THE RAMAN EFFECT AND ITS SIGNIFICANCE FOR SPECTROSCOPIC STUDIES OF MOLECULAR STRUCTURE*

A. Smekal, Halle

Introduction

1. Among the discoveries with which experimental physics has presented us in recent years, the Raman effect occupies an altogether special position. It is an extraordinarily elementary phenomenon. The fact that light scattered by some medium may have a spectral composition different from that of the incident light could easily have been discovered ten, twenty, or more years ago. To convince oneself of this, no special experimental means are required, so that a good student could without difficulty, in the course of one laboratory exercise, obtain a photograph revealing the phenomenon of interest to us. To this it must be added that the possibility and necessity of such a phenomenon had long been known from theoretical considerations, and its quantum theory had been so fully developed that even after the experimental realization of the phenomenon nothing essentially new had to be added to this theory.

Theory of Combination Scattering

1. Since the discovery of the Compton effect it has been known that, in the scattering of monochromatic radiation

* Report at the XXXV Congress of the German Bunsen Society for Applied Physical Chemistry, 28/V—1/VI 1931, in Heidelberg. Z. f. Electroch. 38, 618, 1930.

matter there may occur a change in wavelength. The quantum theory of the phenomenon, given by Compton and Debye, showed that, in the interaction of a light quantum with practically free electrons, part of the energy and momentum of the quantum is transferred to the electron, so that the quanta of the scattered light possess a smaller momentum and a smaller energy and, correspondingly, a greater wavelength. The increase in wavelength is extremely small; it depends on the magnitude of the scattering angle and, for rays scattered at an angle of \(90^\circ\) to the primary beam, is equal to \(0.024\) Å. Such a small change in wavelength could be detected and measured only in the scattering of X-rays. Compton processes accompanied by a decrease in wavelength are also possible, but because of their low probability they have not been observed directly.

If the scattering occurs not on quasi-free electrons but on atomic systems of more complex structure (atoms and molecules in any aggregate state), then the scattering process, according to the quantum theory, remains essentially the same as in the Compton effect. Owing to the larger magnitude of the scattering mass, however, the loss of energy of the quantum will be so insignificant that it is impossible to detect the decrease in wavelength. The resulting molecular scattering corresponds well to the Rayleigh scattering of light known for all spectral regions. Thanks to the equality of the wavelength of the incident and scattered light, interference phenomena occur, which determine the refraction, absorption, and dispersion of light.

3. A new prospect opened only when, in connection with the quantum theory of dispersion phenomena, it was taken into account that the scattering atomic systems are formations which, in themselves, are capable of absorbing and emitting light. The incident rays produce disturbances of the atomic system sufficient to make it, during the scattering process, absorb or emit quantities of energy characteristic of its spectral frequencies.

Let the atomic system before scattering be at the energy level \(E_m\), and after scattering at the level \(E_n\); let \(\nu_0\) be the frequency of the incident light, and \(\nu_s\) the frequency of the scattered light. Then, by the law of conservation of energy, we have for \(h\nu_0\) and \(h\nu_s\) the condition:

\[ h\nu_0 + E_m = E_n + h\nu_s . \tag{1} \]

The frequency \(\nu_s\) of the scattered light must differ from \(\nu_0\), the frequency of the incident light, and by so much the more, the greater the difference between the energy levels \(E_m\) and \(E_n\). According to Bohr’s frequency condition, the system’s own frequency \(\nu_1\) is determined by the indicated difference of energy levels:

\[ E_m - E_n = \pm h\nu_1 . \tag{2} \]

Introducing this expression into (1), we obtain, for \(E_m < E_n\),

\[ \nu_s = \nu_0 - \nu_1 \tag{a} \]

and, for \(E_m > E_n\),

\[ \nu_s = \nu_0 + \nu_1 . \tag{b} \]

The application of the law of conservation of momentum of the atomic system and the light quantum introduces no appreciable change into the effect described, and therefore there is no need to take it into account. In other words, the difference between the frequencies of the scattered and incident light is practically independent of the angle of scattering. Since at our disposal there are atomic systems whose frequencies lie throughout the entire spectral region from the infrared to the X-ray, in the elementary processes (a) and (b) any increase or decrease whatever in the wavelength of the scattered light may be obtained.^1 The choice of the wavelength of the primary radiation also remains, to a sufficient degree, arbitrary. According to (a) and (b), the primary frequencies “combine” in a known manner with the proper frequency of the system \(\nu_1\); therefore the special type of scattering described is called “combination” scattering. Owing to the difference between the frequencies of the primary and scattered light, the possibility of their coherence is in general excluded, so that combination scattering is, in contrast to Rayleigh co-

coherent scattering “incoherent scattering”; in general, it does not lead to interference phenomena and the dispersion processes associated with them. By analogy with the phenomena of fluorescence, the frequency \(\nu_s\) in case (a) is denoted as the Stokes frequency with respect to the primary frequency \(\nu_0\), and in case (b) as the anti-Stokes frequency. For process (a) it is required that \(\nu_0 > \nu_i\); it causes an increase of energy in the system, i.e. excites it. Thus Stokes combination scattering can occur for unexcited systems that are in their normal state.

The appearance of anti-Stokes frequencies is free from any restrictions with respect to \(\nu_0\), but can take place only in the case of interaction of light with excited systems, which thereby pass to a lower energy level, losing part of their energy. By analogy with the two types of electronic collisions (of the 1st and 2nd kind), one could say that Stokes scattering processes constitute, for an atomic system, in a certain sense, collisions of the 1st kind with light quanta; anti-Stokes processes—collisions of the 2nd kind. For \(\nu_0 = \nu_i\) these elementary processes pass into the ordinary absorption and emission of the spectral frequency \(\nu_i\), i.e. the phenomenon of resonance fluorescence.

1. Combination scattering is not a specifically quantum effect. Its necessity could have been predicted while remaining on the ground of classical wave conceptions²: from the well-known laws of motion of periodically perturbed oscillatory systems it follows directly that, along with forced oscillations whose frequency is equal to the frequency of the perturbing force \(\nu_0\) (Rayleigh scattering), there must also appear, in the first approximation, combinations \(\nu_i\) with the frequency of perturbation \(\nu_0\) in the form of sum and difference frequencies \(\nu_0 \pm \nu_i\). We owe an exhaustive consideration of the quantum treatment of the phenomenon to Kramers and Heisenberg³. The formulation of the phenomenon both from the point of view of quantum and of wave mechanics gives, in the main, coinciding results⁴. The significance of these re-

results in that, while confirming the conclusions of an elementary consideration of the problem, they moreover make it possible to draw a conclusion about the intensity of the lines of incoherent scattering. The intensity of absorption or emission of a frequency \(\nu_i\) occurring in an atomic system is determined by the “transition probability,” which has a definite value depending on the energy levels \(E_m\) and \(E_n\). It would seem natural to suppose that, for the intensity of the scattering processes (a) and (b), the value of this transition probability from \(E_m\) to \(E_n\) is also significant. However, this supposition turns out to be incorrect. The theory shows that the intensity of combination scattering is not determined by the probability of the direct transition from \(E_m\) to \(E_n\), but by the probability of all pairs of quantum transitions \(E_m \to E_k\) and \(E_k \to E_n\), where \(E_k\) denotes any energy level different from \(E_m\) and from \(E_n\) and capable of combining with them according to the general rules of quantum theory. It may happen, for example, that the probability of the direct transition \(E_m \to E_n\) is equal to zero, so that the frequency \(\nu_i\) appears neither in absorption nor in emission, and nevertheless the scattering processes (a) and (b) will have a finite probability. This will always be the case if there exists at least one quantum level \(E_k\) for which the probabilities of the transition \(E_m \to E_k\) and of the transition \(E_k \to E_n\) are different from zero. Such cases are possible in which the probability of the transition \(E_m \to E_n\) is different from zero and nevertheless combination scattering does not take place, because there exists no level \(E_k\) capable of combining with \(E_m\) and with \(E_n\).

As an example we mention the strictly harmonic linear oscillator. For such an oscillator only the probabilities of transition between two neighboring levels are different from zero; in accordance with this, combination scattering on such oscillators is excluded. On the other hand, there are very diverse “forbidden” quantum transitions which can be “activated” by processes of combination scattering, i.e., the frequencies \(\nu_i\) of such “forbidden” transitions can enter into the formulae (a) and (b). “Forbidden” quantum transitions can therefore be carried out-

... occur upon collision with the scattering light quanta, just as they can be caused by electronic impacts[^5]. Calculations of the intensities of combination lines have so far been obtained for hydrogen atoms[^6], and also for the rotational and vibrational oscillations of diatomic molecules[^7] and, finally, for the vibrations of a crystal lattice[^8].

Combination Scattering and the Compton Effect.

The First Experimental Proof of Combination Scattering

5. The first experimental proof of incoherent scattering should be seen in the experiments that led to the Compton effect[^9], hence in the X-ray region, which for a long time remained unnoticed. The experimental Compton effect concerns scattering by electrons which at first are still bound to atomic systems. These electrons, in the process of scattering, are torn away from the atoms, as was shown by Compton and Simon through direct photographs with a Wilson chamber. Only after this process of detachment do the electrons pass into that free state which is assumed by the Compton–Debye theory of the Compton effect. For sufficiently hard X-rays this circumstance is of no significance, since the energy of detachment of the electrons is very small in comparison with the magnitude of the X-ray quantum \(h\nu_0\). But in principle it should be noted that scattering takes place on atomic systems, and in this process part of the energy of the primary light goes into the excitation or ionization of the scattering atomic system. The proper frequency \(\nu_k\), corresponding to this loss of energy, belongs to the continuous X-ray absorption spectrum of the atom.

The experimental Compton effect is thus a special case of combination scattering; it is identical with Stokes combination scattering of X-rays by atomic systems, the proper frequencies being borrowed from the region of the continuous spectrum. On the basis of these conceptions, the absence of anti-Stokes ...

of Compton lines is naturally explained by the unavoidably small intensity of the continuous X-ray spectrum corresponding to atoms under natural conditions. In scattering by free electrons possessing sufficiently high velocities (for example, a beam of cathode rays), anti-Stokes Compton lines should also appear. * As the initially bound electrons one may have the electrons of the \(K\), \(L\), \(M\) orbits of our atom. Owing to such a diversity of possible combination processes, a broadening of the Compton lines is to be expected, and this was indeed found by DuMond \(^{10}\).

Proof of Combination Scattering at Discrete Natural Frequencies (Raman Effect)

1. Although, as indicated, the Compton effect should be regarded as a special case of combination scattering, nevertheless there was no proof of the possibility of combination scattering associated with the discrete natural frequencies of the scattering atomic system.

Following the theoretical prediction of scattering of this type, various experiments were undertaken in this direction, chiefly in the region of the visible spectrum, but they yielded no results. Only in February 1928 did the Indian physicist Raman \(^{11}\), in Calcutta, succeed in realizing this phenomenon experimentally. For a number of years Raman had been engaged in investigations of the scattering of light by liquids, so that the combination scattering with discrete natural frequencies discovered by him, which received the name of the Raman effect, worthily crowned his many years of work. A qualitative change in the spectral composition of the scattered light had already long been noticed by Raman, but had been ascribed to fluorescence. The correct interpretation of the phenomenon appeared, however, independently of the theoretical—

* This experiment has not yet been carried out, although it may perhaps not be hopeless.

theoretical formulation of the question. The character of the phenomenon became clear when Raman first made use of primary light of a definite spectral composition and compared with it the spectrum of the scattered light. Illuminating liquid benzene with the light of a mercury lamp, Raman discovered in the scattered light, alongside the principal mercury lines (Rayleigh scattering), new sharp lines which were not present in the primary light. It turned out that for the frequencies of these lines the relations (a) and (b) are exactly satisfied, with \(\nu_0\) referring to different mercury lines, while the characteristic frequencies \(\nu_i\) coincide with the frequencies

Fig. 1. Raman spectrum \((a)\) of liquid ether and \((b)\) of ether vapor. Photograph from the Raman Institute, 1928.

of known infrared vibrations of the benzene molecule\(^ {12}\) (Table I). Further investigations by Raman and his collaborators confirmed the existence of this effect not only in liquids, but also in gases and in crystalline substances\(^ {13}\). Independently of Raman and Krishnan, and only somewhat later, Landsberg and Mandelstam in Moscow found combination scattering with discrete frequencies in crystals and established its connection with their infrared spectra\(^ {14}\).

The following figures refer to Raman’s first results. Figs. 1 and 2 indicate the influence of the state of aggregation on the intensity and width of the Raman lines. In the scattering spectrum of carbon tetrachloride (Fig. 3), alongside the principal mercury lines, both Stokes and anti-Stokes lines are observed especially distinctly asสล็อตออนไลน์

… Raman lines. We leave aside a whole series of questions that are of special interest to physicists,^15 such as, for example, the polarization of Raman lines, coherence,^16 the increase in the intensity of Raman lines with the fourth power of the frequency,^17 dependence on temperature—quantitative measurements leading to the determination of universal constants,^18 and so forth.

Fig. 2. Raman spectrum of water, ice, and quartz. Photograph from the Raman Institute, 1928.

Labels in the figure: water; ice; quartz.

Raman effect and investigations of infrared spectra. Method of investigation.

7. Immediately after Raman’s discovery it became clear that the new phenomenon has great practical importance for determining frequencies of vibrations lying in the infrared region. Since the primary light—for example, the radiation of a mercury lamp—belongs to the visible or ultraviolet region, the Raman lines (a) and (b) also lie in the visible or ultraviolet part and can be conveniently determined with optical accuracy. Hence, using relations (a) and (b), the infrared frequencies \(\nu_i\) are easily obtained, numerically expressed as the difference of frequencies,

\[ \nu_i = \pm(\nu_0 - \nu_s). \]

Thus the Raman effect solves the problem of determining infrared frequencies by transferring them from the difficult-to-access infrared region into the region of the visible or ultraviolet spectrum, where investigations are accurate and convenient.

If the source of the primary light is rich in various lines, as, for example, the light of a mercury lamp, then each frequency \(\nu_i\) gives as many pairs of Raman lines—Stokes and anti-Stokes—as there are lines in the primary source. Owing to the usually low intensity of combination scattering (especially of the anti-Stokes lines) in comparison with the main scattered lines, and to the differences in the intensity of the latter, some weak Raman lines do not appear on photographs even with long exposures. Contradictions in equal determinations of \(\nu_i\) often arise from an incorrect “assignment” of the found Raman line to one or another principal line; in discussing individual results we shall see that even now these discrepancies have not been completely eliminated.

A reliable method for the exact coordination of Raman lines is the use of strictly monochromatic primary light, which can often be achieved by employing appropriate light filters. The requirement of greater intensity makes it possible to use, as the primary source, besides the mercury lamp only the helium lamp constructed by Wood\(^{19}\), which gives practically monochromatic light.

Fig. 3. Raman spectrum of carbon tetrachloride with especially intense anti-Stokes Raman lines. Photograph from the Raman Institute, 1928.

Experimental arrangements employ various devices for better utilization of the primary intensity, and also for the maximum increase of scattering in the direction of observation (reflectors, the shape and arrangement of vessels in the investigation of liquids and gases, etc.). The spectral decomposition of the scattered light can be

achieved with any spectrograph; of course, preference is given to instruments with good luminosity and, if possible, large dispersion, which substantially affects the accuracy of the measurements. Under especially favorable experimental conditions it is possible to observe visually the strongest Raman lines. For measurements, however, a photographic image of the scattering spectrum is required.

The simplest of all are investigations of scattering in liquids. Since the Raman spectrum of a mixture, in the first approximation, is additively composed of the Raman spectra of its constituent parts, the requirement of great purity of the liquid under study is not necessary: the principal Raman lines of the main liquid will not be obscured by the presence of weak lines of impurities. Far more important is the removal of colloidal contaminants (for example, dust), which give rise to non-molecular scattering containing, it is true, only the principal lines of the primary spectrum, but with such intensity that it greatly interferes with observations of the Raman lines. Investigations of gases are carried out at elevated pressure^20, in order to increase the number of scattering centers, which is generally proportional to the density of the substance (see Fig. 1). Since only a few solids occur in the form of large crystals convenient for investigation, it is sometimes necessary to make measurements with crystalline powders^21 placed in a cuvette filled with a liquid having the same refractive index as the crystal being studied, which facilitates the penetration of light into the crystalline grains^22.

1. As a comparison of the preceding data with the methodology of infrared investigations^23 shows, the method based on the Raman effect presents a whole series of advantages. Obtaining scattering spectra is very simple and requires no particularly complex experimental arrangements. The wide use of photographic methods gives a great saving of labor and time; finally, the accuracy of measurements by the new method^24 exceeds the accuracy of infrared determinations. In view of these undoubted methodological advant—

... of the study of Raman spectra assumed, especially at first, that the data of these investigations could also fully replace the data of infrared investigations. On the basis of theoretical considerations (§ 4), these assumptions, however, are justified only under certain conditions, which

Table I

Infrared and Raman vibrations of benzene and toluene

can best be clarified purely empirically, on some well-studied example.

Let us take the classical scattering object—benzene and the related toluene. Table I contains the data, apparently from very careful measurements by Zederkvist[^24], carried out by the scattering method, and the data of Coblenz’s infrared measurements.* Next to the wavelengths, in parentheses, the relative intensities are given. As was already known earlier from infrared measurements, the C—H bond and the benzene ring possess certain characteristic natural frequencies. These frequencies are marked in the table, from which it is evident that they coincide very well for substances of analogous structure, as in our example. All wavelengths found for both substances are arranged in the table in such a way that they, like the other mutually corresponding quantities, stand in one row.

From this comparison it is immediately clear that the wavelengths obtained from the Raman spectra coincide only partially with the infrared data; the values of the relative intensities of the lines in the one and the other spectra differ completely; the accuracy of measurement is considerably higher for the Raman spectra. Alongside the natural frequencies common to both methods, there is still a considerable number of such frequencies which are found either only in the Raman spectra or only in the infrared. All these observed discrepancies agree qualitatively very well with the above-stated

* Toward the shorter-wavelength side, the bands of benzene were studied by T. Dreisch, and also by J. Barnes and W. Fulweiler. See C. Schaefer and Matossi. Die ultrarote Spektrum, p. 271, Berlin, 1930. Concerning the photoluminescence bands in benzene see Reimann, Ann. Physik 80, 43, 1926; on their relation to the Raman spectrum see Schapiro, Nature 124, 372, 1929; I. Black, Nature 125, 274, 1930; Austin and Black, Phys. Rev. 35, 457, 1930; F. Almassÿ and C. Schapiro, ib. 14, 32; Schapiro, Gibbs and Ianson, ib., 1422.

The frequencies of benzene marked in Table 1 with an asterisk appear, according to these latter investigations, in the fine structure of the optical bands. The infrared fine structure of the C—H bands for benzene, toluene, and other organic substances has been studied in detail by Barnes—Phys. Rev. 35, 1524, 1930. Unfortunately, they could not yet be noted in Table 1.

by the theory of combination scattering. Thus infrared investigations and Raman spectra cannot completely replace one another; they supplement one another in a very valuable way. Of special importance is the “appearance” of inactive natural frequencies in Raman spectra. It should further be noted that, for the pair under consideration (benzene–toluene), the asymmetric molecules \((\mathrm{C_6H_5CH_3})\), in comparison with the symmetric molecule \((\mathrm{C_6H_6})\), reveal in the Raman spectrum a large number of lines coinciding with the infrared data. Such a difference should also be expected theoretically.

General Results of the Raman Effect

9. Despite the short period that has elapsed since the discovery of the Raman effect (about three years), the literature concerning the results obtained in this field is quite extensive; for example, the number of substances studied by the new method has exceeded two hundred. A consolidated treatment of all these data has not yet been carried out and would be premature, both because of the low accuracy of most determinations and because of the constant growth of the experimental material and its insufficient coordination with the data of infrared measurements. We shall therefore confine ourselves to a few considerations of a general character.

9a. Combination scattering with discrete natural frequencies on free atoms has not yet been observed. The experimental conditions here are unfavorable; such scattering could most easily have been discovered in the X-ray region \(^{25}\). Here signs of Raman lines have been found repeatedly \(^{26}\), but to this day the question still remains controversial \(^{27}\).

9b. For molecules in the gaseous and liquid states, in combination scattering the vibrations of atomic groups manifest themselves first of all. From considerations of intensity it follows, in full agreement with other optical data, that the fundamental frequencies are chiefly excited, and much more rarely the overtones. The exact determination of the fundamental frequencies in many cases facilitates the establishment of the rotational-

new ones, often occurring in large number in the infrared spectrum. In polyatomic molecules not all fundamental frequencies always appear, but in all cases analyzed so far valence frequencies are always found. Their appearance in Raman spectra indicates (§ 4) the anharmonic character of the valence bonds in molecular structure.

Fig. 4. Rotational Raman spectra of oxygen and nitrogen. From Rasetti, 1930.

Independently of atomic vibrations, but in combination with them, the rotational frequencies of molecules appear in Raman spectra. With a small resolving power of the spectral apparatus, and also in liquids, the rotational motions of molecules manifest themselves in an asymmetric broadening of the Raman lines[^28]. But with sufficient dispersion, typical rotational Raman spectra are obtained, as shown in Fig. 4, obtained by Rasetti for gaseous nitrogen and oxygen. The distances between the lines in Rasetti’s spectra correspond to four times the distance between rotational lines in ordinary band spectra, in complete agreement with theoretical expectations[^29]. In liquid gases (for example, H₂, NH₃), sharp rotational lines are also sometimes observed.

The influence of electronic transitions on Raman spectra has so far been noted only in isolated cases.^30

9c. Whereas in gases and liquids we deal chiefly with unexcited atomic systems, the lattice vibrations in crystals at ordinary temperature correspond to thermally strongly excited states. The smallest excitation of this kind occurs in diamond,^31 where Raman frequencies have actually been observed. Crystals built from ions—for example, crystals of the rock-salt type \((\mathrm{CaF_2}, \mathrm{BaCl_2}, \mathrm{ThCl_4})\)—at room temperature show no Raman lines whatever, so that for them analysis of the infrared spectra is for the time being absolutely unavoidable. Strong Raman lines have been found in crystallized \(\mathrm{HgCl}\), \(\mathrm{HgCl_2}\), \(\mathrm{SbCl_3}\), \(\mathrm{BiCl_3}\); weaker ones—in \(\mathrm{AuCl_3}\), \(\mathrm{ZnCl_2}\), \(\mathrm{CdCl_3}\). The appearance of frequencies characterizing the crystal lattice depends on the kind of chemical bonds.^32 Such frequencies also occur in crystal lattices with the groups \(\mathrm{CO_3}\), \(\mathrm{NO_3}\), \(\mathrm{SO_4}\). But in Raman spectra the “internal” vibrations of these groups also stand out clearly. Some of them coincide with the “active” infrared fundamental frequencies; the rest correspond to the “inactive” frequencies of these groups, which until now could not be detected by direct observations and were introduced from theoretical overtones of infrared spectra. It should be noted that Raman spectra, like absorption measurements of infrared spectra, give the true frequencies of vibrations, so that there is no need for the recalculations to which one must resort if one wishes to determine infrared vibrations by the reflection method.

Special Data of the Raman Effect

1. The great accuracy in determining infrared frequencies attained with the aid of Raman spectra makes it possible to answer a whole series of questions whose resolution in infrared investigations was altogether impossible, or could be carried out only in an imperfect way. This includes the question of the influence of the state of aggregation

on the magnitude of the atomic frequencies in molecules. It turned out that, in a first approximation, atomic vibrations may be regarded as unchanged, so that in different aggregate states it is easy to identify the corresponding frequencies. In a second approximation it is obviously necessary to take account of the intermolecular Stark effect, which can cause not only a change in intensity, but also changes of frequency and even splitting of lines. More careful observations could reveal the existence of these effects; moreover, one may foresee that substances possessing a dipole moment will be of special interest.

Table II

Effect of the aggregate state of a substance on the frequency of atomic vibrations
(frequencies in cm\(^{-1}\))

The great sharpness of the Raman lines in ice as compared with liquid water must be connected with a reduction of molecular rotations; this change can easily be seen in Fig. 2. According to the observations of Rao \(^{33}\), this increase is accompanied by a certain decrease in wavelength and by a characteristic change in intensity \(^{34}\); there are also data on a decrease in the wavelength and intensity of the lines for water when the temperature is raised \(^{35}\). Changes in the intensities of the water bands are observed in electrolytes as the concentration is increased; in this case the bands sometimes become as sharp as in ice.

A noticeable difference between the solid and liquid states

* The frequencies of gaseous NH\(_3\) enclosed in parentheses are not found in the Raman spectrum and are borrowed from the analysis of the optical spectrum of NH\(_3\).

** The frequencies enclosed in parentheses are not found in the Raman spectrum.

shows a Raman frequency relating to the bond \(C=O\) (1650 and \(1657\ \mathrm{cm}^{-1}\)), characteristic of the dipole moment of benzophenone. \({}^{35}\)

Similar, but much more significant in magnitude, changes are found for a whole series of substances upon transition from the liquid to the vapor state. The data collected in Table II concern atomic vibrations and relate chiefly to hydrogen compounds.

Comparison of the data for crystalline quartz, fused quartz, and glasses \({}^{36}\) does not reveal, just as analogous measurements of infrared spectra do not reveal, any substantial differences in the frequencies of the normal vibrations. In the amorphous state the Raman lines are noticeably broadened and show characteristic changes in intensity. Unfortunately, the spectra of \(\mathrm{SiO}_2\) have still been studied too little to permit a conclusion about the molecular structure of glasses, which will probably someday become possible on the basis of such investigations.

An analogous change in the sharpness and intensity of individual Raman lines is observed with a significant increase in the temperature of crystalline quartz. \({}^{37}\) The normal frequencies of ionic groups in crystals and in solutions, as well as the properties of water of crystallization, have been studied many times. Sulfates (Table III) and nitrates (Table XII) have been investigated especially fully. In Table III the data for gypsum, barite, and a solution of ammonium sulfate are compared. \({}^{38}\) The position of the water bands is easily established from their absence in barite, which contains no water of crystallization.

Table III

Raman spectra of crystalline and dissolved sulfates

The lines corresponding to the normal frequencies of the \(\mathrm{SO}_4\) ion prove in gypsum to be split and considerably shifted ...

valuable in comparison with those in solutions. At present we cannot say exactly what accounts for the difference in the spectra of the two solid sulfates: whether it is the difference of the cations or the influence of water of crystallization. We shall return to this question when considering the role of the cations in nitrates (§ 15).

1. Studies of the Raman spectra of mixtures of liquids promise to help elucidate the influence of intermolecular interactions. In this respect, of interest is the appearance of a quasi-continuous Raman spectrum in liquids

Table IV

Raman frequencies (in cm\(^{-1}\)) of sulfuric acid in the mixture H\(_2\)SO\(_4\) + water

with high viscosity, for example in glycerin\(^{39}\), and the effect on this spectrum of adding other liquids that reduce the viscosity. However, in these observations fluorescence, which gives a continuous spectrum\(^{40}\), must be taken into account. A considerable shift of frequencies is shown by the Raman spectra of aqueous solutions of sulfuric acid of various concentrations, given in Table IV\(^{41}\), where the bands corresponding to water have already been excluded. Since here the frequencies of the SO\(_4\) ion play the chief role, it is of interest to compare Table IV with the data of Table VIII.

The simultaneous change in frequency and intensity, as well as the splitting of lines, is well illustrated by the

example of acetic acid (\(\mathrm{CH_3COOH}\)) in mixture with various liquids\(^{41}\) (Table V). The frequency \(1650\ \mathrm{cm}^{-1}\), on which the influence of impurities is especially reflected, belongs to the double bond \(\mathrm{C=O}\), which also shows a special sensitivity in benzoic acid (\(\mathrm{C_6H_5COOH}\)—\(1648\ \mathrm{cm}^{-1}\)) in mixtures with other substances.

Table V

Raman frequencies of acetic acid in binary mixtures

As we have already indicated above, with regard to the \(\mathrm{C=O}\) bond, for liquid and solid benzophenone \(\mathrm{C_6H_5CO\cdot C_6H_5}\) (\(1650, 1658\ \mathrm{cm}^{-1}\)), this special sensitivity is connected with the presence of a dipole moment, on which also depends the known capacity for association characteristic of these substances\(^{42}\).

For the majority of other liquid mixtures investigated up to now, no such considerable changes of the characteristic frequencies as those given in Tables IV and V have been found. Reliable establishment of effects of smaller magnitude requires, correspondingly, greater accuracy of observation. How fruitful this may be is indicated by Gerlach’s\(^{43}\) investigations of changes in the Raman spectra of nitrates as a function of the concentration of the solution. From Table VI it is evident that the inactive frequency of the \(\mathrm{NO_3}\) group, even at small

concentrations exhibits measurable changes depending on dilution,—a circumstance of importance for the theory of the molecular state of electrolytes. The establishment of such fine features by means of the infrared-spectra method may be considered quite impossible.

Table VI

Dependence of the nonactive NO₃-frequency
(in cm⁻¹) on concentration

1. The investigations of Raman spectra indicated above give an idea of the degree of independence of the molecular intrinsic frequencies from intermolecular interactions.

Table VII

Fundamental frequencies for trivalent and tetravalent
halide compounds (in cm⁻¹)

In what follows we shall give several more examples characterizing the dependence of the intrinsic frequencies of molecules on the features of their structure (intramolecular interaction). In Table VII are given the fundamental vibrations of a whole series of homologous trivalent and tetravalent halide compounds⁴⁴. In all cases there are present the four fundamental frequencies that should be expected theoretically.

(according to Dennison)\(^{46}\) for models of molecules having the form of a symmetric pyramid (trivalent compounds) or a tetrahedron (tetravalent compounds), if the atoms in such a model are regarded as material points.

In tetrahedral models the first and third frequencies are “inactive,” and they are indeed not observed in the infrared spectra \((\mathrm{CCl}_4, \mathrm{SiCl}_4)\). In these spectra, in addition to the two active frequencies, there also appears a series of frequencies representing combinations of active and inactive vibrations and their overtones\(^{46}\) (i.e., vibrations with frequencies \(\nu_1 + \nu_2\), \(\nu_1 > 2\nu_2\), etc., if \(\nu_1\) and \(\nu_2\) are the fundamental frequencies). These combination frequencies, generally speaking, do not appear in the Raman spectra (the exceptions are the overtones observed in \(\mathrm{CCl}_4\), \(445\ \mathrm{cm}^{-1}\), \(1539\ \mathrm{cm}^{-1}\))\(^{47}\). From the example of four tetrahedral molecules (Table VII) one can establish precisely that in Raman spectra practically only the fundamental frequencies appear; moreover, both the “active” and the “inactive” frequencies are manifested.

The regular decrease of vibrational frequencies with increasing molecular weight, clearly evident for the substances in Table VII, is determined in the first approximation by the masses of the constituent parts of the molecule. Calculation shows that the distance of the Cl atoms from the central atom increases with increasing atomic number of the latter, so that the strength of the valence bond depends regularly on the size and deformability of the atoms. Vibrations occurring in the direction of the valence bond—“valence vibrations”—correspond, understandably, to the largest of the fundamental frequencies found; conversely, the smallest frequencies are associated with the “deformation vibrations” of the molecular model. The presence of similar phenomena for the fundamental frequencies is illustrated by Table VIII, which compares the characteristic frequencies of complex ions of the type \(\mathrm{XO}_4\)\(^{48}\).

The successful interpretation of the relationships in carbon tetrachloride (Table VII) prompts the extension of the range of consideration to the other halogen derivatives of methane. Although the spectrum of methane has not yet been fully deciphered\(^{49}\), nevertheless from its Raman and infrared spectra it follows that the smallest

the fundamental frequencies of methane must be greater than \(1300\ \mathrm{cm}^{-1}\). In accordance with this, all frequencies of the halogen derivatives of methane lying below this limit have been grouped together as the presumptive fundamental halogen frequencies of these substances \(^{50}\).

Table VIII

Raman frequencies of the \(XO_4\) group in solutions
\((\text{in } \mathrm{cm}^{-1})\)

Methyl halides \((\mathrm{CH_3—X})\) give only one frequency,* which should apparently be identified with the valence vibrations of the carbon–halogen bond. We see that this valence frequency retains its value also for the higher halogen derivatives, although a slight change, as well as splitting, occurs.

Beginning with the methylene halides \((\mathrm{CH_2=X_2})\), additional, appreciably smaller frequencies already appear, connected in all probability with deformation vibrations of the molecules. Such a clearly expressed insensitivity of the halogen valence bond to the number of halogen atoms in the molecule may be regarded as a sign of the constancy of the valence force. With an extension of our model conceptions, which have proved so fruitful in application to \(\mathrm{CCl_4}\), one would in these cases \(^{61}\) have to regard the groups \(\mathrm{CH_3}\) and \(\mathrm{CH}\) as point masses.

Correct numerical values of the fundamental frequencies can be obtained if, as models of compounds of the type \(\mathrm{CH_3—X}\), one takes a linear oscillator; for \(\mathrm{X—CH_2—X}\), a plane triangle; and for \(\mathrm{X—CH—X}\), a symmetric pyramid,

\[ \begin{gathered} \mathrm{X—CH—X}\\ \ \ \ \downarrow\\ \mathrm{X} \end{gathered} \]

similarly to what is accepted for \(\mathrm{PCl_3}\), in Table VII. The calculations confirm (with the exception of bromoform),

* Their values agree with the infrared data only in order of magnitude. Schaefer u. Matossi—Ultrarotspektrum, pp. 271—273.

that the valence forces do indeed have one and the same order of magnitude; moreover, it turns out that the angles between the “valence directions” toward the halogen atoms are always close in magnitude to the angles between the stereochemical valence directions of a tetrahedral carbon atom.

1. According to the data indicated above, similar to those given in Tables VII and VIII, all the expected fundamental frequencies always appear; it is therefore not difficult to identify among them the valence frequencies, which are numerically the largest. Experience shows, however, that for other equally simple molecules, such as NH₃ and CH₄, the appearance of all fundamental frequencies does not always occur. To determine the valence frequencies on the basis of Raman spectra alone, one must assume either that all the fundamental frequencies are in fact present or, at least, that among the Raman frequencies there must necessarily be found valence frequencies.

Table IX

Raman frequencies of halogen derivatives of methane
(in cm⁻¹)

Unfortunately, at the present time it is not yet possible to justify theoretically the necessity of the appearance of valence frequencies in the Raman spectra of any polyatomic molecules. However, the suitability of such a rule for molecules seems probable on the grounds that stereochemical directions must almost always coincide with the direction of the forces of maximum interaction of the constituent parts of the molecule, so that in these directions the greatest deviations from the purely harmonic character of the interaction should be observed. According to theory, such a deviation from the harmonic character, generally speaking, means that the given natural vibration predom—

directly participates in combination scattering. Since on this question there is as yet no final certainty, we are compelled, in interpreting the observed Raman frequencies, to resort to verification by comparison with the data of infrared measurements or by analysis of visible band spectra. Obviously, for the present this is the only way to determine the region of empirical validity of the stated rule.

Table X

Fundamental frequencies of XH molecules
in cm\(^{-1}\), according to Mecke

In the case of methane, the infrared spectrum gives a large number of characteristic frequencies, as may be expected from the tetrahedral model of the molecule. Two or three series of Raman frequencies found for methane gas should be regarded as the fine structure of a single (according to the model, actually triple) frequency, as we had for the valence frequency of PCl\(_3\) and CCl\(_4\) (Table 7). The interpretation of the Raman frequencies as valence frequencies remains here not entirely reliable either, owing to the incompleteness of the infrared measurements. For complete confirmation, a direct optical determination of the fundamental frequency of the unsaturated CH group is required, since the valence frequency of the CH\(_4\) molecule obviously cannot be less than that for CH. The fundamental frequencies of XH molecules are given in Table X, compiled by Mecke \(^{52}\) and compared with the valence fundamental frequencies of saturated molecules, determined independently of the Raman effect. On the basis of these data it is difficult to derive an unambiguous interpretation of the Raman spectra of methane. The frequencies of CH\(_4\) that are smaller in magnitude and absent from the Raman spectra (exactly as are the frequencies of gaseous NH\(_4\), see Table II) should be attributed to deformation vibrations of these molecules.

The establishment of the valence frequency of $\mathrm{CH}_4$ opens the possibility of interpreting the CH frequency ($3000\ \mathrm{cm}^{-1}$), encountered in the infrared and Raman spectra of many organic substances (including benzene and toluene, see Table I), as the fundamental valence frequency of these organic molecules. Thus the supposition, expressed on the basis of infrared-investigation data, that this frequency may be regarded as the octave of a lower frequency,^53 is rejected; such a lower frequency would have to be either a deformation frequency of the $\mathrm{CH}_2$ or $\mathrm{CH}_3$ complex, or have an entirely different interpretation.*

1. The study of Raman spectra of more complex molecules has been undertaken for extensive series of organic substances in a number of numerous investigations. As was already known from infrared studies—though not such extensive ones—the presence of like kinds of bonds or atomic groups is always associated with the appearance of almost identical characteristic natural frequencies.

The establishment of differences between fundamental frequencies and overtones, as well as the separation into deformation and valence frequencies, has hitherto been carried out only very imperfectly and is still far from having become the object of systematic study. Of the large number of already touched-upon and partly rather thoroughly investigated problems of the structure of matter, we shall dwell here only on the question of multiple bonds.

Table 11 brings together the data available on this question, obtained from Raman spectra. Simple, double, and triple bonds between atoms of nearly equal mass (C, N, O) differ by quite definite and strongly different frequency values. If, from these frequencies, one calculates the corresponding mean quasi-elastic forces, then the mean values for these three kinds of bonds give, according to Dadieu and Kohlrausch,^54 the simple numerical ratios indicated in the table (last column). Therefore

* From the data of Raman spectra and from Table XI it follows that only the fundamental valence frequency of this C-bond can be taken into consideration.

is, perhaps, rational to take these quantities as numerical characteristics of the multiplicity of bonds^55.

The natural frequencies of diatomic molecules should be regarded as undoubted valence fundamental frequencies. The data given above, as well as other correlations, indicate that the remaining frequencies also belong to valence fundamental vibrations. However, these conclusions lose all reliability if deformation frequencies also appear in the Raman spectra, which happen by chance to belong to the same frequency region as those given in Table XI. Under these conditions the danger is not excluded of confusing such deformation and valence frequencies, if there is no possibility of resorting to other sources of analysis.

Table XI

Raman frequencies of multiple bonds

Thus, for example, on the basis of the values of the low frequencies of the Raman spectrum observed only in liquid ammonia (Table II), it was erroneously considered possible to draw an unambiguous conclusion about the presence of single and double nitrogen bonds (according to Table XI), i.e. the existence of polymerization processes^56. However, as analysis of the infrared spectrum showed, the same frequencies are also present in gaseous ammonia (Table II),* and must

* For this indication I am indebted to Prof. Mecke, Trans. Faad. Soc. 25, 830, 1929.

therefore be considered as deformation frequencies of the molecule. Table XI shows that the “isosteric” molecules CO and N₂ both have triple bonds and consequently behave identically in this respect as well. The isosteric pair CO and N₂ even possesses almost coincident Raman frequencies (1281.8 cm⁻¹, 1285.1 cm⁻¹, as well as 1264.5, 1387.7, 1408.4 cm⁻¹). The Raman spectra of isomeric substances, just like the infrared spectra, are in general different, at least with respect to the distribution of intensity.

Table XII

Inactive Raman frequencies of crystalline nitrates (in cm⁻¹)

NO₃ → solution — 1045
NaNO + KNO₃ solution — 1042

15. The influence of the structure of the constituent parts of molecules on its own frequencies is most easily observed in ionic compounds, for which one can judge separately the role of the ionic charge and the role of the deformability of the ion. Determination of the inactive frequencies of the NO₃ group in crystalline mono- and divalent nitrates by means of Raman spectra made it possible to study the influence of both of these factors.

Table XII gives Terlakh’s measurement data,⁵⁷ as well as the positions of the intrinsic frequencies in solution and in the molten salt. It is evident from the table that the change in the value of the inactive frequency of the crystal (in comparison with the frequency of the dissolved or molten salt), at constant ionic charge, is the greater the smaller the ordinal number of the cation, i.e., the smaller the distance between the ions and the greater the deforming ability of the cation.

The comparative data for KNO₃ and Ca(NO₃)₂ further show that, with practically the same ordinal number and correspondingly the same mass and identical dispo-

the electrons of the cation a greater charge corresponds to a stronger action. The influence of water of crystallization consists in a diminution of this effect, which, at the highest degree of hydration (for strontium nitrate), turns out to be completely compensated. Thus, in the question under consideration as well, the Raman-effect method, when the accuracy it affords is fully utilized, leads to substantially more complete results than infrared investigations, which could only outline certain initial points in the problem of the regular influence of the cation in the study of carbonates.

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9. Of the three halogen compounds only PCl₃ gives overtones, namely 340, 1 340 cm⁻¹.

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