My answer is as follows (and made more explicit in my ebook "Guidance Waves"). Following Lagrangian formalism, motion in a central field involves two degrees of freedom. In semiclassical Hamiltonian mechanics, see Schiller, Phys Rev. 125, 1100 (1962) for details. What this means is for n = 1, there are two wave components, corresponding to the relation n = (nr + 1/2) + (ℓ + 1/2) where nr and ℓ count the number of angular and radial nodes respectively, and the 1/2 implies a periodic time of two circulations. This, of course, means no classical orbit could possibly describe this, and given that the change of action represents the change of phase in each wave, and we add the condition that action can only change in discrete quanta (the condition for the phase change per period and that the period must be stationary) means that neither of these component waves can lose any further energy. Therefore, the electron cannot radiate. As an aside, a wave function can only have one energy, hence if the electron collides with the proton, it can only deliver 13.6 eV, whereas to convert a proton to a neutron it needs over 1 MeV, so it can't do that either.

So the answer depends on two postulates. There are two separate wave components, and the action in each can only change in discrete quanta. You will note this also gives rise to the spectral selection rules, so the proposition explains more than one thing.

Dear Aman,

This is not what we know from electromagnetism.

The idea that accelerated charges radiate and always loose energy is a long enduring theoretical concept that came from Larmer end of the 1800's before it could be experimentally verified that this does not happen.

From electromagnetism, we know, more specifically from application of the Coulomb force stemming from Maxwell's first equation, which is in fact Gauss's equation for the electric field, that quite the contrary, charges "accumulate" energy as they accelerate, and that this energy can be liberated only if the charge is stopped in some least action equilibrium state in an atomic orbital.

The best documented case is the electron accelerating towards a proton that stabilizes in the hydrogen atom ground state, releasing the 13.6 eV of kinetic energy that it accumulated as it accelerated before being stopped at the mean ground state distance from the proton.

Ian gave a good summary of how this is dealt with from the quantum mechanics perspective.

At our macroscopic level, Coolidge tubes also give us easy means to verify this by emitting x-ray photons coming from the cathode, as electrons accelerate after being liberated from the anode until they are captured by positively charged ions on the cathode, each X-ray photon carrying away the exact amount of energy accumulated by Coulomb acceleration by each electron liberated from the anode.

Best Regards

André.

Dear Aman,

Having gone through the suggestions and explanations offered by Andre and Ian,I think these two views suffice for your questions and read carefully the Physical review as stated by Andre for further insight.

Hope it helps.

Best regards,

Lateef.

Electrons do lose or gain energy when accelerated. This is called synchrotron radiation. It is used in microwave and millimetre tubes (and synchrotrons) to generate power, and is the reason why particle accelerators are made with very large diameter rings, to keep the acceleration, and thus the radiation, as low as possible. An electron beam that is rippled up and down (undulatron) radiates.

Electrons in atoms are not described only by electromagnetism, but by quantum mechanics, which is more fundamental, and means that electrons round an atom are completely prevented from radiating small amounts of energy at a time, even if they are accelerating. The particle going round in a circle is a partly helpful but ultimately poor description of an electron round an atom, so the concept of it continually accelerating is probably not valid anyway.

Dear Malcolm,

Electrons do not produce synchrotron radiation because they are accelerated.

Synchrotron radiation is produced because moving electron beams are forced to suddenly deflect their trajectories sidewise in synchrotrons, and wigglers of all sorts. Each time their trajectory is forced sidewise, synchrotron photons are emitted due to the sudden transverse slowing down.

When electrons are accelerated in straight line without their trajectories being forced sidewise in unnatural ways, no transverse energy emissions are produced.

Also, when accelerated in Betatrons, in which perfectly circular trajectories can be obtained, no synchrotron radiation is produced.

In high energy storage rings, synchrotron radiation is produced because the trajectory cannot be kept perfectly circular due to multiple booster magnetic fields along the ring, which is not the case in Betatrons.

In atoms, electrons are now known not to move on circular orbits but to be stabilized in least action axial electromagnetic equilibrium states in orbitals. They do not radiate energy unless they emit excess energy when returning to least action orbitals after having been excited to further away metastable orbitals by energy coming from outside the atom, which amounts to them being axially wiggled with respect to nuclei, which is what produces the visible light frequencies that allow us to see about us.

https://www.omicsonline.org/open-access/on-adiabatic-processes-at-the-elementary-particle-level-2090-0902-1000177.pdf

Best Regards

André

Andre, you are correct that I was sloppy in what I said, and electrons (or charged particles in general) do not emit radiation when they are accelerated in the direction they are already going (or presumably when they are first moved). However, a quick survey confirms what I thought, which is that all charged particles radiate when accelerated sideways or slowed down, and this includes constant acceleration in a circle. This is referred to variously as Betatron radiation, Cyclotron radiation, Synchrotron radiation and Bremsstrahlun radiation. The only real difference is the energy level, with relativistic effects sharpening the synchrotron beam.

In particular, I found several direct references to Betatron radiation, and to cyclotron radiation from charged particles travelling in constant magnetic fields.

The following are quotes from Wikipedia, which, while not authoratitive, agree with other stuff I found.

Broadly speaking, bremsstrahlung or braking radiation is any radiation produced due to the deceleration (negative acceleration) of a charged particle, which includes synchrotron radiation (i.e. photon emission by a relativistic particle), cyclotron radiation (i.e. photon emission by a non-relativistic particle), and the emission of electrons and positrons during beta decay. However, the term is frequently used in the more narrow sense of radiation from electrons (from whatever source) slowing in matter.

Cyclotron radiation is emitted by all charged particles travelling through magnetic fields, not just those in cyclotrons.

For example

https://physics.aps.org/articles/v8/36;

Cyclotron Radiation from One Electron

• Patrick Huber, Center for Neutrino Physics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

April 20, 2015• Physics 8, 36

Electric charges radiate when moving in circular orbits in a magnetic field, an effect that was first predicted by Oliver Heaviside in 1904. Cyclotron radiation has been observed in astrophysical radio sources and is the basis for generating x rays at synchrotrons. Radiation emitted by these sources comes from the motion of many electrons, but Project 8 is the first experiment to detect cyclotron radiation from a single electron.

Dear Malcolm,

Your documentation does not completely account for the Betatron case.

For the Betatron, when multi electrons beams are used on perfectly circular orbits, synchrotron radiation is produced only because even though the total typical beam is in perfect circular orbit, the individual electrons are not on individual perfectly circular orbits within the beam and oscillate transversally due to mutual interaction.

John P. Blewett observed and documented with the GE Betatron that when the beam was thinned out sufficiently for electrons to be individually moving in a perfect circle, no energy whatsoever was emitted irrespective of the energy of the electrons.

Such perfectly circular orbits of individual electrons cannot be achieved with synchrotrons, which is why it has not been observed since the GE Betatron was decommissioned in the 1940's.

John P. Blewett. Radiation Losses in the Induction Electron Accelerator, Phys. Rev. 69, 87 (1946).

His results are on record and can be verified.

It is a pity that this is not correctly documented in the community and that Blewett's research has not been continued.

Individual electrons do not radiate when moving in really perfect circular orbits as can be achieved only in a betatron in a stable or smoothly varying magnetic field, but the slightest transverse disturbance in their motion will cause transverse bremmsstrahlung photons to be emitted as you say. The same for longitudinal slowing downs.

Best Regards

André

http://www.feynmanlectures.caltech.edu/I_34.html

You will find in this Feynman lecture that synchnotron radiation is emitted by a charge travelling in a circle in a uniform magnetic field. - It actually doesn't travel in a circle because it slows down as it radiates - but it does radiate in a uniform magnetic field.

Dear Malcolm,

I know of Feynman's lectures and most literature referring to the Betatron.

This is why I said that it is a pity that Blewett's results are not well referenced, if at all, and have not been continued in the community.

Sorry to say, but Feynman's comment only means that he did not know about the details of Blewett experiments that were being conducted as he was himself deeply captivated by his contribution in progress to establishing QED.

Electrons travelling in a perfect circle at our macroscopic level do slow down due to energy expended as work due to the 2nd principle of thermodynamics, but without any radiation being emitted, as observed by Blewett.

Blewett's formally published results are not going away and will remain available for whoever cares to become informed about them. Reference already given.

Best Regards

André

The book

Synchrotron Radiation and Its Applications

By Igorʹ Mikhaĭlovich Ternov, V. V. Mikhailin, V. R. Khalilov

includes Blewett's work and also discusses the effect of quantum effects on syncrotron radiation (which is relevant to the original question) and deals with synchrotron radiation in constant uniform magnetic field.

This CERN accelerator school gives these details about Blewett's failure to observe radiation from his betatron.

https://cds.cern.ch/record/235242/files/CERN-94-01-V1.pdf

Then in 1946 Blewett measured the energy loss due to SR in the G.E. 100 MeV betatron and found agreement with theory, but failed to detect the radiation after searching in the microwave region of the spectrum [4]. It was later pointed out by Schwinger however that the spectrum peaks at a much higher harmonic of the orbit frequency and that the power in the microwave region is negligible [5]. The first direct observation of the radiation as a "small spot of brilliant white light" occured by chance in the following year at the G.E. 70 MeV synchrotron, when a technician was looking into the transparent vacuum chamber [6]. Later accounts of the event may be found in Refs. [7,8].

Dear Malcolm,

Thank you so much for this reference by Ternov et al. that I had not come across during my research.

I note however that the "small spot of brilliant white light" was obtained in a sychrotron.

Since synchrotrons by very design are unable to exactly reproduce a "perfect circular orbit" as can be done in a betatron, as I mentioned already, I think that this experiment does not address the issue of the absence of radiation detection in the betatron.

Obviously to me, the slightest transverse oscillation out of perfect circular orbit will obviously generate transverse bremmsstrahlung radiation in a synchrotron, particularly if the energy is cranked up, however "close" to perfect circular orbit electrons can be constrained into in a synchrotron, which I think is what has to have been observed.

Schwinger's argument is valid of course, but I think that a really conclusive test of his observation can only be made in a betatron which to my knowledge is the only design that by structure is able to induce perfect circular orbits of individual charges. In view of these considerations, I will thus reserve my judgment until such a test is conducted.

Thank you again for this very valuable reference.

Best Regards

André

Thank You all.