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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 Like Wien's Law, the Rayleigh-Jeans Law was superseded by Max Planck's quantum mechanical relation for the blackbody now known to be correct.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 θ123. 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 = 4l22

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

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