$. The brightness of an
optically thin ($\tau \ll 1$) radio
emission line may be proportional to the column density of emitting gas
but nearly independent of
its temperature.

Even if a macroscopic system is not in
LTE, we can define its
excitation temperature $T_{\rm x}$ by

$$\bbox[border:3px blue solid,7pt]{{N_{\rm U} \over N_{\rm L}}
\equiv {g_{\rm U} \over g_{\rm L}} \exp \biggl( - { h \nu_0 \over k
T_{\rm x}} \biggr)}\rlap{\quad \rm {(7B8)}}$$
If for some reason the upper level is
overpopulated; that is $${N_{\rm U} \over N_{\rm L}} >
{g_{\rm
U} \over g_{\rm L}}~,$$ then $T_{\rm x}$ is actually negative,
$$\biggl[ 1 -
\exp \biggl( - {h \nu_0 \over k T_{\rm x}} \biggr) \biggr]$$ is
negative, and Equation 7B7 gives a negative net opacity coefficient
$\kappa_\nu$. Negative opacity implies brightness gain
instead of loss. At radio wavelengths this is called
maser
(microwave amplification by
stimulated emission of radiation) amplification. Astrophysical masers
are common at radio frequencies because $h\nu \ll kT$. They can
have brightness temperatures as high as $10^{15}$ K, much higher than
the kinetic temperature of
the masing gas.