The A B C of Atoms

n + n′.[6] Of the three sets of facts which show that elliptical orbits must occur, we shall pass by the Zeeman effect (which shows how magnetism splits one line into three, or sometimes more) and the Stark effect (which shows the influence of an electric field). The third, however, is so interesting that it cannot be omitted, since it shows that the electron, in so far as it obeys ordinary dynamical laws, follows the principles of Einstein in preference to those of Newton. Very careful observation shows that the lines in the spectrum which we have hitherto treated as single really consist of two (and in other cases three or more) separate lines very close together. This suggests that two different orbits which give the same value of n + n′ do not produce exactly the same line in the spectrum when an electron jumps to or from them. The phenomenon is more noticeable in the case of other elements than in that of hydrogen, for reasons which the theory explains. Fortunately on this point our theory is able to tell us a good deal about other atoms; but in what follows we shall confine ourselves to hydrogen. The mathematical argument which shows that the energy of the electron in its orbit only depends upon n + n′ proceeds on Newtonian principles; more particularly, it treats the mass of the electron as constant. But in the modern theory of relativity, the mass of a body is increased by rapid motion. This increase is not noticeable for ordinary velocities, but becomes very great as we approach the velocity of light, which is a limit that no material body can quite reach. Readers may remember that Einstein’s theory of gravitation has been confirmed by two facts which remained inexplicable on Newtonian principles. One is the fact that light bends by a certain amount (double what Newtonian principles allow) when it passes near the sun, which has been verified in two eclipses. The other is the fact which is called the motion of the perihelion of Mercury, which had long been known to astronomers without their being able to find any way of accounting for it. It is the analogue of this fact that concerns us. Mercury, like the other planets, moves in an ellipse with the sun in a focus; it is sometimes nearer to the sun and sometimes further from it. Its “perihelion” is the point of its orbit which is nearest to the sun. Now it has been found by observation that, when Mercury has gone once round the sun from its previous perihelion, it has not quite reached its next perihelion; that is to say, it has to go a little more than once round the sun in passing from one occasion when it is nearest the sun to the next. This of course shows that its orbit is not quite accurately an ellipse. There is supposed to be a similar phenomenon in the motions of the other planets, but it is too small to be observed; in the case of Mercury it is just large enough to be noticeable. Einstein’s theory of gravitation, but not Newton’s, explains why it exists, and why it is just as large as it is; it also explains why the effect in the case of the other planets is too small to be observed. In order to be noticeable, the orbit must depart fairly widely from a circle, but the orbits of all the planets except Mercury are very nearly circular. In the case of the electron in the hydrogen atom, as we have seen the possible orbits which are not circles are very markedly elliptical. This makes the effect which has been noticed in the case of Mercury very much more pronounced in the case of the electron. Moreover, since the velocity of the electron in its orbit is much greater than that of the planets, there is a much more noticeable effect of the increased velocity, when the electron is near the nucleus, in increasing the mass. This causes a quite appreciable effect of the same sort as the motion of the perihelion of Mercury. That is to say, the electron makes a little more than one complete revolution between one occasion when it is nearest to the nucleus and the next occasion when this happens. It is found that this accounts for the fine structure of the spectral lines, though it would be impossible to set forth the explanation in non-mathematical language. It is curious that, although the quantum theory is something quite outside traditional dynamics, everything unaffected by this theory proceeds exactly according to the very best principles of non-quantum dynamics, that is to say, according to Einstein rather than Newton. The proof is in this fact of the fine structure.[7] [6] For a full mathematical treatment of the above topic, see Sommerfeld, op. cit. 286-297. [7] The mathematical theory of the fine structure will be found in Sommerfeld, op. cit., Chap. VIII. The explanation of the motion of the perihelion in the above is not, properly speaking, the same as in the case of Mercury; the latter depends upon the general theory of relativity, and Einstein’s new law of gravitation, while the former depends only upon the special theory of relativity.