The Rayleigh-Jeans Law was a proposed solution to the problem of
blackbody radiation, first proposed by Lord Rayleigh in 1900 and
corrected slightly by James Jeans shortly thereafter. Like
Wien's Law, the Rayleigh-Jeans Law relies on classical physics, though
Rayleigh used statistical mechanics to study standing waves in a cavity
rather than the thermodynamical approach of
Wien. Rayleigh's relation could fit the long-wavelength side
of the blackbody curve, but it failed
catastrophically at short-wavelengths,
predicting that the energy density of a blackbody cavity would be
infinite at infinitely-small wavelengths. Like Wien's Law,
the Rayleigh-Jeans Law was superseded by Max Planck's quantum mechanical
relation for the blackbody now known to be correct.
The blackbody, prior to 1900
Gustav Robert Kirchhoff first outlined his
Laws of Radiation in 1859, describing
how light and matter interact. In his work, he stated that the spectrum
of light emitted by hot, solid objects must be some function of the
temperature of the object, and the wavelength of light emitted, as
Eλ = constant × f
(λ,T)
However, the problem was no one could figure out what that function, f,
should be. Several physicists had tried, and Wilhelm Wien
apparently got close in 1896 with Wien's Law, though that was soon
found to fail experimentally at very long wavelengths. Wien had used purely
thermodynamic arguments for his relation. Lord Rayleigh
would take a different track.
The Rayleigh-Jeans Law
Rayleigh had the
idea of using statistical mechanics to study standing waves of light
inside a blackbody cavity. He derived his relation in the following way.
As with the one-dimensional case of sound waves
in a pipe, light could be described as a series of standing "ether" waves
inside a three-dimensional cavity. Assume the box is a cube with sides
of length l, and that the standing waves intersected the three sides
of the box at angles
θ1,θ2,θ3. For
standing waves, there must be an integer number of wavelengths
within the wavetrain, so the system can be described by the following
equations:
- (2l/λ) cos (&theta1) =
i1
- (2l/λ) cos (&theta2) =
i3
- (2l/λ) cos (&theta3) =
i3
-
cos2θ1 +
cos2θ2 +
cos2θ3 = 1
which then yields
i21 + i22 +
i23 = 4l2/λ2
This is an equation for a sphere in the
i1,2,3-space, with a radius of
2l/λ. The total number of unique integer combinations
which yield standing waves (i.e. the number of modes) corresponds to the
volume of one octant of this
sphere:
n = (1/8) (4π/3) (2l/λ)3
This number n is the number of degrees of freedom in the box, so
the number of degrees of freedom per unit volume is simply
n/l3, which we'll define as φ. Now, to get the
blackbody spectrum, we're interested
in the number of degrees of freedom
within a wavelength increment dλ. This is just the
absolute value of the derivative
of the above with respect to λ,
abs(dφ) = 4π
dλ λ-4
The blackbody emission spectrum is the
number of possible degrees of freedom within a blackbody cavity
multiplied by the average energy per degree of freedom.
We know from thermodynamics (I won't explain,
this is too long already) that each degree of freedom must have an
energy equal to (1/2)kT. So,
when we mush all this together, we find that the monochromatic energy
density, Eλ is
Eλ = C1
Eaverage × φ = C1
kTλ-4
Thus, Rayleigh's approximation to Kirchhoff's blackbody function f is
kT λ-4. Not too shabby.
The only problem now is that it's utterly, utterly wrong!
The Ultraviolet Catastrophe
One thing you should
always do when deriving any physical function is to look at what happens in
the limiting cases. Here, the important thing is when wavelengths are very
short, corresponding to bluer and bluer light. At any fixed temperature,
Rayleigh's relation predicts that the monochromatic energy density becomes
infinite at infinitely small wavelengths! This would mean, for
example, that when you turn on your toaster, you are instantly fried by a
massive gamma ray burst, since your little blackbody toaster should emit
infinite energy at the shortest wavelengths. Clearly, this is absurd, and
poor Rayleigh and Jeans knew it. This conundrum became known as
The ultraviolet catastrophe, meaning that their relation failed
spectacularly at short (e.g. ultraviolet) wavelengths.
The reason anyone paid any attention to their relation at all was because
it worked so nicely at long wavelengths,
precisely where Wilhelm Wien's blackbody radiation law failed.
So it appeared that blackbodies obeyed Wien's Law at short wavelengths,
and Rayleigh-Jeans Law at long ones.
Lord Rayleigh published this derivation around 1900, and James Jeans corrected
it slightly shortly thereafter. However, at right about the same time,
Max Planck published his own law of blackbody
radiation. Planck's Law, which ushered in the age of quantum mechanics
also used statistical mechanics to study the number of states inside the
blackbody cavity, but he treated the blackbody as if it were composed of a
large number
of little quantum oscillators. He also said that light wasn't exactly a
standing wave but was actually composed of little bits of energy called
quanta. Planck's function fit blackbody spectra perfectly at all
wavelengths, and we now know that his is the correct answer.
Like Wien's Law, the Rayleigh-Jeans law is a good approximation to the
blackbody spectrum in the appropriate limit. For the Rayleigh-Jeans Law
this is the long-wavelength limit, which makes it
convenient to use in radio astronomy where all observed wavelengths are
well into the Rayleigh-Jeans regime (even the cosmic microwave background
at a frosty 2.73 kelvins peaks in the microwave).
The Rayleigh-Jeans Law can also be derived from Planck's radiation law by
the use of a Taylor series. Since the Taylor expansion of exp(x) is
1 + x/1! + x2/2! + ..., the case where x is very
small can be approximated quite well by (1 + x). When this is applied to
Planck's Law when hc/λ is much smaller than kT, the
Rayleigh-Jeans Law is obtained.
Sources:
I obtained Rayleigh's derivation from Introduction to Modern Physics
by F.K. Richtmeyer (McGraw-Hill 1934). The rest came from my class notes.
blackbody radiation
Rayleigh-Jeans Law -- Planck's radiation law -- Wien's Law