Radiation in a Cavity (Rayleigh, Planck (1900))
Introduction
A particularly difficult problem arose during
the 19th century – What are the properties
of the radiation from a perfect “black body”
(an object which absorbs or emits radiation
without
favouring
any
particular
frequencies)?
A good approximation to a black body is a
cavity with a small aperture. In the example
shown, a hollow sphere with a small hole in
it is heated – black body radiation emerges
from the aperture.
A number of facts were established experimentally:
1. Power emitted per unit area ∝ T4 (Stefan, 1879)
Intensity
A simple relationship between the wavelength where the peak intensity occurs and the
temperature: λmaxT = Constant
λ m ax
100
1000
10000
W a v e le n g th (n m )
But – No model to explain these observations! (Classical thermodyanmics can predict
Stefan’s law, but not Wien’s displacement law or the form of the spectrum)
The Rayleigh-Jeans Law
Lord Rayleigh attempted a solution in 1900
(see Atkins, chapter 1 and Appendix 1)
Consider an enclosure with perfectly
reflecting sides, with dimensions p, q and r.
Such a cavity can support modes of radiation
provided the wavelength, λ, in each mode
satisfies the following relationships:
p=
lλ
mλ
nλ
where l, m and n are integers.
,q =
,r =
2
2
2
We can then write:
( p 2 + q 2 + r 2 ) = 41 λ2 (l 2 + m 2 + n 2 )
The total number of modes for which the wavelength is greater than or equal to λ is equal
to the number of ways l, m and n can be chosen to satisfy the inequalities:
4( p 2 + q 2 + r 2 )
(l + m + n ) ≤
and l , m, n ≥ 0 (negative modes are meaningless!)
λ2
2
2
2
Imagine a three-dimensional space with the axes labelled l, m and n.
The above inequalities therefore define an ellipsoid in this
space, and the number of modes, N, is then simply the
volume of the octant of this ellipsoid with l , m, n ≥ 0 . Hence,
N = 81 ( Vol. of ellipsoid)
= 81 ( 43 πlmn )
pqr
λ3
V
= 43 π 3 (V = volume of cavity)
λ
= 43 π
The number of modes per unit volume for which the
wavelength is greater than or equal to λ is then simply:
n( λ ) = 43 π
1
λ3
Differentiating with respect to λ gives the mode density (the number of modes per unit
volume per unit wavelength)
D( λ )dλ =
4π
dλ
λ4
Re-writing in terms of frequency, and allowing for the fact that there are two polarisations,
we obtain:
8πν 2
D(ν )dν =
dν
c3
In classical statistical mechanics, every mode carries an average energy kT (k is
Boltzmann’s constant and T is the absolute temperature)
Hence, the energy density is given by:
8πν 2
U (ν )dν = kTD(ν )dν =
kTdν The Rayleigh-Jeans Law.
c3
This result agrees well with experiment at low frequencies, but at high frequencies, a
problem becomes apparent – The Ultraviolet Catastrophe! The Rayleigh-Jeans Law
predicts that the energy density at high frequencies tends to infinity, i.e., all bodies should
radiate infinite amounts of energy! This is shown below for a temperature of 6000 K
Planck’s Law
At about the same time, Max Planck was also working on the problem of black body
radiation. He made the assumption that the energy content of each mode is not a
continuous variable. Rather, each mode may contain only discrete amounts of energy*
(i.e., the energy is quantised) given by E = mhν where m is an integer and h is a
universal constant, now known as Planck’s constant.
In this case, the average energy of a mode may be calculated from the Boltzmann
distribution:
∞
E =
∑ mhνe
− ( mhν kT )
m =0
∞
∑e
− ( mhν kT )
where e
−( mhν kT )
is the Boltzmann factor.
m =0
∞
Using the result that
∑x
m
= (1 − x )
−1
∞
for x<1, and
m=0
∑ mx
m =0
∞
m
= x (∂ ∂x ) ∑ x m
we obtain:
m =0
hν
E =
e
hν
kT
−1
Hence, the energy density as a function of frequency is given by:
U (ν )dν =
8πhν 3
dν
⋅ hν
Planck’s Law
3
c
kT
e
−1
This result gives the correct behaviour at all frequencies, and predicts the properties
previously found experimentally.
(*It is interesting to note that Planck regarded the oscillators producing the radiation as
quantised, rather than the radiation field itself. Later (1931) Planck was to describe his
derivation as “an act of desperation” – but this does not refer to the quantisation of energy,
rather to his use of Boltzmann statistics which was a controversial subject at the end of
the nineteenth century. For more details on Planck’s work on the radiation law, see “The
Genesis of Quantum Theory 1899–1913” (Armin Herman), Library ref. 530.12)
Some properties of black body radiation
1. Radiant emittance (Power emitted per unit area)
c
W (ν,T ) = U (ν ) W / m 2
4
2. Power at the peak of the distribution
Planck’s Law ⇒ Wien’s Law
2.9 × 10 6
λmax =
nm and, emittance at λmax, W max = 129
. × 10 −15 T 5 W / m 2
T
3. Total power output
Planck’s Law ⇒ Stefan’s Law
Integrating over all frequencies gives:
W = 5.68 × 10 −8 T 4 W / m 2
Some examples
1. The Sun (T≈6000K ⇒ λmax ≈ 500nm)
2. Xe flashgun (T ≈ 10000K (transient) ⇒ λmax ≈ 290nm)
3. The universe! (T ≈ 2.7K ⇒ λmax ≈ 1mm)
Limitations of thermal sources
A hot black body radiator may seem like a good idea for a light source. However, there are
a number of problems:
1. Brightness can only be increased by sacrificing monochromaticity or directionality.
2. The radiation cannot be well focused due to poor spatial coherence.
3. Use in interference experiments is limited by poor spatial and temporal coherence.
Coherence ⇒ phase relationship between different parts of the wavefront, in both time
and space. These quantities are simply related to the bandwidth, ∆ν, of the source.
Coherence time, ∆t = 1∆ν . Coherence length, ∆l = c∆t
e.g.
Source
Coherence length
Coherence time
Tungsten lamp
~ 1 µm
~3 fs
Low pressure Hg lamp
3 cm
0.1 ns
Cadmium lamp
20 cm
0.7 ns
Lasers
0.3 m – 1 km
1 ns – 3 µs