The Black Body Radiation
= Chapter 4 of Kittel and Kroemer
The Planck distribution
Derivation
Black Body Radiation
Cosmic Microwave Background
The genius of Max Planck
Other derivations
Stefan Boltzmann law
Flux => Stefan- Boltzmann
Example of application: star diameter
Detailed Balance: Kirchhoff laws
Another example: Phonons in a solid
Examples of applications
Study of Cosmic Microwave Background
Search for Dark Matter
Phys 112 (S06) 5 Black Body Radiation short
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B.Sadoulet
The Planck Distribution***
Photons in a cavity
Frequency !
Angular frequency ! = 2"#
Mode characterized by
Number of photons s in a mode => energy
Occupation number
Massive
Particles
Photons
1st
Quantification
Wave like
Discrete
states
Obvious
2nd
Quantification
Discrete
particles
Obvious
Planck
# photons
/mode
s =
1
# h! &(
exp%
)1
$ " '
! = sh" = sh#
h"
!" =
( )
exp
h"
#
$1
Radiation energy density between ω and ω+dω?
2 different polarizations
h! 3d!
8" hv 3dv
u! d! =
= u+ d+ =
h
!
$
'
$
$
'
$ h+ ' '
" 2 c 3 & exp & ) * 1)
c 3 & exp & ) * 1)
% # ( (
% # ( (
%
%
Phys 112 (S06) 5 Black Body Radiation short
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B.Sadoulet
Cosmic Microwave Radiation
Big Bang=> very high temperatures!
When T≈3000K, p+e recombine =>H and universe becomes transparent
Redshifted by expansion of the universe: 2.73K
Phys 112 (S06) 5 Black Body Radiation short
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B.Sadoulet
Other derivations
Grand canonical method
s = N =
What is µ?
!(" logZ )
=
!µ
! +e" e
1
% h# $ µ (
exp'
*$ 1
& " )
µ ! + µ e " µ e = 0 # µ! = 0
Microcanonical method
To compute mean number photons in one mode, consider ensemble of N
oscillators at same temperature and compute total energy U
We have to compute the multiplicity g(n,N) of number of states with energy U= number of
combinations of N positive integers such that their sum is n
%
1 "#
1
1
1(
%' Nh$ (*
=
=
log 1 +
=
log'1 + *
&
! "U h$
U ) h$
&
s)
Phys 112 (S06) 5 Black Body Radiation short
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s =
1
# h! &(
exp%
)1
$ " '
B.Sadoulet
Fluxes
Energy density traveling in a certain direction
So far energy density integrated over solid angle. If we are interested in energy density
traveling traveling in a certain direction, isotropy implies
h! 3 d!d$
u! (" ,# )d!d$ =
' ' h! * *
4% 3 c 3 ) exp) , - 1,
Note if we use ν instead of ω
( (& + +
2h! 3 d!d$
u! (" ,# )d!d$ =
& & h! ) )
c 3 (exp( + , 1+
' '% * *
Flux density in a certain direction=brightness
(Energy /unit time, area, solid angle,frequency)
opening is perpendicular to
direction
1
2h! 3 d!dAd"
I! d!dAd" = d! u! cdtdA d" =
424
3
$ h! ' '
dt 1
2$
energy in
c &exp& ) * 1)
cylinder
% %# ( (
dA
cdt
Flux density through a fixed opening
1
J! d!dA = d!
dt
$
2%
0
dA
θ
(Energy /unit time, area,frequency)
2%h! 3 d!dA
c
d" $0 u! cdt cos#dA d cos# =
= u! dvdA
14
4244
3
' ' h! * * 4
energy in cylinder
c 2 ) exp) , - 1,
( (& + +
n
1
Phys 112 (S06) 5 Black Body Radiation short
!
5
cdt
B.Sadoulet
Stefan-Boltzmann Law
Total Energy Density
Integrate on ω
!2
! 2k B4
4
4
u=
with aB =
3 3 " = aB T
15h c
15h3c 3
Total flux through a fixed aperture***
multiply above result by c/4
!2
!2 k B4
4
4
$8
J=
with # B =
W/m 2 /K 4
3 2 " = # BT
3 2 = 5.67 10
60h c
60h c
Stefan-Boltzmann constant!
Phys 112 (S06) 5 Black Body Radiation short
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B.Sadoulet
Detailed Balance: Kirchhoff law
Definition:
A body is black if it absorbs all electromagnetic
radiation incident on it
Usually true only in a range of frequency
e.g. A cavity with a small hole appears black to the outside
Detailed balance:
in thermal equilibrium, power emitted by a system=power
received by this system! Otherwise temperature would change!
Consequence: The spectrum of radiation emitted by a black body is the
uBB (! )d! = ucavity (! )d!
“black body” spectrum calculated before
Cavity
Black
Body
Absorptivity, Emissivity:
Black
Body
Absorptivity =fraction of radiation absorbed by body
Emissivity = ratio of emitted spectral density to black body spectral
density.
Kirchhoff: Emissivity=Absorptivity a(! ) = e(! )
Grey
Body
Phys 112 (S06) 5 Black Body Radiation short
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Entropy, Number of photons
Entropy
" 2 d"
! = # !" d x 2 3
$c
! " = # $ ps log ps
3
!=
s
Number of photons
N! =
Proportional
Phys 112 (S06) 5 Black Body Radiation short
4u
4 aB
V=
VT 3
3"
3 kB
V 2" (3) 3 30" (3) aB 3
aB
3
$
=
VT
%
0.37
VT
kB
#2 c3h 3
#4 kB
! " 3.6N#
8
B.Sadoulet
Spectrum at low frequency
log Iv
Rayleigh-Jeans region (= low frequency)
if h! << "
#! $ "
log v
! 2 d!
8$ % 2 d%
u! d! " # 2 3 u% d% " #
$ c
c3
Power / unit area/solid angle/unit frequency
=Brightness
2! 2 d!dAd"
2d!dAd"
I ! d!dAd" # $
=
$
c2
%2
! 2 d!
2%d"
Power emitted / unit (fixed) area
J! d! = J" d" # $ 2 2 = $ 2
4% c
&
2
Ae = !r d Power received from a diffuse source
Ae
dP
2#" e Ae 2# "r Ar
Ar
= I!" e Ae =
=
= 2#
2
2
!e = 2
d!
$
$
!e
d
dP
(1polarization) = "
Detected
Power
(1
polarization)
d
If diffraction limited:
d!
2
1 dP
!r Ar = "
Antenna temperature
!r
TA =
(1polarization)
kB d!
Ar
Phys 112 (S06) 5 Black Body Radiation short
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B.Sadoulet
Applications
Many!
e.g. Star angular diameter
Approximately black body and spherical!
Spectroscopy =>Effective temperature
Apparent luminosity l =power received per unit area
But power output
!e
d
Ar
!e =
Ar
d2
l=
L! e 1
L
=
4" Ar 4"d 2
4
L = 4!r 2" BTeff
r2
4
! l = 2 " BTeff
d
r
l
angular diameter = ! =
d
" B Teff4
=> Baade-Wesserlink distance measurements of varying stars
• Oscillating stars (Cepheids, RR Lyrae)
• Supernova
assuming spherical expansion
Phys 112 (S06) 5 Black Body Radiation short
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B.Sadoulet
Phonons in a solid
Phonons:
Quantized vibration of a crystal
h" ! s = sh"
! = h" p = hk =
cs
1
Velocity
s =
Same as for photons but
#% h! &(
exp
)1
3 modes
$ " '
Maximum energy (minimum wavelength/finite # degrees of freedom)
= Debye Energy
=
described in same way as photons
If s phonons in a mode
Debye approximation:
!
• isotropic
!
"
or p= !with velocity cs = constant
• k=
cs
cs
!D
hc #
N&
k
Introducing
the Debye temperature TD = ! = s % 6" 2 (
kB $
V'
3 4
T4
U = ! kB N 3
5
TD
Phys 112 (S06) 5 Black Body Radiation short
" T %3
12 4
CV = ! k B N $ '
5
# TD &
11
1/ 3
=
h) D
kB
# T &3
12 4
! = " kB N% (
15
$ TD '
B.Sadoulet
Applications
Calorimetry:
!E
Measure energy deposition by temperature rise
!E
!T =
" !need small C
C Heat Capacity
Bolometry: Measure energy flux F by temperature rise
Chopping e.g., between sky and calibration load
Sky
!F
!T =
!F
Load
"!need small G but time constant=
G
Very sensitive!
C
limited by stability " small heat capacity C
G
Heat Conductivity
Heat capacity goes to zero at low temperature
Study of cosmic microwave background
Search for dark matter particles
Phys 112 (S06) 5 Black Body Radiation short
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C ! T3
B.Sadoulet