Chapter V: The Bohr Theory of the Hydrogen Spectrum (1)

The Nuclear Atom and Electrodynamics.

Even if Rutherford had not yet succeeded in giving a complete answer to the first of the questions propounded in the previous chapter, namely, that concerning the positions of the positive and negative particles of the atom, one might at any rate hope that his general explanation of the structure of the atom—that is, the division into the nucleus and surrounding electrons, and the determination of the number of electrons in the atoms of the various elements—would furnish a good foundation for the answer to the second question about the connection between the atomic processes and the physical and chemical properties of matter. But in the beginning this seemed so far from being true that it appeared almost hopeless to find a solution of the problem of the atom in this way.

We shall best understand the meaning of this if we consider the simplest of the elemental atoms, namely, the atom of hydrogen with its positive nucleus and its one electron revolving about the nucleus. How could it be possible to explain from such a simple structure the many sharp spectral lines given by the Balmer-Ritz formula (p. 57)? As has previously been mentioned, the classical electron theory seemed to demand a very complicated atomic structure for the explanation of these lines. According to the electron theory, the atoms may be likened to stringed instruments which are capable of emitting a great number of tones, and in these atoms the electrons are naturally supposed to correspond to the “strings.” But the hydrogen atom has only one electron, and it hardly seems credible that in a mass of hydrogen the individual atoms would be tuned for different “tones,” with definite frequencies of vibration.

Now, it certainly cannot be concluded from the analogy with the stringed instrument that a single electron can emit light of only a single frequency at one time, corresponding to a single spectral line. For a plucked string will, as we know, give rise to a simple tone only if it vibrates in a very definite and particularly simple way; in general it will emit a compound sound which may be conceived as made up of a “fundamental” and its so-called “overtones,” or “harmonies” whose frequencies are 2, 3, ... times that of the fundamental (_i.e._ integral multiples of the latter). These overtones may arise even separately because the string, instead of vibrating as a whole, may be divided into 2, 3, ... equally long vibrating parts, giving respectively 2, 3, ... times as great frequencies of vibration. We call such vibrations “harmonic oscillations.” The simultaneous existence of these different modes of oscillation of the string may be thought of in the same way as the simultaneous existence of wave systems of different wave-lengths on the surface of water. Corresponding to the possibility of resolving the motion of the string into its “harmonic components,” the compound sound waves produced by the string can be resolved by resonators (cf. p. 44) into tones possessing the frequencies of these components.

According to the laws of electrodynamics the situation with the electron revolving about the hydrogen nucleus might be expected to be somewhat similar to that described above in connection with the vibrating string. If the orbit of the electron were a circle, it should emit into the ether electromagnetic waves of a single definite wave-length and corresponding frequency, ν, equal to ω, the frequency of rotation of the electron in its orbit; that is, the number of revolutions per second. But just as a planet under the attraction of the sun, varying inversely as the square of the distance, moves in an ellipse with the sun at one focus, so the electron, under the attraction of the positive nucleus, which also follows the inverse square law, will in general be able to move in an ellipse with the nucleus at one focus. The electromagnetic waves which are emitted from such a moving electron may on the electron theory be considered as composed of light waves corresponding to a series of harmonic oscillations with the frequencies:

ν₁ = ω, ν₂ = 2ω, ν₃ = 3ω ... and so on,

where ω, as before, is the frequency of revolution of the electron. According as the actual orbit deviates more or less from a circle, the frequencies ν₂, ν₃ ... will appear stronger or weaker in the compound light waves emitted. But the actual distribution of spectral lines in the real hydrogen spectrum presents no likeness whatever to this distribution of frequencies.

From this it is evident that no agreement can be reached between the classical electron theory on the one hand and the Rutherford atom model on the other. Indeed, the disagreement between the two is really far more fundamental than has just been indicated. According to Lorentz’s explanation of the emission of light waves, the electrons in a substance (see again p. 75) should have certain equilibrium positions, and should oscillate about these when pushed out of them by some external impulse. The energy which is given to the electron by such an impulse is expended in the emission of the light waves and is thus transformed into radiation energy in the emitted light, while the electrons fall to rest again unless they receive in the meantime a new impulse. We can get an understanding of what these impulses in various cases may be by thinking of them, in the case of a glowing solid, for example, as due to the collisions of the molecules; or in the case of the glowing gas in a discharge tube, from the collisions of electrons and ions. The oscillating system represented by the electron (the “oscillator”) will possess under these circumstances great analogy with a string which after being set into vibration by a stroke gradually comes back to rest, while the energy expended in the stroke is emitted in the form of sound waves. Although the vibrations of the string become weaker after a while, the period of the vibrations will remain unchanged; the string vibrations like pendulum oscillations have an invariable period, and the same will be the case with the frequency of the electron if the force which pulls it back into its equilibrium position is directly proportional to the displacement from this position (the “harmonic motion” force).

Rutherford’s atomic model is, however, a system of a kind wholly different from the “oscillator” of the electron theory. The one revolving hydrogen electron will find a position of “rest” or equilibrium only in the nucleus itself, and if it once becomes united with the latter it will not easily escape; it will then probably become a nuclear electron, and such a process would be nothing less than a transformation of elements (see p. 79). On the other hand, it follows necessarily from the fundamental laws of electrodynamics that the revolving electron must emit radiation energy, and, because of the resultant loss of energy, must gradually shrink its path and approach nearer the nucleus. But since the nuclear attraction on the electron is inversely proportional to the square of the distance, the period of revolution will be gradually decreased and hence the frequency of revolution ω, and the frequency of the emitted light will gradually increase. The spectral lines emitted from a great number of atoms should, accordingly, be distributed evenly from the red end of the spectrum to the violet, or in other words there should be no line spectrum at all. It is thus clear that Rutherford’s model was not only unable to account for the number and distribution of the spectral lines; but that with the application of the ordinary electrodynamic laws it was quite impossible to account for the existence even of spectral lines. Indeed, it had to be admitted that an electrodynamic system of the kind indicated was mechanically unstable and therefore an impossible system; and this would apply not merely to the hydrogen atom, but to all nuclear atoms with positive nuclei and systems of revolving electrons.

However one looks at the matter, there thus appears to be an irremediable disagreement between the Rutherford theory of atomic structure and the fundamental electrodynamic assumptions of Lorentz’s theory of electrons. As has been emphasized, however, Rutherford founded his atomic model on such a direct and clear-cut investigation that any other interpretation of his experiments is hardly possible. If the result to which he attained could not be reconciled with the theory of electrodynamics, then, as has been said, this was so much the worse for the theory.

It could, however, hardly be expected that physicists in general would be very willing to give up the conceptions of electrodynamics, even if its basis was being seriously damaged by Rutherford’s atomic projectiles. Surmounted by its crowning glory—the Lorentz electron theory—the classical electrodynamics stood at the beginning of the present century a structure both solid and spacious, uniting in its construction nearly all the physical knowledge accumulated during the centuries, optics as well as electricity, thermodynamics as well as mechanics. With the collapse of such a structure one might well feel that physics had suddenly become homeless.

The Quantum Theory.

In a field completely different from the above the conclusion had also been reached that there was something wrong with the classical electrodynamics. Through his very extended speculations on thermodynamic equilibrium in the radiation process, Planck (1900) had reached the point of view expressed in his _quantum theory_, which was just as irreconcilable with the fundamental electrodynamic laws as the Rutherford atom.

A complete representation of this theory would lead us too far; we shall merely give a short account of the foundations on which it rests.

By a black body is generally understood a body which absorbs all the light falling upon it, and, accordingly, can reflect none. Physicists, however, denote by the term “perfect black body” in an extended sense, a body which at all temperatures absorbs all the radiation falling upon it, whether this be in the form of visible light, or ultra-violet, or infra-red radiation. From considerations which were developed some sixty years ago by Kirchhoff, it can be stated that the radiation which is emitted by such a body when heated does not depend on the nature of the body but merely on its temperature, and that it is greater than that emitted by any other body whatever at the same temperature. Such radiation is called temperature radiation or sometimes “black” radiation, though the latter term is apt to be misleading, since a “perfect black body” emitting black radiation may glow at white heat. It may be of interest to note here the fundamental law deduced by Kirchhoff, which may best be illustrated by saying that good absorbers of radiation are also good radiators. An instructive experiment illustrating this is performed by painting a figure in lampblack on a piece of white porcelain. The lampblack surface is clearly a better absorber of radiant energy than the white porcelain. When the whole is heated in a blast flame, the lampblack figure glows much more brightly than the surrounding porcelain, thus showing that at the same temperature it is also the better radiator. Following the same law we conclude that highly reflecting bodies are not good radiators, a fact that has practical significance in house heating. The perfect black body, then, being the best absorber of radiation, is also the best radiator.

In actual practice no body is absolutely black. Even a body coated with lampblack reflects about 10 per cent. of the light waves incident on it. The Danish physicist Christiansen remarked long ago that a real black body could be produced only if an arrangement could be made whereby the incident waves could be reflected several times in succession before finally being emitted. To take the case of lampblack, three such reflections would reduce the re-emission from the lampblacked body to only ⁴/₁₀₀₀ of the radiation initially falling on it. This type of black body was finally realized by making a cavity in an oven having as its only opening a very small peep-hole, and keeping the temperature of the wall of the oven uniform. If a ray is sent into the cavity through the peep-hole, it will become, so to speak, captured, because, when once inside it will suffer countless reflections from the walls of the cavity, having more and more of its energy absorbed at each reflection. Very little of the radiation thus entering will ever get out again, and consequently such a body will act very much like a “perfect black body” according to the theoretical definition above. Accordingly, the radiation which is emitted from such a glowing cavity through the peep-hole will be black or practically black radiation.

The cavity itself will be criss-crossed in all directions by radiation emitted from one part of the inner surface of the cavity and absorbed by (and partially reflected from) other parts of the surface. When the walls of the cavity are kept at a fixed and uniform temperature there will automatically be produced a state of equilibrium in which every cubic centimetre of the cavity will contain a definite quantity of radiation energy, dependent only on the temperature of the walls. Further, in the equilibrium state the radiation energy will be distributed in a perfectly definite way (dependent as before on the temperature only) among the various types of radiation corresponding to different wave-lengths and frequencies. If there is too much radiation of one kind and too little of another, the walls of the cavity will absorb more of the first kind than they emit, and emit more of the second kind than they absorb, and so the state will vary until the right proportion for equilibrium is attained.

This distribution of the radiation over widely differing wave-lengths can be investigated by examining spectroscopically the light emitted from the peep-hole in the cavity. Then, by means of a bolometer or some other instrument, the heat development in the different portions of the spectrum can be measured. At a temperature of 1500° C., for example, one will find that the maximum energy is represented for rays of wave-length in the close vicinity of 1·8 μ, _i.e._ in the extreme infra-red. If the temperature is raised, the energy maximum travels off in the direction of the violet end of the spectrum; if the temperature is lowered, it will move farther down into the infra-red.

It is also possible to make a theoretical calculation of the distribution of energy in the spectrum of the black-body radiation at a given temperature. But the results obtained do not agree with experiment. The English physicists, Rayleigh and Jeans, developed on the basis of the classical electrodynamic laws and by apparently convincing arguments a distribution law according to which actual radiation equilibrium becomes impossible, since if it were true the energy in the radiation would tend more and more to go over to the region of short wave-lengths and high frequencies, and this shifting would apparently go on indefinitely. The theory thus leads to results which are not only in disagreement with experiment, but which must be looked upon as extremely unreasonable in themselves.

Planck, however, had vanquished these difficulties and had obtained a radiation law in agreement with experiment by introducing an extremely curious hypothesis. Like Lorentz, he thought of radiation as produced through the medium of small vibrating systems or oscillators, which could emit or absorb rays of a definite frequency ν. But while, according to the Lorentz theory and the classical electrodynamics, radiation can be emitted in infinitely small quantities (_i.e._ small without limit), Planck assumed that an oscillator can emit and absorb energy only in certain definite quantities called _quanta_, where the fundamental quantum of radiation is dependent on the frequency of the oscillator, varying directly with the latter. If thus we denote the smallest quantity of energy which an oscillator of frequency ν can emit or absorb by E, then we can write

E = _h_ν,

where _h_ is a definite constant fixed for all frequencies. Accordingly the cavity can receive radiation energy of frequency ν from the radiating oscillators in its wall, or transfer energy to these in no smaller quantity than _h_ν. The total energy of that kind emitted or absorbed at any given time will always be an integral multiple of _h_ν. Oscillators with a frequency 1½ times as great will emit energy in quanta which are 1½ times larger, and so on.

The quantity _h_ is independent, not only of the wave-length, but also of the temperature and nature of the emitting body. This constant, the so-called _Planck constant_, is thus a universal constant. If one uses the “absolute” units of length, mass and time (see table, p. 210), its value comes out as 6·54 × 10⁻²⁷. For the frequency 750 × 10¹² vibrations per second, corresponding to the extreme violet in the visible spectrum, the Planck energy quantum thus becomes about 5 × 10⁻¹² erg, or 3·69 × 10⁻¹⁹ foot-pounds (note that the “erg” is the “absolute” unit of work, or the amount of work done when a body is moved through a distance of 1 cm. by a force of one dyne acting in the direction of the motion, while the “foot-pound” is the work done when a force of 1 pound moves a body 1 foot in the direction of the force). For light belonging to the red end of the spectrum, the energy is about half as great. If we pass, however, to the highest frequencies and the shortest wave-lengths which are known, namely those corresponding to the “hardest” (_i.e._ most penetrating) γ-rays (see p. 78), we meet with energy quanta which are a million times larger, _i.e._ 2 × 10⁻⁶ erg, although they are still very small compared to any amount of energy measurable mechanically.

This remarkable theory of quanta, which in the hands of Planck still possessed a rather abstract character, proved under Einstein’s ingenious treatment to have the greatest significance in many problems which, like heat radiation, had provided physicists with many difficulties. For by assuming that energy in general could only be given up and taken in in quanta, certain facts about the specific heats of bodies could be accounted for—facts which the older physics had proved powerless to explain. The Planck energy quanta, as Einstein showed, could also explain in a very direct and satisfactory way the _photoelectric effect_, as it is called. This effect consists in the freeing of electrons from a metal plate which ultra-violet rays are allowed to strike. The maximum velocity with which these electrons are propelled from the plate is found to be independent of the intensity of the incident light, but dependent simply on the frequency of the radiation. Careful measurements have indeed shown, as Einstein predicted, that the incident light really does utilize an energy quantum _h_ν to free each electron and give it velocity (cf. p. 172). Of the different methods which nowadays are at hand, the photoelectric effect constitutes one of the best means of determining the value of _h_. It has been applied for that purpose by Millikan, to whose ingenious experiments the most accurate direct determination of _h_ is actually due.

All this lay completely outside the laws of electrodynamics, and pointed to the existence of unknown and more fundamental laws. But, for the time being, physicists had to be satisfied merely with a recognition of the fact that these mysterious energy quanta play a very significant part in many phenomena.

A decade ago physics, as regards radiation problems, was in a very unsettled state; with four separate branches of knowledge, each of which seemed firm and well-founded enough in itself, but which had no common connecting link, indeed, were even to some extent inconsistent with each other. The first of these was the classical electrodynamics surmounted by the felicitous electron theory of Lorentz and Larmor. The second was the empirical knowledge of the spectra resting on the work of Balmer, Ritz and Rydberg. The third was Rutherford’s nuclear atom model. And the fourth was Planck’s quantum theory of heat radiation. It was quite evident that progress in the theory of radiation and the structure of the atom was hopeless as long as these four points of view remained uncorrelated.

Main Outlines of the Bohr Theory.

Such was the situation when, in 1913, Bohr published his atomic theory, in which he was able with great ingenuity to unite the nuclear atom, the Balmer-Ritz formula and the quantum theory. As far as electrodynamics is concerned, the impossibility of retaining that in its classical form was presented in a much clearer way than ever before. But, as will presently be evident, the Bohr theory has a very definite connection with the classical theory, and Bohr’s attempts to preserve and develop this connection have proved to be of the greatest significance for his theory. In spite of the fundamental rupture with the old ideas, the Bohr theory strives to absorb all that is useful in the classical point of view.

At the head of the theory appear the two fundamental hypotheses or postulates on the properties of the atom.

_The first postulate states that for each atom or atomic system there exists a number of definite states of motion, called “stationary states,” in which the atom (or atomic system) can exist without radiating energy. A finite change in the energy content of the atom can take place only in a process in which the atom passes completely from one stationary state to another.

The second postulate states that if such a transition takes place with the emission or absorption of electromagnetic light waves, these waves will have a definite frequency, the magnitude of which is determined by the change in the energy content of the atom. If we denote the change in energy by_ E _and the frequency by ν we may write_

E
E = _h_ν, or ν = -----
_h_

_where h is the Planck constant. In consequence of the second postulate the emission as well as the absorption of energy by the atom always takes place in quanta._

The two postulates say nothing concerning the nature of the motion in the stationary states. In the applications, however, a connection with the Rutherford atomic model is established. Confining our attention first to the hydrogen atom, the system with which we are concerned consists, accordingly, of a positive nucleus and one electron revolving about it. The various states of motion which the electron can assume in virtue of the first postulate are a series of orbits at different distances from the nucleus. In each of these “stationary orbits” the electron follows the general mechanical laws of motion; _i.e._ under the nuclear attraction which is inversely proportional to the square of the distance, the electron describes an ellipse with the nucleus at one focus, as has previously been stated; but in contradiction with the classical electrodynamics it will emit no radiation while moving in this orbit. Fig. 25 shows a series of these orbits, to which the numbers 1, 2, 3, 4 have been attached, and which for simplicity are represented as circular.

If the electron passes from an outer orbit to an inner one; for example, if it goes from number 4 to number 2, or from number 2 to number 1, the electric force which attracts it to the nucleus will do work just as the force of gravity does work when a stone falls to the ground. A part of this work is used to increase the kinetic energy of the electron, making its velocity in the inner orbit greater than in the outer, but the rest of the work is transformed into radiation energy which is emitted from the atom in the form of monochromatic light. In consequence of the second postulate the frequency of the emitted radiation is proportional to the energy loss. When the electron has reached the innermost orbit (the one denoted by 1 in the figure), it cannot get any nearer the nucleus and hence cannot emit any more radiation unless it first is impelled to pass from its inner orbit to an outer orbit again by the absorption of external energy sufficient to bring about this change. Once in the outer orbit again, it is in a state to produce radiation by falling in a second time. The innermost orbit represents thus the electron’s equilibrium state, and corresponds to the _normal state of the atom_.

If we try to illustrate the matter with an analogy from the theory of sound, we can do so by comparing the atom not with a stringed instrument, but with a hypothetical musical instrument of a wholly different kind. Let us imagine that we have placed one over another and concentrically a series of circular discs of progressively smaller radii, and let us suppose that a small sphere can move around any one of these without friction and without emitting sound. In such a motion the system may be said to be in a “stationary state.” Sooner or later the sphere may fall from the first disc on to one lower down and continue to roll around on the second, having emitted a sound, let us assume, by its fall. By passing thus from one stationary state to another it loses a quantity of energy equal to the work which would be necessary to raise it again to the disc previously occupied, and to bring it back to the original state of motion. We can assume that the energy which is lost in the fall reappears in a sound wave emitted by the instrument, and that the pitch of the sound emitted is proportional to the energy sent out. If, moreover, we imagine that the lowermost disc is grooved in such a way that the sphere cannot fall farther, then this fanciful instrument can provide a very rough analogy with the Bohr atom. We must beware, however, of stretching the analogy farther than is here indicated.

It must be specially emphasized here that the frequency of the sound emitted in the above example has no connection with the frequency of revolution of the sphere. In the Bohr atom, likewise, the frequency of revolution ω of the electron in its stationary orbit has no direct connection with the frequency of the radiation emitted when the electron passes from this orbit to another. This is a very surprising break with all previous views on radiation, a break whose revolutionary character should not be under-estimated. But, however unreasonable it might seem to give up the direct connection between the revolutional frequency and the radiation frequency, it was absolutely necessary if the Rutherford atomic model was to be preserved. And as we shall now see, the new point of view of the Bohr theory leads naturally to an interpretation of the Balmer-Ritz formula, which had previously not been connected with any other physical theory.

The quantity of energy E, which the atom gives up when the electron passes from an outer to an inner orbit, or which, conversely, is taken in when the electron passes from an inner to an outer orbit, may, as has been indicated, be regarded as the difference between the energy contents of the atom in the two stationary states. This difference may be expressed in the following way. Let us imagine that we eject the electron from a given orbit (_e.g._ No. 2 in the diagram) so that it is sent to “infinity,” or, in other words, is sent so far away from the nucleus that the attraction of the latter becomes negligible. To bring about this removal of the electron from the atom demands a certain amount of energy, which we can call the _ionizing work_ corresponding to the stationary orbit in question. We may here designate it as A₂. To eject the electron from the orbit No. 4 will demand a smaller amount of ionizing work, A₄. The difference A₂ - A₄ is accordingly the work which must be done to transfer the electron from the orbit No. 2 to the orbit No. 4. This is, however, exactly equal to the quantity E of energy which will be emitted as light when the electron passes from orbit No. 4 to orbit No. 2. If we call the frequency of this light ν, then from the relations E = _h_ν and E = A₂ - A₄, we have

_h_ν = A₂ - A₄

If, now, in place of this specific example using the stationary orbits 2 and 4 we take any two orbits designated by the numbers _n″_ (for the inner) and _n′_ (for the outer), we can write for the frequency of the radiation emitted for a transition between these arbitrary states

_h_ν = Aₙ˶ - Aₙˊ or

Aₙ˶ Aₙˊ
ν = ----- - -----
_h_ _h_

We have now reached the point where we ought to bring in the Balmer-Ritz formula for the distribution of the lines in the hydrogen spectrum. This formula may be written (see p. 59)

K K
_ν_ = ------ - -----
_n″_² _n′_²

We can now see very clearly the similarity between the formula derived from the spectrum investigations and that derived from the two Bohr postulates. In both formulæ the frequency appears as the difference between two terms which are characterized in both cases by two integral numbers, in the first formula, numbers denoting two stationary orbits in the Bohr model for hydrogen, and in the second the two numbers which in the Balmer-Ritz formula for the hydrogen spectrum characterize, respectively, a series and one of the lines of the series. To obtain complete agreement we have merely to equate the corresponding terms in the two formulæ. Thus we have for any arbitrary integer _n_

Aₙ K
----- = ---- or
_h_ _n_²

_h_K
Aₙ = ------
_n_²

For the innermost stationary orbit, for which _n_ = 1, the ionizing work A₁ will accordingly be equal to the product of the constants _h_ and K of Planck and Balmer respectively; and for the orbits No. 2, No. 3, No. 4, etc., the values will be respectively ¼, ¹/₉, ¹/₁₆, etc., of this product. From the charges on the nucleus and the electron, which are both equal to the elementary quantum _e_ of electricity (see p. 90), and from the ionizing energy for a given orbit we can now find by the use of simple mechanical considerations the radius of the orbit. If we denote the radii of the orbits 1, 2, 3 ... by _a_₁, _a_₂, _a_₃ ..., we then obtain for the diameters 2_a_₁, 2_a_₂, 2_a_₃ ... the values 2_a_₁ = 1·056 × 10⁻⁸ cm. (or approximately 2_a_₁ = 10⁻⁸ cm.), 2_a_₂ = 4 × 10⁻⁸ cm., 2_a_₃ = 9 × 10⁻⁸ cm., etc. It is seen that the radii of the orbits are in the proportion 1, 4, 9 ..., or in other words the squares of the integers which determine the orbit numbers. It is in this proportion that the circles in Fig. 25 are drawn. We must remember, however, that we have here for the moment been thinking of the orbits as circles, while in reality they must in general be assumed to be ellipses. The foregoing considerations will, however, still hold with the single change that 2_a_ₙ will now mean, instead of the diameter of a circle, the major axis of an ellipse.

Let us return to the formulæ

Aₙ″ Aₙ′
ν = ----- - ----
_h_ _h_

K K
and ν = ------ - -----
_n″_² _n′_²

Here _n″_ denotes in the first formula the index number for the _inner_ of the two orbits between which the transition is supposed to take place, while in the second formula _n″_ denotes a definite series in the hydrogen spectrum. If _n″_ is 2 while _n′_ takes on the values 3, 4, 5 ... ∞ then in the Bohr model of the hydrogen atom this corresponds to a series of transitions _to_ the orbit No. 2 _from_ the orbits 3, 4, 5 ..., while in the hydrogen spectrum this corresponds to the lines in the Balmer series, namely, the red line (Hα) corresponding to the transition 3-2, the blue-green line (Hβ) to 4-2, the violet line (Hγ) to 5-2 and so on. If we now put _n″_ = 1 while _n′_ takes the values 2, 3, 4 ..., we get in the atom transitions to the orbit No. 1 from the orbits No. 2, 3, 4 ..., corresponding in the spectrum to what is called the Lyman series in the ultra-violet (named after the American physicist Lyman, who has carried on extensive researches in the ultra-violet region of the spectrum). Thus every line in the hydrogen spectrum is represented by a transition between two definite stationary states in the hydrogen atom, since this transition will give the frequency corresponding to the line in question.

At first sight this would seem perhaps to be such an extraordinary satisfactory result that it would prove an overwhelming witness in favour of the Bohr theory. A little more careful thought, on the other hand, would perhaps cause a complete reversion from enthusiasm and lead some to say that the whole thing has not the slightest value, because the stationary states were so chosen that agreement might be made with the Balmer-Ritz formula. This last consideration, indeed, states the truth in so far that the agreement between the formula and the theory, at least as developed here up to this point, is of a purely formal nature. In the Bohr postulates the frequencies of the emitted radiation are determined by a difference between two of a series of energy quantities, characterizing the stationary states, just as in the Balmer-Ritz formula they appear as a difference between two of a series of terms (K, K/4, K/9, ...) each characterized by its integer. Now by characterizing the quantities of energy in the stationary states by a series of integers (in itself a wholly arbitrary procedure) complete agreement between the Bohr stationary states idea and the spectral formulæ can be attained. It is not even necessary to introduce the Rutherford atomic model to attain this end. By bringing in this specific model, one might join the new theory to the knowledge already gained of the atomic structure, and, so to speak, crystallize the hitherto undefined or only vaguely defined stationary states into more definite form as revolution in certain concrete orbits. This would then lead to a more comprehensive conception of atomic structure. But the theory unfortunately would still be rather arbitrary, since there would seem to be no justification for picking out certain fixed orbits with definite diameters or major axes to play a special rôle. One cannot wonder then that many scientists considered the Bohr theory unacceptable, or at any rate were inclined to look upon it simply as an arbitrary, unreasonable conception which really explained nothing.

Naturally, Bohr himself clearly recognized the formal nature of the agreement between the Balmer-Ritz formula and his postulates. But Bohr was the first to see that the quantum theory afforded the possibility of bringing about such an agreement, and he saw, moreover, that the agreement was not merely fortuitous, but contained within it something really fundamental, on which one could build further. That atomic processes on his theory took on an unreasonable character (compared with the classical theory) was nothing to worry about, for Bohr had come to the clear recognition that it was completely impossible to understand from known laws the Planck-Einstein “quantum radiation,” or to deduce the properties of the spectrum from the Rutherford atom alone. He therefore saw that his theory was really not introducing new improbabilities, but was only causing the fundamental nature of the contradictions which had previously hindered development in this field to appear in a clearer light.

But in addition to this the choice of the dimensions of the stationary states was by no means so arbitrary as might appear in the foregoing. In his first presentation of the theory of the hydrogen spectrum, Bohr had derived his results from certain considerations connected with the quantum theory—considerations of a purely formal nature, indeed, just as those developed in the preceding, but leading to agreement with the spectral formulæ. He, moreover, called attention to the fact that the values obtained for the orbital dimensions were of the same order of magnitude as those which could be expected on wholly different grounds. The diameter of the innermost orbit, _i.e._, that which defines the outer limit of the atom in the normal state, was found to be, as has been noted above, about 10⁻⁸ cm., _i.e._, of the same order of magnitude as the values obtained for the diameters of molecules on the kinetic theory of gases (see p. 27). The stationary states corresponding to very high quantum numbers one could expect to meet only when hydrogen was very attenuated, for otherwise there could be no room for the large orbits. We note that the 32nd orbit must have a diameter 32² (or over 1000 times) as great as the innermost orbit. Since, now, lines with high number in a hydrogen series correspond on the Bohr theory to transitions from orbits of high number to an inner orbit, it became understandable why only comparatively few lines of the Balmer series are ordinarily observed in the discharge tube, while many more lines are observed in the spectra of certain stars. For in such stars the possibility is left open for hydrogen to exist in a very attenuated state, and yet in such large masses that the lines in question can become strong enough for observation. In fact, one must assume that in a great mass of hydrogen a very large number of atoms send out simultaneously light of the wave-length corresponding to one line. For the ionizing work, _i.e._, the work necessary to eject the electron completely from the normal state and thus make the atom into a positive ion, the Bohr theory gives a value of the same order of magnitude as the so-called “ionization potentials” which have been found by experiment for various gases. An exact correspondence between theory and experiment could for hydrogen not be attained with certainty, because the hydrogen atoms in hydrogen gas under ordinary conditions always appear united in molecules.

In his very first paper, however, Bohr had studied Balmer’s formula also from another point of view, and had derived in this way an expression for the Rydberg constant K which agreed with experiment. These considerations have reference to the above-mentioned connection of the theory with the classical theory of electrodynamics.

Such a connection had previously been known to exist in the fact that, for long wave-lengths, the radiation formula of Planck reduces practically to the Rayleigh Jeans Law which can be derived from electrodynamics. This is related to the fact that when ν is small (long wave-lengths), the energy quantum _h_ν is very small, and hence the character of the radiation emitted will approach more and more nearly to a continuous “unquantized” radiation. One might then expect that the Bohr theory also should lead in the limit of long wave-lengths and small frequencies to results resembling those of the ordinary electrodynamic theory of the radiation process. On the Bohr theory we get the long wave-lengths for transitions between two stationary states of high numbers (numbers which also differ little from each other). Thus suppose _n_ is a very large number. Then the transition from the orbit _n_ to the orbit _n_ - 1 will give rise to radiation of great wave-length. For in this case Aₙ and Aₙ₋₁ differ very little, and accordingly _h_ν is very small, as must ν be also. According to the electrodynamic theory of radiation, the revolving electron should emit radiation whose frequency is equal to the electron’s frequency of revolution. According to the Bohr theory it is impossible to fulfil this condition exactly, since radiation results from a transition between two stationary orbits in each of which the electron has a distinct revolutional frequency. But if _n_ is a large number, the difference between the frequencies of revolution ωₙ and ωₙ₋₁ for the two orbits _n_ and _n_ - 1, respectively, becomes very small; for example, for _n_ = 100, it is only 3 per cent. For a certain high value of _n_, then, the frequency of the emitted radiation can therefore be _approximately_ equal to the frequency of revolution of the electron in both the two orbits, between which the transition takes place. But even if this proved correct for values of n about 100, one could not be sure beforehand whether it would work out right for still larger values of _n_, for example, 1000.

In order to investigate this latter point we must look into the formulæ for the revolutional frequency ω in a stationary orbit and for the radiation frequency ν. Since, according to the Bohr theory, we can apply the usual laws of mechanics to revolution in a stationary orbit, it is an easy matter to find an expression for ω. From a short mathematical calculation we can deduce that ω = R/_n_³, where R is the frequency of revolution for the first orbit (_n_ = 1). We find ν, on the other hand, by substituting in the Balmer-Ritz formula the numbers _n_ and _n_ - 1, and a simple calculation shows that for great values of _n_, the expression for ν will approach in the limit the simple form ν = 2K/_n_&sup3. For large orbit numbers, ν accordingly varies as ω, _i.e._, inversely proportional to the third power of n, and by equating R and 2K, we find that the values for ν and ω tend more and more to become equal.

In this way the value of K, the Balmer constant, may be computed. It is found that

_m_
K = 2π²_e_⁴-----
_h_³

where _e_ is the charge on the electron, _m_ the mass of the electron, and _h_ is Planck’s constant. Upon the substitution of the experimental values for these quantities, a value of K is determined which agrees with the experimental value (from the spectral lines investigation) of 3·29 × 10¹⁵ within the accuracy to which _e_, _m_ and _h_ are obtainable. This agreement has from the very first been a significant support for the Bohr theory.

One might now object that we have here considered radiation due to a transition between two successive stationary states, _e.g._, No. 100 and No. 99, or the like (a “single jump” we might call it). On the other hand, for transitions between states whose numbers differ by 2, 3, 4 or more (as in a double jump, or a triple jump) the agreement found above will wholly disappear, and doubt be cast on its value. For in such cases of high orbit numbers the frequency of revolution will remain approximately the same even for a difference of 2, 3, 4 or more in orbit number; but the radiation frequency for a double jump will be nearly twice that for a single jump, while that for a triple jump will be nearly three times, etc. Accordingly, for approximately the same revolutional frequency ω we shall have in these cases for the radiation frequency very nearly ν₁ = ω, ν₂ = 2ω, ν₃ = 3ω, etc. We must, however, remember that when the orbit in the stationary states is not a circle, but an ellipse (as must in general be assumed to be the case), the classical electrodynamics require that the electron emits besides the “fundamental” radiation of frequency ν₁ = ω, the overtones of frequencies ν₂ = 2ω, ν₃ = 3ω .... We then also here see the outward similarity between the Bohr theory and the classical electrodynamics. We may say that the radiation of frequency ν, produced by a single jump, _corresponds_ to the fundamental harmonic component in the motion of the electron, while the radiation of frequency ν₂, emitted by a double jump, corresponds to the first overtone, etc.

The similarity is, however, only of a formal nature, since the processes of radiation, according to the Bohr theory, are of quite different nature than would be expected from the laws of electrodynamics. In order to show how fundamental is the difference, even where the similarity seems greatest, let us assume that we have a mass of hydrogen with a very large number of atoms in orbits, corresponding to very high numbers, and that the revolutional frequency can practically be set equal to the same quantity ω. There may take place transitions between orbits with the difference 1, 2, 3 ... in number, and as the result of these different transitions we shall find, by spectrum analysis, in the emitted radiation frequencies which are practically ω, 2ω, 3ω, etc. According to the radiation theory of electrodynamics we should also get these frequencies and the spectral lines corresponding to them. It must, however, be assumed that they are produced by the simultaneous emission from every individual radiating atom of a fundamental and a series of overtones. According to the Bohr theory, on the other hand, each individual radiating atom at a given time emits only one definite line corresponding to a definite frequency (monochromatic radiation).

We can now realize that the Bohr theory takes us into unknown regions, that it points towards fundamental laws of nature about which we previously had no ideas. The fundamental postulates of electrodynamics, which for a long time seemed to be the fundamental laws of the physical world itself, by which there was hope of explaining the laws of mechanics and of light and of everything else, were disclosed by the Bohr theory as merely superficial and only applicable to large-scale phenomena. The apparently exact account of the activities of nature, obtained by the formulæ of electrodynamics, often veiled processes of a nature entirely different from those the formulæ were supposed to describe.

One might then express some surprise that the laws of electrodynamics could have been obtained at all and interpreted as the most fundamental of all laws. It must, however, be remembered that the Bohr theory for large wave-lengths, _i.e._, the slow oscillations, leads to a formal agreement with electrodynamics. It must, moreover, be remembered that the laws of electrodynamics are established on the basis of large-scale electric and magnetic processes which do not refer to the activities of separate atoms, but in which very great numbers of electrons are carried in a certain direction in the electric conductors or vibrate in oscillations which are extremely slow compared with light oscillations. Moreover, the observed laws, even if they can account for many phenomena in light, early showed their inability to explain the nature of the spectrum and many other problems connected with the detailed structure of matter. Indeed the more this structure was studied, the greater became the difficulties, the stronger the evidence that the solution cannot be obtained in the classical way.

If we ask whether Bohr has succeeded in setting up new fundamental laws, which can be quantitatively formulated, to replace the laws of electrodynamics and to be used in the derivation of everything that happens in the atom and so in all nature, this question must receive a negative answer. The motion of the electron in a given stationary state may, at any rate to a considerable extent, be calculated by the laws of mechanics. We do not know, however, why certain orbits are, in this way, preferred over others, nor why the electrons jump from outer to inner orbits, nor why they sometimes go from one stationary orbit to the next and sometimes jump over one or more orbits, nor why they cannot come any closer to the nucleus than the innermost orbit, nor why, in these transitions, they emit radiation of a frequency determined according to the rules mentioned.

It must not be forgotten that in science we must always be patient with the question “Why?” We can never get to the bottom of things. On account of the nature of the problem, answers cannot be given to the questions why the smallest material particles (for the time being hydrogen nuclei and electrons)—the elementary physical individuals—exist, or why the fundamental laws for their mutual relationships—the most elementary relationships existing between them—are of this or that nature; a satisfactory answer would necessarily refer to something even more elementary. We cannot claim more than a complete _description_ of the relative positions and motions of the fundamental particles and of the laws governing their mutual action and their interplay with the ether.

If we examine our knowledge of the atomic processes in the light of this ideal we are tempted, however, to consider it as boundless ignorance. We are inconceivably far from being able to give a description of the atomic mechanism, such as would enable us to follow, for example, an electron from place to place during its entire motion, or to consider the stationary states as links in the whole instead of isolated “gifts from above.” During the transition from one stationary state to another we have no knowledge at all of the existence of the electron, indeed we do not even know whether it exists at that time or whether it perhaps is dissolved in the ether to be re-formed in a new stationary state. But even if we turn aside from such a paradoxical consideration, it must be recognized that we do not know what path the electron follows between two stationary orbits nor how long a time the transition takes. As has been done in this book, the transition is often denoted as a jump, and many are inclined to believe that the electron in its entire journey from a distant outer orbit to the innermost spends the greatest part of the time in the stationary orbits, while each transition takes but an infinitesimally short time. This, however, in itself does not follow from the theory, nor is it implied in the expression “the stationary states.” These states may in a certain sense be considered as way stations; but when we ask whether an electron stays long in the station, or whether the stationary state is simply a transfer point where the electron changes its method of travelling so that the frequency of its radiation is changed, these are other matters, and we cannot here go into the considerations connected with them.