Part 6

The chemical and physical properties of an element depend on its atomic number, i. e., on the positive charge carried by the nucleus of an atom of that element. And this positive charge is, as we have seen, a resultant charge. It is a result of the combination of a number of unit positive charges with a smaller number of unit negative charges. We can obviously reach the same resultant figure in as many ways as we please. If the resultant charge on the nucleus is to be 5, for instance, then we could take the combinations +6 and −1, or +7 and −2, or +8 and −3, and so on. Each of these arrangements would give atoms having identical chemical and physical properties. But their atomic weights would be different. The atomic weights depend, not on the resultant positive charge, but on the actual number of positive charges present in the nucleus, including those that are compensated for by negative charges as well as those that are not. In the above case, for instance, our atoms would have atomic weights 6, 7, 8, and so on. And this is the only difference they would have. By no other chemical or physical properties could they be distinguished one from another.

It is therefore a very interesting fact, and one fitting in beautifully with our theory, that many elements have been shown to consist of a mixture of atoms having different atomic weights, but identical in every other respect. The element chlorine, for example, has the atomic weight 35·46. This number is about as far removed as it could be from being a whole number, and is therefore specially fatal to the theory that all atomic weights are simple multiples of the same unit. But it has been shown that chlorine is really a mixture of two groups of atoms, the atomic weight of the atoms of one group being 35 and the other 37. These groups are mixed together in about the proportion of 3 to 1, and the ordinary measured atomic weight of 35·46 is really the average weight of the mixture. Neon, again, whose atomic weight is ordinarily given as 20·2, is found to consist of two groups of atoms with atomic weights 20 and 22. Much more complicated groupings have been discovered. Thus krypton, whose atomic weight is put as 82·92, is made up of groups of atoms having the weights 78, 80, 82, 83, 84, 86. Such elements are called Isotopes, the name indicating that the groups of atoms belonging to these elements occupy the same place in the periodic table. It will be noticed that the atomic weights of these groups of atoms are all whole numbers. This is on the basis of oxygen taken as 16. On this basis hydrogen is not exactly unity, but is 1·008. It appears probable, then, that all atoms have atomic weights which are nearly, but not quite, whole multiples of hydrogen.

The existence of isotopes definitely destroys the great importance that chemists had always assigned to atomic weights. We have atoms of different atomic weights but of the same properties. Further, as we saw in studying the disintegration of radium into lead, we have elements of the same atomic weight but with wholly different chemical and physical properties. The atomic weight of an element, therefore, by no means suffices to determine its chemical and physical properties. We see once more that the really important quantity to be known about an element is its atomic number, i. e., its position in the periodic table. It is worth noting that the existence of isotopes was not suspected until comparatively recent times, although very delicate determinations of atomic weights have been practised for decades. The most refined measurements customary in such determinations never varied from sample to sample of the same element. Exactly the same mixture of atoms constitutes chlorine, for instance, wherever the chlorine is obtained, and always has done so ever since men began to study chlorine. The groups of atoms which make up chlorine or any other isotope must have been thoroughly and universally mixed long ages ago—in all probability before the formation of the earth’s crust, when such universal and complete diffusion would have been possible.

§ 5. Relativity and the Atom

It is necessary, now, to say a little about the Restricted Principle of Relativity, since certain points about the modern theory of atomic structure cannot be understood without it. But it is not necessary to explain the principle itself. It is only necessary to describe the relation between energy and mass that the theory shows to exist. In pre-Relativity mechanics, it was always assumed that the mass of a body was completely independent of its velocity. There was no reason to suppose otherwise. Whether a body was moving fast or slow, or whether it was at rest, its mass, when measured, was always found to be the same. But the theory of relativity asserts that the mass of a body does vary with its velocity. As the body moves faster its mass increases. The mass increases in such a way that, at the speed of light, it becomes infinite. This can only mean that the velocity of light is a natural limit, that no material body could possibly exceed this speed. If it be true that the mass of a body increases with its velocity, it might be thought that experiment would long ago have led us to suspect that fact. But the law according to which the increase occurs is such that the increase is not measurable except at very great speeds. Now we are not familiar with bodies moving at very great speeds. We know of velocities such as 100 miles per hour and even, in astronomy, of velocities which reach a few miles per second. But velocities which are a considerable fraction of the velocity of light, viz., 186,000 miles per second, are practically unknown. It is the α- and the β-particles which furnish us with examples of bodies moving at speeds comparable with that of light. And experiments on these bodies show that their mass does increase with their velocity, and precisely in the way predicted by Einstein’s theory. So that the ratio e/m, the ratio of charge to mass of an electron, varies with the velocity of the electron. As the velocity increases m increases and therefore e/m grows smaller. The value of e/m usually given, viz., 1·77 × 10^{7} (electromagnetic) units, is the value for low velocities, when m may be taken as the mass of the electron at rest. We will denote this value of m as m{o}; m_{o} is the mass of the stationary electron. At half the velocity of light, the mass of the electron is 1·15 m_{o}, i. e., it is about one-seventh greater. If the electron is moving at nine-tenths the velocity of light its mass is 2·3 m__{o}, or nearly two and a half times greater. At ninety-nine-hundredths of the velocity of light the mass of the electron is seven times its value at rest, and at the velocity of light itself, as we have said, its mass is infinite.

This result of relativity theory must obviously be borne in mind in any attempts to ascertain in detail what goes on inside an atom. If the mass of the rotating electrons is a quantity which enters into our calculations, then obviously we must remember that the mass varies with the velocity of rotation we ascribe to the electrons. Another important aspect of this theory is that it shows we must ascribe mass to energy. This, again, is a very novel conception. We are used to thinking of energy as something to which the property of possessing mass cannot be ascribed. The two things seem to have nothing to do with one another. But it can be shown that energy certainly does possess inertia, and the property of possessing inertia is what we really mean by mass. The mass of a body is, indeed, only another way of measuring the total amount of energy it contains. Every piece of matter possesses a vast store of internal energy. If the piece of matter begins to move its energy is increased in virtue of its motion. Its mass also is increased. But the increase in mass due to increase in energy is usually extremely small. In any chemical combination which is attended by the development of heat, for instance, there is a certain loss of mass due to the energy radiated away during the process of combination. The resultant mass of the compound is less than the sum of the original masses of its constituents. But the loss which occurs in this way during any chemical process is exceedingly small, and the old law of the invariability of mass is, in all such cases, quite good enough. But there appears to be a beautiful and highly interesting exception. We have seen that there is reason to suppose that the helium nucleus, which is shot out of radioactive bodies as an α-particle, is a very stable structure. It is composed, as we have said, of four hydrogen nuclei and two electrons. The great stability of this structure suggests that its formation was attended by a great expenditure of energy, so that an enormous amount of energy would have to be communicated to it to break it up. Now the atomic weight of helium is 4, and the atomic weight of hydrogen is not 1, but 1·008. Four times the mass of the hydrogen atom would give an atomic weight of 4·032. The suggestion is that the difference between this value and the actual measured value of 4, represents the mass of the energy lost in the process of combining the four hydrogen nuclei into the helium nucleus. This gives a measure, also, of the amount of energy that would be required to split up the helium nucleus into its original components. The amount of energy represented by this figure is really enormous. It is sixty-three million times greater than the energy expended in ordinary chemical processes, and this figure is a measure of how much more stable the helium nucleus is than an ordinary chemical compound. A chemical compound can often be dissociated merely by raising its temperature a few degrees, but even the enormous energy possessed by the fastest α-particles is only about a third of that required to dissociate the helium nucleus.

CHAPTER V: Quantum Theory

Chapter V

Quantum Theory

§ 1. The Stability of the Atom

We have seen that the theory we have been describing, called the nuclear theory of the atom, gives a very satisfactory account of a large number of phenomena. The observed scattering of α-particles on passing through thin sheets of metal, the existence of isotopes, the changes which occur in radioactive phenomena, all receive very convincing explanations. There can be no doubt that the nuclear theory of the atom is essentially true, that the atomic models we have imagined correspond closely to actual atoms. But there is a fatal objection to this theory of the atom, as we have presented it hitherto. Such an atom could not continue to exist!

According to the classical theory of electrodynamics every change of motion on the part of an electrically charged body is attended with a radiation of energy. In wireless telegraphy, for instance, it is the rapid oscillations of the electrons in the sending apparatus which produce the electromagnetic waves. Each time an electron suffers a change in the direction or speed of its motion, or in both, it sends out an electromagnetic wave. Such a process cannot be kept up without a continual supply of energy. In the atomic model, as we have presented it, the outer electrons, which we imagine to be continually circulating round the nucleus, would be continually sending out energy. For a circular motion is a perpetually changing motion, and every change of motion on the part of a charged body is accompanied by the emission of energy. For an electron not to radiate energy, according to the classical theory, it must either be at rest, or be moving uniformly in a straight line. It is obvious that the outer electrons of our atom cannot be imagined as at rest. They would be attracted by the nucleus and simply fall into it, just as the planets would fall into the sun if they were robbed of their orbital motion. In order to counterbalance the attraction of the nucleus the outer electrons must have a circular or elliptical or some such motion. And any such motion would be attended by a radiation of energy. As a result of this radiation of energy it can be shown that the orbit of the rotating electron would grow smaller and its velocity of rotation greater. This process would continue until finally the electron fell into the nucleus. That is to say, the atom, as we have depicted it, is, on the classical theory of electrodynamics, essentially unstable. The whole material world, as we know it, ought to have vanished long ago. Further, the spectrum of any element contains perfectly sharp lines which are situated in perfectly definite parts of the spectrum. The radiations from the atoms of a given element are perfectly definite; they do not assume all values. But if the outer electrons, from which these radiations proceed, are continually changing their orbital distances and velocities, then there ought to be a continuous succession of lines in the spectrum of that element, instead of the perfectly distinct and permanent arrangement which exists in fact.

So we see that, when we come to investigate the mathematical theory of our atomic model it turns out to be highly unsatisfactory. Are we, therefore, to abandon our model completely? Before we answer this question we will look at one or two other phenomena where similar extraordinary difficulties have been found. We will consider, in the first place, the phenomena of heat radiation, since it is here that the insufficiency of the old theory of electromagnetic radiation was first demonstrated. Let us consider the heat rays radiated by what is called a “black body.” A black body is defined as one which absorbs the whole of the radiant energy that it receives. There is no substance which exactly satisfies this condition, but it is possible to produce the equivalent of it by artificial means. It was shown by the German physicist Kirchhoff that a space enclosed by an opaque envelope, and maintained at a uniform temperature, is filled with a radiation identical with that which would be emitted by a black body at the same temperature. If, therefore, a small hole be made in the opaque envelope, the rays which escape through it will be the same as those that would be produced by a black body. Several physicists have studied these rays, and they have reached extraordinarily interesting results. The way in which the total amount of energy radiated is distributed amongst the different rays has been the chief object of their researches. The rays which come from the enclosure are of very different wave-lengths; they vary between wide limits. Corresponding to each wave-length is a certain fraction of the total energy radiated, and this fraction depends upon the length of the actual wave concerned in a rather complicated way. It is found, as the result of actual measurements, that the longest waves have very little energy. As the wave-lengths decrease the energy increases until a certain wave-length is reached where the energy has a maximum value. As we go on past this point to shorter and shorter wave-lengths the energy decreases, until for very short wave-lengths it is practically zero. Now this result is in the most flagrant contradiction with the theoretical calculations. According to the mathematical theory, the energy contained in the very short wave-lengths should be very great. As the wave-lengths get shorter and shorter, tending towards zero, the energy contained in them should, according to the calculations, tend towards infinity. Observation shows that it tends towards zero. The contradiction is as striking as it could be.

We can see how extraordinary this observed result is if we consider an analogous case. Suppose that we have a number of corks floating on the surface of a bowl of water. Now suppose that, by some means, we agitate these corks, causing them to oscillate up and down in the water, and then leave them to themselves. We know that the oscillations will, after a time, die down. The whole mass, water and corks, will once again become quiescent. The difference is that the water will be slightly warmer. The energy which was contained in the oscillating corks is ultimately transferred to the molecules of the water and appears as heat energy. Now this result is quite in accord with the calculations. But if the result were to be analogous to the radiation result mentioned above, the corks would have to go on oscillating for ever with undiminished vigour. We should all agree that such a phenomenon was highly mysterious. The results obtained in the radiation experiments are no less mysterious.

Let us turn to yet another phenomenon which is entirely contradictory of our expectations. It is found that light of high frequency, i. e., of short wave-length, when allowed to fall on a metal, liberates electrons from the metal. The old scientific question of “How much?” immediately, of course, comes to the fore. We want to know the number of the electrons liberated and their velocities. And we want to know how these two quantities depend on the light which is used. We find, as the result of careful experiment, that the number of electrons liberated depends on the intensity of the light, but that the velocity of the electrons depends on the frequency of the light. This result is very surprising. We should have expected that the more intense the beam of light the higher the velocity of the liberated electrons. But only the number of electrons is influenced by the intensity. A very weak beam of high frequency light will cause electrons to be shot out of the metal with high velocity. We get a firmer grasp of the paradoxical nature of this result if we first create X-rays by bombarding an anti-cathode with electrons, and then use the X-rays to liberate electrons from a metal. X-rays, as we have said, may be regarded as extremely high frequency light-waves. Now let us suppose that we produce some electrons in a cathode tube, and cause them to bombard the anti-cathode, so producing X-rays. The electrons will have a certain velocity, depending upon the voltage applied to the tube, and they will generate X-rays having a certain frequency. The higher the velocity of the electrons the higher the frequency of the resulting X-rays. These X-rays are now allowed to fall on a sheet of metal. Immediately electrons are liberated from the metal, and the astonishing discovery is made that the electrons so produced have the same velocity as the electrons which generated the X-rays. We may illustrate this result by an analogy used by Sir William Bragg. Imagine that we drop a plank, from the height of a hundred feet, into the ocean. The impact produces waves in the ocean which spread out in circles around the point of impact. As the waves spread out they naturally get feebler and feebler, since the same total amount of energy is distributed over a longer and longer circumference. After travelling, say, two miles, let us suppose that the outermost wave reaches a ship. We are to imagine that, immediately the wave reaches the ship, it causes a plank to be shot up out of the ship to a height of one hundred feet. This case seems precisely analogous to the liberation of electrons by X-rays. The X-rays have spread out in ever increasing spheres from the point of impact, and yet, wherever they touch a metal, they liberate electrons having precisely the energy of the electrons which generated the X-rays.

The key to these extraordinary results is to be found in Planck’s Quantum Theory. It was at the end of the year 1900 that Max Planck published his theory that energy is not emitted in a continuous fashion, but only in little finite packets, as it were. An oscillating atom, for instance, is to be conceived as sending out little doses of energy, one after the other. It does not emit energy continually. And Planck asserted that the size of these little packets depended on the frequency of the oscillation, being greater the greater the frequency. Such an hypothesis is very strange, and is in entire contradiction to the classical dynamical theory on which the whole science of physics had been built. Yet, strange as the theory was, the results it was invented to explain certainly existed, and it could be shown that the old dynamics not only had not explained them, but could not possibly explain them. It was clear that any satisfactory explanation would have to be something quite revolutionary in character. And Planck’s theory did, as a matter of fact, explain the observed radiation phenomena extremely well. Planck calculated, on his theory, how the energy of radiation should be distributed amongst the different wave-lengths, and his calculations precisely agreed with the experimental results. It is possible that, even so, this revolutionary theory would not have obtained general acceptance. But Einstein applied the theory to the phenomena attending the liberation of electrons from metals under the influence of light, and his calculations, also, were shown to be in precise agreement with the evidence. The quantum theory, then, although strange and, in many respects, little understood, has become one of the great arms of modern physical research. It is still attended with very grave difficulties. The phenomena of electron emission from metals, for instance, certainly suggests that light energy exists in small bundles, dotted about round the surface of the sphere which was regarded as forming the old “wave-front.” Each bundle, we may suppose, contains sufficient energy to liberate an electron with the velocity of the electron which gave rise to the bundle. On the other hand, certain well-known phenomena in light, particularly the phenomenon of “interference,” seem utterly irreconcilable with this assumption; they are perfectly well explained on the old wave theory of light, but they seem quite inexplicable on the new quantum theory of light. It depends on which phenomenon we want to explain which theory we employ. Neither of them seem in the least adequate to explain all the known phenomena, and they also seem quite irreconcilable with one another. The physicist must keep both and yet they cannot live together. A compromise has been tried. Sommerfeld and Debye, for instance, have endeavoured to work out a theory whereby the energy brought by the light waves has been regarded as continuous, but as being able, in some way, to accumulate until the amount contained in a quantum “bundle” is reached. Having accumulated to this amount, the energy is then supposed to work suddenly and to shoot out the electron with the requisite velocity. But the period required for this accumulation can be calculated, and it is found that, to explain the effects produced by X-rays, an accumulation period amounting to some years is required. So that the emission of electrons under the influence of X-rays should not take place until some years had elapsed. It is found, however, that the emission takes place immediately the X-rays are applied, and ceases instantly when they are discontinued. The contradiction is complete.

But the quantum theory, however puzzling it may be in certain aspects, has shown itself competent to deal with very baffling phenomena. It was natural, therefore, faced by the great puzzle presented by the stability of the atom, to surmise that here, also, the quantum theory would prove competent to overcome the difficulties. In its original form, the theory could not be applied to the atom. It was first necessary to extend it. This was first done, and the theory successfully applied, by a brilliant young Danish physicist, Niels Bohr.

§ 2. Bohr’s Atom

Before we go on to describe Bohr’s conception of the atom we must make a few remarks about spectra, since the explanation of spectrum lines is one of the most important duties that an atomic model has to fulfil. The whole science of spectrum analysis began with Fraunhofer’s discovery that light from the sun, if spread out in a coloured band by a prism, contained, besides its different colours, a large number of fine dark lines crossing the band at right angles to its length. Kirchhoff found that the light from incandescent gases, when treated in the same way, also gave lines, although in this case the lines were bright lines. But he further found that a gas will absorb the same lines that it emits, so that if light be passed through a gas, dark lines will occur at the same positions as the bright lines occur when the gas is incandescent. Each chemical element was found to have its own appropriate series of lines, and these lines serve, with remarkable delicacy and exactitude, as a means of recognising the presence of these elements. The lines in the sun’s spectrum, for instance, can be disentangled into the groups belonging to each separate chemical element in the sun. A similar analysis, performed on the light from various stars, enables us to say what chemical elements are present in those stars.

Every incandescent substance sends out light of several different wave-lengths. These different rays are, in the ordinary way, jumbled together, but, on being passed through a prism, they are separated out in an orderly manner. The spectrum ranges from the red to the violet. The waves giving red light are the longest waves and those giving violet the shortest. Waves longer than red waves, the so-called infra-red waves, do not affect the retina of our eye as light at all, and the same remark applies to the waves shorter than violet waves, the so-called ultra-violet waves. But such waves, although they do not affect our eyes, can be made to affect certain chemical preparations; with ultra-violet waves, for instance, photographs may be taken of invisible objects, a fact perfectly well known to certain “spirit” photographers. Now each line on a spectrum corresponds to a definite wave-length. Light which is all of one wave-length is called monochromatic light; each line on a spectrum corresponds to a certain wave-length of monochromatic light. Corresponding to each line in the spectrum of a given element the wave-length can be measured, and the interesting question arises as to whether there is any relation between the lengths of the different waves emitted by that element. We shall see that there are such relations, and that Bohr’s theory of the atom takes us some way towards explaining them.

In the first place, we have to assume, in applying quantum theory to the atom, that an electron describes a circle or an ellipse round the nucleus without radiating any energy. This assumption is in flat disagreement with the classical theory of electrodynamics, but it is in agreement with the quantum theory. Another assumption we must make, and which is not in agreement with the old theory, is that the electron can only move in certain orbits. If the orbit be a circular one, for instance, then an electron can only circulate round the nucleus at certain definite distances from it. It could not describe a circle whose radius was intermediate between two of these distances. Whatever one of the possible circles the electron is on, it will continue to traverse that circle indefinitely unless some external force acts on it. If an external force does act on it, then the electron passes directly to another of the possible circles. During this transition from one possible circle to another the electron radiates energy, and this energy is monochromatic, that is, it is energy of a perfectly definite wave-length. And the amount of energy so emitted is a quantum of energy. The quantum of energy belonging to a certain frequency depends upon that frequency. Its amount is, in fact, equal to the frequency multiplied by a certain extremely small figure called Planck’s constant. Thus the quantum, or the atom of energy, is not an invariable thing. Like the atoms of matter, energy atoms are of different sizes. The higher the frequency the greater the atom of energy. As we have said, a monochromatic radiation is emitted by the electron in passing from one possible orbit to another. This radiation has, of course, a definite wave-length and therefore a definite frequency corresponding to it. This frequency, multiplied by the quantity called Planck’s constant, is equal to the total energy emitted by the electron in passing from one orbit to the other.

We have said that certain relations have been found to exist between the lines in the spectrum of a given element. It was in 1885 that Balmer discovered that the lines in the spectrum of hydrogen could be represented by a certain very simple formula. The frequencies corresponding to a certain prominent group of lines in the hydrogen spectrum may be represented by multiplying a certain constant figure by the quantity (1/4 − 1/n^{2}) where n takes on the values 3, 4, 5, 6, 7. These are the five strongest hydrogen lines, and for them the quantity in the brackets becomes (1/4 − 1/9), (1/4 − 1/16), (1/4 − 1/25), (1/4 − 1/36), (1/4 − 1/49). Each of these values is to be multiplied by a certain figure, the same in each case, and the results will be the frequencies corresponding to each of these five lines respectively. Another series of lines in the hydrogen spectrum is obtained by using, instead of the general quantity in brackets given above, the quantity 1/9 − 1/n^{2}, where n takes on the values 4, 5, 6, etc. Still another series can be obtained from the general formula 1/1 − 1/n^{2}, where n has the values 2, 3, 4, etc. It is easy to see that the most general formula, including all these cases, is 1/m^{2} − 1/n^{2}. In the first formula we gave, for instance, m = 2. In the second m = 3, and in the third m = 1. Formulæ which are a trifle more complicated were discovered later, and were found to represent still other series of lines. And these formulæ were applied to other elements besides hydrogen.

On Bohr’s theory, when an electron passes from one orbit to another, it emits a certain quantity of energy, and the energy so radiated has a certain frequency. If, therefore, we subtract the energy possessed by the electron in its second orbit from the energy it possessed in its first orbit, we have the total quantity of energy emitted by it in passing from one to the other. The frequency, therefore, could be calculated from the subtraction of these two quantities. Now it is suggestive that Balmer’s formula for the frequency, given above, is expressed by the subtraction of two quantities. Bohr showed that this was no accident, and that the two quantities in Balmer’s formula do indeed correspond to the energies before and after the transition of the electron from one orbit to the other. In fact, Bohr was able, on his theory, to deduce Balmer’s formula. It no longer appeared as a mere empirical rule, but as a theoretical consequence of the structure of the atom. This result was a most striking success for Bohr’s theory to achieve. He also deduced, from his theory, the value of the constant figure which is used to multiply the different quantities in brackets given above; his calculated figure and the empirically ascertained figure were in precise agreement. The values of the different possible radii on which the electron in a hydrogen atom can move were also deduced by Bohr. The electron is most stable when on its first orbit, the orbit nearest the nucleus. This is the normal condition for a hydrogen atom. The actual diameter of a hydrogen atom in this condition can be calculated on Bohr’s theory, and the value so obtained is found to be in agreement with the value obtained by quite other methods.

The fact that the spectrum of hydrogen possesses a number of lines, therefore, shows us that, in the immense number of atoms present in any specimen of hydrogen, there are always many whose electrons are passing from one orbit to another. In one atom an electron will be passing from the second to the first orbit, or from the third to the second, or from the fourth to the third, and so on. Such transitions must always be going on, for it is only in virtue of them that the hydrogen atoms radiate any energy at all. The state to which all these changes tend is the most stable state, when the electron is on its first orbit. We may say, then, as Bohr puts it, that the spectrum of hydrogen shows us the formation of the hydrogen atom, since the transition to the successively decreasing orbits may be regarded as stages in the process by which the hydrogen atom reaches its normal condition.

§ 3. The Fine Structure of Hydrogen Lines

We may summarise the theory of the hydrogen atom we have given hitherto by saying that the hydrogen atom consists of a positive nucleus carrying one unit of charge, and that a single electron is describing an elliptical orbit about it. We can imagine a number of ellipses, of different sizes, enclosing the nucleus. Each of these ellipses will have a common focus, and it is at this focus that the nucleus is situated. The single electron can move on any one of these ellipses, but only on these; it cannot describe an intermediate orbit. Under the influence of an external force the electron may jump from one of these ellipses to another. During this jump it radiates energy in the form of monochromatic waves. As long as it remains on any one of these ellipses it is not radiating energy. These elliptical motions are called stationary states. The electron only radiates energy, then, in passing from one stationary state to another. This simple theory suffices to explain the positions of the lines in the hydrogen spectrum. We reach a formula which is exactly like Balmer’s formula showing the distribution of these lines.

When we come to look more closely into the matter, however, we find that there is a factor we have neglected in our calculations. We have said that the electron describes an ellipse about the nucleus. We suppose that it describes this ellipse in obedience to the ordinary laws which regulate the motion of a single planet about the sun. It is a peculiarity of such motion that the speed is not uniform. A planet, in its elliptical orbit about the sun, is sometimes moving faster and sometimes slower, depending upon which part of the ellipse it is describing. At those parts of the ellipse which are nearest the sun the planet is moving fastest. At the parts most remote from the sun it is moving most slowly. As precisely the same laws apply to our electron moving round the nucleus, we have to take into account the fact that the speed of the electron in its orbit is continually changing. This is where the theory of relativity comes in. We have seen that it is a consequence of that theory that the mass of an electron varies with its velocity, becoming greater the greater the velocity. Our electron, therefore, is not only moving with a varying speed; it is also moving with a varying mass. What influence will this variation of mass have on the motion?

This problem was solved by Sommerfeld. The result is that the electron continues to move in an almost elliptical orbit, but this orbit itself is slowly and uniformly rotating. The actual motion of the electron in space is a combination of these two motions. The effect of this on the spectrum of hydrogen will be that corresponding to each hydrogen line there will be two or three lines extremely close together. Each hydrogen line will really consist of more than one line. These lines will be so close together that it would be almost impossible to see them separately. Nevertheless, measurements have been made, and these measurements are in agreement with Sommerfeld’s theory. The fine structure, as it is called, of the hydrogen lines, is due to the variations in mass of the electron in describing its orbit about the hydrogen nucleus. The complete explanation of the hydrogen spectrum requires both quantum theory and relativity theory; conversely, the striking agreement between calculation and observation in the hydrogen spectrum greatly supports both these theories.