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
1
B.Sadoulet
The Planck Distribution***
Photons in a cavity
Mode characterized by
Frequency !
Angular frequency ! = 2"#
Number of photons s in a mode => energy
s is an integer ! Quantification
! = sh" = sh#
Similar to harmonic oscillator
Photons on same "orbital" cannot be distinguished. This is a quantum state, not s systems in
interactions!
Occupation number
!
System =1 mode , in contact with reservoir of temperature
sh"
$
!
1
Partition function
Z = %e # =
h" (
s =0
1 ! exp& !
' # )
&( h# )+
Mean number of photons
sh#
exp
"
"
1
%
log
Z
'
$ *
s = ! sp(s ) = ! se $ = "
=
& h# )+
& h# )+
Z s
s
%(
1 " exp( "
' $ *
' $ *
1
s =
# h! &(
!" =
***Planck Distribution
exp%
)1
$ " '
exp
h"
h"
( )
Phys 112 (S06) 5 Black Body Radiation
2
#
$1
B.Sadoulet
2 quantifications
1st
Quantification
Wave like
2nd
Quantification
Discrete
particles
Phys 112 (S06) 5 Black Body Radiation
Massive
Particles
Discrete
states
Photons
Obvious
Cavity modes
Obvious
Planck
3
B.Sadoulet
Photon properties
Maxwell equations
in vacuum
r
r
r
r #
r
'B
!"E =
= 0 !% E = &
r
r
$
't r
r r
r r o &r
%E )
%E 1 %E r
! " B = 0 ! # B = µo ( j + $o
= µ o$ o
=
( j = 0!)
%t +*
%t c 2 %t
'
r
2E
r
1
#
r
r r r r r r
! " 2 E = 2 2 (wave equation)
Using identity ! " ! " E = ! # (! # E) $ ! 2 E
c #t
rr
r r
r r
"
Solutions E = Eo exp i k .x ! "t with k = and k .Eo = 0
c
h! "
p = hk =
=
c
c
=> Photon has zero mass and 2 polarizations
((
))
Quantifications
y
1st quantification: Photons in a cubic perfectly conductive box L
Electric field should be normal to the surfaces
#
& #
& #
&
0
"x ( % "y ( % "z (
%
E x = E x sin !t cos nx
sin n
sin n
%
( % y
( % z
(
L ' $
L ' $
L '
$
#
&
#
& #
&
"x (
"y ( % "z (
%
%
0
E = E sin !t sin nx
cos ny
sin n
%
(
%
( % z
(
y
y
L '
L ' $
L '
$
$
#
& #
&
#
&
0
"x ( % "y (
"z (
%
%
E z = E z sin !t sin nx
sin n
cos nz
%
( % y
(
%
(
L ' $
L '
L '
$
$
2nd
n x ,n y ,n z should be integers
that should satisfy the wave equation
L
!2 2
"2
h2 2
h 2" 2
2
2
2
2
(n x + n y + n z ) = c 2 or 4L2 (n x + n y + n z ) = c 2
L2
This the superposition of momenta
n h
n h
n h
h"
p x = ± x , p y = ± y , p y = ± x such that p x2 + p y2 + p z2 =
2L
2L
2L
c
x
2
#o 2 2 2
# o E 0 L3
1 r r r r
3
3
U = "space (E ! D + B ! H )
d x = U = "space
E +c B )
d x=
= sh% & sum of 4 cos 2
quantification:
(
3
2
2
16n x n y n z $
time
time
r r
r r
0
0
0
Moreover, ! " E = 0 # n " E = 0 # n x E x + n y E y + n z E z =r0
=> for each n x ,n y ,n z ,s, 2 independent ways to choose E 0 (2 polarizations)
Phys 112 (S06) 5 Black Body Radiation
4
B.Sadoulet
Black Body Radiation
Radiation energy density between ω and ω+dω?
Note that the first quantification give the same prescription on how
momentum states are distributed in phase space as for massive particles.
Taking into account that we have 2 different polarizations, the state
(=mode)density in phase space
is
d 3 xd 3 p
d 3 x p 2 dpd! d 3 x # 2 d#d!
2
=
=
3
3 3
h
4" h
4 " 3c3
The density of energy flowing in the direction (θ,ϕ ) is therefore
1
d 3 xd 3 p
h!
d 3 x ! 2 d!d$
3
u! (" ,# )d xd!d$ = h{
!
2
=
3
3 3
&
)
&
)
&
)
&
)
h
!
h
!
h
4
c
energy
24
3 (exp( + , 1+
+ , 1+14
/photon ( exp(
of
' ' % * * density
' '% * *
states
144244
3
or
mean number
of photons
3
h! d!d$
' ' h! * *
4% 3 c 3 ) exp) , - 1,
( (& + +
If we are not interested in the direction of the photons, we integrate on solid
angle and we get
h! 3d!
8" hv 3dv
u! d! =
= u+ d+ =
h
!
$
'
$
$
'
$ h+ ' '
" 2 c 3 & exp & ) * 1)
c 3 & exp & ) * 1)
% # ( (
% # ( (
%
%
u! (" ,# )d!d$ =
Phys 112 (S06) 5 Black Body Radiation
5
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
Conclusions:
• Very efficient thermalization
In particular no late release of energy
• No significant ionization of universe since!
e.g., photon-electron interactions= Sunyaev-Zel'dovich effect
• Fluctuations of T => density fluctuations
Phys 112 (S06) 5 Black Body Radiation
6
B.Sadoulet
WMAP 3 years 3/16/06
Power Spectrum of temperature fluctuations
TT
TT
TE
EE
BB
Fitted Cosmological Parameters
Phys 112 (S06) 5 Black Body Radiation
Correlation Power Spectra of temperature fluctuations and E/B polarization
7
B.Sadoulet
The genius of Max Planck
Before
Physicists only considered continuous set of states (no 2nd quantization!)
Partition function of mode ω
d"
' "* &
Z = $0# exp ) % , =
!
( & + "0
"0
Mean energy in mode ω
# 2 $log Z
!" =
= # independent of ω
$#
Sum on modes => infinity =“ultraviolet catastrophe”
Max Planck
1
!"
$ h" ')
1 ! exp& !
% # (
!
h"
!" =
$ h" '
exp& ) * 1
% # (
!
h!
# h! &
# h! &
# h! &
h!
h! << " exp% ( ) 1 +
* + ) " h! >> " exp% ( >> 1 * + ) h! exp%,
(- 0
!
!
$" '
$
'
$
"
"
" '
Cut off at h! " # The energy of 1 photon cannot be smaller than h!! => finite integral
Quantized!
Z=
Phys 112 (S06) 5 Black Body Radiation
8
B.Sadoulet
A technical remark
Important to sum with series instead of integral
If we had used a continuous approximation
Z! =
+
*
0
#
$ h! x '
dx exp & "
=
instead of
% # )( h!
1
"
h!
#
,h,,
-1
! >> #
1" e
we would have add the same problems as the classical physicists
We would have underestimated zero occupancy for large ! !
$ !xh" '
exp &
% # )(
1
For large ω big drop in first bin
But discrete sum still gives 1!
xh!
Phys 112 (S06) 5 Black Body Radiation
9
B.Sadoulet
Other derivations (1)
Grand canonical method $ µ ! h"
Consider N photons Z = * exp&
%
N
!(" logZ )
s = N =
=
=>
!µ
(
)N ' =
1
)
#
( 1! exp$µ ! h" '
&
)
%
# (
1
% h# $ µ (
exp'
*$ 1
& " )
What is µ? There is no exchange of photons with reservoir
!" R
!N #BB
(only exchange of energy) => entropy of reservoir does not
change with the number of photons in the black body (at constant energy)
Zero chemical potential! =>
1
s =
Microscopic picture
=
U
$µ #
=0
%
# h! &(
exp%
)1
$ " '
Photons are emitted and absorbed by electrons on walls of cavity
! +e" e
We have the equilibrium
Special case of equilibrium (cf Kittel & Kroemer Chap. 9 p. 247)
A +B!C
Equilibrium corresponds to maximum entropy
=>
"!
"!
"!
In particular
d! U,V =
dN A +
dN B +
dNC = 0
"N A
"NB
d! = 0
"NC
! µ A + µ B " µC = 0
µ ! + µ e " µ e = 0 # µ! = 0
this is due to the fact that the number of photons is not conserved
dN A = dN B = !dNC
Phys 112 (S06) 5 Black Body Radiation
10
B.Sadoulet
Other derivations (2)
Microcanonical method
To compute mean number photons in one mode, consider ensemble of N
oscillators at same temperature and compute total energy U
U
!
U
=>
! = lim
s =
= lim
h$ N"# Nh$
N"# N
Microcanonical= compute entropy
! (U ) with U = nh" (n photons in combined system)
1 "#
and define temperature at equilibrium as
=
$!! (U ) $ U (! ) $ s (! )
! "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
N
! ! ni = n =
i =1
U
ni # 0
h"
Same problem as coefficient of tn in expansion (cf Kittel chap. 1)
=>
g( n, N ) =
1 !
n
( )
1
n! !t n 1 " t
# ! m& N
% "t (
$ m= 0 '
N
=
t=0
1
n!
=
( )
1
1)t
N
=
" g(n, N )t n
n
N # ( N + 1) # ( N + 2 ) # # # ( N + n " 1) =
( ) ( )
( N + n " 1)!
n! ( N " 1)!
(
(
Using Stirling approximation
n
N
U
U
Nh#
! (n) = log( g(n, N )) " N log 1 +
+ n log 1 +
= N log 1 +
+
log 1 +
N
n
h#N
h#
U
=>
%
1 "#
1
1
1(
% Nh$ (*
=
=
log'1 +
=
log'1 + * or
&
! "U h$
U ) h$
&
s)
Phys 112 (S06) 5 Black Body Radiation
11
)
s =
1
# h! &(
exp%
)1
$ " '
)
B.Sadoulet
Counting Number of States
From first principles (Kittel-Kroemer)
In cubic box of side L (perfectly conductive=>E perpendicular to surface)
#
& #
& #
&
#
&
#
&
#
&
0
0
"x ( % "y ( % "z (
"x (
"y (
"z (
%
%
%
%
E x = E x sin !t cos nx
sin n
sin n
B = B x cos !t sin nx
cos ny
cos nz
%
( % y
( % z
( x
%
(
%
(
%
(
L ' $
L ' $
L '
L '
L '
L '
$
$
$
$
#
&
#
& #
&
#
& #
&
#
&
"x (
"y ( % "z (
"x ( % "y (
"z (
%
%
%
%
0
0
E = E sin !t sin nx
cos ny
sin n
B = B c cos !t cos nx
sin n
cos nz
%
(
%
( % z
(
%
( % y
(
%
(
y
y
y
y
L '
L ' $
L '
L ' $
L '
L '
$
$
$
$
#
& #
&
#
&
#
&
#
& #
&
0
0
"x ( % "y (
"z (
"x (
"y ( % "z (
%
%
%
%
E z = E z sin !t sin nx
sin n
cos nz
B z = B z cos !t cos nx
cos ny
sin n
%
( % y
(
%
(
%
(
%
( % z
(
L ' $
L '
L '
L '
L ' $
L '
$
$
$
$
(
=> ω has specific values! (“1st quantification”)
2 degrees of freedom (massless spin 1 particle)
("2nd quantification")
U=
"
space
2
2 2
!2 2
"2
2
2
2 " L
n
+
n
+
n
=
#
n
=
x
y
z
L2
c2
! 2 c2
with the wave equation constraint
1 r r r r
(E ! D + B ! H )
2
r
! r r0
B0 =
n#E
"L 2 2
r 2 n ! r
$ B0 = 2 2 B0
" L
d x =U =
3
time
"
space
)
r r
r r
! " E = 0 # n " E = 0 # n x E x0 + n y E y0 + n z E z0 = 0
#o 2 2 2
(E + c B )
2
2
# o E 0 L3
d x=
= sh% & sum of 4 cos 2
3
16n x n y n z $
time
3
=> number of states in mode ω is 2xnumber of integers satisfying wave equation constraints
2 2
Sum on states can be replaced by integral on quadrant of sphere of radius squared nx2 + n2y + n2z = ! 2 L2
" c
Energy in ω ω+dω
$
U! d! = 2
n x ,n y ,n z
U d! = 2
!
4"
h!
2
n dn
8
exp
( )
h!
#
Phys 112 (S06) 5 Black Body Radiation
"! L
2
=
$1
3
"c
h!
h!
( )
( )
exp
h! d!
3
3
exp
h!
#
"
=2
#1
%quadrant
h!
( )
L
$1
12
3
d n
h!
exp
#1
"
U!
1
u! d! = 3 d! = 2 3
h! 3 d!
" c exp$& h! ') * 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
3
u! (" ,# )d!d$ =
h! d!d$
' h! * *
3 3'
4% c ) 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,
( (& + +
Phys 112 (S06) 5 Black Body Radiation
n
1
!
13
cdt
B.Sadoulet
Stefan-Boltzmann Law***
Total Energy Density
Integrate on ω
"
u d!
0 !
#
Change of variable
"
$4
" x 3dx
#0 u! d! = h 3% 2c 3 #0 e x & 1
" 1
0 $ 2 c3
=#
! "x=
h!
#
h! 3
d!
&( h! )+
exp
,1
' % *
" x 3 dx
=
0 e x !1
#
$4
15
!2
! 2k B4
4
4
u=
with aB =
3 3 " = aB T
15h c
15h3c 3
Total flux through an aperture
Integrate Jn
= multiply above result by c/4
!2
!2 k B4
4
4
$8
2 4
J=
"
=
#
T
with
#
=
=
5.67
10
W/m
/K
B
B
3 2
3 2
60h c
60h c
Stefan-Boltzmann constant!
Phys 112 (S06) 5 Black Body Radiation
14
B.Sadoulet
Detailed Balance
Definition:
on it
A body is black if it absorbs all electromagnetic radiation incident
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
“black body” spectrum calculated before
Cavity
Black
Body
Proof :
c
c
AuBB = Aucavity ! uBB = ucavity
4
4
We could have inserted a filter
Phys 112 (S06) 5 Black Body Radiation
15
uBB (! )d! = ucavity (! )d!
B.Sadoulet
Kirchhoff law
Absorptivity, Emissivity:
Absorptivity =fraction of radiation absorbed by body
Emissivity = ratio of emitted spectral density to black body spectral
density.
Kirchhoff law: Emissivity=Absorptivity
Black
Body
received power = a(! )uBB (! )d! = emitted power = e(! )u BB (! ) d!
Grey
Body
Applications:
a(! ) = e(! )
Johnson noise of a resistor at temperature T
We will come back to this !
Rayleigh ! Jeans =>Spectral density = k BT "
Superinsulation (shinny)
Phys 112 (S06) 5 Black Body Radiation
16
2
d Vnoise
d#
= 4kB TR
B.Sadoulet
Entropy, Number of photons
Entropy
• Method 1 (Kittel) use thermodynamic identity+ 3rd law
dU
du
at constant volume
V
2
d!
d"
V
= d" = d"
"
"
" d!
Third law : ! = #0
d"
=
s
Remembering density of states
s h"
$
4# V
3
15h c
"
3
2
4$2V 3 4 aB 3
" =
VT
3 kB
45h 3c3
• Method 2 : sum of entropy of each mode
! " = # % ps log ps = log Z" +
=
dU = !d"
( ( )) ( ( ) )
= # log 1 # exp
#h"
$
h"
+
" 2d"
& 1 u u )+
! = # !" d x 2 3 = V (
+
(integrating first term by part)
'3 % % *
$ c
$ exp
h"
$
#1
3
!=
Finally
Number of photons
Starting from
s =
1
# h! &(
exp%
)1
$ " '
4u
4 aB
V=
VT 3
3"
3 kB
N! =
" x s !1
dx
0 e x !1
V ,
# 2 d#
"2 c3 -0 %' exp%' h# (* + 1(*
& & $ ) )
= (s !1)!$ ( s) with $ (3) % 1.22
Classical integral
#
=>
V 2" (3)
30" (3) aB 3
aB
N! = 2 3 3 $ 3 =
VT
%
0.37
VT 3
4 k
kB
# ch
#
B
Note that entropy is proportional to number of photons!
Phys 112 (S06) 5 Black Body Radiation
17
! " 3.6N#
B.Sadoulet
Spectrum at low frequency
log Iv
Rayleigh-Jeans region (= low frequency)
! 2 d!
8$ % 2 d%
u! d! " # 2 3 u% d% " #
$ c
c3
Power / unit area/solid angle/unit frequency =Brightness
2h! 3d! dAd"
2! 2 d! dAd"
2d! dAd"
I! d! dAd" = c
+#
=
#
c2
,2
$ h! ' '
3$
c & exp & ) * 1)
! 2 d!
2%d"
% # ( (
%
Power emitted / unit (fixed) area
J! d! = J" d" # $ 2 2 = $ 2
4% c
&
2#"e Ae 2#"r Ar
Power received from a diffuse source dP
Ae
= I! "e Ae =
=
2
2
d
!
$
$2
Ae = !r d
dP
2
!e
= 2"
If telescope is diffraction limited: !r Ar = "
d!
A
Detected Power (1 polarization)
dP
!e = 2r
(1polarization) = "
!r
d
d!
Antenna temperature
1 dP
TA =
(1polarization)
Ar
kB d!
Phys 112 (S06) 5 Black Body Radiation
18
B.Sadoulet
if h! << "
d
log v
#! $ "
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
d! d
l
1 dr#
Luminosity/temperature variation
=
=
dt dt " BTeff4
d dt
Doppler
dr
!"
velocity v// =
c = //
dr//
"
dt
If spherical expansion dr
dr
//
! d = ddt"
= !
dt
dt
dt
Phys 112 (S06) 5 Black Body Radiation
19
B.Sadoulet
Phonons in a solid
Phonons:
Quantized vibration of a crystal
h"
!
=
h
"
p
=
hk
=
described in same way as photons
cs
1
If s phonons in a mode
s =
# h! &(
mean occupancy
exp%
)1
$
'
"
Same as for photons but
•3
! s = sh"
Velocity
modes (3 polarizations)
d 3 xd 3 p
d 3 x p 2 dp
3" 2 d"
3
3
integrating over angles 3
=d x
3
2 3
h
2! h
2! 2 cs 3
12
4 4
3
• Maximum energy = Debye Energy
D (" ) d"
But half-wave length smaller than lattice spacing is meaningless
k
=
x
!
<
where a is the lattice spacing " maximum energy #
a
D
<=> solid is finite: If N atoms each with 3 degrees of freedom, at most 3N modes
N atoms =
()
L
3
(cubic lattice) !
a
#
"
0
D
VD ( " )d" = 3N where D ( " ) is the density of states
Debye
approximation:
!
• isotropic
!D
•
k
Phys 112 (S06) 5 Black Body Radiation
k=
!
cs
or p=
20
"
!with velocity cs = constant
cs
B.Sadoulet
Phonons in a Solid
Debye law
"
!D
0
U=
VD (! )d! = "
!D
0
! d x!
3
kD
0
3!
V 2 3 d! = 3N $ ! D
2# cs
2
h"
3 k 2 dk
=V
$
' 2+ 2
$ h" '
*1
&% exp &%
# )( )(
for ! << " D
!
"D
0
%
='
'&
1
6N 2 3 ( 3
# cs *
*)
V
( )
= 6#
1
2 3
cs
a
h"
3 " 2 d"
$
' 2+ 2 cs 3
$ h" '
*1
&% exp &%
# )( )(
3! 4
U = 3 2 3V
2h # cs
&
%
0
x 3 dx
#2
=
V kB4T 4
x
3
3
e $ 1 10h cs
hc #
N&
Introducing the Debye temperature TD = ! = s % 6" 2 (
kB $
V'
T4
3
U = ! 4 kB N 3
5
TD
Phys 112 (S06) 5 Black Body Radiation
3
"
%
12
T
CV = !4 k B N $ '
5
# TD &
21
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
Sky
Chopping e.g., between sky and calibration load
!F
!T =
!F
Load
"!need small G but time constant=
G
Very sensitive!
C
limited by stability " small heat capacity C
G
Heat Conductivity
3
Heat capacity goes to zero at low temperature
C
!
T
5 1
Fluctuations
! " E = kB T 2 C # T 2 M 2
Wide bandwidth (sense every frequency which couples to bolometer)
Study of cosmic microwave background
4K ! 300mK ! 100mK
Bolometers
170g at 20mK
Search for dark matter particles
Phys 112 (S06) 5 Black Body Radiation
22
B.Sadoulet