New Developments in the Problem of Light Scattering

Gr. Landsberg, Moscow.

§ 1. Light Scattering and Fluorescence

According to our usual conceptions, it is not difficult to establish the difference between fluorescence and the scattering of light. In the case of fluorescence we are dealing with the excitation of radiation characteristic of the fluorescing substance. The spectrum of this radiation is determined in its main features by the nature of the fluorescing substance, and the wavelength of the exciting light plays a secondary role. It goes without saying that only such light as is absorbed by our substance can be exciting. In this case, the absorption band, as a rule, has a maximum somewhat shifted toward shorter waves in comparison with the maximum of the fluorescence spectrum (Stokes’ rule). The scattering of light, however, is a process as a result of which the primary beam of light incident upon a body undergoes only a change in the direction of propagation, without undergoing any change in the frequency of its oscillations: the spectrum of the scattered light repeats the spectrum of the exciting light. What changes, to be sure, is the distribution of light intensity with wavelength, according to Rayleigh’s law (1), according to which the intensity of scattered light is inversely proportional to the fourth power of the wavelength.

The distinctions established above require clarification. From the point of view of the mechanism by which fluorescence and scattering arise, the distinction between them may be formulated in terms of the classical theory as follows. Fluorescence is the excitation of oscillations characterizing

molecules and atoms of the fluorescing substance. These oscillations occur with periods determined by processes taking place in the substance itself, and are therefore natural oscillations. Conversely, scattering is caused by the excitation of oscillations in which the atom is involved in an oscillatory process with the period of the external wave, i.e. by forced oscillations. In accordance with this, a change in wavelength should not occur in scattering. But it may also fail to be observed in fluorescence. Indeed, in the gaseous state isolated atoms are characterized by sharply defined periods, so that they are capable of absorbing only a very narrow monochromatic region of the spectrum, and the natural oscillations excited in them have periods coinciding with the period of the absorbed light. Fluorescence in this case must have a resonance character, i.e. the wavelength of the fluorescence light coincides with the wavelength of the exciting (absorbed) light. This is precisely the fluorescence of Na vapor discovered by Wood (2). The same phenomenon of resonance fluorescence was later observed also in the vapors of other substances (Hg, J₂) (3). In liquids or solutions we are dealing with molecules that are under the strong influence of the surrounding molecules. Therefore the natural optical periods of liquids prove to be not so definite as is the case for monatomic vapors and gases. Accordingly, the absorption spectrum of liquids is characterized by comparatively broad bands, and one need not speak of strict resonance between light and atoms. For the same reasons—the interaction of molecules—the fluorescence spectrum of these objects is banded. The noncoincidence of the absorption and fluorescence spectra does not admit of a simple interpretation from the standpoint of classical conceptions. Formally, Stokes’ law was obtained, as is known, by Einstein (4) from the conception of light quanta.

Thus, we may define fluorescence as the phenomenon of the appearance, under the action of light, of natural oscillations (or quantum processes leading to emission)

... proper frequencies); in contrast to this, the scattering of light represents a process of the occurrence of forced oscillations (or of the corresponding quantum processes).

In connection with the indicated difference, another circumstance should also be noted, one especially important from the point of view of the question that forms the subject of the present article.

In the excitation of fluorescence—ordinary or resonance—we are dealing with the proper frequencies of the substance. The act of emission of light is a secondary act, separated from the primary one—the action of light—by the processes leading to the assimilation of the supplied energy. We do not know exactly what these processes consist in; they reduce to certain internal perturbations, as a result of which the atom finds itself in another quantum state. Experience makes it probable, however, that these processes require a certain time, the order of which may be estimated as \(10^{-9}\) sec. (5), (6). The separateness of the two acts mentioned is manifested also in the fact that the emitted light is not phase-related to the exciting light. Therefore the initial phase of the light emitted by each center is determined chiefly by its internal processes. Thus even centers separated by a distance of the order of a wavelength, i.e. excited by an evidently coherent aggregate of light waves, become sources of secondary waves that are not coherent with one another. For this reason fluorescence light is propagated uniformly in all directions, like the light of independent sources. The impossibility of interference of waves emitted by different centers excludes the preferential significance of any directions.

In contrast to fluorescence light, the light of scattering we treat as secondary waves caused by forced oscillations of electrons arising in atoms or molecules under the action of the primary light and proceeding with the period of this light. At the very high frequency that corresponds to ordinary light (of the order of \(10^{15}\)), the process of forced rocking of the electron must very quickly attain a stationary state. In all probability, it is attained after the lapse of several...

tens of oscillations, so that after a time interval of the order of \(10^{-13}\) sec. the forced oscillations already become a source of established secondary waves. This time interval is a thousand times shorter than the interval separating the act of excitation from the act of emission in fluorescence and in other processes connected with the excitation of natural oscillations1. The forced character of the oscillations that determine the scattering of light leads, of course, to the fact that the period of the secondary waves generated by these oscillations coincides with the period of the exciting light. Moreover, the initial phase of the secondary waves is determined by the phase of the primary wave. Therefore all centers situated so close to one another that their excitation can be produced by waves still belonging to the same coherent train become sources of secondary waves coherent with one another and with the primary wave. Thus, the observed wave arises as the result of the interference of these secondary waves superposed on the primary one. In other words, the phenomenon of the scattering of light must be accomplished by means of the same process as all other phenomena of the propagation of light through a material medium (rectilinear propagation, dispersion, reflection, and refraction). Entering a material medium, the electromagnetic wave propagates at the first instant as in emptiness, with velocity \(c\). But after a very short interval of time it sets into oscillation the electrons composing the medium, which become sources of secondary waves coherent with the primary one. On being combined, these waves give, as the resultant, a wave whose velocity of propagation depends on the properties of the medium. As is known, these considerations lead to the dispersion for—

to the usual type of formulae (7), (8), (38). Thus the refractive index of the medium must at the first instant be equal to 1, and only by the time the stationary secondary waves are established must its magnitude reach the usual value. However, the establishment of the stationary state occurs so rapidly that there is no possibility of detecting experimentally a deviation of the refractive index from the norm.

§ 2. The physical cause of molecular scattering of light.

However, the ability of the secondary waves to interfere with one another, which underlies the above-mentioned theory of dispersion, leads, as is known from Fresnel’s arguments, to the rectilinear propagation of light. In other words, a plane primary wave must remain plane also inside a material medium, if the latter is optically homogeneous. Optical homogeneity presupposes that the number of centers falling within small volumes (i.e. volumes whose linear dimensions are comparable with the wavelength) is proportional to the volumes selected. In such a case an infinitely thin layer adjoining the wave front, drawn through any place in the medium, will be uniformly filled with centers serving as sources of secondary (Huygens) waves. Dividing this layer by Fresnel’s method into zones, we shall be able, by repeating the usual arguments, to show that the resulting wave propagates rectilinearly, i.e. the secondary waves in all lateral directions are destroyed by mutual interference.

Rayleigh, too, in discussing the question of the scattering of light, showed that forced coherent oscillations cannot give rise to scattering of the primary plane wave to the sides. However, Rayleigh (1) assumed that these arguments are applicable only to the case of stationary particles composing the medium. He believed that in the presence of disordered motion of the particles there can be no question of a constant phase difference between the oscillations arising in them, which would be due merely to the retardation of the exciting wave passing through them.

with a finite velocity from one layer to another. In such a case, according to Rayleigh, particles moving chaotically (thermal motion) become sources of incoherent secondary waves and, consequently, can cause the scattering of a plane wave in all directions. It is known that, on the basis of these considerations, Rayleigh arrived at a formula determining the intensity of light scattered by a homogeneous medium (molecular scattering).

Rayleigh’s reasoning, however, must be supplemented. As was shown by Mandelstam (9), the presence of disordered motion of particles by itself does not lead to the formation of incoherent secondary waves if the number of these particles is very large. In such a case we can always divide the medium into small volumes, fixed in space, and with respect to these volumes all our considerations about the random distribution of the phases of the secondary waves remain valid. In order that Fresnel’s construction should again lead to rectilinear propagation of a plane wave, it is only necessary that these small volumes contain a number of centers proportional to the volumes, i.e. that the medium satisfy the above-stated condition of optical homogeneity. Thus an optically homogeneous medium constructed of randomly moving molecules likewise should not cause scattering of plane waves. However, a medium constructed of a large number of moving molecules cannot be optically homogeneous, even with respect to waves as long as light waves. Smoluchowski (10) was the first to point out that density fluctuations, caused by the molecular structure of the medium, must lead to a violation of optical homogeneity. This phenomenon and the light scattering connected with it become especially noticeable near the critical point, where the compressibility of the liquid increases greatly.

Indeed, “opalescence” near the critical point becomes so intense that it serves as one of the well-observed signs of the approach to the critical state. Thus Smoluchowski gave an explanation of this “critical opalescence,” showing at the same time

time, where one must seek the physical cause of any scattering of light by homogeneous media.

Since these density fluctuations are a direct consequence of the molecular nature of matter, it is natural to call this type of fluctuation scattering of light “molecular scattering.” It must be remembered, however, that the presence of a granular structure, determined by the molecular constitution of matter, is in itself insufficient to cause scattering of waves lying in the optical region. This “granularity” is coarse enough for the scattering of short (X-ray) waves to take place, since the distance between molecules is of the order of \(10^{-8}\) cm—a quantity comparable with the wavelength of X-rays. For the optical region, however, this structure may be regarded as homogeneous. Only when density fluctuations disturb the uniformity in the distribution of the centers of oscillation will light waves be diffracted by these inhomogeneities in exactly the same way as a disturbance of homogeneity at the wave front (screen, diffraction grating) causes diffraction, i.e. a deviation of the wave from its initial direction of propagation.

Relying on Smoluchowski’s idea of the role of fluctuation inhomogeneities, Einstein (11) gave a complete derivation of the intensity of scattered light from the indicated point of view. His derivation may be called thermodynamic, since it makes no explicit use of molecular notions, but uses the fact that deviations from the mean density, associated with the expenditure of work, may occur at the expense of the kinetic energy of molecules. In other words, the most probable deviations will be those for the formation of which work of the order of \(RT\) per gram-molecule is required. To obtain a general formula expressing the dependence of the intensity of scattered light on the absolute temperature, Avogadro’s number, and the constants of the medium (compressibility, dependence of the refractive index on density), these considerations are sufficient. A more detailed study of the question (the influence of molecular shape on polarization and intensity of radiation, etc.), however, requires special assumptions about the nature of the electro-

magnetic fields created within the medium by the interaction of molecules, i.e. a more detailed use of molecular representations. These molecular-kinetic supplements to Einstein’s thermodynamic derivation constitute the content of a number of later works (Cabannes, Born, Gans, Raman with collaborators, etc.).

For gases the formula obtained by Einstein coincides exactly with Rayleigh’s original formula. This circumstance is connected with the fact that, for gases, Rayleigh’s assumption of the complete incoherence of the secondary waves is mathematically equivalent to the idea of the diffraction of light by those quite randomly distributed condensations and rarefactions of the medium which constitute density fluctuations. This mathematical equivalence, however, should not make us close our eyes to the profound physical difference between the conceptions of Rayleigh and of Einstein—Smoluchowski.

§ 3. Possible Changes in the Wavelength of Scattered Light.

Thus molecular scattering of light, in contrast to fluorescence, is due to secondary waves that are mutually coherent and are generated by forced oscillations of the electrons of the medium arising under the action of the primary light wave. As a result of density fluctuations the optical homogeneity of the medium is disturbed, and part of the energy of the primary wave is carried off by these secondary waves in all directions; moreover, of course, the wavelength of the scattered waves remains unchanged.

There arises, however, the question: cannot a change of wavelength be observed in the scattering of light, and what may be the physical causes of this change?

From the standpoint of those classical ideas about the oscillations of electrons which we have used up to now, it is natural to see one of such causes in the thermal motion of atoms or molecules; we may associate the effect with the Doppler effect. However, in the case considered by us the effect

New Developments in the Question of Light Scattering

the Doppler effect acquires an extremely distinctive character, on which it is necessary to dwell.

Indeed, as was shown above, scattering to the sides is fundamentally connected with the formation of fluctuation inhomogeneities in the medium, so that scattering may be regarded as the reflection of light waves from these inhomogeneities. Therefore, in calculating the Doppler effect one should have in view the velocity of motion of these inhomogeneities, and not the molecular velocities themselves. As was shown by L. I. Mandelshtam (12), the expected change in wavelength does not depend on the form of the disturbance that has violated homogeneity, but only on the ratio of the velocity of its propagation to the velocity of light and on the angle formed by the direction of observation with the direction of the light wave.

For gaseous media the velocity of a disturbance propagating in the medium by virtue of its elasticity (the speed of sound) is of the same order of magnitude as the velocity of the thermal motion of the molecules. Therefore, for gaseous media we obtain nothing special; but for solids the matter is different. From the point of view of Debye’s theory of heat capacity (13), every thermal motion in a solid body1 may be regarded as an aggregate of elastic waves propagating with the speed of sound in all directions. From this point of view, molecular scattering of light may be regarded as the process of reflection of light waves from inhomogeneities created in the medium by the indicated elastic waves. In other words, we decompose the temperature fluctuations of the density of the medium, which are the physical cause of the molecular scattering of light, into an aggregate of elastic waves, arising at random in different places of the medium with different intensities and running off in all directions with the speed of sound. For solids this speed is considerably greater than the molecular velocities,

especially at low temperatures. The totality of such elastic waves turns, at each instant, a homogeneous solid body into a totality of spatial gratings of all possible periods. Diffraction by these gratings is precisely the molecular scattering of light. For a given wavelength of light and a chosen direction of observation (for example, at an angle \(\theta\) to the primary beam), the diffraction will be due to some one system of our spatial gratings, moving in two directly opposite directions with the velocity of sound \(v\). The application of the Doppler principle leads in this case to the conclusion that, instead of a monochromatic wave of frequency \(\nu\), two waves must be formed, whose frequencies are determined by the formula

\[ \nu\left(1 \pm 2\frac{v}{V}\sin\frac{\theta}{2}\right), \]

where \(v\) is the velocity of sound and \(V\) the velocity of light in our medium, while \(\theta\) is the angle between the primary and the scattered wave. This peculiar Doppler effect was formulated by L. I. Mandelstam in 1918. L. Brillouin (14), in a paper that appeared in 1922, also considers the question of the scattering of light by sound waves and likewise arrives, among other things, at the phenomenon indicated above.

The question of a possible change in the wavelength of the scattered light may also be approached from another side, guided by conceptions dictated by the quantum theory.

Any exchange of energy between a quantum of light undergoing scattering and the medium is, according to the quantum theory, equivalent to a change in the wavelength of the scattered light. Since the discovery of the Compton effect, which revealed the possibility of such an exchange in the scattering of X-rays, attempts have repeatedly been made to discover an analogous phenomenon in the optical region, but invariably with an unsuccessful result. It is not difficult to indicate the reason for these failures, which lies in the fundamental error made by the authors in arranging these experiments. As often happens when drawing conclusions dictated by the quantum theory, we lack

knowledge of the details of the process. Owing to this, we are not in a position to draw conclusions about the probability of one process or another, the possibility of which is predicted by quantum theory. Since the establishment by Bohr of the correspondence principle (and up to the epoch of the new quantum mechanics), this gap has been filled by investigating the question from the classical point of view and then transferring the results obtained to the corresponding quantum phenomena. In the present case, too, it will be useful to consider, from the classical point of view, those processes in which an investigation of a quantum change of wavelength was assumed. Thus Ross (15) attempted to detect the change of wavelength produced by multiple reflection from a mirror or by scattering in paraffin, which in essence is also reflection from paraffin crystallites, whose dimensions are still quite considerable in comparison with the wavelength. In Franck’s laboratory in Göttingen, works were carried out in which the matter likewise reduced to the reflection of light from a mirror. The authors proceeded from the theory of the Compton effect, but supposed that the numerical value of the expected change of wavelength should be considerably smaller, since the exchange of energy must take place between the quantum and an atom, and not an electron (as in the Compton effect), for the processes of absorption of light require an atomic mechanism. But since the mass of the atom greatly exceeds the mass of the electron, the conditions for energy exchange (according to the law of elastic impact) will be far less favorable than in the case of Compton. Therefore attention was directed toward strengthening the effect (multiple reflection—Ross), or toward employing more delicate optical methods [an absorption method for observing the change of wavelength—Rump (17) in Franck’s laboratory]. However, the fundamental error consists in the fact that light that had undergone regular reflection was observed. But regular reflection, as is known, can occur only under the condition that coherence between the reflected rays is preserved. Only under this condition do they remain capable of interference. As a result, in one direction, determined by the law of reflection, there will be observed

a bright light will appear, corresponding to an interference maximum. In all other directions, owing to interference, the light waves mutually extinguish one another. As was explained above, the condition of coherence of the secondary waves is satisfied if the centers of these secondary waves are distributed sufficiently densely, i.e., if in a volume whose linear dimensions are comparable with the wavelength there are still very many molecules. Then, even despite the intense thermal motion of our molecules, the conditions of coherence are not violated, and consequently regular reflection will take place1. In any solid mirror these conditions are satisfied, and reflection is possible. Those same light waves which, as a result of an exchange of energy with the molecules of the mirror, would undergo a change in wavelength, thereby become mutually incoherent and thus lose the ability to interfere. Because of this they will not be regularly reflected, but will be scattered in all directions and will escape the observer who is studying the reflected light2. Moreover, one must bear in mind that in the interference of waves it is their amplitudes that are added, so that the total brightness proves proportional to the square of the number of radiating centers. If, however, the emitted waves do not interfere, then their intensities are added, so that in this case the total brightness will be proportional to the number of centers \((N)\). For any appreciable \(N\), the brightness of the reflected and of the scattered light will prove to be entirely incommensurable. The possibility is not excluded, however, of observing a change in wavelength upon reflection from a mirror. It is only necessary to bear in mind that such changed waves must be sought among

waves scattered in all possible directions, and not in an intense beam of regularly reflected rays.

Such surface scattering is again due to fluctuation phenomena, by virtue of which the smooth surface of the mirror is continuously rippled by small waves of molecular origin. This ripple leads to an increase in the surface of the mirror and, consequently, will be the less intense the greater the capillary constant of the surface material. Surface scattering is especially easy to observe at the interface of two liquids near the critical mixing temperature [Mandelstam, (17)]. It also proves accessible to observation on the surface of mercury [Raman and Ramdas (18)].

Thus waves whose lengths are changed in interaction with matter and which, by virtue of this, become incoherent can be observed and studied only under conditions of molecular scattering (surface or volume), and not of regular reflection. Molecular scattering, caused by density fluctuations, sends out, as explained above, waves in all directions, notwithstanding the fact that the individual secondary waves are mutually coherent. Incoherent radiation, if it is formed in the interaction of light and a medium, will likewise propagate in all directions. And indeed, the occurrence of incoherent scattering accompanied by a change in wavelength was discovered precisely in the study of molecular scattering, and at that simultaneously in two places: by Raman and Krishnan (19, 20) in Calcutta, who were engaged in investigations of molecular scattering in liquids, and by L. I. Mandelstam and the author (21, 22) of the present article in Moscow, who were studying molecular scattering in solids.

§ 4. Change of wavelength observed in the scattering of light

The change of wavelength observed by the authors mentioned above may be regarded as the optical analogy of the Compton effect, if by the latter one understands

act of energy exchange between light quanta and material systems (atoms, electrons), far from resonance with the acting light. However, the mechanism of the phenomenon in these two cases is essentially different. In particular, in optical experiments only the energy of the light quantum plays a role. In accordance with this, the result of the observation proves to be independent of the direction of scattering, as was found in the experiments of Mandelstam and of the author, which showed that the picture remains unchanged if the light is observed at an angle of 60°, 90°, or 120° to the primary beam.

Fig. 1. Experimental arrangement.

On the contrary, in the Compton effect, as is known, the change in wavelength depends substantially on the direction of scattering.

The phenomenon itself was discovered in the study of the spectrum of molecular scattering of light. With sufficient intensity of the latter, the spectrograms revealed not only the lines of the primary source (a mercury lamp), but each of them was accompanied by a group of satellites corresponding to a changed wavelength.

The scheme of the experimental arrangement is analogous in the experiments of Raman—Krishnan and Mandelstam—Landsberg. Fig. 1 shows the apparatus used by the latter. Here \(Q\) is the light source (a quartz

lamp), \(L_1\) and \(L_2\)—lenses concentrating the light on the object under investigation, \(D_1\) and \(D_2\)—diaphragms protecting against stray lateral rays, \(L_3\)—a lens projecting the trace of the beam in the scattering body onto the slit of the spectrograph \(Sp\). The tubes \(R_1\) and \(R_2\), blackened on the inside, serve respectively as a black background and as an absorber of rays that have passed through the body. The measures listed above, protecting the spectrograph from stray rays, are important for quantitative measurements. For detecting the phenomenon itself, one need not take care over all these precautions.

Wood (23) proposed a somewhat different method of investigation, which offers considerable advantages, especially in the study of liquids and gases. According to Wood (see Figs. 2a and b), the mercury lamp is brought as close as possible to a tube placed parallel to the lamp and containing the substance under investigation. Two additional aluminum reflectors intensify the illumination. Water cooling serves as protection against heating. The flat wall on the left (Fig. 2a) serves for observation; the opposite blackened end plays the role of a black background. Blackening near the flat wall protects against the direct action of lateral light. With a sufficient width of the investigated layer of liquid and with accurate adjustment of the spectrograph, whose collimator tube is directed parallel to the axis of the tube containing the object under investigation, it is possible to shield the spectrograph from the action of accidentally reflected rays. The advantage of this

Scheme of Wood’s arrangement.

Fig. 2a. General view of the vessel with the substance under investigation.
Fig. 2b. Cross section of the arrangement of the instruments.

method in a considerably greater intensity of the incident light, and also in a considerable thickness of the layer sending the scattered light (the length of the tube). This latter circumstance applies especially to liquids and gases and enabled Wood to reduce the exposure to a few minutes. Spectral pictures investigated by such methods have now been obtained for crystals [Landsberg and Mandelstam (24), Ramakrishna-Rao (25), Wood (23), for liquids Raman and Krishnan (26), Cabannes (27), Pringsheim (28), and others] and for gases [Ramdass (30)]. Several typical spectra are shown in Figs. 3, 4, 5.

Fig. 3. Spectrogram of light scattered by quartz. 1. Comparison spectrum. 2. Spectrum of light scattered by quartz at \(t = 20^\circ\) C. 3. Spectrum of light scattered by quartz at \(t = 210^\circ\) C. \(\alpha\) — red trabants; \(\beta\) — violet trabants.

Already in the first studies it was found that the appearing trabants can be grouped into several systems. All lines of each system are characterized by the fact that the difference of the frequencies of this line (trabant) and of the corresponding principal line remains constant over the entire extent of the spectrum. For each system of trabants this difference \(\Delta\nu\) has its own value (21, 26, 27). Moreover, it was found that, in addition to satellites lying on the side of the longer wavelengths (red trabants), there are considerably weaker satellites situated symmetrically to the first on the side of the shorter wavelengths (violet trabants). Thus each system consists of the aggregate of trabants corresponding to a constant value of the difference

frequencies \(\pm \Delta \nu\) (26, 30). Table I, given below and relating to quartz, may serve to illustrate what has been said.

The constancy of \(\Delta \nu\) makes it possible to give a simple theoretical interpretation of the phenomenon. In the language of light quanta, which

Fig. 4. Spectrogram of light scattered by benzene. Above is the comparison spectrum.

is here extremely convenient, we may express the observed fact as follows. If the energy of the incident quantum is \(h\nu\), and the energy of the scattered one is \(h\nu'\), then \(h\nu - h\nu'\)

Fig. 5. Scattering spectrum in \(CCl_4\) (after Wood). \(\alpha_1, \alpha_2, \alpha_3, \alpha_4\) are the red satellites at the line \(4358\,\text{\AA}\); \(\beta_2, \beta_3, \beta_4\) are the corresponding violet satellites.

represents the energy given to the scattering substance or borrowed from it.

TABLE I.

$\Delta \nu_{\mathrm{av}} = 1.395 \cdot 10^{13}$

$\Delta \nu_{\mathrm{av}} = 1.394 \cdot 10^{13}$

Mean value $\nu_k = (1.395 \pm 0.005)\cdot 10^{13}$
corresponding wavelength $\lambda_k = 21.50\,\mu$.

Naturally the question arises whether the indicated difference $h(\nu-\nu')$ is a quantity in some way connected with the nature of the scattering substance.

§ 5. Relation of the observed phenomenon to the natural infrared frequencies of substances.

If the process under consideration is indeed to be interpreted as an exchange of energy between the incident light and the scattering substance, then $h(\nu-\nu') = h\nu_k$ is the quantum given up (or borrowed) by this substance. Consequently, $\nu_k = (\nu-\nu')$ is the frequency of some periodic processes,

...characteristic of our substance. The numerical value \((\nu-\nu')\) lies in the region of infrared frequencies. Therefore, first of all an attempt was made to compare the value \((\nu-\nu')\) with the infrared frequencies of the scattering substance (21, 26, 27).

At the present time we have extensive material, leaving no room for doubt as to the correctness of the conjecture made: the frequencies of the emitted (perceived) quanta are indeed the infrared frequencies of the scattering substances.

The following table presents a collection of several examples from the results obtained up to the present. In it, to facilitate comparison, the values of the quantity

\[ \lambda_k=\frac{c}{\nu_k}, \]

are given, where \(c\) is the speed of light and \(\nu_k\) is the change in frequency of the scattered light, and \(\lambda_i\) are the wavelengths corresponding to the infrared frequencies observed by the ordinary method. It should be noted that \(\lambda_i\) for most bodies is determined on the basis of absorption methods, so that \(\lambda_i\) corresponds to the absorption maximum. As the theory of dispersion shows, the position of the absorption maximum does not coincide quite exactly with the natural frequency, but is somewhat shifted into the region of shorter waves. For the case of crystalline bodies the necessary correction was indicated by Försterling (31); subsequently Gavelok (32) showed that this correction must be considerably smaller.

The data collected in the preceding table clearly show that it is precisely the proper infrared vibrations characteristic of the scattering substance that determine the change in wavelength observed in the experiments described. The small discrepancies between different authors lie within the limits of observational errors. The discrepancy between the quantities \(\lambda_k\), determined in the indicated manner, and \(\lambda_i\), found by the absorption method of infrared rays, is determined by two factors. First, errors in the absorption measurements of course play a role, often giving a rather blurred maximum, which does not always make it possible to determine \(\lambda_i\) with the necessary accuracy. In the case of very strong absorption, the merging of two separate close bands is possible

TABLE II.

absorptions into one. An example may be the band between \(12.5\,\mu\) and \(13.5\,\mu\), noted by Koblenz in carbon tetrachloride, which according to Wood’s data, found by the new method, corresponds to two lines: \(12.2\,\mu\) and \(13.2\,\mu\).

Moreover, the discrepancy, noted above, between the absorption maxima and the proper frequencies may also be of some significance. Therefore one may think that well-executed measurements by the scattering method can yield more reliable and direct data for determining the proper infrared frequencies of a substance than investigations by the absorption method. To this it should also be added that the measurement technique in this new procedure is considerably simpler and more universal. In infrared measurements the question of monochromatizing the light, especially that corresponding to a large wavelength, is extremely difficult. Rubens’ residual-rays method—almost the only method giving reliable results—is extremely difficult, for it permits monochromatization not of any arbitrary wavelength, but only of those for which the corresponding reflectors have been found. Moreover, for each new wavelength a new setup is required. The extreme paucity of measurements relating to the region of long waves (50 μ and more) is connected with the indicated difficulty. Conversely, by the new method all infrared frequencies are accessible to detection, and, moreover, in a single experiment. It is necessary only, in order to find the long-wave proper frequencies, to have at one’s disposal a spectral apparatus with sufficient dispersion and, consequently, if possible, intense scattered light. In connection with this last requirement, the Wood method described above is of considerable interest.

Despite the extreme novelty of the proposed method, it has already yielded some new results in the study of infrared frequencies. As is seen from Table II, with its aid many long-wave frequencies have been detected. Some of these data will, of course, be confirmed by corresponding absorption measurements. In one case, reported by Professor P. Pringsheim1, this has already occurred. Namely, special absorption measurements

infrared rays in quartz, carried out at Pringsheim’s suggestion in Rubens’s laboratory in Berlin, led to the discovery of a new, very intense band near \(38\,\mu\), to which data obtained by the scattering method had pointed. This band had until then escaped observation, because a quartz window was used in the measuring instrument, which wholly absorbed this region of the spectrum and therefore made its investigation impossible. When this window was replaced by a paraffin plate, it became possible to measure the absorption, which revealed in quartz a new long-wave band. It should be noted, however, that the line at \(38\,\mu\), found by Pringsheim in quartz, is considerably weaker than the other line at \(48\,\mu\), which was likewise previously unknown to investigators of the infrared region. Yet all attempts to detect an absorption band corresponding to \(48\,\mu\) have not led to positive results. There is no such band or, at least, it is extremely weak.

The discrepancy between the intensity of the observed satellites and the corresponding absorption bands, noted in the preceding lines, is not exceptional. As a comparison of the experimental data shows, very many of the infrared frequencies determined by the absorption method are not obtained in the scattering spectra. In this case the most intense absorption bands often cannot be detected (for example \(7.0\,\mu\) in Iceland spar). In other cases the most intense absorption maxima correspond to very weakly expressed satellites in the spectra (for example \(8.5\,\mu\) in quartz). On the other hand, a few satellites have been found that are sometimes very intense, whose connection with infrared absorption lines presents certain difficulties (for example \(9.1\,\mu\) in Iceland spar). It seems to me difficult at present to give a complete interpretation of these phenomena. The non-appearance or weak intensity of some satellites indicates the difficulty of exciting them, although our system proves capable of the corresponding natural vibrations. In some cases, however, absorp-

New in the Question of the Scattering of Light

tional maxima may correspond to overtones or combination tones caused by the presence of definite fundamental vibrations. In such a case the corresponding satellites should not appear or, more precisely, the probability of their appearance is extremely small, for this would mean that one and the same incident quantum yields two or several infrared quanta—an event whose probability is vanishingly small. It is possible that a change in the conditions of excitation may alter the state of affairs (for example, a change in the wavelength of the exciting light; see below). It is also possible, especially with respect to crystalline bodies, that excitation of vibrations characteristic of molecules occurs more readily than excitation of lattice vibrations, as might have been expected on the basis of certain theoretical considerations. This is indicated, among other things, by the failure to obtain satellites in the scattering spectrum of rock salt and fluorite (24). Being built of ions, these crystals do not possess molecular infrared frequencies. The absorption bands known for them (60 μ for rock salt and 35 μ for fluorite) characterize vibrations of the lattice as a whole. These conclusions cannot yet be regarded as final. Against them speak, perhaps, also the long-wave satellites obtained in quartz and Iceland spar, which are hardly likely to characterize vibrations of the molecules of these crystals. From the indicated point of view, the study of crystals with the same molecular frequencies (for example, the calcite group), as well as a parallel study of crystals and the corresponding solutions, is of considerable interest.

The appearance of satellites not corresponding directly to infrared frequencies also cannot at present be explained unambiguously. This includes, first of all, the satellite corresponding to \(\lambda = 9.1\,\mu\), found in Iceland spar. True, it may be brought into correspondence with one of the rather numerous infrared frequencies observed in Iceland spar, although not with a very intense one (for example \(\lambda = 8.8\,\mu\)) (36), especially if the Fersterling correction is taken into account. On the other hand,

on the other hand, $\lambda = 9.1\,\mu$ agrees well with one of the fundamental frequencies of the $\mathrm{CO}_3$ group, which can be calculated from Born’s theory as an optically inactive frequency.

This was pointed out by Kornfeld (37), who carried out such calculations for the calcite group (39). The existence of such optically inactive frequencies is also supported by experimental considerations. Numerous measurements of infrared absorption bands in various calcites, performed by C. Schaefer (40) and his collaborators, led him to the conclusion that the entire set of these bands can be represented as a system of combination vibrations, constructed from four fundamental vibrations characteristic of the $\mathrm{CO}_3$ group and capable of being precalculated on the basis of Born’s theory. Three of these fundamental vibrations correspond to the three actually observed absorption maxima; the fourth must be chosen so that, with its aid, all the experimental results can be represented. For all the calcites investigated (dolomite, cerussite, witherite, magnesite, siderite, and Iceland spar) this assumed fundamental vibration should correspond to a wavelength of $9.2\,\mu$, i.e. to a value very close to that calculated by Kornfeld (about $8\,\mu$) and almost exactly coinciding with the measurements by the new method.

The optical inactivity of one of the frequencies is explained by the fact that the vibrations corresponding to it are not connected with a change in the electric moment of the molecule. Therefore such a frequency does not reveal itself in absorption measurements, since the corresponding vibration is not excited by the alternating field of light. It may be assumed, however, that if the molecule has been set by the action of light into vibration with another, active frequency, then an uninterrupted vibration can already be excited by mechanical means; as a result, combination vibrations arise, which are detected by absorption methods. Something similar may perhaps also occur in the scattering of light. The action produced by quanta whose energy is considerably greater than that necessary for the excitation of the proper vibrations may cause significant perturbations in the molecule. As a secondary effect

In this case the appearance of optically inactive, but molecularly characteristic, fundamental vibrations is possible. Of course, the indicated explanation cannot be regarded as entirely satisfactory, all the more so since, in the case of Iceland spar, the satellite corresponding to this optically inactive frequency proves to be especially intense. Nevertheless, relying on these results obtained for Iceland spar, Pringsheim makes an attempt to explain the appearance of satellites corresponding to \(\lambda=9.6\,\mu\), discovered in nitric-acid compounds. Pringsheim (33) assumes that the group \(\mathrm{NO}_3\), in addition to the three proper frequencies known from absorption experiments, also has a fourth, optically inactive one, the value of which can be interpolated by analogy with \(\mathrm{CO}_3\). Table III shows that the value \(\lambda=9.6\,\mu\), assigned to this inactive frequency, agrees rather well with the data for \(\mathrm{CO}_3\), justified, as we have seen, quite well.

TABLE III.

“Fundamental” frequencies corresponding to infrared bands
(\(\lambda\) in microns).

On the basis of the still rather small amount of material available concerning the infrared frequencies of molecules, collected by the new method, it would be difficult to make any generalizations. The study of a group of organic liquids has, it is true, led to the conclusion that a definite frequency, corresponding to \(\lambda=3.27\,\mu\), characterizes the \(\mathrm{C—H}\) bond (28), as, incidentally, follows also from absorption data. But one may hope that the new method will make it possible considerably to expand the amount of necessary factual material, since it represe...

has a number of advantages over the previous methods. Among these are considerable simplicity and universality, the possibility of obtaining more accurate data, and moreover by a more direct method; finally, one may think that the frequencies obtained in this way really characterize the proper frequencies of the substance, and do not correspond to any overtones or combination tones of the fundamental proper vibrations.

§ 6. Violet (anti-Stokes) satellites and their significance for the theory of the phenomenon.

As the law illustrated by Table I shows, the frequency of the scattered light $\nu'$ is obtained from the frequency of the incident light by means of the relation

\[ \nu'=\nu\pm\nu_k, \]

where $\nu_k$ is the proper frequency characterizing the scattering system. Thus the observed frequency is obtained as the frequency of a combination tone arising from the interaction of the proper vibration of the system and a vibration arriving from outside.

From the indicated point of view, it is natural to call this new type of scattering, in distinction from the classical one, combination scattering (Kombinationsstreuung)—a term that we shall use hereafter.

Experiment (see Fig. 3 and Fig. 5) shows, however, that with complete symmetry of the violet and red satellites in the sense of their position relative to the principal line, their intensities differ considerably from one another, and the more so the farther they are from the principal line, i.e. the larger the $\nu_k$ to which they correspond. This difference in intensity cannot be explained from the point of view of classical conceptions and will require the introduction of the quantum point of view. Indeed, the appearance of red and violet satellites can be classically interpreted in two ways.

Let us imagine our molecule in the form of an electric dipole rotating with a constant angular velocity

$\omega_k = 2\pi\nu_k$ about the axis $y$, perpendicular to the direction of its moment. Let a plane polarized wave of frequency $\nu$ be incident on our dipole, propagating along the axis $x$, so that the electric vector is directed along the axis $z$. This wave $z = A\cos 2\pi\nu t$ induces in our dipole an electric moment proportional to $A\cdot \cos 2\pi\nu t\cdot \cos\varphi$, where $\varphi = \omega_k\cdot t = 2\pi\nu_k\cdot t$ is the angle of the dipole with the electric field of the wave (assuming that at the initial moment the dipole is directed along $z$). In this case our dipole will become a source of radiation determined by the formula:

\[ I = k\cdot A\cos 2\pi\nu t \cos 2\pi\nu_k t = \frac{kA}{2}\cos 2\pi(\nu+\nu_k)t+ \]

\[ +\frac{kA}{2}\cos 2\pi(\nu-\nu_k)\cdot t. \]

In other words, the radiated wave will indeed contain waves with frequencies $\nu \pm \nu_k$, but the amplitudes and intensities of both shifted waves must prove to be equal.

Another classical interpretation of the change in the wavelength of the scattered light may be obtained from the following consideration.

Let a sinusoidal wave $A\sin 2\pi\nu t$ be incident on a system performing its own oscillations with frequency $\nu_k$. In the case of large amplitudes, the oscillations cease to obey a simple sinusoidal law, for the equation of oscillations will no longer be linear. In the superposition of such oscillations there arise, as is known, combination tones, whose frequencies are $\nu+\nu_k$ and $\nu-\nu_k$. These tones are analogous to a certain degree to the tones known in acoustics under the name of summation and difference tones, and obtained when two waves with periods $\nu$ and $\nu_k$ act upon one and the same resonator. As in acoustics, the necessary condition for their formation is the nonlinearity of the equations. It may take place in the experiments described, since even at ordinary temperature the intensity of infrared oscillations may have quite large amplitudes. Thus the combination tones could explain the existence of red and violet

trabants symmetric with respect to the principal line. However, even in this interpretation the question of intensity is not resolved satisfactorily. The relative intensity of the two trabants should be the same. Experiment shows, however, that the violet trabants are considerably less intense than the red ones, and can be observed only for the brightest lines.

Thus the classical interpretation of the red and violet trabants does not lead to a satisfactory result.

It is therefore natural to turn to a quantum interpretation of the observed phenomenon. In terms of the hypothesis of light quanta, the appearance of scattered light of lower frequency means that part of the energy of the incident quantum \((h\nu - h\nu')\) is given up in scattering to the substance in the form of a quantum \(h\nu_k\) characteristic of it. On the other hand, the reverse process is also possible, when a quantum is added to the incident quantum \(h\nu\), leading to the formation of a violet trabant with frequency \(\nu+\nu_k\). These simple considerations were advanced in their time by Smekal (41), who from them drew the conclusion that modified components can appear in the scattered light. Smekal’s arguments referred to isolated atoms, but they can obviously also be retained for our systems. In the case of crystal lattices, the frequency \(\nu_k\) may represent the frequency of the natural vibrations of atomic complexes, or it may characterize the vibrations of the lattices themselves.

It is not difficult to show that, from the quantum point of view, the question of the relative intensity of the red and violet trabants does not lead to a contradiction with experiment. We shall consider it while adhering to the hypothesis of light quanta. The same results can be reached without resorting to such an extreme modification of quantum ideas as the hypothesis of light quanta. Heisenberg and Kramers (42) gave a quantum theory of the scattering of radiation by atoms, adjoining Kramers’s quantum theory of dispersion, i.e. based entirely on the correspondence principle. They arrive, just as Smekal did, at the conclusion that

scattered radiation, along with the original frequency, will contain waves corresponding to the frequencies $\nu+\nu_a$ or $\nu-\nu_e$, if $\nu_a$ and $\nu_e$ are the frequencies corresponding to the emission and absorption of the atom under consideration. Their method is essentially a translation into the language of quanta of conclusions and formulas obtained from classical notions. In other words, it is based on the assumption of nonlinearity in the equations of oscillatory processes, leading to the formation of combination tones. However, on the question of the relative intensity of the sum and difference tones, quantum considerations give a different result than classical theory. The intensities of classical radiation are defined in quantum theory as the probabilities of transitions from one stationary state to another and, consequently, depend on the excited stationary states.

From the point of view of Schrödinger’s wave mechanics, the explanation of the difference in intensity between the red and violet companions again encounters certain difficulties. According to Schrödinger, the appearance of one or another line of the type $\nu \mp \nu_k$ is possible only in the presence of molecules, both in the first and in the second states $A$ and $B$, the transition between which determines the frequency $\nu_k$. The intensity of the corresponding line should be determined by the product of the concentrations of molecules in the one and the other state, i.e., by the same product $N_A \cdot N_B$, whether we have in mind the transition from $A$ to $B$ or the reverse, i.e., whether we obtain the combination line $\nu+\nu_k$ or $\nu-\nu_k$ (49). Born (59), however, noted that the indicated initial conception of Schrödinger is not the only possible one. Moreover, on the basis of a number of other considerations Born defends the statistical interpretation of the new quantum mechanics. From the standpoint of this interpretation, the question of the relative intensity of the red and violet companions is resolved just as satisfactorily as from the standpoint of light quanta.

The following elementary reasoning can give, although imprecisely, an idea of the expected intensity of the red and violet companions. The shifted frequencies of the type $\nu \mp \nu_k$ are obtained in the scattering of light quanta,

accompanied by an exchange of energy with the scattering centers. Therefore, the intensity of the red and violet satellites may, other conditions being equal, be regarded as proportional to the number of centers capable of accepting or giving up a quantum \(h\nu_k\). To the former belong all unexcited centers. In addition, some of the excited centers will prove capable of accepting an additional portion of energy \(h\nu_k\), i.e. also of giving rise to red satellites. The other part of the excited centers will give up its energy to the radiation in the form of a portion \(h\nu_k\), passing to a lower degree of excitation and thus producing violet satellites.

Of the two periodic processes that can occur in molecules—rotation and vibration—the first is of no significance, since the rotational frequencies are too small to explain the observed effects: they could give only a broadening of the lines comparable, however, with the thermal broadening of the lines. The vibrations of molecular groups may, to a first approximation, be considered harmonic, so that excitation levels are possible corresponding to an energy difference \(h\nu_k\).1 According to Boltzmann’s formula, the numbers of centers in the indicated states will be

\[ N_1 = N_0 \cdot e^{-\frac{h\nu}{kT}};\quad N_2 = N_0 \cdot e^{-\frac{2h\nu}{kT}};\quad N_3 = N_0 \cdot e^{-\frac{3h\nu}{kT}}. \]

The processes leading to the formation of the red and violet satellites are nothing other than the positive and negative radiation (positive und negative Einstrahlung) of Einstein’s well-known derivation (42). We shall not make a large error if we assume that the probabilities of one or the other transition are equal to each other (the equality of the coefficients \(B_m^n\) and \(B_n^m\) in Einstein’s formulas). Thus the intensity of the red satellites will prove proportional to the number of all unexcited centers plus one-half of all excited—

...and the intensity of the violet ones is proportional to one half of the excited ones, i.e.,

\[ J_r=K\left\{N_0+\frac{1}{2}N_1+\frac{1}{2}N_2+\ldots\right\} =K\cdot N_0\left\{1+\frac{1}{2}e^{-\frac{h\nu}{kT}}+\right. \]

\[ \left.+\frac{1}{2}e^{-\frac{2h\nu}{kT}}+\ldots\right\} =K\cdot N_0\left\{1+\frac{1}{2}\frac{e^{-\frac{h\nu}{kT}}}{1-e^{-\frac{h\nu}{kT}}}\right\} \]

\[ J_v=K\left\{\frac{1}{2}N_1+\frac{1}{2}N_2+\ldots\right\} =K\cdot N_0\left\{\frac{1}{2}e^{-\frac{h\nu}{kT}}+\frac{1}{2}e^{-\frac{2h\nu}{kT}}+\ldots\right\} = \]

\[ =KN_0\left\{\frac{1}{2}\cdot \frac{e^{-\frac{h\nu}{kT}}}{1-e^{-\frac{h\nu}{kT}}}\right\} \]

Whence the required ratio of intensities is

\[ \frac{J_v}{J_r} = \frac{\frac{1}{2}e^{-\frac{h\nu}{kT}}} {1-\frac{1}{2}e^{-\frac{h\nu}{kT}}} = \frac{1}{2e^{\frac{h\nu}{kT}}-1}. \]

The formula obtained shows that the intensity of the violet and red satellites may differ considerably from one another. It goes without saying, of course, that our formula, like every formula based on the theory of quanta, leads in the limiting case to conclusions coinciding with the classical ones. Indeed, for small \(\nu\) or large \(T\) we have \(J_v=J_r\), as could also be expected according to the classical theory. The interest of the phenomenon under study lies in the fact that for the \(\nu\) with which we have to deal in experiment, already at ordinary temperature \(T\), \(h\nu\) is close to \(kT\), i.e., the intensity of the violet satellites becomes quite appreciable. As \(\nu\) decreases, the intensity of the violet satellites must increase rapidly. Experiment indeed confirms this conclusion. An excellent photograph, borrowed from

Budd’s work (23), reproduced in Fig. 5, may serve as a clear illustration of what has been said.

It follows further from our formula that the intensity of the violet Trabants depends strongly on temperature. Thus, for Trabants corresponding to $\lambda_k = 21.5\,\mu$ (quartz), the intensity of the violet Trabants at ordinary temperature ($T = 300^\circ$) is about $5\%$ of the intensity of the red ones. When the temperature is raised to $T = 500^\circ$, the intensity of the violet Trabant should increase threefold, reaching $15\%$ of the intensity of the red one. Strictly speaking, the intensity of the red Trabants should decrease somewhat with increasing temperature, since the number of unexcited centers decreases and excited centers appear at their expense. However, this decrease is negligibly small in comparison with the unexcited centers present. Thus one may expect that, as the temperature is raised, the intensity of the red Trabants will remain unchanged, whereas the intensity of the violet ones will increase considerably.

In carrying out the corresponding experiment, however, the following must be borne in mind. Measurement of such weak intensities is possible only by the method of photographic photometry. In order to eliminate the possible influence of fluctuations in the intensity of the light source, it is best to measure the ratio of the intensities of the two Trabants to each other and to the principal line. However, the intensity of the principal line does not remain unchanged when the temperature is raised. Indeed, the intensity of the unshifted lines depends on the intensity of the fluctuations of the medium, which increases rapidly with temperature. As the Einstein–Rayleigh formula shows, the intensity of classical scattering is proportional to the absolute temperature. This dependence was in fact found experimentally in measurements with quartz made by the author (44). The intensity of the Trabants, however, should not depend on fluctuations, since the change in wavelength that characterizes them leads to incoherence of this combination scattering. Therefore it can propagate in all directions,

even if the medium is optically homogeneous, and its intensity is not connected with the degree of violation of the homogeneity of the medium, with the intensity of the fluctuations.

Thus the complete picture of the dependence of the spectrum of scattered light on temperature must be as follows. The intensity of the principal lines increases in proportion to the absolute temperature, the intensity of the red satellites remains unchanged, and the intensity of the violet satellites grows considerably faster than the intensity of the principal lines.

All these conclusions, predicted by the theory, proved possible to verify and confirm experimentally. In the experiments of L. I. Mandelstam, M. A. Leontovich, and the author (24) (45), the light scattered by a quartz crystal at temperatures of 20°C and 210°C was photographed on one and the same plate, onto which spectral intensity marks were also applied. Thus the development conditions for all the photographs were absolutely identical. The exposure times for hot and cold quartz were chosen to be the same (105 hours). The operating regime of the lamp was monitored. The unavoidable fluctuations of brightness over the course of 5 days were distributed more or less uniformly over both exposures. By photometering the photographs with a microphotometer, one can compare with one another the intensities of the corresponding lines. To determine the influence of temperature on the relative intensity of the violet and red satellites, there is of course no need to be concerned about the identity of the conditions of the photographs of hot and cold quartz. But under the conditions described it becomes possible to verify all the conclusions given above. The results of the measurements are quite satisfactory. The intensity of the red satellites remains unchanged. The ratio of the intensities of the principal lines lies within the limits from 1.40 to 1.87 (for different wavelengths), with an average value of 1.61, while the ratio of the absolute temperatures is 1.65. The intensity of the violet satellites increases so sharply that this increase can be noticed even in the reproduction (see Fig. 3 b and c). Quantitative measurement of this increase is difficult, since, because of the weakness of the violet satellites, blackening ...

...blackening of the plate corresponding to them, even with a 100-hour exposure, is far below the region of normal blackening of the plate, so that the connection between blackening and light intensity cannot be reliably established. There is no doubt, however, that their intensity increases according to a law much more rapid than a linear one, in agreement with the formula derived above.

Recently Krishnan (45) determined the influence of temperature on the intensity of the violet satellites in carbon tetrachloride, where they are very well expressed (cf. Fig. 5). The increase in the temperature of carbon tetrachloride cannot be made considerable (Krishnan’s experiment was carried out at temperatures of \(34^\circ\mathrm{C}\) and \(81^\circ\mathrm{C}\)). Accordingly, the increase in the intensity of the violet satellites cannot be considerable. Krishnan measured the spectrograms obtained by him with a self-recording microphotometer and considers, on the basis of these curves, that a certain increase in the intensity of the violet satellites has been proved.

§ 7. Various questions connected with combination scattering.

Experiments on the influence of temperature on the intensity of combination scattering make it possible to give some experimental confirmation of the repeatedly expressed idea of the incoherence of the secondary waves giving rise to combination scattering. Indeed, the independence of the intensity of the red satellites from temperature shows that the light corresponding to them is propagated in all directions with unchanged intensity, irrespective of whether the fluctuation disturbances of the homogeneity of the medium are large or small. On the basis of what was set forth at the beginning of the article, this means, however, that the radiation corresponding to our satellites represents an aggregate of waves mutually incoherent with one another, in agreement with our conception of the nature of their origin. A well-known confirmation of the incoherence of combination scattering may also be considered the experiment of Bogro and Rocard (47)

Recent Developments in the Question of Light Scattering

and Martin (48), who showed that in a mixture of water and phenol at the critical temperature of dissolution, when the classical (coherent) scattering increases greatly (opalescence), no increase in the intensity of the combination lines can be observed. In the experiments described, the principal lines appeared on the spectrogram in a few minutes, whereas the trabants could not be detected even with a three-hour exposure. The conclusion that combination scattering is incoherent is contradicted by an observation described by Raman in one of his first communications (20). According to Raman, the intensity of combination scattering increases parallel to the intensity of the classical scattering observed in CO₂, when the formation of a cloud is produced in it by sudden expansion. This observation, however, seems erroneous (49) and has not yet been confirmed in other work. It may be, nevertheless, that the conclusion as to the incoherent character of combination scattering is still somewhat premature. The observed facts say only that the fluctuations of density and concentration which determine the intensity of classical scattering are not directly related to combination scattering.

On the contrary, there are a number of indications that the intensity of combination scattering is determined, for a given substance, by its density, i.e. by the number of molecules per unit volume, and not by their random distribution. Thus Ramdas (29) studied scattering in liquid and vaporous ether and came to the conclusion that the intensity of the observed trabant is about 300 times weaker for the vapor phase than for the liquid, whereas the ratio of the densities of the liquid and the vapor is 250. The estimate of the intensity was made, approximately, from the duration of exposures capable of giving comparable blackening, and therefore the agreement attained may be considered quite satisfactory. To the same conclusion—that the intensity of combination scattering is proportional to the density—Dore (50) came on the basis, it is true, of a very rough estimate of the intensities.

The question of the intensity of the lines of combination scattering is of considerable interest in itself, and investi-

its investigation may shed light on the mechanism of formation of combination lines. As was already mentioned above, at the present time we are not in a position to answer why the intensities of some satellites prove to be considerably greater than the intensities of others, whereas in the absorption intensities of the corresponding infrared vibrations inverse ratios may be observed.

According to the approximate estimate of most authors who have worked with liquids, the intensity of the brightest satellites amounts to \(1—2\%\) of the intensity of the principal line (23, 27, 36, 50). For a quartz crystal, the measurements made by M. A. Leontovich and the author (45) give a considerably larger value. These measurements, carried out by the method of photographic photometry, gave for the brightest satellite in quartz a value of about \(40\%\). The result obtained does not contradict the observations on liquids cited above. Indeed, since the intensity of combination scattering depends on density, it must be of the same order for liquids and crystals. The intensity of the principal lines, however, caused by fluctuations, is hundreds of times greater for liquids than for crystalline bodies. The value obtained for the intensity of the red satellite in quartz sheds light on one circumstance noted by the author in studying the dependence of the intensity of light scattered by quartz on temperature (44). It was observed that only about \(3/4\) of the scattered light increases linearly with temperature, while \(1/4\) remains constant, and it was supposed that this fraction is due to accidental inhomogeneities of the crystal, and not to molecular fluctuations. The present investigations show, however, that the effect of combination scattering, unknown at that time, constitutes a noticeable fraction of the total effect. If the other satellites are also taken into account, and if one also allows for the decrease of their relative intensity toward the visible part of the light, then the temperature-independent part should amount to about \(40\%\) of the varying part, or about \(28\%\) of the total intensity, which agrees very well with the above-cited observation. The noted

The above-mentioned certain decrease in the relative intensity of the trabants as the wavelength of the principal line increases gives grounds for investigating this phenomenon over a wider spectral interval. It is possible that investigations of this kind will shed some light on the mechanism of the phenomenon itself. Making use of the excellent analogy given by Pringsheim (49), the phenomenon of combination scattering may be likened to an inelastic collision of electrons of the first and second kind with atoms, a collision accompanied by an exchange of energy between the colliding systems. Classical scattering, on the other hand, corresponds (in the first approximation) to an elastic collision, as a result of which only the direction of flight of the bombarding electrons changes, without a change in energy. It is known, moreover, that collisions of an inelastic character are possible only on condition that the energy of the electrons reaches a certain critical value characteristic of the atoms under investigation; and the probability of this process is a certain function of the electron energy. This excitation function has not yet been sufficiently investigated. The question of a similar function as applied to the phenomena that interest us is likewise next in line.

Another path toward the study of the mechanism of excitation of these or those vibrations is afforded by the study of the polarization of the shifted and unshifted lines of the scattered light. As is known, the simple theory of Rayleigh predicts that the scattered light must be completely polarized in the plane passing through the primary and secondary rays. Let us, for definiteness, take this plane to be horizontal. The forced vibrations in the scattering medium must be directed in the same way as the vibrations in the exciting light. Even if this light is natural, nevertheless all its vibrations are situated in the vertical plane, perpendicular to the primary ray. If the primary beam is parallel, and the direction of observation makes a right angle with it, then, by virtue of the transverse character of light rays, in the direction of observation there will propagate only waves corresponding to vertical vibrations of the electric

of the vector, i.e., polarized toward the horizontal plane. This conclusion was well confirmed in experiments on classical scattering. It is true that it subsequently became clear that, for certain substances, the polarization does not reach 100%. The reason for this must be sought in the anisotropy of the molecules composing the scattering medium. In the preceding arguments it was tacitly assumed that the scattering molecule is isotropic, and that in it an electric moment of any direction, coinciding with the electric vector of the exciting field, is excited with equal intensity. Since all the electric vectors of the exciting field lie in a plane perpendicular to the primary beam, the induced electric moments will also be situated in the same plane; i.e., in a direction perpendicular to the primary beam, rectilinearly polarized light will propagate. In the case of anisotropic molecules, however, the induced electric moments will be different for different directions in the molecule and, consequently, their direction must depend on the orientation of the molecule. Thus the induced electric moments will be situated not only in the plane perpendicular to the primary beam. In accordance with this, the secondary scattered light will prove to be only partially polarized. Such an explanation was given by Rayleigh (51) for the fact of incomplete polarization of light scattered by gases, first observed by Strutt (52). The phenomenon of partial depolarization was subsequently also observed in liquids, where it can attain very considerable magnitudes. In numerous works by Cabannes, Raman, Gans, and others, a method was developed for estimating the anisotropy of molecules on the basis of the value of the depolarization factor.

Nevertheless, scattered light is always polarized to a more or less considerable degree. A natural question arises: to what extent does this conclusion apply to the new type of scattering. Raman and Krishnan, in their first publications (20, 21), noted the strong polarization of the new radiation and saw in this proof that the observed phenomenon is distinct from fluorescence.

This proof is not convincing, for, as noted above, the scattered light can be only partially polarized (up to 50% and below); on the other hand, the phenomenon of polarization of fluorescence not only in gases but also in liquids has been discovered and measured in numerous recent works (53, 54, 55). From the point of view of the mechanism of formation of combination scattering that we have set forth, complete polarization of this scattering by no means appears inevitable. The process of energy exchange between light and molecules is naturally to be likened to a new act of emission, so that the degree of polarization of this light must depend significantly on the properties of the excited (emitting) molecule. It would not be at all unexpected if different lines of combination scattering possessed different degrees of polarization. This would mean that, for anisotropic molecules, forced oscillations in one direction more readily excite some proper infrared oscillations, while with another direction of the forced oscillations other infrared frequencies are apt to be excited. Indeed, further observations have shown that the degree of polarization of different combination lines is different and may be greater or less than the degree of polarization of the principal scattered lines. But the combination lines corresponding to one and the same infrared oscillation (i.e., to one and the same change in the incident frequency) turn out to be polarized identically, no matter for which principal line of the incident light the observations are made. Thus, Cabannes (56) found for benzene that, while the principal lines are polarized by 40%, the satellites corresponding to \(\Delta \nu = 2.98 \cdot 10^{13}\) are polarized almost completely (90%), whereas the satellites corresponding to \(\Delta \nu = 9.2 \cdot 10^{13}\) prove to be polarized only by 25%. The same observations were made in the subsequent works of Raman and Krishnan (57) for benzene and amyl alcohol, and of Pringsheim with collaborators (33) for benzene, toluene, carbon tetrachloride, and a solution of nitric acid. Here it also became clear that different degrees of polarization (from 0% to 90%) correspond-

correspond to different satellites (different $\Delta \nu$). But the degree of polarization of some definite satellite (a given $\Delta \nu$) is one and the same, whichever of the principal lines is taken as the object of study. Moreover, for CCl$_4$, which is distinguished by extremely bright red and violet satellites, the polarization of both was estimated and proved to be identical, as can be seen from Table IV.

TABLE IV.

In considering this table, compiled on the basis of data of Pringsheim and co-workers (33), it must be borne in mind that the error in estimating the polarization, especially of comparatively weak violet satellites, may be very considerable, so that the figures given should be regarded as indicative.

The measurement of the polarization of the principal lines and satellites in crystals would be of special interest in view of the regularity in the arrangement and orientation of the molecules.

The material obtained so far is not sufficiently extensive to permit any conclusions. It is not excluded, however, that the difference in the degree of polarization of the various satellites will help to clarify in greater detail the mechanism of their appearance. Thus Pringsheim notes that the satellite corresponding to that $\Delta \nu$ which corresponds, in a series of organic compounds, to the C—H bond, also exhibits the same character of polarization in different compounds.

In general, it would hardly be erroneous to assert that in this new field there are far more questions posed than questions resolved. True, the fundamental phenomenon, its physical meaning and interpretation, do not arouse doubt. But there are still exceedingly numerous, varied controversial and unclear points, the resolution of which will be possible only as further factual material is accumulated. As I have tried to show on the preceding pages, besides the study of various materials that may enrich our information about the proper infrared frequencies of molecules and, perhaps, of crystal lattices, there remains a series of questions concerning the conditions for exciting these infrared vibrations, the elucidation of the facts determining their intensity, the investigation of the polarization of combination lines, etc.

§ 8. Conclusion.

It seems appropriate to me to conclude the present article with a discussion of the very same question with which I began it.

From the point of view of the new type of combination scattering described here, the distinction between fluorescence and scattering ceases to be so sharply delineated. We see that scattering of light is possible in which the forced vibrations of the system combine with its proper vibrations, and, consequently, the opposition between proper vibrations and forced ones loses its meaning. The phenomenon of combination scattering may be regarded as additional fluorescence. In ordinary fluorescence, what is readily observed is the proper radiation to which the molecule is excited. The remainder of the energy (“Stokes shift”) is not studied directly, and one can only make more or less plausible assumptions about its fate. In the phenomenon of combination scattering, on the contrary, it is precisely this remainder that is the object of observation. The energy of the proper vibrations, however, is not directly accessible to observation, and only from numerical data can one have no doubt that the excited proper emis-

there is infrared radiation. It is not excluded, of course, that such objects may be selected for which both the excited proper luminescence and the “residue” will prove to lie in a region convenient for observation. What is essentially new, however, is the fact of the excitation of intense infrared oscillations by means of light whose frequency lies in a far region of the spectrum (the ultraviolet). The observed exchange of energy between light and matter does not at all fit within the framework of the usual classical notions of resonance. Here we have processes very close, if not identical, with those phenomena of positive and negative absorption which Einstein (60) postulated in his well-known derivation of the formula for black radiation. From the indicated point of view, the phenomenon described is one more, and not unimportant, argument in favor of the quantum character of light.^1

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^1 Note in proof. During the time elapsed since the writing of the article (December 1928), about two dozen new works devoted to combination scattering have appeared. They can no longer find reflection either in the text or in the appended list of literature. Some of these works are very interesting, but they add nothing especially essential that would require alteration of the text.

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1. The index \(k\) is hereafter omitted for simplicity of notation. ↩↩↩↩↩

2. In connection with this it will not be superfluous to note that the Compton phenomenon should not prevent exact X-ray spectroscopic measurements of wavelength, as might seem at first glance. Indeed, in X-ray spectroscopy one observes X-rays regularly reflected from the planes of a crystal, i.e., those that certainly have not undergone a change in wavelength. ↩