The Model That Fit Too Well — What Bohr Could Not Explain in 1913

In 1913 a 27-year-old Niels Bohr published a model of the atom that got the wavelengths of hydrogen's light right. More than that, it calculated a number that until then could only be measured, using nothing but basic quantities such as the mass and the charge of the electron. And yet why it worked was something Bohr himself could not explain. Part 3 of this series on the old quantum theory is about a model that fit too well for comfort (Part 2 is here).

All he could say was that something sits at the centre

In May 1911 Ernest Rutherford, in Manchester, published a paper. Geiger and Marsden had fired alpha particles — heavy, positively charged particles, the same thing as a helium nucleus — at a sheet of gold foil, and about one in 20,000 came back deflected by 90 degrees or more on average. On the picture then current, in which the positive charge is smeared thinly through the atom, a bounce that large simply cannot happen. Rutherford showed that the deflections made sense if the charge was concentrated into a very small region at the centre of the atom.

Here is a detail textbooks tend to leave out. The word "nucleus" does not appear anywhere in that paper. What Rutherford writes is "central charge". And he says plainly that he cannot even decide whether that charge is positive or negative: "It has not so far been found possible to obtain definite evidence to determine whether it be positive or negative." The story that the nucleus was discovered in 1911 was tidied up afterwards.

There was a worse problem. If a heavy charge sits at the centre and electrons circle it, then by classical electromagnetism a circling electron radiates light continuously, loses energy, and spirals into the centre in a very short time. As it stood, the model could not even explain why atoms do not fall apart.

The "markings on a butterfly's wing" that nobody took seriously

Meanwhile, somewhere else entirely, a formula was sitting unused.

In 1885 Johann Balmer, a schoolteacher in Basel, noticed that the wavelengths of the light given off by hydrogen fall neatly onto a simple formula built from whole numbers. Take a reference length and multiply it by m×m divided by (m×m − 4). Put 3, 4, 5 and 6 in for m and you get the wavelengths of the four observed lines, correct to several decimal places. But there was no reason behind the formula at all. It was a formula that merely fit.

That was how physicists of the day regarded the pattern of spectral lines. Bohr himself put it this way in a 1962 interview: "Just as if you have the wing of a butterfly then certainly it is very regular with the colors and so on, but nobody thought that one could get the basis of biology from the coloring of the wings of a butterfly." Beautiful, but no help in finding out what an atom is made of. That was the common sense of the time. In the same interview Bohr also says that during his Manchester period he was not aware of Balmer's formula.

Two rules were all it took to produce the numbers

Bohr laid down two audacious rules.

First: an electron has certain fixed states in which it can sit without giving off light. Classical electromagnetism insists that a circling electron must radiate; Bohr simply declared that while it is in one of these states, it does not. Second: light is given off only in the instant an electron jumps from one state to another, and the energy of one grain of the light emitted is exactly equal to the difference in energy between the two states. The "step" in energy handed down from Planck in Part 1 and Einstein in Part 2 had now been carried inside the atom.

Work through those two rules and Balmer's formula drops out as it stands. And not only that: the constant sitting in front of the formula — a number that until then could only be read off from measurements — turns out to be writable as a combination of the electron's mass, its charge and Planck's constant, quantities that have nothing to do with spectral lines.

The numbers Bohr printed are honest ones. The value that comes out of the theory is 3.1×10¹⁵ per second; the observed value is 3.290×10¹⁵ per second. That is about 6 per cent off. Bohr put the gap down to the still-crude measured values of the electron's mass and charge, and wrote that the agreement held within the uncertainty of those constants. Not an exact hit — but not a coincidence of the wrong order of magnitude either. That was the delicate position from which July 1913 started.

With helium, it agreed to five significant figures

The clinching evidence came not from hydrogen but from helium.

Among the lines in starlight there was a group that Pickering had found, and which at the time were taken to belong to hydrogen. Bohr argued that they were not hydrogen at all but helium that had lost one of its two electrons — a helium ion. The arrangement is the same, one electron circling a heavy central charge; the only difference is that the central charge is twice as large.

The spectroscopist Alfred Fowler objected. On Bohr's formula the ratio of the constants for the helium ion and for hydrogen should be exactly 4.0000, he said, but experiment gave about 4.0016.

Bohr's reply appeared in Nature in October 1913. The heavy central charge is not a fixed post: it too wobbles slightly along with the electron. Allow for that, and the ratio becomes 4.00163 — agreeing with Fowler's measurement to five significant figures. Fowler himself acknowledged the correction as a valuable addition.

In September of the same year, Hevesy passed the story on to Einstein, and his reaction survives in a letter to Bohr. Einstein was astonished, and said that in that case Bohr's theory must be right. As we saw in Part 2, this is the man whose own light quantum was still struggling to be accepted.

What had been achieved by this point, and what had not

What had been achieved was this: the spectral lines of hydrogen, and of its one-electron relatives such as the helium ion, could now be calculated from more basic quantities. And as long as one accepts that fixed states exist, there is at least a consistent reason why atoms do not fall apart.

The list of what had not been achieved is longer. Helium itself — an atom with merely two electrons — would not come out right. The model gives the wavelengths of the lines, but says nothing about which lines come out strong. And above all, why an electron in a fixed state gives off no light, the very heart of the matter, remained a rule with no reason behind it. Bohr had not explained it; he had simply decided it.

One more caution when reading this in modern language. In this model the electron travels a definite "orbit", but in today's quantum mechanics the very picture of an electron following a path does not hold. Treat the word orbit as a convenience for reading the theories of this period.

All the same, the model fit. And precisely because it fit, physicists set about finding how far it could be stretched. Allow ellipses instead of circles, add the effects of relativity, put the atom in a magnetic field. Next time we follow how far that extension went, and where it tore.