Chapter 6. Radiation and Thermal Equilibrium
Chapter 6.
Radiation and Thermal Equilibrium
Chapter 6. Radiation and Thermal Equilibrium
In the previous three chapters, we examined the energy levels of various gaseous,
liquid, and solid materials, and discussed the spontaneous radiative properties
associated with transitions between those levels. Radiation rates were described in
terms of transition probabilities and relative oscillator strengths of the transitions.
In this chapter, we will consider the collective properties of a large number of
radiative atoms or molecules. This will lead to the concept of radiation in
thermodynamic equilibrium, from which we can deduce the principles of stimulated
emission.
Many of the concepts discussed in this chapter were introduced in PC344. You may
wish to review your notes from that course.
Chapter 6. Radiation and Thermal Equilibrium
Outline
Table of Contents
1
Equilibrium
2
Radiating Bodies
Stefan-Boltzmann Law
Wien’s Law
3
Cavity Radiation
Counting the Number of Cavity Modes
Rayleigh-Jeans Formula
Planck’s Law for Cavity Radiation
Relation between Cavity and Blackbody Radiation
4
Absorption and Stimulated Emission
The Principle of Detailed Balance
Chapter 6. Radiation and Thermal Equilibrium
Equilibrium
Thermal Equilibrium §6.1
In the study of thermodynamics, thermal equilibrium describes the case where
each individual mass within a closed system has the same temperature. If two
masses (M1 and M2) with different temperatures (T1 and T2) are brought into close
proximity, they will after some time reach the same temperature, somewhere
between T1 and T2. The duration of time over which this temperature change occurs
can range over orders of magnitude, depending on the thermal properties of the
masses and the intervening medium.
The transfer of thermal energy occurs by one or more of three processes:
conduction, convection, and radiation (Fig. 1). The only one of these that is relevant
to understanding laser operation is radiation, so we will ignore the other two
processes in this course. It should be noted however, that conduction (and to a
lesser extent convection) are critically important in the actual engineering of lasers,
as they produce a large amount of waste heat.
Fig. 1:
Examples of energy transfer via (a) conduction, (b) convection, and (c) radiation - see WTS Fig. 6-1
Chapter 6. Radiation and Thermal Equilibrium
Radiating Bodies
Radiating Bodies §6.2
In Fig. 1(c), if we remove all of the air from the space between M1 and M2, then
there is no material contact between the bodies. In this case, radiation is the only
process that will bring the two masses into equilibrium. For this process to occur, the
masses must be capable of radiating energy, and each must be capable of
absorbing radiation from the other body.
If a collection of atoms is at temperature T then, by the Boltzmann equation, the
probability distribution function fi that any atom has an energy Ei is given by
fi (Ei ) = C1 gi e −Ei /kT .
(6.1)
−5
Here, k = 8.6164 × 10 eV/K is Boltzmann’s constant and gi is the statistical weight
of level i , the number of states that have the same energy (refer back to the end of
chapter 3). The term C1 is a normalizing constant that is the same for all energy
levels, and is determined by the constraint that
X
X
fi =
C1 gi e −Ei /kT = 1
(6.2)
i
i
(meaning that the atom must exist in one of the i energy levels). If N is the total
number of atoms per unit volume of this atomic species
and Ni is the population
P
density occupying a specific energy level i , then i Ni = N , indicating that Ni can be
expressed as Ni = fi N = C1 gi e −Ei /kT N .
Chapter 6. Radiation and Thermal Equilibrium
Radiating Bodies
Radiating Bodies cont’
§6.2
In the case of a high-density (i.e. solid) material, the energy levels are usually
continuously distributed. Exceptions to this guideline are ion-doped insulators which we will encounter frequenctly in PC482 - and quantum-confined
semiconductors. For continuously-distributed energy levels, the distribution function
is expressed as a probability per unit energy, g (E ), such that the probability of
finding a fraction of that material excited to a specific energy E within an infinitesimal
“slice” dE along the energy axis is
g (E )dE = C2 e −E /kT dE ,
(6.3)
where we have ignored the statistical weights for reasons that will be explained
shortly. As before, the constant C2 exists to normalize the probability distribution:
Z ∞
Z ∞
g (E )dE =
C2 e −E /kT dE = 1.
(6.4)
0
0
Using this relation, it is easy to show that C2 = 1/kT , and thus
g (E ) =
1 −E /kT
.
e
kT
(6.5)
Chapter 6. Radiation and Thermal Equilibrium
Radiating Bodies
Radiating Bodies cont’
§6.2
Again, if N is the number of atoms per unit volume in the solid, and if we refer to the
number of atoms per unit volume within an infinitesimal
R ∞ energy range dE as N (E ),
then the normalization condition requires that N = 0 N (E )dE . The number of
atoms at energy E within an energy range dE is then
N (E )dE =
N −E /kT
dE .
e
kT
(6.6)
It is fundamental in laser studies to know the ratio of population densities
(number of particles per unit volume) that exist at two specific energy levels.
For discrete energy levels, the ratio of densities at the upper energy level Eu and
lower energy level Eℓ is expressed as
gu −(Eu −Eℓ )/kT
gu −∆Euℓ /kT
Nu
e
=
e
,
=
Nℓ
gℓ
gℓ
∆E u ℓ = E u − E ℓ .
(6.7)
Similarly, for continuous energy distributions,
N (Eu )dE
= e −∆Euℓ /kT .
N (Eℓ )dE
(6.8)
Chapter 6. Radiation and Thermal Equilibrium
Radiating Bodies
Radiating Bodies cont’
§6.2
These two formulae for population density ratio are identical except for the ratio of
the statistical weights, gu /gℓ . In dense materials, there are so many energy
sublevels within small ranges of energy that the statistical weights for most levels are
essentially the same; thus, they would cancel out when taking the ratio.
To summarize, when a collection of atoms are assembled together and brought to
equilibrium, not only will the kinetic energies related to their motion be in thermal
equilibrium, but the distribution of their internal energies associated with the specific
energy levels they occupy will also be in thermal equilibrium.
Chapter 6. Radiation and Thermal Equilibrium
Radiating Bodies
Stefan-Boltzmann Law §6.2
The Stefan-Boltzmann Law states that the total radiated intensity (W/m2 ) emitted
from a body at temperature T is proportional to the fourth power of the temperature:
I = eM σT 4 ,
(6.9)
where σ = 5.67 × 10−8 Wm−2 K−4 and eM is the emissivity that is specific to the
radiating material. eM (which lies between zero and one) represents the ability of a
body to efficiently absorb and radiate energy. It is generally a safe assumption that
eM is independent of wavelength.
Wien’s Law §6.2
Wien’s Law describes the wavelength λm at which the maximum emission occurs
for any given temperature:
λm T = 2.898 × 10−3 m · K.
We see that λm varies inversely with temperature. This is shown in Fig. 2.
(6.10)
Chapter 6. Radiation and Thermal Equilibrium
Radiating Bodies
Fig. 2:
Spectrum of radiation vs. wavelength of a heated mass (blackbody) for three different temperatures. Wien’s law
specifies the wavelength of peak emission as a function of wavelength - see WTS Fig. 6-3
Chapter 6. Radiation and Thermal Equilibrium
Cavity Radiation
Cavity Radiation §6.3
In the case of an ideal radiating blackbody, it will soon prove useful to know the
amount of radiated energy - and in particular, the spectral density of the
radiation (that is, the frequency distribution) - at a particular temperature.
Electromagnetic theory tells us that in order for EM waves to be supported in a
cavity, the electric field vector must be zero at the cavity walls. This leads to the
development of discrete “standing waves” as shown in Fig. 3, referred to as cavity
modes. To satisfy the boundary conditions, there are an integral number of
half-wavelengths occurring along the wave direction within the cavity. If we calculate
the number of such modes at each frequency within a cavity and multiply by the
average energy of each mode at these frequencies, this gives us the emission
spectrum for cavity radiation, from which we calculate the number of modes that
exist within a cavity of a given size.
Fig. 3:
Several modes of electromagnetic radiation within a confined cavity - see WTS Fig. 6-4
Chapter 6. Radiation and Thermal Equilibrium
Cavity Radiation
Counting the Number of Cavity Modes §6.3
Assume that we have a cavity that is rectangular in shape with dimensions Lx , Ly , Lz
and volume V = Lx Ly Lz . As we are considering only standing waves, we assume
that the spatial dependence of the EM waves is an oscillatory function of the form
e −i (kx Lx +ky Ly +kz Lz ) .
In order to satisfy the boundary condition that the field is periodic in each direction
and also zero at the boundaries, the exponential phase factor must be an integral
multiple of π. This is accomplished by specifying that
kx Lx = nx π, ky Ly = ny π, kz Lz = nz π, nx , ny , nz = 0, 1, 2, · · · .
(6.11)
Each of the modes in this cavity will have a specific value of k , and can be identified
using the three mode numbers nx , ny , nz :

!2 
 nx π 2
ny π
nz π 2 
2
2
2
2

k = kx + ky + kz = 
+
+
(6.12)
.
Lx
Ly
Lz 
The total number of modes in any volume V up to a given value of k can then be
counted by visualizing a 3D space whose axes are defined as the number of modes,
as in Fig. 4.
Chapter 6. Radiation and Thermal Equilibrium
Cavity Radiation
Counting the Number of Cavity Modes cont’
§6.3
In this figure, the number of modes in the x direction is the length of the cavity in that
direction (Lx ), divided by one half-wavelength; i.e. nx = Lx /(λ/2). In total, the
number of modes along each axis of the volume for a specific wavelength is
nx = 2Lx /λ, ny = 2Ly /λ, nz = 2Lz /λ. Since the mode numbers must be positive, we
need only to consider one octant of a full ellipsoid, as shown in the figure.
Fig. 4:
3D diagram of the cavity mode volume - see WTS Fig. 6.5
Chapter 6. Radiation and Thermal Equilibrium
Cavity Radiation
Counting the Number of Cavity Modes cont’
§6.3
Each mode can be considered as a cube of unit dimensions within the octant. For
volumes with dimensions significantly larger than the wavelength, the total number of
modes can be expressed as
!
4πη3 ν3
1 4π 2Lx 2Ly 2Lz
M= ·
V,
(6.13)
=
·
·
·
8 3
λ
λ
λ
3c 3
where we have used the relationship λν = c /η. When the cavity dimensions are on
the same scale as the wavelength, this equation becomes less accurate due to the
“voxelization” of the modes.
Now, this calculation defines the number of modes within a volume V for all
frequencies up to and including the frequency ν. This number must then be
doubled to allow for the fact that each spatial mode can exist in two
orthogonal polarizations. Thus, for frequencies up to the value ν, the mode
density ρ (that is, the number of modes per unit volume) is
ρ(ν) = 2M /V = 8πη3 ν3 /3c 3 .
(6.14)
This equation can then be differentiated to obtain the number of modes per unit
volume within a given frequency interval between ν and ν + d ν,
d ρ(ν)
8πη3 ν2
=
.
dν
c3
(6.15)
Chapter 6. Radiation and Thermal Equilibrium
Cavity Radiation
Rayleigh-Jeans Formula §6.3
If the average energy per mode is on the order of kT - which you may recall from
PC344 is a good approximation of any entity’s thermal energy - then we can obtain
the Rayleigh-Jeans formula for the energy density u(ν) of radiation per unit volume
within the frequency interval ν to ν + d ν; it is simply the energy per mode kT times
the mode density per unit frequency (eq. 6.15):
u(ν) =
d ρ(ν)
8πη3 ν2
kT =
kT .
dν
c3
(6.16)
This result suggests that - at a given temperature T - there is a continuous increase
in energy density with frequency. It agrees with experimental evidence for lower
frequencies, but does not predict the experimentally observed maximum value as
shown in Fig. 2. Rather, it suggests that energy density approaches infinity as
frequency is increased. This clearly incorrect result is known as the ultraviolet
catastrophe.
Chapter 6. Radiation and Thermal Equilibrium
Cavity Radiation
Planck’s Law for Cavity Radiation §6.3
The ultraviolet catastrophe was resolved by Einstein using Planck’s law (not by
Planck himself, as the textbook indicates). Planck’s law suggests that the energy of
a cavity mode is quantized ; than an oscillator of freqency ν can only have one of a
discrete set of energies Em = mh ν, where m = 0, 1, 2, · · · .
In this case, the Boltzmann distribution of energies at a given temperature T
becomes
fm = Ce −Em /kT = Ce −mh ν/kT .
(6.17)
The normalization condition becomes
∞
X
m=0
fm =
∞
X
m=0
Ce −mh ν/kT = C
∞
X
e −mh ν/kT = 1,
(6.18)
m=0
which can be solved to obtain C = 1 − e −h ν/kT . Then, the average mode energy Ē
is not kT , but rather
P∞
hν
m=0 Em fm
(6.19)
Ē = P
= h ν/kT
∞
e
−1
f
m=0 m
(see p. 211 of the text for the intermediate steps in this derivation).
Chapter 6. Radiation and Thermal Equilibrium
Cavity Radiation
Planck’s Law for Cavity Radiation §6.3
Now if we return to eq. (6.16) and use this value of Ē in place of kT, we find the
following relationship for the energy density per unit frequency:
u(ν) =
8πh η3 ν3
.
c 3 (e h ν/kT − 1)
(6.20)
This equation no longer suffers from the ultraviolet catastrophe. Rather, it exhibits
the same functional form as the experimental distributions as shown in Fig. 2. The
expression is known as Planck’s law for cavity radiation.
We must be reminded that u(ν) describes the emitted energy density per unit
frequency. This quantity will be useful toward the end of this chapter. We can also
calculate the total emitted energy density by integrating over all frequencies:
Z ∞
u=
u(ν)d ν.
(6.21)
0
Carrying out this integration (hey, that would make a good assignment problem)
leads to the Stefan-Boltzmann law.
Chapter 6. Radiation and Thermal Equilibrium
Cavity Radiation
Relation between Cavity and Blackbody Radiation §6.3
The previous 7 slides showed how to calculate the spectral energy density for
radiation traveling in all directions within a cavity. In a laser, we are interested in
directed radiation; that is, the radiation that is emitted along a particular direction.
We can examine this by exploring the relationship between cavity and blackbody
radiation.
Refer back to Fig. 1(c). If we were to make a small hole of unit area through mass
M2 into the cavity, we would observe a small amount of radiation of spectral intensity
I(ν) emerging from the hole. I(ν) is related to the energy density u(ν) within the
cavity. We can then calculate the total flux of radiation emerging from the cavity at
frequency ν, traveling in all directions within a solid angle of 2π (a hemisphere). We
refer to this radiation as the blackbody radiation intensity per unit frequency, IBB (ν).
Chapter 6. Radiation and Thermal Equilibrium
Cavity Radiation
Relation between Cavity and Blackbody Radiation cont’
§6.3
We first need to obtain a relationship between u(ν) (energy per unit volume per unit
frequency) and the flux I (ν) (energy per unit time per unit frequency flowing through
a unit area).
With the aid of Fig. 5, we consider a beam of light at I (ν) with cross-sectional area
dA and traveling in a direction z that passes through dA in a time dt . The energy
density of this radiation would be I (ν)dAdt . This would be equivalent to considering
the energy density u(ν) of a beam existing within a volume dV = dA · z , where
z = vdt and v is the velocity of the beam. Therefore,
I(ν)dAdt = u(ν)dV = u(ν)dA · z = u(ν)dA · vdt .
(6.22)
If the volume element is in a vacuum, then v ≡ c , and thus
I(ν) = u(ν)c
(6.23)
for a beam traveling in a specific direction. When the volume element is in a medium
with index of refraction η, then we have
I(ν) = u(ν)c /η.
(6.24)
Chapter 6. Radiation and Thermal Equilibrium
Cavity Radiation
Fig. 5:
Beam of light intensity I (ν) incident upon a volume of thickness z = cdt - see WTS Fig. 6.6
Chapter 6. Radiation and Thermal Equilibrium
Cavity Radiation
Relation between Cavity and Blackbody Radiation cont’
§6.3
We can now convert the expression for energy density of radiation within a cavity of
temperature T to an expression for the blackbody radiation emerging from the hole
of unit surface area, as shown in Fig. 6.
We start by realizing that radiation from any point within the cavity is traveling in all
directions - that is, within a solid angle of 4π. We apply the coordinates shown in Fig.
7, where the fraction of radiation traveling within a solid angle d Ω/4π at a particular
angle θ with respect to the normal to the plane of the hole is described by
I(ν) cos θd Ω/4π. Here, we have assumed that the intensity emitted from the source
is independent of angle; the factor cos θ accounts for the fact that when you look at
the hole from an oblique angle, its area appears smaller.
In spherical polar coordinates, the solid angle can be written d Ω = sin θd θd φ, and
therefore the radiation flux along direction θ is
I(ν) cos θ sin θd θd φ/4π.
(6.25)
Chapter 6. Radiation and Thermal Equilibrium
Cavity Radiation
Fig. 6: Blackbody radiation escaping from a cavity with
mass M at equilibrium temperature T - see WTS Fig. 6.7(a)
Fig. 7: Coordinates for analyzing blackbody radiation
escaping from a confined cavity - see WTS Fig. 6.7(b)
Chapter 6. Radiation and Thermal Equilibrium
Cavity Radiation
Relation between Cavity and Blackbody Radiation cont’
§6.3
The total blackbody flux is obtained by integrating this component of the flux over the
entire hemisphere (a solid angle of 2π):
Z 2π Z π/2
I(ν) cos θ sin θd θd φ
I(ν)
u(ν)c
IBB (ν) =
=
=
.
(6.26)
4π
4
4
0
0
Finally, using the expression for u(ν) that we derived in eq. (6.20), we have (in a
vacuum)
IBB (ν, T ) =
2πh ν3
1
.
2
h
ν/
kT
c
e
−1
(6.27)
This represents the spectral radiance of a blackbody as a function of frequency and
temperature. In terms of wavelength (derivation is on p. 214 of the text),
IBB (λ, T ) =
1
2πc 2 h
.
λ5 e ch /λkT − 1
(6.28)
Chapter 6. Radiation and Thermal Equilibrium
Absorption and Stimulated Emission
Absorption and Stimulated Emission §6.4
In previous chapters, we investigated the spontaneous emission process, in which
electrons decay radiatively from a higher to a lower energy level. We also saw that
radiative emission is not necessarily the dominant decay process; collisions with
other particles (in a gas) or phonons (in a solid) can sometimes depopulate the
upper level faster than the radiative process.
Excitation or de-excitation of the upper level can also occur by way of photons.
Excitation in this case is nothing more than absorption (a photon is absorbed, its
energy raising an electron to a higher energy level). It sometimes helps to refer to
this process as “stimulated” absorption, since it requires electromagnetic energy to
stimulate the electron into the upper level. In 1917, Einstein suggested that perhaps
stimulated radiative de-excitation - i.e. stimulated emission - is also possible.
Chapter 6. Radiation and Thermal Equilibrium
Absorption and Stimulated Emission
The Principle of Detailed Balance §6.4
The principle of detailed balance (sometimes referred to as the principle of
microscopic reversibility) states that in equilibrium, the total number of particles
leaving a certain quantum state per unit time equals the number arriving in that state
per unit time. Furthermore, when there are various pathways (radiative,
non-radiative, etc.), the number leaving by a particular pathway equals the number
arriving by that pathway.
This principle suggests that if a photon can stimulate an electron to move from a
lower energy state ℓ to a higher state u by means of absorption, then a photon
should also be able to stimulate an electron to move from the same upper state u to
the same lower state ℓ. In the case of absorption, the photon disappears, with the
energy being transferred to the absorbing species. In the case of stimulated
emission, the species gives up a quantum of energy to produce a photon. This
process must occur in order to keep the population of the two energy levels in
thermal equilibrium.
In the QM picture that we developed in chapter 3, the photon that initiates stimulated
emission produces an oscillation of the excited atom’s dipole moment, which results
in the emission of the second photon. This requires that the phase and direction of
the emitted photon is identical to that of the stimulating photon. Coherence
and directionality!
Chapter 6. Radiation and Thermal Equilibrium
Absorption and Stimulated Emission
Fig. 8:
The three fundamental radiative processes associated with the interaction of light with matter. Top: spontaneous
emission. Middle: stimulated emission. Bottom: absorption - see WTS Fig. 6.8
Chapter 6. Radiation and Thermal Equilibrium
Absorption and Stimulated Emission
Absorption and Stimulated Emission Coefficients §6.4
Here, we derive the absorption and stimulated emission coefficients associated with
these processes by considering radiation in thermal equilibrium.
Consider a group of atoms having electrons which occupy either energy levels u or ℓ
with population densities Nu and Nℓ . Since the atoms are in thermal equilibrium,
these populations are related by the Boltzmann distribution
gu −∆Euℓ /kT
Nu
e
.
=
Nℓ
gℓ
(6.29)
Now we consider photons interacting with this collection of atoms. We assume that
the photons have frequencies νuℓ = ∆Euℓ /h ; that is, the photon energies are exactly
equal to the difference in energy between levels u and ℓ.
We previously defined Auℓ as the spontaneous transition probability (the rate at
which spontaneous transitions occur). Thus, the number of spontaneous transitions
per unit time per unit volume is Nu Auℓ .
Chapter 6. Radiation and Thermal Equilibrium
Absorption and Stimulated Emission
Absorption and Stimulated Emission Coefficients cont’
§6.4
As for the stimulated processes, these are proportional not only to the population of
the appropriate level, but also to the photon energy density u(ν) at frequency νuℓ . If
we label the proportionality constant of stimulated transitions as B , then the “upward
flux” (the number of stimulated upward transitions per unit volume per unit time per
unit frequency) is Nℓ Bℓu u(ν). Similarly, the “downward flux” would be Nu Buℓ u(ν). The
constants Auℓ , Buℓ , and Bℓu are called the Einstein A and B coefficients; bear in
mind that we have yet specify the formulas for the B coefficients.
For the populations Nu and Nℓ to be in thermal equilibrium, and for the principle of
detailed balance to apply, the downward and upward radiative flux between the
two levels must be equal:
Nu Auℓ + Nu Buℓ u(ν) = Nℓ Bℓu u(ν).
(6.30)
This equation can be manipulated in several ways to provide useful information
about the equilibrium. First, we can rearrange it to solve for the photon energy
density,
"
#
!−1
Nu Auℓ
Auℓ gℓ Bℓu h νuℓ /kT
u(ν) =
e
−1 ,
=
(6.31)
Nℓ Bℓu + Nu Buℓ
B u ℓ gu B u ℓ
where we used eq. (6.29) to eliminate the population densities and made the
substitution ∆Euℓ = h νuℓ .
Chapter 6. Radiation and Thermal Equilibrium
Absorption and Stimulated Emission
Absorption and Stimulated Emission Coefficients cont’
§6.4
Recall, however, that we previously calculated u(ν) for radiation in thermal
equilibrium, eq. (6.20). These two representations of u(ν) will be equivalent if
gℓ B ℓ u
= 1 or gℓ Bℓu = gu Buℓ
gu B u ℓ
(6.32)
and
Bu ℓ =
c3
Au ℓ .
8πh η3 ν3
(6.33)
Thus, we have derived the relationship between the stimulated emission and
absorption coefficients Buℓ and Bℓu , along with their relationship to the spontaneous
emission coefficient Auℓ .
Chapter 6. Radiation and Thermal Equilibrium
Absorption and Stimulated Emission
Absorption and Stimulated Emission Coefficients cont’
§6.4
You will possibly recall that in chapter 4, we derived an expression for Auℓ in terms of
Auℓ (ν) (the transition probability per unit frequency for a Lorentzian emission profile).
We substitute this in the previous equation to obtain
Bu ℓ =
(ν − ν0 )2 + (γuTℓ /4π)2
c3
Auℓ (ν).
3
3
8πh η ν
γuTℓ /4π2
If we now define
Buℓ (ν) =
γuTℓ /4π2
Bu ℓ ,
(ν − ν0 )2 + (γuTℓ /4π)2
which describes the frequency dependence of Buℓ , then we can write
8πh η3 ν3
Auℓ (ν)
Au ℓ
=
.
=
Buℓ (ν)
Bu ℓ
c3
(6.34)
(6.35)
(6.36)
Hence, the expressions for Buℓ (ν) and Bℓu (ν) will satisfy eq. (6.31) if
gℓ Bℓu (ν)
= 1 or gℓ Bℓu (ν) = gu Buℓ (ν).
gu Buℓ (ν)
(6.37)
We have thus derived the stimulated emission coefficients (and their
frequency-dependent counterparts) that define the way a photon beam interacts with
a two-level system of atoms.
Chapter 6. Radiation and Thermal Equilibrium
Absorption and Stimulated Emission
Absorption and Stimulated Emission Coefficients cont’
§6.4
It is intersting to compare the magnitude of the stimulated to spontaneous emission
rates from level u to level ℓ. From eqns. (6.20) and (6.33),
Buℓ u(ν)
1
= h ν /kT
.
Au ℓ
e uℓ − 1
(6.38)
Thus, stimulated emission is only significant for temperatures in which kT is of the
same magnitude as νuℓ . The ratio equals 1 when h νuℓ /kT = ln 2 = 0.693.
As an example, visible transitions in the green region of the spectrum have a photon
energy of about 2.5 eV. In this region of the spectrum, Buℓ u(ν)/Auℓ = 1 for
temperatures around 33,500 K. In nature, such temperatures only occur in stars. In
lasers however, the ratio can be significantly greater than unity. This is because we
use mirrors to contain the photons within a small volume. That is, we engineer a
very large value of u(ν).