VII.REFINEMENTS OF THE HYDROGEN SPECTRUM
IN Bohr’s theory, the electron always moves round the hydrogen nucleus in a circle. But according to Newtonian principles, the electron ought also to be able to move in an ellipse, and the generalized quantum-principle can be applied to elliptic orbits as well as to those that are circular. It is natural to inquire whether it is possible to work out a theory that allows for elliptic orbits, and, if so, whether it will fit the facts better or worse than Bohr’s original theory. It is found that, as regards the broad facts, it makes no difference whether we admit or reject elliptic orbits; in either case, the facts will accord with observation to a first approximation. There are, however, three delicate phenomena which are observed to occur, which cannot be accounted for if all the possible orbits are circles, but are to be expected if ellipses also occur. These are the following: First, there is what is called the Zeeman effect, which is an alterationproduced by a strong magnetic field. Secondly, there is the Stark effect, which is produced by a strong electric field. Thirdly, there is what is called the “fine structure,” which is the fact that each single line of the spectrum, when very closely examined, is found to consist of a number of almost identical lines. The explanation of the Zeeman effect is still in part incomplete, but the explanation of the other two by means of Sommerfeld’s methods is as perfect as could be desired. We shall not attempt to set forth the explanation, which would be impossible without a good deal of mathematics. We shall only attempt to describe the orbits which Sommerfeld admits as possible, in addition to Bohr’s circles.
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If there is any reader who does not know what an ellipse looks like, he can construct one for himself by the following simple device. Tiea piece of string to two pins, and stick them into a piece of paper at two points,′ near enough together for the string to remain loose. Then take a pencil, and with its point draw the string taut. Any placethat the pencil will reach is on a certain ellipse, and by moving the pencil round, the whole ellipse can be drawn. The points,′ are called “foci.” An ellipse may be defined as a curve such that, ifand′ are any two points on it the sum of the distance offromand′ (the foci) is equal to the sum of the distances of′ fromand′. In our construction, both are equal to the length of the string that we tied to the two pins. The ratio of the distance between the pins to the length of the string is called the “eccentricity” of the ellipse. It is obvious that if we were to stick the two pins into the same place we should get a circle, so that a circle is a particular case of an ellipse, namely an ellipse which has zero eccentricity. All the planets move in ellipses which are very nearly circles, whereas the comets move in ellipses which are very far removed from circles. In each case the sun is in one of the foci, and there is nothing particular in the other focus. An ellipse which is very far from being a circle can be drawn by making the distance′ between the two pins not very much shorter thanthe length of the string.
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There is another way of thinking of an ellipse which is also useful; it may be thought of as a circle which has been squashed. Suppose for instance that you took a wooden hoop and stood it up and put a weight on the top of it; the hoop would get squashed into more or less the shape of an ellipse. In the figure, the hoop is drawn circular, as it is before the weight is put on; then a heavy weight is put on the highest point, and the hoop takes more or less the form of the dotted curve in the figure. The weight, which was put on at,has made the top of the hoop sink to.The hoop is supposed to be fastened, like a wheel, on to an axle in the middle,.An ellipse can be obtained from a circle which is standing upright by diminishing allvertical distances in a certain fixed proportion; that is to say, ifis any point of the circle, which is at a heightabove the level of the axle, we go down to a pointbelow,such that the heightbears a fixed ratio, to,the same, of course, as the ratio ofto.The ratio oftois also of course the same for any point of the curve, and equal to the ratio ofto.We will call this the amount of “flattening” of the ellipse. This is not a recognized expression, but will prove convenient for our purposes. To state the whole thing precisely: Given a circle, imagine it to be stood upright, like a wheel, with an axle through the centre. Then lower each point in the top half of the wheel by a fixed proportion of its height above the level of the axle, and raise each point in the bottom half in the same proportion of the lowering to the final height (or of the raising to the final depth, in the lower half) we will call the amount of “flattening” in the ellipse. That is to say, ifis half of(andhalf of), the amount of flattening is a half; ifis a third of,the amount of flattening is a third; and so on.
We can now explain what are the ellipses which are possible for the electron in a hydrogen atom.
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In the figure,represents the electron,represents the nucleus, which is in a focus of the ellipse.is the point where the electron is nearest to the nucleus,the point where it is farthest from it,the centre of the ellipse, which is half way betweenand.There are now two periodic characteristics of the orbit, instead of only one, as in the case of the circle. The first periodic characteristic is, as before, the angle whichmakes with.The other is,the distance of the electron from the nucleus. This grows continually smaller while the electron is travelling fromto,and then continually larger while it is travelling fromto.As there are two periodic characteristics of the orbit, the general quantum-theory will give two conditions that the orbit must fulfil, instead of only one. It is impossible to explain the process by which the results are obtained,but the results themselves are fairly simple.
The first quantum condition is very much the same as in the case of circular orbits. Take the mass of the electron, its velocity at(where it is nearest to the nucleus), and the circumference of the circle whose radius is,and multiply these three together; the result must be an exact multiple of,say.The second quantum condition determines how much the ellipse departs from a circle; it states that there is a second whole number(the second quantum number), such thatis the amount of flattening in the sense defined a moment ago. The second number′ may be zero; we then obtain Bohr’s case of circular orbits. If it is not zero, the electron moves in a more or less eccentric orbit.
It turns out that, apart from niceties, the energy of an electron in its orbit, and therefore the spectral lines corresponding to jumps from one orbit to another, do not depend upon the separate numbersand′, but only upon their sum'. The result is that the lines to be expected, apart from niceties, are the same as on Bohr’s original theory of circular orbits. If the matter were to end here, we might seem to have had a lot of trouble for nothing. Eventhen, however, we could have drawn a useful lesson from the theory of elliptic orbits. There are, as we shall see, certain facts which are explained by elliptic orbits and not by circular orbits, but these facts are mostly recent discoveries, and might easily have remained unknown for some time longer. In that case, Bohr’s original theory would have accounted admirably for all the known facts, and there would have seemed to be very strong grounds for accepting it. Yet the theory of elliptic orbits would have accounted for the facts just as well, so that there would have been no way of deciding between them. This illustrates what is sometimes forgotten, that a theory which explains all the known relevant facts down to the minutest particular may nevertheless be wrong. There may be other theories, which no one has yet thought of, which account equally well for all that is known. We cannot accept a theory with any confidence merely because it explains what is known. If we are to feel any security, we must be able to show that no other theory would account for the facts. Sometimes this is possible, but very often it is not. Poincaré advanced a proof that the facts of temperature radiation cannot be explained if we assume that radiation is a continuous process, and that any possible explanationmust involve sudden jumps such as we have in the quantum theory. His argument is difficult, and it is possible that it may not ultimately prove wholly cogent. But it affords an instance of that further step without which scientific hypothesis must remain hypothetical. In our case, fortunately, there is evidence that elliptic orbits actually do occur when an electron moves round a hydrogen nucleus. That is to say, there is evidence that this hypothesis explains certain facts which the hypothesis of circular orbits cannot explain. But although the agreement between theory and observation is astonishingly close, it cannot be said that we have yet reached the stage where we can be quite certain that no other theory would account for the facts.
All the broad facts in the spectrum depend upon the sum of the two quantum-numbers,,not on either separately. We therefore classify orbits by this sum. We thus arrive at the following possible orbits:
1st case..Sincecannot be zero (because if it were the electron would fall into the nucleus), this gives only one possibility, namely,.When,there is no flattening, and the orbit is a circle. Thus this first case is that of Bohr’s minimum circle.
2nd case..Here there are two possibilities, namely,and,The first of these gives Bohr’s second circle; the second gives an ellipse in which there is a unit amount of flattening, that is to say, the ellipse is half as high as it is broad.
3rd case..Here there are three possibilities, namely: (a),;this gives Bohr’s third circle (b),;this gives an ellipse in which the amount of flattening is a half, that is to say, the ellipse is two-thirds as high as it is broad, (c),;this gives an ellipse in which the amount of flattening is two, that is to say, the ellipse is a third as high as it is broad.
In the fourth case,,there are four possibilities; and so on. The breadth of the ellipse depends only upon,so that all the possible ellipses under one head have the same breadth. The energy also, apart from niceties, depends only upon.[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 ofdo not produceexactlythe 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 uponproceeds 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. Readersmay 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 theoryof 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 quantumtheory 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.
[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.
[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.