Two New Phenomena in Second-Order Phase Transformations

I. A. Yakovlev and T. S. Velichkina

In recent years, new acoustic and optical phenomena occurring in second-order phase transformations in liquid helium, in Rochelle salt, and in quartz have been studied experimentally and theoretically. In the first two substances, anomalous absorption of sound near the $\lambda$-points was discovered; in quartz, strong molecular scattering of light during a phase transformation was found. The study of these phenomena reveals new aspects of the molecular-statistical picture of a phase transition. The theoretical treatment of the new questions can be carried out on the unified basis of L. D. Landau’s theory of second-order phase transitions.

In the present review we devote principal attention to phenomena in solids, as being better known to us. Since in all the cases considered there are experimental results, for greater clarity of exposition we begin each part of our review with a description of the actually observed course of the phenomena, without following the chronological sequence of the reports by individual authors.

§ 1. The Phenomenon of Opalescence in the Phase Transformation of Quartz

As is known, in crystalline quartz, as well as in other transparent crystals, liquids, and gases, the phenomenon of molecular scattering of light takes place. The cause of this phenomenon in pure substances is optical inhomogeneities, continuously arising in any medium as a result of thermal fluctuations. Molecular scattering of light in solids was first discovered precisely in crystalline quartz by G. S. Landsberg. This work of G. S. Landsberg was the beginning of that extensive cycle of his joint investigations with L. I. Mandelstam which culminated in the discovery of combination scattering of light.

Experiment shows that at $20^\circ\mathrm{C}$ molecular scattering of light in quartz is very weak: the intensity of the light scattered in all directions by illuminated quartz amounts at $20^\circ\mathrm{C}$ to only $10^{-7}$ of the intensity of the light incident upon it*).

The intensity of the scattered light $I$ in the temperature interval from room temperature to $+250^\circ\mathrm{C}$ depends practically linearly on temperature. The scattered light is partially depolarized in accordance with the anisotropy

*) To illustrate the figure cited, we note that quartz scatters light only 7 times more strongly than clean air at atmospheric pressure.

of the crystal; the quantity \(I \sim \lambda^{-4}\) (Rayleigh’s law), where \(\lambda\) is the wavelength of light.

The brief summary given of the necessary information on molecular scattering of light in solids will be needed in order to compare previously known facts with the new phenomenon to the description of which we now turn directly.

In the works of G. S. Landsberg\(^1\), and subsequently in the works of G. P. Motulevich\(^2\), the phenomenon of light scattering was investigated in the \(\alpha\)-modification of a quartz crystal, also called the low-temperature phase of quartz. In this modification the quartz crystal belongs to the trigonal crystal system and has the symmetry elements \(A_3 3A_2\). At a temperature of \(573^\circ\)C the crystal lattice of quartz rearranges: it becomes hexagonal; the symmetry of the lattice increases (\(A_6 6A_2\)). The high-temperature modification of quartz, at \(T > 573^\circ\)C, is called \(\beta\)-quartz. On the basis of measurements of heat capacity\(^3\), thermal

Fig. 1. Dependence of the refractive index of quartz on temperature (according to the data of K. N. Baranskii).

expansion of quartz\(^4\), and on the basis of the nature of the change in the symmetry of its crystal lattice during the phase transformation, the \(\alpha \rightleftarrows \beta\) phase transition should be regarded as a \(\lambda\)-transition, close to a Curie critical point.*)

A characteristic feature, and one very essential for the phenomena described below, of the phase transformation of quartz is the sharp change in the refractive index \(n\) of this crystal near its \(\lambda\)-point. Namely, \(\Delta n = 0.0012\) in a temperature interval \(\sim 0.1^\circ\) near the transition point. The dependence of \(n\) on \(T\), according to the data of K. N. Baranskii\(^5\), who investigated the immediate vicinity of the transition temperature, is given in Fig. 1.

Another feature of the quartz transformation under consideration, which will also be essential for us in what follows, is the dependence on temperature, on approaching the \(\lambda\)-point, of the isothermal elastic constants of this crystal\(^6\) (Fig. 2).

*) We recall that a Curie critical point is customarily understood to mean that point on the \((p,T)\)-diagram of a substance at which the line of phase transitions of the second kind passes into the line of phase transitions of the first kind.

In the work of I. A. Yakovlev, T. S. Velichkina, and L. F. Mikheeva⁷ the problem was posed of studying the molecular scattering of light in a quartz crystal undergoing a rearrangement of the crystal lattice during its transition from the α- to the β-modification at \(+573^\circ\text{C}\)*). The essence of the experiment consisted in measuring, as a function of temperature, the intensity of the light scattered by a quartz single crystal at an angle of \(90^\circ\) to the direction of propagation of the primary light beam. The experimental arrangement

Fig. 2. Dependence of the elastic modulus of quartz \(\perp\) to the optical axis on temperature.

is shown in Fig. 3; the captions to it sufficiently explain the optical aspect of the experiment.

In the study of phase transformations, the thermal regime of the experiment is, naturally, of decisive importance. Therefore we shall quote from⁷, in addition to the optical scheme, also the data on the temperature field in the furnace at the location of the quartz block under investigation \((20 \times 20 \times 40\ \text{mm})\). The horizontal temperature gradient along the direction of the primary light beam was, in different experiments, from \(0.03\) to \(0.1^\circ/\text{mm}\). The vertical temperature gradient was equal to \(0.01^\circ/\text{mm}\). Such a temperature distribution in the furnace and in

*) The reasons determining the expediency of carrying out such an experiment are set forth in § 2.

quartz provided an arrangement in which the flat isothermal layers of the crystal were practically normal to the propagation of the primary light beam. It is obvious that, as the temperature of the furnace was raised, the layer of quartz situated opposite the observation window of the furnace (Fig. 3) at some moment reached the temperature of the phase transition. Then, along the path of the primary light beam, on both sides of this quartz layer there were two different modifications of the crystal, and the region of contact of the two solid phases of the substance became accessible for investigation.

Fig. 3. General layout of the apparatus (horizontal projection): \(Л\)—mercury SVD lamp; \(O_1, O_2, O_3\)—objectives; \(\Phi_1\)—heat-protective light filters; \(\Phi_2\)—colored light filters; \(H_1\) and \(H_2\)—polarizing prisms; \(\Phi Э\)—photoelement; \(Ф.Э.У.\)—photomultiplier; \(З\)—mirror shutter; \(T\)—telelens; \(П\)—furnace.

After the explanations given concerning the course of the experiment, we may present its results, which are shown in Fig. 4. In the temperature interval \(20\text{--}450^\circ\mathrm{C}\) a linear dependence of \(I_T/I_{20^\circ C}\) on temperature was established. Earlier, in the works of G. S. Landsberg and G. P. Motulevich \(^{1,2}\), this linear dependence had been studied up to \(250^\circ\mathrm{C}\). A further increase in the temperature of the crystal brings the value of \(I_T/I_{20^\circ C}\) to a sharp maximum near the temperature of the phase transformation. The steep fall of the curve on the high-temperature side corresponds to a decrease in the scattering of light in the high-temperature phase of quartz. In another figure, given in \(^{7}\), it is shown that, after the minimum of the intensity of the scattered light reached in the high-temperature phase at \(600^\circ\mathrm{C}\), a new slight rise of the value of \(I_T/I_{20^\circ C}\) begins at still higher temperatures.

The vertical arrow with the number \(1.4\cdot10^4\), placed at the top of the peak in Fig. 4, symbolically indicates the intensity of that special phenomenon of opalescence which constitutes the principal result of work \(^{7}\).

Photographic and visual observations of the quartz crystal reveal that, when its temperature approaches \(573^\circ\mathrm{C}\), within the crystal, from its “hot” end toward the “cold” end, there moves a strongly light-scattering band of haze. On one side of this band of haze lies the \(\alpha\)-phase of quartz, on the other side its \(\beta\)-phase. The speed of motion of the band of haze along the crystal (in the direction of the primary light beam (Fig. 3)) is set by the rate of heating of the furnace.

The geometrical width of the fog band, depending on the horizontal temperature gradient produced in the quartz, varied from 0.5 to 3 mm. The temperature width of the fog band was always \(\sim 0.1^\circ\). Photographs of the phenomenon at different temperature gradients are given in Figs. 5 and 6. For comparison, Fig. 7 gives a photograph of a light beam in a single-phase quartz crystal far from its phase-transition temperature.

Measurements of the intensity of the scattered light, made with a photoelectron multiplier, showed that the fog band scatters light \(1.4 \cdot 10^4\) times more strongly than quartz at room temperature or than the high-temperature phase of this crystal at \(600^\circ\mathrm{C}\) *).

Experience shows that, upon cooling the crystal, the process described above of the motion of the fog band in quartz develops reversibly in the opposite direction. The phenomenon can be reproduced many times both on one and the same crystal specimen and on specimens of different origin and

Fig. 4. Dependence of the intensity of scattered light on temperature: ●—heating of quartz; ○—cooling; ×—repeated experiment; — theoretical curve (for the temperature interval \(573\text{–}600^\circ\), it coincides completely with the experimental one).

Fig. 5. Photograph of the intersection of the fog band in quartz by the primary light beam (exposure 1 sec.; horizontal temperature gradient \(\simeq 1\ \mathrm{deg/mm}\)).

Fig. 6. Photograph of the intersection of the fog band in quartz by the primary light beam (exposure 1 sec.; horizontal temperature gradient \(\simeq 0.03\ \mathrm{deg/mm}\)).

*) For a visual illustration of this quantitative result, let us note that the light scattering of the fog in quartz is 10 times more intense than in liquid benzene, in which the phenomenon is already quite sufficient in intensity for lecture demonstration.

of various orientations. X-ray photographs of the quartz crystals taken before and after the above-described experiments are identical. The intensity of the light scattered by the turbidity obeys Rayleigh’s law \((\sim \lambda^{-4})\). The scattered light is partially depolarized, \(I_x/I_y = 6\%\). (The depolarization of light scattered by quartz at temperatures adjacent to the \(\lambda\)-point is \(\sim 18\%\). In both cases the primary light is linearly polarized.)

Fig. 7. Photograph of the primary light beam in quartz (exposure 1 hour).

Thus, investigations of molecular light scattering in a solid (in quartz) at a temperature close to its \(\lambda\)-point have shown that in this case intense opalescence occurs in the crystal.

Turning to the further discussion of this optical phenomenon, we note that the observations described already make it possible to draw a certain thermodynamic conclusion concerning the character of the phase transition in quartz \(^{8}\). Namely, the presence between the two modifications of quartz of a turbidity band of finite width leaves no possibility for interpreting the \(\alpha \rightleftarrows \beta\) transformation of quartz as a first-order phase transition or, more precisely, as a first-order transition far from the Curie critical point. Thus, light scattering can in some cases serve as an independent criterion for establishing the thermodynamic characteristic of a phase transition in a solid.

§ 2. THEORY OF LIGHT SCATTERING

IN PHASE TRANSITIONS OF THE SECOND ORDER

The first in time and so far the only general consideration of the question of light scattering in phase transitions of the second order belongs to V. L. Ginzburg \(^{9*}\). The formulation of the problem and the prediction of a new effect in V. L. Ginzburg’s work preceded the report of the discovery of the phenomenon of opalescence in quartz \(^{7}\), which, however, was investigated on the basis of the independent considerations set forth below.

As is known, light scattering in isotropic media is described by Einstein’s theory \(^{10}\), which established a quantitative relation between the intensity of the scattered light \(I\) and the fluctuations of the optical dielectric constant \(\Delta \varepsilon\) of the medium. The problem of light scattering in crystals was solved in the works of M. A. Leontovich and L. I. Mandelstam \(^{11}\) and, in its final form, by G. P. Motulevich \(^{2}\). At the present stage we shall confine ourselves to considering quartz as an isotropic medium.

According to Einstein,

\[ I \sim \frac{I_0(\Delta \varepsilon)^2}{\lambda^4}, \tag{1} \]

where \(I_0\) is the intensity of the primary light beam; \((\Delta \varepsilon)^2 = \overline{(\varepsilon - \varepsilon_0)^2}\) is the mean square fluctuation of the dielectric constant \(\varepsilon = n^2\) around its mean value \(\varepsilon_0\). Using as statistically independent thermodynamic variables the density \(\rho\) and the temperature \(T\),

* We do not consider it necessary to dwell here on earlier theoretical works concerning light scattering in He. These works proceeded from a specific, and experimentally incorrect, interpretation of the \(\lambda\)-point of He as a “Bose–Einstein condensation” point in the momentum space of a gas. The results of these works were experimentally refuted.

TWO NEW PHENOMENA IN PHASE TRANSITIONS OF THE SECOND KIND

we can write:

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

expressing the fluctuations of \(\varepsilon\) in terms of the fluctuations of \(\rho\) and \(T\).

Experience shows that in all previously known cases of light scattering

\[ \left(\frac{\partial \varepsilon}{\partial \rho}\right)_T^2 \overline{(\Delta \rho)^2} \gg \left(\frac{\partial \varepsilon}{\partial T}\right)_\rho^2 \overline{(\Delta T)^2}. \tag{3} \]

Then

\[ \overline{(\Delta \varepsilon)^2} = \left(\rho \frac{\partial \varepsilon}{\partial \rho}\right)_T^2 kTb_T\,\frac{1}{v}, \tag{4} \]

since

\[ \overline{(\Delta \rho)^2} = \rho^2 \frac{kTb_T}{v}, \]

where \(k\) is Boltzmann’s constant, \(b_T\) is the isothermal compressibility of the medium, and \(v\) is the volume of the fluctuation. Taking into account inequality (3) and using (4), we find:

\[ I_\rho \sim I_0 k b_T \rho^2 \left(\frac{\partial \varepsilon}{\partial \rho}\right)_T^2 T. \tag{5} \]

The last expression gives the intensity of the light, \(I_\rho\), scattered by inhomogeneities of the refractive index of the medium produced by fluctuations of its density.

Expression (5), together with the temperature dependence of the elastic modulus of quartz shown in Fig. 2, already made it possible to expect an anomaly in the scattering of light at the \(\lambda\)-point. Indeed, the minimum of the elastic modulus evidently corresponds to a maximum of the compressibility and, consequently, of the intensity of the scattered light. If this circumstance is taken into account and Einstein’s formula is applied to quartz, then we obtain the dependence of \(I\) on \(T\) shown in Fig. 4 by the dashed curve. The high-temperature branch of this dashed curve coincides so closely with the experimental graph in the interval \(573\)—\(600^\circ\)C that, on the scale of the present drawing, they cannot be shown separately. Thus, the steep approach to the maximum of the scattered light and the steep fall of the curve after this maximum can be represented by Einstein’s formula, provided that the temperature dependence of the compressibility of quartz is taken into account.

However, the phenomenon of opalescence at the \(\lambda\)-point itself, i.e., the strong scattering of light by the fog band lying between the \(\alpha\)- and \(\beta\)-phases of quartz, does not find an explanation in the considerations set forth. Even if one assumes that, in measurements of the elastic moduli, their minimum values were missed, there is still no basis for supposing that the compressibility of quartz \(b_T\) could increase by \(10^4\) times at the \(\lambda\)-point. It must be remembered that even in the process of the phase transition quartz remains a solid body and an optically uniaxial crystal\(^5\). Consequently, the phenomenon of opalescence requires a special explanation.

An attempt to find this explanation, while remaining within the framework of relation (2), is made in work\(^7\). In this work it is indicated that the dependence, shown in Fig. 1, of the refractive index of quartz \(n \equiv \varepsilon^2\) on temperature near the phase-transition point compels one to reconsider the validity of inequality (3). Denoting by \(I_T\) the intensity of the scattered light associated with the presence of the term
\(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_\rho^2(\Delta T)^2\) in formula (2), we can

write:

\[ \frac{I_T}{I_\rho}=-\frac{4n^2T\left(\dfrac{\partial n}{\partial T}\right)^2_\rho} {\rho^2\left(\dfrac{\partial \varepsilon}{\partial \rho}\right)^2_T b_T C_v}, \tag{6} \]

taking into account that

\[ \left(\frac{\partial \varepsilon}{\partial T}\right)^2_\rho = \left[2n\left(\frac{\partial n}{\partial T}\right)_\rho\right]^2 \quad \text{and} \quad \overline{(\Delta T)^2}=\frac{kT^2}{C_v}, \]

where \(C_v\) is the heat capacity of a unit volume.

If one forms the ratio \(I_T/I_\rho\) at two different temperatures, putting \(T=\theta\) in the expression for \(I_T\), and \(T=T_{\mathrm{room}}\) in the expression for \(I_\rho\), then

\[ \frac{I_\theta}{I_{\rho(T_k)}}= -\frac{4n^2\theta^2\left(\dfrac{\partial n}{\partial T}\right)^2_\rho} {\rho^2\left(\dfrac{\partial \varepsilon}{\partial \rho}\right)^2_T b_T C_v T_k}. \tag{6'} \]

This is convenient for comparison with experiment, since under normal conditions all the light scattering is determined by the term \(I_\rho\), and the parameters of the medium are known with comparative reliability.

The numerical values of the factors entering into (6′) are as follows:

\[ 4n^2=10,\quad b_T=2\cdot 10^{-12}\ \mathrm{cm}^2/\mathrm{dyn};\quad \theta=846^\circ\mathrm{K};\quad \left(\rho\frac{\partial \varepsilon}{\partial \rho}\right)_T\sim 1; \]

\[ C_v=7\cdot 10^7\ \mathrm{erg}/\mathrm{deg}\cdot \mathrm{cm}^3. \]

The difficulty arises in substituting into (6′) the value of \(\left(\dfrac{\partial n}{\partial T}\right)^2_\rho\). Directly measured in experiment is \(\left(\dfrac{\partial n}{\partial T}\right)_p\), where \(p\) is pressure, and not \(\left(\dfrac{\partial n}{\partial T}\right)_\rho\). The relation between these two different derivatives of \(n\) with respect to \(T\) may be written in the form:

\[ \left(\frac{\partial \varepsilon}{\partial T}\right)_\rho = \left(\frac{\partial \varepsilon}{\partial T}\right)_p + \left(\frac{\partial \varepsilon}{\partial p}\right)_T \frac{a}{b_T}, \tag{7} \]

where \(a\) is the thermal coefficient of volume expansion of quartz. For \(\left(\dfrac{\partial n}{\partial T}\right)_p\), the experiment of K. N. Baranskii\(^5\) gives the value \(1.2\cdot 10^{-2}\ 1/\mathrm{deg}\). The values of \(a\) may be taken from the above-cited work of P. G. Strelkov et al.\(^4\): \(a_{\max}=1.2\cdot 10^{-3}\ \mathrm{deg}^{-1}\). Further, it is easy to show that

\[ \left(\frac{\partial \varepsilon}{\partial p}\right)_T = \rho\left(\frac{\Delta \varepsilon}{\Delta p}\right)_T b_T \sim b_T. \]

After these substitutions into (7) it turns out that \(\left(\dfrac{\partial \varepsilon}{\partial T}\right)_\rho=\left(\dfrac{\partial \varepsilon}{\partial T}\right)_p\) to an accuracy of 10%.

On the basis of these estimates, formula (6) gives for \(I_T/I_\rho\sim 10^4\), i.e., a figure close to the experimental results. Thus, the opalescence of quartz near the \(\lambda\)-point may be regarded as scattering of light by optical inhomogeneities caused by temperature fluctuations. Small volumes of the scattering medium have different (fluctuating) refractive indices, lying within the limits given by the steep drop of the curve of the dependence of \(n\) on \(T\) (see Fig. 1).

Here attention should be paid to the special meaning, in the present case, attached to the concept of the dependence of the refractive index on

temperature. Usually such a dependence is understood to mean a change in the refractive index of a substance as a result of a temperature shift of its absorption bands, occurring with an unchanged molecular structure of the medium. Under normal conditions such a dependence of \(n\) on \(T\) is negligible\({}^{12}\). But in our case temperature changes (at \(p=\mathrm{const}\)) near the \(\lambda\)-point correspond, in essence, to changes in the structure of the crystal, which are responsible for the sharp temperature behavior of its refractive index. Therefore the above derivation of formula (6) is only a method of phenomenological estimation of the scattering of light during a structural rearrangement of the crystal—a calculation based on data on the temperature dependence of the refractive index of the crystal near the phase-transition point. These considerations can be supported by the following estimate: a simple calculation shows that temperature fluctuations in volumes of order \((0.1\lambda)^3\) have a magnitude of \(0.1^\circ\mathrm{C}\), i.e. such volumes may, as a result of fluctuations, undergo as it were transformations from the \(\beta\)- to the \(\alpha\)-phase of quartz and back. The size of the volume is indicated in accordance with the requirements of Rayleigh’s law \((\sim \lambda^{-4})\), which holds for light scattered by a mist. The calculated value \(\sqrt{\overline{(\Delta T^2)}}=0.1^\circ\) coincides with the temperature width of the mist zone estimated in work\({}^{7}\).

In view of the fact that large values of \(\dfrac{\partial n}{\partial T}\) occur only on the low-temperature side of the \(\lambda\)-point, anomalous light scattering should also be considered to occur only at \(T<\theta\), where \(\theta\) is the temperature of the phase transition.

The foundations of Einstein’s theory set forth above, and the special case of its application to quartz opalescence, make it possible to present clearly the essence of V. L. Ginzburg’s theory\({}^{9}\), which preceded both experiment\({}^{7}\) and the quantitative treatment of the phenomenon only just carried out.

The calculation by V. L. Ginzburg\({}^{9}\) is based on the application of L. D. Landau’s theory of second-order phase transitions\({}^{13}\). Since in V. L. Ginzburg’s paper the theory of second-order phase transitions is applied for the first time to an optical statistical problem, it is necessary to recall, as briefly as possible, those aspects of this theory that are directly relevant to the calculation.

According to L. D. Landau’s theory, the thermodynamic potential \(\Phi\) of a system near the \(\lambda\)-point may be represented as a function of three quantities: \(\Phi=\Phi(\varphi_1,\varphi_2,\eta)\). In addition to two ordinary thermodynamic variables, for example \(\varphi_1=p\) and \(\varphi_2=T\), the expression for the potential contains one more variable \(\eta\), called the characteristic parameter of the system. By means of the quantity \(\eta\), one takes into account quantitatively those changes of \(\Phi\) near the \(\lambda\)-point that are connected with the appearance in the system (as a result of a phase transition) of certain new qualities: ordering of the arrangement of atoms of different kinds in an alloy, a change in lattice symmetry, the emergence of a superfluid component in helium or of spontaneous polarization in a ferroelectric, etc. Various quantities may serve as the parameter \(\eta\), for example, the probability of finding atoms of a certain kind at specified sites of a crystal lattice (the case of an alloy), the magnitude of the displacement of atoms (or molecules) from their initial positions during a lattice rearrangement, the magnetic moment of spontaneous magnetization, etc. Such parameters, even for one and the same case of a phase transition, can be chosen in many ways. Naturally, each choice of \(\eta\) will correspond to its own functional expression of \(\Phi\) in terms of \(p\), \(T\), and \(\eta\). In contrast to the quantities \(p\) and \(T\), the parameter \(\eta\) is not one of the independent thermodynamic variables connected with the others only by an equation of state. Under equilibrium conditions the value of \(\eta\) is determined

quantities \(p\) and \(T\), associated with \(\eta\), by the condition of minimality of the thermodynamic potential at given \(p\) and \(T\):

\[ \left(\frac{\partial \Phi}{\partial \eta}\right)_{p,T}=0. \]

According to the meaning of the parameter \(\eta\), its value is determined as follows: the equilibrium value is \(\eta=0\) for \(T>\theta\) and \(\eta \ne 0\) for \(T<\theta\), where \(\theta\) is the Curie temperature at the given pressure \(p\).

In view of the fact that near the point of a second-order transition the quantity \(\eta\) assumes arbitrarily small values, the potential \(\Phi\) under these conditions may be written in the following form:

\[ \Phi=\Phi_0+\alpha\eta^2+\frac{\beta}{2}\eta^4, \tag{8} \]

i.e., expanded in a series in powers of the small parameter \(\eta\). The expansion coefficients \(\alpha=\alpha(p,T)\) and \(\beta=\beta(p,T)\). At the \(\lambda\)-point, as shown in the theory of L. D. Landau, \(\alpha=0\); therefore, for \(p=\mathrm{const}\) one may put \(\alpha=\alpha_0'(T-\theta)\). The coefficient \(\beta\) vanishes at the Curie critical point. Then the expansion of \(\Phi\) in powers of \(\eta\) cannot be restricted to the terms written in (8). For the equilibrium values of \(\eta\), from the condition

\[ \frac{\partial \Phi}{\partial \eta}=0 \]

we obtain, for \(T<\theta\):

\[ \eta_0^2=-\frac{\alpha}{\beta}=\frac{\alpha'(\theta-T)}{\beta}, \]

and for \(T>\theta\), \(\eta=0\).

The values of the quantity \(\eta\), as well as of other thermodynamic parameters, are subject to statistical fluctuations. The magnitude of these fluctuations can be calculated by general methods. Namely, the probability of a fluctuation \(\Delta\eta\) at constant \(p\) and \(T\) is proportional to

\[ \exp\left(-\frac{\Delta\Phi_v}{kT}\right), \]

where \(v\) is the volume of the fluctuation, \(k\) is Boltzmann’s constant, and \(\Delta\Phi\) is the increase in the thermodynamic potential caused by the fluctuation \(\Delta\eta\).

The investigation of fluctuations of the parameter \(\eta\) is, as L. D. Landau showed, of special interest in the immediate neighborhood of \(\lambda\)-points. Let us explain this using the case considered in his work, namely the calculation of fluctuations \(\Delta\eta\) in an alloy. Here the quantity \(\eta\) characterizes the degree of order in the distribution of atoms of the two components of the alloy over the sites of its crystal lattice. Above the Curie point, where at equilibrium \(\eta=0\), the fluctuation \(\Delta\eta=\eta\). From (8) we find, to accuracy up to \(\eta^2\),

\[ \Delta\Phi=\Phi-\Phi_0=\alpha\eta^2=\alpha(\Delta\eta)^2. \]

The probability of a fluctuation is proportional to the expression indicated above. Therefore the mean value

\[ \overline{(\Delta\eta)^2}=\frac{kT}{2\alpha v}. \]

But at the Curie point \(\alpha\to 0\), and therefore

\[ \overline{(\Delta\eta)^2}\to\infty, \]

which must manifest itself in those physical phenomena for which the homogeneity of the crystal is essential. The conclusion just given concerning the value of \(\overline{(\Delta\eta)^2}\) near the Curie point applies, of course, only to fluctuations in each small part of the body. In the entire volume of the body the fluctuation of the ordering is nonuniform. Since the term \(\alpha\eta^2\) in the expansion of \(\Phi\) near the Curie point is small, when considering fluctuations that are functions of the coordinates we must take into account not only the dependence of \(\Phi\) on \(\eta\), but also on the gradient of \(\eta\):

\[ \Phi=\Phi_0+\alpha\eta^2+\frac{\beta}{2}\eta^4+\gamma(\Delta\eta)^2. \tag{9} \]

The question of fluctuations of the parameter \(\eta\) and their consequences for the scattering of light occupies a central place in the theory of V. L. Ginzburg.

After this quite summary account of the information from the theory of phase transitions of the second kind, it is easy to understand the essence of the general theory of molecular light scattering near the \(\lambda\)-point belonging to V. L. Ginzburg.

V. L. Ginzburg naturally relates the changes in the optical properties of substances near their \(\lambda\)-points to the appearance of that new property which is characteristic of the given phase transition: rearrangement of the crystal structure, the appearance of spontaneous polarization, superconductivity, etc. In other words, he takes into account a definite correspondence between the values of the refractive index and of the characteristic parameter of the substance near the transition point.

But for questions of light scattering it is not the value of the refractive index itself that is essential, but its fluctuations; therefore, for the further calculation only the relation between the fluctuations of the optical dielectric permittivity \(\Delta \varepsilon\) and the fluctuations of the characteristic parameter \(\Delta \eta\) is needed. In a first approximation it is natural to assume that this relation is expressed by the simple relation

\[ \Delta \varepsilon = a \Delta \eta^{2}, \tag{10} \]

where \(a\) is a certain coefficient of proportionality *). In order to obtain concrete results from the assumption made, it is necessary to calculate the fluctuations of the characteristic parameter and, expressing through them the fluctuations \(\Delta \varepsilon\), to apply Einstein’s theory of light scattering to the optical part of the problem. Thus, the essence of the matter will be to take into account in the phenomenon of light scattering a new cause of fluctuations of \(\varepsilon\). As we shall see from what follows, an explicit disclosure of the meaning of the characteristic parameter will in this case even prove unnecessary, since the quantity \(\eta\) will not enter into the final formula for the intensity of light “scattered by fluctuations of the characteristic parameter.”

Taking into account light scattering caused by the new reason makes it possible to generalize formula (2), which will now be written as follows:

\[ \overline{(\Delta \varepsilon)^{2}} = \left(\frac{\partial \varepsilon}{\partial \rho}\right)^{2}_{T,\eta} \overline{(\Delta \rho)^{2}} + \left(\frac{\partial \varepsilon}{\partial T}\right)^{2}_{\rho,\eta} \cdot \overline{(\Delta T)^{2}} + \left(\frac{\partial \varepsilon}{\partial \eta^{2}}\right)^{2}_{\rho,T} \overline{(\Delta \eta^{2})^{2}}, \tag{11} \]

where \(\left(\dfrac{\partial \varepsilon}{\partial \eta^{2}}\right)^{2}_{\rho,T}\) corresponds to the coefficient \(a\) in relation (10). The form of the expression (11) written down shows that fluctuations of density, temperature, and of the characteristic parameter \(\eta\) are considered as independent of one another. The second term in the new expression \(\overline{(\Delta \varepsilon)^{2}}\), on the basis of the usual considerations, is always assumed to be small, while the last, on the contrary, predominates near the point of phase transition.

The value \(\overline{(\Delta \varepsilon)^{2}}_{\eta}\) must be calculated with the aid of the adopted expression for the thermodynamic potential. Carrying out the required calculation for the case \(T < \theta\) is sufficiently simple and gives:

\[ \overline{(\Delta \varepsilon)^{2}}_{\eta} = a^{2}\frac{k \theta}{\beta_{0} v}. \tag{12} \]

The case \(T > \theta\), as the more complicated one, has recently been re-examined by V. L. Ginzburg and A. P. Levanyuk \(^{14}\). Since we have in mind here only to set forth the basic propositions of V. L. Ginzburg’s theory, we shall not dwell on this point.

\[ \text{*) The proportionality between } \Delta \varepsilon \text{ and } \Delta \eta^{2}, \text{ and not } \Delta \eta, \text{ must hold because the sign preceding } \eta \text{ must not be significant for the value of the quantity } \Delta \varepsilon = \varepsilon - \varepsilon_{0}. \]

The value found for \(\overline{(\Delta \varepsilon)^2}\) can be substituted into the general expression of the form (1) for the intensity of the scattered light, which we shall now denote by \(I_\eta\). However, for subsequent estimates it is more convenient to form the ratio \(I_\eta/I_\rho\), where \(I_\rho\), as above, denotes the intensity of light scattered by density fluctuations. Then we find:

\[ \frac{I_\eta}{I_\rho} = \frac{a^2\theta} {\beta_\theta\left(\rho\frac{\partial \varepsilon}{\partial \rho}\right)^2 b_T T}, \tag{13} \]

where \(\theta\) and \(T\) are the temperatures at which the values of \(I_\eta\) and \(I_\rho\), respectively, are taken. The use of the ratio \(I_\eta/I_\rho\), just as earlier of the ratio \(I_T/I_\rho\) (6), makes it possible to avoid the difficulty of determining the absolute value of \(I_\eta\).

Expression (13) is very significant. It shows that in phase transitions close to the Curie critical point, at which \(\beta_\theta=0\), the scattering of light by fluctuations of the characteristic parameter becomes very intense, analogous to critical opalescence. It is true that at the Curie critical point itself expression (13) loses its meaning when \(\beta_\theta=0\). This indicates that in that case one must use an expansion of the thermodynamic potential of the form (9). But for approximate estimates one may apply formula (13) also near the Curie point, proceeding in the manner set forth below.

The value

\[ a^2=\left(\frac{\partial \varepsilon}{\partial \eta^2}\right)^2_{\rho,T} \]

is expressed from condition (10) as follows:

\[ a^2 = \frac{(\Delta \varepsilon)^2}{\eta_1^4} = \frac{4n^2(\Delta n)^2}{\eta_1^4}. \tag{14} \]

The change in refractive index \(\Delta n\) is the difference between the values of \(n\) at the transition point, where \(\eta_0^2=0\), and at the point with the considered value \(\eta_1^2\).

The coefficient \(\beta_\theta\), entering into (13), from the expansion of the thermodynamic potential in powers of \(\eta\), can be represented, according to the general theory of L. D. Landau, through the difference of the heat capacities \(\Delta C_p\) of the two phases of the crystal,

\[ \frac{1}{\beta_\theta} = \frac{\eta_1^4\theta} {\Delta C_p(\theta-T)^2}. \tag{15} \]

Substituting (14) and (15) into (13), we find:

\[ \frac{I_\eta}{I_\rho} = \frac{4n^2(\Delta n)^2_{\rho,T}\cdot\theta^2} {\Delta C_p(\theta-T)^2\left(\rho\frac{\partial \varepsilon}{\partial \rho}\right)^2_T b_T T}. \tag{16} \]

Further, for the numerical calculation the quantity \((\Delta n)^2_{\rho,T}\) is identified with the value \((\Delta n)_p\sim 1.2\cdot 10^{-3}\) measured by K. N. Baranskii,\(^5\) and the difference \((\theta-T)\) with the temperature width of the fog band, \(\sim 0.1^\circ\mathrm{C}\). The jump in heat capacity is

\[ \Delta C_p=4.2\cdot 10^7\ \text{erg}/\text{deg}\cdot\text{cm}^3. \]

The values of the quantities \(\left(\rho\frac{\partial \varepsilon}{\partial \rho}\right)\) and \(b_T\) have already been discussed above, in connection with the quantitative estimate of the results of the calculation by formula (6′). Finally, \(I_\eta\sim 10^4 I_\rho\), which is in good agreement with experiment.\(^7\)

Thus, the new concept of light scattering by fluctuations of the characteristic parameter in second-order phase transitions, consistently developed by V. L. Ginzburg, has proved quite fruitful. Comparing expressions (6′) and (16), we see that they differ only in that in the denominator of (6′) there is the factor \(C_v\), while in (16) there is \(\Delta C_p\). Such a similarity of results obtained by different methods is not accidental.

by chance, since in both calculations the thermodynamic method was used, and the fluctuations of the characteristic parameter are related to temperature fluctuations by the relation

\[ \overline{(\Delta \eta^2)^2}=\frac{C_v(\Delta T^2)}{b_T v}. \]

Thus, the result of the direct phenomenological use of the dependence of \(n\) on \(T\) to explain the new phenomenon and the application of the general theory of second-order phase transitions are not in contradiction.

Since \(C_p \sim C_v \sim \Delta C_p\), the question of the accuracy of the agreement between the results of applying both formulas and the experimental data is, in essence, resolved in the same way. In the numerical interpretation of both formulas it is not possible to identify all the required parameters with the results of their direct experimental measurements. If special measurements of such quantities as \(\left(\frac{\partial \varepsilon}{\partial T}\right)_\rho\) or \(\left(\frac{\partial \varepsilon}{\partial \eta}\right)_{\rho,T}\) are at the limit of experimental technique, then the matter is simpler with finding \(C_p\) and \(b_T\) near the transition points.

The discovery of intense fluctuations near the \(\lambda\)-point, the geometric localization of the boundary region of the two phases of the crystal, and the successful calculation, by general methods, of the phenomena occurring in it—all this, taken together, advances the task of a more detailed investigation of phenomena near second-order phase transitions.

The role of fluctuations in second-order phase transitions is also repeatedly emphasized by V. K. Semenchenko. From this point of view he also considers the phenomenon of opalescence of quartz in his work on second-order phase transitions in anisotropic media \(^{15}\).

§ 3. ABSORPTION OF SOUND AND SECOND-ORDER PHASE TRANSITIONS

In 1947 Pellaam and Squire \(^{16}\), studying the propagation of first sound in liquid helium, discovered anomalous values of its velocity and absorption coefficient near the \(\lambda\)-point. It turned out that near the point of the phase transition (\(\theta=2.19^\circ\text{K},\ p=1\ \text{atm}\)) the velocity of sound in He passes through a shallow minimum, and the absorption coefficient through a very sharp maximum. Both phenomena were later investigated in greater detail and more carefully by Chase \(^{17}\), whose absorption data we present here (Fig. 8).

We see that over a temperature interval of \(\sim 0.03^\circ\) the absorption coefficient increases by more than a factor of one hundred. The experimentally found coincidence of the values \(\chi/\omega^2\) for two frequencies (2 and 12.1 MHz) makes it possible to conclude that the absorption coefficient of first sound in He, near the \(\lambda\)-point, is proportional to the square of the frequency. The author notes that, by changing the temperature of the experiment and smoothly passing through the \(\lambda\)-point, it was possible continuously to record the sound signal that had passed through the helium. Its amplitude did not go to zero at the \(\lambda\)-point, but passed through a clear minimum. The latter result should not be regarded as contradicting the form of the graph shown in Fig. 8. The break in the curve after the \(\lambda\)-point is connected with the inaccessibility, for precise measurements, of the temperature interval—\(0.002^\circ\) above the transition point. We shall not dwell here on this circumstance, which has no bearing on the acoustic properties of helium.

L. D. Landau and I. M. Khalatnikov \(^{18}\) considered in general form the question of sound absorption near points of a second-order phase transition. They came to the conclusion that anomalous sound absorption must occur near the \(\lambda\)-point.

For understanding the course of development of their considerations, we must return to the expansion of the thermodynamic potential in the parameter \(\eta\), cf. (8), and to the properties of the coefficients of this expansion, \(\alpha\) and \(\beta\).

In addition, since in this case the discussion will, of course, concern relaxation absorption of sound, it is necessary to consider the question of how, in time, the processes of establishment of thermodynamic equilibrium proceed in the system under consideration.

Fig. 8. Dependence of the coefficient of absorption of sound in helium on temperature.

Let a system described by a single parameter \(\eta\), being in an asymmetric state, be taken out of equilibrium. The rate of approach of the parameter \(\eta\) to the equilibrium value \(\eta_0\) is determined by the kinetic equation

\[ \frac{\partial \eta}{\partial t}=\gamma \frac{\partial \Phi}{\partial \eta}, \tag{17} \]

where \(\gamma\) is a kinetic coefficient, about which we shall assume that it has no special features near \(T=\theta\).

Expanding the derivative \(\frac{\partial \Phi}{\partial \eta}\) in a series in the difference \((\eta-\eta_0)\), with the aid of (17) we find

\[ \frac{d\eta}{dt}=4\gamma \beta \eta_0^2(\eta-\eta_0)= \]

\[ =-\frac{1}{\tau}(\eta-\eta_0). \tag{18} \]

It follows from this that the relaxation time \(\tau\), characterizing the establishment of equilibrium in the asymmetric phase, is equal to

\[ \tau=-\frac{1}{4\beta\gamma\eta_0^2}. \tag{19} \]

Near the \(\lambda\)-point we have:

\[ \tau=-\frac{1}{4\gamma\alpha'_0(\theta-T)}. \tag{20} \]

Consequently, the relaxation time near the transition point increases rapidly as the \(\lambda\)-point is approached. The establishment of equilibrium, therefore, near the \(\lambda\)-point takes place extremely slowly, and this must lead to appreciable relaxation damping of sound. It is quite obvious that such anomalous damping of sound will be observed only in the asymmetric phase, i.e. below \(\theta\) (at a given pressure). In the symmetric phase the parameter \(\eta\) is identically equal to zero in all states, both equilibrium and nonequilibrium, and consequently there will be no anomalies of the described type in the absorption of sound.

Having obtained the fundamentally important conclusion on the dependence of the relaxation time on temperature, L. D. Landau and I. M. Khalatnikov use

Next, the relaxation theory of sound absorption of L. I. Mandelstam and M. A. Leontovich \(^{19}\) is used. The expression found in this way for the coefficient of sound absorption in a liquid has the form

\[ \chi=\frac{\omega^{2}\tau}{1+\omega^{2}\tau^{2}}\,\frac{1}{2c_{\mathrm I}^{3}}\left(c_{\mathrm{II}}^{2}-c_{\mathrm I}^{2}\right). \tag{21} \]

Here \(\omega\) is the sound frequency. The quantities \(c_{\mathrm I}\) and \(c_{\mathrm{II}}\) are the equilibrium speeds of sound in the high-temperature and low-temperature phases, respectively\(^*\).

The relaxation time \(\tau\) near the \(\lambda\)-point is determined by formula (20). It increases rapidly as \(T\) approaches the \(\lambda\)-point. According to (21), as the \(\lambda\)-point is approached (at a given sound frequency \(\omega\)), the absorption coefficient also increases. In the immediate vicinity of the \(\lambda\)-point \((\omega\tau\sim 1)\), the quantity \(\chi\) reaches a maximum and then begins to decrease. Such is the general picture of the phenomenon of anomalous sound absorption near the \(\lambda\)-point in the low-temperature phase.

The magnitude of the jump in the speed of sound at the \(\lambda\)-point, \(c_{\mathrm{II}}-c_{\mathrm I}\), entering the formula for \(\chi\), can be expressed in terms of the usually well-known quantity of the jump in heat capacity at the \(\lambda\)-point \(^{13}\).

A comparison of the developed theory with experimental results was carried out with the aid of Chase’s data \(^{17}\), relating to helium. By comparing the experimental values of the absorption coefficient at various temperatures (the frequencies used in \(^{17}\) were 2 and 12.1 megacycles per second), the authors find the value of the relaxation time. The experimentally obtained value of the relaxation time follows very well the temperature dependence given by formula (23):

\[ \tau=\frac{4\cdot 10^{-13}}{\theta-T}\ \text{sec.} \tag{22} \]

Fig. 9. Dependence of the relaxation time in helium on temperature.

In Fig. 9 the graph of the dependence of \(\tau\) on \((\theta-T)\), constructed according to formula (22), is shown. The values of \(\tau\) calculated by formula (21) from Chase’s data for \(\chi\) are plotted on this graph by crosses.

§ 4. SOUND ABSORPTION IN A SECOND-ORDER PHASE TRANSITION IN A FERROELECTRIC

The thermodynamic generality of the calculation of L. D. Landau and I. M. Khalatnikov \(^{18}\), and the successful application of their results to the case of helium, advance the problem of finding anomalous sound absorption near \(\lambda\)-points in solids.

On the basis of the above considerations, I. A. Yakovlev, T. S. Velichkina, and K. N. Baranskii \(^{20}\) undertook an attempt to find an analogous effect in the second-order phase transition in a crystal of Rochelle salt,

\(^*\) A formula analogous to (21) was used by P. E. Stepanov \(^{24}\) in analyzing data on the absorption of elastic vibrations in \(\beta\)-brass near the Curie point. The author, however, did not take into account the substantial proximity of the phase-transition point in the temperature dependence of the relaxation time.

the general properties of which are relatively well known. A consistent treatment of its upper Curie point \((+24^\circ C)\) as a second-order phase transition has been given in the works of V. L. Ginzburg\(^{22}\), outlined, in particular, in this journal. Therefore we do not discuss here the question of the nature of this phase transformation in Rochelle salt. We note only that its ferroelectric properties and the mechanism of its polarization under the propagation of sound in it require, of course, special theoretical consideration. Therefore the theory of L. D. Landau and I. M. Khalatnikov, in the form set forth above, determined only the general direction of the experimental investigation.

Following the order of exposition adopted above, we shall first present the experimental facts relating to this question.

As indicated in \(^{20}\), ultrasonic pulses with a frequency of 5 MHz, a duration of 1.5 μsec, and a duty factor of 0.002 sec were applied from transmitting piezoquartz to a single crystal of Rochelle salt. From receiving quartz located on the opposite side of the crystal, the signals passed through an amplifier to the input of a pulse oscillograph used to measure the amplitude of the signals. The Rochelle-salt crystal and the piezoquartz plates were placed in a thermostat. The investigations were carried out near the upper Curie point.

In a phase transformation there may be a change in the acoustic resistance of the crystal and, consequently, in the boundary conditions for double passage of sound through the quartz—Rochelle-salt boundary. Therefore, in order to exclude boundary conditions, the absorption coefficient was determined from simultaneous measurements of the attenuation of sound in two crystals of different thickness.

a)

b)

c)

Fig. 10. Oscillograms of ultrasonic pulses passed through a Rochelle-salt crystal:
a) \(T>\theta\); b) \(T\sim\theta\); c) \(T<\theta\).

The experiments revealed a close connection between the conditions of sound absorption and the ferroelectric and piezoelectric properties of the crystal. It turned out that the absorption of longitudinal acoustic waves propagating along the \(z\) axis of the crystal has no essential peculiarities at the Curie point. The same applies to transverse waves propagating along the \(z\) axis and polarized along the \(x\) axis. But matters are quite different with transverse waves propagating along the \(z\) axis and polarized along the \(y\) axis. On passing through the crystal, the amplitude of the pulses of such waves decreases sharply as the temperature of the Rochelle salt approaches the Curie point. The qualitative aspect of this phenomenon can be judged from the oscillograms of the signals at the receiving piezoquartz shown in Fig. 10. The conditions for obtaining all three oscillograms were identical in every respect except for the values of the crystal temperature indicated in the caption to the figure.

The quantitative side of the phenomenon is presented in Fig. 11. Here the temperature dependence of the amplitude absorption coefficient \(\chi\ \text{cm}^{-1}\) of transverse waves of the polarization indicated above is given. The graph of the dependence \(\chi(T)\) shows that, near the Curie point, the values of \(\chi\) exceed by \(\sim 12\) times the value \(\chi_{35^\circ\mathrm{C}} \simeq 0.5\ \text{cm}^{-1}\).

Thus it was established that in Rochelle salt, near its upper Curie point, the phenomenon of anomalous sound absorption occurs.

The connection of this phenomenon with the electrical properties of the crystal becomes clear if we take into account the well-known correspondence between the electrical polarization of Rochelle salt and the deformations acting in it\(^{23}\). Specifically, a shear deformation along the \(y\)-axis produces an electrical polarization of the crystal along its ferroelectric axis \(x\). Precisely this type of deformation is produced in the crystal by the transverse wave for which the new phenomenon was found. In order to clarify the role of the domain structure formed in Rochelle salt below the Curie point, experiments were carried out with single-domain crystals.

Fig. 11. Dependence of the sound absorption coefficient in Rochelle salt on temperature.

Obtaining such specimens is possible, as was shown by M. A. Chernysheva\(^{25}\), by applying to the crystal an electrostatic field of sufficient strength along the ferroelectric axis of the crystal. The general scheme of the experiment remained the same in this series of experiments. The experiments showed that, at temperatures above the Curie point, sound absorption does not change under the action of the field. At temperatures \(10\text{–}12^\circ\) below the Curie point, the amplitude of the sound increases under the action of the field by \(10\text{–}15\%\)

Fig. 12. Oscillograms of ultrasonic pulses that passed through a Rochelle-salt specimen:
a) unpolarized crystal, \(T \sim \theta\); b) polarized crystal, \(T \sim \theta\).

in comparison with its value at the same temperature but outside the field. The field has its maximum influence on sound absorption at temperatures somewhat lower than the Curie temperature. The qualitative picture

this phenomenon can be seen in Fig. 12. In this figure two oscillograms are presented of sound signals that have passed through a crystal of Rochelle salt. The temperature of the experiment in both cases is the same and lies below the Curie point by \(\sim 0.2^\circ\). In the oscillogram taken without a field, the acoustic signal is barely noticeable. The next oscillogram corresponds to an experiment with a crystal polarized by a constant voltage of \(600\ \mathrm{V/cm}\).

The photographs shown clearly demonstrate the strong influence of an electrostatic field on the absorption of sound in Rochelle salt near its upper Curie point. Measurements show that, under the action of the field, the amplitude of the sound signal increases by tens of times. In Fig. 13 we give typical graphs of the temperature dependence of the amplitude of a signal that has passed through Rochelle salt. The solid curve

Fig. 13. Dependence of the amplitude of the sound signal that has passed through a Rochelle-salt crystal on temperature. (The dashed graph corresponds to the polarized crystal.)

corresponds to the unpolarized crystal; the dashed curve, to the polarized crystal. The high-temperature branches of both curves are identical.

Thus, in the case of a second-order phase transition in a ferroelectric, as in helium, strong attenuation of sound is observed near the \(\lambda\)-point.

Now, after presenting the experimental results concerning Rochelle salt, we can proceed to the theoretical consideration of the problem due to L. D. Landau. The solution of the new problem was carried out by L. D. Landau in accordance with the general considerations on the character of relaxation processes in phase transitions of the second kind that were set forth above. In the specific application of these considerations to the case of Rochelle salt, it is necessary to take into account the following circumstances: the anisotropy of the medium, the presence of the piezoelectric effect in the propagation of an acoustic wave, the dependence of the dielectric permittivity \(\varepsilon\) of the crystal on temperature, and the possible polarization of the crystal by an external electrostatic field.

Therefore the thermodynamic potential of the crystal \(\Phi\) must be written as follows:

\[ \Phi=\Phi_{0}+\frac{\alpha D_{x}^{2}}{8\pi}+\frac{\beta D_{x}^{2}}{16\pi}-\lambda D_{x}Y_{z}-\frac{\mu}{2}Y_{z}^{2}-\frac{1}{4\pi}D_{x}E_{x}^{\mathrm{external}}. \tag{1} \]

Here \(D_x\) is the component of the induction vector along the crystallographic axis \(x\); \(E_x\) is the corresponding component of the vector of the external field strength; \(Y\) is the shear stress along the axis \(y\); \(\mu = \dfrac{1}{c^D_{44}} = S^D_{44}\) is the shear modulus at constant induction \(D\); \(\lambda = \dfrac{d_{14}}{\varepsilon_x}\) is the piezoconstant of Rochelle salt; \(\alpha\) and \(\beta\) are coefficients, dependent on temperature and pressure, in the expansion of the thermodynamic potential in powers of \(D_x\). The anisotropy of the crystal is taken into account in the expression for the thermodynamic potential in the form required for considering a shear wave propagating along the \(z\) axis. The indices on the elastic and piezoelectric constants of the crystal have the meaning generally accepted in crystal physics; below we omit them.

The changes of \(D\) in time are related to \(\Phi\) by the kinetic equation

\[ \frac{\partial D}{\partial t}=\gamma\frac{\partial \Phi}{\partial D}. \tag{II} \]

Here \(\gamma\) is a kinetic coefficient, about which we shall assume that it has no special features near the Curie point.

The right-hand side of the equation can be represented by the series

\[ \frac{\partial \Phi}{\partial D} = \left(\frac{\partial \Phi}{\partial D}\right)_{D=D_0} + \left(\frac{\partial^2 \Phi}{\partial D^2}\right)_{D=D_0} (D-D_0)+\cdots . \tag{23} \]

Let us consider the case of temperature \(T>\theta\), where \(\theta\) is the Curie temperature:

\[ \frac{\partial \Phi}{\partial D} = \frac{\alpha D}{4\pi} - \lambda Y, \tag{24} \]

where we have neglected the term \(\dfrac{\beta D^3}{4\pi}\).

Equating this expression to zero, we find the equilibrium value \(D_0\) corresponding to the stress \(Y\),

\[ D_0^2=\frac{4\pi\lambda Y}{\alpha}. \tag{25} \]

Calculating \(\left(\dfrac{\partial^2\Phi}{\partial D^2}\right)_{D=D_0}\) and using (25), we find that equation (II) can be written as

\[ \frac{\partial D}{\partial t} = \gamma\left( \frac{\alpha}{4\pi}D-\lambda Y \right). \tag{26} \]

In the case \(T<\theta\), the spontaneous polarization of the ferroelectric must be taken into account. Therefore,

\[ D=D_{\text{spont}}+D_{\text{sound}}, \]

where \(D_{\text{spont}}\gg D_{\text{sound}}\).

Now the equilibrium value is \(D^2_{0\,\text{spont}}=-\dfrac{\alpha}{\beta}\), and correspondingly

\[ D_0=-\frac{4\pi\lambda Y}{2\alpha}. \]

Therefore equation (26) will now be written as follows:

\[ \frac{\partial D}{\partial t} = \gamma\left(-\frac{2a}{4\pi}\dot D-\lambda Y\right). \tag{27} \]

We must establish the relation between the induction \(D\) and the elastic stress \(Y\) in a plane sound wave. We can do this by putting \(D\sim e^{i(\omega t-kx)}\). Then expressions (26) and (27) are written, respectively, as

\[ D=\frac{4\pi\lambda Y}{a-\dfrac{i4\pi\omega}{\gamma}} \quad \text{for } \quad T>\theta, \]

\[ D=\frac{4\pi\lambda Y}{-2a-\dfrac{i4\pi\omega}{\gamma}} \quad \text{for } \quad T<\theta. \]

Introducing, for uniformity of the last two expressions, the notation \(a=\varphi\) for \(T>\theta\) and \(2a=-\varphi\) for \(T<\theta\), we find:

\[ D=\frac{4\pi\lambda Y}{\varphi-\dfrac{i4\pi\omega}{\gamma}}. \tag{28} \]

Turning now to the relation between the deformation and the elastic stress

\[ y=-\frac{\partial \Phi}{\partial Y} \tag{III} \]

and using (I), we find:

\[ y=\mu Y+\lambda D \tag{29} \]

and

\[ Y=\frac{y-\lambda D}{\mu}. \tag{30} \]

Substituting (28) into (30) and using (29), we have:

\[ D= \frac{\dfrac{4\pi\lambda}{\mu}\,y} {\varphi+\dfrac{4\pi\lambda^2}{\mu}-\dfrac{i4\pi\omega}{\gamma}} \tag{31} \]

and

\[ y=\left(\mu+\frac{4\pi\lambda^2}{\varphi-\dfrac{i4\pi\omega}{\gamma}}\right)Y. \tag{32} \]

Thus, using expressions (I—III) and the general properties of the thermodynamic potential near the \(\lambda\)-point, we have found the relations we need, (28), (31), (32), between the induction, the deformation, and the elastic stress in an acoustic wave propagating in Rochelle salt, located in a state close to the Curie point. The relations obtained show that in this case the acoustic wave will undergo relaxation absorption. Indeed, the complex denominators of all three expressions have the form typical for describing relaxing process-

of kinds. For example, expression (32), relating the deformation to the elastic stress, directly indicates that the work done in each periodic cycle in the acoustic wave will be nonzero. The cause of the dissipation of energy will be its losses to the relaxing polarization of the crystal (28). Further, from the expressions under discussion it follows that the relaxation time in the establishment of thermodynamic equilibrium in a ferroelectric is determined near the Curie point by the expression

\[ \tau=\frac{4\pi}{\varphi\gamma}. \tag{33} \]

We shall return to these questions, but first let us finish our calculation by finding the coefficient of sound absorption. The wave vector of the acoustic wave can be found from the wave equation

\[ \rho \ddot{y}=\frac{\partial Y}{\partial z}, \tag{IV} \]

the relation (32), and the conditions \(y\sim e^{i(\omega t-kx)}\) and \(Y\sim e^{i(\omega t-kx)}\). Then

\[ k^2=\rho\omega^2\left(\mu+\frac{4\pi\lambda^2}{\varphi-\dfrac{i4\pi\omega}{\gamma}}\right), \tag{34} \]

and the amplitude absorption coefficient \(\chi\) will be expressed by the relation

\[ \chi^2=\frac{\omega^2\rho}{2} \left( \mu\sqrt{ 1+\frac{8\pi\lambda^2\left(\mu\varphi+2\pi\lambda^2\right)} {\varphi^2+\dfrac{16\pi^2\omega^2}{\gamma^2}} } -\mu -\frac{4\pi\lambda^2\varphi} {\varphi^2+\dfrac{16\pi^2\omega^2}{\gamma^2}} \right). \tag{35} \]

Using subsequently the numerical values of the parameters entering into (35), it is easy to show that the second term in the radical expression is much less than unity. Therefore we finally obtain:

\[ \chi^2=\frac{\omega^2\rho}{2}\, \frac{8\pi^2\lambda^4} {\mu\left(\varphi^2+\dfrac{16\pi^2\omega^2}{\gamma^2}\right)}. \tag{36} \]

It now remains only to determine the values of \(\varphi\) and \(\gamma\). As for \(\gamma\), this quantity, which entered the calculation from the kinetic equation (II), cannot be calculated within the framework of the developed quasithermodynamic theory. The quantity \(\gamma\) can be found only by using for this purpose at least one experimental value of \(\chi\), or, in other words, by tying the theoretical function at one point to the experimental results. This is what we shall do below.

The situation is quite different with the values \(\varphi=\alpha\) for \(T>\theta\) and \(\varphi=-2\alpha\) for \(T<\theta\). As we know from the general propositions of the theory of second-order phase transitions set out above\(^{13}\), the coefficient \(\alpha\to 0\) as \(T\to\theta\). This circumstance immediately indicates to us the growth of the sound absorption coefficient \(\chi\) and of the relaxation time \(\tau\) as \(T\to\theta\), in full agreement with the general proposition formulated in \(^{18}\). But it is essential that the quantity \(\alpha\) has a clear physical meaning and tends to zero according to a quite definite and known law. Namely, if one refers to the work of V. L. Ginzburg\(^{22}\) cited above, in which the Curie point of a ferroelectric is considered as a second-order phase transition, then it is easy to prove that

\[ \alpha=\frac{1}{\varepsilon}\ \text{for } T>\theta \quad \text{and} \quad \alpha=-\frac{1}{2\varepsilon}\ \text{for } T<\theta . \]

But, as is well known, \(\varepsilon\) near the Curie point obeys the Curie–Weiss law:

\[ \varepsilon=\frac{4\pi C}{T-\theta}\quad \text{for } T>\theta \quad \text{and} \quad \varepsilon=-\frac{2\pi C}{T-\theta}\quad \text{for } T<\theta, \]

where \(C=180\) is the Curie constant. Consequently, we can now write:

\[ \varphi=\frac{T-\theta}{4\pi C};\qquad \tau=\frac{16\pi^{2}C}{(T-\theta)\gamma} \quad \text{for } T>\theta, \]

\[ \varphi=\frac{\theta-T}{2\pi C};\qquad \tau=\frac{8\pi^{2}C}{(\theta-T)\gamma} \quad \text{for } T<\theta. \]

Thus, we see that near the Curie point both \(\chi\) and \(\tau\) increase.

Using the experimental data for \(\chi\), we find the values of \(\gamma\) and \(\tau\):

\[ \tau=\frac{5.2\cdot 10^{-8}}{T-\theta} \quad \text{for } T>\theta;\qquad \tau=\frac{2.6\cdot 10^{-8}}{\theta-T} \quad \text{for } T<\theta. \]

Figure 14 gives the computed theoretical dependence of \(\chi\) on \(T-\theta\).

Fig. 14. Temperature dependence of the coefficient of sound absorption in Rochelle salt according to theoretical and experimental data.

On the same figure, the experimentally found values of \(\chi\) are plotted as points.

We see that the high-temperature branch of the theoretical curve is in excellent agreement with the experimental data. The low-temperature branch of the theoretical graph for \(\chi\) passes substantially below the experimental values of the absorption coefficient. However, this circumstance is easily explained. In fact, L. D. Landau’s theory is developed for a single-domain crystal, whereas below the Curie point a ferroelectric, as a rule, breaks up as a result of spontaneous polarization into separate domains. At the boundaries of these domains sound scattering must occur. In addition, below the Curie point hysteresis phenomena occur in a ferroelectric. This is where the causes of the excess of the experimental values of \(\chi\) over the theoretical ones should be seen. The experiments described above on the investigation of sound propagation in a polarized crystal confirm this point of view. Indeed, as is seen from Fig. 13, the values of the amplitudes of the sound signal increase substantially under the action of the field at temperatures \(T<\theta\). But the application of an electric field to the crystal near the Curie point leads precisely to the formation of a single-domain crystal, as was convincingly shown by M. A. Chernyshëva \(^{25}\). Therefore,

the symmetry of both branches of the amplitude curve for a monodomain crystal confirms the validity of the theory both above and below the \(\lambda\)-transition.

The good agreement between the theoretical and experimental results also means that the quantity \(\gamma\) is constant, and this, in turn, makes reliable the definition given above of the relaxation time in the establishment of the state of thermodynamic equilibrium in the system of a Rochelle-salt crystal.

Thus, L. D. Landau’s theory has made it possible to elucidate the relaxation nature of anomalous sound absorption in a ferroelectric near the Curie point and in helium near the \(\lambda\)-point. At the same time, in agreement with experiment, the theory shows that anomalous sound absorption in helium should occur only in the low-temperature phase, whereas in Rochelle salt it occurs on both sides of the Curie point.

The general result of the study of sound absorption in these two cases is proof of the temperature dependence of the relaxation time in a system that is in a state close to a second-order phase transformation.

In conclusion to the acoustic part of our review, we should repeat the considerations expressed above concerning the advisability of developing further studies in the immediate vicinity of second-order phase transitions.

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