Time to return to my long-delayed series of posts on the history of invisibility physics! The first two posts were:

- Acceleration without radiation (1910), describing Ehrenfest's arguments suggesting acceleration without radiation could be possible,
- Schott's radiationless orbits (1933), describing G.A. Schott's analytical demonstration that a charged spherical shell could move in a periodic orbit without producing radiation.

Our next stop in the study of invisibility physics is a pair of results, one by G.A. Schott in 1937 and another by D. Bohm and M. Weinstein in 1948, in both of which it is suggested that under the right circumstances, not only can an extended charged particle oscillate without radiating, but that it can also oscillate under the influence of its own electromagnetic field, without the application of an external force!

It is worth mentioning again the basic ideas behind these 'radiationless orbits', thanks to my long invisibility hiatus. It is well understood in the physics community that accelerating point charges, such as the electron, will produce radiation. This is the basis of radio transmitters, which use oscillating electrical currents to produce radio waves, and it also results in high-energy radiation being emitted from particle accelerators such as the synchrotron at Fermilab, a process known as synchrotron radiation.

In the early days of atomic physics, before the structure of the atom was well-understood, the radiation of accelerating charges posed a bit of a theoretical problem. It was generally accepted that electrons were present in the atom, and moving about, but it seemed that such electrons would have to radiate away all of their energy almost immediately. This was especially a problem for nucleo-planetary models of the atom, in which electrons were thought to orbit around a central, positive nucleus.

In 1910, Ehrenfest demonstrated that, classically, one could construct *extended* (i.e. non-point) oscillating charge distributions which do not produce radiation. Very shortly afterwards, however, Bohr produced his quantum model of the atom which seemed to make the question rather superfluous: Bohr's result suggested that the behavior of the atom fell outside of classical physics. In other words, the old rules of acceleration and radiation did not necessarily apply.

Some scientists were not ready to accept this new, rather mysterious, quantum theory of the atom nor give up on a classical explanation. The most prominent of these was G.A. Schott, who I have described previously as the last of the respectable 'anti-quantum' scientists. In 1933, Schott published a remarkable calculation, in which he demonstrated that* a uniformly charged spherical shell of radius , undergoing periodic motion of period , will not produce radiation if the following condition is satisfied:*

,

*where is an integer (1,2,3,…), and is the speed of light*.

A slowed-down movie of such a system might appear as follows:

Schott's result is a beautiful and complicated piece of mathematics. He himself, however, realized that it could not supplant the quantum theory at that point, which was quite firmly established. However, the neutron had been discovered only a year before, and Schott suggested that his sphere might be part of a model of the neutron.

For the next forty years, in fact, this would be the general pattern of research into radiationless motions: a new researcher would rediscover and expand upon the theory of radiationless motions. Then, reasoning that the theory is so elegant and unusual that it must be able to explain *something* in nature, the researcher suggests it as an explanation of the physics mystery du jour!

The next researcher to make a significant contribution to the theory (not including Schott, again, who we will return to shortly) was none other than the remarkable quantum physicist David Bohm, together with his Princeton co-author M. Weinstein, in 1948. Bohm and Weinstein^{1} made the remarkable observation that "certain charge distributions can oscillate without radiation even when no forces are present, other than their own retarded fields."

In other words, in Schott's original paper, the charged sphere was implicitly assumed to be kept in its orbital motion by an external force. Bohm and Weinstein demonstrate quite generally that one can construct charge distributions that will oscillate on their own, being kept in their oscillation only by their own local electromagnetic fields!

This observation is the application of a branch of classical electromagnetic theory known as *radiation reaction*. The idea is simple enough: because a charged particle produces its own electromagnetic field, when the particle is in motion it 'runs into' its own field, and can be affected by it. In macroscopic (day-to-day) electromagnetic interactions the radiation reaction is typically a negligible effect, but it can in principle be large for subatomic particles. Initial calculations of radiation reaction, however, were plagued with seemingly impossible solutions, including "runaway solutions" (in which a particle accelerates without limit only under the influence of its own field) and "non-causal solutions" (in which a particle starts moving *before* an electromagnetic field reaches it). These problems were part of the initial impetus that made researchers consider "extended electron theories", in which the electron is treated as an extended charged object instead of a point particle.

Schott himself had in fact already shown in an earlier series of papers ^{2,3} that self-oscillations were possible; however, his calculations were restricted to a spherical shell of charge, as in his earlier papers. Schott found that the spherical shell could oscillate without external forces, but had one rather serious caveat: the result only holds if the sphere has no *material mass *(referred to as 'intrinsic mass').

This was not such an odd idea at the time as it might appear. It had been noted as early as 1881 that it is more difficult to move a charged particle than an uncharged particle, which can be equated with an 'electromagnetic mass' associated with the electromagnetic energy of the particle. Some scientists went so far as to suggest that *all* mass might be the result of the electromagnetic energy of a particle. Such electromagnetic mass was still being discussed in 1933 by no less an esteemed scientist than Max Born^{4}. Schott's spherical shell of charge could in principle oscillate without external forces, if its entire mass was of the electromagnetic type.

Bohm and Weinstein made Schott's results more palatable by showing that, if one considers charge distributions more general than a spherical shell, it is possible to find 'self-oscillating' modes even when the intrinsic mass is nonzero.

Bohm and Weinstein give due credit to Schott's work; it is unclear, though, whether they extended his research after discovering his original papers, or whether they rediscovered radiationless orbits and later found Schott's work. It is also worth noting that Schott managed to add spin to his radiationless calculations; i.e. he showed that a radiationless particle could be both spinning on its axis as well as orbiting around a central point.

To understand how a particle might be able to self-oscillate, I like to think of the charge distribution and the electromagnetic field as forming a two-body system which can orbit or vibrate around one another. This can only work, however, if the electromagnetic field is localized, i.e. does not radiate. This is (*very* crudely) illustrated below:

Can such self-oscillating charge distributions actually exist? Though the mathematics is rigorous, it is still limited by a number of unrealistic assumptions:

- The calculations do not take into account relativistic effects, though the velocities that arise in the theoretical calculations are fast enough to make such effects significant.
- The charge distribution itself is assumed to be perfectly rigid, which is inconsistent with Einstein's theory of special relativity. To see this, imagine a very long, infinitely rigid pole which cannot bend or compress. If one were to push one end of the pole, the other end would have to move instantaneously; by special relativity, however, we know that such impulses cannot move faster than the speed of light.
- The model used to describe the oscillating charge distribution does not use any quantum mechanical effects, which would be very important if the distribution were an elementary particle.

It is safe to say that the question of radiationless, self-oscillating charge distributions is still quite an open one!

Like Schott and Ehrenfest before them, Bohm and Weinstein thought that these phenomena might shed light on as-yet unexplained fundamental physics:

The idea then suggests itself that perhaps some kinds of mesons are really excited states of the electron. The decay from one kind of meson to another, or from meson to electron would then correspond simply to the loss of this excitation energy. The exact values of the energy are probably not very significant, first, because they depend on the shape assumed for the charge distribution, and second, because the theory given thus far is not relativistically invariant. The essential point of this work is simply to suggest that the same step which makes the theory finite can also bring in the idea of unifying a whole spectrum of particles into a single particle.

This idea did not pan out, as mesons were instead found to be a combination of more elementary particles, namely quarks. However, it would still not be the last time that a researcher would suggest that nonradiating sources might have a role in fundamental fields of physics.

There is a bit of a sad historical note associated with the timing of Bohm's paper. Shortly after its publication, in 1949, Bohm was called before the House Committee on Un-American Activities because of his previous ties to Communism. He was arrested and, though exonerated later, lost his post at Princeton and left the United States.

It seems that McCarthyism ended Bohm's investigations into radiationless motions, though he would still make major contributions to physics: in 1959, he jointly discovered the Aharonov-Bohm effect. Fortunately, Bohm would not be the last to study these strange objects, as we will see in future posts.

************************************

^{1} D. Bohm and M. Weinstein, "The self-oscillations of a charged particle," Phys. Rev. 74 (1948), 1789-1798.

^{2} G.A. Schott, "The general motion of a spinning uniformly and rigidly electrified sphere, III," 159 (1937), 548-570.

^{3} G.A. Schott, "Uniform circular motion with invariable normal spin of a rigidly and uniformly electrified sphere, IV," 159 (1937), 570-591.

^{4} M. Born and L. Infeld, "Electromagnetic mass," Nature 132 (1933), 970.

I love finding out this sort of thing, because it really adds depth to what they teach us in school. In my semiconductor physics class, they just glanced over the whole topic, saying "if the atom were classical, the electrons would radiate their kinetic energy away and spiral into the nucleus". Which is true, but there are many fascinating ideas behind that reasoning, as you've pointed out. In our textbook those ideas didn't even rate an asterisk.

On the subject of self-oscillating charge distributions (and the difficulty of treating them mathematically without unrealistic assumptions), I've actually seen some numerical simulations of the complex Ginzburg-Landau equation and the generalized nonlinear Schrödinger equation that show some similar behaviors. Perhaps if they had had greater computing power back in the thirties they could have explored these sorts of classical models more fully without some of the assumptions.

Wade: Yeah, in principle radiationless excitations exist for pretty much any wave equation with a source term, so it wouldn't surprise me to see similar behaviors in quantum systems!

"Perhaps if they had had greater computing power back in the thirties they could have explored these sorts of classical models more fully..."

I consider these radiationless motions a somewhat big "what if...?" of physics: if the idea had been proposed much earlier, it seems like it could have become a serious contender for explaining atomic structure.

Hmm... that makes me think about Kelvin's vortex ring atomic model from the mid-1800s in a slightly new light. Since his proposed fluid was inviscid and incompressible, I guess his vortices couldn't radiate, so his type of classical atoms wouldn't have had the "radiative death spiral" problem.

His theory had other conceptual problems though, like the question of how atoms could interact at long range with finite propagation speed in an incompressible medium (essentially the same special relativity problem you mentioned above).

Kelvin's model would have seemed even more relevant if experimental particle physics had been around in his time. The pair creation/annihilation of vortices and their discrete nature cries out for some sort of atomistic interpretation. I'm planning to discuss in a future post some more. I've also got to dig up the references for Kelvin's model...

Kelvin's model would have seemed even more relevant if experimental particle physics had been around in his time. The pair creation/annihilation of vortices and their discrete nature cries out for some sort of atomistic interpretation. I'm planning to discuss in a future post some more. I've also got to dig up the references for Kelvin's model...