Blackbody Radiation
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Blackbody Radiation
The intensity and spectrum of blackbody radiation depends only on the temperature of the black
body or cavity. The same is true for the electrical noise generated by a warm resistor, which
plays an important role in radio astronomy. You may already be familiar with the standard
derivation of the Rayleigh-Jeans and Planck radiation laws, but it is worth repeating because
blackbody radiation is so fundamental and because its one-dimensional analog yields the
spectrum of electrical noise generated by a resistor.
The Rayleigh-Jeans Law
Consider a large (meaning side length a µ Õ, where Õ is the longest wavelength of interest)
cubical cavity filled with radiation in thermodynamic equilibrium. The purpose of the cavity is to
confine the radiation long enough for it to reach equilibrium. The walls of a cavity capable of
confining radiation must have nonzero conductivity because walls of zero conductivity would be
transparent; they would generate no currents in response to the electric field of the incoming
radiation, so the radiation would pass through unaffected. For walls having nonzero
conductivity, the equilibrium electric field strength at the walls is E = 0 because modes with
E=
= 0 at the walls are lossy. Only those standing waves with E = 0 at the walls will persist
after some time t µ a =c .
We can enumerate all possible standing-wave modes in the cavity. For example, consider all
standing waves whose wave normals point in the x direction.
Standing waves corresponding to nÕ x =2 = a for n = 1; 2; 3. Abscissa: x-axis of the cavity,
bounded by walls at x = 0 and x = a . Ordinate: Electric field strengths of the longest standing
waves satisfying the boundary condition E = 0 at the walls.
The boundary conditions E = 0 at x = 0 and at x = a mean that only the waves having the
discrete wavelengths
Õx
= a;
2
1 of 10
2Õx
= a;
2
3Õx
= a; :::;
2
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Blackbody Radiation
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can have nonzero amplitudes. Thus
nÕx
=a
2
where n = 1, 2, 3, :::. Similarly,
ny Õ y
nx Õ x
nz Õ z
=
=
=a
2
2
2
for wave normals in the x; y; z directions.
What about a wave whose normal is in some arbitrary direction? Let Ë; Ì; Í be the angles
between the wave normal and the x; y; z axes, respectively.
This two-dimensional figure illustrates standing waves propagating in a cavity with wave
normals at angles Ë and Ì from the x and y axes, respectively. Examples of wave nodes,
where jEj = 0, are indicated by dashed lines for the case nx = 3, ny = 2 .
From the figure it is clear that
Õ = Õx cos Ë
where Õ is the wavelength measured in the direction of the wave normal and Õ x Õ Õ is the
spacing between the wave nodes measured along the x-axis. Thus
Õx =
2 of 10
Õ
cos Ë
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Blackbody Radiation
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Similarly,
Õ
Õ
and Õz =
cos Ì
cos Í
Õy =
Our boundary conditions become
2a
2a
2a
; ny =
; nz =
Õx
Õy
Õz
nx =
nx =
2a cos Ë
2a cos Ì
2a cos Í
; ny =
; nz =
Õ
Õ
Õ
Squaring and summing the last three equations gives
n2x
+
n2y
+
nz2
=
Ò
2a
Õ
Ó2
(cos2 Ë + cos2 Ì + cos2 Í ) :
2
2
2
The Pythagorean theorem implies (cos Ë + cos Ì + cos Í ) = 1 so
n2x
+
n2y
+
nz2
=
Ò
2a
Õ
Ó2
The permitted frequencies · = c=Õ of these standing waves are:
·=
cq 2
n x + n2y + nz2
2a
for all positive integers nx, ny, and nz .
The permitted standing-wave modes can be represented as a lattice in the positive octant of the
space whose axes are nx, ny, and nz . Each point of the lattice represents one possible mode of
electromagnetic radiation. The space density of points in this lattice is unity, so the number of
points in any volume is equal to that volume.
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Blackbody Radiation
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The (x,y) plane in the imaginary space whose axes are (nx ; ny ; nz ). Permitted standing waves
in the (nx ; ny ) plane are indicated by dots at positive integer values of these axes.
Let Ú be the radial coordinate in (nx ; ny ; nz )-space. Then
Ú2 = n2x + n2y + nz2
and
·=
cÚ
2a
The number N(·)d· of modes having frequencies in the range · to · + d· is the volume of the
spherical octant shell between Ú and Ú + dÚ, multiplied by 2 (two polarizations):
N(·)d· =
4ÙÚ2 dÚ
Â2
8
Ò
2a·
N(·)d· = Ù
c
Ó2
2a
d·
c
In thermodynamic equilibrium at temperature T , each mode has average energy hEi = kT
according to the classical Boltzmann law, but not according to quantum mechanics. If we
proceed using kT for the average energy per mode, the spectral energy density u· (T ) of
radiation in the frequency range · to · + d· is the total energy of all modes in that frequency
range divided by the volume a 3 of the cavity:
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Blackbody Radiation
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u· (T ) =
u· (T ) =
N(·)d·
kT
a3
8Ùa3 · 2
·2
kT
=
8
ÙkT
a 3 c3
c3
The spectral energy density of the radiation is spectral energy per unit volume. It equals the
total flow of spectral power per unit area divided by the flow speed c :
1
u· =
c
Z
I· dÊ
(2C1)
Calling the specific intensity of blackbody radiation B· and making use of the fact the blackbody
radiation is isotropic, we get for blackbody radiation:
1
u· =
c
Z
B·
B· dÊ =
c
4Ù
Z
dÊ =
4Ù
4Ù
B·
c
Thus
4Ù
8ÙkT· 2
B· =
c
c3
and we obtain the Rayleigh-Jeans law for the spectral brightness of blackbody radiation
B· =
2kT· 2
c2
(2C2)
that is valid only in the low-frequency limit h· Ü kT .
Notice that:
1. Brightness is proportional to frequency squared because the volume of a spherical shell in
three dimensions is proportional to frequency squared.
2. We assumed all modes have hEi = kT , the classical assumption that breaks down at high
frequencies.
3. B· diverges at high · . This is called the "ultraviolet catastrophe".
4. B· is independent of direction.
The Planck Radiation Law
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Blackbody Radiation
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The only flaw in the derivation of the Rayleigh-Jeans law is the classical assumption that each
radiation mode can have any energy E . Then the continuous Boltzmann probability distribution
Ò
Ó
ÀE
P (E) / exp
kT
implies an average energy per mode
R1
EP (E)dE
hEi = R0 1
= kT :
P
(E)dE
0
Planck postulated that possible mode energies are not continuously distributed, but rather they
are quantized and must satisfy
E = nh·;
n = 1; 2; 3; :::
where h Ù 6:63 Â 10À27 erg s is Planck's constant and n is the number of photons (particles of
light each having energy E = h· ) in that mode. Then
Ò
Ó
Ành·
P (E) = P (nh·) / exp
kT
and the average energy per mode is calculated by summing over only the discrete energies
permitted instead integrating over all energies.
hEi =
P1
n=0 nh·P (nh·)
=
P1
n=0 P (nh·)
P1
À
Ành·
n=0 nh· exp kT
À Ành· Á
P1
exp
n=0
kT
Á
Evaluating this sum yields
hEi =
h·
À h· Á
exp kT À 1
Thus the correct radiation law becomes
Ô
Õ
h·=(kT )
2kT· 2
B· =
;
c2
exp[h·=(kT )] À 1
where the first factor is the Rayleigh-Jeans law and the quantity in brackets is the quantum
correction factor.
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Blackbody Radiation
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Planck's law for blackbody radiation is usually written in the form
B· =
2h· 3
1
À
Á
c2 exp h· À 1
kT
(2C3)
This eliminates the ultraviolet catastrophe because integrating over all frequencies gives a finite
integrated brightness a blackbody radiator at temperature T :
B(T ) Ñ
Z
1
B· (T )d· =
0
ÛT 4
Ù
(2C4)
where
ÛÑ
2Ù5 k4
erg
À5
Ù
5
:67
Â
1
0
15c2 h3
cm2 s K4 sr
is called the Stefan-Boltzmann constant. Beware that the dimensionless sr in the
denominator is dropped by many authors.
The spectral energy density per unit solid angle of radiation is just the specific intensity I· divided
by the flow rate c . For isotropic radiation, the spectral energy density is
u· = 4ÙI· =c
R1
so the total radiation energy density u = 0 u· d· of blackbody radiation is
4ÛT 4
u=
c
(2C5)
The frequency ·max at which B· , the brightness per unit frequency of a blackbody, is maximum
is the solution of
@B·
= 0;
@·
it is
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Blackbody Radiation
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·max Ù
59 GHz
:
T (K)
Note that ·max isn't quite the same as c=Õ max Ù 103 GHz, where
Õmax Ù
0:29 cm
T (K)
is the wavelength at which BÕ , the brightness per unit wavelength, is highest. The latter
equation is the familiar form of Wien's Law used by optical astronomers, whose spectrometers
measure wavelengths instead of frequencies.
The electrical power generated by a warm resistor
A resistor is any electrical device that absorbs all of the electrical power applied to it; it is the
"black body" of electric circuits. A warm resistor at temperature T > 0 K generates electrical
noise. The amount of power generated depends only on T , and the frequency spectrum of the
noise is independent of the resistor material, obeying laws analogous to the blackbody radiation
laws. Also, the electrical noise generated is indistinguishable from the noise coming from an
antenna observing a blackbody radiator. Warm resistors are useful in radio astronomy
as standards for calibrating receiver gains, and the power received by a radio telescope is often
described in terms of the resistor temperature required to generate the same power thermally.
Thus the gain of a radio receiver is normally calibrated by connecting it alternately to hot and
cold loads having known temperatures.
The derivation of the electrical power per unit bandwidth P· generated by a resistor is essentially
a one-dimensional version of the three-dimensional argument used for blackbody radiation
[Nyquist, H. 1928, Phys. Rev. 32, 110; Barrett, A. H. 1970, Fundamentals of Radio Astronomy,
p. 22-23 (unpublished)]. For radio frequencies, h· Ü kT , the Rayleigh-Jeans limit. Recall that
the Rayleigh-Jeans derivation starts with a large cube of side length a containing standing waves
of thermal radiation. The average energy in each standing-wave mode is hEi = kT , and the
number of modes with frequency · to · + d· is proportional to · 2 , so B· / · 2 . For the
resistor, consider two identical resistors at temperature T connected by a lossless transmission
line (e.g., a pair of parallel wires) of length a much larger than the longest wavelength of
interest.
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Blackbody Radiation
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We assume that the transmission line has a characteristic impedance matched to the resistance
of the resistors so that power is efficiently coupled between them. In equilibrium, the
transmission line can only support those wave modes satisfying
a=
nÕ
;
2
n = 1; 2; 3;:::;
where Õ is the wavelength. Electrical signals do not travel at exactly the speed of light on a
transmission line, but at some slightly lower velocity v < c . Thus
n=
2a·
v
and, for a µ Õ, the number of modes per unit frequency is
Án
2a
=
:
v
Á·
In thermal equilibrium, the classical Boltzmann law says that each mode has average energy
hEi = kT , so the average energy per unit frequency E· in the transmission line is
E· =
Án
2akT
kT =
v
Á·
This energy takes a time Át = a =v to flow from one end of the transmission line to the other,
so the classical power (energy per unit time) per unit frequency flowing on the transmission line
is
P· =
E·
= 2kT
Át
Note that the velocity v has dropped out. The spectral power P· generated by the two identical
resistors must be 2kT and, by symmetry, the spectral power generated by each resistor is
P· = kT
(2C6)
in the limit h· Ü kT . This equation is called the Nyquist formula and is the electrical equivalent
of the Rayleigh-Jeans law for radiation. Since the "space" of the transmission line has only one
dimension instead of three, the frequency dependence is · 0 instead of · 2 .
Still, the Nyquist formula suffers from an "ultraviolet catastrophe" of its own: the total power
integrated over all frequencies diverges. Planck's quantization rule is the cure for this problem
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Blackbody Radiation
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also: the electrical energy in each mode must be an integer multiple of h· , and the corrected
version of the Nyquist formula is
Ô
Õ
h·=(kT )
P· = kT
;
exp[h·=(kT )] À 1
where the quantity in large brackets is the same quantization correction previously found for
blackbody radiation. This equation is usually written in the form
P· =
h·
À h· Á
exp kT À 1
(2C7)
In many radio-astronomical applications, h· Ü kT and the simpler P· = kT (Eq. 2C6) can be
used.
At low frequencies · Ü kT=h , the specific intensity B· of blackbody radiation in three
dimensions (solid curve) is proportional to · 2 . Its one-dimensional analog, the spectral power
density P· of noise generated by a resistor (dashed curve), is proportional to · 0 . Quantization
causes both curves to cut off exponentially at high frequencies.
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