Black body radiation and particle nature of light.

[americast]

It is given in my book that the phenomenon of black body radiation can be used to prove the particle nature of light. They have also mentioned that the wavelength-intensity relationship "cannot be explained satisfactorily on the basis of wave theory of light." But why?

Thanx in advance...

 

[Born2bwire]

A theory that explained blackbody radiation over the broad spectrum required that light be emitted in energy quanta, which turns out to be the photon. A wave theory allows for energy to be emitted in a continuous range of values. And models that allowed this to happen were only valid for specific cases of the spectrum.

 

[americast]

Can you be a little more specific?

Which theory says that light should be emitted only as photons?
Why can we not say that "waves" of different intensities are emitted when a black body (or anybody for that matter) is heated?

Why can atoms emit energies only in discrete quantities?...

[ If required, borrow Gargantua's mouth :D ]...
Thanx a lot...

 

[Nugatory]

Which theory says that light should be emitted only as photons?
Why can we not say that "waves" of different intensities are emitted when a black body (or anybody for that matter) is heated?

Google for "ultraviolet catastrophe".

At a hand-waving level... If the light were not quantized, then the black body would be able to radiate at all frequencies, all the way up to hard x-rays. But that's not what actually happens, because light is quantized; if there's not enough energy to produce at least one quantum of radiation at a given frequency, then you get zero radiation at that frequency.

 

[jtbell]

The classical prediction for the blackbody radiation spectrum is the Rayleigh-Jeans Law:

http://en.wikipedia.org/wiki/Rayleigh–Jeans_law

A Google search will turn up a lot more information about its derivation and limitations.

 

[Jano L.]

It is given in my book that the phenomenon of black body radiation can be used to prove the particle nature of light. They have also mentioned that the wavelength-intensity relationship "cannot be explained satisfactorily on the basis of wave theory of light." But why?
Thanx in advance...

This is a common and mistaken view, perpetrated in textbooks and physicists' myth flora based on poor understanding of the Rayleigh-Jeans calculation.

This calculation basically assumes that

I) energy of EM radiation is given by the Poynting formula

$$
\int \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0}B^2 dV
$$

and

II) each Fourier component of the total field carries average energy $k_B T$ (equipartition).

None of these are necessary in wave theory of light; they are additional assumptions.

From the above assumptions, Rayleigh and Jeans derived that the spectral curve of equilibrium radiation should behave as square of frequency. This is confirmed by measurements only for low frequencies, while in reality there is a maximum intensity and for high enough frequencies the measured spectral curve decays to zero.

The correct "cannot" statement is thus

the wavelength-intensity relationship cannot be explained satisfactorily on the basis of Rayleigh-Jeans calculation.

The Rayleigh-Jeans curve is wrong experimentally, but it was clear already then that it is wrong even theoretically from the point of view of classical physics: according to R-J, total energy of EM radiation in equilibrium comes out infinite, which is invalid result and means some error was made in the calculation.

Rayleigh, Lorentz and Planck at that time in the beginning of 20th century were well aware of the fact that using equipartition for EM oscillators leads to result that is wrong for high frequencies and is already theoretically unacceptable. But when you read what they wrote on the subject you will hardly get the impression they thought there is a problem with wave theory of radiation at all. Rayleigh and Lorentz correctly pointed out that using equipartition for EM radiation is a long shot and since it does not work, it cannot be considered valid.

Max Planck explained the experimental spectrum with another theory. He never used equipartition, but assumed that material oscillators radiate in packets of energy $h\nu$. Still, he maintained that EM radiation is to described by Maxwell equations and by the wave picture just as it was before. See his excellent book, The theory of heat radiation:

https://archive.org/details/theheatradiation00planrich

Although his calculation gave good result - the Planck spectral curve - it had some issues and did not gain popularity. Instead, another explanations of the same curve were proposed and these two gained more popularity:

* Debye's calculation, replacing equipartition value $k_B T$ by modified formula for average energy of harmonic oscillator

$$
\frac{h\nu}{e^{\frac{h\nu}{k_B T}}-1}
$$

* Einstein's derivation based on kinetic equations for probabilities of discrete states of atoms in the equilibrium radiation and the Bohr formula $E_2 - E_1 = h\nu_{12}$

Both of these discard the idea of equipartition and replace it with quantization of energy. But even if they give spectral curve that corresponds to the experimental curve, they in no way prove that wave theory is wrong or that quantization of energy is necessary. They only derive Planck's formula, that is all.

They too have their issues, just as Planck's original derivation had them. In Debye's calculation, one has to add energy $-h\nu/2$ to the average energy of harmonic oscillator calculated in quantum theory. This is an unclear ad-hoc operation motivated by the desire to obtain the already known Planck function.

Einstein's derivation has different problem: it is based on the old quantum theory, where radiation consists of buckshots of energy $h\nu$ and atoms jump in between discrete states. This is very simplistic view of radiation and atoms and is inconsistent with both Schroedinger's and Maxwell's equations that are regarded as basic physical laws today.

Back to the wave (non-quantum) theory of radiation, it may very well still be able to explain the spectral curve of equilibrium radiation if Poynting formula is replaced by another one. Without Poynting formula, there is no infinity of harmonic oscillators in the energy and thus it may not be divergent.

For example, if matter consists of point charged particles (like in Frenkel's or Feynman-Wheeler electromagnetic theory), the Poynting formula is invalid. Another formula for energy holds then:

$$
\int \frac{1}{2}\sum_{a}\sum_{b}{}^{'} \mathbf E_a \cdot \mathbf E_b + \mathbf B_a \cdot \mathbf B_b ~dV
$$
where the index denotes to which particle the field belongs. This is not quadratic in field and does not allow transformation into sum of squares; there are no harmonic oscillators in energy expression here.

 

[americast]

This is a common and mistaken view, perpetrated in textbooks and physicists' myth flora based on poor understanding of the Rayleigh-Jeans calculation.

This calculation basically assumes that

I) energy of EM radiation is given by the Poynting formula

$$
\int \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0}B^2 dV
$$

and

II) each Fourier component of the total field carries average energy $k_B T$ (equipartition).

None of these are necessary in wave theory of light; they are additional assumptions.

From the above assumptions, Rayleigh and Jeans derived that the spectral curve of equilibrium radiation should behave as square of frequency. This is confirmed by measurements only for low frequencies, while in reality there is a maximum intensity and for high enough frequencies the measured spectral curve decays to zero.

The correct "cannot" statement is thus

the wavelength-intensity relationship cannot be explained satisfactorily on the basis of Rayleigh-Jeans calculation.

The Rayleigh-Jeans curve is wrong experimentally, but it was clear already then that it is wrong even theoretically from the point of view of classical physics: according to R-J, total energy of EM radiation in equilibrium comes out infinite, which is invalid result and means some error was made in the calculation.

Thanx a lot for your replies... So, here I am drawing a short inference:

The particle nature of light was not known when Reyleigh-Jeans derived a formula on the basis of the Poynting formula. It worked for low frequencies but in higher frequencies (such as for UV), they fail and this is what is UV catastrophe. Now, Planck came out with a formula (assuming light to be particles or quantized) which could well-explain the phenomenon, but Planck himself could not justify the reason why he had considered light to be made of packets. Therefore, people did not give it much importance. When Einstein discovered photo-electric effect (and thus the particle nature of light), people realized that may be it is the dual nature of light which made Planck's formula successful.
The only thing which I cannot understand here is that how according to R-J, total energy of EM radiation in equilibrium comes out infinite. In Poynting's formula, what does μ₀ stand for?

Thanx a lot...!

 

[Jano L.]

Now, Planck came out with a formula (assuming light to be particles or quantized) which could well-explain the phenomenon, but Planck himself could not justify the reason why he had considered light to be made of packets.

No! Planck had two theories of the equilibrium spectrum. In both of them, quantization of energy applied only to the material oscillators that radiate EM waves. He assumed that change of energy of this oscillator happens in a novel way, where the amount $h\nu$ plays role, but the change was not necessarily instant. It was not quantum theory in today's sense. Energy of the EM radiation was still assumed to be changing continuously in time, not by sudden steps. Planck did not assume that light is made of particles, quanta of energy, photons or anything like that.

The idea of quantum of light of energy $h\nu$ was introduced, I believe, by Einstein, in his paper on quantum theory of radiation and photoelectric effect. He was able to derive the Planck function using this assumption and the idea that atoms jump in between discrete states, while releasing the quantum of light with energy $h\nu$, which was assumed to equal the difference of energies of the atoms in the two states: $h\nu = E_2-E_1$.

 

[Jano L.]

The only thing which I cannot understand here is that how according to R-J, total energy of EM radiation in equilibrium comes out infinite. In Poynting's formula, what does μ₀ stand for?
Thanx a lot...!

$\epsilon_0$ is called permittivity of vacuum. $\mu_0$ is called permeability of vacuum. They are constants introduced by the SI system of units.

The energy comes out infinite because the integral of the R-J spectral function is infinite:

$$
\int_0^\infty \frac{8\pi k_B T}{c^3} \nu^2 d\nu = +\infty.
$$

 

[americast]

$\epsilon_0$ is called permittivity of vacuum. $\mu_0$ is called permeability of vacuum. They are constants introduced by the SI system of units.

The energy comes out infinite because the integral of the R-J spectral function is infinite:

$$
\int_0^\infty \frac{8\pi k_B T}{c^3} \nu^2 d\nu = +\infty.
$$

Great! Thanx...

 

[americast]

No! Planck had two theories of the equilibrium spectrum. In both of them, quantization of energy applied only to the material oscillators that radiate EM waves. He assumed that change of energy of this oscillator happens in a novel way, where the amount $h\nu$ plays role, but the change was not necessarily instant. It was not quantum theory in today's sense. Energy of the EM radiation was still assumed to be changing continuously in time, not by sudden steps. Planck did not assume that light is made of particles, quanta of energy, photons or anything like that.

The idea of quantum of light of energy $h\nu$ was introduced, I believe, by Einstein, in his paper on quantum theory of radiation and photoelectric effect. He was able to derive the Planck function using this assumption and the idea that atoms jump in between discrete states, while releasing the quantum of light with energy $h\nu$, which was assumed to equal the difference of energies of the atoms in the two states: $h\nu = E_2-E_1$.

Okay. So, Planck himself did not know that he was assuming the particle nature of light, right?

thanx...

 

[Jano L.]

Okay. So, Planck himself did not know that he was assuming the particle nature of light, right?

thanx...

He was not assuming particle nature of light. He assumed that light was EM radiation described by Maxwell's equations just as before him most physicists did.

 

[Nugatory]

Okay. So, Planck himself did not know that he was assuming the particle nature of light, right?

"not know he was assuming" is a sort of weird formulation; better to say that he was assuming that light was the classical EM waves described by Maxwell's classical EM equations.

Planck's work was one of several threads (some others would be Bohr and atomic stability, the photoelectric effect and spectral lines) that started around the turn of the century and came together as quantum mechanics over the next few decades. The pop-sci histories tends to oversimplify this history by making statements such as "Planck's work with black-body radiation proved that..." or "Einstein used the photoelectric effect to show that..."; there's a fair amount of hindsight in these statements.