How does this experimental result show photon emission?

[rtareen]

TL;DR

This experiment (not sure what it is called) is supposed to show that light is emitted as photons. However, the results can be explained in terms of Maxwell's model. Book section attached. Book is Sears & Zemansky University Physics 14th edition.

First I'll explain my understanding, because I'm not very confident in it. The main point is that the electrons are ejected and then accelerated to a very high kinetic energy. Then they start smashing into the anode. Most will go through a series of collisions before completely stopping, so that the decelerations will vary, and thus the emitted frequency will vary. But the "lucky" ones will get stopped immediately, so they emit the max frequency, since they undergo the max deceleration.

The book says the fact that there is a max frequency shows that these x-rays are being emitted as photons. However, if we use Maxwell's model, there should still be a max frequency since there is a limit to the initial potential energy, so that there is a limit to the max deceleration. So what I am not understanding?

(PS It was never really explained, but I'm assuming frequency is proportional to acceleration).

 

[vanhees71]

You are right, this doesn't at all show that the em. field has to be quantized, but you get the emission of em. radiation also from Maxwell's theory. What you describe is "bremsstrahlung".

[EDIT: This was wrong! See posting #4 by @PeterDonis ]

In fact, it is not so easy to really show that the em. field has to be quantized. A lot can be understood from the semiclassical theory, where the charges (electrons and/or atomic nuclei) are treated by quantum theory and the em. field as a classical field. The photoeffect, e.g., can be understood by first-order perturbation theory where classical em. radiation excites a quantum-mechanically described electron bound to an atom to the continuous spectrum, such that it escapes the atom. The same holds for the Compton effect.

The most simple hint at the fact that one indeed needs also to quantize the em. field was found by Einstein in 1917 when explaining the Planck black-body radiation formula from a kinetic argument, i.e., by the emission and absorption rates of em. radiation by the walls of the container. The thermal-equilibrium condition is that the emission rate equals the absorption rate such that the total number of photons in each wave modes stays constant in time (on average). It turned out that Einstein had to assume not only absorption and "induced emission" but also spontaneous emission. This latter contribution is due to the quantum fluctuations of the electromagnetic field and does not exist in the classical description of the em. field.

 

[PeterDonis]

if we use Maxwell's model, there should still be a max frequency since there is a limit to the initial potential energy, so that there is a limit to the max deceleration

No. In Maxwell's model, a limit on the energy carried by a single pulse of radiation does not imply any limit on the frequency of the radiation; a pulse carrying a given energy $E$ can contain radiation of any frequency (and would be expected to--the Maxwell model of the pulse would be a wave packet with no frequency cutoff). Only in the photon model does a limit on energy imply a limit on frequency.

 

[PeterDonis]

this doesn't at all show that the em. field has to be quantized

I disagree. See below.

you get the emission of em. radiation also from Maxwell's theory

But you don't get a limit on the frequency of the radiation from Maxwell's theory.

 

[vanhees71]

That's true! I stand corrected!

 

[PeterDonis]

It was never really explained, but I'm assuming frequency is proportional to acceleration

The energy carried by the radiation is proportional to the acceleration (actually deceleration). But in the Maxwell model, as noted, this does not imply anything about the frequency of the radiation. Only in the photon model does the energy carried by the radiation imply anything about its frequency.

 

[rtareen]

No. In Maxwell's model, a limit on the energy carried by a single pulse of radiation does not imply any limit on the frequency of the radiation; a pulse carrying a given energy $E$ can contain radiation of any frequency (and would be expected to--the Maxwell model of the pulse would be a wave packet with no frequency cutoff). Only in the photon model does a limit on energy imply a limit on frequency.

The energy carried by the radiation is proportional to the acceleration (actually deceleration). But in the Maxwell model, as noted, this does not imply anything about the frequency of the radiation. Only in the photon model does the energy carried by the radiation imply anything about its frequency.

So in Maxwell's model the deceleration says nothing about the frequency. How is frequency determined in Maxwell's model? The book says that, according to Maxwell, the deceleration will cause all frequencies to be emitted, specifically all x-ray frequencies. But if there's no limit wouldn't some gamma rays be emitted as well? I don't really understand that.

 

[PeterDonis]

How is frequency determined in Maxwell's model?

It isn't. All frequencies have an equal chance to be radiated in Maxwell's model.

if there's no limit wouldn't some gamma rays be emitted as well?

Under Maxwell's model, yes, we would expect gamma rays to be emitted. In other words, this is a wrong prediction of Maxwell's model.

 

[rtareen]

It isn't. All frequencies have an equal chance to be radiated in Maxwell's model.
Under Maxwell's model, yes, we would expect gamma rays to be emitted. In other words, this is a wrong prediction of Maxwell's model.

Makes sense! Thanks!

 

[vanhees71]

So in Maxwell's model the deceleration says nothing about the frequency. How is frequency determined in Maxwell's model? The book says that, according to Maxwell, the deceleration will cause all frequencies to be emitted, specifically all x-ray frequencies. But if there's no limit wouldn't some gamma rays be emitted as well? I don't really understand that.

In Maxwell's theory you can calculate the electromagnetic wave emitted by an accelerated point charge by using the socalled Lienard-Wiechert potential (retarded potential) or equivalently the Jefimenko-equations. What you get is a continuous spectrum of em. waves.

https://en.wikipedia.org/wiki/Bremsstrahlung

Due to quantum theory you must at least create one photon. If the electron has moved through an accelerating electrostatic potential $V$ it has an energy of $E=e V$. Thus for the frequency of the bremsstrahlung photons you get $\hbar \omega_{\text{max}}=e V$.

 

[vanhees71]

It isn't. All frequencies have an equal chance to be radiated in Maxwell's model.

The classical bremsstrahlung spectrum is not a uniform frequency distribution of course. You find the classical bremsstrahlung spectrum, e.g., for scattering at a Coulomb potential calculated in Landau and Lifshitz vol. 2.

 

[PeterDonis]

The classical bremsstrahlung spectrum is not a uniform frequency distribution of course.

Yes, you're right, I was being sloppy. But the classical spectrum does include an unbounded range of frequencies.