The Theory of Spectra and Atomic Constitution: Three Essays

by Niels Bohr

II. GENERAL PRINCIPLES OF THE QUANTUM THEORY SPECTRA

In order to explain the appearance of line spectra we are compelled to assume that the emission of radiation by an atomic system takes place in such a manner that it is not possible to follow the emission in detail by means of the usual conceptions. Indeed, these do not even offer us the means of calculating the frequency of the emitted radiation. We shall see, however, that it is possible to give a very simple explanation of the general empirical laws for the frequencies of the spectral lines, if for each emission of radiation by the atom we assume the fundamental law to hold, that during the entire period of the emission the radiation possesses one and the same frequency

, connected with the total energy emitted by the frequency relation

Here

and

represent the energy of the system before and after the emission.

If this law is assumed, the spectra do not give us information about the motion of the particles in the atom, as is supposed in the usual theory of radiation, but only a knowledge of the energy changes in the various processes which can occur in the atom. From this point of view the spectra show the existence of certain, definite energy values corresponding to certain distinctive states of the atoms. These states will be called the stationary states of the atoms, since we shall assume that the atom can remain a finite time in each state, and can leave this state only by a process of transition to another stationary state. Notwithstanding the fundamental departure from the ordinary mechanical and electrodynamical conceptions, we shall see, however, that it is possible to give a rational interpretation of the evidence provided by the spectra on the basis of these ideas.

Although we must assume that the ordinary mechanics cannot be used to describe the transitions between the stationary states, nevertheless, it has been found possible to develop a consistent theory on the assumption that the motion in these states can be described by the use of the ordinary mechanics. Moreover, although the process of radiation cannot be described on the basis of the ordinary theory of electrodynamics, according to which the nature of the radiation emitted by an atom is directly related to the harmonic components occurring in the motion of the system, there is found, nevertheless, to exist a far-reaching correspondence between the various types of possible transitions between the stationary states on the one hand and the various harmonic components of the motion on the other hand. This correspondence is of such a nature, that the present theory of spectra is in a certain sense to be regarded as a rational generalization of the ordinary theory of radiation.

Hydrogen spectrum. In order that the principal points may stand out as clearly as possible I shall, before considering the more complicated types of series spectra, first consider the simplest spectrum, namely, the series spectrum of hydrogen. This spectrum consists of a number of lines whose frequencies are given with great exactness by Balmer's formula

where

is a constant, and

and

are whole numbers. If we put

and give to

the values

,

, etc., we get the well-known Balmer series of hydrogen. If we put

or

we obtain respectively the ultra-violet and infra-red series. We shall assume the hydrogen atom simply to consist of a positively charged nucleus with a single electron revolving about it. For the sake of simplicity we shall suppose the mass of the nucleus to be infinite in comparison with the mass of the electron, and further we shall disregard the small variations in the motion due to the change in mass of the electron with its velocity. With these simplifications the electron will describe a closed elliptical orbit with the nucleus at one of the foci. The frequency of revolution

and the major axis

of the orbit will be connected with the energy of the system by the following equations:

Here

is the charge of the electron and

its mass, while

is the work required to remove the electron to infinity.

The simplicity of these formulae suggests the possibility of using them in an attempt to explain the spectrum of hydrogen. This, however, is not possible so long as we use the classical theory of radiation. It would not even be possible to understand how hydrogen could emit a spectrum consisting of sharp lines; for since

varies with

, the frequency of the emitted radiation would vary continuously during the emission. We can avoid these difficulties if we use the ideas of the quantum theory. If for each line we form the product

by multiplying both sides of (5) by

, then, since the right-hand side of the resulting relation may be written as the difference of two simple expressions, we are led by comparison with formula (4) to the assumption that the separate lines of the spectrum will be emitted by transitions between two stationary states, forming members of an infinite series of states, in which the energy in the

th state apart from an arbitrary additive constant is determined by the expression

The negative sign has been chosen because the energy of the atom will be most simply characterized by the work

required to remove the electron completely from the atom. If we now substitute

for

in formula (6), we obtain the following expression for the frequency and the major axis in the

th stationary state:

A comparison between the motions determined by these equations and the distinctive states of a Planck resonator may be shown to offer a theoretical determination of the constant

. Instead of doing this I shall show how the value of

can be found by a simple comparison of the spectrum emitted with the motion in the stationary states, a comparison which at the same time will lead us to the principle of correspondence.

We have assumed that each hydrogen line is the result of a transition between two stationary states of the atom corresponding to different values of

. Equations (8) show that the frequency of revolution and the major axis of the orbit can be entirely different in the two states, since, as the energy decreases, the major axis of the orbit becomes smaller and the frequency of revolution increases. In general, therefore, it will be impossible to obtain a relation between the frequency of revolution of the electrons and the frequency of the radiation as in the ordinary theory of radiation. If, however, we consider the ratio of the frequencies of revolution in two stationary states corresponding to given values of

and

, we see that this ratio approaches unity as

and

gradually increase, if at the same time the difference

remains unchanged. By considering transitions corresponding to large values of

and

we may therefore hope to establish a certain connection with the ordinary theory. For the frequency of the radiation emitted by a transition, we get according to (5)

If now the numbers

and

are large in proportion to their difference, we see that by equations (8) this expression may be written approximately,

where

represents the frequency of revolution in the one or the other of the two stationary states. Since

is a whole number, we see that the first part of this expression, i.e.

, is the same as the frequency of one of the harmonic components into which the elliptical motion may be decomposed. This involves the well-known result that for a system of particles having a periodic motion of frequency

, the displacement

of the particles in a given direction in space may be represented as a function of the time by a trigonometric series of the form

where the summation is to be extended over all positive integral values of

.

We see, therefore, that the frequency of the radiation emitted by a transition between two stationary states, for which the numbers

and

are large in proportion to their difference, will coincide with the frequency of one of the components of the radiation, which according to the ordinary ideas of radiation would be expected from the motion of the atom in these states, provided the last factor on the right-hand side of equation (10) is equal to

. This condition, which is identical to the condition

is in fact fulfilled, if we give to

its value as found from measurements on the hydrogen spectrum, and if for

,

and

we use the values obtained directly from experiment. This agreement clearly gives us a connection between the spectrum and the atomic model of hydrogen, which is as close as could reasonably be expected considering the fundamental difference between the ideas of the quantum theory and of the ordinary theory of radiation.

The correspondence principle. Let us now consider somewhat more closely this relation between the spectra one would expect on the basis of the quantum theory, and on the ordinary theory of radiation. The frequencies of the spectral lines calculated according to both methods agree completely in the region where the stationary states deviate only little from one another. We must not forget, however, that the mechanism of emission in both cases is different. The different frequencies corresponding to the various harmonic components of the motion are emitted simultaneously according to the ordinary theory of radiation and with a relative intensity depending directly upon the ratio of the amplitudes of these oscillations. But according to the quantum theory the various spectral lines are emitted by entirely distinct processes, consisting of transitions from one stationary state to various adjacent states, so that the radiation corresponding to the

th "harmonic" will be emitted by a transition for which

. The relative intensity with which each particular line is emitted depends consequently upon the relative probability of the occurrence of the different transitions.

This correspondence between the frequencies determined by the two methods must have a deeper significance and we are led to anticipate that it will also apply to the intensities. This is equivalent to the statement that, when the quantum numbers are large, the relative probability of a particular transition is connected in a simple manner with the amplitude of the corresponding harmonic component in the motion.

This peculiar relation suggests a general law for the occurrence of transitions between stationary states. Thus we shall assume that even when the quantum numbers are small the possibility of transition between two stationary states is connected with the presence of a certain harmonic component in the motion of the system. If the numbers

and

are not large in proportion to their difference, the numerical value of the amplitudes of these components in the two stationary states may be entirely different. We must be prepared to find, therefore, that the exact connection between the probability of a transition and the amplitude of the corresponding harmonic component in the motion is in general complicated like the connection between the frequency of the radiation and that of the component. From this point of view, for example, the green line

of the hydrogen spectrum which corresponds to a transition from the fourth to the second stationary state may be considered in a certain sense to be an "octave" of the red line

, corresponding to a transition from the third to the second state, even though the frequency of the first line is by no means twice as great as that of the latter. In fact, the transition giving rise to

may be regarded as due to the presence of a harmonic oscillation in the motion of the atom, which is an octave higher than the oscillation giving rise to the emission of

.

Before considering other spectra, where numerous opportunities will be found to use this point of view, I shall briefly mention an interesting application to the Planck oscillator. If from (1) and (4) we calculate the frequency, which would correspond to a transition between two particular states of such an oscillator, we find

where

and

are the numbers characterizing the states. It was an essential assumption in Planck's theory that the frequency of the radiation emitted and absorbed by the oscillator is always equal to

. We see that this assumption is equivalent to the assertion that transitions occur only between two successive stationary states in sharp contrast to the hydrogen atom. According to our view, however, this was exactly what might have been expected, for we must assume that the essential difference between the oscillator and the hydrogen atom is that the motion of the oscillator is simple harmonic. We can see that it is possible to develop a formal theory of radiation, in which the spectrum of hydrogen and the simple spectrum of a Planck oscillator appear completely analogous. This theory can only be formulated by one and the same condition for a system as simple as the oscillator. In general this condition breaks up into two parts, one concerning the fixation of the stationary states, and the other relating to the frequency of the radiation emitted by a transition between these states.

General spectral laws. Although the series spectra of the elements of higher atomic number have a more complicated structure than the hydrogen spectrum, simple laws have been discovered showing a remarkable analogy to the Balmer formula. Rydberg and Ritz showed that the frequencies in the series spectra of many elements can be expressed by a formula of the type

where

and

are two whole numbers and

and

are two functions belonging to a series of functions characteristic of the element. These functions vary in a simple manner with

and in particular converge to zero for increasing values of

. The various series of lines are obtained from this formula by allowing the first term

to remain constant, while a series of consecutive whole numbers are substituted for

in the second term

. According to the Ritz combination principle the entire spectrum may then be obtained by forming every possible combination of two values among all the quantities

.

The fact that the frequency of each line of the spectrum may be written as the difference of two simple expressions depending upon whole numbers suggests at once that the terms on the right-hand side multiplied by

may be placed equal to the energy in the various stationary states of the atom. The existence in the spectra of the other elements of a number of separate functions of

compels us to assume the presence not of one but of a number of series of stationary states, the energy of the

th state of the

th series apart from an arbitrary additive constant being given by

This complicated character of the ensemble of stationary states of atoms of higher atomic number is exactly what was to be expected from the relation between the spectra calculated on the quantum theory, and the decomposition of the motions of the atoms into harmonic oscillations. From this point of view we may regard the simple character of the stationary states of the hydrogen atom as intimately connected with the simple periodic character of this atom. Where the neutral atom contains more than one electron, we find much more complicated motions with correspondingly complicated harmonic components. We must therefore expect a more complicated ensemble of stationary states, if we are still to have a corresponding relation between the motions in the atom and the spectrum. In the course of the lecture we shall trace this correspondence in detail, and we shall be led to a simple explanation of the apparent capriciousness in the occurrence of lines predicted by the combination principle.

The following figure gives a survey of the stationary states of the sodium atom deduced from the series terms.

Diagram of the series spectrum of sodium.

The stationary states are represented by black dots whose distance from the vertical line a—a is proportional to the numerical value of the energy in the states. The arrows in the figure indicate the transitions giving those lines of the sodium spectrum which appear under the usual conditions of excitation. The arrangement of the states in horizontal rows corresponds to the ordinary arrangement of the "spectral terms" in the spectroscopic tables. Thus, the states in the first row (

) correspond to the variable term in the "sharp series," the lines of which are emitted by transitions from these states to the first state in the second row. The states in the second row (

) correspond to the variable term in the "principal series" which is emitted by transitions from these states to the first state in the

row. The

states correspond to the variable term in the "diffuse series," which like the sharp series is emitted by transitions to the first state in the

row, and finally the

states correspond to the variable term in the "Bergmann" series (fundamental series), in which transitions take place to the first state in the

row. The manner in which the various rows are arranged with reference to one another will be used to illustrate the more detailed theory which will be discussed later. The apparent capriciousness of the combination principle, which I mentioned, consists in the fact that under the usual conditions of excitation not all the lines belonging to possible combinations of the terms of the sodium spectrum appear, but only those indicated in the figure by arrows.

The general question of the fixation of the stationary states of an atom containing several electrons presents difficulties of a profound character which are perhaps still far from completely solved. It is possible, however, to obtain an immediate insight into the stationary states involved in the emission of the series spectra by considering the empirical laws which have been discovered about the spectral terms. According to the well-known law discovered by Rydberg for the spectra of elements emitted under the usual conditions of excitation the functions

appearing in formula (14) can be written in the form

where

represents a function which converges to unity for large values of

.

is the same constant which appears in formula (5) for the spectrum of hydrogen. This result must evidently be explained by supposing the atom to be electrically neutral in these states and one electron to be moving round the nucleus in an orbit the dimensions of which are very large in proportion to the distance of the other electrons from the nucleus. We see, indeed, that in this case the electric force acting on the outer electron will to a first approximation be the same as that acting upon the electron in the hydrogen atom, and the approximation will be the better the larger the orbit.

On account of the limited time I shall not discuss how this explanation of the universal appearance of Rydberg's constant in the arc spectra is convincingly supported by the investigation of the "spark spectra." These are emitted by the elements under the influence of very strong electrical discharges, and come from ionized not neutral atoms. It is important, however, that I should indicate briefly how the fundamental ideas of the theory and the assumption that in the states corresponding to the spectra one electron moves in an orbit around the others, are both supported by investigations on selective absorption and the excitation of spectral lines by bombardment by electrons.

Absorption and excitation of radiation. Just as we have assumed that each emission of radiation is due to a transition from a stationary state of higher to one of lower energy, so also we must assume absorption of radiation by the atom to be due to a transition in the opposite direction. For an element to absorb light corresponding to a given line in its series spectrum, it is therefore necessary for the atom of this element to be in that one of the two states connected with the line possessing the smaller energy value. If we now consider an element whose atoms in the gaseous state do not combine into molecules, it will be necessary to assume that under ordinary conditions nearly all the atoms exist in that stationary state in which the value of the energy is a minimum. This state I shall call the normal state. We must therefore expect that the absorption spectrum of a monatomic gas will contain only those lines of the series spectrum, whose emission corresponds to transitions to the normal state. This expectation is completely confirmed by the spectra of the alkali metals. The absorption spectrum of sodium vapour, for example, exhibits lines corresponding only to the principal series, which as mentioned in the description of the figure corresponds with transitions to the state of minimum energy. Further confirmation of this view of the process of absorption is given by experiments on resonance radiation. Wood first showed that sodium vapour subjected to light corresponding to the first line of the principal series—the familiar yellow line—acquires the ability of again emitting a radiation consisting only of the light of this line. We can explain this by supposing the sodium atom to have been transferred from the normal state to the first state in the second row. The fact that the resonance radiation does not exhibit the same degree of polarization as the incident light is in perfect agreement with our assumption that the radiation from the excited vapour is not a resonance phenomenon in the sense of the ordinary theory of radiation, but on the contrary depends on a process which is not directly connected with the incident radiation.

The phenomenon of the resonance radiation of the yellow sodium line is, however, not quite so simple as I have indicated, since, as you know, this line is really a doublet. This means that the variable terms of the principal series are not simple but are represented by two values slightly different from one another. According to our picture of the origin of the sodium spectrum this means that the

states in the second row in the figure—as opposed to the

states in the first row—are not simple, but that for each place in this row there are two stationary states. The energy values differ so little from one another that it is impossible to represent them in the figure as separate dots. The emission (and absorption) of the two components of the yellow line are, therefore, connected with two different processes. This was beautifully shown by some later researches of Wood and Dunoyer. They found that if sodium vapour is subjected to radiation from only one of the two components of the yellow line, the resonance radiation, at least at low pressures, consists only of this component. These experiments were later continued by Strutt, and were extended to the case where the exciting line corresponded to the second line in the principal series. Strutt found that the resonance radiation consisted apparently only to a small extent of light of the same frequency as the incident light, while the greater part consisted of the familiar yellow line. This result must appear very astonishing on the ordinary ideas of resonance, since, as Strutt pointed out, no rational connection exists between the frequencies of the first and second lines of the principal series. It is however easily explained from our point of view. From the figure it can be seen that when an atom has been transferred into the second state in the second row, in addition to the direct return to the normal state, there are still two other transitions which may give rise to radiation, namely the transitions to the second state in the first row and to the first state in the third row. The experiments seem to indicate that the second of these three transitions is most probable, and I shall show later that there is some theoretical justification for this conclusion. By this transition, which results in the emission of an infra-red line which could not be observed with the experimental arrangement, the atom is taken to the second state of the first row, and from this state only one transition is possible, which again gives an infra-red line. This transition takes the atom to the first state in the second row, and the subsequent transition to the normal state then gives rise to the yellow line. Strutt discovered another equally surprising result, that this yellow resonance radiation seemed to consist of both components of the first line of the principal series, even when the incident light consisted of only one component of the second line of the principal series. This is in beautiful agreement with our picture of the phenomenon. We must remember that the states in the first row are simple, so when the atom has arrived in one of these it has lost every possibility of later giving any indication from which of the two states in the second row it originally came.

Sodium vapour, in addition to the absorption corresponding to the lines of the principal series, exhibits a selective absorption in a continuous spectral region beginning at the limit of this series and extending into the ultra-violet. This confirms in a striking manner our assumption that the absorption of the lines of the principal series of sodium results in final states of the atom in which one of the electrons revolves in larger and larger orbits. For we must assume that this continuous absorption corresponds to transitions from the normal state to states in which the electron is in a position to remove itself infinitely far from the nucleus. This phenomenon exhibits a complete analogy with the photoelectric effect from an illuminated metal plate in which, by using light of a suitable frequency, electrons of any velocity can be obtained. The frequency, however, must always lie above a certain limit connected according to Einstein's theory in a simple manner with the energy necessary to bring an electron out of the metal.

This view of the origin of the emission and absorption spectra has been confirmed in a very interesting manner by experiments on the excitation of spectral lines and production of ionization by electron bombardment. The chief advance in this field is due to the well-known experiments of Franck and Hertz. These investigators obtained their first important results from their experiments on mercury vapour, whose properties particularly facilitate such experiments. On account of the great importance of the results, these experiments have been extended to most gases and metals that can be obtained in a gaseous state. With the aid of the figure I shall briefly illustrate the results for the case of sodium vapour. It was found that the electrons upon colliding with the atoms were thrown back with undiminished velocity when their energy was less than that required to transfer the atom from the normal state to the next succeeding stationary state of higher energy value. In the case of sodium vapour this means from the first state in the first row to the first state in the second row. As soon, however, as the energy of the electron reaches this critical value, a new type of collision takes place, in which the electron loses all its kinetic energy, while at the same time the vapour is excited and emits a radiation corresponding to the yellow line. This is what would be expected, if by the collision the atom was transferred from the normal state to the first one in the second row. For some time it was uncertain to what extent this explanation was correct, since in the experiments on mercury vapour it was found that, together with the occurrence of non-elastic impacts, ions were always formed in the vapour. From our figure, however, we would expect ions to be produced only when the kinetic energy of the electrons is sufficiently great to bring the atom out of the normal state to the common limit of the states. Later experiments, especially by Davis and Goucher, have settled this point. It has been shown that ions can only be directly produced by collisions when the kinetic energy of the electrons corresponds to the limit of the series, and that the ionization found at first was an indirect effect arising from the photoelectric effect produced at the metal walls of the apparatus by the radiation arising from the return of the mercury atoms to the normal state. These experiments provide a direct and independent proof of the reality of the distinctive stationary states, whose existence we were led to infer from the series spectra. At the same time we get a striking impression of the insufficiency of the ordinary electrodynamical and mechanical conceptions for the description of atomic processes, not only as regards the emission of radiation but also in such phenomena as the collision of free electrons with atoms.