I expected to quickly finish up the blackbody radiation discussion and move on to more important subjects. But there's so much connected with this subject I can't give it up without trying to get a better understanding of it. It's more important and relevant than I thought.
For one thing, it's connected with the use of "normal modes" that crop up so often in quantum mechanics and in classical vibration theory. I discussed the significance of normal modes in quantum electrodynamics in the middle of my recent (12 Dec 2014) post called Illuminating Quantum Vacuum Quotes. Planck's 2nd paper from 1900 on the blackbody radiation formula he discovered is titled, "On the theory of the Energy Distribution Law of the Normal Spectrum." (He first presented it as a talk or colloquium for his colleagues.)
Normal modes are often called characteristic modes, and both terms refer to the description of a vibrating or oscillating system in terms of its independent or non-interacting vibrations or oscillations. Finding the normal modes means describing the system in the simplest way possible. But what exactly is a "mode"? It's almost the same thing as a single frequency of oscillation, but not quite.
Normal modes are often called characteristic modes, and both terms refer to the description of a vibrating or oscillating system in terms of its independent or non-interacting vibrations or oscillations. Finding the normal modes means describing the system in the simplest way possible. But what exactly is a "mode"? It's almost the same thing as a single frequency of oscillation, but not quite.
Let's look at one of the simplest systems that can be described in terms of normal modes of vibration:
|~~~~~~~~~~~~~~~0~~~~~~~~~~~~~0~~~~~~~~~~~~~~|
The curvy lines are springs, the 0s are masses, and the system is confined between two walls and can oscillate along the horizontal direction only, like it's sitting on a frictionless table. The normal modes of this system are single-frequency motions of the masses in unison: one where both masses are moved the same distance to the right or the left and let go (they move in phase with each other), and one where they are moved toward (or away from) each other and let go (out of phase with each other). More complicated motions are linear superpositions (addition of amplitudes of) these two normal modes.
This system is confined to a one dimensional box. You can imagine making it two dimensional by thinking of springs attached to the top and bottom of each mass. This would be enclosed in a square. Next, make the leap to three dimensions, with springs also attached to the back and front of each mass, and the system enclosed in a cube. Finally, imagine more masses and more springs forming a large grid or lattice in this cubic box.
Well, that's pretty much what Planck described in his Energy Distribution Law of the Normal Spectrum, but the only thing in the box was radiant energy: light waves of many different frequencies. Well, most of the frequencies were not in the visible region of the spectrum, so let's say electromagnetic radiation of many different frequencies. What kind of box was it? A platinum box! It was a platinum-walled oven, coated on the inside with soot (or maybe iron oxide). There was a little hole in the oven for experimenters to be able to measure the frequencies and intensities of light in it as the temperature was gradually changed.
Such a box, more often called a cavity, is a perfect emitter of thermal radiation, or as perfect as can be achieved. Remember that a blackbody is a perfect emitter and a perfect absorber? With radiation coming from a tiny hole in a heated, insulated, metal box, absorption of light from outside is not something that needs to be considered. No radiation is assumed to be absorbed through the small hole, and the emission and absorption occurring inside the box is due to its temperature only. The absorption part of the required equilibrium between emission and absorption is produced simply from heating the box.
Two more related comments about boxes containing vibrations. An empty box full of sound waves whose frequencies are multiples of the dimensions of the box is another type of model on which the blackbody-box (cavity radiation) spectrum calculation is based. The "box of sound" has its acoustical normal modes of vibration, just as the "box of light" has its electromagnetic normal modes. Lord Rayleigh (William Strutt), an older contemporary of Planck's, did both these calculations, the box-of-sound then then the box-of-light, before anybody else, but the formula he found in the case of light didn't fit the experimental data. Planck came along just months later and got it right by using the quantization of energy of the emitters and absorbers (models for atoms) in the walls of the box. These are the sources of the radiation in the box, and the radiation they produce is due entirely to the temperature of the box.
Also, the 3-dimensional box I mentioned above, with a lattice of equal masses and equally springy springs, is the simplest model of matter in the solid state. This model was first proposed by Einstein in 1907. He used Planck's idea of quantized emission and absorption of energy, but it was not in this case electromagnetic energy. It was acoustical or sound wave energy!
Einstein's model was oversimplified in that he assumed all the atoms vibrated at one frequency. He didn't do a normal-mode calculation! His model nevertheless predicted that the heat capacity of a solid would go to zero as the temperature of the solid approached zero, something that had been observed experimentally but that classical physics didn't predict. The problem was Einstein's model didn't predict the correct rate at which the heat capacity decreased. As described in the Wikipedia entry on the Einstein Solid:
In Einstein's model, the specific heat approaches zero exponentially fast at low temperatures. This is because all the oscillations have one common frequency. The correct behavior is found by quantizing the normal modes of the solid in the same way that Einstein suggested. Then the frequencies of the waves are not all the same, and the specific heat goes to zero as a
power law, which matches experiment. This modification is called the Debye Model, which appeared in 1912.
The "normal modes" link is a good one to follow to get educated on that subject.
Two more related comments about boxes containing vibrations. An empty box full of sound waves whose frequencies are multiples of the dimensions of the box is another type of model on which the blackbody-box (cavity radiation) spectrum calculation is based. The "box of sound" has its acoustical normal modes of vibration, just as the "box of light" has its electromagnetic normal modes. Lord Rayleigh (William Strutt), an older contemporary of Planck's, did both these calculations, the box-of-sound then then the box-of-light, before anybody else, but the formula he found in the case of light didn't fit the experimental data. Planck came along just months later and got it right by using the quantization of energy of the emitters and absorbers (models for atoms) in the walls of the box. These are the sources of the radiation in the box, and the radiation they produce is due entirely to the temperature of the box.
Also, the 3-dimensional box I mentioned above, with a lattice of equal masses and equally springy springs, is the simplest model of matter in the solid state. This model was first proposed by Einstein in 1907. He used Planck's idea of quantized emission and absorption of energy, but it was not in this case electromagnetic energy. It was acoustical or sound wave energy!
Einstein's model was oversimplified in that he assumed all the atoms vibrated at one frequency. He didn't do a normal-mode calculation! His model nevertheless predicted that the heat capacity of a solid would go to zero as the temperature of the solid approached zero, something that had been observed experimentally but that classical physics didn't predict. The problem was Einstein's model didn't predict the correct rate at which the heat capacity decreased. As described in the Wikipedia entry on the Einstein Solid:
In Einstein's model, the specific heat approaches zero exponentially fast at low temperatures. This is because all the oscillations have one common frequency. The correct behavior is found by quantizing the normal modes of the solid in the same way that Einstein suggested. Then the frequencies of the waves are not all the same, and the specific heat goes to zero as a
power law, which matches experiment. This modification is called the Debye Model, which appeared in 1912.The "normal modes" link is a good one to follow to get educated on that subject.
