Molecular Structure of Liquids*

P. Debye, Leipzig

For a long time the question of the structure of liquids was regarded as solved by Van der Waals’s theory of the continuity of the gaseous and liquid states. According to this theory there is no fundamental difference between the two aggregate states, and we considered a liquid to be nothing other than a strongly condensed gas; however, it is maintained in the liquid state not only by external pressure, but also by molecular forces of attraction, which give rise to the liquid state and automatically create an additional pressure of the order of many hundreds of atmospheres. Even at the present time there is no reason to doubt the correctness of this theory; nevertheless, it should be pointed out that within the framework of the theory there remains room for many questions whose purpose is precisely to clarify the differences between the molecular states of gas and liquid. In this sense, attention must first of all be directed to the state of motion of the molecules. Further, one may ask whether the high density of liquids causes some tendency of the molecules toward spatial ordering, to some extent analogous to the crystalline state of solids. In the present article we shall consider a number of experiments that make it possible to illuminate both these questions.

As Raman’s experiments have shown, when molecules are illuminated with monochromatic light, the incident energy quantum \(h\nu\) of the primary light may change in two ways: either part of its energy is transferred to the molecule, which enters into vibrational or rotational motion, or else the energy of the scattered quantum increases, with part of the energy being taken from the molecule. We shall not concern ourselves here with these changes of frequency. In many cases the primary energy quantum will be scattered without any change at all. In such cases one speaks of ordinary scattering of light, called Rayleigh scattering in contrast to Raman scattering, since Rayleigh first considered this process in studying the causes producing the blue color of the sky. Let us now ask whether the assertion that in Rayleigh scattering the wavelength remains unchanged is strictly valid. Let us first consider this

* Chapter from the book by P. Debye, Struktur der Materie, Leipzig 1933, translated by N. N. Malov.

phenomenon for a gas. If one takes into account that gas molecules move with velocities whose order coincides with the speed of sound—several hundred meters per second—then one should expect that, owing to the motion of the molecules, a Doppler effect must arise, as a result of which even in Rayleigh scattering there must exist a change in wavelength. For example, if one looks at a molecule in the direction of the incident light, and the molecule itself is moving away from the observer, then the scattered light arriving at the observer must have an altered wavelength for the following two reasons. Since the molecule is moving away from the light source, it will be excited by a frequency smaller than the frequency of the primary light perceived by the observer. This reduced frequency, scattered by the molecule, will appear to the observer still further reduced, since the molecule is moving away from him. If, however, the observer looks in the opposite direction, i.e. in the direction of the light source, then the two frequency changes will have opposite signs, so that the frequency he perceives will not differ from the original one. If one recalls that in a gas there exist all possible velocities and directions, then it becomes clear, first of all, that the primary strictly monochromatic ray will be perceived as a spectral line of finite width; it is further clear that the magnitude of the broadening will depend on the angle made by the direction of observation with the direction of the primary ray. If one observes at an angle to the direction of the primary ray, then a broadening of the spectral line must always be observed; moreover, the distribution of intensity within it will be characterized by the distribution of velocities of the gas molecules, and the line width will be the greater the closer the directions of observation and of the primary ray coincide.

So far as I know, this effect has not yet been investigated. One may think, however, that at the present time, when the distribution of velocities in a gas can be studied by a whole series of other methods, such experiments would not be of special interest.

Quite a different state of affairs exists in the question of the change of primary light when it is scattered by a liquid. In this case we are not so well acquainted with the state of motion of the molecules as in the case of a gas. In order to judge whether characteristic differences from gases can be expected here at all, it is useful to examine the question of the scattering of light by solids. This scattering is the second well-known limiting case, and one may suppose that the solid state has much in common with the liquid, just as the liquid and gaseous states continuously pass into one another. Let us consider a definite case, for example, the reflection of light from a crystal of rock salt, which is known to be constructed of chlorine and sodium atoms. The simplest conception of the motion of these atoms was given by Einstein in deriving the formula for specific heats. According to Einstein’s conception,

stein, each atom oscillates about its equilibrium position independently of its neighbors. If this picture corresponds to reality, then it is easy to show that at sufficiently high temperatures, when the influence of quantization becomes negligible, the same broadening of the primary beam should be obtained as in the case of a gas. We know, however, that this picture of the motion of molecules is too simplified. In reality each atom is bound to its neighbors, so that its motion depends on the motion of the surrounding atoms. To express this dependence, the thermal motion of an atom may be regarded as the result of the interference of a large number of sound waves propagating in the body in all possible directions and having various frequencies, filling the entire spectrum. At one time I used this interference in order to prove that Einstein’s formula for specific heats must be replaced by another, giving—especially at low temperatures—a considerably slower decrease of the specific heat with temperature (proportionality to the third power of the absolute temperature). The question naturally arises whether the interaction of the motions of neighboring atoms will in any way affect the scattering of light. Brillouin dealt with this question over a number of years and resolved it completely. He assumes, as Einstein had done earlier, that the scattering of light is caused by thermal fluctuations of density and by the directly related changes in the refractive index; at the same time he takes into account the connection between fluctuations in neighboring regions, regarding thermal motion as the result of the interference of sound waves. He obtains a result substantially different from the scattering of light by a gas, which may be formulated as follows.

If one considers the primary beam and a scattered beam of a definite direction, then first of all it may be asserted that the energy scattered in each direction will be determined only by those of all possible thermal sound waves* penetrating the body whose front is a mirror reflecting the beam according to the ordinary optical laws (equality of the angles of incidence and reflection). However, there exists a very large number of waves of different length possessing such a front. Brillouin further showed that the scattering will be determined not by all such waves, but only by waves of a quite definite length. Rays reflected from planes separated from one another by one wavelength must coincide in phase, and only they can create an appreciable energy of scattered light. This condition is analogous to the well-known Bragg condition for the reflection of X-rays from the atomic planes of a crystal. Instead of the length of the X-ray—

* The term “thermal sound waves” refers to fictitious sound waves determining the thermal motion of atoms.

novsk wave here is the wavelength of light, and the distance between atomic planes is replaced by the wavelength of a sound wave. From Brillouin’s point of view, the intensity of scattering observed in a definite direction makes it possible to judge the intensity of the thermal sound wave of the corresponding direction and wavelength existing in the body. If one changes the direction in which the observation is made, then one can successively study the influence of thermal sound waves of different directions and different wavelengths.

But we know, moreover, that when light is reflected from a moving mirror there arises a change in wavelength, obeying Doppler’s principle. In the scattering of light this phenomenon must also occur, and taking it into account leads to interesting conclusions. It turns out that the sound waves that cause the scattering and that we considered above exist in two kinds, one part of them moving upward, while the other part moves downward (Fig. 1). The first waves will increase the frequency of the primary light, whereas the second will decrease it. The order of magnitude of this change in frequency is determined by the ratio of the velocity of sound to the velocity of light and, moreover, as in the case of a gas, depends on the direction in which the observation is made. What is essential in this phenomenon is the circumstance that we should observe not a broadening of the spectral line, but its splitting into two narrow components.

Fig. 1. Reflection from thermal sound waves.

The material set forth above makes it possible to turn again to experiments with a liquid. The question arises whether, in the scattering of light, there will be observed a broadening of the original spectral line or its splitting into two components. In the first case, the thermal motion of the molecules of the liquid corresponds to molecular motion in gases; in the second case—to that in solids.

The first experiments on the fine spectral structure in Rayleigh scattering were carried out by Gross in Leningrad. He found that side lines arise which can be photographed in the spectrographic analysis of scattered light with the aid of a step grating. However, he found several such components, which are extremely difficult to interpret. Rafalovsky in Warsaw, who repeated Gross’s experiments, did not find a splitting of the primary spectral lines. Therefore I suggested to Meyer and Ram in Leipzig that they carry out these experiments again, making use at first of the mercury line 4358 Å. The very first experiments, also carried out with a step grating, revealed

splitting, but not into a large number of components, only into a triplet. The middle line had the frequency of the primary light, and the two new components were situated symmetrically to the left and right of it. However, the effect was not as distinct as was desired. Toluene was chosen as the scattering liquid. The distance of the new components from the middle line was

\(0.06\ \text{Å}\); however, at the same distances several satellites were observed for this mercury line. Of course, this primary fine structure made the picture of scattering very inconclusive; therefore a new light source was chosen, which had to be sufficiently bright, since the scattered light, after passing through the monochromator and the echelon grating, had an intensity amounting to only a small fraction of the intensity of the primary light. In addition, the source of the primary light had to be strictly monochromatic and not give satellites of the principal radiation. These requirements were satisfied by a quartz lamp filled with pure zinc. It was ignited like an ordinary mercury lamp, but with much greater difficulties. As the primary light there was used the zinc line \(4680\ \text{Å}\), and the appearance of a triplet due to the scattering of light in toluene was established quite distinctly and reliably.

The distance of the side components from the middle line in the case of toluene was \(0.06\ \text{Å}\). When calculated from the speed of sound it also comes out to be \(0.06\ \text{Å}\). Thus there can be no doubt that we are dealing here with the effect predicted by Brillouin for solids; but here this effect is observed in a liquid. We may therefore conclude that the thermal motion of molecules in a liquid is similar to the motion of molecules in a solid. There is a strong bond between each molecule and its neighbors, and one should imagine that the molecule performs chiefly an oscillatory motion. But whereas in a solid this motion takes place about a point which on the average does not change its position, in a liquid each center of oscillation moves with a relatively small velocity.

If our conclusions have general significance, then we should expect, for example, that the atomic heat capacity of a monatomic liquid should be close to the value indicated by Dulong and Petit (\(6\ \text{cal}\)), since in this case an increase in the average potential energy of the oscillatory motion with rising temperature will require almost the same supply of heat as the kinetic energy, which alone exists in gases and amounts to about \(3\ \text{cal}\) per degree. This is confirmed in the case of liquid mercury, whose atomic heat capacity (recalculated to constant volume) is \(5.87\ \text{cal}\).

However, there is no exact agreement between Brillouin’s calculations and the experimental data for liquids; namely, according to the calculations, not a triplet but a doublet should be formed, while the average unshifted line should be completely absent. In fact, however, this is not fulfilled; moreover, apparently the unshifted line cannot be ascribed to the influence of dust present in the liquid. It is possible that this phenomenon is due to differences that undoubtedly exist between a liquid and a solid body. But it is also possible that scattering of unchanged frequency is due to the fact that the vibrations of oriented molecules of the liquid create something resembling an oscillating aggregate of small regions with crystalline properties. At present neither theory nor experiment makes it possible to decide this question precisely.

At the end of his theoretical work Brillouin notes that it would be very interesting to prove the presence of the reflection, calculated by him, from thermal sound waves also in the case of artificially created sound waves. In connection with a report on the experiments of Meier and Ram, made at the Massachusetts Institute of Technology, I had a similar idea, which I carried out together with Sears. With the aid of a piezoquartz excited at high frequency, ultrasonic vibrations with a wavelength of several tenths of a millimeter were obtained in a liquid. We wanted to investigate the reflection of light from the wave fronts, choosing angles corresponding to Brillouin’s theory. We immediately succeeded in discovering new and interesting interference phenomena. When the light emerging from the liquid was collected by means of a lens and focused on a screen, there appeared not only an image of the central slit that served as the source of light, but to the left and to the right several very intense spectra arose, quite similar to the spectra obtained from an ordinary grating. It was perfectly obvious that the periodic condensations and rarefactions arising in the liquid played the role of a diffraction grating. It was further found that the occurrence of the interference is not substantially affected by the motion of the ultrasonic grating with the ultrasonic velocity (about \(1\,000\ \mathrm{m/sec}\)) relative to the lens. The phenomenon was very bright, so that it could be demonstrated on a screen in a large auditorium without any difficulty. Following us and quite independently of us, Lucas and Biquard discovered this same phenomenon. Observation of the angles at which the spectral lines arise when monochromatic light is passed through an ultrasonic grating easily makes it possible to determine the velocity of propagation of the ultrasonic vibrations and the compressibility of the liquid. I think that, because of its simplicity, this method may have practical significance, for example, for measuring the compressibility of gases.

Although Brillouin’s theory did not fully foresee the occurrence of the interference indicated above, but assumed only reflection at Bragg angles, the results of these experiments do not contra-

contradict the theory. It must be kept in mind that the dimensions of ultrasonic waves cannot be regarded as infinitely large in comparison with the wavelength of light; if this circumstance is taken into account, then the phenomenon can easily be explained with the aid of the same assumptions that Brillouin used, but in doing so one must not remain at the first approximation, which regards scattering as a very weak effect. Otherwise the appearance of higher-order spectra, readily obtained in practice, remains incomprehensible. Another possibility—the emission by quartz of overtones of sufficiently high intensity—practically plays no role.

Up to now we have considered only the fine structure of scattered light with respect to its wavelength; but every light also has another structure, which is likewise accessible to observation and permits us to obtain further information about the structure of liquids. I mean the volume structure, i.e., the distribution of the scattering intensity over different spatial directions. With ordinary visible light nothing interesting can be observed here. The point is that the wavelength of visible light is too large in comparison with the dimensions of the volume inhomogeneities caused by the molecular structure of the liquid. For these observations one must use X-ray light with a wavelength of the order of \(1\,\overset{\circ}{\mathrm A}=10^{-8}\,\mathrm{cm}\). We shall now consider the question of what should be observed here and what conclusions may be drawn from the results of such observations.

The simplest case allowing such observations to be made is the scattering of light by a monatomic liquid, for example, liquid mercury. But even in this case it is necessary first to ascertain how primary X-ray light will be scattered by individual mercury atoms. This scattering will be determined not by the heavy nucleus of the atom, but practically only by the atomic shell, consisting of light electrons. But this shell occupies a volume whose radius is of the order of \(1\,\overset{\circ}{\mathrm A}\), i.e., it coincides in order of magnitude with the wavelength used. Owing to this, when light is scattered by different parts of one and the same atom, a phase difference arises, and the observed scattering intensity has an interference effect caused by the scattered rays arriving from different regions of the electron shell. As a consequence, the scattering of an atom forward, i.e., in the direction of the primary ray, proves to be more considerable than in the backward direction. But this change in intensity occurs along a smooth curve, indicating that there are no preferred lateral directions of scattering. This theoretical result is also confirmed by experiment. The corresponding experiments were carried out by Scherrer and Steger, who worked with mercury vapor; the results of their experiments are shown in Fig. 2, where the scattering angle is plotted along the abscissa axis, and the scattering intensity along the ordinate axis. The points show the experimental data, while the solid ...

the curve has been calculated on the basis of modern ideas about the distribution of electrons in the atom.

Since we know the action of a single atom, we can proceed to investigations with liquid mercury. In these investigations the primary beam falls at a small angle on the free surface of the liquid, and then the secondary scattered radiation is photographically recorded in various directions. Although the primary beam penetrates into the liquid to a depth of only a few hundredths of a millimeter, the experimental results nevertheless characterize the internal structure of the liquid, since the features due to the presence of a surface film manifest themselves only at depths of the order of millionths of a millimeter. This experiment was carried out by Menke in Leipzig. In the case of liquid mercury the result obtained differed sharply from the result found in work with the gas. In the case of the liquid a whole series of intensity maxima was observed. The results of the experiment can be corrected if one takes into account the influence of absorption in the liquid, of the polarization arising in scattering, and of the structure of the individual atoms, whose influence is already known from experiments with the gas. After these corrections one obtains an intensity curve whose course depends only on the peculiarities of the distribution of atoms in the liquid. This curve is shown in Fig. 3, and along the abscissa axis there is plotted not the angle itself but its function \(\frac{\sin \vartheta/2}{\lambda}\), since this function is more convenient for a quantitative discussion of the results obtained.

Fig. 2. Scattering of light by mercury vapors.

The intensity curve has sharply expressed maxima and minima, obviously indicating that, despite the liquid state and the considerable mobility of the mercury atoms associated with it, there nevertheless exists a certain average regularity in their arrangement.

Let us consider how the distribution of atoms can be determined from our experimental data.

Suppose that we are able to see atoms, and let us focus our attention on two definite atoms, which we shall denote by \(A\) and \(B\). We shall now, for a long time, observe atom \(B\) from atom \(A\), i.e., suppose that we move together with atom \(A\). Atom \(B\) will now approach atom \(A\), perhaps even collide with it, then move away again; in short, it will be located successively at all possible distances

from atom \(A\). We are interested in whether, among the distances existing between \(A\) and \(B\), there are any preferred ones, or whether all distances are equally probable.

It is also possible to imagine that some volume element is rigidly connected with \(A\) (which may be situated at any distance from \(A\)) and to observe, during some time, whether the center of gravity \(B\) will be found inside this volume during the motion of the atoms. Repeating this imaginary experiment at different distances, we obtain for each distance a certain time proportional to the probability of finding atom \(B\) in the volume element under consideration. If the curve of this probability is plotted as a function of the distance between \(A\) and \(B\), this curve will make it possible to draw essential conclusions about the structure of the liquid.

One might think that all distances are equally probable, except, of course, for the very small ones, which cannot exist, since at these distances the atoms would have to penetrate into one another. If we obtain some probability curve as a function of distance, then we shall be able to judge the character of the scattering, i.e. the course of the scattering intensity as a function of the scattering angle, without making any additional assumptions. If the corresponding calculation is carried out, assuming equiprobability of all distances, then a scattering curve is obtained that is completely unlike the experimental curve shown in Fig. 3. Instead of sharply expressed maxima, only weak hints of elevations in individual portions of the curve are obtained.

Fig. 3. Scattering of light by liquid mercury.

Therefore it is necessary to seek another probability curve, the consequence of which should be the intensity curve observed experimentally. In doing so one may use the method developed by Prins and Zernike and used by Menke. I have already indicated that knowledge of the probability curve (as a function of the distance between atoms) is sufficient for calculating the intensity curve (as a function of the scattering angle). But this proposition can be reversed. If the scattering curve is known, then there exists an unambiguous mathematical method for calculating the probability curve from the given experimental scattering curve. This method was applied to the experimental curve shown in Fig. 3, as a result of which

a curve was obtained, shown in Fig. 4. From this curve we can see, for example, that two mercury atoms are preferentially located at a distance of \(3.3\ \text{Å}\), that a distance of \(6\ \text{Å}\) is also preferred, but less sharply expressed. Between these distances lies the distance \(4.4\ \text{Å}\), which the atoms avoid. The same thing is repeated at larger distances, but the difference here is smaller and in the end becomes quite imperceptible.

In this preference for certain distances we can, with full justification, discern a certain analogy with the crystalline structure of the solid state. Indeed, a probability curve such as that shown in Fig. 4 recalls the corresponding curve determining the atomic lattice of a crystal, calculated by Laue and Bragg.

Since the results of our consideration have proved rather unexpected, it seems interesting to investigate the existence of preferred distances in a liquid by means of a certain model. A box with glass walls was taken, filled with steel balls, while, however, free space remained in the box; two balls were blackened; then the box was shaken, after which the distance between the two black balls was measured. This experiment was carried out 6,000 times, after which, on the basis of the numerical material obtained, the statistics of the recurrence of different distances were studied. The result obtained corresponded exactly to what had been found in the study of the scattering of light by liquid mercury. The experiment with the model also gave preferred distances and a distribution curve resembling the curve of Fig. 4. Prins carried out a similar experiment, in which lycopodium was used as the model, the distribution of which on the surface of a glass plate was studied under a microscope. The result of his experiment corresponded to the results set forth above.

Fig. 4. Distribution of atoms in liquid mercury.

Undoubtedly, the case of monatomic mercury is one of the simplest. If a liquid consists not of atoms but of molecules, the phenomenon becomes considerably more complicated. First, the molecules themselves can

give rise to the appearance of interference maxima. Secondly, in the case of a liquid, such characteristic fluctuations of intensity may be caused not only by the distribution of interatomic distances but, perhaps, also by the circumstance that the molecules too have a certain ordered distribution. After the interference phenomena in liquids were first studied by Scherrer and myself in 1916, the American investigator Stewart devoted much work to these investigations and established, although only qualitatively but quite definitely, the presence of a certain molecular order in a liquid. The essential importance of the interference of neighboring atoms for the scattering of light by a liquid became quite evident when Keesom established the presence of interference maxima upon illuminating liquefied monatomic gases.

The material set forth above permits the conclusion that the study of the scattering of light by liquids clearly proves that, in addition to the undoubtedly existing continuity of the gaseous and liquid states, a liquid also has something in common with the solid state, both with respect to the conditions of motion and in the sense of the spatial distribution of molecules.