Lecture 27 — The Planck Distribution
Chapter 8, Friday March 21st
•Quick review of exam 2
•Black-body radiation
•Before Planck: Wien and Rayleigh-Jeans
•The ultraviolet catastrophe
•The Planck distribution
Reading:
All of chapter 8 (pages 161 - 186)
Homework 8 not due until Mon. Mar. 31st
Assignment will be handed out on Monday
Exam 2 – question 1
  n1, n2 , n3     3/ 2  n1  n2  n3 
Exam 2 – question 2
The Planck Distribution
A. A. Michelson (late 1900s): “The grand underlying principles (of
physics) have been firmly established... ...the future truths of
physics are to be looked for in the sixth place of decimals.”
Planck credited with the birth of quantum mechanics (1900)
- developed the modern theory of black-body radiation
Quantum nature of radiation
1st evidence from spectrum emitted by a black-body
What is a black-body?
An object that absorbs all incident radiation, i.e. no reflection
A small hole cut into a cavity is the most popular
and realistic example.
None of the incident radiation escapes
What happens to this radiation?
•The radiation is absorbed in the walls of the cavity
•This causes a heating of the cavity walls
•Atoms in the walls of the cavity will vibrate at frequencies characteristic of
the temperature of the walls
•These atoms then re-radiate the energy at this new characteristic frequency
The emitted "thermal" radiation characterizes the
equilibrium temperature of the black-body
Black-body spectrum
Black-body spectrum
•Black-bodies do not "reflect" any incident radiation
They may re-radiate, but the emission characterizes the black-body only
•The emission from a black-body depends only on its temperature
We (at 300 K) radiate in the infrared
Objects at 600 - 700 K start to glow
At high T, objects may become white hot
Stefan-Boltzmann Law
Power per unit area radiated by black-body
R = sT 4
Found empirically by Joseph Stefan (1879); later calculated by Boltzmann
s = 5.6705 × 10-8 W.m-2.K-4.
A black-body reaches thermal equilibrium when the incident radiation power is
balanced by the power re-radiated, i.e. if you expose a black-body to radiation, its
temperature rises until the incident and radiated powers balance.
Wien's displacement Law
lm T = constant = 2.898 × 10-3 m.K, or lm  T-1
Rayleigh-Jeans equation
Consider the cavity as it emits blackbody radiation
The power emitted from the blackbody is proportional to the radiation energy
density in the cavity.
One can define a spectral energy distribution such that u(l)dl is the fraction of
energy per unit volume in the cavity with wavelengths in the range l to l + dl.
Then, the power emitted at a given wavelength, R(l)  u(l)
u(l) may be calculated in a straightforward way from classical
statistical physics.
u(l)dl = (# modes in cavity in range dl) × (average energy of modes)
# of modes in cavity in range dl,nldl8pl-4dl
Average energy per mode is kBT, according to kinetic theory

u(l) = kBTnl8pkBTl-4
Wien, Rayleigh-Jeans and Planck distributions
8p kBT
e -  / lT
8p hc
uRJ  l  
; uW  l  
; uP  l  
4
5
l
l
l 5  ehc / lkBT - 1
Wilhelm Carl Werner Otto Fritz Franz Wien
The ultraviolet catastrophe
There are serious flaws in the reasoning by Rayleigh and Jeans
Furthermore, the result does not agree with experiment
Even worse, it predicts an infinite energy density as l  0!
(This was termed the ultraviolet catastrophe at the time by Paul Ehrenfest)
Agreement between
theory and
experiment is only
to be found at very
long wavelengths.
The problem is that
statistics predict an infinite
number of modes as l0;
classical kinetic theory
ascribes an energy kBT to
each of these modes!
Planck's law (quantization of light energy)
In fact, no classical physical law could have accounted for measured blackbody spectra
Max Planck, and others, had no way of knowing whether the
calculation of the number of modes in the cavity, or the average
energy per mode (i.e. kinetic theory), was the problem. It turned
out to be the latter.
Planck found an empirical formula that fit the data, and then made
appropriate changes to the classical calculation so as to obtain the
desired result, which was non-classical.
The problem is clearly connected with u(l)  , as l  0
The problem boils down to the fact that there is no connection between the energy
and the frequency of an oscillator in classical physics, i.e. there exists a continuum
of energy states that are available for a harmonic oscillator of any given frequency.
Classically, one can think of such an oscillator as performing larger and larger
amplitude oscillations as its energy increases.
Maxwell-Boltzmann statistics
Define an energy distribution function f  E   A exp  - E / kBT  ,such that
Then,


0
0

 f E  1
0
E   E f  E  dE   EA exp(- E / kBT )dE  k BT
This is simply the result that Rayleigh and others used, i.e. the average
energy of a classical harmonic oscillator is kBT, regardless of its frequency.
Planck postulated that the energies of harmonic oscillators could
only take on discrete values equal to multiples of a fundamental
energy  = hf, where f is the frequency of the harmonic oscillator,
i.e. 0, , 2, 3, etc....
Then,
En = nnhf
n = 0, 1, 2...
Here, h is a fundamental constant, now known as Planck's constant.
Although Planck knew of no physical reason for doing this, he is
credited with the birth of quantum mechanics.
The new quantum statistics
f n  A exp  - En / kBT   A exp  -nhf / kBT 
Replace the continuous integrals with a discrete sums:

f
n 0

 A exp  -nhf / k BT   1
n
n 0


n 0
n 0
E   En f n   nhf  A exp  -nhf / k BT 
Solving these equations together, one obtains:

hf
hc / l
E


exp  / kBT  - 1 exp  hf / kBT  - 1 exp  hc / lk BT  - 1
Multiplying by D(l), to give....
hcl -5
u (l ) 
exp  hc / l k BT  - 1
This is Planck's law
The results of Planck's law
Note: the denominator [exp(hc/lkBt)] tends to infinity faster than the numerator (l-5),
thus resolving the infrared catastrophe, i.e. u(l)  0 as l 0.
Note also: for very large l:
exp  hc / l k BT  - 1 
hc
l k BT
 u  l   l -4k BT
From a fit between Planck's law and experimental data, one obtains
Planck's constant to be:
h = 6.626 × 10-34 J.s
Planck's restriction of the available energies for radiation gets around the
ultraviolet catastrophe in the following way: the short wavelength/high frequency
modes are now limited in the energy they can have to either zero, or E  hf; in the
calculation of the average energy, these modes with high energy are cut off by the
Boltzmann factor exp(-E/kBT), i.e. these modes are rarely excited and, therefore,
contribute nothing to the average energy in the limit l 0.