QUASICRYSTALLINE STRUCTURE OF LIQUIDS

P. Debye1

At the present time the opinion is still widely held that a liquid may be regarded as a condensed gas. However, one must not forget that a liquid can not only evaporate but also solidify, and therefore it is often expedient to regard a liquid as a state of matter that has already come considerably closer to the crystalline state.

We know that the atomic heat capacity of a monatomic gas is equal to 3 cal.; for a solid it is approximately 6 cal. The atoms of an ideal gas possess only kinetic energy; for the atoms of a solid the additional 3 cal. are due to potential energy. The heat capacity of monatomic liquids is also close to that of a solid. Thus, for example, liquid mercury at 0° has an atomic heat capacity \(C_p = 6.72\); if we reduce this quantity to constant volume, we obtain \(C_v = 5.90\). As a second example one may cite argon, which behaves in an analogous manner. Liquid argon near its boiling point at constant pressure has an atomic heat capacity \(C_p = 10.5\). But, as A. Eucken and P. Hauck showed, its heat capacity reduced to constant volume is \(C_v = 5.50\). Thus the heat capacities of monatomic liquids almost reach the value 6 cal. On this basis we may conclude that atoms in monatomic liquids possess considerable average potential energy, and therefore their behavior should not differ greatly from the behavior of atoms in solids.

In gases we picture atoms as moving from one collision to another along straight lines, and we assume that during these intervals of time the atoms possess only kinetic energy. Potential energy appears only at the very moment of collision. On the contrary, in liquids, as in a solid, each atom oscillates, and the intervals of time during which the kinetic or potential energy of the atom predominates are comparable in duration. Nevertheless, the difference between a solid and a liquid must, of course, manifest itself in some way. In a liquid an atom performs oscillations about a point which itself in turn

in turn, slowly and nonuniformly, as in Brownian motion, moves in the liquid. In a crystal, on the contrary, the center of oscillation of an atom is fixed in space, if one does not take diffusion phenomena into account.

As is well known, in order to determine experimentally the state of motion of an atom in a gas, it is sufficient to observe the distribution of energy in the lines of the emission spectrum. It is quite conceivable to carry out, in principle, a completely analogous experiment in scattered light. Purely monochromatic light, after being scattered by a monatomic gas, would have to possess an intensity distribution represented by a Gaussian curve. According to the Doppler principle, scattering by moving atoms must involve a change in the wavelength of the light, and as a result we obtain an intensity distribution that is a reflection of the distribution of atomic velocities. The order of magnitude of the values obtained for the half-width of the spectral lines is determined by the ratio of the velocity of the molecules to the velocity of light, and therefore the broadening is always very small. However, for the observed scattering effect it is characteristic that the broadening also depends on the direction of observation; namely, calculation shows that the line width is proportional to \(\sin \frac{\vartheta}{2}\), where \(\vartheta\) is the angle between the primary and secondary ray. It is essential that in a gas we are dealing with broadening of the original spectral line, and not with its splitting into components.

Brillouin considered the question of the scattering of light by an ideal solid body. In this case each atom in its motion is elastically bound to neighboring atoms, and therefore the state of motion of an atom may be regarded as a superposition of oscillations, each of which separately belongs to an elastic wave. These waves pass through the body in all directions, and their frequencies fill an entire band of wave numbers from 0 to some limiting frequency. This representation proved quite suitable for calculating the specific heat; it gives the \(T^3\) law for low temperatures and was taken by Brillouin as the basis for his calculation of scattering\(^{1}\). Brillouin arrives at the conclusion that scattering in such a solid body may be regarded as Bragg reflection from thermal waves. For such reflection, only those waves are of significance whose front is oriented in such a way that the incident and scattered ray can be regarded as a ray incident on and reflected from the front in the sense of geometrical optics. But there exists a multitude of elastic waves differing from one another in wavelength. In this respect light is also given only a limited choice; namely, for the scattering of light at the angle under consideration, only elastic waves having such a wavelength \(\lambda\) are of significance that the light rays reflected from planes of equal phase, possessing—

\(^{1}\) Independently of Brillouin, analogous considerations were expressed by L. I. Mandelstam. Ed.

give a path difference exactly equal to the wavelength of light \(\lambda\) and, reinforcing each other, create an interference maximum. On this basis we spoke above of Bragg reflection. The condition for the possibility of reflection has the form

\[ 2\Lambda \sin \frac{\vartheta}{2}=\lambda . \tag{1} \]

In the special case considered by us one may speak only of a path difference \(\lambda\), and not \(2\lambda, 3\lambda\), etc. Brillouin’s calculations confirmed the fact, known earlier as well, that in the case of a sinusoidal distribution of the scattering mass one obtains reflection (or diffraction) only of the first order, while the intensities of all higher orders vanish. Therefore one may say that scattering at an angle \(\vartheta\) to the original direction of the light gives a measure of the intensity of those thermal elastic waves whose wavelength is related by relation (1) to the wavelength of the incident light. Thus observations in directions from \(\theta=0\) to \(\theta=\pi\) can cover the range of wavelengths from \(\Lambda=\infty\) to \(\Lambda=\frac{\lambda}{2}\). The wavelength \(\frac{\lambda}{2}\) of ordinary visible light is approximately 1000 times greater than the distance between atoms, which in turn is a quantity of the same order as the limiting wavelength of the elastic spectrum; hence it is clear that for the scattering of light only a very small part of all possible thermal waves can have significance, namely: only elastic waves with sufficiently small frequencies. According to Brillouin’s calculations, other relations can be obtained only by taking X-rays instead of visible light; in his work Brillouin attempts, on the basis of his calculations, to predict certain results for this case as well.

One detail in Brillouin’s calculations is of special significance for us. Thermal waves of a selected direction and length, causing the scattering of light, may travel in two mutually opposite directions with the velocity of sound. As a consequence of this there will appear a change in the frequency of the light in the spirit of the Doppler effect. Instead of the initial frequency of the light \(\nu_0\), the scattered light will now contain 2 frequencies \(\nu_1\), calculated by the formula

\[ \frac{\nu-\nu_0}{\nu_0}=\pm 2 n \frac{q}{c}\sin\frac{\vartheta}{2}. \tag{2} \]

Here \(n\) is the refractive index, \(\frac{c}{n}\) is the phase velocity of light, and \(q\) is the velocity of sound.

The scattering effect will now be entirely different than in gases. Instead of a broadening of the original spectral line, according to Brillouin’s calculations there should occur its doublet splitting. This splitting is very small. If we assume that the order of magnitude of the speed of sound corresponds to \(1000\ \text{m/sec}\), then

the maximum splitting (for \(\theta=\pi\)) will be of the order of \(0.1\) Å at a wavelength of \(5000\) Å.

After we had established that there is a qualitative difference in the behavior of a gas and of a solid in the region of scattering, it would be interesting to investigate experimentally the intermediate region, namely liquids. In recent years such experiments were carried out first by Gross and then by Rapp. These experiments clarified the behavior of liquids in the case under consideration, and later investigations of the Indian school headed by Raman only confirmed the results obtained earlier.

In liquids, as in solids, according to Brillouin’s theory there occurs a splitting of the original spectral line. The magnitude of the splitting can be calculated from formula (2). But the occurrence of a doublet is, generally speaking, not confirmed. We obtain a triplet, whose middle line represents the unchanged frequency \(\nu_0\), while the two outer lines evidently represent what would have been expected, according to the calculation for a solid.

But it is quite obvious that between a liquid and an ideal solid there must nevertheless exist a greater difference. On the other hand, since in a liquid splitting is observed, and not broadening, we must conclude that, for describing the behavior of a liquid in scattering, it is expedient to take as the starting point not a gas but a solid. An ideal solid is characterized by the fact that its potential energy can be represented in the form of a quadratic function of the atomic displacements. As is known, such an idealization leads to the result that the body under consideration cannot undergo thermal expansion, and therefore its specific heat at constant pressure \(C_p\) must be equal to its specific heat at constant volume \(C_v\). Planck, in a brief communication, gave his results, according to which in liquids, for the intensity of the middle line, it is precisely the difference \(C_p-C_v\) that is important. Let us denote the intensity of the middle line by \(I_0\), and the intensity of one of the outer components by \(I_a\); then the relation

\[ \frac{I_0+2I_a}{2I_a}=\frac{C_p}{C_v}. \tag{3} \]

must be satisfied.

On the basis of Rapp’s experiments one cannot judge the validity of this relation. Rapp set himself the task of obtaining the maximum splitting, and therefore always observed in the direction \(\theta=\pi\), opposite to the original direction of the beam. But precisely in this direction there is an especially great probability that secondary light will be added to the scattered light, and therefore the observed intensity of the middle line will not be the intensity caused exclusively by the scattering process. Therefore Birus carried out new experiments on the scattering of light at a right angle to the original beam. These experiments were published in Physikalischen Zeitschrift. Birus found that for toluene completely

the linearly polarized part of the light scattered at an angle of \(90^\circ\), and caused by density fluctuations, gives the following distribution of intensities of the three triplet lines \(5:4\pm0.5:5\). Since for toluene
\[ \frac{C_p}{C_v}=1.364, \]
according to formula (3) one should have expected an intensity ratio of \(5:3.64:5\). The correctness of the results obtained by Placzek is especially clearly confirmed by scattering for water. Water has its maximum density at \(4^\circ\); at a temperature close to \(4^\circ\), \(C_p\) and \(C_v\) are almost equal to each other. In this case, according to formula (3), the middle line should not appear at all. Water presents well-known difficulties for carrying out scattering experiments, since its scattering power is very small. However, Birus succeeded in obtaining good photographs (with exposure times from 2 to 4 days), and he came to the conclusion that the middle line is indeed absent.

In general we may state the following.
The splitting proves that the motion of a molecule in a liquid, just as in a solid body, is closely connected with the motion of neighboring molecules.

After experiments on the scattering of visible light had helped to clarify definitively the question of molecular motion, it became possible to study the question of the mean relative position of molecules, and one could attempt to find certain regularities in this domain. For this purpose one can likewise make use of scattering experiments, only in the present case these experiments must be carried out not with visible light but with X-rays, whose wavelengths are of the same order of magnitude as the details of the structure under investigation.

Before proceeding to the investigation of the problem posed, it is necessary to determine the scattering power of individual molecules. The scattering-intensity curve (as a function of the angle with the initial direction) is especially simple for atoms. Experiments carried out with monatomic gases show a continuous decrease of the scattering intensity as the angle increases, but nevertheless in the neighborhood of \(90^\circ\) there is a maximum, or a hint of one. This latter effect is connected with the polarization due to scattering, which is complete in the direction perpendicular to the initial ray. As a consequence, every scattering function corresponding to natural unpolarized primary radiation always contains the factor
\[ \frac{1+\cos^2\theta}{2} \]
(where \(\theta\) is the angle with the direction of the initial ray). In what follows, when speaking of scattering curves, we shall mean that the polarization effect in them has already been allowed for unpolarized primary radiation by multiplying the intensity values by the reciprocals of the above coefficient.

If we now turn to scattering in liquids and again restrict ourselves only to monatomic liquids, we obtain not a smooth-

QUASICRYSTALLINE STRUCTURE OF LIQUIDS

an intensity curve characteristic of a single atom, but a curve with a whole sequence of broad, yet clearly expressed maxima. This was first observed by Debye and Scherrer.

These maxima are undoubtedly caused by regularities in the mutual arrangement of the centers of the atoms. These regularities may be described qualitatively as follows. Let us observe, from the center of an atom, making its vibrations together with it, the center of some other atom. Then one can speak of the probability that the distance between the centers \(r\) lies between \(r\) and \(r + dr\). If there is no regularity, then this probability will simply be proportional to \(4\pi r^2 dr\). Any regularity must be characterized by the fact that an additional density coefficient \(D(r)\) appears, so that the probability takes the form \(4\pi r^2 D(r)dr\). Thus the problem consists in deriving, from the arrangement and magnitude of the maxima of the intensity curve observed as a function of the scattering angle \(\theta\), the density coefficient \(D\) as a function of \(r\). For this there exists a single mathematically rigorous method, which was first set forth by Prins and Zernike and was first applied by Menke to the case of scattering in liquid mercury. It turns out that \(D\) goes to zero for very small distances, since atoms cannot penetrate into one another. Further, it turns out that the density coefficient has maxima for certain distances, and these maxima become less and less sharp as \(r\) increases. Thus in a liquid some distances between atoms are preferred, while others, intermediate ones, are avoided. This result allows us to consider that the arrangement of atoms in a monatomic liquid is quasicrystalline, and we may establish the following.

The character of the interference of X-rays in liquids shows that in liquids there also exist regularities in the arrangement of atoms, very close to the regularities observed in solid crystals.

If the individual particles are not atoms but molecules, then the picture is somewhat complicated by the fact that the individual molecule itself can give well-marked wave maxima on the intensity curve. These maxima will be caused by the interference among themselves of scattered rays emanating from the individual atoms of one and the same molecule. Experiments carried out with polyatomic gases in X-rays and electron rays have fully confirmed this circumstance. At the same time all experiments carried out with molecular liquids have shown that in this case, as also for monatomic liquids, new additional interference maxima appear, caused by the dense packing of the molecules, and therefore here too there is a certain spatial system of arrangement of atoms. However, at the present time we are not in a position, in all details, to determine unambiguously the mutual arrangement of molecules from the observed intensity curve. Namely, the structure cannot be represented as a density function of

of one parameter, since for a given distance \(r\) the spatial direction of the molecule’s axis will also have significance. It is obvious that from the intensity curve, which is a function of one parameter (the scattering angle), it is impossible to determine uniquely the desired density coefficient as a function of many parameters. Nevertheless, one may try to draw certain conclusions from the photographs about the mutual arrangement of the molecules. Stewart, to whom we owe a whole series of experiments in this field and who has always pointed to the great importance of studying the structure of liquids, comes to the conclusion that, for example, molecules elongated in length exhibit a sharply expressed tendency toward a parallel arrangement, approximately like nails in a sack.

Evidently, the structures of liquids may provide the key to understanding one remarkable phenomenon observed in ultrasonic waves. It has been shown that in liquids ultrasonic waves are absorbed very differently and that the absorption is greater, often even much greater, than could have been expected on the basis of the known value of the viscosity. In this case, in full agreement with one remark of Bernal’s, it may be considered that the structure of a liquid corresponds to a definite relaxation time. One may try to explain this in the following way. When a liquid is compressed, the density function considered earlier, which determines the average mutual arrangement of the molecules, must necessarily also change. But since this change requires time, the reaction will proceed differently depending on whether the compression occurs rapidly or slowly. If we denote by \(s\) the deviation of the density \(\rho\) from its equilibrium value \(\rho_0\), then the pressure \(P\) may be represented by the expression

\[ P=\frac{1}{x}\left(s+\tau \frac{\partial s}{\partial t}\right), \]

where \(x\) is the compressibility and \(\tau\) the relaxation time. With the aid of the classical equations of hydrodynamics we obtain in this case that the intensity of plane sound waves propagating with velocity \(q\) in the direction of the \(X\)-axis decreases as \(e^{-\alpha X}\), where the absorption coefficient \(\alpha\), for not too high frequencies, is calculated from the formula

\[ \alpha=\frac{\omega^2}{q}\left(\tau_r+\tau\right). \]

The relaxation time \(\tau_r\) is usually determined by internal friction and is equal to \(\tau_r=\frac{4}{3}\mu x\), where \(\mu\) is the coefficient of internal friction. The order of magnitude of this interval of time is \(10^{-12}\) sec. But here there is added another quantity, arising from the relaxation of the structure and proportional to \(\tau\). Suppose, for example, as indeed happens, that the absorption is 100 times greater than could have been expected from the value of the viscosity; then one would have to take \(\tau\) to-

order of \(10^{-10}\) sec. It is remarkable, and probably not accidental, that we have a quantity of the same order in the dispersion and absorption of electric waves in normal polar liquids; this, as is known, is explained by the relaxation of the arrangement of the dipole axes.

We shall now see what property of liquids is the most important for the scattering of X-rays, and that it is precisely this that we can always determine unambiguously from interference experiments, if we recall that the coherent scattered radiation issuing from a volume element is determined by the electron density in the given volume element. Let us consider a spatially fixed volume element. In it the electron density will change with time as a consequence of the motion of molecules. We may speak of the time-average density \(\rho_0\) and of the fluctuations \(\delta\) of this density. Let us denote these fluctuations in two spatially fixed volume elements \(dV_1\) and \(dV_2\), situated at a distance \(r\) from one another, by \(\delta_1\) and \(\delta_2\). It can be shown that the density function which determines the occurrence of interference is a measure of the time-average value of the product \(\delta_1\delta_2\). It is obvious that, owing to the isotropy of the liquid, \(\overline{\delta_1\delta_2}\) can depend only on the distance \(r\) between the chosen volume elements. If \(\delta_1\) and \(\delta_2\) were completely independent of each other, we would have \(\overline{\delta_1\delta_2}=0\). In reality, however, this condition is not satisfied: on the one hand because of molecular structure, and on the other hand because of the structure of the liquid. The condition is satisfied to a lesser degree the smaller \(r\) is. With the aid of the integral inversion (Integralumkehrung) proposed by Prins and Zernike, the mean \(\overline{\delta_1\delta_2}\) can always be derived unambiguously as a function of \(r\) from the observed intensity curve. However, without introducing some hypothesis, on the basis of the obtained result one cannot acquire further information about the position and orientation of the molecules, which determine the fluctuations \(\delta\) owing to the motion of their centers of gravity and to rotation. Moreover, in all these considerations one must not forget that coherent scattered radiation is always accompanied by a certain amount of incoherent scattered radiation. This latter radiation constitutes a background, which must first of all be eliminated.

Such a consideration prompts the search for other methods by which one could determine the mutual arrangement of molecules in a liquid. Since we are dealing with the rotation of molecules, it would be natural to turn to experiments in which the orientation is changed under an external influence. Among such experiments one should first mention the measurement of the dielectric constant in liquids with polar molecules, which owe their dielectric action chiefly to the partial parallel arrangement of the dipole axes produced by the field. Alongside this, one should note the measurement of electric double refraction, which also arises in molecules without an initial dipole moment, if there is a sufficient difference in polariz-

ization of molecules in different directions. In this case a static field can create a parallel arrangement of the axes of greatest polarizability of the molecules.

By measuring the temperature dependence of the dielectric constant of gases one can determine the dipole moment of a molecule; by measuring the Kerr effect in gases one can find the ellipsoid of polarizability of the molecule. If, on the basis of these measurements, we try by the usual formulas to predict the dielectric constant or the Kerr constant of gases condensed into a liquid, then we obtain, generally speaking, values that are too large. In this case two possibilities are conceivable. Either the dipole moment and, at the same time, the difference of the polarizabilities of the molecule in different directions decrease, i.e. the structure of the molecule in the liquid differs from the structure of the gaseous molecule, or the liquid hinders the rotation of the molecule, preventing it from taking up the required direction under the action of the field. A number of circumstances indicate that the second assumption is the correct one. This again leads to an analogy between liquids and solid crystals; the potential energy increases not only upon displacement of the molecules, but also upon their rotation.

Having set myself the task of characterizing the resistance to rotation quantitatively, even if only in a rough approximation, in calculating the arrangement of the dipole axes under the action of an external electric field \(F\) I made the following assumption. The dipole axes of the molecules of a liquid do not rotate freely in space, as in gases, but require an energy \(-E\cos\theta\) for a rotation through an angle \(\theta\) relative to an axis determined by the molecular action of their surroundings. Under this assumption we find that the mean moment of a molecule with dipole moment \(\mu\) in the direction of the field strength \(F\) is not equal to \(\dfrac{\mu F}{3kT}\), as for gases, but differs from this quantity by a factor \(R\). This factor is a function of the ratio \(\dfrac{E}{kT}\) and is calculated from the formula

\[ R = 1 - L^2\left(\frac{E}{kT}\right), \]

where \(L(x)\) is the Langevin function: \(L(x)=\operatorname{cth} x-\dfrac{1}{x}\). Thus, for example, in liquid water the dipole part of the molecular polarization calculated by the Clausius–Mossotti formula amounts to only \(1/5\) of the value that one would expect from the magnitude of the dipole moment. This corresponds to a binding energy of about \(E=10\,kT\). This means that in such a liquid the rotational motion of the molecules is much more like oscillations about a slowly moving axis than free rotation.

In order to establish whether such an estimate of the binding energy is close to reality, attempts were made, with the aid of hindrance, to explain the fact that the observed saturation effect

in liquids often proves to be much smaller than would have been expected on the basis of the usual formulas. Thus, for example, according to Miles’s measurements, the dielectric constant of water, measured after the application of a strong electric field simultaneously with an alternating field of small amplitude, decreased appreciably. The relative decrease (proportional to the square of the strength of the constant field) proved to be 3500 times smaller than the calculated value. A new calculation for this case, taking account of the energy of opposition, shows that the effect for freely rotating molecules must be calculated with allowance for a certain coefficient, which for large values of \(\frac{E}{kT}\) is equal to \(3\left(\frac{kT}{E}\right)^4\). This corresponds, at \(E = 10 kT\), to a diminution of the effect by a factor 3300 times smaller, which comes close to the experimental results.

Thus the use of observations of dielectric polarization makes it possible, on the basis of such rough assumptions as those formulated above, to obtain information concerning the braking of rotation and the corresponding forces.

Analogous results are obtained in the study of the Kerr effect; and new experiments and calculations carried out by Raman compel one to suppose that, in electrical dispersion and absorption, the opposition to free rotation likewise plays an important role. But even apart from quantitative data, we may assert with confidence that, with respect to the rotation of molecules as well, liquids exhibit a great similarity to solid crystals.

1. P. Debye, Die Quasikristalline Struktur von Flüssigkeiten, translated by N. I. Polyakova and M. V. Volkenstein. ↩