# Models of the Atom

Michael Fowler, University of Virginia

## Early String Theory

The first attempt to construct a physical model of an atom was made by William Thomson (later elevated to Lord Kelvin) in 1867. The most striking property of the atom was its permanence. It was difficult to imagine any small solid entity that could not be broken, given the right force, temperature or chemical reaction. In contemplating what kinds of physical systems exhibited permanence, Thomson was inspired by a paper Helmholtz had written in 1858 on vortices. This work had been translated into English by a Scotsman, Peter Tait, who showed Thomson some ingenious experiments with smoke rings to illustrate Helmholtz' ideas. The main point was that in an ideal fluid, a vortex line is always composed of the same particles, it remains unbroken, so it is ring-like. Vortices can also form interesting combinations—a good demonstration is provided by creating two vortex rings one right after the other going in the same direction. They can trap each other, each going through the other in succession. This is probably what Tait showed Thomson, and it gave Thomson the idea that atoms might somehow be vortices in the ether.

Of course, in a non-ideal fluid like air, the vortices dissipate after a while, so Helmholtz' mathematical theorem about their permanence is only approximate. But Thomson was excited because the ether was thought an ideal fluid, so vortices in the ether might last forever! This was very aesthetically appealing to everybody—"Kirchhoff, a man of cold temperament, can be roused to enthusiasm when speaking of it." (Pais, Inward Bound, page 177, source for this material). In fact, the investigations of vortices, trying to match their properties with those of atoms, led to a much better understanding of the hydrodynamics of vortices—the constancy of the circulation around a vortex, for example, is known as Kelvin's law. In 1882 another (unrelated) Thomson, J. J., won a prize for an essay on vortex atoms, and how they might interact chemically. After that, though, interest began to wane—Kelvin himself began to doubt that his model really had much to do with atoms, and when the electron was discovered by J. J. in 1897, and was clearly a component of all atoms, different kinds of non-vortex atomic models evolved.

It is fascinating to note that the most exciting theory of fundamental particles at the present time, string theory, has a definite resemblance to Thomson's vortex atoms. One of the basic entities is the closed string, a little loop, which has fields flowing around it reminiscent of the swirl of ethereal fluid in Thomson's atom. And it's a very beautiful theory—Kirchhoff would have been enthusiastic!

## Floating Magnets

In 1878, Alfred Mayer, at the University of Maryland, dreamed up a neat demonstration of how he imagined atoms might be arranged in molecules (Note: not electrons in atoms, that idea came a bit later). He took a few equally magnetized needles and stuck them through corks so that they would float with their north poles all at the same height above the water, all repelling each other equally. He then held the south pole of a more powerful magnet some distance above the water, to attract the needles towards this central point. The idea was to see what equilibrium patterns the needles would form for different numbers of needles. He found something remarkable—the needles liked to arrange themselves in shells. Three to five magnets just formed a triangle, square and pentagon in succession. but for six magnets, one went to the center and the others formed a pentagon (well, usually—see picture).  For more magnets, an outer shell began to form.

Kelvin's immediate response to Mayer's publication was that this should give some clues about the vortex atom. Apparently it didn't, but twenty-five years later it guided his thinking on a new model.

## Plum Pudding

Kelvin, in 1903, proposed that the atom had the newly discovered electrons embedded somehow in a sphere of uniform positive charge, this sphere being the full size of the atom. (Of course, the sphere itself must be held together by unknown non-electrical forces—which is still true of the positive charge in our modern model of the atom.) This picture was taken up by J. J. Thomson too, and was dubbed the plum pudding model, after traditional English Christmas fare, a large round pudding (rich with suet) with raisins embedded in it. In 1906, J. J. concluded from an analysis of the scattering of X-rays by gases and of absorption of beta-rays by solids, both of which he assumed were effected by electrons, that the number of electrons in an atom was approximately equal to the atomic number. This led to a picture of electron arrangements in an atom reminiscent of Mayer's magnets. Perhaps by analyzing possible modes of vibration of electrons in these configurations, the spectra could be calculated.

The simplest case to consider was clearly hydrogen, now assumed (correctly) to contain just one electron.

## How does an atom's color depend on its size?

By "color" we mean here the spectral colors emitted when the atom is excited. In Thomson's plum pudding model, there is a clear relationship between the size of the pudding and the frequency at which the electron will oscillate, and hence presumably radiate, when excited. The two are related because the assumption is that the total positive charge—which is uniformly spread throughout the sphere—is just equal to the electron's negative charge. At rest in its lowest state, the electron just sits in the middle of this sphere of charge. When bumped somehow, it will oscillate about that point. If the electron is at distance $x$ from the center, it will feel a restoring force towards the center equal to the attraction from that part of the positive charge it is "outside" of—that is, the charge within a sphere of radius x about the center. Therefore, the larger the whole atom—the pudding—the more thinly spread the positive charge is, and the smaller the amount of charge within the small sphere of radius $x$ that is attracting the electron back towards the center. So, the bigger the atom is, the slower the electron's oscillation is, and the lower frequency the radiation emitted.

It is straightforward to give a quantitative estimate of the size of the atom based on the observation that when excited it emits radiation in the visible range.

Let us assume that the positively charged sphere has radius ${r}_{0}$ (this is then the size of the atom, which we know is about 10-10 meters).

If the electron is displaced from the center of the atom in the $x$ -direction an amount $x,$ it is attracted back by all the charge that is now closer to the center than itself, that is, an amount of charge equal to $e{x}^{3}/{r}_{0}^{3}.$ (Recall $e$ is the total amount of charge on the sphere, and ${x}^{3}/{r}_{0}^{3}$ is the fraction of the sphere closer to the center than $x.$ ) This charge acts as if it were a point charge at the origin, so the inverse-square law gives a $1/{x}^{2}$ factor, and the equation of motion for the electron is therefore:

$m\frac{{d}^{2}x}{d{t}^{2}}=-\frac{{e}^{2}}{4\pi {\epsilon }_{0}{r}_{0}^{3}}x.$

Provided it stays within the sphere, the electron will execute simple harmonic motion with a frequency

${\omega }^{2}=\frac{1}{4\pi {\epsilon }_{0}}\cdot \frac{{e}^{2}}{m{r}_{0}^{3}}.$

Notice that, as we discussed above, as the size of the atom increases the frequency goes down. And we know the frequency, at least approximately—it corresponds to visible light. Therefore, this model will predict a size of the atom, which we can compare with the size from other predictions, such as Brownian motion (plus the assumption that in a liquid, the atoms are fairly close packed—they take up most of the room available).

If we take visible light, say with a frequency 4.1015 radians per second, we find ${r}_{0}$ must be about 2.10-10 meters, a little on the large side, but encouragingly close to the right answer for a first attempt.

Sad to report, though, no real progress was made beyond this in predicting spectra using Thomson's pudding.  Many attempts were made to find stable arrangements of electrons in atoms, not just hydrogen, using models like Mayer's magnets, and also having the electrons going around in circles. It was hoped that if certain numbers of magnets formed a very stable arrangement, that might model a chemically nonreactive atom, etc.—but nobody succeeded in making any real predictions along these lines, the models could not be connected with the properties of real atoms.

Evidently, then, the theorists were stuck—and the experimental challenge was to find some way to look inside an atom, and see how the electrons were arranged. This is what Rutherford did, as we shall discuss in the next lecture. He was very surprised by what he saw.