METHODS FOR ANALYZING COMPLEX SPECTRA, IN PARTICULAR THE SPECTRUM OF IRON1

V. Grotrian.

The activity of the natural scientist is often compared with the task of solving the riddles of nature. We shall not discuss here the question of whether this comparison is applicable to all areas of scientific research, but it seems to us that it gives a correct idea of the task that confronts the spectroscopist engaged in seeking regularities in the structure of spectra. For the characteristic feature of riddles consists in the deliberate obscuring of the true relations between objects and phenomena, and indeed, in studying spectra, one involuntarily gets the impression that nature has consciously and intentionally tried carefully to conceal the regular relations existing between spectral lines. Specializing our analogy somewhat, we may compare the task of analyzing, or “untangling,” spectra with the well-known problem of the “knight’s move.” In puzzles of this type the separate syllables appear to be arranged in complete disorder. Similarly, in spectra adjacent lines (i.e., lines corresponding to almost identical wavelengths) often have nothing in common with one another. The very first task of the spectroscopist is to find the general rule by means of which it would be possible to discover connections between lines, i.e., to find a rule analogous to the rule of the knight’s move in the chess problem. This rule has long since been found and bears the name of the Rydberg–Ritz combination principle. In the interpretation of the modern theory of the atom, this rule corresponds to Bohr’s “frequency condition,” which states that the emission of each of the spectral lines is caused by the transition of an atom from one quantum state to another (the knight’s move), and that the frequency of the radiation \(\nu\) is proportional to the change in energy \(E_1 - E_2\) corresponding to this jump—in other words, that

\[ \nu = \frac{E_1 - E_2}{h}. \]

The second part of the problem consists in applying this rule to the search for regular relations among the lines of some definite spectrum. As is known, for many spectra this problem too has been successfully solved. This was possible because nature was so considerate that it included in its collection of spectral problems both difficult and simple riddles, so that a person, passing from the easy to the difficult, could gradually perfect himself in solving these riddles. One may say that at the present time both simple and not overly complicated spectra have been analyzed. The laws found in this way are called series laws of line spectra. The characteristic feature of these laws comes down to the fact that the energy levels of atoms, between which quantum jumps occur, can be arranged in separate series, which we shall call energy ladders. However, the distances between the steps of each such ladder are not equal to one another, as in an ordinary ladder, but decrease more and more as one goes upward, as is schematically shown in Fig. 1. When an atom jumps from different steps of some energy ladder, for example the ladder \(k = 2\), to one and the same step of some other ladder, for example to the first step of the ladder \(k = 1\), i.e. when the outer electron of an atom jumps from a series of quantum orbits to some other definite quantum orbit, a series of spectral lines is obtained, forming the so-called spectral series. The frequencies of these spectral lines are proportional to the lengths of the vertical segments between the steps \(^{1}\).

Fig. 1. Schematic representation of atomic energy levels with simple spectral lines. The lowest level 1 corresponds to the normal state of the atom. Levels with identical azimuthal quantum numbers \(k\) form series tending to the same limit. This limit, marked by a dashed line, corresponds to the state of separation of the electron, i.e. to the state of ionization.

\(^{1}\) To make the following easier to understand, let us recall that each quantum state of an atom is determined first of all by two quantum numbers \(n\) and \(k\). The principal quantum number \(n\) increases as one rises along each of the energy ladders, whereas the “additional” or azimuthal quantum number \(k\), equally

In all those spectra which have been successfully analyzed up to the very latest time, similar series have been found; series regularities are the principal characteristic of these spectra. It has been possible to distribute by series the spectral lines of the elements of the 1st, 2nd, and 3rd and partly the 4th group of the periodic system, as well as of the noble gases helium and neon. However, all these solved riddles belong either to easy problems or to problems of medium difficulty. An approximate measure of the complexity of a spectrum is the number of lines belonging to it. The spectra of the elements listed, although they do not belong to spectra with few lines, nevertheless, with the exception of the spectrum of neon, cannot also be classed among those spectra abundant in lines in which the lines, figuratively speaking, lie close up against one another. Such quite complex spectra belong predominantly to elements standing in the first halves of the long periods of the system of elements, i.e. to those elements in which, from the standpoint of atomic physics, according to Bohr, the filling of the inner groups of electrons is taking place; such are, for example, the elements from scandium to nickel. Quite recently we were completely helpless before the task of analyzing these spectra, but at the very latest time a key has been found to the solution of these complex spectra. The purpose of this article is to relate the successes achieved in this field.

Fig. 2. Schematic representation of the energy levels of an atom with doublet spectra. Each rung of the ladders with \(k \geq 2\) splits into two levels close to one another.

Whereas the characteristic feature of the spectra analyzed earlier was the presence of series, in complex spectra the serial regularities recede into the background, although this, of course, does not mean,

for all levels of each ladder. In Fig. 1, at each rung of a ladder there stands the Bohr symbol \(n_k\), characterizing the corresponding state of the atom. According to the selection principle, under normal conditions only such quantum jumps can take place as correspond to a change of the number \(k\) by \(\pm 1\). In other words, a jump occurs not between the levels of one and the same ladder, but always from a level of one ladder to a level of a neighboring ladder. Ed.

that in these spectra series are altogether absent. In complex spectra another characteristic feature comes to the fore, one encountered in a less general form also in the spectra analyzed earlier. It consists in the fact that spectral lines are either simple, or form doublets and triplets, i.e. groups consisting of two or three lines, the frequency difference of the lines entering such a group being repeated in other groups as well. Recently it has become clear that, besides doublets and triplets, quartets, quintets, sextets, etc. may also occur, i.e. more complex groups of lines, which have received the general name of multiplets. This generalization has proved extremely fruitful and, when applied to the analysis of complex spectra, has in a short time led in a whole series of cases to remarkable successes.

To describe the characteristic features of these multiple, or multiplet, spectra, we must begin with the simplest case of a doublet spectrum. The twofoldness of the lines of doublet spectra is explained by the fact that all the energy levels of an atom for which \(k \geq 2\) turn out to be split into two (see Fig. 2). Each step of the ladder shown in Fig. 1 (with the exception of that ladder for which \(k = 1\)) splits into two steps close to one another. The distances between two such neighboring steps decrease as \(n\) increases, i.e. as one ascends the ladder. To transitions of the atom, for example, from one of the two lower steps of the ladder \(k = 2\) to the first step of the ladder \(k = 1\), there correspond two lines whose frequency is proportional to the distance between the steps, i.e. proportional to the length of the vertical arrows drawn in the figure. Two such lines form a doublet, for example the double line \(D\) of sodium.

Fig. 3.
Fig. 4.
Diagram of the energy levels of Rydberg’s complete doublet. The thickness of the vertical lines is an approximate measure of the intensity of the corresponding spectral line. The spectral line corresponding to the dashed vertical line does not arise under normal excitation conditions.

A somewhat more complicated case is that of transitions of an atom from any two steps of the ladder \(k = 3\) to two steps of the ladder \(k = 2\). This case is shown separately in Fig. 3. It turns out here that, of the four possible transitions, only three are actually realized. These three lines form the so-called complete Rydberg doublet, although it would be better to call them an incomplete doublet. The absence of one of the four possible lines is of fundamental significance. It ...

makes us suppose that we are dealing with a prohibition imposed by a selection principle, analogous to the known prohibition of those quantum jumps in which \(k\) changes not by \(\pm 1\). And indeed, Sommerfeld showed that the absence, in the complete doublet, of the fourth line can be explained if one introduces (at first in a purely formal way) a third quantum number \(j\), which he called the inner quantum number1. For there is no doubt that the multiplicity of atomic states which occurs in the splitting of energy levels cannot be described with the aid of only two quantum numbers \(n\) and \(k\). To the different levels produced by the splitting of the levels of the energy ladders in Fig. 1 there are assigned different values of the inner quantum number \(j\), so that each level is uniquely determined by three quantum numbers \(n\), \(k\), and \(j\), or by the symbol \(n_{kj}\).

In Fig. 2, beside each level, the corresponding value of this symbol is placed. In the case of a complete doublet, the individual levels correspond to the values of the number \(j\) indicated in Fig. 3. The absence of the fourth line will become understandable if one assumes the validity of the selection rule: under normal conditions of excitation of spectra, only those quantum transitions occur in which the number \(j\) either remains unchanged or changes by \(\pm 1\), i.e.

\[ \begin{array}{c} j+1\\[-2mm] \nearrow\\[-1mm] j\to j\\[-1mm] \searrow\\[-2mm] j-1 \end{array} \]

And indeed, with the aid of inner quantum numbers it proved possible, in systematic order, to number the energy levels both in doublet and in triplet spectra in such a way that both the existence of actually observed lines and the absence of some of the possible lines can be predicted with astonishing accuracy on the basis of two selection rules (for \(k\) and for \(j\)). Fig. 4 relates to the case of triplets, in which the levels of all ladders for which \(k\ge 2\) split into three small levels close to one another; the figure shows transitions between the three levels of the ladder \(k=3\) and the three levels of the ladder \(k=2\). Of the 9 possible transitions only 6 occur (the forbidden transitions are shown by a dotted line), and precisely those which are to be expected according to the selection rule, guided by the values of the quantum number \(j\) given in the figure. These six lines form the so-called complete Rydberg triplet.

One may try to generalize these regularities, found for doublets and triplets, to the case of quartets, quintets, and multiplets in general. It is natural to suppose that, for example, in the case of quintets each step in Fig. 1 will split into five little steps. However, the laws of multiple spectra are not so simple; this is evident if only from the fact that the steps of the ladder \(k=1\) remain unsplit both in doublets and in triplets. But before turning to these questions, we shall briefly consider those methods by means of which it proved possible to find, in the chaos of lines of complex spectra lying close to one another, groups of lines belonging to one and the same multiplet, i.e. groups of lines analogous to the complete doublets and triplets shown in Figs. 3 and 4.

The first auxiliary means, which has rendered valuable service in finding related lines, is experimental in character. Over a number of years the American physicist A. King investigated the conditions of purely temperature excitation of the spectra of a large number of elements, chiefly metal vapors. The element under investigation was placed in an electric furnace, the essential arrangement of which is clear from Fig. 5. An alternating current of low voltage, with a strength of several hundred or thousand amperes, is supplied through two thick, water-cooled copper tubes \(Z_1\) and \(Z_2\), and through two graphite sleeves \(G\), to a graphite tube \(R\), which is heated by the currents to about \(3000^\circ\). Into the tube are introduced those metals whose vapors are to be investigated. The entire furnace is placed in an iron cylinder \(M\), in which either a vacuum or an increased pressure may be produced, depending on the exact conditions under which the emission of the spectra is to be investigated. The radiation of the gases and vapors located inside the graphite tube is observed in the longitudinal direction through two glass or quartz windows \(F_1\) and \(F_2\), set into the side walls of the iron cylinder. Fig. 6 gives a photograph of King’s furnace of the latest design, specially intended for observations in vacuum. The cover, shown raised, is lowered during operation of the furnace and hermetically fits against the lower plate.

Fig. 5. Diagram of King’s electric furnace. \(Z_1\) and \(Z_2\)—copper tubes for running water (cooling); \(G\)—graphite sleeves; \(R\)—graphite tube; \(M\)—water-cooled iron cylinder; \(F_1, F_2\)—glass or quartz windows.

A systematic study of the dependence of the intensities of the lines of some spectrum on temperature leads to the subdivision of these lines into temperature classes. To class I belong those lines,

which appear already at low temperatures; to classes II, III, and IV belong lines appearing at medium, high, and the highest temperatures. In Fig. 7 photographs are given of a small region of the spectrum of iron, where a is the arc spectrum, and b, c, and d are spectra taken in the furnace at 2600°, 2000°, and 1600° C. In these photographs it is clearly seen how, from the extremely numerous lines of the arc spectrum, as the temperature is lowered an ever greater and greater number of lines disappears. In the end, comparatively few lines remain, and not all of the remaining lines are distinguished by especially great intensity in the arc spectrum. The simultaneous disappearance of a number of lines indicates that they belong to one and the same group of multiplets. Indeed, under thermal excitation of spectra, part of the atoms of the element being studied passes into a state of greater energy—in other words, a certain fraction of the atoms is in states corresponding to the higher steps of our energy ladders; moreover, as the temperature is raised, ever higher steps are reached. In the reverse jump downward, the atoms emit spectral lines corresponding to this step. Since lines belonging to one and the same multiplet originate from steps very close to one another, these lines must first appear at one and the same temperature and will, consequently, be assigned to one and the same temperature class.

Fig. 6. Spectral furnace of King’s new design for observations in vacuum. The supply of current and water is carried out through the stands. The cover, during the experiment, is lowered and hermetically adjoins the lower plate. In the middle on the lower plate lies the graphite tube for the specimen; its length is 30 cm, bore 13 mm.

Thus, in the search for multiplets one should always seek regular connections between lines of one and the same temperature class. This indication, of course, is not yet a solution

problem, but nevertheless it substantially facilitates what is still only a beginning and still quite painstaking task. Regular relations are recognized by the presence, between various lines, of identical intervals (on the frequency scale). Multiplying these frequency differences by Planck’s constant \(h\), we obtain nothing other than the energy difference between neighboring steps of the energy ladder. To explain exactly how the belonging of lines to one and the same multiplet is recognized, and what the structure of multiplets is, it will be best to turn to a concrete example.

But first let us give a brief historical note on the course of development in the study of multiplets. The first multiplets were found by A. Catalan in the spectrum of manganese and, almost simultaneously with him but independently of him, by H. Gieseler in the spectrum of chromium (in 1922). The name “multiplet” was introduced by Catalan. Sommerfeld gave the Catalan multiplets a theoretical interpretation. Very important works on the general structure of multiple spectra and especially on the Zeeman effect in them belong to Landé and Back; we shall return to these works later. In the search for multiplets in complex spectra, besides those named, American physicists in particular also took part, especially the staff of the Bureau of Standards in Washington, who achieved a number of record results. Here one must name Meggers (W. F. Meggers), Walters (F. M. Walters), and the Kiesses (C. C. and H. K. Kiess). It was possible to analyze the spectrum of iron in particularly great detail. The spectrum

Fig. 7. Part of the spectra of iron from photographs by G. King. \(a\)—arc spectrum; \(b\), \(c\), and \(d\)—spectra obtained with King’s furnace at temperatures of 2600, 2000, and 1600° C. It is evident that, as the temperature is lowered, only a few characteristic lines remain from the arc spectrum, rich in lines.

iron is one of the most interesting of all spectra, owing to the abundance in it of bright, characteristic lines. Moreover, the lines of this spectrum have been measured most accurately. Many of the iron lines are so-called secondary standards, whose wavelengths have been measured by interference methods with an accuracy of many thousandths of an Angstrom. Until quite recently we were powerless before this chaos of lines; no regularities in the spectrum of iron possessing the features of reality were known at all, until, finally, F. M. Walters succeeded in finding 20 multiplets in this spectrum. Kayser and Konen, in the 7th volume of their Handbuch der Spektroskopie, noted this success in the following words: “Nevertheless Walters finally succeeded in lifting a corner of the curtain lying over the spectrum of iron.” And indeed, these 20 multiplets were the beginning and the basis of the further disentangling of the iron spectrum, which by the present time has advanced so far that one can speak no longer of a corner, but of a considerable part of the curtain, for about 600 iron lines have already been arranged into multiplets.

The quantum-theoretical interpretation of the Walters multiplets was given independently by Catalán, H. Gieseler, and the author of this article, and finally, in especially detailed form, by O. Laporte, who found 11 new multiplets and arrived at a number of interesting conclusions about the general spectrum of iron. Recently Walters and Laporte have again found a number of new multiplets. In all those cases in which the aforementioned investigators deal with the same questions, they arrive at completely identical conclusions. Both the distribution of lines into multiplets and the quantum-theoretical interpretation of them are almost entirely devoid of any arbitrariness, so that there can be no doubt that the regularities found are truly real in character. It is true that, with the great abundance of lines in complex spectra, one might fear that numerical coincidences are accidental. But precisely in the case of the iron spectrum the wavelengths are known with such extraordinary accuracy that the possibility of such an error is excluded. The numerical regular relations between the lines are fulfilled with an accuracy fully corresponding to the accuracy of the measurements. In view of this, and also in view of the fact that the iron spectrum is in general of special interest, we shall choose from the iron spectrum one of the multiplets and, using its example, explain the essence of the newly found spectral laws.

In Table I a numerical scheme of one of the iron multiplets is given, from which it is easy to discern the numerical relations between the frequencies of the individual lines. (See table, p. 195.)

Strictly speaking, the table gives not the frequencies of the lines, but the so-called “wave numbers,” i.e. the reciprocal values of the wavelengths.

METHODS OF ANALYSIS OF COMPLEX SPECTRA

TABLE I

wavelengths, measured in centimeters1. For each line there is given: in the first line its intensity (for example, 80 $R$), in the second—the wavelength in Angstrom units, and in the third—the wave number. The symbol $R$ means, as usual, that the line is to a considerable degree reversed in the arc spectrum; the symbol $r$ means that it is weakly reversed2. Between the wave numbers of neighboring lines there are given (printed in italics) the differences of these numbers. From the table it is clear that one and the same difference of wave numbers occurs twice, between two different pairs of lines (for example, the differences 415,95 and 415,97). These repeated values of the differences create a connection among the individual lines of the whole group. Only two lines, 3745,900 and 3719,938, hang, so to speak,

in air, for the differences 292.29 and 89.91 occur only once. However, the belonging of these lines to the group under consideration follows directly from the results of experiments in an electric furnace (all the lines of this group belong to King’s 1st temperature class), and also from the existence of certain other regular relations, to the consideration of which we shall soon pass.

What is the quantum-theoretical interpretation of the group of lines considered? It is easy to see that this group of lines arises by the combination of two fivefold energy levels and that therefore we are dealing with a system of quintets. Further, we note that in square Table I a number of places marked with primes remain.

TABLE II.

unshifted, i.e., the corresponding lines are absent. We shall try to explain the existence of the observed lines and the absence of the unobserved ones with the aid of the selection principle applied to the internal quantum numbers. This proves possible if, in the vertical and horizontal rows, one writes the values of the internal quantum number \(j\) that are given at the edges of Table I. Then each of the existing lines will correspond to one of the three possible combinations: \(j \to j+1\), \(j \to j\), \(j \to j-1\); for example, \(4 \to 5\), \(4 \to 4\), \(3 \to 3\).

In order to understand the meaning of the numerical scheme given in Table I, let us construct the corresponding scheme of energy levels (Fig. 8), analogous to the schemes shown in Figs. 3 and 4 for doublets and triplets. The repeated values of the differences of wave numbers in Table I will now correspond to differences between energy levels. Individual lines correspond to different combinations, allowed by the selection principle, of two fivefold groups of levels, the values of the internal quantum numbers characterizing these levels having already been fixed in Table I.

After we have in this way assigned to each level a definite value of the internal quantum number \(j\), the question arises as to what values of the azimuthal quantum number \(k\) correspond to each group of levels. This question can be resolved only in connection with the general question of the structure of multiplets. This problem was solved empirically by Landé. We have seen that the energy levels are determined by three quantum numbers \(n\), \(k\), and \(j\), or, in other words, by the symbol \(n_k^j\). However, this symbol proves insufficient, since it does not determine the multiplicity of the corresponding energy level; that is, the question remains open as to whether the given level belongs to a system of doublets or triplets or to some multiplets. We therefore need one more, fourth, number, which, following Landé, we shall denote by \(r\); for doublets \(r=2\), for triplets \(r=3\), and so on.\(^1\) Then each energy level will be uniquely determined by the symbol \(n_k^{jr}\). The question of the general structure of multiplets reduces to establishing how many different values, and precisely which values, of the number \(j\) correspond to given values of the numbers \(k\) and \(r\). It might seem, for example, that for quintets, i.e. for \(r=5\), each level should be fivefold and should accordingly have five different values of the number \(j\). This, however, is not so, as may be seen if only from the fact that for both doublets and triplets the steps of the energy ladder with \(k=1\) are always strictly single.

\(^1\) The number \(r\) is also a quantum number; in all probability, it determines the magnitude of the total angular momentum of the atomic core, i.e. of that internal part of it which, after the removal of the outer electron, constitutes the corresponding ion.

The true multiplicity of the various energy levels for all values of \(r\) and \(k\) is indicated in Table 2, which we have taken from Landé’s work. In this table are given the symbols \(\eta_k^j\) of all lines actually existing in the various multiplet systems. At the same time, all symbols of one and the same horizontal row correspond to one and the same value of the number \(k\), while all symbols of each vertical row correspond to an identical value of the number \(j\). It is evident from the table that, as \(k\) increases, the number of symbols in the horizontal rows also increases, i.e. the number of subdivisions of the energy levels, until the number of these levels becomes equal to the number \(r\). Thus, for example, in quintets the number of levels grows from 1 at \(k=1\) to 3 at \(k=2\), then to 5 at \(k=3\), and thereafter already remains constant and equal to 5 at \(k=4, 5, 6,\) etc. Thus the number \(r\) determines the permanent multiplicity of the levels, which is attained at a definite value of \(k\).

If we now ask ourselves what values of the azimuthal quantum number \(k\) must be assigned, in the example of the multiplet considered by us, to both fivefold groups of levels, then with the aid of Table II we shall convince ourselves that in the system of quintets the inner quantum numbers \(j=0, 1, 2, 3, 4\) correspond to \(k=3\), and the numbers \(j=1, 2, 3, 4, 5\) correspond to \(k=4\).

Thus we have arrived at an exhaustive interpretation of our multiplet. Unfortunately, it must be noted that this interpretation is not the only possible one. The point is that there exists a certain freedom in the choice of the absolute value of the inner quantum numbers corresponding to the various energy levels. If we increased all values of \(j\) by 1 or by 2, then, with the aid of the selection principle, we would obtain the same set of lines. Some indications of the proper choice of the absolute values of the number \(j\) may be extracted from Landé’s “interval rule,” which states that the frequency differences are approximately proportional to the corresponding inner quantum numbers. Thus, for example, in our example the ratios between the frequency differences are

\[ 415{,}9:288{,}1:184{,}1:89{,}9=4:2{,}7:1{,}8:0{,}9 \]

(whereas theoretically they should be \(4:3:2:1\));

and

\[ 292{,}3:227{,}9:164{,}9:106{,}7=5:3{,}9:2{,}8:1{,}8, \]

whereas theoretically these ratios should have been equal to

\[ 5:4:3:2. \]

Thus, the interval rule is approximate in character, but nevertheless from it one can extract important indications concerning the proper choice of the absolute values of the number \(j\). With the aid of this rule one can establish with a certain confidence whether the values were correctly chosen

included in the multiplet of lines considered by us as an example, \(\lambda = 3745.900\) and \(\lambda = 3719.938\), connected with the remaining lines by only one single frequency difference.

The question of whether a given line belongs to this or that multiplet can be decided with complete definiteness and quite unambiguously by studying the magnetic splitting of this line. On the basis of very precise measurements of the anomalous Zeeman effect, made by H. Gieseler in the spectrum of chromium and by E. Back in the spectrum of manganese, Landé succeeded, empirically, in finding the laws of the anomalous Zeeman effect. An exposition of these very complex and remarkable rules would lead us too far afield. The principal result, besides its purely theoretical interest and also of primary importance for the analysis of spectra, consists in the fact that, if the values of the symbol \(n_k r\) are known for the initial and final states of the atom, the transition between which corresponds to the given spectral line, then the character of the magnetic splitting of this line can be predicted theoretically. Thus, if by the method set forth above we have arrived at a definite interpretation of a series of lines, the correctness of our interpretation can be checked by comparing the Zeeman splitting of these lines, theoretically calculated by us, with that actually observed. In principle this method is ideal and perfectly exact; the difficulty consists only in the fact that it is necessary, first, to observe the Zeeman splitting and, second, to measure it with irreproachable precision. It is known that the precise measurement of Zeeman splitting is a very difficult problem and can be carried out only with the aid of the best spectral instruments (large concave diffraction gratings). The old measurements are for the most part imperfect and often unusable. Really good measurements of the splitting were made at the Tübingen Physical Institute, chiefly by F. Paschen and E. Back. Recently, at the Mount Wilson Observatory, Babcock has begun extensive measurements of the Zeeman effect; however, these measurements have only partly been published, so that it is impossible to judge how good they are. In any case, the further accumulation of experimental data on

Fig. 8. Diagram of the energy levels of a quintet from the spectrum of iron. Between the individual levels of both groups the values of the distance between levels in \(\mathrm{cm}^{-1}\) are given; the lengths, in the form of spectral lines, arising in the corresponding jumps are marked along the vertical lines.

in the Zeeman effect, since it is a prerequisite for the successful analysis of complex spectra.

And yet, even if the Zeeman splittings of the lines under investigation have been measured, it is still not always possible to decide with complete certainty the question of the origin of these lines. The fact is that, according to Landé’s theory, the splitting of many lines is of an extremely complicated character (20 or more components, often lying very close to one another). In such cases, even with the aid of the best experimental equipment presently available, it is impossible to achieve separation of these components, and one has to confine oneself to comparing the general picture of the observed Zeeman splitting with the theoretical one, assuming in doing so that some components merge into one.

Although we refrain from a detailed exposition of the laws of the Zeeman effect, nevertheless, using as an example two lines of the multiplet considered by us, we shall show with what striking accuracy theory agrees with experiment. Let us note that simple (single) lines ($r = 1$) give the normal Zeeman effect, i.e. in transverse observation they give the normal Zeeman triplet. The distance of the outer components, polarized perpendicular to the direction of the magnetic field, from the middle component, polarized parallel to the direction of the field, is called the normal splitting. An anomalous splitting is characterized by the fact that, if the magnitude of the normal splitting is chosen as unity, then the distances of the separate components from the position occupied by the unsplit line will be expressed by rational fractions.

These fractions can be calculated theoretically on the basis of Landé’s formulae. In the following table are given the results of the measurement of two lines of our multiplet, made by Babcock:

Thus, theory agrees excellently with experiment. The same is true for many other lines as well.

With the aid of the methods described above, as has already been indicated, it has been possible to find in the spectrum of iron a large number of multiplets and to give them a theoretical interpretation, i.e. to characterize them by defi–

...separate values of the quantum numbers. In this connection it turned out that those frequency differences by means of which, in the multiplet considered by us, the distances between the individual energy levels are expressed, are also repeated in other multiplets. Especially often repeated are the differences 415.9, 288.1, 184.1, and 89.9 (in the 16 multiplets known at the present time). From the point of view of the scheme of energy levels this means that all these multiplets correspond to the transition of the atom from various new groups of levels to a group of levels already known to us from the study of our multiplet, the distances between the individual steps of which are equal to these differences. Thus one can determine the distances of these new groups of levels from the former ones.

Fig. 9. Graphical representation of the presently known energy steps in the iron atom.

In this way there is created a system of groups of levels which, as the number of known multiplets increases, becomes more and more complex. The energy levels known at the present time in the spectrum of iron are shown schematically in Fig. 9. In order not to make this scheme excessively confusing, each group of levels is represented on it by only one horizontal line. Thus, for example, the lowest step of this diagram (designated by \(3d\)) is identical with the lower group of levels of the multiplet considered by us. Correspondingly, all the numerous lines of each individual multiplet are represented by a single vertical arrow connecting the two corresponding horizontal steps. Beside each such arrow is given the wavelength of the brightest line of the corresponding mul-

triplet, for example, in our multiplet 3720 (rounded, instead of 3719.938).

All the levels fall, first of all, into three main groups, corresponding to the three multiplets found in the spectrum of iron. In addition to quintets, whose lines predominate both in number and in brightness, a number of triplets was also found, and recently Laporte found several more septets. Moreover, combinations occur between quartet and triplet levels, and also between quartet and septet levels. In our diagram the individual levels are arranged so that, within each system of multiplets, the levels corresponding to the same value of the number \(k\) stand one above another. To the left of each level stands its spectroscopic symbol according to Laporte (the reader who is not familiar with these designations should not for the time being pay attention to them); on the right is indicated the greatest frequency difference occurring in the group of levels denoted by one horizontal line—415.9 in our group \(3d\).

With the aid of Fig. 9 one can understand certain characteristic features of the spectrum of iron, which occur mutatis mutandis also in other complex spectra. We have pointed out that one of the essential characteristics of a serial spectrum consists in the fact that the energy levels corresponding to one and the same value of \(k\) form a ladder, the distances between whose rungs decrease according to a definite law. From our diagram it is evident that in the spectrum of iron the situation is quite different. The distances between the levels lying one above another apparently obey no regularity. It must be noted, however, that by a proper selection of levels one can pick out those among them which belong to one definite ladder. This can be done, for example, with the levels of the quintet system for which \(k=3\). In all probability, those of these levels which are denoted by \(3d\), \(4d\), \(5d\) are the initial levels of a definite ladder. If this ladder is mentally continued, using for this purpose the known law of series, then one can determine its upper limit, indicated in the diagram by the horizontal line \(\infty d\). This limit corresponds to the removal of an electron from the atom; from its height one can calculate the ionization potential of the iron atom, which proves to be equal to 8.15 volts. Between the levels of this ladder lie the levels \(d^1\), \(d^2\), \(d^3\), belonging probably to an entirely different class and being, perhaps, the initial levels of a number of other energy ladders, whose limits differ considerably from the limit \(\infty d\) just calculated by us. That these levels belong to another class is evident at least from the fact that transitions are known between them and the levels \(3d\), \(4d\) of our ladder. Although these transitions are forbidden by the principle

selection (for in the present case $\Delta k=0$), nevertheless all the corresponding lines possess considerable intensity. The question of the interpretation of these transitions is still far from being solved. According to the views of Bohr and Wentzel, these groups of lines correspond to such changes in the state of the atom in which there is a change in the quantum orbit not of one only, but of two or several of its electrons. By generalizing Laporte’s selection principle it has been possible to divide all energy levels into two groups in such a way that within each group the transitions between individual levels obey the ordinary selection rule ($\Delta k=\pm 1$), whereas, among the possible transitions between two levels belonging to two different groups, only those occur for which $\Delta k=0$. To the first group belong, for example, the already mentioned levels $3d$, $4d$, $5d$. The levels of the second group are marked by a stroke drawn above their spectroscopic symbol, placed in our diagram to the left of the levels; such, for example, is the level $d^1$. That these “stroked” levels do not obey the ordinary series laws is evident if only from the fact that the distances between the separate sublevels into which each level is split do not decrease as the height of the level increases. We would also like to note that even for high levels these distances between the separate sublevels are unusually large (for example, for $f^4$ $\Delta \nu=587\ \mathrm{cm}^{-1}$). Whereas in normal series spectra it is only in rare cases possible to prove the very fact of a splitting of the levels corresponding to $k \geqq 4$, in the present case this splitting is very large.

It is also remarkable that, quantitatively, the levels corresponding to large values of the number $k$ predominate. Whereas in the previously known spectra transitions between levels $k=1$ and $k=2$ predominated, the brightest lines in the spectrum of iron correspond to transitions between $k=3$ and $k=4$. In addition, levels with azimuthal quantum number $k=5$ are encountered here, previously known only from Paschen’s measurements in spark spectra. The height of these levels, and also in part the height of the levels $k=3$ and $k=4$, shows that the outer electron, while on orbits with a high value of the azimuthal quantum number, is bound to the atom much more strongly than is the case on the corresponding hydrogen-like orbits. Thus, for example, an electron on the orbit $k=3$, corresponding to the lower energy level $3d$, is bound to the atom 5 or 6 times more strongly than an electron on the corresponding hydrogen orbit. Such especially strong bonds have hitherto been interpreted on the basis of the assumption that the corresponding orbits penetrate deeply into the atom, inside the orbits of the other electrons surrounding it (see, for example, the well-known drawings in Bohr’s issue of the journal Naturwissenschaften). However, it is doubtful whether the electron orbits can be explained in a similar way.

in the iron atom. In general, it must be admitted that a number of individual remarkable features which have emerged in the study of complex spectra can only with difficulty be explained on the basis of the modern model of the atom. The analysis of spectra makes it possible to determine with great accuracy the energy of an electron in its various orbits; but the very form of these orbits is undoubtedly very complex, especially since precisely in the group of elements from scandium to nickel the newly captured electrons fill the inner group of three-quantum electrons, whereas four-quantum electrons already appear at earlier stages of the periodic system of the elements. Moreover, one may even doubt whether the quantum numbers in complex spectra have the same physical significance as in simple ones. It is true that, if one makes use of empirically generalized rules, the analysis of spectra permits an almost completely unambiguous determination of the numerical values of the quantum numbers corresponding to the various energy levels of the atom. But it must always be borne in mind that these rules, precisely because they have been obtained empirically, have to a considerable extent a formal character. A broad field of activity is opened here to the theoretical investigator. Unfortunately, the results achieved in this field are still very slight. Even in the simplest cases the calculations become very complicated and do not lead (for example, in the case of the helium-atom model) to satisfactory results. Apparently a number of difficulties of a deeply fundamental character still have to be overcome here—difficulties whose cause may lie in the fact that the laws of mechanics, in their present form, are not applicable to the calculation of electronic orbits in those cases where there is interaction of several electrons moving about the nucleus.

In conclusion we shall briefly, without going into details, dwell on those regularities that have been found in the spectra of other elements. The spectra of all elements, insofar as they have a multiplet character at all, are distinguished by the same characteristic properties with which we became acquainted in the example of iron. Besides iron, the elements titanium, vanadium, chromium, and manganese have been studied especially thoroughly; for scandium, too, a number of multiplets has been found, so that in the group of elements from potassium to copper only cobalt and nickel remain unknown. However, according to the information available to us, the investigation of these elements is already being carried out at the present time. The systems of multiplets found in the spectra of these elements are collected in Table 3 (after Catalan).

It follows from the table that, in passing from element to element, even systems of multiplets alternate with odd ones. This is the generalization of the so-called Rydberg alternation law (Wechselsatz), according to which, in the periodic system, doublets alternate with triplets.

TABLE III.

Further, we see that, in passing from the left-hand part of the table to the right, the multiplicity of the highest multiplet system occurring in the spectrum of the given element increases continuously. In parentheses are enclosed those systems of multiplets which have not yet been observed. We leave open the question whether Catalan was right in including these hypothetical systems in the table. In any case, the regular increase in the multiplicity of the multiplets is striking. The recent work of Landé and Heisenberg is an important step toward an understanding of this regularity. This work suggests that we are gradually beginning to make sense of the mystery of such complex spectra as do not at all possess the ordinary multiplet character—for example, the spectra of the noble gases and of the elements of groups IV and V of the periodic system. The splitting of the energy levels in these spectra does not obey Landé’s interval rule; the Zeeman splitting of the lines of these spectra likewise cannot be interpreted by means of the previous Landé rules. According to Landé and Heisenberg, these spectra must be assigned to the so-called multiplet spectra of higher orders; the appearance of such spectra can be understood on the basis of the same “branching principle” which leads to an explanation of the regularities we noted in considering Table III. It would be premature to enter into a more detailed discussion of these questions, which are still in the stage of development. The unusual successes achieved in the field of spectral research within a very short time allow us to hope that, in the not too distant future, we shall succeed in solving the spectral riddles that have so far remained undeciphered, the number of which is still quite large.

1. As is known, the wave number $\frac{1}{\lambda}$ is connected with the frequency $\nu$, i.e. with the number of oscillations per second, by the following relation $\nu = c \cdot \frac{1}{\lambda}$, where $c$ is the velocity of light. ↩↩↩

2. A line is called reversed if it is noticeably absorbed in the very gas or vapor that emits it when glowing. At high vapor density the intensity of the central part of reversed lines, owing to absorption, noticeably falls (an analogy with Fraunhofer lines). Translator’s note. ↩