The atom and the Bohr theory of its structure

Introduction.

We have dwelt at length upon the theory of the hydrogen spectrum because it was particularly in this relatively simple spectrum that the Bohr theory first showed its fertility. Moreover, by studying the case of the hydrogen atom with its one electron, it is easier to gain insight into the fundamental ideas of the Bohr theory and its revolutionary character. Naturally, the theory is limited neither to the hydrogen atom nor to spectral phenomena, but has a much more general application. As has already been said, it takes, as its problem, the explanation of every one of the physical and chemical properties of all the elements, with the exception of those properties known to be nuclear (cf. p. 94). This very comprehensive problem can naturally, even in its main outlines, be solved but gradually and by the co-operation of many scientists, and it is quite impossible to go very deeply into the great work which has already been accomplished, and into the difficulties which Bohr and the others working on the problem have overcome. We must be content with showing some especially significant features.

Different Emission Spectra.

While the neutral hydrogen atom consists simply of a positive nucleus and one electron revolving about the nucleus, the other elements, in the neutral state, have from two up to 92 electrons in the system of electrons revolving around the nucleus. Even 2 electrons, as in the helium atom, make the situation far more complicated, since we have in this case a system of 3 bodies which mutually attract or repel each other. We are thus confronted with what, in astronomy, is known as the three-body problem, a problem considered with respect by all mathematicians on account of its difficulties. In astronomy, the difficulties are restricted very much when the mass of one body is many times greater than that of the others, as in the case of the mass of the sun in relation to that of the other planets. Here, by comparatively simple methods, it is possible to calculate the motions inside a finite time-interval with a high degree of approximation even when there are not two but many planets involved.

We might now be tempted to believe that in the atom we had to deal with comparatively simple systems—solar systems on small scale—since the mass of the nucelus is many times greater than that of the electrons. But even if the suggested comparison illustrates the position of the nucleus as the central body which holds the electrons together by its power of attraction, the comparison in other respects is misleading. While the orbits of the planets in the solar system may be at any distance whatsoever from the sun, and the motions of the planets are everywhere governed by the laws of mechanics, the atomic processes, according to the Bohr theory, are characterized by certain stationary states, and only in these can the laws of mechanics possibly be applied. But in addition, the forces between nucleus and electrons are determined not at all by the masses, but rather by the electric charges. In the helium atom the nuclear charge is only double that of an electron, and the attraction of the nucleus for an electron will therefore be only twice as large as the repulsions between two electrons at the same distance apart. This repulsion under these circumstances will, therefore, also have great influence on the ensuing motion. In elements with higher atomic numbers the nuclear charge has greater predominance over the electron charges; but, on the other hand, there are then more electrons. The situation is in each case more complicated than in the hydrogen atom.

Nevertheless, the line spectra of the elements of higher atomic number show how the lines, as in the hydrogen spectrum, are arranged in series although in a more complicated manner (cf. p. 59); in any case in many instances there is great similarity between the radiation from the hydrogen atom and that from the more complicated atoms. Thus in the line spectra of many elements, just as in that of hydrogen, the frequency ν of every line can be expressed as a difference between two terms, involving certain integers which can pass through a series of values. From the combinations of terms, two at a time, the values of ν corresponding to the different spectral lines can be derived. This so-called combination principle enunciated by the Swiss physicist, Ritz, can evidently be directly interpreted on the basis of Bohr’s postulates, since the different combinations may be assumed to correspond to definite atomic processes, in which there is a transition between two stationary states, each of which corresponds to a spectral term.

Moreover, the terms (cf. p. 59) may often be approximately given by the Rydberg formula

where K has about the same value as in hydrogen, and α can take on a series of values α₁, α₂ ... αₖ, while n takes on integer values. Since we thus determine the different lines by assigning values to the two integers n and k in each term, we have in this respect something like the fine structure in the hydrogen spectrum, where the stationary states are determined by a principal quantum number and an auxiliary quantum number. The spectra of which we are speaking here, and for which the terms have the form given above, are often called arc spectra, because they are emitted particularly in the light from the electric arc or from the vacuum tube. We must expect that the similarity which exists in the law for the distribution of spectral lines will correspond to a similarity in the atomic processes of hydrogen and the other elements.

The hydrogen atom emits radiation corresponding to the different spectral lines when an electron from an outer stationary orbit jumps, with a spring of varying size, to an orbit with lower number, and at last finds rest in the innermost orbit in a normal state, where the energy of the atom is as small as possible. Similarly, we must assume that the electrons in other atoms, during processes of radiation, may proceed in towards the nucleus until they are collected as tightly as possible about the nucleus, corresponding to the normal state of the atom, where its energy content is as small as possible: “capture” of electrons by the nucleus. The region in space which, in the normal state, includes the entire electron system, must be assumed to be of the same order of magnitude as the dimensions of the atom and molecule which are derived from the kinetic theory of gases. This normal state may be called a “quiescent” state, since the atom cannot emit radiation until it has been excited by the introduction of energy from without. This excitation process consists of freeing one (or more) electrons, in some way or other, from the normal state and either removing it out to a stationary orbit farther away from the nucleus or ejecting it completely from the atom. Not all electrons can be equally easily removed from the quiescent state. Those moving in small orbits near the nucleus will be tighter bound than those moving in larger orbits farther from the nucleus. The arc spectrum is now caused by driving one of the most loosely bound electrons out into an orbit farther from the nucleus or removing it completely from the atom. In the latter case the rest of the atom, which with the loss of the negative electron becomes a positive ion, easily binds another electron, which, with the emission of radiation, corresponding to lines of the series spectrum, can approach closer to the nucleus.

Let us now assume, first, that this radiating electron moves at so great a distance from the nucleus and the other electrons that the entire inner system can be considered as concentrated in one point; then the situation is quite as if we had to deal with a hydrogen atom. If the atomic number is as high as 29 (copper), for instance, the nuclear charge will consist of twenty-nine elementary quanta of positive electricity; but since there are twenty-eight electrons in the inner system, the resultant effect is that of only one elementary quantum of positive electricity, as in the case of a hydrogen nucleus. The spectral lines which are emitted in the jumps between the more distant paths will be practically the same as hydrogen lines. But, since in the jumps between these distant orbits, very small energy quanta will be emitted, the frequencies are very small, the wave-lengths very great, i.e., the lines in question lie far out in the infra-red.

When the electron has come in so close to the nucleus that the distances in the inner system cannot be assumed to be small in comparison to the distance of the outer electron from the nucleus, the situation is changed. The force with which the nucleus and the inner electrons together will work upon the outer electron will be appreciably different from the inverse square law of attraction of a point charge. The consequence of this difference is that the major axis in the ellipse of the electron rotates slowly in the plane of the orbit as described in case of the theory of the fine structure of the hydrogen lines (cf. p. 146), and even if the cause is different the result is the same; the orbit of the outer electron in the stationary states will be characterized by a quantum number n and an auxiliary quantum number k. If the electron comes still closer to the nucleus, its motion is even more complicated. When the electron in its revolution is nearest the nucleus it will be able to dive into the region of the inner electrons, and we can get motions like those shown in Fig. 29 for one of the eleven sodium electrons. The inner dotted circle is the boundary of the inner system which is given by the nucleus and the ten electrons remaining in the “quiescent” state—little disturbed by the restless No. 11. In the figure we can see greater or smaller parts of No. 11’s different stationary orbits with principal quantum numbers 3 and 4. We shall not account further for the different orbits and the spectral lines produced by the transitions between orbits, but shall merely remark that the yellow sodium line, which corresponds to the Fraunhofer D-line (cf. p. 49), is produced by the transition 3₂-3₁, between two orbits with the same principal quantum number. The sketch shows to a certain degree how fully many details of the atomic processes can already be explained. The theory can even give a natural explanation of why the D-line is double.

We have restricted ourselves to the case where only one electron is removed from the normal state of the neutral atom. It may, however, happen that two electrons are ejected from the atom so that it becomes a positive ion with two charges. When an electron from the outside is approaching this doubly charged ion it will, at a distance, be acted upon as if the ion were a helium nucleus with two positive charges. The situation, in other words, will be as in the case of the false hydrogen spectrum (cf. p. 142), where the constant K in the formula for the hydrogen spectrum is replaced by another which is very close to 4K. But if the atom is not one of helium, but one with a higher atomic number, the stationary orbits of the outer electron which approach closely to the nucleus will not coincide exactly with those in the ionized helium atom, corresponding to the fact that the terms in the formula for the spectrum, instead of the simple form 4K/n², have the more complicated form 4K/(n + α)². Spectra of this nature are often called spark spectra, since they appear especially strong in electric sparks; they appear also in light from vacuum tubes, when an interruptor is placed in the circuit, making the discharge intermittent and more intense.

An atom with several electrons can, however, be much more violently excited from its quiescent state when an electron in the inner region of the atom is ejected by a swiftly moving electron (a cathode ray particle or a β-particle from radium) which travels through the atom. Such an invasion produces a serious disturbance in the stability of the electron system; a reconstruction follows, in which one of the outer, more loosely bound electrons takes the vacant position. In the transitions, in which these outer electrons come in, rather large energy quanta are emitted. The emitted radiation has therefore a very high frequency; monochromatic X-rays are thus emitted. Since these have their origin in processes far within the atom, it can be understood that the different elements have different characteristic X-ray spectra, which can give very valuable information about the structure of the electron system (cf. p. 91).

Between these X-ray spectra and the series spectra previously mentioned there lie, as connecting links, those spectra which are produced when electrons are ejected from a group in the atom which does not belong to the innermost group, but does not, on the other hand, belong in the outermost group in the normal atom. We have very little experimental knowledge about such spectra, because the spectral lines involved have wave-lengths lying between about 1·5 μμ and 100 μμ. Rays with these wave-lengths are absorbed very easily by all possible substances; they have very little effect on photographic plates, where they are absorbed by the gelatine coating before they have an opportunity to influence the molecules susceptible to light. But there can be scarcely any doubt that, in the course of a few years, experimental technique will have reached such efficiency that this domain of the spectrum, so important for the atomic theory, will also become accessible to experiment. In individual cases, wave-lengths as small as 20 μμ have already been obtained by Millikan.

Of entirely different character from these spectra are the band spectra. They are in general produced by electric discharges through gases which are not very highly attenuated (cf. p. 55); they are not due to purely atomic processes, but can be designated as molecular spectra. Their special character is due to motions in the molecule, not only motions of the electrons, but also oscillations and rotations of the nuclei about each other. We shall not go into these problems here; in what follows we shall investigate a certain type of band spectra somewhat more closely in connection with the absorption of radiation.

While the band spectra with a spectroscope of high resolving power can be more or less completely resolved into lines, this is not the case with the continuous spectra. They are emitted not only by glowing solids (cf. p. 54), but also by many gaseous substances. When such gases are exposed to electric discharges they emit, in addition to the line spectra and band spectra, continuous spectra which in certain parts of the spectrum furnish a background for bright lines which come out more strongly. It might seem impossible to correlate these with the Bohr theory; but in reality a spectrum does not always have to consist of sharp lines. This can at once be seen from the correspondence principle. If the motions in the stationary states are of such nature that they can be resolved into a number of discrete harmonic oscillations each with its own period (for instance the orbit of an electron in a rotating ellipse; cf. p. 149), then, according to the correspondence principle, in the transition between two such stationary states there are produced sharp spectral lines “corresponding” to these harmonic components. But not all motions of atomic systems can be thus resolved into a number of definite harmonic oscillations. When this cannot be done, the stationary states cannot be expected to be such that transitions between them produce radiation which can be resolved into sharp lines.

A simple example, where it is easily intelligible that the Bohr theory will not lead to sharp lines, is obtained in a simple consideration of the hydrogen atom. Let us examine the lines belonging to the Balmer series which are produced when an electron passes to the No. 2 orbit from an orbit with higher orbit number, which is farther from the nucleus. As has been said, we obtain here an upper limit for the frequency corresponding to a value of the outer orbit number which is infinite; this means, in reality, that the electron in one jump comes in from a distance so great that the attraction of the nucleus is infinitely small. The energy released by such a jump is the same as the ionizing energy A₂ which is required to eject the electron from the orbit No. 2 and drive it from the atom. It is here assumed, however, that the electron out in the distance was practically at rest. If the captured electron has a certain initial velocity outside, it will have a corresponding kinetic energy A. When in one jump this electron comes from the outside into orbit No. 2, the energy lost by the electron and emitted in the form of radiation will be the sum of the ionizing energy A₂ and the original kinetic energy A. The frequency ν will then become greater than that corresponding to A₂; and since the velocity of the electron before it is captured is not restricted to certain definite values, neither is the value of ν. The radiation from a great quantity of hydrogen atoms which are binding electrons in this way will, in the spectrum, not be concentrated in certain lines, but will be distributed over a region in the ultra-violet which lies outside of the limit calculated from the Balmer formula; still in a certain sense this continuous spectrum is correlated with the Balmer series. In the spectra from certain stars there has actually been discovered a continuous spectrum, which lies beyond the limits of the Balmer series and may be said to continue it.

Also the X-rays, which are generally used in medicine, have varying frequencies; this is caused by the fact that some of the electrons which, in an X-ray tube, strike the atoms of the anticathode and travel far into it at a high speed, lose a part or all of their velocity without ejecting inner electrons. The lost kinetic energy then appears directly as radiation. These remarks ought to be sufficient to show that the radiation, for instance, from a glowing body, where the interplay of atoms and molecules is very complicated, can give a continuous spectrum.

Electron Collisions.

The excitation of an atom in the normal state (cf. p. 157), by which one of its electrons is removed to an outer stationary orbit, may be caused by a foreign electron which strikes the atom. A study of collisions between atoms and free electrons is therefore of the greatest importance when investigating more closely the conditions by which series spectra are produced.

These investigations can be carried out by giving free electrons definite velocities by letting them pass through an electric field, where the “difference of potential” is known in the path traversed by the electrons. When an electron moves through a region with a difference of potential of one volt (the usual technical unit), the kinetic energy of the electron will be increased by a definite amount (of 1·6 × 10⁻¹² erg). If its initial velocity is zero, its passage through this field will make the velocity 600 km. per second; if the potential difference were 4 volts, 9 volts, etc., the velocity obtained by the electron would be 2, 3, etc., times larger. For the sake of brevity we shall say that the kinetic energy of an electron is, for instance, 15 volts, when we mean that the kinetic energy is as great as would be given by a difference of potential of 15 volts.

In 1913 the German physicist Franck began a series of experiments by methods which made it possible to regulate accurately the velocity of the electrons, and to determine the kinetic energy before and after collisions with atoms. He first applied the methods to mercury vapour, where the conditions are particularly simple, since the mercury molecules consist of only one atom. Franck bombarded mercury vapour with electrons all of which had the same velocity. He then showed that if the kinetic energy of the electrons was less than 4·9 volts the collisions with the atoms were completely “elastic,” i.e., the direction of the electron could be changed by the collision, but not its velocity. If, however, the velocity of the impinging electrons was increased so much that it was somewhat larger than 4·9 volts, there was an abrupt change in the situation, since many of the collisions became completely inelastic, i.e., the colliding electron lost its entire velocity and gave up its entire kinetic energy to the atom. If the initial velocity was even greater, so that the kinetic energy of the colliding electron was 6 volts, for instance, then when the collision took place there would always be lost a kinetic energy of 4·9 volts, since the electrons would either preserve their kinetic energy intact or have it reduced to 1·1 volt (cf. Fig. 30).

This remarkable phenomenon can be understood from the Bohr theory if we assume that to send the most loosely bound electron in the mercury atom out to the nearest outer stationary orbit there is required an energy of 4·9 volts, since in that case, according to the first postulate, an energy of less than this magnitude cannot be absorbed by the atom. The use of the word “understanding” must here be qualified; if the forces which influence the free electron as it comes into the electron system of the mercury atom are no other than the usual repulsion from the electrons and the attraction from the nucleus, the conduct of the colliding electron can in no way be explained by the laws of mechanics. But what happens is in agreement with the characteristic stability of the stationary states, and Bohr had prophesied how it would happen. Curiously enough Franck believed in the beginning that his experiment disagreed with the Bohr theory because he made the mistake of supposing that what happened was merely ionization, i.e., complete disruption of a bound electron from a mercury atom.

Franck’s experiments showed, moreover, that mercury vapour, as soon as the inelastic collisions appeared, began to emit ultra-violet light of a definite wave-length, namely, 253·7 μμ. The product of the frequency ν of this light and Planck’s constant h agrees exactly with the energy quantum possessed by an electron which has passed a potential difference of 4·9 volts; but this also agrees with what might be expected, according to the Bohr theory, from the radiation the removed electron would emit upon returning to the normal state. The energy which is respectively absorbed and emitted in the two transitions must be indeed hν.

Since an electron can not only be driven out to the next stationary orbit, but also to an even more distant one (or entirely ejected) and thence can come in again in one or more jumps, it is evident that a far more complicated situation may arise. The Franck experiment, which now has been extended to many other elements, clearly gives extraordinarily valuable information in such cases. In mercury it has been found that the energy a free electron must have in order to eject an electron from an atom and turn the atom into a positive ion, corresponds to a difference of potential of 10·8 volts, a value which Bohr had predicted. At the same time that Franck’s experiments, in this respect and in others, have strengthened the Bohr theory in the most satisfactory way, they have also advanced its development very much. Indeed it may be said that they have been of the greatest help in atomic research. Even if the spectroscope has greater importance, the investigations on electron collisions make the realities in the Bohr theory accessible to study in a more direct and palpable manner.

The peculiarities in the electron collisions appear most clearly in an old and well-known phenomenon of light, namely, the stratification of the light in a vacuum tube (Fig. 31). This stratification, which previously seemed so incomprehensible, agrees exactly with the feature so fundamental in the atomic theory that a free electron cannot give energy under a certain quantum to an atom. We can imagine that, in the non-illuminated central space between the bright strata, the electrons each time under the influence of the outer electric field obtain the amount of kinetic energy which must serve to excite the atoms of the attenuated vapour.

As has been said (p. 161), electron collisions may cause the emission of characteristic X-rays; but to produce them very great energy is required. Therefore the electrons which are to produce this effect must have an opportunity to pass freely through a certain region under the influence of a proportionately strong electric field (with potential of from 1000 to 100,000 volts and more). The electrons find such a field in a highly exhausted X-ray tube, where the electrons under strong potential are driven from the cathode against the anticathode, into which they penetrate deeply.

Absorption.

In the experiments previously described it was the electron collisions which furnished the energy required to excite the atoms, i.e., to carry them from the normal state over into a stationary state with greater energy. This “excitation energy” may, however, also be furnished to the atoms in the form of radiation energy; we shall now examine this case more closely.

Let us assume that to transfer an atom from the normal state to another stationary state, or, in other words, to transfer one of the electrons to an outer stationary orbit, a certain quantity of energy E is demanded; then the radiation emitted by the atom when it returns to the normal state will have a frequency ν depending upon the relation E=hν or ν=E/h, where h, as usual, is the Planck constant. But just as the atom in the transition from the stationary state to the normal state can emit radiation only with the definite frequency ν, then the opposite transition can only be performed by absorption of radiation with the same frequency; when this happens the absorbed radiation energy has exactly the value E=hν.

This reciprocity, which may be considered as a direct consequence of the Bohr postulates, agrees with what has been said (cf. p. 50) about the correspondence between the lines in the line spectrum of an element and the dark absorption lines of that element—e.g., the Fraunhofer lines in the solar spectrum. Let us examine, as an example, the yellow sodium line, the D-line. Light with the corresponding frequency, 526 × 10¹² vibrations per second, is emitted by a sodium atom, when the loosest bound electron goes over from a stationary orbit with quantum numbers 3₂ to the orbit 3₁, which belongs to the normal state of the sodium atom. The transition in the opposite direction, 3₁ to 3₂, can take place under absorption of radiation only when in the light from some other source of light, which passes the sodium atoms, there are found rays with the frequency 526 × 10¹². Even if there is present radiation energy with some other frequency, the sodium atoms take no notice of this energy; they absorb only rays with the frequency stated, and every time an atom absorbs energy from a ray the energy taken is always an energy quantum of the magnitude hν, i.e. about 6·54 × 10⁻²⁷ × 526 × 10¹² = 3·44 × 10⁻¹² ergs (1 erg is the unit of energy used in the determination of h). When there are present a large number of sodium atoms (as, for instance, in the previously mentioned common salt flame), the transition 3₁ to 3₂ can take place in some atoms, the transition 3₂ to 3₁ in others; therefore, at the same time there can be absorption and radiation of the light in question. Whether absorption or radiation at any given time has the upper hand depends upon various conditions (temperature, etc.).

For the sake of simplicity we have here tacitly understood that there can be but one definite transition (from the normal state) corresponding to the assumption that the sodium spectrum had no other lines than the D-line. In reality this is not the case, and there can equally occur absorption of rays with larger frequencies belonging to other spectral lines in the sodium atom and corresponding to other possible transitions between stationary states in the sodium atom. If the temperature of the sodium vapour is sufficiently low, in which case almost all the atoms are in the normal state, it is evident that in the absorption only those lines will appear which correspond to transitions from the normal state, and which therefore form only a part of all the lines of the sodium spectrum. We thus obtain an explanation of the previously enigmatical circumstance that not all spectral lines which can appear in emission will be found in absorption. At the same time we get, in absorption experiments, valuable information about the structure of the atom beyond what the observations in the emission spectra are able to give.

Interesting phenomena may arise owing to the fact that the jumps between the stationary states of the atom sometimes, as we know, take place in single jumps, sometimes in double or multiple jumps, so that the intermediate stationary states are jumped over. There is then evidently a possibility that absorption can take place, for instance, with a double jump of an electron, which may later return to the original stationary orbit in two single jumps. The absorbed radiation energy will then appear in emission with two frequencies which are entirely different from the frequency of the absorbed rays (this latter in this case will be the sum of the other two). When an element is illuminated with a certain kind of rays, it can, in other words, emit in return rays of a different nature. Such changes of frequencies have also been observed in experiment; they contain, in principle, an explanation of the characteristic phenomenon called fluorescence.

We shall not go further into this problem, but dwell for a time on the characteristic phenomenon of absorption which is known as the photoelectric effect. In this phenomenon (cf. p. 116) a metal plate, by illumination with ultra-violet light, is made to send out electrons with velocities the maximum value of which is independent of the strength of the illumination, but depends only on the frequency of the rays. What happens is that some of the electrons in the metal which otherwise have, as their function, the conduction of the electric current, by absorbing radiation energy, free themselves from the metal and leave it with a certain velocity. The reason why the rays for most metals must be ultra-violet (i.e. have a high frequency and consequently correspond to a proportionately large energy quantum) depends upon the fact that the energy quantum absorbed by the electrons must be large enough to carry out the work of freeing the electrons. But as long as the frequency of the rays (and therefore their energy quantum) is no less than what is needed for the freeing process, it does not need to have certain fixed values. If the energy quantum hν which the rays can give off is greater than is required to free the electrons, the surplus becomes kinetic energy in the electrons, which thus acquire a velocity which is the greater the greater the frequency ν, and which coincides with the maximum velocity observed in the experiments. What happens here is evidently something which can be considered as the reverse of the process which leads to the production of the continuous hydrogen spectrum (described on p. 163). In the latter case, electrons with different velocities are bound by the hydrogen atoms, which thus emit rays with frequencies increasing with increasing velocity, while, vice versa, in the photoelectric effect rays with different frequencies free the electrons and give them velocities increasing with increasing frequencies.

It must be acknowledged that there is something very curious in this effect. If the electromagnetic waves, as has been assumed, are distributed evenly over the field of radiation, it is not easy to understand why they give energy to some atoms and not to others, and why the selected ones always—with a given frequency—acquire a definite energy quantum, independent of the intensity of the radiation. For small intensities of the incident radiation, the atom, in order to acquire the proper quantum, must absorb energy from a greater part of the field of radiation (or for a longer time) than for large intensities. When the atoms acquire energy in electron collisions, the situation is apparently easier to understand, since in this case the colliding electrons give their kinetic energy to definite atoms, namely, those which they strike.

Einstein, in 1905, when there was not yet any talk of the nuclear atom or the Bohr theory, enunciated his theory of light quanta, according to which the energy of radiation is not only emitted and absorbed by the atoms in certain quanta, with magnitudes determined by the frequency ν, but is also present in the field of radiation in such quanta. When an atom emits an energy quantum hν, this energy will not spread out in waves on all sides, but will travel in a definite direction—like a little lump of energy, we might say. These light quanta, as they are called, can, like the electrons, hit certain atoms.

But even if in this theory the difficulties mentioned are, apparently, overcome, far greater difficulties are introduced; indeed it may be said that the whole wave theory becomes shrouded in darkness. The very number ν which characterizes the different kinds of rays loses its significance as a frequency and the phenomena of interference—reflection, dispersion, diffraction, and so on—which are so fundamental in the wave theory of the propagation of light, and on which, for instance, the mechanism of the human eye is based, receive no explanation in the theory of light quanta.

For instance, in order to understand that grating spectra can be produced at all, we must think of a co-operation of the light from all the rulings (cf. Fig. 10, p. 47), and this co-operation cannot arise if all the slits at a given moment do not receive light emitted from the same atom. In a bundle of rays which comes in at right angles to a grating, we must, in order to explain the interference, assume that the state of oscillation at a given moment is the same in all slits, that, for instance, there are wave crests in all at the same time, if we borrow a picture from the representation of water waves. Only in this case there can behind the grating at certain fixed places—for which the difference in the wave-length of the distances from successive slits is a whole number of wave-lengths—steadily come wave crests from all the slits at one moment and wave troughs from all at another moment (the classical explanation of the “mechanism” of a grating). If we imagine, however, that some slits are hit by light quanta from one atom and others from a second atom, it is pure chance if there are wave crests simultaneously in all slits, because the different atoms in a source of light emit light at different times, depending purely on chance. An understanding of the observed effect of a grating on light seems then out of question.

The theory of light quanta may thus be compared with medicine which will cause the disease to vanish but kill the patient. When Einstein, who has made so many essential contributions in the field of the quantum theory, advocated these remarkable representations about the propagation of radiation energy he was naturally not blind to the great difficulties just indicated. His apprehension of the mysterious light in which the phenomena of interference appear on his theory is shown in the fact that in his considerations he introduces something which he calls a “ghost” field of radiation to help to account for the observed facts. But he has evidently wished to follow the paradoxical in the phenomena of radiation out to the end in the hope of making some advance in our knowledge.

This matter is introduced here because the Einstein light quanta have played an important part in discussions about the quantum theory, and some readers may have heard about them without being clear as to the real standing of the theory of light quanta. The fact must be emphasized that this theory in no way has sprung from the Bohr theory, to say nothing of its being a necessary consequence of it.

In the Bohr theory, absorption and radiation must be said to be completely reciprocal processes, i.e. processes of essentially the same nature, but proceeding in opposite directions. In itself it cannot be said to be more incomprehensible that an atom absorbs energy from a field of radiation in agreement with the Bohr postulates than that it emits energy into the field; but in both cases we naturally encounter the great difficulties mentioned in Chap. V.

We have hitherto restricted ourselves to the purely atomic processes. But just as in the emission of radiation we meet spectra which owe their characteristics to molecular processes (band spectra, cf. p. 162), we have also absorption spectra with characteristics depending essentially upon motions of the atomic nuclei in the molecules. A particularly interesting and instructive example of this nature is met with in the infra-red region of the spectrum in certain broad absorption lines or absorption bands, which are due to gases having molecules containing several atoms. In hydrogen chloride, for instance, there is found, in the region of the spectrum which corresponds to a wave-length of about 3·5 μ, such an absorption band, which by more accurate investigation has been shown to consist of a great number of absorption lines.

The explanation of this collection of lines must be sought in the motions which the hydrogen nucleus and the chlorine nucleus perform, as they in part vibrate with respect to each other and in part rotate about their common centre of gravity. Just as in the case of the motions of the electrons in the atom, there are also certain stationary states for the nuclear motions. When the molecule absorbs radiation energy it will go from one of these states to another, where the energy content is greater. This absorption of energy proceeds according to the quantum rule, i.e., the product of the Planck constant h and the frequency ν for the absorbed radiation must be equal to the difference in energy between the two stationary states; only those rays which have frequencies fulfilling this condition are absorbed.

In hydrogen chloride, at standard temperature, the molecules will be in different stationary states of rotation (cf. the remarks on p. 27), corresponding to different definite values of the rotation frequency, while the nuclei, on the other hand, must be assumed to be at rest with reference to each other, i.e., they preserve their mutual distance. In Fig. 32, H and Cl indicate the circles which the two nuclei will describe about the centre of gravity; here, however, it must be remarked that the hydrogen circle is drawn too small in comparison with that of chlorine. If heat rays with all possible wave-lengths around 3·5 μ are sent through the hydrogen chloride, that radiation energy will be absorbed which can in part set the nuclei in oscillation and in part change the state of rotation. Let us for a moment assume that only the former change could happen. Then a ray with wave-length 3·46 μ would be absorbed, this frequency corresponding to the energy in the stationary state of oscillation into which the molecule goes; this frequency is very nearly equal to the frequency with which the nuclei vibrate relatively to each other. In reality, at the same time that the nucleus is set in oscillation, there will always be a change in the state of rotation—consisting either in an increase or in a decrease in the velocity of rotation. The energy absorbed, and therefore the frequency for the radiation absorbed, is thereby changed a little, so that in the spectrum of the rays sent through we do not obtain an absorption line corresponding to 3·46 μ, but a line somewhat removed from that. Since there are, however, many stationary states of rotation to start from, and since in some molecules there is one transition, in others another, we get many absorption lines on each side of 3·46 μ.

Even before Bohr propounded his theory, at a time when the quantum theory did not yet have a clarified form, the Danish chemist, Niels Bjerrum, had predicted that the infra-red absorption lines ought to have such a structure. This structure must be interpreted in the above way which differs somewhat from Bjerrum’s ideas, but his prediction was essentially strengthened by investigations, and it was one of the most significant features in the development of the quantum theory prior to 1913. The first to detect the structure of the infra-red absorption bands was the Swedish physicist, Eva von Bahr. Her experiments were later extended in a most significant way by the work of Imes and other American investigators. They enable us to calculate exactly the distance between the two nuclei in the molecule.

It may be asked what becomes of the energy which the hydrogen chloride molecule thus absorbs, and whether it necessarily after a longer or shorter time must be re-emitted as radiation. The latter is not the case. In a collision between molecules or atoms, the energy which one molecule (or atom) has absorbed by radiation can undoubtedly be transferred to another molecule, the velocity of which is thereby increased. The theoretical necessity of the occurrence of such collisions was clearly shown for the first time in a very significant investigation by two of Bohr’s students, Klein and Rosseland. Without collisions of this nature the radiation energy absorbed could never be transformed into heat energy. Here we come to a very great and important field, which has a very close connection with the theory of the chemical processes and to a better explanation of which the more recent experiments of Franck and his co-workers have made important contributions.