# Why is black-body radiation curve smooth without a sharp cutoff?

Planck's law is able to predict a graph that is consistent with experimental observation: In essence, unlike Rayleigh-Jeans law that assumes equipartition theorem to hold (that each mode of motion shares equal energy at thermal equilibrium, so all modes excited at thermal equilibrium ⇒ ultraviolet catastrophe), Planck's hypothesis proposed that oscillators of frequency v will be excited only if they can acquire energy of at least hv , where h is Planck constant.1

So based on this argument, I am not sure why the shaded region of the graph will slope in a smooth manner when wavelength is being decreased. Say if you are only able to supply energy of hv, then all oscillators with frequency > v should not be excited and hence will not contribute to energy density ⇒ ρ = 0: I expect the behaviour to be same as that in photoelectric effect: if frequency of incidence wave is below threshold frequency, you will never observe photoemission. But the actual observed graph is the first one. Where did I go wrong? I am not a physicist so a not-too-complicated explanation will be appreciated.

1 Peter Atkins, Julio de Paula. Physical Chemistry (8th Edition). OUP. 2006. Page 247.

• Long story short, not all molecules in a sample have the same energy. Aug 13 '20 at 5:06
• Consider the thermal energy of molecules at given temperature does not have a sharp cut off either, so there is always some nonzero probability the energy hν can be obtained. In contrary, photons in photoemission contexts have the particular energy level, so there is the sharp cut-off edge. Aug 13 '20 at 7:30
• What made you think there's just one single way to radiate for any substance? Your argument would make sense only perhaps in very close vicinity of absolute zero! Aug 14 '20 at 22:18

The key to understanding the curve is thinking about how energy is distributed among components of the system

The mathematics that produce the curve involves some not-very-simple statistical mechanics some of which Planck didn't understand when he first developed his theory.

But it isn't that hard to get the intuitive idea. Consider the case of a gas where molecules have some kinetic energy. The individual molecules don't all have the same energy: some are moving faster than others. The temperature of the gas is function of the average kinetic energy of the molecules. But individual molecules are exchanging energy very frequently as they bang into each other. But those random collisions exchange random amounts of the kinetic energy between the molecules. Sometimes one molecule will gain a great deal of energy so it ends up with far more than the average amount. But the probability of a series of collisions giving a single molecule a very, very large kinetic energy is low and the higher that energy, the lower the probability. There is no sharp cut-off just an exponentially decreasing probability of getting a higher and higher energy.

Take that picture and apply the appropriate statistical probability theory and you get the overall Planck distribution: a curve where the average kinetic energy is given by the temperature but where individual molecules have some probability of having much lower energy (with slowly decreasing probability) and others have some probability of having much higher energy (but with a sharply decreasing probability). There is no sharp cutoff at high energies just sharply decreasing probability of reaching those levels.

In reality the picture is far more complex as molecules have vibrational and electronic energy as well as kinetic energy, but these details don't matter much for the intuitive picture.

Textbooks like Atkins and others do a severe injustice to science by re-writing fictitious history as if it were a nice and smooth story. You can see below in the quote that the thought process of Planck is an "act of desperation" as he called it himself. He had to derive a formula that will fit the experimental black body curve by all means even if it required violating classical physics rules. The original derivation in Planck's paper used very advanced mathematics and statistics (and I do not fully follow his arguments as a lowly chemist). You can roughly think at that you are heating a body which is glowing white and there is a distribution of frequencies of oscillators (hence advanced statistics). I remember reading that Planck was walking with his son and told him that I have discovered something which will be as important as Newtons. It was a feat!

In his time, electrons or modern atomic structure was not fully known and hence photoelectric effect was another story. It was Einstein who explained the photoelectric effect which earned a a Nobel Prize. In photoelectric effect there is no distribution of energies of incident photons. Roughly, you can say that is why the photoelectric effect is abrupt.

See Planck, the Quantum, and the Historians* by Clayton A. Gearhart Phys. Perspect. 4 (2002) 170–215.

letter from Planck to the American physicist Robert W. Wood, October 7, 1931: In this letter, Planck responded to Wood’s request for a description of the ‘‘considerations which had led me to propose the hypothesis of energy quanta.’’ Planck spoke of his work as an ‘‘act of desperation,’’ and said: I also knew the formula that expresses the energy distribution in the normal spectrum. A theoretical interpretation therefore had to be found at any cost, no matter how high. It was clear to me that classical physics could offer no solution to this problem, and would have meant that all energy would eventually transfer from matter to radiation. …This approach was opened to me by maintaining the two laws of thermodynamics. The two laws, it seems to me, must be upheld under all circumstances. For the rest, I was ready to sacrifice every one of my previous convictions about physical laws. …[One] finds that the continuous loss of energy into radiation can be prevented by assuming that energy is forced at the outset to remain together in certain quanta. This was purely a formal assumption and I really did not give it much thought except that no matter what the cost, I must bring about a positive result.