SOME ISSUES OF MOLECULAR LIGHT SCATTERING IN LIQUIDS

I. L. Fabelinskii

CONTENTS

§ 1. Theory of molecular light scattering in pure liquids . . . . . . . . . . 356
    a) Thermodynamic theory . . . . . . . . . . . . . . . . . . 356
    b) Nonthermodynamic theory . . . . . . . . . . . . . . . . 360
    c) Intensity of light scattered by fluctuations of anisotropy . . . . 364
§ 2. Absolute measurements of the intensity of molecular light scattering in liquids. Comparison with the conclusions of theory . . . . . . . . . . . 365
§ 3. Spectral composition of Rayleigh-scattered light . . . . . . . . . . . 378
§ 4. Dispersion of the speed of sound and the speed of hypersonic waves . . . 381
§ 5. Polarization of the Mandelstam–Brillouin components in liquids . . . . . 391
§ 6. Ratio of intensities in the components of the fine structure of the line of scattered light . . . . . . . . . . . . . . . . . . . . . . . . . . 396
§ 7. Width of the components of the fine structure of the line of scattered light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

Molecular light scattering is one of the central problems of molecular optics. The experimental and theoretical study of this phenomenon over the last 30 years has led to results of great significance for statistical physics in general and for the most important questions of molecular optics, molecular dynamics, and the kinetics of thermal fluctuations in particular.

The internal connection between the phenomena of light scattering and the Debye concept of thermal motion, pointed out by L. I. Mandelstam, made it possible to penetrate deeply into the problem of the propagation of elastic waves in a molecular medium, leading to the creation of a relaxation theory of ultrasonic propagation and providing the only optical method so far for studying the propagation of ultrasonic waves with a frequency of \(\sim 10^{10}\) Hz (hypersound) in a condensed medium.

Despite the major advances achieved in the study of molecular light scattering, many important questions remain unclear up to the present time. This applies, in particular, to the problem of molecular light scattering in liquids.

In the present review the principal results are reported that have been obtained in the study of total Rayleigh scattering of light and of the fine structure of the line of scattered light in pure liquids.

The wide range of questions arising in the study of depolarized light scattering in liquids (the wing of the Rayleigh line), light scattering in solutions and glasses, and light scattering near the critical temperature are not included in this review at all. All these questions deserve separate consideration.

In presenting the problems touched upon in the article, emphasis has in a number of places been placed on results obtained in the optical laboratory named after Academician G. S. Landsberg of the Physics Institute of the Academy of Sciences of the USSR.

The beginning of the fruitful study of the problem of molecular scattering in a condensed medium in general, and of the investigation of the fine structure of the Rayleigh line in particular, is associated with the name of G. S. Landsberg. The development of these works also led Landsberg and Mandelstam to the discovery of combination scattering of light. The works of G. S. Landsberg on the molecular scattering of light and his personal participation in an extensive cycle of works on the molecular scattering of light in condensed media contributed greatly to the progress of this field as a whole.

§ 1. THEORY OF MOLECULAR SCATTERING OF LIGHT IN PURE LIQUIDS

a) Thermodynamic theory.

The statistical character of the thermal motion of the molecules of a medium leads to the appearance in such a medium of fluctuations of density and orientation in volumes small in comparison with the wavelength of light. Both kinds of fluctuations are the cause of the optical inhomogeneity of the medium that produces light scattering.

This optical inhomogeneity is characterized by the fact that, in a fluctuation volume \(v^*\), the dielectric permittivity differs from the mean value \(\varepsilon_0=n^2\) by an amount \(\Delta \varepsilon\). Taking into account that the fluctuations create anisotropy in small volumes \(v^*\), it is necessary to regard this deviation of the dielectric permittivity from the mean value as a tensor \(\Delta \varepsilon_{ik}\).

Following V. L. Ginzburg\(^2\), we divide \(\Delta \varepsilon_{ik}\) into two parts:

\[ \Delta \varepsilon_{ik}=\Delta \varepsilon \cdot \delta_{ik}+\Delta \varepsilon'_{ik}, \tag{1.1} \]

where \(\delta_{ik}\) is the unit Kronecker tensor and \(\sum \Delta \varepsilon_{ii}=0\). Fluctuations of density and temperature, or of pressure and entropy, do not lead to a violation of the isotropy of substances in the fluctuation volume, and therefore only the first term on the right-hand side of (1.1) can be associated with them. Light scattered by these fluctuations will be completely polarized. The fluctuations \(\Delta \varepsilon'_{ik}\) must be determined by the internal parameters of the liquid characterizing it in a nonequilibrium state and as yet remaining unknown. It is evident that light scattered by fluctuations \(\Delta \varepsilon'_{ik}\) will be depolarized.\(^3\) The total intensity of the light scattered by fluctuations of density and anisotropy is determined by \(\overline{\Delta \varepsilon_{ik}^{2}}\), i.e., by quantities of the type

\[ \overline{\left|\Delta \varepsilon\cdot \delta_{ik}+\Delta \varepsilon'_{ik}\right|^2} = \overline{\left|\Delta \varepsilon\right|^2} + \overline{\left|\Delta \varepsilon^{(s)}_{ik}\right|^2} + \overline{\left|\Delta \varepsilon^{(a)}_{ik}\right|^2}, \tag{1.2} \]

where \(\Delta \varepsilon'_{ik}\) is naturally divided into a symmetric \(\Delta \varepsilon^{(s)}_{ik}\) and an antisymmetric \(\Delta \varepsilon^{(a)}_{ik}\) part. The part \(\Delta \varepsilon_{ik}\) caused by fluctuations of density and temperature is amenable to thermodynamic calculation*).

The intensity of light scattered by fluctuations \(\Delta \varepsilon\), when a liquid volume \(V\) is illuminated by natural light, can be expressed as follows\(^1\):

\[ I=I_0\frac{\omega^4 v^* V}{(4\pi c^2 r)^2}\,\overline{\Delta \varepsilon^2}\,(1+\cos^2\theta), \tag{1.3} \]

where \(I_0\) is the intensity of the incident light, \(\omega\) is the frequency of the light, \(c\) is the velocity of light, \(r\) is the distance from the scattering volume \(V\) to the point of observation, and \(\theta\) is the scattering angle.

*) According to\(^2\), \(\Delta \varepsilon=\Delta \varepsilon(p,s)+\Delta \varepsilon'\), where \(\Delta \varepsilon'\) is the part of \(\Delta \varepsilon\) that does not depend on \(p\) and \(s\). In the absence of special reservations we shall take \(\Delta \varepsilon'=0\).

From (1.3) it is clear that the intensity of the scattered light can be calculated only in the case where \(\overline{\Delta \varepsilon^{2}}\) is expressed in terms of medium parameters that are amenable to experimental determination, or in terms of universal constants. The results of such calculations can be compared with experiment. A complete molecular theoretical calculation has so far been carried out only for gases. In the case of liquids, a detailed molecular theory of the phenomenon under consideration is as yet impossible, which, as is known, is connected with the general difficulty of constructing a molecular theory of the liquid state. Therefore one has to resort to a phenomenological treatment of the problem.

One of the possible ways of carrying out a thermodynamic calculation is as follows:

Let \(\Delta \varepsilon\) be a function of the pressure \(p\) and the entropy \(s\) (the choice of these variables corresponds to the natural conditions of the problem); then

\[ \Delta \varepsilon(p,s)=\left(\frac{\partial \varepsilon}{\partial p}\right)_s \Delta p+ \left(\frac{\partial \varepsilon}{\partial s}\right)_p \Delta s . \tag{1.4} \]

Owing to the independence of \(p\) and \(s\), \(\overline{\Delta p \Delta s}=0\). Upon statistical averaging of \(\overline{\Delta \varepsilon(p,s)^2}\) we obtain

\[ \overline{\Delta \varepsilon(p,s)^2}= \left(\frac{\partial \varepsilon}{\partial p}\right)_s^2 \overline{\Delta p^2}+ \left(\frac{\partial \varepsilon}{\partial s}\right)_p^2 \overline{\Delta s^2}. \tag{1.5} \]

Thus, \(\overline{\Delta \varepsilon(p,s)^2}\) immediately decomposes into two parts. The first,

\[ \left(\frac{\partial \varepsilon}{\partial p}\right)_s^2 \overline{\Delta p^2}, \]

characterizes the intensity of light scattered by adiabatic density fluctuations; the second,

\[ \left(\frac{\partial \varepsilon}{\partial s}\right)_p^2 \overline{\Delta s^2}, \]

characterizes the intensity of light scattered by isobaric density fluctuations.

The calculation of the intensity of light scattered by adiabatic and isobaric density fluctuations can be carried out in the following way: from a simple calculation it follows that

\[ \left. \begin{aligned} \left(\frac{\partial \varepsilon}{\partial s}\right)_p &= \left(\frac{\partial \varepsilon}{\partial T}\right)_p \left(\frac{\partial T}{\partial s}\right)_p ; \qquad \left(\frac{\partial s}{\partial T}\right)_p = \frac{C_p \rho v^*}{T},\\ \left(\frac{\partial \varepsilon}{\partial p}\right)_s &= \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_s \beta_s , \end{aligned} \right\} \tag{1.5a} \]

on the other hand\(^4\),

\[ \overline{\Delta p^2}=\frac{kT}{\beta_s v^*}. \tag{1.6} \]

and

\[ \overline{\Delta s^2}=k C_p \rho v^* . \tag{1.6a} \]

Here \(\beta_s\) is the adiabatic compressibility, \(k\) is Boltzmann’s constant, \(C_p\) is the heat capacity at constant volume, \(T\) is the absolute temperature, and \(\rho\) is the density. Substituting (1.5a), (1.6), and (1.6a) into (1.5), we obtain, upon statistical averaging:

\[ \overline{\Delta \varepsilon(p,s)^2} = \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_s^2 \beta_s^2 kT \frac{1}{v^*} + \left(\frac{\partial \varepsilon}{\partial T}\right)_p^2 \frac{kT^2}{C_p \rho v^*} = \]

\[ = \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_s^2 \beta_s kT \frac{1}{v^*} + \left(\frac{1}{b}\frac{\partial \varepsilon}{\partial T}\right)_p^2 \frac{b^2 kT^2}{C_p \rho v^*}, \tag{1.7} \]

where \(b\) is the expansion coefficient. Substituting (1.7) into formula (1.3),

we obtain (putting \(\Delta \varepsilon'_{ik}=0\)):

\[ I = B\left[\left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_s^2 \beta_s kT +\left(\frac{1}{b}\frac{\partial \varepsilon}{\partial T}\right)_\rho^2 \frac{b^2 kT^2}{\rho C_p}\right], \tag{1.8} \]

where

\[ B = I_0 \frac{\omega^2 V}{(4\pi c^2)^2}(1+\cos^2 \theta). \tag{1.9} \]

From (1.8) it is clear that the intensities of light scattered by isobaric \((I_{is})\) and adiabatic \((I_{ad})\) density fluctuations, and the corresponding scattering coefficients \(R=\dfrac{I}{I_0}\dfrac{r^2}{V}\), are equal to

\[ \left. \begin{aligned} I_{is} &= B\left(\frac{1}{b}\frac{\partial \varepsilon}{\partial T}\right)_\rho^2 \frac{b^2 kT^2}{\rho C_p},\\ R_{is} &= \frac{\pi^2}{2\lambda^4} \left(\frac{1}{b}\frac{\partial \varepsilon}{\partial T}\right)_\rho^2 \frac{b^2 kT^2}{\rho C_p}, \end{aligned} \right\} \tag{1.10} \]

\[ \left. \begin{aligned} I_{ad} &= B\left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_s^2 \beta_s kT,\\ R_{ad} &= \frac{\pi^2}{2\lambda^4} \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_s^2 \beta_s kT. \end{aligned} \right\} \tag{1.11} \]

In Einstein’s theory\(^5\) the intensity of light scattered by isobaric and adiabatic density fluctuations is expressed through the isothermal compressibility and the isothermal value \(\left(\rho\dfrac{\partial \varepsilon}{\partial \rho}\right)_T\). According to Einstein,

\[ I = B\left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)\beta_T kT. \tag{1.12} \]

In our derivation of the formula for the intensity of scattered light (1.8), equivalent to Einstein’s derivation of formula (1.12), no approximations have been made, and therefore formula (1.8) is more complete than (1.12), in deriving which Einstein assumes \(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_\rho=0\). Under this assumption formula (1.8) goes over into (1.12).

Indeed, as is easily obtained on the basis of a simple thermodynamic calculation\(^6,7\),

\[ \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_T - \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_s = -\left(\frac{\partial \varepsilon}{\partial T}\right)_\rho \frac{bT}{C_v\beta_T\rho}, \tag{1.13} \]

or, what is the same thing,

\[ \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_T - \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_s = -\frac{b^2T}{\rho C_p\beta_s} \left[ \left(\frac{1}{b}\frac{\partial \varepsilon}{\partial T}\right)_\rho + \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_T \right]. \tag{1.14} \]

Putting \(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_\rho=0\) in formula (1.13), we obtain:

\[ \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_s = \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_T, \]

and on the basis of (1.14) we obtain

\[ \left(\frac{1}{b}\frac{\partial \varepsilon}{\partial T}\right) = -\left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_T. \]

Thus, neglecting $\left(\dfrac{\partial \varepsilon}{\partial T}\right)_{\rho}$, we find from (1.13), (1.14), and (1.8) that the intensity of light scattered by adiabatic and isobaric density fluctuations is equal to

\[ I = B\left(\rho\frac{\partial \varepsilon}{\partial \rho}\right)_{T}^{2} kT \left[\beta_s + \frac{b^{2}T}{\rho C_p}\right]. \tag{1.15} \]

Using the well-known thermodynamic relation\(^8\)

\[ \beta_T = \beta_s + \frac{b^{2}T}{\rho C_p}, \tag{1.16} \]

we arrive at Einstein’s formula (1.12), written above. Thus, it is clear that, to within the accuracy of $\left(\dfrac{\partial \varepsilon}{\partial T}\right)_{\rho}$, Einstein’s formula describes the intensity of light scattered by adiabatic and isobaric density fluctuations. Further, it follows from the derivation that formula (1.12) must indeed contain the isothermal compressibility of the liquid. This indisputable circumstance,\(^2\) however, was called into question in the work of Sundal and Bay.\(^9\) She came to the conclusion that formula (1.12) should contain only the adiabatic compressibility. We shall not go into the details of C. Bay’s reasoning, since V. L. Ginzburg\(^2\) has already shown that she arrived at her conclusion owing to errors in the calculation.

In the derivation given, we did not resort to any model conceptions of the structure of liquids, but based ourselves on the most general propositions of thermodynamics and statistical physics; therefore there are no grounds for doubting the correctness of the relations obtained within the framework of thermodynamics.

From relations (1.13) and (1.14) there follows the condition of applicability of Einstein’s formula. This condition is as follows: Einstein’s formula is applicable if the inequality

\[ \left(\frac{\partial \varepsilon}{\partial T}\right)_{\rho} \frac{bT}{C_v\beta_T\rho} \ll \left(\rho\frac{\partial \varepsilon}{\partial \rho}\right)_{T} \tag{1.16'} \]

is satisfied, or the equivalent inequality, following from (1.14),

\[ k'\left[\left(\frac{1}{b}\frac{\partial \varepsilon}{\partial T}\right)_{\rho} + \left(\rho\frac{\partial \varepsilon}{\partial \rho}\right)_{T}\right] \ll \left(\rho\frac{\partial \varepsilon}{\partial \rho}\right)_{T}, \tag{1.17} \]

where

\[ k'=\frac{b^{2}T}{C_p\beta_s\rho}. \]

All quantities entering into the criterion (1.17) can be determined experimentally, and thus in any case one can judge the applicability of Einstein’s formula.\(^5\)

The left-hand side of inequality (1.17) is not difficult to calculate. As an example, let us give some numerical data. For water,

\[ k'\left[\left(\frac{1}{b}\frac{\partial \varepsilon}{\partial T}\right)_{\rho} + \left(\rho\frac{\partial \varepsilon}{\partial \rho}\right)_{T}\right] \]

is $\sim 1.6\cdot 10^{-2}$; for ether $\sim 1.9\cdot 10^{-2}$, for benzene $\sim 5\times 10^{-2}$, and for methyl alcohol $\sim 10^{-3}$.

The right-hand side of inequality (1.17) is close to unity. Therefore, on the basis of the estimate made, one may assert that, to an accuracy of approximately 2%, Einstein’s formula (1.12) and formula (1.8) give results that agree with one another.

This difference becomes, however, substantial when comparing the ratios of the intensities of light scattered by adiabatic and isobaric density fluctuations. In this case the ratio, calculated from (1.10) and (1.11), is greater than that calculated when the quantity \(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_p\) is neglected by a factor \(N\), where

\[ N=\frac{\left(\dfrac{1}{b}\dfrac{\partial \varepsilon}{\partial T}\right)_p^2} {\left(\rho\dfrac{\partial \varepsilon}{\partial p}\right)_s^2}. \]

For a number of liquids this ratio is noticeably greater than unity.

It should also be noted that the quantity \(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_p\), entering into criterion (1.13), cannot be found directly from experiment\(^{10,11}\). This quantity is calculated, for example, from relation (1.13). Such a calculation gives only the order of magnitude of \(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_p\), since the values \(\left(\rho\dfrac{\partial \varepsilon}{\partial p}\right)_s\) and \(\left(\rho\dfrac{\partial \varepsilon}{\partial p}\right)_T\) are very close to one another, and therefore small errors in the value of each of them lead to a large error in the difference.

An estimate of \(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_p\) by order of magnitude, made from (1.13), gives the following values for various liquids: water \(\sim 6\cdot 10^{-3}\), ether \(\sim 8\times 10^{-5}\), benzene \(\sim 2\cdot 10^{-4}\), toluene \(\sim 7\cdot 10^{-5}\), methyl alcohol \(\sim 3\cdot 10^{-6}\), and ethyl alcohol \(\sim 10^{-4}\). It must, however, be borne in mind that when the calculation of the total intensity is concerned, it is not the derivative \(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_p\), in comparison with a quantity of order 1, that is neglected, but the considerably larger quantity:

\[ \left(\frac{\partial \varepsilon}{\partial T}\right)_p \frac{bT}{C_v\beta_T\rho}. \]

b) Nonthermodynamic theory. Above, thermodynamic formulas were obtained expressing the absolute intensity of light scattered by adiabatic and isobaric density fluctuations as a function of thermodynamic parameters measured experimentally.

Strictly speaking, Einstein’s formula and the expression obtained above for the absolute intensity cannot be applied to problems of light scattering, since thermodynamics presupposes very slow (infinitely slow, reversible) changes of parameters, whereas in the scattering phenomenon the changes occur extremely rapidly (in any case, for adiabatic fluctuations). So long as it had not been established that certain parameters determining the scattering are functions of frequency, the application of thermodynamic relations could nevertheless be regarded as legitimate, although in principle even then one could assume that a dependence on frequency, for example in the compressibility, should occur\(^{2}\), since adiabatic density fluctuations “dissipate” with the velocity of elastic waves of frequency \(10^{10}\) Hz. However, after dispersion of the sound velocity was discovered in a number of liquids\(^{6,12,13}\), the application of thermodynamic formulas becomes unjustified.

In deriving relation (1.8), thermodynamics is used in determining the quantities \(\overline{\Delta p^2}\) and \(\overline{\Delta s^2}\). The corresponding expressions for these quantities are obtained under the natural assumption that the specification of the quantities \(p, s\) (in contrast to \(\bar p, \bar s\)) can be characterized

state of incomplete equilibrium. In other words, the relaxation time for these quantities must be large in comparison with the relaxation time of the other processes of approach to equilibrium that are possible in the system under consideration.^4 Under this assumption we obtain an expression for the complete fluctuation of \(p\) or \(s\) or another thermodynamic quantity.

For the scattering of light, knowledge of the spectral density of the corresponding fluctuation \(\overline{\Delta p_\omega^2}\) and \(\overline{\Delta s_\omega^2}\) is essential, and these quantities are expressed through parameters that may be functions of frequency. The complete solution of this problem must be the subject of a special theoretical investigation. Such an investigation has been carried out very recently by S. M. Rytov. It has not yet been published and has not been compared with experimental data. Therefore we shall not present the results of this general theory, but shall use another method of calculation, which was applied earlier.^6,7

In this connection one should point out another approach, different from that set forth above, to the consideration of the phenomenon of light scattering in matter. This other approach, in the sense of obtaining a general formula for the total intensity of light scattered by adiabatic density fluctuations, is completely equivalent to the one considered earlier.

Debye^14, in calculating the heat capacity of a solid, regarded the latter as a continuum, and represented the thermal motion in it as an aggregate of elastic waves propagating in all possible directions. The lower limit of the wavelength of Debye elastic waves is determined by the distance between the atoms or molecules of the substance. Density fluctuations in the Debye interpretation of thermal motion will be represented as the result of interference of thermal waves.^1

The representation of thermal motion as an aggregate of elastic waves, used by L. I. Mandelstam^15 and Brillouin^16 for the consideration of important physical questions, proved very fruitful. It makes it possible to understand and explain a whole series of phenomena, including the scattering of light.

This approach makes it possible to obtain much valuable information about the kinetics of fluctuations. In this case the phenomenon of light scattering is regarded as diffraction of light by elastic thermal waves. In the body under study there exists a set of elastic waves of every possible wavelength and direction of propagation. If the scattering volume is illuminated by a parallel beam of light with wavelength \(\lambda\), and observed at an angle \(\theta\) to the direction of propagation of the incident light, then the light scattered in the direction \(\theta\) is the result of diffraction of light by another thermal wave of length \(\Lambda\), satisfying the Bragg condition^17:

\[ 2\Lambda n \sin \frac{\theta}{2}=\lambda, \]

where \(n\) is the refractive index of the medium. Hence

\[ \Lambda=\frac{\lambda}{2n\sin \frac{\theta}{2}}. \]

When illuminated by the blue line of the mercury spectrum, \(\lambda=4358\) Å, and when observed at the angle \(\theta=90^\circ\), \(\Lambda=2.3\cdot 10^{-5}\) cm \((n=1.5)\), or the frequency \(f_{90}=0.75\cdot 10^{10}\) cps (assuming the speed \(v=1.5\cdot 10^5\) cm/sec). When observing in the direction of propagation of the incident light \(\theta=0^\circ\), \(f_0=0\), and when observing in the opposite direction \(\theta=180^\circ\), \(f_{180}=1.05\cdot 10^{10}\) cps. Thus, \(1.05\cdot 10^{10}\) cps is the limiting high frequency of the elastic thermal wave that affects the process of light scattering.

with wavelength \(\lambda = 4358\,\text{\AA}\). The study of shorter elastic waves can be carried out by going over to shorter wavelengths of light, but even here one cannot advance very far. For some substances (for example, quartz crystals) \(f_{180}\) may turn out to be greater than the value cited here by only about a factor of five.

In a real experiment, because of the nonmonochromaticity of the exciting light and the finite aperture of the light beam incident on the scattering volume, a certain set of elastic waves \(\Delta\Lambda\) is used. However, \(\Delta\Lambda\) is a very small quantity, and in practice one may speak of a definite frequency of the elastic waves studied in a given experiment.

From what has been said it is clear that the scattering process is determined by frequencies \(f \sim 10^{10}\,\text{cps}\), while all the other frequencies of elastic waves in the scattering of visible light play no role (when observed at an angle of \(90^\circ\)).

Relying on Debye’s conception of thermal motion, L. I. Mandelshtam, at one of his seminars in the Physics Faculty of Moscow State University, calculated the intensity of the scattered light. The idea of this derivation is as follows.

Let us suppose that the change of the dielectric permittivity \(\varepsilon\) due to the propagation of an elastic thermal wave is small; the dielectric permittivity in the presence of a sound wave is then written as the sum of the value \(\varepsilon=\varepsilon_0\) and the change \(\Delta\varepsilon\) caused by hypersound: \(\varepsilon=\varepsilon_0+\Delta\varepsilon\). Accordingly, the electric and magnetic fields of the light wave may be represented in the form \(\mathbf{E}=\mathbf{E}_0+\mathbf{E}_1\) and \(\mathbf{H}=\mathbf{H}_0+\mathbf{H}_1\), where \(\mathbf{E}_1\) and \(\mathbf{H}_1\) are the field of the scattered light wave. Restricting ourselves to the first approximation, we obtain Maxwell’s equations in the following form:

\[ \left. \begin{aligned} c\,\operatorname{rot}\mathbf{H}_1&=\varepsilon_0\frac{\partial \mathbf{E}_1}{\partial t}+\frac{\partial}{\partial t}(\Delta\varepsilon\mathbf{E}_0),& \operatorname{div}\mathbf{E}_1&=-\frac{1}{\varepsilon_0}\operatorname{div}(\Delta\varepsilon\mathbf{E}_0),\\ -c\,\operatorname{rot}\mathbf{E}_1&=\frac{\partial \mathbf{H}_1}{\partial t},& \operatorname{div}\mathbf{H}_1&=0. \end{aligned} \right\} \tag{1.18} \]

Let us prescribe the incident wave in the form \(\mathbf{E}_0=\mathbf{a}e^{i(\omega t-\mathbf{k}\mathbf{r}')}\) (\(\mathbf{r}'\) is the distance from the center of the scattering volume to any of its points), \(\mathbf{k}\) is the optical wave vector. The deviation from the mean value of the dielectric constant in the hypersound wave with cyclic frequency \(\Omega\) is written as a function of time and coordinate in the following form:

\[ \Delta\varepsilon=\frac{\Delta\varepsilon_0}{2}\{e^{i(\Omega t-\mathbf{K}\mathbf{r}')}+e^{-i(\Omega t-\mathbf{K}\mathbf{r}')}\}, \]

where \(\mathbf{K}\) is the acoustic wave vector.

From Maxwell’s equations (1.18), for prescribed \(\mathbf{E}_0\) and \(\Delta\varepsilon\), we find the field of the scattered light wave, polarized in the plane of scattering, for the satellite with frequency \(\omega+\Omega\):

\[ \mathbf{E}_1=\frac{\Delta\varepsilon_0\mathbf{a}}{8\pi c^2}(\omega+\Omega)^2\frac{e^{-ikr}}{r}e^{i(\omega+\Omega)t}V. \]

An analogous expression is obtained for the satellite of frequency \(\omega-\Omega\). A plane wave polarized perpendicular to the plane of scattering (the vector \(\mathbf{E}\) oscillates in the plane of scattering), in the direction under consideration \((\theta=90^\circ)\), will not be scattered, which follows directly from the calculation. The intensity of the scattered light is \(I_{ad}=|\mathbf{E}_1|^2\), while \(I_0=a^2\), whence

\[ I=I_0\frac{\Delta\varepsilon_0^2}{64\pi^2c^4}\frac{V^2}{r^2}(\omega+\Omega)^4. \]

The total scattering coefficient \(R=\dfrac{I}{I_0}\dfrac{r^2}{V}\) for both satellites (assuming that their wavelengths differ very little from one another) will be equal to

\[ R=\frac{\pi^2 V}{2\lambda^4}\Delta\varepsilon_0^2 . \tag{1.19} \]

The amplitude of the change in the dielectric constant may be represented as follows:

\[ \Delta\varepsilon_0=\left(\rho\frac{\partial\varepsilon}{\partial\rho}\right)_s\frac{\Delta\rho_0}{\rho}. \tag{1.20} \]

The adiabatic value \(\left(\rho\dfrac{\partial\varepsilon}{\partial\rho}\right)_s\) has been taken because the process of propagation of sound of frequency \(f\sim 10^{10}\ \text{cps}\) is adiabatic. Propagation of sound in a liquid becomes isothermal at a frequency \(f\sim \dfrac{v^2}{\chi}\) (\(\chi\) is the coefficient of thermal conductivity, \(v\) is the speed of sound); thus \(f\sim 10^{12}\ \text{cps}^{18}\). The hypersonic waves participating in the scattering process have frequencies \(\sim 10^{10}\ \text{cps}\), and therefore are adiabatic.

According to the hydrodynamic theory of sound\(^{19}\)

\[ \left(\frac{\Delta\rho_0}{\rho}\right)^2=\frac{u^2}{v^2}=\frac{V\rho u^2}{V\rho v^2}, \]

where \(u\) is the velocity of oscillations of a particle of the liquid. Equating the energy of the sound wave to \(kT\) and taking into account that \(\beta_s=\dfrac{1}{\rho v^2}\), we obtain the final expression for the scattering coefficient

\[ R=\frac{\pi^2}{2\lambda^4}\left(\rho\frac{\partial\varepsilon}{\partial\rho}\right)_s^2 kT\beta_s . \tag{1.21} \]

From comparison of formula (1.21) with formula (1.11), obtained by thermodynamic means (if the latter is expressed through the scattering coefficient at \(\theta=90^\circ\)), it is seen that these expressions formally coincide, but in essence they differ.

Here, from the very nature of the derivation it is clear that \(\left(\rho\dfrac{\partial\varepsilon}{\partial\rho}\right)_s\) and \(\beta_s\) are quantities corresponding to the frequency of that sound wave whose diffraction of light is being considered. Thus, for determining the intensity of the scattered light, a formula has been obtained that is free from thermodynamic restrictions. To calculate the intensity by formula (1.21), as \(\left(\rho\dfrac{\partial\varepsilon}{\partial\rho}\right)_s\) and \(\beta_s\) one should take their values corresponding to the corresponding frequency of the process. The difference in the values calculated from (1.21) and (1.11) appears especially strongly when an appreciable dispersion of the speed of sound is found in the liquid. Examples may be such liquids as benzene, carbon disulfide, and carbon tetrachloride, in which an appreciable dispersion of the speed of sound of \(\sim 10\%\) was found (see § 4), and in which, therefore, the integral intensity of the shifted components of the fine structure should be approximately \(20\%\) lower than that calculated from thermodynamic parameters.

Since entropy fluctuations arise and dissipate much more slowly than pressure fluctuations, it is possible, with a known approximation, for the scattering coefficient on entropy fluctuations to use—

to use a thermodynamic calculation (formula (1.10)). (For this remark we are indebted to V. L. Ginzburg.) The extent to which such an assumption is admissible will be considered in more detail in § 6.

c) Intensity of light scattered by fluctuations of anisotropy. As a result of the calculations carried out above, relations have been obtained which express the intensity of light scattered by adiabatic and isobaric density fluctuations as functions of parameters measured experimentally. Relations (1.8) and (1.12) cannot be subjected to experimental verification, since the intensity of light scattered by fluctuations of anisotropy still remains unaccounted for.

Experiment shows that the intensity of light scattered by fluctuations of anisotropy in a large number of cases makes a substantial contribution to the total flux of scattered light.

It is impossible to calculate \(\overline{|\Delta \varepsilon'_{ik}|^2}\) in the same way as was done for \(\overline{\Delta \varepsilon^2}\), since the parameters characterizing \(\overline{\Delta \varepsilon_{ik}^{\prime\,2}}\) in a nonequilibrium state of a liquid are unknown.

A number of attempts to perform such calculations are known in the literature\(^{20,21,22,23,24,25}\). Of particular interest are the theories of M. A. Leontovich\(^{23}\) and A. I. Anselm\(^{25}\); however, at the present time these theories also cannot be used for exact quantitative calculations of the intensity. Therefore, in calculating the intensity of light scattered by fluctuations of anisotropy, we shall proceed differently.

It has already been mentioned above that scattering by density fluctuations in the case of a liquid leads to completely polarized scattered light. This result is obvious, since density fluctuations associated with pressure and entropy fluctuations do not disturb the isotropy of the medium.

Consequently, the observed depolarization of light scattered in a liquid is always due to depolarized scattered light caused by fluctuations \(\Delta \varepsilon'_{ik}\).

To take account of the intensity of light scattered by fluctuations of anisotropy, we proceed as follows: denote the intensity of the total light scattered by density fluctuations and anisotropy by \(J_1 = J + I\), where \(J\) is the intensity of light scattered by density fluctuations, and \(I\) is the intensity of light scattered by fluctuations of anisotropy.

If we now assume that the incident natural light propagates along the \(x\)-axis, the observation is made in the direction of the \(y\)-axis, and the \(z\)-axis is perpendicular to the scattering plane, then, obviously, \(J = J_z\) and \(I = I_x + I_z\). Thus, the total intensity of the scattered light will be:

\[ J_1 = J_z + I_x + I_z. \tag{1.22} \]

On the other hand, by definition the depolarization of the total scattered light is equal to

\[ \Delta = \frac{I_x}{J_z + I_z} \tag{1.23} \]

and the depolarization of the light scattered by fluctuations of anisotropy is

\[ r = \frac{I_x}{I_z}. \tag{1.24} \]

Let us now express \(J_1\) in terms of the experimentally measured quantity \(\Delta\), the calcula-

earlier value of \(J=J_z\) and \(r\). Eliminating \(I_x\) and \(I_z\) from (1.22), (1.23), and (1.24), we obtain:

\[ J_1=J_z+J_z\left[\frac{\Delta(1+r)}{r-\Delta}\right]. \tag{1.25} \]

Thus, an expression has been obtained for the total intensity of light scattered in a liquid.

G. Placzek\(^3\) carried out a detailed study of the scattering tensor and showed that, for purely “anisotropic scattering,” the depolarization \(r\) will be equal to \(6/7\) under the condition that \(\Delta\varepsilon_{ik}=\Delta\varepsilon_{ki}\), and that the scattering volume is illuminated by natural light. This conclusion of the theory is in good agreement with experiment\({}^{26,27,28}\). It should also be noted that in the above-mentioned works of M. A. Leontovich\({}^{23}\) and A. I. Anselm\({}^{25}\) one also obtains \(r=6/7\). We shall use this well-substantiated theoretical and experimental conclusion and substitute into formula (1.25) the value of \(r\) equal to \(6/7\). We obtain:

\[ J_1=J\left(\frac{6+6\Delta}{6-7\Delta}\right), \tag{1.26} \]

where the factor \(\dfrac{6+6\Delta}{6-7\Delta}\) is called the Cabannes factor\({}^{29}\), who introduced it for the first time. From (1.26) and (1.22) it also follows that the intensity of light scattered on anisotropy fluctuations is equal to

\[ I=J\frac{13\Delta}{6-7\Delta}. \tag{1.27} \]

Thus, the formula for the intensity of light scattered on density and anisotropy fluctuations, subject to experimental verification, has the form

\[ J=B\left[\left(\rho\frac{\partial\varepsilon}{\partial\rho}\right)_S^{2}\beta_S\rho kT+ \left(\frac{1}{b}\frac{\partial\varepsilon}{\partial T}\right)_p^{2} \frac{b^{2}kT^{2}}{C_p}\right]\cdot \left[\frac{6+6\Delta}{6-7\Delta}\right], \tag{1.28} \]

where

\[ B=J_0\frac{\omega^{2}V}{(4\pi r c')^{2}}\left(1+\cos^{2}\theta\right). \]

If condition (1.17) is satisfied with sufficient accuracy, then formula (1.28) goes over into the Einstein—Cabannes formula

\[ J=B\left(\rho\frac{\partial\varepsilon}{\partial\rho}\right)_T^{2} \beta_T kT\left(\frac{6+6\Delta}{6-7\Delta}\right). \tag{1.29} \]

Formula (1.29) has been subjected to experimental verification many times. However, before proceeding to compare the results of calculation with the experimental results, let us briefly consider the methods and results of measuring the intensity of scattered light or the scattering coefficient.

§ 2. ABSOLUTE MEASUREMENTS OF THE INTENSITY OF MOLECULAR LIGHT SCATTERING IN LIQUIDS. COMPARISON WITH THE CONCLUSIONS OF THE THEORY

The very difficult experimental measurements of the absolute intensity of scattered light, begun more than thirty years ago, continue up to the present time.

The first measurements of the absolute intensity of scattered light were carried out by Martin and Lermatov 30 (1922); these were followed by measurements by Cabannes and Dor 31, Raman and Rao 32 (1929), and Peyrot 33 (1938).

Owing to great experimental difficulties, absolute measurements could not encompass a large number of liquids, and therefore the authors listed above confined themselves to the study of one, two, and, in rare cases, a larger number of substances.

Particularly great attention is now being devoted to measurements of the absolute intensity of scattered light, since this question has acquired not only scientific but also great practical significance. The practical significance of measuring \(R\) consists in the fact that, as Debye 34 showed, the masses of macromolecules (polymers) can be determined by measuring the absolute intensity of the light scattered in solutions of these molecules. Debye’s method became effective and sufficiently simple because the measurement of the absolute intensity of light scattered in solutions was replaced by relative measurements. The intensity of the light scattered by the solution was measured relative to the intensity of the light scattered by a standard liquid (benzene, ether), for which the absolute value of the intensity of scattered light had to be reliably determined. Therefore new measurements of the absolute intensity of scattered light were undertaken, chiefly in those liquids that were used as standards in Debye’s method.

Among the new measurements one should mention the measurements of Bai 35 (1948), Voukuler 36 (1951), Carr and Zimm 37 (1950), Brice, Halwer, and Speiser 38 (1950), Doty and Steiner 39 (1950), Stamm and Bouton 40 (1953), Calvert, Nottinga 41 and Brice (1953), Maron and Lou 42 (1954), Kraut and Dondliker 43 (1955), Harrand 60, and others.

In all works carried out before 1939, as well as in works 35, 36, a photographic method was used to measure the scattering coefficient and the degree of depolarization of the scattered light.

In all the remaining, later works, a photoelectric method was used to measure these quantities. The photographic and photoelectric methods differ mainly in that in the former, in order to determine \(R\), the brightnesses of the exciting source and of the scattered light are compared, whereas in the photoelectric method light fluxes are compared.

In the photographic version of the method, the scattering coefficient 33, 36, 123 is equal to

\[ R = k_1 \frac{e_1}{e_2}, \tag{2.1} \]

where \(e_1\) and \(e_2\) are the illuminations of the photographic plate caused respectively by the scattered and exciting light; \(e_1\) and \(e_2\) are proportional to the brightnesses of the scattered light and of the exciting source; \(k_1\) is determined by known parameters of the apparatus.

In the photoelectric method \(R\) is expressed through light fluxes as follows:

\[ R = k_2 \frac{F_1}{F_2}, \tag{2.2} \]

where \(F_1\) and \(F_2\) are, respectively, the fluxes of scattered and exciting light. \(k_2\), like \(k_1\), is determined by the parameters of the apparatus. Thus, formulas (2.1) and (2.2) contain measured quantities.

Figure 1 gives a diagram of an apparatus for measurement by the photographic method 36. When a vessel with a scattering substance is placed in the apparatus, on the photographic plate there are obtained simultane-

…namely the blackening caused by the light scattered by the liquid and by the light branched off from the incident beam and attenuated a known number of times.

Next, instead of the vessel with the scattering liquid, a plate coated with magnesium oxide is placed at an angle of \(45^\circ\) to the direction of the incident light, as is done in the installation diagram shown in Fig. 1 (lower right), or a vessel with the same liquid is used, into which there is immersed

Fig. 1. Diagram of the apparatus for measuring the absolute intensity of light scattering by the photographic method \((B)\) (after J. Vaucouleurs). \(L\)—light source; \(1'\)—condenser; \(O_1\)—objective forming the image of \(S\) at the center of the vessel located in the thermostat; \(\Delta, \Delta', \Delta'', \Delta_1, \Delta_2\), and \(\Delta_3\)—diaphragms; \(K_1\) and \(K_2\)—cubes separating the light beams; \(Q\)—photometric wedge; \(T\)—diffusing surface; \(L_1, L_2\), and \(L''\)—standard attenuators; \(E\)—standard diffuser; \(S\)—slit; \(S'\)—slit image; \(W\)—Wollaston prism; \(G\)—analyzer with limb \(C\); \(Ch\)—camera.

a prism of total internal reflection, rotating the incident ray through \(90^\circ\) \({}^{33}\). Before this ray reaches the photographic plate, it is strongly attenuated a known number of times. On the same photographic plate there is then photographed the directly attenuated light and part of the light branched off from the incident beam, as in photographing the scattered light. From these two photographs one can obtain the ratios of the illuminations of the photographic plate caused by the scattered and the exciting light.

Figure 2 gives a diagram of a typical photoelectric setup \({}^{37}\) for determining the scattering coefficient \(R\). Figure 2a shows the diagram of the illuminating part of the setup; Fig. 2b shows the scattering vessel, the diaphragm limiting the aperture, and the system \(s\) sending the light to the photomultiplier \(PM\). Figure 2c shows that the photometer can occupy position \(I\), in which the scattered light or the light reflected from the magnesium plate \(DR\) is measured, or position \(II\), in which the photometer records the directly transmitted light.

In both types of apparatus, of course, the geometric conditions for comparing brightnesses and luminous fluxes must be strictly observed.

Fig. 2. Diagram of an apparatus for measuring the absolute intensity of scattered light by the photoelectric method \((F)\) (after Carr and Zimm).
\(H\)—light source; \(L, R, S\)—objectives and condensers; \(A\)—aperture for illuminating the photomultiplier \(PM\); \(D\)—diaphragm; \(F\)—light filter; \(h\) and \(W\)—cross sections of the illuminating beam; \(DR\)—magnesian plate (comparison body).

Tables I and II give the results of absolute measurements of \(R\) by photoelectric and photographic methods. The results of measurements of \(R\) obtained by comparing brightnesses \((B)\), and the results obtained by comparing luminous fluxes \((F)\), in most cases show a very considerable difference. Benzene is almost always used as the standard in determinations of macromolecular masses by the Debye method \(^{34}\), and therefore especially numerous measurements have been devoted to it. The indicated discrepancies in the value of \(R\) for benzene amount to approximately 50%, which cannot be explained by random errors of the two experiments. Evidently, what is involved here is some gross systematic error.

Recently a discussion of this problem has arisen in the literature \(^{44,45,46}\). Carr and Zimm \(^{37}\) believe that the “high” values of \(R\) in benzene are correct \((R = 48.5 \cdot 10^{-6}\ \mathrm{cm}^{-1},\ t = 25^\circ \mathrm{C})\), and not the “low” ones \((R = 31.8 \cdot 10^{-6}\ \mathrm{cm}^{-1},\ t = 20^\circ \mathrm{C})\). Criticizing the earlier measurement methods, these authors point out that in works using the photographic technique, where the “low” values of \(R\) were obtained, corrections for the refractive index and the apparent volume of the scattering liquid were not taken into account.

It was shown in \(^{36,42}\) that the corresponding corrections will have the following form:

correction to the refractive index

\[ C_n=n^2\left[1-\frac{r'(n-1)}{rn}\right]. \tag{2.3} \]

and correction to the visible volume

\[ C_v=1-\frac{r'(a+b)/(r-r')}{2[nl+r'(a+b)/(r-r')]} . \tag{2.4} \]

The meaning of the quantities entering into (2.3) and (2.4) is clear from Fig. 3. According to \(^{34,42}\), the directly measured quantity must be multiplied by \(C_n \cdot C_v\).

Table I

Measured and calculated scattering coefficients \(R\) for benzene, reduced to \(20^\circ\mathrm{C}\) and \(\lambda\,4358\,\text{\AA}\).

Authors of the work	Method	Experimental values \(R\cdot 10^6\ \mathrm{cm}^{-1}\)	Calculated \(R\cdot 10^6\ \mathrm{cm}^{-1}\): static formula (2.6)	Calculated \(R\cdot 10^6\ \mathrm{cm}^{-1}\): dynamic, formula (2.6)	Calculated \(R\cdot 10^6\ \mathrm{cm}^{-1}\): dynamic, formula (2.5)	Calculated \(R\cdot 10^6\ \mathrm{cm}^{-1}\): formula (2.6)	Calculated \(R\cdot 10^6\ \mathrm{cm}^{-1}\): formula (2.5)
Martin and Lehrman \(^{30}\)	Method of comparison of brightness \((B)\)	32.8	37.6	36.0	31.8	38	42.4
Raman and Rao \(^{32}\)	Method of comparison of brightness \((B)\)	55.1	37.6	36.0	31.8	38	42.4
Cabannes and Dor \(^{31}\)	Method of comparison of brightness \((B)\)	35.6	37.6	36.0	31.8	38	42.4
Peyro \(^{33}\)	Method of comparison of brightness \((B)\)	33.8	37.6	36.0	31.8	38	42.4
Vaucouleur \(^{36}\)	Method of comparison of brightness \((B)\)	31.8	37.6	36.0	31.8	38	42.4
Sunaida Bai \(^{35}\)	Method of light streams \((F)\)	29.14	37.6	36.0	31.8	38	42.4
Bleker, Badger and Gilman \(^{120}\)	Method of light streams \((F)\)	46.8	37.6	36.0	31.8	38	42.4
Carr and Zimm \(^{37}\)	Method of light streams \((F)\)	46.8	37.6	36.0	31.8	38	42.4
Brice, Halwer, Sponer \(^{38}\)	Method of light streams \((F)\)	46.8	37.6	36.0	31.8	38	42.4
Doty and Steiner \(^{39}\)	Method of light streams \((F)\)	46.5	37.6	36.0	31.8	38	42.4
Destriau \(^{121}\)	Method of light streams \((F)\)	46.7	37.6	36.0	31.8	38	42.4
Stamm and Bouton \(^{40}\)	Method of light streams \((F)\)	44.4	37.6	36.0	31.8	38	42.4
Trapp and Hermans \(^{61}\)	Method of light streams \((F)\)	40.0	37.6	36.0	31.8	38	42.4
Maron and Lou \(^{42}\)	Method of light streams \((F)\)	48.1	37.6	36.0	31.8	38	42.4
Kallberg, Notting and Brice \(^{41}\)	Method of light streams \((F)\)	48.3	37.6	36.0	31.8	38	42.4
Oster \(^{122}\)	Method of light streams \((F)\)	45.5	37.6	36.0	31.8	38	42.4
Harrand \(^{60}\)	Method of light streams \((F)\)	27.02	37.6	36.0	31.8	38	42.4

We are not speaking here of other corrections that must be introduced in the final determination of \(R\) and that do not give rise to perplexities or objections. The correction \(C_n\) under ordinary experimental conditions, when \(r'\ll r\), tends to the value \(C_n\to n^2\), whereas under the same condition \(C_v\to 1\).

In the discussion mentioned, Rousset and Loup \(^{44,46}\) showed that, in the photographic method of measuring \(R\), when brightness quantities are compared,

no correction \(C_n\) need be introduced. Their reasoning is approximately as follows. Suppose that the scattering volume is bounded by a diaphragm \(D\) (Fig. 4), placed in a liquid with refractive index \(n\). Let us denote the brightness of the diaphragm \(D\) in the direction \(OA\) by \(B_1\). The brightness of the image of this diaphragm will no longer be \(B_1\), but \(B_1/n^2\), and the illuminance

Fig. 3. On corrections for the refractive index (3a) and for the effective volume (3b) (after Carr and Zimm).

Fig. 4. On the discussion of corrections for the refractive index (after Rousse and Loppe).

of the photographic image will be reduced by a factor of \(n^2\). If now a magnesium plate is immersed in the scattering liquid, then the brightness of the diaphragm in the direction \(OA\) will be \(B_2/n^2\), and the ratio of the illuminances of the two images on the photographic plate will still be equal to \(B_1/B_2\), and we shall obtain the correct value of \(R\). However, in some photographic setups the comparison body (the magnesium plate) was not placed in the same liquid, but was in air. If, instead of the scattering vessel, one places the magnesium plate in air, then the source will not be focused on it, and no definite conclusions can be drawn about its illuminance. To focus the source at the point \(O\), one must change the position of the condenser or of the light source.

If, as a result of the adjustment, the solid angle at the point \(O\) now turns out to be the same as in the case when, instead of the plate, there was a scattering liquid, then the brightness of the magnesium plate is reduced by a factor of \(n^2\), while the brightness of the image of the diaphragm placed in air is increased by a factor of \(n^2\) in comparison with the brightness of the image of the diaphragm \(D\) illuminated by the magnesium plate immersed in the liquid. Thus, in this case as well the result remains the same. Analysis of the setups described in \(^{31,33,36}\) leads authors \(^{44,46}\) to the conclusion that the measurements obtained on these setups do not need any

Table II

Measured and calculated values of the scattering coefficient for different liquids, \(R \cdot 10^6\ \mathrm{cm}^{-1}\)

Authors of measurements	Water 3650 Å	Water 4047 Å	Water 4358 Å	Water 5641 Å	Ether 4358 Å	Ether 5641 Å	Carbon tetrachloride 4358 Å	Carbon tetrachloride 5461 Å	Carbon disulfide 4358 Å	Carbon disulfide 5461 Å	Toluene 4356 Å	Toluene 5461 Å
Martin and Lehrman \(^{30}\)	—	—	2	—	—	9,2 (20°)	—	—	—	—	—	—
Raman and Rao \(^{32}\)	—	—	3,08 (32°)	—	—	—	—	—	—	—	—	—
Voukler \(^{36}\)	—	—	—	—	10,1 (20°)	4,1 (20°)	—	—	—	—	—	—
Krout and Dandliker \(^{42}\)	6,8	4,05	2,89	1,05	—	—	—	—	—	—	—	—
Stamm and Bouton \(^{40}\)	—	—	—	—	—	—	15,40 (25°)	5,79 (25°)	—	—	32,2 (28°)	11,8 (25°)
Karr and Zimm \(^{57}\)	—	—	—	—	—	—	15,5 (30°)	5,78 (30°)	—	—	—	—
Maron and Lou \(^{42}\)	—	—	—	—	—	—	14,9 (25°)	5,32 (25°)	—	—	60,3 (25°)	20,5 (25°)
Blecker, Bodger, and Gilmont \(^{120}\)	—	—	—	—	—	—	—	—	151 (25°)	47 (25°)	—	—
Formula (2,6)	—	—	2,3	—	11,6	—	—	—	166	—	41,9	—
Formula (2,6)	—	—	2,0	—	9,75	—	—	—	—	—	44,1	—
Formula (2,5)	—	—	2,1	—	10,3	—	—	—	147,11	—	46,0	—

The calculation of the theoretical values of \(R\) was made for \(20^\circ\mathrm{C}\) and \(\lambda = 4358\) Å. The temperature indicated in parentheses is in degrees Celsius.

additional corrections, and the result of these measurements must be treated with complete confidence.

Thus, the discrepancies observed cannot at present find a convincing explanation, and new photographic and photoelectric measurements with a detailed discussion of the experiment as a whole are undoubtedly necessary.

Of substantial importance is the comparison of the measured values of \(R\) with those calculated from theoretical relations (in particular, for establishing which value of \(R\) should be given preference). However, in calculating \(R\) from the formulas given in § 1, certain difficulties are encountered.

Formula (1.8), written for the scattering coefficient at an observation angle of \(90^\circ\), has the following form:

\[ R=\frac{\pi^2}{2\lambda^4} \left[ \left(\rho\frac{\partial\varepsilon}{\partial\rho}\right)_s^2 \beta_s kT\rho+ \left(\frac{1}{b}\frac{\partial\varepsilon}{\partial T}\right)_\rho^2 \frac{b^2 kT^2}{C_p} \right] \left[\frac{6+6\Delta}{6-7\Delta}\right]. \tag{2.5} \]

When conditions (1.16) and (1.17) are fulfilled, formula (2.5) becomes Einstein’s formula

\[ R=\frac{\pi^2}{2\lambda^4} \left(\rho\frac{\partial\varepsilon}{\partial\rho}\right)_T^2 \beta_T kT \left(\frac{6+6\Delta}{6-7\Delta}\right). \tag{2.6} \]

This latter expression, as was said, has been subjected many times to experimental verification.

Here the question arises as to which of the measured values of the quantities \(\beta_s\) or \(\beta_T\) and \(\left(\rho\dfrac{\partial\varepsilon}{\partial\rho}\right)_s\) or \(\left(\rho\dfrac{\partial\varepsilon}{\partial\rho}\right)_T\) should be used in calculating \(R\). From what was set forth in § 1 it is clear that \(\beta\) and \(\left(\rho\dfrac{\partial\varepsilon}{\partial\rho}\right)\) must be measured at the frequency of the thermal waves that are significant for the scattering of light, i.e. at a frequency \(f\sim 10^{10}\) cps. Formerly, dispersion of the velocity of sound had not been detected, and therefore static values of \(\beta\) were used. However, the use of the static values \(\left(\rho\dfrac{\partial\varepsilon}{\partial\rho}\right)_T\) and \(\beta_T\) for calculating \(R\) led to a discrepancy between the measured and calculated values\(^{28}\). This discrepancy between theory and experiment compelled the assumption that the quantities \(\left(\rho\dfrac{\partial\varepsilon}{\partial\rho}\right)_s\) and \(\left(\rho\dfrac{\partial\varepsilon}{\partial\rho}\right)_T\), measured under static conditions, cannot be used to calculate the intensity of scattered light\(^{29}\), for which processes changing with a frequency \(\sim 10^{10}\) cps are responsible. However, for a change in the optical dielectric permittivity \((\varepsilon=n^2)\), for which processes with a frequency \(\sim 10^{15}\) cps are significant, the static value \(\left(\rho\dfrac{\partial\varepsilon}{\partial\rho}\right)\) can hardly differ appreciably from this quantity measured at a frequency \(\sim 10^{10}\) cps, unless, of course, there are as yet unknown processes relaxing at a frequency lower than \(10^{10}\) cps.

This problem has been little discussed\(^{29,7}\). One may point, in particular, to Kaban’s remark\(^{29}\) that a static measurement of \(\left(\rho\dfrac{\partial\varepsilon}{\partial\rho}\right)\), performed in a large volume, cannot serve to characterize \(\left(\rho\dfrac{\partial\varepsilon}{\partial\rho}\right)\) at a point (the volume of fluctuations). The sense of this remark may be seen in the fact that, in principle, the values of \(\left(\rho\dfrac{\partial\varepsilon}{\partial\rho}\right)\) measured statically and dynamically may differ if one considers that this quantity is a function of some-

second region.^48 This region must have an order of magnitude determined by intermolecular interaction. Taking this circumstance into account, we may write that, when measuring \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)\) at a sound frequency corresponding to the wavelength \(\Lambda\),

\[ \rho \frac{\partial \varepsilon}{\partial \rho} = \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_0 \left\{1+k\left(\frac{a}{\Lambda}\right)^2\right\}, \tag{2.7} \]

where \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_0\) is the static value of this quantity, \(a\) is a constant having the dimension of length and equal, in order of magnitude, to \(10^{-7}\ \mathrm{cm}\), \(k\) is a dimensionless constant \(\sim 1\), whose sign is not known in advance. For frequencies \(50\), \(10^7\), and \(10^{10}\ \mathrm{Hz}\), the additional term in the braces is of the order of \(\sim 10^{-20}\), \(\sim 10^{-10}\), and \(\sim 10^{-4}\), respectively. Thus, taking this circumstance into account does not explain the effect, and the question requires further consideration.

Cabannes and a number of other authors, after comparing \(R\), calculated on the basis of static data, with experiment, refrain from using the static values of \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)\) and turn to calculating this quantity from formulas of the Lorentz—Lorenz^49 or Gladstone and Dale^50 type.

The essential point in calculating the dependence of \(\varepsilon\) on \(\rho\), or, what is the same thing, the quantity \(\rho \dfrac{\partial \varepsilon}{\partial \rho}\), consists in finding the magnitude of the internal, or molecule-acting, field. The solution of this question by A. H. Lorentz^49 is well known. He arrived at the expression

\[ \frac{\varepsilon-1}{\varepsilon+2}=c'\rho, \tag{2.8} \]

where \(c'\) is a constant. Einstein used relation (2.8) to calculate \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_T\), which, along with other parameters, determines the intensity of the scattered light according to formula \((2.6')\). From formula (2.8) we obtain:

\[ \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_T = (\varepsilon-1)\left(\frac{\varepsilon+2}{3}\right). \tag{2.9} \]

Then the formula for the coefficient of light scattering takes the form

\[ R= \frac{\pi^2}{2\Lambda^4} (\varepsilon-1)^2 \left(\frac{\varepsilon+2}{3}\right)^2 kT\beta_T \left(\frac{6+\Delta}{6-7\Delta}\right). \tag{2.10} \]

The scattering coefficients calculated from (2.10) turn out to be considerably larger than those measured experimentally by method \((B)\), and in many cases larger than those calculated from (2.6) using the static values of \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_T\).

Rocard^51 criticized Einstein’s method of calculating \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_T\) from formula (2.8) by simple differentiation. He pointed out that the expression \(\varepsilon+2\) in formula (2.8) appears as a result of taking into account the action of molecules situated outside the physically infinitely small sphere (the Lorentz sphere) surrounding the molecule under consideration. Fluctuations of \(\varepsilon\) in the substance outside the selected sphere will have only a very small influence on the change of the field.

inside the sphere; therefore \(\varepsilon\) in the term \(\varepsilon+2\) must be regarded as practically constant. Consequently, only the factor \(\varepsilon-1\), due to the molecules contained inside the small sphere, where the density fluctuations prove to be substantial, need be differentiated.

In this case, putting \(\varepsilon+2=\mathrm{const}\), we obtain:

\[ \varepsilon-1=c'\rho . \tag{2.11} \]

Then

\[ \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_T=\varepsilon-1 \tag{2.12} \]

and formula (2.6) will take the following form:

\[ R=\frac{\pi^2}{2\lambda^4}(\varepsilon-1)^2 kT\left(\frac{6+6\Delta}{6-7\Delta}\right). \tag{2.13} \]

Calculations by (2.13) for certain substances are in satisfactory agreement with experiment; however, for a whole series of other substances the discrepancy still remains very large. The discrepancy between the experimental data and those calculated from (2.13) is interpreted even to the present time as a shortcoming of Einstein’s theory of light scattering in liquids. In particular, Vaucouleurs\(^{52}\) and Rocard\(^{53}\) assume that the theory does not take into account slow thermodynamic density fluctuations, and they attempt to calculate these fluctuations from molecular considerations. It seems to us that from the derivation of the scattering formula given in § 1 it is clear that all fluctuations \(\Delta\varepsilon\) (slow and fast) of a “thermodynamic” character are fully taken into account in formulas (1.8) and (1.12).

The only thing that could be at issue here is the scattering of light by fluctuations \(\Delta\varepsilon'\), which, according to V. L. Ginzburg\(^{2}\), can in principle occur and are not reducible to fluctuations of anisotropy and density. However, from the discrepancy of formula (2.13) with experiment it is hardly possible to draw any conclusions about the imperfection of the thermodynamic theory of light scattering in liquids, primarily because the application of formulas (2.8) and (2.11) to the calculation of \(\rho \dfrac{\partial \varepsilon}{\partial \rho}\) in a liquid is completely unjustified, despite certain refinements introduced\(^{51}\) into formula (2.8). The inapplicability of formulas (2.8) and (2.11) to liquids follows from the inadmissibility of the assumption that the molecules of a liquid are arranged completely chaotically. All the more incorrect, in the case of a liquid, is the neglect of molecular dimensions (as is assumed in the derivation of (2.8)), since the distance between molecules is of the order of their dimensions.

Therefore, the discrepancy or agreement of formulas (2.10) and (2.13) with experiment can serve neither as a refutation nor as a confirmation of the statistical theory of light scattering in a liquid under consideration. There exists also a whole series of empirical relations of the type of relations (2.8) and (2.11); however, they too are just as unsatisfactory as those just considered, and therefore we shall not discuss them in greater detail.

The present state of the theory of the liquid state is such that solving the problem of the relation of the optical dielectric permittivity to the density of the medium does not appear possible. On the other hand, formulas (2.8) and (2.11), which in application to a liquid must be regarded as empirical, prove unsatisfactory in application to problems of light scattering.

Therefore, it appears that, in the present situation, it is best to use the experimental values of \(\rho \dfrac{\partial \varepsilon}{\partial \rho}\), choosing, however, such a method for determining this quantity which is based—

is based on a phenomenon which, by its nature, is close to the phenomenon of light scattering.

Proceeding from this, a method was developed for determining $\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_s$ from the phenomenon of diffraction of light by ultrasound ^54,55,56. At present it is not possible to generate artificially sound waves at a frequency of $10^{10}$ cps; therefore sound of frequency $10^7$ cps was used.

The obtained results of measurements of $\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_s$ were used to calculate $R$ from formula (2.61) and were published in papers ^57,58. Subsequent ultrasonic investigations ^59 made it possible to estimate, rather roughly, the influence of the acoustic wind on the measured quantity $\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_s$. As a result of this estimate, corrected values of $\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_s$ at a frequency of $10^7$ cps were obtained. These values are given in Table III, from which it is seen that for benzene and toluene the difference between the static and dynamic values of $\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)$ is practically absent. Further refinements are needed of the estimate made for the influence of the acoustic wind. In the other cases studied, the difference between the static and dipole quantities has noticeably decreased, but has remained. Since, as has been pointed out more than once ^57,58, the physical cause of the dispersion of $\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)$ at frequencies $\sim 10^7$ cps is not clear, it may be thought that the remaining difference is determined by unaccounted-for systematic errors of the static and dynamic experiments or by an inaccurate allowance for the influence of the acoustic wind.

Calculation of $R$ by formula (2.6) for benzene using static values of $\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_T$, $\beta_T$ gives values appreciably larger than follows from the most accurate photographic measurements of the scattering coefficient ^36. However, if for calculating $R$ one uses the more exact formula (2.5) and takes into account the dispersion of the velocity of sound (see § 4), which appreciably changes the value of $\beta_s$ at frequencies $\sim 10^{10}$ cps, it turns out that the result of the calculation is in very good agreement with the result of the photographic measurements.

Below we discuss the results of various measurements and calculations of the scattering coefficient $R$ for benzene and some other liquids.

Table I gave all the measurements known to us of the absolute value of the scattering coefficient $R$ in benzene. On the basis of data ^33 on the temperature dependence $R\left(\dfrac{1}{R}\dfrac{\Delta R}{\Delta T}=0.007\right)$, all results have been reduced to $20^\circ$ C and to the wavelength 4358 Å (on the basis of Rayleigh’s law). The results of measuring $R$ by the method of comparison of brightnesses ($B$) are grouped around the value 32—34, with the exception of the data of Raman and Rao ^32, apparently connected with an experimental error ^33. Most of the results obtained by the method of comparison of light fluxes ($F$), with surprisingly small deviation, lead to the value 46.5. Exceptions are the data of C. Bay ^35, Trapp and Hermans ^61*) and Harrand ^60, which in magnitude are closer to the results obtained by the method ($B$). The results of absolute measurements of $R$ for other liquids were given in Table II.

*) The author of this review is acquainted with the results of ^61 from paper ^47.

Table III

Some parameters for calculating scattering coefficients for \(20^\circ\mathrm{C}\), \(\lambda 4358\,\text{Å}\)

Substance	\(\rho,\ \mathrm{g/cm^3}\)	\(n\)	\(10^5 b=\dfrac{1}{V}\dfrac{dV}{dT},\ \mathrm{deg}^{-1}\)	\(C_p\cdot 10^{-7},\ \mathrm{erg/deg}\)	\(\beta_s\cdot 10^{12},\ \mathrm{bar}^{-1}\)	\(\beta_T\cdot 10^{12},\ \mathrm{bar}^{-1}\)	\(\left(\dfrac{1}{\varepsilon}\dfrac{\partial \varepsilon}{\partial T}\right)_\rho\)	\(\left(\rho\dfrac{\partial \varepsilon}{\partial \rho}\right)_s\) dynamic	\(\left(\rho\dfrac{\partial \varepsilon}{\partial \rho}\right)_T\) dynamic	Static **) \(\left(\rho\dfrac{\partial \varepsilon}{\partial \rho}\right)_T\)	\(\Delta\cdot 10^9\)
Water	0.997	1.340	20.70	4.18	45.7	45.7	−1.075	0.82	0.82	0.868	8.8
Methyl alcohol	0.791	1.338	119	2.56	100.5	127.1	−0.875	0.82	0.83	0.902	6.0
Ethyl alcohol	0.789	1.369	110.1	2.39	92.8	109.5	−1.031	0.95	0.96	1.041	6.7
Ether	0.713	1.364	162	2.27	138.7	184.2	−0.98	0.89	0.91	0.989	9.2
Benzene	0.879	1.527	121.5	1.70	52.6*)	93.6	−1.635	1.56	1.58	1.62	42
Carbon disulfide	1.262	1.697	118	1.0	49.5*)	92.0	−2.33	2.39***)	—	2.37	62
Toluene	0.865	1.521	106	1.68	70.0	88.4	−1.78	1.60	1.645	1.602	48

*) Calculated from the velocity of hypersound.

**)
\[ \left(\rho\frac{\partial \varepsilon}{\partial \rho}\right)_T = 2n\left(\rho\frac{\partial n}{\partial \rho}\right)_T \]
—the value is taken for the sodium \(D\)-line.

***) \(\left(\rho\dfrac{\partial \varepsilon}{\partial \rho}\right)_s\) for carbon disulfide was obtained from static measurements by recalculation.

A comparison of the experimental results obtained with the theoretical calculation (formulas (2.5) and (2.6)) is possible only in the case where the quantities known from experiment are \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_S\), \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_T\), \(\beta_S\), \(\beta_T\), and other parameters determining \(R\).

It has already been said above that the calculation of \(R\) using the value \(\rho \dfrac{\partial \varepsilon}{\partial \rho}\), computed from the Lorentz–Lorenz, Gladstone–Dale, or other similar relations, cannot be justified. The value \(\rho \dfrac{\partial \varepsilon}{\partial \rho}\) must be taken from static or, still better, dynamic measurements of this quantity.

A number of authors\(^{37,42}\), however, together with \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_T\), use the value \(\left(\dfrac{1}{b}\dfrac{\partial \varepsilon}{\partial T}\right)_\rho\). As is evident from relation (1.14), this can be done if the difference between \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_S\) and \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_T\) is very small, or else, for a rough estimate of \(R\), if there are no other possibilities. Strictly speaking, neither \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_T\) nor \(\left(\dfrac{1}{b}\dfrac{\partial \varepsilon}{\partial T}\right)_\rho\) determines the intensity of the scattered light. As is clearly seen from formula (2.5), the intensity \(J_p\) of the light scattered by isobaric density fluctuations is determined by \(\left(\dfrac{1}{b}\dfrac{\partial \varepsilon}{\partial T}\right)_\rho\) and \(c_p\), while the intensity \(J_s\) of the light scattered by adiabatic density fluctuations is determined by the quantities \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_S\) and \(\beta_S\).

Thus, on the basis of formula (2.5), one may write separately the expressions for \(R_s\) and \(R_p\) (the scattering coefficients corresponding to \(J_s\) and \(J_p\)), and also \(R_a\), the scattering coefficient due to fluctuations of the orientation of anisotropic molecules:

Table IV

Scattering coefficients of light on adiabatic and isobaric density fluctuations and anisotropy fluctuations for several liquids

Substance	\(R_s \cdot 10^6\ \mathrm{cm}^{-1}\)	\(R_p \cdot 10^6\ \mathrm{cm}^{-1}\)	\(R_a \cdot 10^6\ \mathrm{cm}^{-1}\)	\(R \cdot 10^6\ \mathrm{cm}^{-1}\)
	1	2	3	4
Water	1.70	0.02	0.365	2.08
Methyl alcohol	3.68	0.96	0.65	5.19
Ethyl alcohol	3.90	1.105	0.79	5.80
Ether	6.04	2.59	1.67	10.30
Benzene	7.05	4.39	20.4	31.8
Toluene	9.66	4.15	32.6	46.48

\[ R_s=\frac{\pi^2}{2\lambda^4}\left(\rho\frac{\partial \varepsilon}{\partial \rho}\right)_S^2 \beta_S kT, \tag{2.14} \]

\[ R_p=\frac{\pi^2}{2\lambda^4}\left(\frac{1}{b}\frac{\partial \varepsilon}{\partial T}\right)_\rho^2 kT\,\frac{b^2T}{\rho C_p}, \tag{2.15} \]

\[ R_a=(R_s+R_p)\cdot \frac{13\Delta}{6-7\Delta}. \tag{2.16} \]

In Table IV, by way of example, the results of calculating \(R_s\), \(R_p\), and \(R_a\) are presented

for some liquids for which \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_S\) was measured, and for carbon disulfide, for which \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_S\) was calculated from \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_T\) and \(\left(\dfrac{1}{b}\dfrac{\partial \varepsilon}{\partial T}\right)_p\), according to formula (1.14).

The values of the constants used for calculating \(R\), as well as those used subsequently, were given in Table III.

It must be specially noted that the speed of sound in benzene, carbon disulfide, and carbon tetrachloride, measured from the components of the fine structure (see § 4), differs appreciably from the values obtained in ultrasonic measurements. This also affects the values of the adiabatic compressibility for these liquids.

It is clear that, for calculating \(R\), one must use the adiabatic value of the compressibility, determined from the hypersonic velocity \(v\) \(\left(\beta_s=\dfrac{1}{\rho v^2}\right)\). Therefore the part of the total \(R\) expressed in \(R_s\) will, for benzene, be 20% smaller than when the static value \(\beta_s\) is used.

The calculation of \(R\) for benzene (Table I) was carried out according to formula (1.6), using the static values \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_T\) and \(\beta_T\) (column 4). Column 5 gives the calculation by the same formula using the dynamic value \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_T\) and the static value \(\beta_T\). Column 6 gives the calculation by formula (2.5) using the dynamic value \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_S\) and the value \(\beta_s\) calculated from the hypersonic velocity in benzene. The following two columns give, respectively, the calculations of Carr and Zimm \(^{37}\) and of Maron and Lou \(^{42}\). The calculation by the more accurate formula (2.5), using the value of \(\beta_s\) corresponding to the light-scattering experiment, gives a result in excellent agreement with the most accurate measurements by method \((B)\). The calculation by (2.6) gives values close to the data of method \((B)\). Strongly differing values of \(R\) were obtained in calculation \(^{42}\), where the quantities \(\left(\dfrac{1}{b}\dfrac{\partial \varepsilon}{\partial T}\right)_p\) were used, which are greatly overestimated \(^{62}\) in comparison with the data \(^{11,63,64,65}\), and the calculation was carried out according to formula (2.6) with the replacement of \(\left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_T\) by \(\left(\dfrac{1}{b}\dfrac{\partial \varepsilon}{\partial T}\right)_p\).

There is no doubt that the calculated data for \(R\) placed in column 6 of Table I are the most accurate. Therefore one should conclude that the measurements obtained by method \((F)\) do not agree with the results of the calculation. This conclusion, however, cannot serve as unequivocal proof of the erroneousness of the data obtained by method \((F)\).

Measurements of \(R\) in benzene simultaneously by methods \((B)\) and \((F)\) showed that both methods give mutually consistent values \(^{60}\). However, these values are so low (see Table I) that they are puzzling.

§ 3. SPECTRAL COMPOSITION OF RAYLEIGH-SCATTERED LIGHT

The spectral composition of Rayleigh-scattered light is much richer than the spectral composition of the exciting light. If the scattering volume of a liquid is illuminated with monochromatic light, then in the spectrum of the scattered light, besides the exciting frequency, there will be present shifted frequencies and a continuous spectrum extending over \(100\text{–}150\ \mathrm{cm}^{-1}\) on both sides of the exciting line. In the spectrum of the scattered light there appear

also the lines of combination scattering. The continuous spectrum and the lines of combination scattering are not considered at all in this review.

The complex composition of the Rayleigh-scattering spectrum owes its origin to the changes undergone in time by thermal fluctuations; as a result of the change of fluctuations in time (their appearance and dissipation), a light wave interacting with a fluctuation proves to be modulated^15,70; moreover, depending on the character of the fluctuations, the modulated light wave will have a different spectral composition.

Each kind of fluctuation imposes its own characteristic imprint on the frequency change of the scattered wave. This circumstance makes it possible to study subtle questions in the kinetics of various fluctuations. L. I. Mandelstam^15 and L. Brillouin^16 showed theoretically, in particular, that adiabatic density fluctuations (thermal elastic waves) in a crystal give in the spectrum of scattered light a doublet, each line of which is separated from the unshifted frequency by a distance \(\Delta \nu\), where

\[ \frac{\Delta \nu}{\nu} = 2n\,\frac{v}{c}\,\sin \frac{\theta}{2}, \tag{3,1} \]

where \(\nu\) is the frequency of the incident light. Thus, the magnitude of the splitting \(\Delta \nu\) coincides with the frequency of the modulating elastic wave.

In work^15 the question of light scattering by isobaric density fluctuations was also considered. In this case an unshifted component will be observed in the scattered light.

The width of the components due to light scattering by adiabatic fluctuations, and the width of the unshifted component, will be determined respectively by the absorption coefficient of sound of the frequency under consideration and by the coefficient of thermal diffusivity^66,67,2.

The width of the spectrum caused by the change in time of orientation fluctuations is determined by the relaxation time of the anisotropy^23,68,69. The magnitude of the splitting \(\Delta \nu\) is very small (\(0.1\)—\(1.5\ \mathrm{cm}^{-1}\)), but quite detectable on instruments of high resolving power, although experiments of this kind are accompanied by difficulties, mainly because of the insignificant intensity of the scattered light.

The effect of the splitting of the Rayleigh line in crystals (quartz), predicted by L. I. Mandelstam^15, was discovered in the experiments of G. S. Landsberg, L. I. Mandelstam, and in the experiment of E. F. Gross^71, who, at the suggestion of G. S. Landsberg and L. I. Mandelstam and in parallel with them^70, investigated the fine structure of the Rayleigh line on a spectroscope of high resolving power.

In subsequent experiments by E. F. Gross^71,72 the fine structure of the scattering line in liquids was discovered. The result of these experiments shows that the damping of thermal waves in a liquid is not so great as to make observation of the fine structure impossible. In a liquid a triplet was found, in which the central component was distinguished by considerable intensity in comparison with the shifted lines.

L. D. Landau and G. Placzek^73 found a formula expressing the ratio of the integral intensities of the central component to the intensity of the Mandelstam—Brillouin components through the thermodynamic value of the heat capacities.

Owing to great difficulties, the number of experimental works devoted to the investigation of fine structure is comparatively small. Their number does not exceed 15. The main theoretical investigations of the fine ...

structures were carried out by L. I. Mandelstam\(^ {15,66,74}\), G. S. Landsberg\(^ {74,75}\), M. A. Leontovich\(^ {23,66,67,74}\), V. L. Ginzburg\(^ {2,76,77}\), L. D. Landau\(^ {73}\), V. V. Vladimirsky\(^ {78,79,80}\), G. P. Motulevich\(^ {81}\), G. S. Plachek\(^ {3,73}\), and L. Brillouin\(^ {16}\). Experimental investigations on the qualitative side confirmed the principal predictions of the theory. In those cases, however, where even somewhat reliable quantitative data were required from experiment, the experiments mentioned, with rare exceptions, could not provide such data. As a result of quantitative comparison of theory with experiment, such considerable discrepancies were obtained that they cast doubt either on the theory, or on the experiment, or on both together.

Such a situation arises, for example, in comparing with experiment the Landau–Plachek relation; to some extent this also applies to the verification of relation (3.1) and to the measurement of the polarization of fine-structure lines.

Fig. 5. Schematic of an apparatus for photographing the fine structure of a scattering line in two mutually perpendicular polarization directions. \(C\)—vessel for the scattering liquid; \(L_1, L_2\)—light sources (mercury lamps of low pressure); \(O_1\)—condenser; \(O_2\) and \(O_3\)—objectives; \(F\!-\!P\)—Fabry–Perot interferometer; \(P\)—photographic plate.

The analysis carried out of experimental and theoretical investigations of this question showed that the experimental installations and methods of processing experimental data, as well as the theory of the phenomenon, require substantial improvement\(^ {6,7}\). Therefore new investigations, both experimental and theoretical in character, were undertaken, which in certain points brought clarification to the question\(^7\).

Figure 5 shows the principal scheme of a high-aperture apparatus\(^ {6,7}\) allowing simultaneous photographs to be taken of the fine structure of the Rayleigh line for two mutually perpendicular polarization directions. In photographs obtained with such an apparatus, the ratio of the integral intensities in the components of the fine structure and a number of other questions can be studied (see below).

From the splitting of the components of the fine structure of a scattered-light line, on the basis of L. I. Mandelstam’s formula (3.1), the velocity of an elastic thermal wave \(v\) of frequency \(\sim 10^{10}\) cycles/s (hypersound) can be determined. Sufficiently accurate determinations of the velocity of hypersound are of undoubted interest, since their comparison with the velocity of ultrasound \((10^6\!-\!10^8\) cycles/s)

makes it possible to judge the magnitude of the dispersion of the speed of sound, if it can be detected at all.

In the following sections of this review the principal results obtained in the study of the dispersion of the speed of sound, and other questions connected with the problem of fine structure, will be presented.

§ 4. DISPERSION OF THE SPEED OF SOUND AND THE VELOCITY OF HYPERSONIC WAVES

A large number of works have been devoted to the search for dispersion of the speed of sound in the ultrasonic region; they are presented with sufficient completeness in the book by L. Bergmann3. In the majority of investigations in the frequency interval from several kilohertz to 200 Mc/s, negative results were obtained. In some investigations positive results were obtained: dispersion of the speed of sound was found. In a number of cases it turns out that the small values of the dispersion of the speed of sound found are explained not by a dependence of the speed of sound on frequency, but by local heating of the liquid in the sound field[^83] or by other causes3.

For the first time in the ultrasonic frequency range, dispersion of the speed of sound in acetic acid was found by B. G. Shpakovskii[^84], whose results were confirmed by Lemb and co-workers[^85]. Lemb[^86] discovered dispersion in propionic acid. The magnitude of the dispersion of the speed of sound in these substances was

\[ \frac{\Delta v}{v} \sim 1\%. \]

Yung-Te-Chao[^87] found a decrease in the speed of sound in carbon disulfide on passing from a frequency of \(4 \cdot 10^5\) c/s to \(1.36 \cdot 10^6\) c/s by \(2.2\%\). Negative dispersion was also found by Shellomakh[^88] for geranium oil and di-di[[unclear: gridolimononic]] ether in the temperature interval from \(+15\) to \(-75^\circ\)C. At frequencies from \(6.89 \cdot 10^5\) to \(2.943 \cdot 10^6\) c/s a negative dispersion

\[ \frac{\Delta v}{v} \sim 0.65\% \]

was observed at \(-50^\circ\)C. Upon further lowering of the temperature the dispersion changes sign. In the investigations of V. L. Ginzburg[^89] and V. V. Vladimirskii[^78] it is shown that negative dispersion

\[ \left(\frac{\partial v}{\partial f}<0\right) \]

is theoretically possible. However, in subsequent experimental investigations the results of Yung-Te-Chao were not confirmed. Before making a final conclusion about the reality of negative dispersion, these results must be carefully checked.

As has been said, very many measurements of the speed of sound at ultrasonic frequencies have been performed, and with sufficient accuracy. Meanwhile, only for acetic and propionic acids was a small positive dispersion of the speed of sound found. In all other cases the experimental results proved negative or were doubtful and in need of careful verification. It therefore seems expedient to transfer the investigation of the dispersion of the speed of sound to the region of very high frequencies, where the dispersion effect should appear especially distinctly.

The highest frequency of artificial ultrasonic vibrations attained at the present time is \(\sim 10^9\) c/s. The difficulties of generation and of working at a frequency \(\sim 10^9\) c/s are so great that up to the present time at this frequency it has been possible to measure only the absorption in liquid mercury[^90]. The application of the frequency \(\sim 10^9\) c/s to the study of other liquids at present encounters great difficulties, chiefly because of the enormous absorption of sound in liquids at high frequencies. On the other hand, as has already been mentioned, the method of investigating the fine structure of the Rayleigh scattering line opens up the possibility of studying the velocity of propagation of elastic vibrations of frequency \(\sim 10^{10}\) c/s. The study of hypersound from the splitting of the components of the fine structure of scattered light is so far the only...

a method of studying elastic vibrations of such high frequencies, which therefore should expediently be realized.

The first attempt to detect dispersion of the sound velocity from the shift of the components of the fine structure of the Rayleigh line belongs to Rao^91. He found that in acetone there exists negative dispersion \(\Delta v/v \sim 20\%\), while in carbon tetrachloride there is approximately the same, but positive, dispersion of the sound velocity. If the results relating to carbon tetrachloride could be regarded as natural, then the negative dispersion \(\sim 20\%\) for acetone seemed completely inexplicable. Attempts to explain this result by the fact that at frequencies \(\sim 10^{10}\) cps the propagation of sound is no longer an adiabatic but an isothermal process^92 are untenable, as was shown in § 1 of this work, and also in^89.

Thus, Rao’s results caused perplexity. The matter became still more complicated when Venkateswaran^93 published a special paper devoted to measuring the velocity of hypersound in 17 liquids. In none of the liquids he investigated, including acetone and carbon tetrachloride, did the velocity of hypersound determined from the components of the fine structure differ from the corresponding velocity measured in the ultrasonic frequency range.

Analyzing the situation that had arisen here, the author of the present article came to the conclusion that, at any rate for some liquids, the velocity of hypersound measured from the fine structure of the scattering line should differ noticeably from the sound velocity in these liquids^6,7. This conclusion was based on the following considerations.

Hydrodynamic theory^19 leads to the following expression for the coefficient of sound absorption \(\alpha\) (neglecting the term determined by the thermal-conductivity coefficient and insignificant for the liquids under study)

\[ \alpha = \alpha_{\eta} + \alpha_{\eta'} = \frac{\omega^2}{2\rho v_0^3}\left\{\frac{4}{3}\eta+\eta'\right\}, \tag{4.1} \]

where \(\omega\) is the cyclic frequency of sound, \(v_0\) is the sound velocity in the liquid measured at low frequency, \(\eta\) and \(\eta'\) are the first and second viscosity coefficients, \(\alpha_{\eta}\) is the part of the total absorption due to shear viscosity, and \(\alpha_{\eta'}\) is the part of the absorption coefficient caused by volume viscosity.

Formula (4.1) has repeatedly been subjected to experimental verification. It has been established that, with the exception of two cases, \(\alpha/\omega^2\) remains a constant quantity up to the highest ultrasonic frequencies. If the measured absorption coefficients \(\alpha\) for such liquids as benzene, carbon disulfide, and carbon tetrachloride are extrapolated to a frequency \(\sim 10^{10}\) cps according to the quadratic law (formula (4.1)), it turns out that \(\alpha\Lambda \gg 1\) (\(\Lambda\) is the sound wavelength). Meanwhile, distinct lines of the fine structure can be observed only in the case when the attenuation of sound over the length of a wavelength is small (many fringes in the diffraction grating), i.e., under the condition that \(\alpha\Lambda \ll 1\).

For benzene, for example, extrapolation of the absorption coefficient according to the quadratic law to a frequency of \(10^{10}\) cps gives \(\alpha\Lambda = 12\). On the other hand, in benzene under the same conditions a distinct fine structure of the Rayleigh scattering line is observed; consequently, one must conclude that \(\alpha\Lambda \ll 1\). This contradiction served as the basis for creating the relaxation theory of sound absorption in liquids by M. A. Leonto-

… by Vigem and L. I. Mandelstam1. According to the relaxation theory,

\[ \alpha_{\eta'}=\frac{\omega^2\tau\left(v_\infty^2-v_0^2\right)} {2v_0^3\left(1+\omega^2\tau^2\right)}, \tag{4.2} \]

where \(v_\infty\) is the velocity at infinitely high frequency, and \(\tau\) is the relaxation time of the second viscosity coefficient.

It is evident from formula (4.2) that at frequencies satisfying the condition \(\tau^2\omega^2 \gg 1\), \(\alpha_{\eta'}\) ceases to depend on frequency and becomes a constant quantity. Since in such substances as benzene, carbon disulfide, and carbon tetrachloride the main absorption is due to \(\alpha_{\eta'}\), it is clear that in the range of frequencies lying beyond the relaxation region the absorption will be considerably smaller than the value obtained by extrapolation according to the quadratic law. In this way, naturally, one explains the existence of fine structure in liquids with a large absorption coefficient, in which the dependence of \(\alpha\) on \(\omega\) remains quadratic up to the highest ultrasonic frequencies reached at the present time. The relaxation theory does not make it possible to obtain a numerical value of the dispersion of the sound velocity unless \(\alpha\), \(\alpha_{\eta'}\), and \(\tau\) have been measured independently. It is possible, however, to estimate in what case a significant dispersion should be expected, using the relaxation theory, measurements of \(\alpha\), and the dependence of \(\alpha\) on frequency. Indeed, in the ultrasonic region, or, more precisely, in the frequency region below the relaxation of \(\eta'\), the hydrodynamic and relaxation theories give the same dependence of \(\alpha\) on frequency. In this region \(\omega^2\tau^2 \ll 1\). Combining expressions (4.1) and (4.2), we find:

\[ \tau=\frac{\eta'}{\rho\left(v_\infty^2-v_0^2\right)}. \tag{4.3} \]

Here \(\eta'\) is the second viscosity coefficient, first determined experimentally by Bazhulin2 from the excess of the measured value of \(\alpha\) over that calculated by formula (4.1) when only \(\eta\) is taken into account.

It follows directly from (4.3) that the dispersion of the velocity \(\dfrac{\Delta v}{v}\) is determined as follows:

\[ \frac{\Delta v}{v}=\frac{1}{2v_0^2\rho}\,\frac{\eta'}{\tau}. \tag{4.4} \]

The speed of sound \(v_0\) varies little for different liquids; consequently, the dispersion will be determined chiefly by the ratio \(\dfrac{\eta'}{\tau}\). The larger \(\eta'\) is and the smaller the relaxation time \(\tau\), the more significant the dispersion that should be expected.

An estimate of the magnitude of the expected dispersion in benzene from formula (4.4) can be made by using the determination of \(\eta'\) and the fact that \(\alpha\) increases with frequency proportionally to the square of the frequency, up to \(f=1.65\cdot 10^8\) cps3. This means that from the condition \(\tau^2\omega^2 \ll 1\) the relaxation time

\[ \tau \ll \frac{1}{2\pi f}=9.6\cdot 10^{-10}\ \text{sec.}\sim 10^{-9}\ \text{sec.} \]

On the other hand, as was already indicated earlier, a distinct fine structure is observed in benzene \((a\Lambda \ll 1)\). Consequently, the frequency \(10^{10}\) cps, responsible for the formation of the fine structure, lies beyond the relaxation region of \(\eta'\), i.e.,

\[ \tau \gg \frac{1}{2\pi 10^{10}}\simeq 10^{-11}\ \text{sec.} \]

Taking into account that for benzene \(\eta'=0.866\) poise, we find \(\frac{\Delta v}{v} > 3\%\), or \(\Delta v > 40\ \mathrm{m/sec}\). Consequently, measurements of the velocity of sound from the fine structure of the scattering line must in any case reveal a dispersion of the velocity of sound greater than 3%.

An even more reliable estimate of the dispersion of the velocity of sound can be made for carbon disulfide. According to the data of papers \(^{96,97}\), the dependence of \(a\) on \(f\) in the frequency interval \(f=10^{7}-2\cdot 10^{8}\ \mathrm{cps}\) departs from the quadratic and follows formula (4.2) with \(\tau=\frac{1}{2\pi f_c}=2.2\cdot 10^{-10}\ \mathrm{sec}\) \((f_c=7.2\cdot 10^{7}\ \mathrm{cps})\). Taking into account that for carbon disulfide \(\eta'=5.5\) poise, we find, according to (4.4),

\[ \frac{\Delta v}{v} \approx 8\% \quad \text{or} \quad \Delta v \approx 104\ \mathrm{m/sec}. \]

A noticeable dispersion should apparently also be expected in the case of carbon tetrachloride. In the same way one can obtain an estimate of the dispersion of the velocity of sound for a number of other liquids. It must be noted, however, that among the liquids studied, few have a large coefficient of volume viscosity and a short relaxation time. Therefore, the dispersion of the velocity of sound from the components of the fine structure can, so long as this method gives comparatively low accuracy, be investigated in a limited number of cases.

The estimate carried out above shows that, in any case, in carbon disulfide and benzene the expected effect is so large that it must be detected from the components of the fine structure, if the relaxation theory and the measurements of \(a\) on which the estimate made is based are correct.

Therefore new measurements of the velocity of hypersound in liquids \(^{12,13}\) were undertaken, in which, according to the estimate made, a noticeable dispersion of the velocity of sound was to be expected.

The negative result obtained by Venkateswaran \(^{93}\) is probably explained purely by experimental conditions, which have already been partly discussed in the literature \(^{98,99}\). The results of measurements of the velocity of hypersound carried out by the author, O. A. Shustin, and V. A. Molchanov \(^{12,13}\) are presented in Tables V–VIII. Photographs of the fine structure of the scattering line

Table V

Toluene

Wavelength of the exciting light \(\lambda\ 4358\ \text{Å}\).
Scattering angle \(90^\circ\), refractive index \(n=1.519\).

Temperature coefficient of the velocity

\[ \frac{\Delta v}{\Delta t}=-4.5\ \mathrm{cm/sec^{-1}\cdot deg}. \]

Velocity of ultrasound at \(20^\circ\mathrm{C}\): \(v_{20}=1324\ \mathrm{m/sec}\).

No. of photographs	\(t,\ ^\circ\mathrm{C}\)	\(\Delta\nu,\ \mathrm{cm}^{-1}\), Stokes	\(\Delta\nu,\ \mathrm{cm}^{-1}\), anti-Stokes	\(v_t,\ \mathrm{m/sec}\), Stokes	\(v_t,\ \mathrm{m/sec}\), anti-Stokes	\(v_{20},\ \mathrm{m/sec}\), Stokes	\(v_{20},\ \mathrm{m/sec}\), anti-Stokes	\(v_{20},\ \mathrm{m/sec}\), mean of Stokes and anti-Stokes
1	7	0.223	0.217	1360	1295	1307	1242	1314±34
3	17	0.226	0.214	1380	1310	1368	1292	1314±34
8	16	0.227	0.217	1390	1330	1374	1314	1314±34
10	16	0.213	0.224	1310	1375	1294	1359	1314±34
12	6	0.228	0.224	1395	1370	1311	1284	1314±34
				Average	Average	1330	1298	—

Table VI

CARBON DISULFIDE

Wavelength of the exciting light \(\lambda\,4358\ \text{Å}\).
Scattering angle \(90^\circ\), refractive index \(n=1.675\).

Temperature coefficient of velocity

\[ \frac{\Delta v}{\Delta t}=-3.2\ \text{m/sec}\cdot\text{deg}. \]

Ultrasonic velocity at \(20^\circ\text{C}\): \(v_{20}=1153\ \text{m/sec}\).

No. of photographs	\(t,\ ^\circ\text{C}\)	\(\Delta\nu,\ \text{cm}^{-1}\) Stokes	\(\Delta\nu,\ \text{cm}^{-1}\) anti-Stokes	\(v_t,\ \text{m/sec}\) Stokes	\(v_t,\ \text{m/sec}\) anti-Stokes	\(v_{20},\ \text{m/sec}\) Stokes	\(v_{20},\ \text{m/sec}\) anti-Stokes	\(v_{20},\ \text{m/sec}\), mean from Stokes and anti-Stokes
2	23	0.233	0.217	1285	1200	1295	1210	1265±22
4	21	0.232	0.223	1280	1230	1283	1233	1265±22
5	23	0.229	0.229	1265	1265	1275	1275	1265±22
6	23	0.223	0.234	1230	1290	1240	1300	1265±22
9	20	0.227	0.232	1255	1280	1255	1280	1265±22
				Mean . . . .		1270	1260	—

\[ \frac{\Delta v}{v}=0.088 \]

Table VII

BENZENE

Wavelength of the exciting light \(\lambda\,4358\ \text{Å}\).
Scattering angle \(90^\circ\). Refractive index \(n=1.524\).
Ultrasonic velocity at \(20^\circ\text{C}\): \(v_{20}=1324\ \text{m/sec}\).

Temperature coefficient of velocity

\[ \frac{\Delta v}{\Delta t}=-4.6\ \text{m/sec}\cdot\text{deg}. \]

No. of photographs	\(t,\ ^\circ\text{C}\)	\(\Delta\nu,\ \text{cm}^{-1}\) Stokes	\(\Delta\nu,\ \text{cm}^{-1}\) anti-Stokes	\(v_t,\ \text{m/sec}\) Stokes	\(v_t,\ \text{m/sec}\) anti-Stokes	\(v_{20},\ \text{m/sec}\) Stokes	\(v_{20},\ \text{m/sec}\) anti-Stokes	\(v_{20},\ \text{m/sec}\), mean from Stokes and anti-Stokes
1	17	0.249	0.232	1510	1415	1407	1402	1470±20
2	20	0.250	0.232	1520	1410	1520	1410	1470±20
4	14	0.250	0.233	1520	1420	1498	1397	1470±20
5	12	0.259	0.241	1575	1470	1528	1423	1470±20
6	12	0.255	0.243	1550	1480	1528	1443	1470±20
13	10	0.262	0.244	1590	1490	1544	1444	1470±20
14	10	0.252	0.235	1535	1430	1490	1384	1470±20
15	10	0.248	0.248	1510	1510	1464	1464	1470±20
16	10	0.255	0.252	1550	1535	1504	1489	1470±20
22	12	0.256	0.242	1560	1470	1514	1424	1470±20
				Mean . . . .		1507	1423	—

\[ \frac{\Delta v}{v}=0.10 \]

Table VIII

CARBON TETRACHLORIDE

Wavelength of the exciting light \(\lambda\ 4358\ \text{Å}\).
Scattering angle \(90^\circ\). Refractive index \(n = 1.472\).
Velocity of ultrasound at \(20^\circ\text{C}\), \(v_{20} = 920\ \text{m/sec}\).
Temperature coefficient of the velocity
\[ \frac{\Delta v}{\Delta t} = -3.1\ \text{m/sec}\cdot\text{deg}. \]

No. of photographs	\(t,\ ^\circ\text{C}\)	\(\Delta \nu,\ \text{cm}^{-1}\) Stokes	\(\Delta \nu,\ \text{cm}^{-1}\) anti-Stokes	\(v_t,\ \text{m/sec}\) Stokes	\(v_t,\ \text{m/sec}\) anti-Stokes	\(v_{20},\ \text{m/sec}\) Stokes	\(v_{20},\ \text{m/sec}\) anti-Stokes	\(v_{20},\ \text{m/sec}\), average from Stokes and anti-Stokes
3	10	0.173	0.156	1090	1015	1059	984
4	20	0.170	0.153	1070	1025	1070	1025
5	26	0.176	0.160	1110	1010	1119	1119	\(1040 \pm 27\)
1	17	0.166	0.161	1045	1015	1036	1006
Average . . .						1071	1008	—

\[ \frac{\Delta v}{v} = 0.12 \]

in various liquids, corresponding to the \(z\)-component of polarized light, were obtained on the apparatus whose schematic is shown in Fig. 5. The experimental details of these measurements are set forth in papers 7, 12, 13.

Especially careful study was made of liquids in which, according to the estimates made above, a considerable dispersion of the velocity of sound was expected. Three such liquids were investigated: carbon disulfide, benzene, and carbon tetrachloride. In the remaining cases studied, involving low-viscosity liquids, the dispersion expected from formula (4.4) proved to be so small that it lay within the limits of measurement error.

The scattering spectra were photographed at different temperatures. To reduce the velocity of hypersound to the temperature \(20^\circ\text{C}\), the temperature coefficients of the velocity of ultrasound\(^82\) were used. There was no particular doubt that the temperature coefficient of the velocity determined for ultrasound coincides with the temperature coefficient of the velocity of hypersound. However, in order to have complete certainty, the temperature coefficient of the velocity of hypersound in acetone was determined\(^13\).

The spectrum of the fine structure of the scattering line was photographed at room temperature and at a temperature of \(-31.5^\circ\text{C}\). The results of processing the photographs of the fine structure at different temperatures are given in Table IX.

Table IX

Temperature change of the velocity of hypersound in acetone
\(\theta = 90^\circ,\ \lambda = 4358\ \text{Å}\). Velocity of ultrasound \(v = 1190\ \text{m/sec}\)

\(t = 20^\circ\text{C}\) \(\Delta \nu,\ \text{cm}^{-1}\)	\(t = 20^\circ\text{C}\) \(n\)	\(t = 20^\circ\text{C}\) \(v,\ \text{m/sec}\)	\(t = -31.5^\circ\text{C}\) \(\Delta \nu,\ \text{cm}^{-1}\)	\(t = -31.5^\circ\text{C}\) \(n\)	\(t = -31.5^\circ\text{C}\) \(v,\ \text{m/sec}\)	\(\Delta v/v\) hypersound	\(\Delta v/v\) ultrasound
0.175	1.367	1190	0.223	1.39	1483	\(-5.7 + 0.5\)	5.6

Good agreement between the hypersonic and ultrasonic temperature coefficients of velocity indicates the validity of using the ultrasonic temperature coefficient of velocity to reduce the hypersonic velocity measurements to a common temperature.

Columns 3 and 4 of Tables V–VIII contain the values of \(\Delta \nu\), determined on the long-wavelength (Stokes) and short-wavelength (anti-Stokes) sides of the undisplaced line at the temperature of the experiment. In the following two columns of the table are given the hypersonic velocities calculated at the experimental temperature for the Stokes and anti-Stokes satellites. In columns 7 and 8 the hypersonic velocities are reduced to a temperature of \(20^\circ\)C by means of the temperature coefficient of velocity borrowed from ultrasonic measurements. From Tables V–VIII it is seen that between the velocities calculated from the Stokes and anti-Stokes satellites there is a small, but systematic, difference. This asymmetry cannot be explained by the asymmetry of the exciting line \(4358\) Å of the mercury spectrum. Indeed, the asymmetry of the exciting line is such that the anti-Stokes side of the line is less intense than the Stokes side. As a result of the superposition of the “tails” of the central line, reproducing the shape of the exciting line, on the shifted components, the latter change the apparent position of their maxima. The change in the apparent position should lead to \(\Delta \nu\) measured on the Stokes side being smaller than that on the anti-Stokes side. In reality the opposite picture is observed. Since in other liquids studied\(^{7,12,13}\) this effect is absent, it cannot be attributed to defects of the apparatus. Moreover, benzene was investigated on another apparatus, with other objectives and another spectrograph (constructed specially to check this effect), than toluene. The result obtained was the same. Consequently, the observed phenomenon is not connected with the method of investigation. What the true cause of the asymmetry is remains as yet unclear.

Table X presents the results of ultrasonic and hypersonic measurements for eight of the liquids studied\(^{6,12,13}\).

Table X

Ultrasonic and hypersonic velocities measured in various liquids, reduced to a temperature of \(20^\circ\)C

Substance	Ultrasonic velocity \(v_s\), m/sec	Hypersonic velocity \(v\), m/sec	Dispersion of sound velocity \(\dfrac{\Delta v}{v}\cdot 10^2\)
Benzene	1324	\(1470 \pm 20\)	10
Carbon disulfide	1158	\(1265 \pm 22\)	8.8
Carbon tetrachloride	920	\(1040 \pm 27\)	12
Toluene	1324	\(1314 \pm 34\)	\(\sim 0\)
Acetone	1190	\(1190 \pm 40\)	\(\sim 0\)
Acetic acid	1144	\(1140 \pm 35\)	\(\sim 0\)

Noticeable dispersion, as was to be expected, was observed in three liquids; in the remaining cases the dispersion effect apparently exists, but it was not detected because of the insufficient accuracy of the measurement and the smallness of the effect itself. In acetic acid, for example, \(\dfrac{\Delta v}{v} \sim 1\%\), which lies within the error of the described experiment. In other cases (alcohols, hydrocarbons, etc.), according to theoretical estimates, this effect is still smaller.

From the measured value of the dispersion of the speed of sound, \(\Delta v/v\), and determinations of the second viscosity coefficient, performed by P. A. Bazhulin and other authors\(^{95,97}\), the relaxation time of the second viscosity coefficient \(\tau\) can be calculated from formula (4.4). Table XI gives the results of calculating \(\tau\) and \(f_c\) for benzene, carbon tetrachloride, and carbon disulfide.

Table XI

Calculation of the relaxation time and critical frequency

Substance	\(t,\,^\circ\mathrm{C}\)	Density \(\rho,\ \mathrm{g/cm^3}\)	\(\eta'\), poise	Speed of sound \(v_0,\ \mathrm{m/sec}\)	\(\dfrac{\Delta v}{v}\cdot 10^2\)	\(\tau\cdot 10^{10}\ \mathrm{sec}\)	\(f_c=\dfrac{1}{2\pi\tau}\cdot 10^{-8}\ \mathrm{Hz}\)
Benzene	20	0.879	0.866	1324	10	2.44	6.58
Carbon tetrachloride	20	1.594	0.355	920	12	0.78	20.5
Carbon disulfide	20	1.263	6.01	1158	8.8	26.4	0.603
Carbon disulfide	20	1.263	5.52	1158	8.8	21.8	0.72

From the results obtained it is clear that only in the case of carbon disulfide is the critical frequency so low that the relaxation curve in this liquid is accessible to direct experimental investigation at ultrasonic frequencies. Such investigations were carried out, and the entire relaxation curve is presented in the work of Lamb and Andrea\(^{97}\) (Fig. 6).

Fig. 6. Relaxation curve obtained from absorption data in carbon disulfide in the ultrasonic frequency range (after Lamb and Andrea).

If one uses the value of \(\eta'\) obtained from ultrasonic measurements\(^{97}\), then the dispersion proves to be equal to 9%, which is in good agreement with direct measurements of the speed of sound\(^{13}\). Unfortunately, it is impossible to compare measurements of the dispersion of the speed of sound in benzene and carbon tetrachloride in the same way as was done for carbon disulfide. Measurements of absorption in benzene and \(\mathrm{CCl}_4\) up to frequencies \(\sim 10^8\ \mathrm{Hz}\) show a quadratic dependence of the absorption coefficient \(\alpha\) on the frequency \(\omega\). This result, however, is in qualitative agreement with the results obtained\(^{12}\). Indeed, the critical frequency in benzene is \(f_c=6.53\cdot 10^{-8}\ \mathrm{Hz}\), and in \(\mathrm{CCl}_4\), \(f_c=14.5\cdot 10^8\ \mathrm{Hz}\). Ultrasonic measurements have not yet been advanced to such frequencies.

Using \(\Delta v/v\) and \(f_c\), and rewriting formula (4.2) in the form

\[ \frac{\alpha_{\eta'}}{f^2} = \frac{\dfrac{\pi}{f_c}\left(v_\infty^2-v_0^2\right)} {v_0^3\left[1+\left(\dfrac{f}{f_c}\right)^2\right]}, \tag{4.5} \]

we constructed curves of \(\alpha/f^2\) as a function of frequency \(f\) for benzene and carbon tetrachloride (Figs. 7 and 8). The curve for carbon disulfide, naturally, almost exactly coincides with the curve in Fig. 6.

The vertical line on the left marks the limiting ultrasonic frequencies at which the given substance was studied. The vertical line

Fig. 7. Relaxation curve constructed from data on the absorption of ultrasound and hypersound in benzene.

Fig. 8. Relaxation curve constructed from data on the absorption of ultrasound and hypersound in carbon tetrachloride.

on the right marks the high frequency whose study can be carried out from the fine structure of the Rayleigh line. The interval between the ultrasonic and hypersonic regions, as well as frequencies higher than \(10^{10}\) Hz, remain as yet unexplored. However, from relaxation theory, which agrees well with experiment, and from measurements of the dispersion of the speed of sound according to (4.5), one can estimate the absorption at any

frequency, into the region of relaxation of the shear (ordinary) viscosity. This latter, for ordinary viscosities, should occur at frequencies much greater than \(10^{10}\) Hz. The study of the relaxation of shear viscosity is of unquestionable interest, but as yet it is impossible to indicate a ready path for such a study in ordinary low-viscosity liquids. It may be that certain possibilities will present themselves with further study of the wing of the Rayleigh line[^100]. In liquids of high viscosity, relaxation of shear viscosity has been observed, manifested in the dispersion of the sound velocity in glycerin and castor oil, observed from the fine structure of the scattering line in these liquids[^101]. It was impossible to detect a deviation from quadratic behavior in the variation of the absorption coefficient with frequency in carbon tetrachloride, as shown by the curve in Fig. 8, up to the highest frequency used, \(1.05\cdot 10^8\) Hz. In benzene, a deviation from quadratic behavior could in principle be detected at the frequency \(1.65\cdot 10^8\) Hz (Fig. 7); however, this deviation is so small that it lies within the limits of measurement error and, naturally, has not been established with the necessary certainty.

One may hope that small changes in velocity at the beginning of curve relaxation in benzene and in a number of other liquids can be detected by means of the very sensitive method developed by Zverev[^102] specially for measuring small dispersion of the sound velocity.

The experimental values obtained for the dispersion of the sound velocity in benzene, carbon tetrachloride, and carbon disulfide can be used for experimental verification of the correctness of theoretical calculations of the dispersion, as well as of the relaxation times of the second viscosity coefficient.

K. Herzfeld[^103] attempted to calculate the relaxation times of the second viscosity coefficient from relaxation concepts that take into account processes of a Knudsen character[^104]. As a result of the calculation, a formula was obtained expressing the relaxation time as a function of the thermodynamic heat capacities \(C_p\) and \(C_v\), the internal heat capacity \(C_i\), and the absorption coefficient due to the second viscosity coefficient:

\[ \tau=\frac{v_0 C_p (C_p-\Delta)\,2\alpha_{\eta}}{C_i\cdot \Delta\cdot \omega^2}, \tag{4.6} \]

where \(\Delta=C_p-C_v\).

Calculation of \(C_i\) from spectroscopic data makes the estimate of \(\tau\) for liquids very approximate. However, apparently the simplifications made do not greatly affect the results of calculating the value of \(\tau\) for benzene and carbon tetrachloride, carbon disulfide, and some other liquids.

Table XII

Calculation of \(\tau\) by formula (4.6)

Liquid	\((\alpha/f^2)_{\mathrm{meas}}-(\alpha/f^2)_{\mathrm{class}}\)	\(C_p\)	\(C_p-C_v\)	\(C_i\)	\(\tau\cdot 10^{10}\) sec
Benzene	850	32	9.8	10.6	4
Toluene	75	38	7.9	16.8	0.4
Acetone	27	30	9.4	9.7	0.19
Carbon tetrachloride	400	31	9.4	12.6	1.0
Carbon disulfide	3500	18	6.4	3.9	15
Water	17.8	18	0.087	0.05	1000

The results of the calculation by formula (4.6) and the data of Table XI are in satisfactory agreement with our data for benzene, carbon tetrachloride, and carbon disulfide. However, the limited character of Herzfeld’s theory is especially clearly seen in the example of water. The calculation leads to roughly erroneous results, which contradict the ultrasonic experiment.

An estimate of the dispersion by (4.4), using the relaxation-time data obtained by Herzfeld^103, gives for toluene \(\Delta v/v \sim 7\%\) and for acetone \(\Delta v/v \sim 6\%\), which is clearly greatly overestimated. K. Herzfeld^105 calculated the relaxation time of the second coefficient of viscosity in benzene, starting from the Lennard-Jones potential, and obtained \(\tau = 1.2 \cdot 10^{-10}\) sec. The values obtained from such a calculation are in good agreement with the data of Table XI, but this is the only calculation of this kind, and it is quite possible that the coincidence is accidental. Recently Nomoto^106 compared our results for \(\tau\) in \(\mathrm{CCl}_4\) and benzene with the results of his ultrasonic experiments and calculations and found that they are in good agreement with one another. Table XIII gives the values of the absorption coefficients at frequencies \(\sim 10^{10}\) c/s for various liquids, with the determination in the first three substances being made from fine-structure data.

In acetic acid, the dispersion of the speed of sound was detected in the ultrasonic region by direct measurement of the velocity.

Table XIII

Absorption coefficients of sound in liquids at frequencies \(\sim 10^{10}\) c/s

Substance	\(t,\ ^\circ\mathrm{C}\)	Density \(\rho,\ \mathrm{g/cm^3}\)	\(\eta \cdot 10^3\), poise	\(\alpha_{\eta'},\ \mathrm{cm^{-1}}\ 10^{-3}\)	\(\alpha_{\eta},\ \mathrm{cm^{-1}}\ 10^{-3}\)	\(\alpha,\ \mathrm{cm^{-1}}\ 10^{-3}\)
Benzene	20	0.879	6.5	3.65	4.4	8.05
Carbon tetrachloride	20	1.594	9.5	19.6	6.4	26
Carbon disulfide	20	1.263	3.6	0.377	2.26	2.64
Acetic acid	20	1.049	12.2	0.006	11.3	11.3

Very good agreement of the experimental results with the conclusions of relaxation theory, for example in the case of carbon disulfide and acetic acid, allows one to think that the variant of relaxation theory with one relaxation time is quite sufficient for describing the real phenomena occurring in low-viscosity liquids during the propagation of a sound wave whose frequency does not exceed \(10^{10}\) c/s.

On the basis of measurements of the dispersion of the speed of sound and the formulas of relaxation theory, the coefficients \(\alpha_{\eta'}\) and \(\alpha_{\eta}\) at the frequency \(10^{10}\) c/s can be calculated separately. The results of such a calculation are given in Table XIII.

§ 5. POLARIZATION OF THE MANDELSHTAM–BRILLOUIN COMPONENTS IN LIQUIDS

As has already been indicated, adiabatic and isobaric density fluctuations do not violate the isotropy of the substance in the volume of fluctuation. Therefore light scattered by such fluctuations will be completely polarized. A more rigorous consideration of the question of the polarization of the components

Mandelstam–Brillouin shows that, under certain conditions, one may nevertheless expect a small depolarization of the shifted components. A quantitative investigation of this question was given in the work of M. A. Leontovich,^23 from which it follows that the depolarization of the Mandelstam–Brillouin components even at \(\Omega_l \tau = 10\) (where \(\Omega_l\) is the frequency of the Mandelstam–Brillouin components, and \(\tau\) is the relaxation time of the anisotropy) is somewhat less than \(0.5\%\) (Fig. 9). For most of the liquids investigated by various authors, \(\Omega_l \tau\) lies between 0.7 and 0.5. Thus, from the theoretical point of view, no noticeable depolarization of the Mandelstam–Brillouin components can be expected in these liquids.

Fig. 9. Spectral distribution of the intensity of scattered light in two mutually perpendicular directions of polarization (according to M. A. Leontovich): 1 — \(I_x\)-component, 2 — \(I_z\)-component. a) \(\Omega_L \tau = 0.1\). b) \(\Omega_L \tau = 10\).

The experimental data existing in the literature on this question are not in agreement with one another. The first measurement of the polarization of the fine-structure components, carried out by Birus,^107 convincingly showed that the Rayleigh triplet in toluene is completely polarized. The same

the result for toluene had been obtained still earlier by Raman and Rao^108. Qualitative studies by E. F. Gross and A. A. Syromyatnikov^109 also showed that there is no depolarization of the Mandelstam–Brillouin components even in such viscous liquids as p-cresol and phenol. Venkateswaran^110 and Sunanda Bai^111 came to the same conclusions. In subsequent works, however, these results were called into question. Rank, MacCartney, and Zatz^98 reported that they had found appreciable depolarization of the Mandelstam–Brillouin doublet, reaching, for example, 15% for acetone. After the work^98, Venkateswaran^112 reported that a re-examination of his old materials (on these materials the paper^110 was published) showed that there is a depolarization of the Mandelstam–Brillouin components which he had not noticed earlier, and which the author attempts to explain on the basis of the hole theory of viscosity developed by Ya. I. Frenkel^18.

Consideration of the above-mentioned works from the methodological point of view shows that at various points they are not free from shortcomings, sometimes very substantial ones. Thus, for example, Rank^98 and his coworkers regard their polarization measurements as unreliable. They see the shortcoming of their measurements in the fact that parasitic light was not completely eliminated in the apparatus. Rank estimates the accuracy of his best polarization measurements of the shifted fine-structure components at 25%. Apparently this accuracy is exaggerated. It seems to us that one of the most substantial sources of error in this work lies in the polarization method developed by Douglas and Rank^114, which requires measurements of spectra photographed at different times. Although such photographs are made on one and the same photographic plate, the possible instability of the light source may strongly affect the result, especially at the small exposures used in this work and the small depolarizations to be measured.

In the detailed work of Venkateswaran^110, polarization measurements were carried out on an apparatus with a Lummer–Gehrcke plate; the paper states that the Mandelstam–Brillouin components were not separated from the central line because of the low resolving power of the apparatus. From this experiment, however, the conclusion is drawn that the shifted components are completely polarized.

In the subsequent publication Venkateswaran^112, on the basis of the same materials, arrives at the directly opposite conclusion. Such different statements concerning one and the same materials are perplexing. It is not possible to discuss them in detail.

In the works^98,^110, data are also given on the depolarization of the central line of the fine structure. In all cases considered by the authors, the depolarization coefficient of the central line of the Rayleigh triplet proves to be very substantial.

Table XIV gives the results of measurements, by various authors, of the depolarization coefficients of the Mandelstam–Brillouin components and of the central component of the triplet. The data of Table XIV exhaust all quantitative measurements of the depolarization of fine-structure components, with the exception of the data of the author of the present article, which will be given below.

In the works of E. F. Gross^109 and his coworkers, a depolarized central component and its strong dependence on temperature were observed. In the cited works no quantitative measurements were made; but from the presentation of the experimental results it is quite clear that the depolarization coefficient of the central line of p-cresol and phenol is very large. The same results follow from the work of Raman and Rao^108,^113.

Table XIV

Depolarization coefficient of fine-structure components

Substance	Depolarization coefficient \(\Delta \cdot 10^2\), Mandelstam–Brillouin components	Depolarization coefficient \(\Delta \cdot 10^2\), central component	Authors
Water	4	46	Rank \(^{98}\)
Ethyl alcohol	2.6	15	Rank \(^{98}\)
Acetone	15	25	Rank \(^{98}\)
Benzene		10	Venkateswaran \(^{110}\)
Tetralin		38	Venkateswaran \(^{110}\)
Phenol		42	Venkateswaran \(^{110}\)
Glycerin		25	Venkateswaran \(^{110}\)
Formic acid		39 \((20^\circ\mathrm{C})\) 24 \((75^\circ\mathrm{C})\)	Suanda Bai \(^{111}\)
Acetic acid		28 \((45^\circ\mathrm{C})\) 18 \((75^\circ\mathrm{C})\) 14 \((120^\circ\mathrm{C})\)	Suanda Bai \(^{111}\)
\(n\)-Butyric acid		20 \((45^\circ\mathrm{C})\)	Suanda Bai \(^{111}\)
Iso-butyric acid		11 \((120^\circ\mathrm{C})\)	Suanda Bai \(^{111}\)
Cyclohexanol		6 \((45^\circ\mathrm{C})\)	Suanda Bai \(^{111}\)
Carbon disulfide		21 \((45^\circ\mathrm{C})\)	Suanda Bai \(^{111}\)
Chloroform		7 \((45^\circ\mathrm{C})\)	Suanda Bai \(^{111}\)
Benzene		9 \((70^\circ\mathrm{C})\)	Suanda Bai \(^{111}\)
Toluene		0	Birus \(^{107}\)

From what has been said one may conclude that almost all authors arrive at the conclusion that the central line of the fine structure, both in low-viscosity liquids and, especially, in liquids with considerable viscosity, is depolarized to a greater or lesser degree.

This state of the question concerning the polarization of the central component and of the Mandelstam–Brillouin components compelled the author to undertake a new experiment \(^{6,7}\).

The scheme of the apparatus used in these experiments is shown in Fig. 5. In designing it, the methodological shortcomings of earlier work were taken into account. In particular, the apparatus provided: 1) simultaneous photographing of the \(I_x\)- and \(I_z\)-components, 2) high resolving power, 3) large linear dispersion, and 4) high luminosity. Photographs of the fine structure of the Rayleigh light-scattering line in various liquids, obtained with this apparatus, are given in Fig. 10. On the left in the photograph is the \(I_x\)-component, and on the right the \(I_z\)-component of the scattered light.

In carbon disulfide, benzene, toluene, acetone, and carbon tetrachloride, a broad wing is observed, caused by fluctuations of anisotropy. The half-width of the wing in the liquids named exceeds the dispersion range of the Fabry–Perot etalon. Therefore, in the \(I_x\)-component a uniform background is observed. Next are presented photographs of the fine structure of the scattering line in acetic acid, triacetin, glycerin, salol, and benzophenone. In these liquids an entirely different picture is observed. In acetic acid

in the presence of a sharp fine structure of the scattering line corresponding to the \(J_z\)-component, a depolarized “central line,” corresponding to the \(J_x\)-component, is clearly visible. The same phenomenon can be observed in triacetin, although the shifted components here are less distinct than in the preceding case, while the depolarized line is even more pronounced. The scattering spectrum of the last three liquids (glycerin, salol, and benzophenol) practically reproduces the spectrum of the exciting light, with the only

Fig. 10. Fine structure of the scattered-light lines for certain liquids in two mutually perpendicular directions of polarization \(J_x\) and \(J_z\).
1 — spectrum of the exciting line \(\lambda\ 4358\) Å, 2 — carbon disulfide, 3 — benzene, 4 — toluene, 5 — acetone, 6 — carbon tetrachloride, 7 — acetic acid, 8 — triacetin, 9 — glycerin, 10 — salol, 11 — benzophenone.

difference that the exciting light is natural, whereas the scattering spectrum is characterized by depolarization different from unity. Thus, in the photographs of the scattering spectra of ten different liquids shown in Fig. 10, one can see the entire range of transition from a sharp fine structure to its disappearance when observing the \(J_z\)-component, and from a uniform continuous background to a narrow intense line when observing the \(J_x\)-component.

Careful examination of the \(J_x\)-component in the spectrum of such liquids as benzene, carbon disulfide, toluene, acetone, and carbon tetrachloride shows that, against the continuous background of anisotropic scattering, a very weak Rayleigh triplet “extends.” Photometric processing of many photographs of the fine structure in these liquids and calculation of the aperture show that traces of the Rayleigh triplet in the \(J_x\)-component

are caused not by its depolarization in the scattered light, but by the finiteness of the aperture of the exciting and scattered light.

This conclusion applies to all three components of the fine structure in benzene, toluene, carbon disulfide, acetone, and carbon tetrachloride, and to the Mandelstam–Brillouin components in acetic acid and triacetin.

It should be noted, however, that, when observing the \(J_z\)-component in the last two liquids, it is not possible to find the displacement of the component. The explanation of this we see in the fact that in acetic acid and, especially, in triacetin the displaced components are somewhat broader and, most importantly, closer to the undisplaced line than, for example, in benzene.

In glycerin and salol, in our photograph at a temperature of \(20^\circ\text{C}\), no displaced components corresponding either to the \(J_z\)- or to the \(J_x\)-components are observed at all. In glycerin at a temperature of \(0^\circ\text{C}\), Raman and Venkateswaran \(^{93}\) observed a weak fine structure upon excitation of the scattered light by the line \(\lambda = 4722\ \text{Å}\) of the zinc-arc spectrum. In our experiment weak displaced lines in glycerin were not detected, possibly owing to the somewhat greater width of the exciting line (\(\lambda = 4358\ \text{Å}\) of the mercury-arc spectrum).

Thus, the results of our experiments on determining the depolarization of the triplet lines lead to the conclusion that, under the conditions of the experiment described, all three lines of the triplet are completely polarized to within an accuracy of \(1\%\). As for the depolarization of the central component of the fine structure obtained in the work of Venkateswaran \(^{110}\) and Sunanda Bai \(^{111}\), who worked by one method and, apparently, on one apparatus, it is quite understandable. Their apparatus had low resolving power and a sufficient range of dispersion. Therefore the narrowest and most intense part of the wing was superposed on the central component and, undoubtedly, depolarization of the central part of the Rayleigh triplet must have been found. However, it does not at all follow from this experiment that the part of the scattering associated with isobaric density fluctuations is depolarized. The observed depolarization of the “central component” is the superposition on the central component (isobaric fluctuations) of a more or less narrow wing of anisotropic scattering. This question was considered by us in detail in papers \(^{6,7}\).

§ 6. RATIO OF INTENSITIES IN THE COMPONENTS OF THE FINE STRUCTURE OF THE SCATTERED-LIGHT LINE

A quantitative expression of the role of adiabatic and isobaric density fluctuations manifested in light scattering is contained in the Landau–Placzek formula \(^{73}\), according to which the ratio of the integral intensities of the central component \(J_c\) and of the intensities of both Mandelstam–Brillouin components \(2J_{\mathrm{M-B}}\) is expressed as follows *):

\[ \frac{J_c}{2J_{\mathrm{M-B}}}=\gamma-1=\frac{C_p-C_v}{C_v}. \tag{6.1} \]

In deriving (6.1), the fluctuation of the dielectric permittivity \(\Delta \varepsilon\) is considered as a function of two independent thermodynamic variables \(p\) and \(T\).

*) This important result was presented in the article of L. Landau and G. Placzek \(^{73}\) without derivation. Relation (6.1) was derived in the work of V. V. Vladimirskii \(^{76}\) with allowance for intermolecular interaction. Subsequently, a somewhat different derivation of (6.1) was given in the works of V. L. Ginzburg \(^{2}\) and E. F. Gross \(^{115}\).

Then

\[ \overline{\Delta \varepsilon^2} = \left(\frac{\partial \varepsilon}{\partial \rho}\right)_T^2 \overline{\Delta \rho^2} + \left(\frac{\partial \varepsilon}{\partial T}\right)_\rho^2 \overline{\Delta T^2}. \tag{6.2} \]

On the basis of the approximate data of Raman and Venkataraman\(^{11}\) and L. M. Levin\(^{10}\), in (6.2) one usually neglects the term
\(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_\rho^2 \overline{\Delta T^2}\) in comparison with
\(\left(\dfrac{\partial \varepsilon}{\partial \rho}\right)_T^2 \overline{\Delta \rho^2}\). In § 1 it was shown that such a neglect leads to an error in the total intensity of light scattered by isobaric and adiabatic density fluctuations of the order of 2% or even less. In calculating the intensity ratio \(J_c/2J_{M-\mathrm{B}}\), the error may turn out to be much larger, since neglect of the quantity
\(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_\rho^2 \overline{\Delta T^2}\) leads to a change only in the intensity of the central component, which will have an especially strong effect for small \(J_c\) (for example, in water).

Let us make use of the calculation already carried out in § 1.

On the basis of (1.10) and (1.11) one can immediately write that the ratio of the integral intensities in the Mandelstam–Brillouin components is equal to

\[ \frac{J_c}{2J_{M-\mathrm{B}}} = \frac{J_{is}}{J_{ad}} = \frac{ \left(\dfrac{1}{b}\dfrac{\partial \varepsilon}{\partial T}\right)_\rho^2 }{ \left(\rho\dfrac{\partial \varepsilon}{\partial \rho}\right)_s^2 } \cdot \frac{b^2 T}{C_p \rho \beta_s}, \tag{6.3} \]

where the values \(\left(\rho\dfrac{\partial \varepsilon}{\partial \rho}\right)_s\) and \(\beta_s\) must correspond to the frequencies of elastic thermal oscillations.

Strictly speaking, \(J_{is}\), just like \(J_{ad}\), should be calculated with account taken of the rate of “relaxation” of the isobaric fluctuations, as was said in § 1. But if the pressure fluctuations change rapidly, which is expressed in a displacement of the Mandelstam–Brillouin components by an amount \(\sim 10^{10}\) Hz, then the isobaric density fluctuations change incomparably more slowly. In the scattered light they appear in the form of an undisplaced line, whose maximum corresponds to \(f=0\). The part of the isobaric fluctuations contributing to the neighborhood of the maximum of the undisplaced line should, strictly speaking, follow thermodynamics. However, the undisplaced line has a finite width, which varies somewhat for different liquids, but on the average differs little from \(10^7\) Hz. Thus, in the formation of the central component there participate processes almost three orders of magnitude slower than the processes responsible for the appearance of the displaced lines. Generally speaking, \(b\), \(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_\rho\), and \(C_p\), which enter formula (6.3), may depend on frequency. According to relaxation theory, the change in \(C_p\) will be determined by the same relaxation time as the sound velocity. Then, at least for benzene, carbon disulfide, and carbon tetrachloride, one can indicate (see § 4) what the value of the relaxation frequency \(f_c\) is. It turns out that for these liquids \(f_c\) is one or two orders of magnitude higher than the value of the half-width of the central component. Consequently, within the limits of the half-width of the central component, where the main part of the intensity is contained, one may use the static value of \(C_p\) for calculation by (6.3). Only at very large distances from the maximum of the central component (\(\sim 10\) half-widths) can the static value of \(C_p\) turn out to be noticeably greater than the dynamic one, and the corresponding small part of the total intensity will prove to be underestimated.

Relaxation of the coefficient of expansion \(b\) is connected with the relaxation of elasticity. Therefore the corresponding \(f_c\) here is the same as for \(C_p\).

The situation may be different with the frequency behavior of \(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_p\). It is not excluded that the part of this quantity caused by a change in density may differ from its static value, but the error caused by this frequency dependence will be very small.

Relation (6.3) will reduce to the Landau–Placzek formula if one sets \(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_p \dfrac{Tb}{C_p\beta_T\rho}=0\) and assumes that the dispersion of \(\beta_s\) is absent. Indeed, under these assumptions, according to (1.13) and (1.14),

\[ \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_s = \left(\frac{1}{b}\frac{\partial \varepsilon}{\partial T}\right)_p \]

and, consequently,

\[ \frac{ \left(\dfrac{1}{b}\dfrac{\partial \varepsilon}{\partial T}\right)_p^2 }{ \left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_s^2 } =1. \]

On the other hand, on the basis of (1.16) we find that

\[ \frac{b^2T}{C_p\beta_s\rho} = \frac{\beta_T}{\beta_s}-1 = \gamma-1. \]

For a number of liquids the ratio

\[ \frac{ \left(\dfrac{1}{b}\dfrac{\partial \varepsilon}{\partial T}\right)_p^2 }{ \left(\rho \dfrac{\partial \varepsilon}{\partial \rho}\right)_s^2 } \]

differs substantially from unity. Thus, for example, for water this ratio is equal to 1.70; for other liquids it varies between unity and 1.50. Likewise, if appreciable dispersion of the sound velocity in a liquid is observed, \(\dfrac{b^2T}{C_p\beta_s\rho}\) differs substantially from \(\gamma-1\). For example, for benzene these quantities differ almost by a factor of two.

K. Birus\(^{107}\) was the first to subject the Landau–Placzek relation to experimental verification. His experiment was carried out on an installation with a Michelson echelon. In view of the great complexity of this experiment, which lasted almost two years, only one liquid—toluene—was studied quantitatively. The value he obtained for \(J_c/2J_{\mathrm{M}-\mathrm{B}}=0.36\), which differs from the value calculated by formula (6.1), (0.40), by only 12%. The author believes that he obtained a figure in good agreement with the Landau–Placzek theory.

A more extensive investigation, also touching on a number of other questions connected with fine structure, was carried out by Venkateswaran\(^{110}\) and S. Bai\(^{111}\). Venkateswaran investigated several liquids; according to his data, in all cases the experimental results do not agree with (6.1).

It should be emphasized that in all the preceding works formula (6.1) was used in calculating \(J_c/2J_{\mathrm{M}-\mathrm{B}}\). To determine \(\gamma\), Teyrer’s data were taken, which in a number of cases differ from modern data for this quantity.

The most recent measurements of the intensity ratio in the fine-structure component of the Rayleigh line were made by Rank, McCartney, and Sait\(^{98}\). The work was performed on a Fabry–Perot interferometer. For increasing

luminous intensity and increasing the resolving power of the apparatus. Rank and his collaborators, in investigating the fine structure, used a standard of thickness 10.5 mm. In this case the long-wavelength component of the fine structure of one order of interference coincides with the short-wavelength component of another order. Superposition doubles the intensity and shortens the exposure. It is puzzling, however, that the authors of this work assert that, for the indicated thickness of the standard, the components are superposed in water, benzene, acetone, and methyl alcohol. It is known that in these liquids the speeds of sound and the refractive indices are different; consequently, if in one case there is coincidence in the positions of the short-wavelength and long-wavelength components, then in the other cases there will be no coincidence or, at best, it can be only partial. Such a situation greatly complicates the investigation and can lead to large errors in the photometric decoding of line contours. The photographs, moreover, were taken with a small focal length of the camera objective and insufficient limitation of the aperture. Measurement by intensity maxima in the case of incomplete coincidence of displaced lines is impossible. Table XV gives the results of measurements of the ratio of the intensity of the central component of the fine structure to the sum of the two side components.

Table XV.

Measured and calculated values of \(\dfrac{J_c}{2J_{M-\mathrm{B}}}\), according to data of various authors

Substance	Calculated according to Landau–Placzek	Observed	Author
Toluene	0.36	0.40	Birus^707
Carbon tetrachloride	0.48	0.84	Venkateswaran^110
Ethyl alcohol	0.18	0.39	Venkateswaran^110
Ether	0.34	0.45	Venkateswaran^110
Cyclohexane	0.47	0.65	Venkateswaran^110
Water	0.01	0.36	Venkateswaran^110
Acetone	0.39	0.79	Venkateswaran^110
Benzene	0.32	0.97	Venkateswaran^110
Methyl alcohol	0.21	0.285	Sunanda Bai^111
n-propyl alcohol	0.34	0.60	Sunanda Bai^111
Iso-propyl alcohol	0.24	0.45	Sunanda Bai^111
n-butyl alcohol	0.30	0.70	Sunanda Bai^111
Iso-butyl alcohol	0.40	0.66	Sunanda Bai^111
Water	0.01	0.14	Rank et al.^98
Ethyl alcohol	0.18	0.15	Rank et al.^98
Acetone	0.39	0.42	Rank et al.^98

According to Venkateswaran^110 and Bai^111, in all cases the theory differs substantially from experiment. According to Rank and collaborators, in two or three of the cases cited in their work the theory is in satisfactory agreement with experiment.

Let us also point out that H. E. Sterin^99 investigated the ratio of the intensities of the components of the fine structure in benzene. Without giving numerical

...his result, he reported that his data, although lower than the data of [110], are nevertheless still above the theoretical value (formula (6.1)). Thus, all the authors of the earlier measurements of the intensity ratio \(J_c/2J_{M-Б}\), except Birus [107], come to the conclusion that the theory is not in agreement with the experimental results. The theoretical relations that they used in many cases differ substantially from the refined formulas [6, 7, 115], but the methodological side of their work also suffers from certain defects that may very substantially distort the character of the true intensity ratio of the fine-structure components.

Some of these methodological shortcomings have been mentioned above. Other essential methodological deficiencies of these works are as follows: the theoretical Landau–Placzek ratio is written for the integral intensities of the fine-structure components. How this ratio was measured in Birus’s work cannot be established, but in Venkateswaran’s work it is clearly stated that the ratio was measured not by integral values, but by maximum values and by the density of blackening, not by intensity. It had already been noted by V. L. Ginzburg [2] that, generally speaking, comparison of such measurements with theory is illegitimate. By their nature, the central and side components are different and have different widths; therefore Venkateswaran’s method of measurement requires a special justification, which the author did not provide. It will be shown below that, under certain experimental conditions, measurement of maximum intensities instead of integral intensities is permissible and does not introduce significant errors. But it is completely impermissible to use blackenings instead of intensities. Such measurements, especially at low intensities, strongly distort the picture of the phenomenon. The same remarks apply to the work of C. Bai [111]. Rank [98] and his collaborators likewise do not convert blackenings into intensities.

Fig. 11. Intensity distribution in the exciting line \(\lambda 4358\) Å.

However, the most serious methodological shortcoming of all the preceding investigations is that the “background line” on the photometric curves was drawn arbitrarily. In a photograph of the fine structure there is never a “complete” separation of the components. As a result of the finite width of the exciting line and the closeness of the components to one another, the lines are always, to a greater or lesser degree, superposed by their “tails.” It must also be taken into account that a continuous lamp background is superposed on the Rayleigh triplet, and, most importantly, the “continuous spectrum” of depolarized scattering. In the spectrogram the Rayleigh triplet rises above a continuous spectrum of greater or lesser intensity. When processing such a photograph, the experimenter is faced with the task of drawing the “background line,” from which one should count...

to take the intensities of the lines of the Rayleigh triplet into account. In the works mentioned, this problem is in fact not solved, since the method employed gives no objective criterion for the correct drawing of the “background line.” The extent to which, in one case or another, the “background line” has been drawn correctly is determined by the experimenter’s intuition. It is difficult to agree that assertions about agreement or disagreement between theory and experiment can be made in this way. On the basis of what has been said we conclude that the experimental data obtained as a result of these investigations cannot be regarded as fully reliable.

On the other hand, the ratio of the intensities in the components of the Rayleigh triplet, measured experimentally, must be compared with the refined formula of Landau—Placzek. Below we shall briefly describe a method of processing experimental data which made it possible to avoid arbitrariness and to obtain sufficiently reliable experimental data.

Reliable experimental data for the ratio of the intensities of the fine-structure components were obtained after a method had been developed that made it possible to avoid arbitrariness in processing the experimental data. The method is based on the fact that, in an apparatus whose scheme is shown in Fig. 5, the spectrum of scattered light is photographed simultaneously in two mutually perpendicular directions of polarization, \(z\) and \(x\) (Fig. 10; for the \(z\)-component the electric-field vector of the light wave \(E\) is perpendicular to the scattering plane).

Besides the contour of the triplet, the contour of the exciting line is constructed (the mercury spectrum line \(\lambda 4358\) Å (Fig. 11)), from which the part of the light intensity due to the continuous background of the source is determined. Hence the intensity of the continuous background of the light source \(i_z^{+}\) for the \(z\)-component of the scattered-light line can be determined (Fig. 12).

From an independent experiment it was found that the depolarization coefficient of the “continuous background,” caused by scattering from anisotropy fluctuations, is equal to \(6/7\) and is independent of frequency, which agrees with theory. This result makes it possible, by determining the component of the intensity of anisotropic scattering \(i_x\), to find the fraction of the intensity of light scattered by anisotropy fluctuations in the \(z\)-component, \(i_z\) (Fig. 12).

Fig. 12. Distribution of intensity in the fine-structure components. \(i^{+}\)—intensity of the continuous background produced by the excitation source; \(i_z\) and \(i_x\)—intensity of the \(z\)- and \(x\)-components of depolarized light scattering; dashed line—visible distribution of intensities; solid line—contour of the triplet components.

Between the horizontal section of the “tail” of the visible contour of the \(z\)-component and the line corresponding to \(i_z\), there remains a small excess of intensity due to the superposition of the “tails” of adjacent components of neighboring interference orders. Therefore it is possible to determine the intensity attributable to one order and, by the method of successive approximations\({}^{117}\), to extract from the total contour all three components of the fine structure. In Fig. 12 they are shown by thin

lines. The integral intensity of the components is determined planimetrically.

Measurement of integral intensities can be used directly to determine the intensity ratio in a triplet. Measurement of maximum intensities, generally speaking, is still insufficiently accurate for comparing theory with experiment. It is necessary, in addition, to know the widths of the shifted and unshifted lines². However, if one considers a real case, for example benzene, then one may come to the conclusion that calculations of the intensity ratio from integral and maximum values differ little from one another¹¹⁵. Indeed, if the width of the exciting line is \(\delta\nu_b = 0.17\ \mathrm{cm}^{-1}\), and the widths of the Mandelstam–Brillouin components \(\delta\nu_{\mathrm{M-B}}\) and of the central component \(\delta\nu_c\) are, respectively, \(8.5\cdot 10^{-3}\ \mathrm{cm}^{-1}\) and \(6.5\cdot 10^{-4}\ \mathrm{cm}^{-1}\), then the difference between the integral and maximum intensities is determined by the factor

\[ \frac{\delta\nu_b+\delta\nu_{\mathrm{M-B}}}{\delta\nu_b+\delta\nu_c} = \]

\[ =1.046, \]

i.e. amounts to only \(4.6\%\), which is much less than the photometric errors of the experiment.

Experiment shows⁷ that, for such a width of the exciting line, the results of measuring integral and maximum intensities lead to the same values of \(J_c/2J_{\mathrm{M-B}}\). Thus, one may draw the conclusion, important in the experimental respect, that with a broad exciting line the ratio of integral intensities practically does not differ from the ratio of maximum intensities. The fact that the data for maximum and integral intensities agree with one another within the photometric error clearly shows that the proper width of the triplet lines is much less than the width of the exciting line (together with the instrumental function). This conclusion agrees with the data on the widths of the triplet lines obtained on the basis of the measured dispersion of the speed of sound (see Tables V–VIII and X).

In calculating \(\dfrac{J_c}{2J_{\mathrm{M-B}}}\) by formula (6.3) for benzene and water, our measurements of

\[ \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_s \]

(Table IV) were taken into account, while for the other liquids this coefficient was determined from the formula for the total intensity of scattered light found by method (B), § 2. In the case of benzene, carbon disulfide, and carbon tetrachloride, the quantity \(\beta_s\) was calculated from the speed of sound measured from the fine structure of the Rayleigh line.

Comparison of the experimental data (Table XVI, column 2) with the results of calculation by formulas (6.1) and (6.3) shows that (6.3) agrees much better with the experimental data than (6.1). Taking into account the complexity of the procedure of photometric processing and the approximate character of relation (6.3), the agreement with experiment must be regarded as satisfactory.

It is interesting to note that in the fine-structure spectrum of the Rayleigh scattering line in water there is no clearly expressed central component. If measurements of the maximum intensities are carried out, and the intensity midway between the side components is taken as the intensity of the central component, then one obtains \(J_c/2J_{\mathrm{M-B}}=0.11\), i.e. a value close to Rank’s data and strongly differing from Venkateswaran’s data (Table XV). If, however, the contours of the side components are resolved, as was described earlier, then no intensity remains in the field between the Mandelstam–Brillouin components. That intensity between the shifted

Table XVI

Ratio of the intensities of the fine-structure components

Liquid	\(\dfrac{J_C}{2J_M - B}\), measured	\(\dfrac{J_C}{2J_M - B}\), calculated by formula (6.1)	\(\dfrac{J_C}{2J_M - B}\), calculated by formula (6.3)
Benzene	0.98	0.46	0.62—0.70 *)
Acetone	0.59	0.40	0.57
Carbon disulfide	0.93	0.56	0.98
Carbon tetrachloride	1.10	0.46	0.84
Water	\(\sim 0\)	0.0068	0.0117

*) The difference in the values of the ratio for benzene is explained by the scatter of the values of
\[ \left(\frac{1}{b}\frac{\partial \varepsilon}{\partial T}\right)_\rho . \]

components, which was found in measuring the maximum intensities, is, within the limits of accuracy, entirely due to the overlap of the “tails” of the Mandelstam—Brillouin lines.

Thus, the results obtained make it possible to speak of satisfactory agreement between theory and experiment.

It deserves mention that comparatively recently J. Vaucouleur \(^{52}\), from the discrepancy of his accurate data for the absolute intensity of scattered light in benzene and ether with formula (2.6), on the one hand, and the data of Venkateswaran \(^{110}\), which do not agree with the inaccurate formula (6.1), on the other hand, concludes that the theory of Rayleigh scattering underestimates the intensity of the central component. At this point Vaucouleur considers the theory of Rayleigh scattering imperfect and proposes to improve it. Rocard \(^{53}\) accepts this proposal and attempts to find an additional source of light scattering, taking into account the correlation of fluctuations between neighboring volumes containing one or two molecules. In § 2, however, it was shown that the contradiction between theory and experiment is caused not by the imperfection of the phenomenological theory of light scattering, but by an incorrect estimate of the values of the parameters entering into formula (1.29).

The situation is exactly the same with the verification of the intensity ratio in the fine-structure components. Refinement of the theoretical relations and new measurements show that our theoretical ideas are in satisfactory agreement with experiment.

It should be pointed out that there is a special case in which direct measurements of the intensity ratio in the fine-structure components will obviously diverge from the data of formula (6.3). We have in mind such liquids as triacetin, in which a very narrow, strongly depolarized wing is observed. Such liquids also include acetic acid, benzophenol, salol, and some other liquids.

In deriving relation (6.3), fluctuations of anisotropy were not taken into account, and therefore (6.3) cannot claim to describe the phenomenon as a whole.

§ 7. WIDTH OF THE FINE-STRUCTURE COMPONENTS OF THE SCATTERED-LIGHT LINE

The Mandelstam—Brillouin components in a liquid have a finite width owing to the appreciable damping of the acoustic thermal wave modulating the monochromatic light wave.

If we imagine that the scattering volume of the liquid can be illuminated by monochromatic light of frequency \(\omega\), and that the direction of observation is chosen so that the light is modulated by a thermal elastic wave of frequency \(\Omega\) and temporal damping coefficient \(\delta\), then the result of the modulation can be represented by the following simple relation:

\[ E(t)=A_0E_0e^{-\delta t}\,[\cos(\omega-\Omega)t+\cos(\omega+\Omega)t]. \]

To find the intensity distribution in the Mandelstam—Brillouin component with frequency \(\omega \pm \Omega\), we choose:

\[ E'(t)=A_0E_0e^{-\delta t}\cos\omega' t, \]

where \(\omega'=\omega-\Omega\) or \(\omega'=\omega+\Omega\).

Expansion of \(E'(t)\) into a Fourier integral in monochromatic waves gives:

\[ E'(t)=\int_{-\infty}^{+\infty} E(\omega)e^{i\omega t}d\omega, \]

whence the required Fourier coefficients are

\[ E(\omega)=\frac{1}{\pi}\int_{-\infty}^{+\infty} E'(t)e^{-i\omega t}; \tag{7.1} \]

integrating (7.1), we obtain for the intensity

\[ (E(\omega))^2=\frac{A_0E_0}{16\pi^2} \left[ \frac{1}{(\omega'-\omega)^2+\delta^2} + \frac{1}{(\omega'+\omega)^2+\delta^2} \right]. \tag{7.2} \]

The second term in (7.2) may be neglected, taking into account that it is of order \(\sim \dfrac{1}{\omega'^2}\). Then for the intensity distribution over frequencies in the Mandelstam—Brillouin components we obtain the expression

\[ J(\omega)=\frac{J_0}{(\omega'-\omega)^2+\delta^2}. \tag{7.3} \]

Here \(J_0=\dfrac{H_0E_0}{16\pi^2}\). From relation (7.3) it follows that the intensity distribution in the shifted component will have a dispersion character with total half-width \(\Delta \omega=\delta\). Taking into account that \(\delta=\alpha v\), where \(\alpha\) is the coefficient of sound absorption (in \(\text{cm}^{-1}\)), and \(v\) is the speed of sound, we see that the half-width of the shifted component, expressed in reciprocal centimeters, is written as follows:

\[ \delta_{\mathrm{M-B}}=\frac{\alpha v}{\pi c} \tag{7.4} \]

(\(c\) is the speed of light).

The contour of the central component of the Rayleigh triplet, as well as the contour of the shifted components near the center of each line, has a dispersion character\({}^{67}\). The half-width of the central component is determined by the coefficient of thermal diffusivity \(\chi\) and is equal to

\[ \delta \nu'_c = \left|\frac{2\pi}{\Lambda}\right|^2 \frac{\chi}{\pi c}\ \mathrm{cm}^{-1}. \tag{7.5} \]

Here \(\Lambda\) is the wavelength of hypersound and \(\chi = \dfrac{\varkappa}{c_p \rho}\), where \(\varkappa\) is the coefficient of thermal conductivity.

A study of the intensity distribution in the exciting line showed\({}^{116}\) that the contour of the exciting line is very close to a dispersion contour. From the general relation between the visible contour of the investigated line \(J(\omega)\) and the contour of the excitation line, including the instrumental function \(i(\omega)\),

\[ J(\omega) = \int_{-\infty}^{+\infty} I(\omega) i(\omega - x)\, dx, \tag{7.6} \]

it follows that the given contour of the investigated line \(I(\omega)\) will also be dispersion-like. Therefore, if the visible width of one of the lines of the Rayleigh triplet is denoted by \(\delta \nu'\), and the half-width of the exciting line by \(\delta \nu'_b\), then the required true half-width \(\delta \nu\) will be determined by the expression

\[ \delta \nu = \delta \nu' - \delta \nu_b. \tag{7.7} \]

Let us note that \(\delta \nu'_b\), in turn, is composed of the true half-width of the exciting line and the instrumental half-width\({}^{118}\), and the relation between the visible contour of the exciting line and its true half-width \(\delta \nu_b\) is determined by a relation of the form (7.7). Taking into account that near the maximum the “instrumental contour” has a dispersion character\({}^{118}\), we obtain:

\[ \delta \nu'_b = \delta \nu_b + \delta \nu_a. \tag{7.8} \]

Thus, all the quantities entering into expression (7.7) are known and, it would seem, the half-width of any line of the triplet can easily be determined. In practice, when attempting to determine the width of the components, a very substantial difficulty arises which has not yet been overcome. This difficulty consists in the fact that the true widths of the lines of the triplet are very small, \(\sim 0.01\ \mathrm{cm}^{-1}\) and even less. Meanwhile, the half-width of the exciting line together with the instrumental half-width is \(\sim 0.15\)–\(0.18\ \mathrm{cm}^{-1}\). Therefore the visible width of the components of the Rayleigh triplet practically does not differ from the visible width of the exciting line. The complicated procedure of deciphering the contour of each component of the Rayleigh triplet increases the error, and thus the true half-width of the triplet components lies within the experimental error. Reducing the half-width of the exciting line by a factor of two or three in the case under consideration will not noticeably improve the result. A more radical reduction of the half-width of the exciting line is needed. Apparently, the way out may be sought in using a light source operating on the isotope of mercury or zinc, where the half-width of the exciting line can be made very small. The calculation of the half-width of the Mandelstam–Brillouin components can nevertheless be made for liquids in which dispersion of the speed of sound is observed. In the remaining cases, for the time being only a rough estimate is possible.

Indeed, Table XIII (§ 4) gives the results of calculating the absorption coefficient \(\alpha\) of hypersound waves at a frequency of \(\sim 10^{10}\) Hz. Using these data, one can determine from formula (7.4) the half-widths of the shifted components of the Rayleigh triplet. Formula (7.5) makes it possible to calculate the half-width of the central component. For this calculation, literature data for the coefficient of thermal conductivity were used.

The calculated values of the widths of the Mandelstam–Brillouin components (column 6 of Table XVII) show that, practically for all

Table XVII

Half-width of the Mandelstam–Brillouin components

Substance	\(\tau\cdot 10^{10}\) sec.	\(\alpha'_{\eta}\cdot 10^{-3}\ \mathrm{cm}^{-1}\)	\(\alpha_{\eta}\cdot 10^{-3}\ \mathrm{cm}^{-1}\)	\(\alpha\cdot 10^{-3}\ \mathrm{cm}^{-1}\)	\(10^3\,\delta\nu_{\mathrm{M-B}}\ \mathrm{cm}^{-1}\)
Benzene	2,44	3,65	4,4	8,05	8,5
Carbon disulfide	21,3	0,377	2,26	2,64	8,58
Carbon tetrachloride	0,78	19,6	6,4	26	28,7
Acetic acid	397	0,006	11,3	11,306	13,7

liquids, the magnitude \(\delta\nu_{\mathrm{M-B}}\) lies beyond the limits of possible experimental determination of this quantity, which has been qualitatively confirmed by experiment.

Unfortunately, it is not yet possible to estimate the width of the central component in the same way. Taking into account the considerations stated earlier concerning the slowness of the processes leading to the appearance of the central component, supported by experimental data, one may hope that the use of thermodynamic quantities for calculating the width by formula (7.5) gives results close to reality.

The results of such a calculation for several liquids are given in Table XVIII.

Table XVIII

Width of the central component of the fine structure

Substance	\(\rho,\ \mathrm{g}/\mathrm{cm}^3\)	\(\varkappa\cdot 10^5,\ \dfrac{\mathrm{cal}}{\mathrm{cm}\cdot \mathrm{sec}\cdot \mathrm{deg}}\)	\(C_p,\ \dfrac{\mathrm{cal}}{\mathrm{g}\cdot \mathrm{deg}}\)	\(\chi\cdot 10^4,\ \mathrm{cm}^2\cdot \mathrm{sec}^{-1}\)	\(\Lambda\cdot 10^5\ \mathrm{cm}\)	\(\delta\nu_C\cdot 10^4\ \mathrm{cm}^{-1}\)
Benzene	0,879	33,3	0,417	9,1	2,02	6,5
Carbon disulfide	1,262	38,5	0,24	12,6	1,9	3,1
Carbon tetrachloride	1,598	28	0,202	8,8	1,98	6,3
Acetic acid	1,049	43	0,483	8,5	2,26	3,75
Acetone	0,792	42,3	0,514	10,5	2,27	4,26
Toluene	0,866	34,5	0,400	9,7	2,03	4,26

In absolute value, the width of the central component is approximately an order of magnitude smaller than the width of the shifted components.

The relation between the ratio of the integrated intensities and the ratio of the intensities at the maxima of the fine-structure components is expressed as follows\(^2\):

\[ \frac{I_c}{2I_{\mathrm{M-B}}} = \left( \frac{I_c}{2I_{\mathrm{M-B}}} \right)_{\max} \cdot \frac{\delta\nu_c}{\delta\nu_{\mathrm{M-B}}}. \tag{7.9} \]

On the basis of the data in Tables XVI–XVIII and relation (7.9), one can easily obtain that for benzene, for example, the true ratio

\[ \left(\frac{J_c}{2J_{M-B}}\right)_{\max}=15.5, \]

whereas, for a width of the exciting line of \(0.17\ \text{cm}^{-1}\), the experimentally measured value is

\[ \frac{J_c}{2J_{M-B}}=0.97. \]

The distortion thus revealed has practically no significance for determining the ratio of the intensities of the fine-structure components, but it makes the experimental determination of the half-widths of the components of the Rayleigh triplet practically impossible. From this point of view, the result of the work of H. E. Sterin \(^{99}\), who attempted to detect broadening of the Mandelshtam–Brillouin components upon adding small amounts of toluene to benzene, becomes understandable.

In the experiments of P. A. Bazhulin \(^{95}\), a strong decrease in the sound absorption coefficient \(\alpha\) was observed when toluene was added to benzene. M. A. Leontovich \(^{119}\) explains this effect by a decrease in the relaxation time \(\tau\). A decrease in \(\tau\) in the hypersonic region acts in such a way as to increase \(\alpha\) and, consequently, should lead to a broadening of the shifted components.

In a benzene–toluene mixture, at a concentration of the latter equal to 20% (the maximum toluene content in the mixture investigated in \(^{99}\)), a threefold decrease in the absorption coefficient \(\alpha\) was observed \(^{95}\). According to what was said above, one should expect a change by the same factor in the width of the Mandelshtam–Brillouin components.

The true half-width of these components in benzene (see Table XVII) is \(0.008\ \text{cm}^{-1}\); consequently, the expected width of the components in a 20% benzene–toluene mixture should be \(0.024\ \text{cm}^{-1}\). On the other hand, the width of the exciting line was about \(0.17\ \text{cm}^{-1}\). Thus, the apparent width of the lines in pure benzene is \(0.178\ \text{cm}^{-1}\), while in the mixture it should become equal to \(0.194\ \text{cm}^{-1}\). It follows that the effect of the impurity on the apparent width of the Mandelshtam–Brillouin components is 8–10%. Such a small change in width is very difficult to detect; it is probably precisely this difficulty that explains the result of the work \(^{99}\).

In the present review, a certain summary has been given of investigations of the total Rayleigh scattering and the fine structure of the spectral line of scattered light. The broad range of questions relating to the study of depolarized scattered light (the wing of the Rayleigh line), to the scattering of light in viscous liquids and glasses, as was mentioned, has not been discussed here and deserves separate consideration.

Intensive investigation, especially in recent times, of the questions considered in this review has led to the clarification of a whole series of problems. On the other hand, new questions of importance in both theoretical and practical respects have arisen and still await solution.

As a result of a careful study of the fine structure of the Rayleigh scattering line, a large dispersion of the velocity of sound has been discovered in a number of low-viscosity liquids, not yet accessible to detection by other methods; the same investigation has raised the question of replacing the thermodynamic theory of light scattering by a new theory, free from thermodynamic restrictions and allowing for the dispersion of the parameters that determine the intensity of the scattered light (compressibility, etc.).

Recently there has arisen a purely experimental problem, namely that photographic and photoelectric methods of measuring the scattering coefficient give quantitative results that are very ...

strongly differing from one another. What the reason is for such judgments still remains unclear.

Meanwhile, the solution of this question is important not only scientifically, but also in practical terms, since for the method of determining the masses of macromolecules by light scattering in polymer solutions a standard is necessary, for which the coefficient of light scattering must be precisely determined.

The further development of the field of molecular optics considered here will undoubtedly prove essential not only for the problem of light scattering in liquids, but also for the physics of polymers and molecular acoustics.

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