CURRENT STATE OF THE THEORY OF DIELECTRIC POLARIZATION

Ya. I. Frenkel and A. I. Gubanov, Leningrad

INTRODUCTION

The present article gives a critical review of various theories of dielectric polarization. The discussion concerns, chiefly, the polarization of dipolar dielectrics, since precisely here there has still not been complete clarity, and in recent years a number of new theories have appeared, differing in their various ways of taking into account the interaction between molecular dipoles.

At the beginning of the first section the classical Clausius–Mossotti and Lorentz theory for nonpolar dielectrics is set forth, as well as a critique of Debye’s attempt to extend this theory to dipolar liquids. Next, the theory of Onsager is presented; he showed that the Lorentz formula for the effective field acting on an individual molecule is inapplicable to dipolar dielectrics.

In the second section Debye’s new theory is presented, concerning the polarization of dipolar liquids in a constant field and taking into account the interaction of dipoles by means of the concept of a local field. A simplified form of this theory is also given, along with its combination with Onsager’s theory.

In the third section Kirkwood’s theory is presented, which is a development and refinement of the theories of Onsager and Debye.

In the fourth section1 Onsager’s theory is generalized to the case of elastic polarization of a liquid dielectric with anisotropic molecules; next, attempts to extend the theory to crystals are considered (in particular, crystals of the Rochelle-salt type).

The last (fifth) section is devoted to the polarization of dielectrics in an alternating field. Here the new theory of Debye and Ramm is set forth, a critique of it is given, and, in conclusion, the formal theory of dielectric losses in an alternating field is presented.

I. ONSAGER’S THEORY

§ 1. On the effective field in a dielectric

Every dielectric, when placed in an electric field, is polarized in it, i.e., acquires an electric dipole moment. This polarization may occur both through the orientation

already present but chaotically oriented permanent dipoles (in so-called polar dielectrics), and also through the formation of elastic dipoles by displacement of charges of opposite sign within an individual molecule (elastic polarization). To explain elastic polarization, Clausius and Mossotti treated the dielectric as a completely nonconducting medium filled with conducting spheres. In an electric field, charges of opposite sign are induced on opposite sides of these spheres, the centers of which are displaced relative to one another, which determines the polarization of the dielectric.¹˒²

If by \(g\) we denote the ratio of the volume of all conducting spheres to the total volume of the dielectric, then the electric susceptibility of the latter turns out to be equal to

\[ \chi=\frac{3g}{4\pi(1-g)}. \tag{1} \]

The dielectric constant \(\varepsilon\) is related to the electric susceptibility by the well-known formula:

\[ \varepsilon = 1 + 4\pi\chi. \tag{2} \]

Substituting (1) into (2), we obtain

\[ \varepsilon=\frac{1+2g}{1-g}; \qquad g=\frac{\varepsilon-1}{\varepsilon+2}. \tag{3} \]

But \(g=\frac{4\pi}{3}a^{3}N\), where \(a\) is the radius of a conducting sphere, and \(N\) is the number of spheres per unit volume of the dielectric. If we assume that the number of dielectric molecules is equal to the number of conducting spheres, i.e., that the dielectric molecules are these conducting spheres, and if we take into account that the polarizability of an individual molecule \(\alpha\) is, under these conditions, equal to \(a^{3}\), then from (3) we obtain the well-known Clausius–Mossotti formula

\[ \frac{\varepsilon-1}{\varepsilon+2}=\frac{4\pi}{3}N\alpha. \tag{4} \]

The representation of Clausius and Mossotti of a dielectric as a system of conducting spheres is not without meaning, since according to more modern theories the polarizability of a molecule is, in order of magnitude, equal to the cube of its radius. This corresponds to an approximately vanishing mean field inside the molecule, just as in the conducting spheres of the Clausius–Mossotti theory.

Formula (4) was later obtained from entirely different considerations, namely by introducing the so-called “effective field” acting on a particle of the dielectric.

In a dielectric one should distinguish two fields: the mean macroscopic field \(E\) and the effective field \(E_e\). The mean field acts in the dielectric on a certain test free charge; it should not be confused with the external field \(E_0\), into which the dielectric under consideration is introduced. The mean field \(E\) is equal to the sum of the field \(E_0\) and the field produced by all dipoles of the dielectric, including

and a dipole located in that volume element for which we determine \(E\) (i.e., \(E_0 = E + 4\pi P\)). The effective field \(E_e\) acting on the molecule of the dielectric under consideration is composed of the field \(E_0\) and the field of all the other polarized molecules.

To calculate this field, Lorentz used the following device: around the particle under consideration a sphere of some radius \(a\) is described, small in comparison with the dimensions of the dielectric, but at the same time so large in comparison with the distances between the particles that the dielectric outside this sphere may be regarded as a uniformly polarized continuum (if the macroscopic mean field is homogeneous). The field created by such a polarized body is equivalent to the field of surface charges that have appeared on its boundaries, i.e., on the outer boundary of the dielectric and on the surface of the spherical cavity\(^1\).

In the case of a plane capacitor, the field of the bound charges appearing on the outer surfaces is, as is well known, equal to \(-4\pi P\) (where \(P\) is the polarization, i.e., the dipole moment per unit volume of the dielectric); together with \(E_0\) it gives \(E\). In the case of a spherical cavity, a charge \(\sigma = P \cos \theta\) appears on the surface, where \(\theta\) is the angle between the positive direction of the field and the outward normal to the sphere. At the center of the sphere it creates a field (equal to the sum of the projections onto the direction of the external field)

\[ E'=-\oiint \frac{P\cos\theta\, dS}{a^2}\cos\theta =-\int_0^\pi \frac{P\cos^2\theta}{a^2}\cdot 2\pi a^2\sin\theta\, d\theta =-\frac{4\pi P}{3}. \tag{5} \]

It is not difficult to show that this field, called the Lorentz field, has the same magnitude and direction throughout the entire volume of the spherical cavity. In the case of isotropic bodies, the field acting on the central molecule from all the other molecules located inside the sphere is equal to zero; thus the effective field acting on a molecule in the dielectric is equal to

\[ E_e = E+\frac{4\pi}{3}P. \tag{6} \]

Born expressed the opinion\(^3\) that this derivation of the effective field is valid only for a variable electric field, provided that the wavelength is small in comparison with the linear dimensions of the dielectric, as is the case, for example, with light waves. Otherwise, the field from the charges on the inner surface of the spherical cavity is more or less compensated by the field of the charges that have appeared on the outer surface of the body. In the case of short waves, however, no such compensation occurs, since charges distributed at different points of the outer surface oscillate in different phases not coordinated with one another.

\(^1\) If one assumes that bound volume charges are absent, i.e.,

\[ \operatorname{div} \mathbf{P}=0. \]

These considerations of Born are based on a misunderstanding consisting in the confusion of the external field \(E\) with the mean field inside the dielectric \(E\). The circumstance that in some cases the Lorentz correction turns out to be valid only for light waves is explained quite differently, as we shall show below.

One of the authors \(^{4}\) proposed another derivation of the Lorentz correction, a more logical one, not requiring the introduction of Lorentz’s fictitious sphere and leading to the same result as Lorentz’s derivation. This derivation is free from Born’s objections, which, incidentally, is their indirect refutation. The idea of the derivation is as follows: in order to obtain the effective field acting on a given molecule, it is necessary to subtract from the mean field inside the dielectric \(E\) the mean value of the field created by the molecule under consideration itself, since, obviously, the molecule cannot act on itself, whereas its field is included in the composition of the field \(E\) acting on an externally introduced “test” charge.

Let us imagine this molecule in the form of a sphere \(K\) of radius \(a\), inside which the bound electrons are arranged in a completely arbitrary manner. This arrangement may be replaced by an equivalent distribution of charges on the surface \(K\), leading to the same value of the electric moment. Let us calculate the mean value of the electric intensity inside a sphere \(S\) concentric with \(K\), whose radius \(b>a\) is determined from the condition that the volume

\[ S=\frac{4\pi}{3}b^{3} \]

coincide with the volume belonging on the average to one molecule of the given body. If, therefore, the number of molecules per unit volume is equal to \(N\), then \(b\) is determined from the equality

\[ \frac{4\pi}{3}b^{3}=\frac{1}{N}. \tag{7} \]

The potential of the molecule outside \(K\) is equal to \(\dfrac{\mathbf{p}\mathbf{R}}{R^{3}}\), and inside \(\varphi^{i}=\dfrac{\mathbf{p}\mathbf{R}}{a^{3}}\); the corresponding field intensities are expressed by the formulas

\[ \mathbf{E}^{a}=\frac{1}{R^{3}}\{3\mathbf{R}_{0}(\mathbf{R}_{0}\cdot\mathbf{p})-\mathbf{p}\};\quad \mathbf{E}^{i}=-\frac{\mathbf{p}}{a^{3}}. \]

The mean value of the field intensity in the volume is

\[ \overline{\mathbf{E}}= \frac{1}{\dfrac{4\pi}{3}b^{3}}\int_{0}^{b}\mathbf{E}\,dv = \frac{1}{\dfrac{4\pi}{3}b^{3}}\int_{0}^{a}\mathbf{E}^{i}\,dv, \]

since the mean value of \(\mathbf{E}^{a}\) is equal to zero for different directions \(\mathbf{R}_{0}\) at constant \(\mathbf{R}\). We obtain, therefore,

\[ \overline{\mathbf{E}}= -\frac{1}{\dfrac{4\pi}{3}b^{3}}\frac{\mathbf{p}}{a^{3}}\frac{4\pi}{3}a^{3} \]

and according to (7)

\[ \mathbf{E}=-\frac{4\pi}{3}N\mathbf{p}=-\frac{4\pi}{3}\mathbf{P}, \tag{8} \]

i.e. the same result as by Lorentz’s method.

The intrinsic field of a molecule can also be calculated by the following, simpler and more general method\(^5\).

Suppose that, as a result of polarization of the dielectric, one of its charges \(e\), belonging to the molecule under consideration, has been displaced by a distance \(l\) relative to its initial position. Around this initial position let us draw a sphere of radius \(b\), and then through the displaced position of the charge let us draw a plane perpendicular to its displacement. This plane cuts the sphere into two unequal parts, the smaller of which we shall reflect in it, singling out from the larger part an unshaded lune (Fig. 1). The mean value of the field produced by the charge \(e\) in the shaded region is obviously equal to zero. To determine the mean value of this field in the volume of the sphere, it is therefore sufficient to consider only the volume of the lune.

Fig. 1

For small \(l\) in comparison with \(b\), the integral of \(\mathbf{E}\) over the volume of the lune may be calculated as follows. From the center of the sphere draw a radius vector at an angle \(\theta\) to the direction opposite to the displacement \(l\). The segment of this radius contained within the lune, i.e. the thickness of the latter, is evidently equal to \(2l\cos\theta\). Multiplying this thickness by the projection of the field produced by the charge in the direction \(l\),

\[ -\frac{e}{r^{2}}\cos\theta \simeq -\frac{e}{b^{2}}\cos\theta \]

and integrating with respect to \(\theta\) from \(0\) to \(\frac{\pi}{2}\) (i.e. over the surface of a hemisphere), we obtain

\[ \overline{\mathbf{E}} = -\frac{1}{\frac{4\pi}{3}b^{3}} \cdot \frac{2el}{b^{2}} \int_{0}^{\frac{\pi}{2}} \cos^{2}\theta \cdot 2\pi b^{2}\sin\theta\, d\theta = -\frac{el}{b^{3}} \]

or

\[ \overline{\mathbf{E}}=-\frac{4\pi}{3}\frac{el}{v}, \tag{9} \]

where \(v=\frac{4\pi}{3}b^{3}\).

The total mean field of all charges belonging to the molecule under consideration is therefore equal to

\[ \overline{\mathbf{E}} = -\frac{4\pi}{3}\sum \frac{el}{v} \]

or, since

\[ \sum \frac{el}{v}=P, \]

\[ \overline{\mathbf{E}}=-\frac{4\pi}{3}P, \]

which agrees with the result obtained above.

The introduction of the effective field (6) leads to the Clausius—Mosotti equation. According to the definition of the dielectric constant, we have

\[ \varepsilon-1=\frac{4\pi P}{E}. \tag{10} \]

The elastic moment created in a molecule of a dielectric is proportional not to the mean field \(E\), but to the effective field \(E_e\). Thus, if \(\alpha\) denotes the polarizability of a molecule, then \(P=N\alpha E_e\), or, by (6):

\[ P=N\alpha \left(1+\frac{4\pi P}{3E}\right)E, \]

whence

\[ P=\frac{N\alpha E}{1-\frac{4\pi N\alpha}{3}} . \]

Substituting this expression into (10), we obtain the Clausius—Mossotti equation

\[ \frac{\varepsilon-1}{\varepsilon+2}=\frac{4\pi N\alpha}{3}. \]

Debye extended this equation to the case of polar dielectrics, assuming that the same effective field which causes the elastic polarization of the molecules of the dielectric acts orientingly on the permanent dipoles. The theory of Onsager presented below is devoted to a critique of this incorrect assumption of Debye and of the consequences following from it. In addition to elastic polarization, an orientational term appears in Debye’s formula. For weak fields, when

\[ \frac{\mu_0 E_e}{kT}\ll 1 \]

(here \(\mu_0\) is the permanent dipole moment of the molecule, \(k\) is Boltzmann’s constant), Debye’s formula in the first approximation gives

\[ \frac{\varepsilon-1}{\varepsilon+2}=\frac{4\pi N}{3}\left(\alpha+\frac{\mu_0^2}{3kT}\right). \tag{11} \]

In the general case the mean dipole moment, depending on orientational polarization, is, as is known, expressed by the Langevin function

\[ L(x)=\operatorname{cth}x-\frac{1}{x}. \tag{12} \]

Debye’s equation takes the form:

\[ \frac{\varepsilon-1}{\varepsilon+2}=\frac{4\pi N}{3}\left[\alpha+L\left(\frac{\mu_0 E_e}{kT}\right)\frac{\mu_0}{E_e}\right]. \tag{13} \]

Restricting ourselves to only two terms in the expansion of the function \(L\) in a series, we obtain

\[ \frac{\varepsilon-1}{\varepsilon+2}=\frac{4\pi N}{3}\left[\alpha+\frac{\mu_0^2}{3kT}-\frac{\mu_0^4 E_e^2}{45(kT)^3}\right]. \tag{14} \]

The correction term on the right-hand side characterizes the so-called “saturation effect,” expressed in a decrease of the dielectric constant in strong fields.

§ 2. The Question of Infinite Polarizability and Residual Polarization

The equations derived indicate the possibility of an arbitrarily large dielectric constant for certain substances. Thus, for example, when \(\frac{4\pi}{3}N\alpha = 1\), the left-hand side of equation (4) must also become unity, which is possible only for \(\varepsilon \to \infty\). In the case of elastic polarization this means an infinitely large increase in the dipole moments of the molecules, i.e., an infinite removal of electrons from the nuclei. Ultimately the electrons must lose their connection with definite positive ions, i.e., they must as it were “collectivize.” In Herzfeld’s opinion this condition corresponds to the transformation of a dielectric into a conductor (metal).

In the case \(\frac{4\pi}{3}N\alpha > 1\), it no longer has physical meaning to consider the polarization proportional to the field, so that equation (4) must be replaced by another one, taking into account the nonlinear dependence of \(P\) on \(E_e\).

The right-hand side of formula (11), at a sufficiently low temperature \(T\), can assume arbitrarily large values. It follows from this that the dielectric constant \(\varepsilon\) of any polar dielectric at an absolute temperature of the order of hundreds of degrees ought to tend to infinity. True, if saturation (nonlinearity) is taken into account and one considers not equation (11) but equation (13), then \(\varepsilon\) cannot tend to infinity; nevertheless, in the temperature region below a certain critical temperature the dielectric constant ought to take anomalously large values at small \(E\). This phenomenon of “high initial electric susceptibility” is connected with another phenomenon—the preservation of “residual electric polarization” after the electric field has been removed.

Both phenomena characterize the behavior of “ferroelectrics”—substances that are electrical analogues of ferromagnetic bodies.

The temperature \(T_1\), below which \(\varepsilon\) assumes anomalously large values in ferroelectrics (in this same region residual polarization must also be observed), is called the Curie temperature, by analogy with the Curie point for ferromagnets.

Let us note that high initial electric susceptibility (in small fields) and the preservation of residual polarization, although connected with one another, nevertheless are essentially different phenomena. This becomes immediately clear if one uses the analogy with ferromagnets. Indeed, there exist grades of soft iron with very large magnetic susceptibility in small fields and at the same time with small residual magnetization; on the other hand, there exist steels with comparatively small initial permeability but with large residual magnetization. However, attempts were made to prove that the introduction of the Lorentz correction into the expression for the effective field provides the possibility of the appearance of residual electric polarization. In ...

this was reasoned as follows. The polarization of a body, caused by the orientation of molecules in the direction of the applied field, is equal to

$$ P = P_0 L\left(\frac{\mu_0 E_e}{kT}\right), $$

where $E_e = E + \dfrac{4\pi}{3}P$, and $P_0 = \mu_0 N$ is the greatest possible value of $P$ (at saturation). In the absence of an applied field, i.e., at $E = 0$,

$$ E_e = \frac{4\pi}{3}P \quad \text{and} \quad P = P_0 L\left(\frac{4\pi \mu_0 P}{3kT}\right). $$

This equation, as is known, admits nonzero solutions corresponding to spontaneous or residual polarization when $T < T_c$, where $T_c$ is the Curie temperature, equal to $\dfrac{4\pi \mu_0 P_0}{9k}$.

It should, however, be borne in mind that the “applied” field $E$ by no means coincides with the external field $E_0$, as is sometimes assumed, representing the average “macroscopic” field in the dielectric. In order to establish the behavior of a dielectric in the absence of an external field, it is necessary to set equal to zero not $E$, but $E_0$. It is not difficult to see that when $E_0 = 0$ the field $E$ must have a negative direction, owing to the depolarizing action of the bound charges distributed over the surface of the dielectric and creating a field opposite to the direction of polarization. In the case of a plane capacitor this depolarizing field is equal to $-4\pi P$, so that it more than covers the Lorentz field; in the case of a spherical dielectric it is equal to $-\dfrac{4\pi}{3}P$, i.e., it exactly compensates it.

Only in the case of an elongated specimen polarized along its axis will the Lorentz field predominate over the depolarizing field. Thus, the preservation of residual polarization (in the absence of an external field) depends on the shape of the specimen, and not only on the internal properties of the material under consideration1.

It follows from Debye’s theory that a high initial electric susceptibility in the region of low temperatures and a Curie point should exist practically in all polar dielectrics, i.e., every polar dielectric should behave as a ferroelectric; in reality, however, the phenomenon of ferroelectricity is observed only in very few substances and, moreover, only in the solid state. This circumstance was explained by the fact that for most dielectrics the crystallization temperature is higher than the Curie point, while high electric susceptibility is possible only for liquid dielectrics, since in crystals the molecules cannot rotate freely, as required by Debye’s theory.

In reality, however, it is precisely crystalline substances that are ferroelectrics. Therefore one is forced to conclude that the Clausius—Mossotti or Lorentz equation cannot be extended to substances with polar molecules, i.e. that the expression \(E + \dfrac{4\pi}{3}P\) for the effective field is inapplicable to the case of orientation of permanent dipoles. This question will be examined in detail in the next paragraph.

In any case, one must state that ferroelectricity cannot be explained on the basis of Debye’s theory, as, for example, I. V. Kurchatov attempted to do.\(^6\) The causes of ferroelectricity should be sought in certain specific properties of ferroelectrics, especially in their crystalline structure, which ensure both a large initial \(\varepsilon\) and the preservation of the residual dipole moment; ordinary polar dielectrics do not exhibit ferroelectric phenomena.

§ 3. Analysis of the effective field in an isotropic polar dielectric according to Onsager

On the basis of the considerations already noted, Onsager\(^7\) analyzes Debye’s effective field, showing that only part of it plays a role in the orientation of dipoles. In doing so, Onsager treats the molecule as a cavity of radius \(a\), determined from the condition \(\dfrac{4\pi}{3}a^3 = v\), where \(v\) is the volume per molecule. The molecules surrounding it, however, are regarded as a continuous medium with dielectric constant \(\varepsilon\), equal to the macroscopic dielectric constant of the given substance. At the center of the sphere representing the molecule there is placed a dipole with moment \(\mathbf{m}\), which is composed of its own permanent moment \(\mu_0\) and the moment induced by the electric field. This model representation is the weak point of Onsager’s theory; in particular, a liquid in the immediate vicinity of any molecule can in no way be regarded as a continuous medium, and the local dielectric constant must differ from the macroscopic one.

Therefore Onsager’s theory is not entirely rigorous. This circumstance, however, is not essential.

The essence of Onsager’s theory consists in decomposing the effective field \(\mathbf{E}_e\) into two parts according to the following scheme.

The first part \(\mathbf{G}\) is the field that would be obtained in a spherical cavity containing the molecule under consideration if the latter, with its moment \(\mathbf{m}\), were removed from this cavity, on condition that the field (and the polarization) at a large distance remained unchanged (in the immediate vicinity of the cavity the macroscopic field must then change and, in particular, lose its homogeneity).

The second component of the effective field \(\mathbf{R}\) is the field that is caused by the polarization of the medium when the molecule under consideration is introduced into the cavity it occupies (more precisely, its

moment m at the center of the latter). Onsager calls this field “reactive,” since it characterizes the action exerted by the molecule on itself through the surrounding medium. The reaction field is analogous to the field created by an electric charge situated above a plane surface of a metal or dielectric owing to the polarization of the latter, which reduces to the effect of an “electric image.”

Thus, according to Onsager,

\[ \mathbf{E}_c=\mathbf{G}+\mathbf{R}. \tag{15} \]

To calculate \(\mathbf{R}\), let us find the solution of Laplace’s equation \(\nabla^2\psi=0\) (where \(\psi\) is the potential characterizing the reaction field), which tends, for small \(r\) (i.e., near the center of the cavity), to the expression

\[ \psi=\frac{m\cos\theta}{r^2}, \tag{16} \]

corresponding to a point dipole, and which satisfies the boundary conditions

\[ \psi_{r=a-0}=\psi_{r=a+0},\quad \left(\frac{\partial\psi}{\partial r}\right)_{r=a-0} =\varepsilon\left(\frac{\partial\psi}{\partial r}\right)_{r=a+0}. \tag{17} \]

This solution is expressed by the formulas

\[ \psi=\frac{m\cos\theta}{r^2}-Rr\cos\theta \qquad (r<a), \tag{18} \]

\[ \psi=\frac{m^*\cos\theta}{\varepsilon r^2} \qquad (r>a), \tag{19} \]

where the vector

\[ \mathbf{m}^*=\frac{3\varepsilon}{2\varepsilon+1}\,\mathbf{m} \tag{20} \]

may be defined as the “external” moment of a dipole placed at the center of a spherical cavity (allowing for the polarization of the surrounding medium), and

\[ \mathbf{R}=\frac{2(\varepsilon-1)}{2\varepsilon+1}\,\frac{\mathbf{m}}{a^3}. \tag{21} \]

Let us now determine the change in the homogeneous average field \(E\) caused by the presence of the spherical cavity that is obtained when the molecule under consideration is removed. Mathematically this problem is completely equivalent to the calculation of the reaction field; however, instead of (16) we have the following condition for the potential \(\psi\) characterizing this field:

\[ \psi(r,\theta)\to -Er\cos\theta \quad \text{as } r\to\infty . \tag{22} \]

This condition means that far from the cavity the average field remains undistorted.

The solution of the problem under consideration has the following form:

\[ \psi=-Er\cos\theta-\frac{M}{r^{2}}\cos\theta \quad \text{for } r>a, \tag{23} \]

\[ \psi=-Gr\cos\theta \quad \text{for } r<a, \tag{24} \]

where the vector

\[ \mathbf{M}=\frac{\varepsilon-1}{2\varepsilon+1}\,\mathbf{E}a^{3} \tag{25} \]

is the electric moment of the cavity, while the vector

\[ \mathbf{G}=\frac{3\varepsilon}{2\varepsilon+1}\,\mathbf{E} \tag{26} \]

is the resultant (uniform) field arising in it under a given macroscopic field \(\mathbf{E}\). Let us note that

\[ \frac{G}{E}=\frac{m^{*}}{m}=\frac{3\varepsilon}{2\varepsilon+1}. \tag{27} \]

The total field acting on a spherical molecule in a polarized dielectric is therefore equal to

\[ \mathbf{E}_{e}=\mathbf{G}+\mathbf{R} = \frac{3\varepsilon}{2\varepsilon+1}\,\mathbf{E} + \frac{2(\varepsilon-1)}{(2\varepsilon+1)a^{3}}\,\mathbf{m}. \tag{28} \]

The field of the cavity \(\mathbf{G}\) coincides in direction with the mean (macroscopic) electric field. The time average, or the mean statistical value, of the reactive field \(\mathbf{R}\) also coincides in direction with \(\mathbf{E}\), i.e. with the direction of preferential orientation of the dipoles. In Debye’s theory it is tacitly assumed that the orienting action is determined by the mean value of the total field acting on the molecule:

\[ \overline{\mathbf{E}}_{e}=\mathbf{G}+\overline{\mathbf{R}}. \tag{29} \]

This, however, is incorrect. The reactive field \(\mathbf{R}\) is always directed along the total moment of the molecule and, consequently, cannot exert an orienting action on it, since the torque exerted by this field on the dipole, equal to the vector product of the field strength and the dipole moment, is zero. Attributing an orienting action to the mean value \(\overline{\mathbf{R}}\) is the result of an erroneous averaging. In reality one must first calculate the torque and only then perform the averaging; it then turns out that the orienting action is produced only by the cavity field \(\mathbf{G}\).

The foregoing applies to isotropic dielectrics—liquids. In an anisotropic body (a crystal), the field \(\mathbf{R}\), generally speaking, does not coincide in direction with \(\mathbf{m}\); consequently, the torque exerted by it on the dipole is not zero.

The total dipole moment of the molecule \(\mathbf{m}\) is composed of the permanent dipole moment \(\boldsymbol{\mu}_{0}\) and the moment formed by elastic polarization by the effective field \(\mathbf{E}_{e}\),

\[ \mathbf{m}=\mu_{0}\mathbf{u}+\alpha\mathbf{E}_{e}. \]

Here u is the unit vector characterizing the direction in which the moment \(\mu_0\) is oriented. Substituting the expression (28) for \(E_e\) into the preceding formula, we obtain

\[ \left[1-\frac{2(\varepsilon-1)a}{(2\varepsilon+1)a^3}\right]\mathbf{m} =\mu_0\mathbf{u}+\frac{3\varepsilon}{2\varepsilon+1}\alpha\mathbf{E}. \tag{30} \]

For convenience in calculation we introduce the refractive index of the given substance \(n\). It may be assumed that the refraction of light rays is due only to the elastic part of the polarization, since permanent dipoles do not have time to orient themselves in an electric field with the frequency of light waves. The quantity \(n^2\) represents the elastic part of \(\varepsilon\), related to the polarizability \(\alpha\) by the Clausius—Mossotti equation (4), which can be rewritten in the form:

\[ \alpha=\frac{n^2-1}{n^2+2}a^3, \tag{31} \]

where \(a\) is the radius of the molecule (i.e., of the cavity it occupies). Substituting (31) into (30), we obtain

\[ \mathbf{m} = \frac{(n^2+2)(2\varepsilon+1)}{3(2\varepsilon+n^2)} \mu_0\mathbf{u} + \frac{\varepsilon(n^2-1)}{2\varepsilon+n^2}a^3\mathbf{E} = \mu\mathbf{u} + \frac{\varepsilon(n^2+2)}{2\varepsilon+n^2}\alpha\mathbf{E}. \tag{32} \]

The effective field determining the elastic polarization is obtained, following Onsager, by substituting (32) into (28):

\[ E_e = E + \mathbf{I} - \left[ 1+ \frac{n^2(\varepsilon-1)}{2\varepsilon+n^2} \right]\mathbf{E} + \frac{2(\varepsilon-1)}{2\varepsilon+1} \frac{\mu_0}{a^3}\mathbf{u}. \tag{33} \]

The field \(\mathbf{I}\), by analogy with the Lorentz field, may be called the Onsager field. In the case of a nonpolar liquid (\(\mu_0=0\) and \(\varepsilon=n^2\)) formula (33) reduces to

\[ E_e = E+ \frac{E(\varepsilon-1)}{3} = \frac{\varepsilon+2}{3}E, \tag{34} \]

which coincides with the Lorentz formula (6), if in it \(4\pi P\) is replaced by \((\varepsilon-1)E^1)\).

The extension of the Lorentz formula to the orientational part, carried out by Debye, thus leads to an overestimated value of the effective field responsible for orientation (according to Debye, as \(\varepsilon\to\infty\), also \(E_e\to\infty\)). Elimination of this error at the same time means eliminating the possibility of ferroelectric phenomena for ordinary polar liquids.

§ 4. Statistical Theory of Onsager

In order to calculate the orientational part of the polarization, it is necessary to determine the effective energy of interaction between the molecules and the field. As was already noted above, for this one should consider

\(^{1}\) This circumstance was noted by E. V. Kuvshinskii.

orienting couple of forces for each direction $\mathbf{u}$. The moment of this couple is equal to

\[ \mathbf{M}=[\mathbf{E}_e,\mathbf{m}]=[\mathbf{G},\mathbf{m}] =\frac{3\varepsilon}{2\varepsilon+1}[\mathbf{E},\mathbf{m}], \tag{35} \]

whence, on the basis of (32), we obtain

\[ \mathbf{M}=\mu[\mathbf{G},\mathbf{u}]=\mu^*[\mathbf{E},\mathbf{u}]. \tag{36} \]

Here the following notations have been introduced:

\[ \mu=\frac{(n^2+2)(2\varepsilon+1)}{3(2\varepsilon+n^2)}\,\mu_0, \tag{37} \]

\[ \mu^*=\mu\frac{3\varepsilon}{2\varepsilon+1} =\frac{\varepsilon(n^2+2)}{2\varepsilon+n^2}\,\mu_0. \tag{38} \]

The quantity $\mu$ represents the sum of the molecule’s intrinsic moment $\mu_0$ and the moment produced by elastic polarization in the reactive field. In direction it always coincides with $\mu_0$; therefore the action of the reactive field reduces to increasing the intrinsic moment of the molecule $\vec{\mu}_0$ to the value $\vec{\mu}$; $\mu^*$ is the effective dipole moment characterizing the action of the molecule outside the sphere occupied by it; its ratio to $\vec{\mu}$ is equal to the ratio of the cavity field $\mathbf{G}$ to the mean field in the dielectric $\mathbf{E}$. To avoid confusion, let us also note that the total moment of the molecule $\mathbf{m}$ (in contrast to $\vec{\mu}$) includes not only the elastic polarization caused by the reactive field $\mathbf{R}$, but also the elastic polarization produced by the cavity field $\mathbf{G}$.

From (36) we obtain $M=\mu^*E\sin\theta$, where $\theta$ is the angle between the direction of the mean field $\mathbf{E}$ and the direction of the dipole $\mathbf{u}$.

The rotating moment is connected with the effective interaction energy $W$ by the relation $\dfrac{\partial W}{\partial\theta}=M$. Hence we obtain

\[ W=-\mu^*E\cos\theta. \tag{39} \]

The mean orientation of molecules in the field is determined by Boltzmann’s formula

\[ \overline{\cos\theta}= \frac{\iint \cos\theta\, e^{-\frac{W}{kT}}\sin\theta\,d\theta\,d\varphi} {\iint e^{-\frac{W}{kT}}\sin\theta\,d\theta\,d\varphi}. \]

If the integration is carried out, the well-known Langevin function is obtained:

\[ \overline{\cos\theta} =L\left(\frac{\mu^*E}{kT}\right) =\operatorname{ctgh}\left(\frac{\mu^*E}{kT}\right) -\frac{kT}{\mu^*E}. \tag{40} \]

For a weak field $E$, one may restrict oneself to the first term of the expansion of $L$ in a series:

\[ \overline{\cos\theta}=\frac{\mu^*E}{3kT}. \tag{41} \]

In this case the polarization per unit volume turns out to be equal to

\[ P=Nm=N\left[\frac{\mu\mu^*}{3kT}+\frac{\varepsilon(n^2+2)}{2\varepsilon+n^2}\alpha\right]E . \tag{42} \]

Using the expressions obtained for the polarization, one can derive equations analogous to Debye’s equations (11) and (14), but now with the proper allowance for the forces orienting the molecular dipoles.

In the case of a pure liquid one may assume that the molecules occupy the entire volume of the dielectric, and put

\[ N\cdot\frac{4\pi a^3}{3}=1 . \tag{43} \]

Strictly speaking, this means that we have taken for \(a\) (the radius of the cavity) not the radius of the molecule itself, but the radius of a sphere whose volume is equal to the volume of the dielectric falling to the share of each molecule (cf. the beginning of § 3).

From (43) we have

\[ 4\pi N(n^2+2)\alpha=4\pi N(n^2-1)a^3=3n^2-1 . \tag{44} \]

Formulas (10), (42), and (44) give

\[ \varepsilon-1=4\pi N\left\{\frac{\mu\mu^*}{3kT}+\frac{3\varepsilon(n^2-1)}{2\varepsilon+n^2}\right\} \]

or

\[ \frac{(2\varepsilon+1)(\varepsilon-n^2)}{2\varepsilon+n^2} =4\pi N\frac{\mu\mu^*}{3kT}. \tag{45} \]

Taking into account (37) and (38), we obtain from (45) Onsager’s final formula for the dielectric constant of pure polar liquids:

\[ \frac{(\varepsilon-n^2)(2\varepsilon+n^2)}{\varepsilon(n^2+2)^2} = \frac{4\pi N\mu_0^2}{9kT}. \tag{46} \]

Let us consider what form formula (46) takes in two limiting cases: when \(\varepsilon\) is large and when it differs little from \(n^2\), i.e., when the polarization of the dielectric is mainly due to the elastic polarization of the molecules.

If \(\varepsilon \gg n^2\), then

\[ \varepsilon \simeq \frac{4\pi N\mu_0^2}{9kT}\,\frac{(n^2+2)^2}{2}. \tag{47} \]

This equation gives no Curie point and in its form is not at all similar to the Clausius—Mossotti equation.

On the other hand, if \(\varepsilon-n^2\ll n^2\), equation (46) assumes a form similar to the Clausius—Mossotti equation:

\[ \frac{\varepsilon-1}{\varepsilon+2}-\frac{n^2-1}{n^2+2} = \frac{3\varepsilon(n^2+2)}{(2\varepsilon+n^2)(\varepsilon+2)} \cdot \frac{4\pi N\mu_0^2}{9kT}. \tag{48} \]

It follows from this that, in the case of weakly polar liquids, Onsager’s theory differs little from the old Lorentz—Debye theory.

Onsager confines himself to a formula containing only the first term of the orientational part of the polarization, i.e., he does not take into account the saturation effect, which is valid only for weak fields. Formula (42), however, can easily be generalized to arbitrary fields and replaced by the exact formula

\[ \mathbf{P} = N\left[ \frac{\mu}{E}L\left(\frac{\mu^{*}E}{kT}\right) + \frac{\varepsilon(n^{2}+2)}{2\varepsilon+n^{2}}\alpha \right]\mathbf{E}. \tag{49} \]

Restricting ourselves, in taking account of the saturation effect, to the second term of the expansion of \(L\) in a series (it will be of third order with respect to \(\frac{\mu^{*}E}{kT}\)), we obtain

\[ \mathbf{P} = N\left[ \frac{\mu\mu^{*}}{3kT} - \frac{\mu\mu^{*3}E^{2}}{45(kT)^{3}} + \frac{\varepsilon(n^{2}+2)}{2\varepsilon+n^{2}}\alpha \right]\mathbf{E}. \tag{50} \]

Formula (46), with exact account of saturation, is replaced by the following:

\[ \frac{\varepsilon-n^{2}}{n^{2}+2} = \frac{4\pi N\mu_{0}}{3kTE} L\left[ \frac{\varepsilon(n^{2}+2)}{2\varepsilon+n^{2}}\frac{\mu_{0}E}{kT} \right], \tag{51} \]

and if, in doing so, one confines oneself only to the third-order term,

\[ \frac{(\varepsilon-n^{2})(2\varepsilon+n^{2})}{(n^{2}+2)^{2}\varepsilon} = \frac{4\pi N\mu_{0}^{2}}{9kT} \left\{ 1 - \frac{1}{15} \left[ \frac{\varepsilon(n^{2}+2)}{2\varepsilon+n^{2}} \frac{\mu_{0}E}{kT} \right]^{2} \right\}. \tag{52} \]

Let us now consider the case of a mixture of several polar or nonpolar liquids. Suppose that the mixture contains, per unit volume, \(N_i\) molecules of each kind, having spherical shape with radii \(a_i\), polarizabilities \(\alpha_i\), and permanent dipole moments \(\mu_{0i}\).

We introduce the refractive index \(n_i\) separately for each component of the mixture by means of the relation

\[ \alpha_i = \frac{a_i^{3}(n_i^{2}-1)}{n_i^{2}+2}. \tag{53} \]

Let \(\vartheta_i\) denote the fraction of the unit volume corresponding to the \(i\)-th component. Analogously to (43), one may write

\[ \vartheta_i = N_i\frac{4\pi a_i^{3}}{3}. \tag{54} \]

From (10), (42), and (54) we obtain

\[ \varepsilon-1 = 4\pi\sum N_i \left[ \frac{\varepsilon(n_i^{2}+2)}{\varepsilon+n_i^{2}}\alpha_i + \frac{\mu_i\mu_i^{*}}{3kT} \right] = \]

\[ = \sum \vartheta_i \frac{3\varepsilon(n_i^{2}-1)}{n_i^{2}+2\varepsilon} + 4\pi\sum N_i\frac{\mu_i\mu_i^{*}}{3kT}. \tag{55} \]

Using the identity

\[ 3\varepsilon(n_i^{2}-1) = (2\varepsilon+n_i^{2})(\varepsilon-1) - (2\varepsilon+1)(\varepsilon-n_i^{2}), \]

we transform (55) to the form:

\[ (1-\sum \vartheta_i)(\varepsilon-1)+(2\varepsilon+1)\sum \frac{\vartheta_i(\varepsilon-n_i^2)}{2\varepsilon+n_i^2} = 4\pi\frac{\sum N_i\mu_i\mu_i^*}{3kT}. \tag{56} \]

Taking into account that

\[ \sum \vartheta_i=1, \tag{57} \]

and taking (37) and (38) into consideration, we rewrite (56) in the form:

\[ \sum \vartheta_i \frac{(\varepsilon-n_i^2)3\varepsilon}{2\varepsilon+n_i^2} = 4\pi\sum \frac{N_i\mu_i^{*2}}{3kT}. \tag{58} \]

If \(\varepsilon \gg n_i^2\), then, using once more relation (38), we obtain from (58)

\[ \varepsilon = \sum \frac{(n_i^2+2)^2}{1}\cdot \frac{4\pi}{3}\,N_i \frac{\mu_{0i}^{2}}{3kT} +O(n^2). \tag{59} \]

In conclusion it should be noted that, despite the approximations made and the adopted inexact model of the molecule, Onsager’s theory in most cases agrees well with experimental data. Only for sharply polar liquid dielectrics are significant discrepancies observed; namely, the theory gives values for the dielectric constant that are too small. Thus, for example, in the case of water, if it is assumed that the permanent dipole moment of the molecule has the same value as for water vapor, one obtains \(\varepsilon=31\) instead of the experimental value \(\varepsilon=81\).

Onsager’s theory eliminated the fundamental difficulty connected with the absence of ferroelectric properties in polar liquids, but nevertheless did not resolve all the contradictions of the old Debye theory with experimental data; therefore its further improvement is required, with account taken of some additional circumstances not considered by Onsager.

II. DEBYE’S NEW THEORY

§ 5. Difficulties of the old Debye theory and the idea of the local field

In application to strongly polar liquids with a large dielectric constant, the old Debye theory encountered a number of difficulties. First of all, the following fact should be noted: if, from the experimental values of the dielectric constant of one and the same substance in the liquid and gaseous states, the proper dipole moments of the molecules are determined by the formulas of Debye’s theory, then in the liquid state they turn out to be considerably (by 2–3 times) smaller than in the gaseous state. Thus, for example, in the case of water vapor the dipole moment is equal to \(\mu_0=1.84\cdot 10^{-18}\), whereas liquid water with dielectric constant \(\varepsilon=81\), according to Debye’s theory, corresponds to the value \(\mu_0=0.9\cdot 10^{-18}\).

This discrepancy was explained by the assumption that in a liquid the dipoles form “astatic pairs” of two oppositely directed and rigidly coupled dipoles, which leads to a decrease in the mean effective dipole moment of the molecules.

However, there is another internal contradiction in Debye’s theory of the polarization of liquids, which cannot be explained by this hypothesis. Namely, the values of the dipole moment determined from the linear term of Debye’s theory (which takes account of polarization in weak fields) and from the cubic term (which characterizes the saturation effect) also turn out to be entirely different.

In a note published in 1935,^(8, 9) Debye expressed the idea that the cause of these contradictions is rooted in an incorrect conception of the character of the thermal motion of molecules in liquids.

Previously it had been assumed that in a liquid, as in a gas, the molecules can rotate quite freely. This conception does not correspond to reality, since in a liquid the molecules are situated close to one another and must exert on one another influences that hinder their free rotation.

Thus, the character of molecular rotation in a liquid must be different from that in a gas. It is natural to suppose that free rotation must in this case be replaced by rotational vibrations of the molecules about disorderly distributed equilibrium orientations, which change from time to time for each molecule. Such a conception is entirely analogous to that developed earlier by Ya. I. Frenkel for the translational motion of particles in a liquid.^(10) This translational motion is regarded as an oscillation about an equilibrium position that remains unchanged during some interval of time \(\tau\) (the relaxation time) and then is displaced by a small segment \(\delta\). In the absence of external forces this displacement occurs with equal frequency in all directions, causing a kind of diffusional or Brownian motion of the molecules.

The time during which a molecule oscillates about an unchanging position, \(\tau\), is related to the period of oscillation \(\tau_0\) by the relation

\[ \tau = \tau_0 e^{\frac{U}{kT}}, \]

where \(U\) is the activation energy for the transition from one equilibrium position to a neighboring one. A completely analogous picture should also be observed in the case of rotational motion of molecules, with the only difference that abrupt rotations of the molecules through considerable angles are then possible in the transition from one equilibrium orientation to another (see below).

Debye^(8, 9, 11) proposed taking into account this interaction of the molecules of a dipolar liquid, which prevents their free rotation, by introducing a certain local electric field acting on each molecule from its neighbors in addition to the averaged effective field \(E_e\). The magnitude and direction of the local field are determined by the random orientation of the neighbors of the given molecule. We shall, however, assume that in magnitude the local field is—

...everywhere identical; this is permissible, since in the final analysis we are interested only in macroscopic quantities, and, in averaging over a large number of molecules, deviations of the local-field magnitude for individual molecules in one direction or another mutually compensate, and only its mean value plays a role.

It must be said that this mean value of the local field \(F\) in Debye’s theory is not calculated directly, but is determined by comparing the results of the theory with experimental data.

Since the local field is due mainly to neighboring molecules, it can, in order of magnitude, be estimated as the field of a point dipole at a distance equal to the mean distance between neighboring molecules in a liquid. For most polar substances the dipole moment is of the order of \(10^{-18}\) CGS; the distance between molecules may be taken equal to \(10^{-8}\) cm. Under these conditions the local field is found to be of the order of \(10^8\) V/cm, i.e., it is much greater than the external electric fields that can practically be applied to a dielectric.

The influence of the local field, however, is determined not by its ratio to the external (or to the mean) field in the dielectric, but by the ratio of the potential energy of a dipole in this field, \(\mu F\), to the energy of thermal motion, \(kT\), which tends to disturb the orientation of the dipoles in the local field. From this point of view the influence of the local field begins to make itself felt at ordinary temperatures only in the case when its strength reaches or exceeds \(10^7\) V/cm, whereas at \(F = 10^6\) V/cm it plays no role (i.e., practically does not hinder the free rotation of the molecules). Both the magnitude and the direction of the local field acting on each molecule change abruptly with time when the position or orientation of neighboring molecules changes abruptly. However, for our calculations this circumstance proves inessential. Indeed, when averaging over a large number of molecules, the changes of the local field with time, in magnitude and direction, for different molecules mutually compensate. In the case of a constant external field, one may therefore assume that the local field at the location of each molecule remains constant in magnitude and direction.

We shall consider all possible directions of the local field at any point to be equally probable. To show the validity of this independently of the presence of the external field \(E_0\) and of the mean field in the dielectric \(E\), it is necessary to clarify the fundamental difference between Debye’s local field \(F\) and that part of the effective field \(E_e\) which is created by the mean orientation of the permanent dipoles of the dielectric. Randomly oriented dipoles participate in creating the field \(F\); therefore its direction is different at different points of space. In calculating the field \(E_e\), however, the mean value of the projection of the dipole moment onto the direction \(E_e\), due to the presence of the external field \(E_0\), is taken into account; moreover, in the case of a homogeneous mean field, the direction of \(E_e\) is the same at all points. The preferential orientation of the molecules as a whole is taken...

is caused by the field \(E_e\), whereas the field \(F\) determines random deviations of the orientation of the molecules from the average, in view of which all its directions may be regarded as equally probable.

§ 6. Derivation of Debye’s formula\(^1\)

Consider in a liquid dielectric a dipole \(\mu\), on which there acts an effective field \(E_e\) and a local field \(F\) directed at some angle \(\theta\) to it (Fig. 2). The dipole is oriented in the resultant field

\[ F' = E_e + F. \]

Fig. 2

The mean projection of the dipole onto the direction \(F'\) is determined in exactly the same way as if \(F'\) were the effective field acting on the molecule. In the presence of thermal motion of the molecules, as above, this projection is determined by the Langevin function:

\[ \bar{\mu}_{F'}=\mu L\left(\frac{\mu F'}{kT}\right). \tag{60} \]

We are interested, however, in the mean projection of the dipole moment onto the direction of the mean field \(E\), i.e., of the effective field \(E_e\). We obtain it by projecting \(\bar{\mu}_{F'}\) onto \(E_e\) and averaging over all possible directions of \(F\):

\[ \bar{\mu}_{E}=\overline{\bar{\mu}_{F'}\cos\theta'}=\mu L\left(\frac{\mu F'}{kT}\right)\cos\theta'. \tag{61} \]

In averaging, all directions of \(F\) (but not \(F'\)!) should be considered equally probable, i.e., \(dN\) is proportional to \(\sin\theta\,d\theta\).

From Fig. 2 we have

\[ F'=\left(F^2+E_e^2+2E_eF\cos\theta\right)^{\frac12}. \tag{62} \]

The condition of equality of the torques acting on the equilibrium-oriented dipole from the fields \(F\) and \(E_e\) is expressed by the equation

\[ E_e\sin\theta'=F\sin\varphi, \tag{63} \]

where

\[ \cos\theta'=\cos\theta\cos\varphi+\sin\theta\sin\varphi. \tag{64} \]

As was already noted above, \(E_e\ll F\); therefore all expressions may be expanded in powers of the ratio \(\dfrac{E_e}{F}\).

To clarify better the idea of the derivation, let us first carry out the calculation to terms of first order with respect to \(\dfrac{E_e}{F}\). In this approximation, from (63) and (64) we have

\[ \left. \begin{aligned} \sin\varphi &\simeq \frac{E_e}{F}\sin\theta;\quad \cos\varphi=1,\\ \cos\theta'&=\cos\theta+\frac{E_e}{F}\sin^2\theta \end{aligned} \right\} \tag{65} \]

\(^1\) Below we give a detailed derivation of this formula, since it has nowhere been published, either by anyone else or by Debye himself.

and then from (62):

\[ F' \simeq F\left(1+\frac{E_e}{F}\cos\theta\right). \tag{66} \]

Introduce the notation:

\[ a_0=\frac{\mu F}{kT};\qquad a=\frac{\mu F'}{kT}. \tag{67} \]

We expand the Langevin function in a series in powers of \(\dfrac{E_e}{F}\):

\[ L(a)=L\left[a_0\left(1+\frac{E_e}{F}\cos\theta\right)\right] = L_0+\left(\frac{dL}{da}\right)_0 a_0\frac{E_e}{F}\cos\theta, \tag{68} \]

where

\[ L_0=L(a_0)\quad \text{and}\quad \left(\frac{dL}{da}\right)_0=\frac{dL_0}{da_0}. \]

Substituting (65) and (68) into (61), we obtain

\[ \overline{L\cos\theta} = L_0\frac{E_e}{F}\,\overline{\sin^2\theta} +\left(\frac{dL}{da}\right)_0 a_0\frac{E_e}{F}\,\overline{\cos^2\theta}. \tag{69} \]

All the remaining terms either vanish upon averaging, or will be of a higher order of smallness.

Since \(L(a)=\operatorname{ctgh} a-\dfrac{1}{a}\), it follows that

\[ \frac{dL}{da}=1-\frac{2L}{a}-L^2. \tag{70} \]

For equal probability of all directions,

\[ \overline{\sin^2\theta}=\frac{2}{3} \tag{71} \]

and

\[ \overline{\cos^2\theta}=\frac{1}{3}. \tag{72} \]

Substituting (70), (71), and (72) into (69), we obtain

\[ \overline{L\cos\theta} =\frac{E_e}{3F}\left[2L_0+a_0\left(1-\frac{2L_0}{a_0}-L_0^2\right)\right] =\frac{E_e a_0}{3F}(1-L_0^2). \]

or, on the basis of (61) and (67),

\[ \overline{\mu_E} =\frac{\mu^2 E_e}{3kT}(1-L_0^2);\qquad L_0=L\left(\frac{\mu F}{kT}\right). \tag{73} \]

We see that, when the local field is taken into account, the usual expression for the mean projection of the dipole moment onto the direction of the field is multiplied by the factor \(1-L^2\). This is precisely Debye’s correction for the linear term in the expression for the orientational polarization. Since \((1-L^2)<1\), the local field reduces the orientational polarization of a liquid, and does so the more strongly, the larger the ratio \(\dfrac{\mu F}{kT}\). From the point of view of the old Debye theory, this result corresponds to a decrease in the dipole moment of the molecules of the liquid in comparison with the dipole moment of the same molecules in the gaseous state, according to the relation

\[ \mu_{\text{liq}}=\mu_{\text{gas}}\sqrt{1-L^2}. \]

We shall carry out the same calculation with accuracy up to quantities of the next (third) order of smallness relative to \(\dfrac{E_e}{F}\). From (63) we have:

\[ \left. \begin{aligned} \sin\varphi &\simeq \frac{E_e}{F}\sin\theta -\left(\frac{E_e}{F}\right)^2\sin\theta\cos\theta + \left(\frac{E_e}{F}\right)^3\sin\theta \left(\cos^2\theta-\frac{1}{2}\sin^2\theta\right),\\ \cos\varphi &= 1-\frac{1}{2}\left(\frac{E_e}{F}\right)^2\sin^2\theta,\\ \cos\theta' &= \cos\theta+\frac{E_e}{F}\sin^2\theta -\left(\frac{E_e}{F}\right)^2 \left(\sin\theta\cos\theta+\frac{1}{2}\sin^2\theta\right)\cos\theta\\ &\quad +\left(\frac{E_e}{F}\right)^3\sin^2\theta \left(\cos^2\theta-\frac{1}{2}\sin^2\theta\right). \end{aligned} \right\} \tag{74} \]

From (62) we further have

\[ F' \simeq F\left(1+\frac{E_e}{F}\cos\theta +\frac{E_e^2}{2F^2}\sin^2\theta -\frac{E_e^3}{F^3}\cos\theta\sin^2\theta\right). \tag{75} \]

Thus,

\[ \begin{aligned} L\Bigg[a_0\Bigg(&1+\frac{E_e}{F}\cos\theta +\frac{E_e^2}{2F^2}\sin^2\theta -\frac{E_e^3}{2F^3}\cos\theta\sin^2\theta\Bigg)\Bigg] &= L_0+\left(\frac{dL}{da}\right)_0 a_0\frac{E_e}{F}\cos\theta +\left(\frac{E_e}{F}\right)^2 \left[\frac{1}{2}\left(\frac{d^2L}{da^2}\right)_0 a_0^2\cos^2\theta +\frac{3}{2}\left(\frac{dL}{da}\right)_0 a_0\sin^2\theta\right]\\ &\quad+\left(\frac{E_e}{F}\right)^3 \left[-\frac{9}{2}\left(\frac{dL}{da}\right)_0 a_0\cos\theta\sin^2\theta +\frac{1}{2}\left(\frac{d^2L}{da^2}\right)_0 a_0^2\cos\theta\sin^2\theta\right.\\ &\qquad\left.+\frac{1}{6}\left(\frac{d^3L}{da^3}\right)_0 a_0^3\cos^3\theta\right]. \end{aligned} \tag{76} \]

Substituting (74) and (76) into (61), we obtain:

\[ \begin{aligned} \overline{L\cos\theta'} &= L_0\left[ \frac{E_e}{F}\overline{\sin^2\theta} -\frac{E_e^2}{F^2}\overline{\sin^2\theta\cos\theta} +\frac{E_e^3}{F^3}\overline{\sin^2\theta \left(2\cos^2\theta-\frac{1}{2}\sin^2\theta\right)} \right]\\ &\quad+\frac{E_e}{F}a_0\left(\frac{dL}{da}\right)_0 \left[\overline{\cos^2\theta} +\frac{E_e}{F}\overline{\sin^2\theta\cos\theta} -\frac{3}{2}\frac{E_e^2}{F^2}\overline{\sin^2\theta\cos^2\theta}\right]\\ &\quad+\left(\frac{E_e}{F}\right)^2 \left[\frac{1}{2}\left(\frac{d^2L}{da^2}\right)_0 a_0^2 \left(\overline{\cos^3\theta} +\frac{E_e}{F}\overline{\cos^2\theta\sin^2\theta}\right)\right.\\ &\qquad\left. +\frac{3}{2}\left(\frac{dL}{da}\right)_0 a_0 \left(\overline{\sin^2\theta\cos\theta} +\frac{E_e}{F}\overline{\cos^2\theta\sin^2\theta}\right) \right]\\ &\quad+\left(\frac{E_e}{F}\right)^3 \left[ -\left(\frac{dL}{da}\right)_0 a_0\frac{9}{2}\overline{\cos^2\theta\sin^2\theta} +\frac{3}{2}\left(\frac{d^2L}{da^2}\right)_0 a_0^2 \overline{\cos^2\theta\sin^2\theta} +\frac{1}{6}\left(\frac{d^3L}{da^3}\right)_0 a_0^3 \overline{\cos^4\theta} \right]. \end{aligned} \tag{77} \]

The bar everywhere denotes averaging over all directions (under the assumption of their equal probability).

Since

\[ \begin{aligned} \overline{\sin^2\theta\cos\theta}&=0;\qquad \overline{\cos^3\theta}=0,\\ \overline{\cos^4\theta}&=\frac{1}{5};\quad \overline{\cos^2\theta\sin^2\theta}=\frac{2}{15};\quad \overline{\sin^4\theta}=\frac{8}{15} \end{aligned} \tag{78} \]

and furthermore

\[ \begin{aligned} \frac{d^2L}{da^2}&=2L^3+\frac{6}{a}L^2+\left(\frac{6}{a^2}-2\right)L-\frac{2}{a},\\ \frac{d^3L}{da^3}&=-6L^4-\frac{24}{a}L^3+\left(8-\frac{36}{a^2}\right)L^2\\ &\quad+\left(\frac{16}{a}-\frac{24}{a^3}\right)L+\frac{8}{a^2}-2, \end{aligned} \tag{79} \]

then, substituting (70), (71), (72), (78), and (79) into (77), we obtain after a number of simplifications

\[ \overline{L\cos\theta'}=\frac{E_e a}{F}\,\frac{1}{3}(1-L^2) -\left(\frac{E_e}{F}\right)^3 \left( \frac{a^3}{5}L^4-\frac{4}{15}a^3L^3+\frac{1}{15} +\frac{8}{15}a^2L^3-\frac{4}{15}a^2L+\frac{2}{15}aL^2 \right). \]

If (61) and (67) are taken into account, then from the last expression there follows the Debye formula with allowance for the saturation effect:

\[ \overline{\mu}=\frac{\mu E_e}{3kT}\,R(a)-\frac{1}{45}\left(\frac{\mu E_e}{kT}\right)^3 R^*(a), \tag{80} \]

where \(R\) and \(R^*\) are Debye correction factors:

\[ R=1-L^2(a) \tag{81} \]

and

\[ R^*(a)=3\left[(1-4L^2+3L^4)+4\frac{L}{a}(2L^2-1)+6\frac{L^2}{a^2}\right], \tag{82} \]

\[ a=\frac{\mu F}{kT}. \tag{83} \]

For small \(a\), the factors \(R\) and \(R^*\) tend to unity, and formula (80) reduces to the corresponding formula of the old Debye theory.

To estimate the values of the Debye correction factors, let us compare the relative decrease \(-\Delta\varepsilon\) of the dielectric constant \(\varepsilon\) due to the second term (saturation), calculated theoretically and determined experimentally. From formula (80) it is easy to derive, just as is done in the old Debye theory:

\[ \frac{\Delta\varepsilon}{\varepsilon} =-\frac{4\pi}{15}N\frac{\mu^4}{k^3T^3}\frac{1}{\varepsilon} \left(\frac{\varepsilon+2}{3}\right)^4 R^*(a)E^2. \tag{84} \]

If here one sets \(R^*=1\), which corresponds to the old theory, which did not take the local field into account, then for water at \(E=10^5\ \mathrm{V/cm}\) one obtains

\[ \frac{\Delta\varepsilon}{\varepsilon}=-3.87. \]

Experimentally, however, at the same field it was found that

\[ \frac{\Delta\varepsilon}{\varepsilon}=-1.1\cdot10^{-3}, \]

i.e., a discrepancy of three and a half thousand times.

Formula (84) gives a value of \(\dfrac{\Delta \varepsilon}{\varepsilon}\) that agrees with experiment if one takes \(a = 10\). The same value for \(a\) is obtained from experiments with weak fields if the moments of water molecules are assumed to be the same in the liquid and in the gaseous state. Thus, both of the contradictions of the old theory noted above disappear.

§ 7. A simplified form of Debye’s theory and its correction in the spirit of Onsager’s theory

We shall present a simplified form of Debye’s theory, applicable in the case when \(kT\) is small in comparison with \(\mu F'^{2}\). The latter condition holds for strongly polar dielectrics, for example, for water. In this case, in the first approximation, one may completely neglect the thermal rotational oscillations of the dipoles and assume that all dipoles are set along the resultant field \(\mathbf F'\) (Fig. 2). We shall carry out the calculation to first order with respect to the quantity \(\dfrac{E_e}{F}\). Then, on the basis of (65), the mean value of the projection of the dipole moment on the direction \(\mathbf E_e\) (averaging over all possible directions of the local field \(\mathbf F\)) proves to be equal to

\[ \bar{\mu}_E = \mu \overline{\cos \theta'} = \frac{2}{3}\mu \frac{E_e}{F}. \tag{85} \]

This formula is an approximation to Debye’s formula (73) for \(\dfrac{\mu F}{2kT} \gg 1\). Indeed, in this case one may replace \(1 - L^2\) by \(\dfrac{2kT}{\mu F}\), and (73) passes into (85).

The result obtained may be regarded as proof that Debye’s theory ascribes to the local field a purely static effect, without taking into account the jump-like changes of the equilibrium orientations of individual molecules with time that are characteristic of a liquid.

We shall now take into account the thermal rocking of the molecules. Let \(\delta\) denote the angle between \(\mathbf F'\) and the moment \(\vec{\mu}\) of the oscillating dipole. The projection of \(\vec{\mu}\) on the direction \(\mathbf E\) is equal to

\[ \mu \cos(\theta' - \delta) = \mu(\cos \theta' \cos \delta + \sin \theta' \sin \delta). \]

The statistical mean of \(\sin \delta\) is zero, whereas for small thermal oscillations (at sufficiently low temperatures) we have

\[ \overline{\cos \delta} = 1 - \frac{\overline{\delta^2}}{2}. \]

Since the expression

\[ \mu F'(1 - \cos \delta) \simeq \mu F' \frac{\delta^2}{2} \]

is the corresponding potential energy, whose statistical mean for a two-dimensional oscillator is \(kT\), neglecting the difference between \(F\) and \(F'\), we obtain

\[ \frac{\overline{\delta^2}}{2} = \frac{kT}{\mu F} \]

or, using (65),

\[ \bar{\mu}_E = \mu \overline{\cos \theta'}\,\overline{\cos \delta} = \frac{2}{3}\mu \frac{E_e}{F}\left(1 - \frac{kT}{\mu F}\right). \tag{86} \]

This formula is also an approximation of Debye’s formula (73), but for higher temperatures.

In Debye’s theory it is tacitly assumed that the effective field acting on a molecule is the Lorentz field \(E+\dfrac{4\pi}{3}P\), i.e., the considerations set forth in the preceding section by Onsager\(^1\) are not taken into account at all. On the other hand, Debye’s local field is not taken into account in Onsager’s theory. It seems natural to combine both theories by replacing the Lorentz field \(E_e=E+\dfrac{4\pi}{3}P\) of Debye’s theory by Onsager’s cavity field \(G\), and the molecule’s own dipole moment \(\mu_0\) by the moment \(\mu\), which takes into account the influence of the reaction field.

All the calculations carried out in the preceding paragraph remain valid in this case. If the calculation is made in the first approximation, then, analogously to (83), we obtain

\[ \bar{\mu}_E=\frac{\mu^2G}{3kT}\left[1-L^2\left(\frac{\mu F}{kT}\right)\right] =\frac{3\varepsilon}{2\varepsilon+1}\frac{\mu^2E}{3kT} \left[1-L^2\left(\frac{\mu F}{kT}\right)\right]. \tag{87} \]

Here \(\mu\) should be understood not as the molecule’s own dipole moment, denoted in Onsager’s theory by \(\mu_0\), but as the intrinsic moment increased by elastic polarization in the reaction field \(R\). The value of \(\mu\) is given by expression (37).

If, in addition, the elastic polarization by the cavity field \(G\) is taken into account, then the following expression is obtained for the polarization of the dielectric, analogous to expression (42), but with Debye’s correction for the local field:

\[ P=N\left\{\frac{3\varepsilon}{2\varepsilon+1}\frac{\mu^2}{3kT} \left[1-L^2\left(\frac{\mu F}{kT}\right)\right] +\frac{\varepsilon(n^2+2)}{2\varepsilon+n^2}\alpha\right\}E. \tag{88} \]

It is not difficult to combine Onsager’s and Debye’s theories also when the saturation effect is taken into account, characterized by the cubic term in the expansion of \(L\left(\dfrac{\mu E_e}{kT}\right)\) in powers of \(\dfrac{\mu E_e}{kT}\). Instead of (44) we obtain

\[ P=N\left[\frac{3\varepsilon}{2\varepsilon+1}\frac{\mu^2}{3kT} R\left(\frac{\mu F}{kT}\right) +\left(\frac{3\varepsilon}{2\varepsilon+1}\right)^3 \frac{\mu E_e^2}{45(kT)^3} R^*\left(\frac{\mu F}{kT}\right) +\frac{\varepsilon(n^2+2)}{2\varepsilon+n^2}\alpha\right]E, \tag{89} \]

where Debye’s correction factors are expressed by the formulas:

\[ R=1-L^2; \]

\[ R^*=3\left[(1-4L^2+3L^4)+4L\frac{kT}{\mu F}(2L^2-1) +6L^2\left(\frac{kT}{\mu F}\right)^2\right]. \tag{90} \]

Let us introduce Debye’s correction also into Onsager’s final formula (46) for polar liquids:

\[ \frac{(\varepsilon-n^2)(2\varepsilon+n^2)} {\varepsilon(n^2+2)^2} = \frac{4\pi N\mu_0^2}{9kT} \left[1-L^2\left(\frac{\mu F}{kT}\right)\right]. \tag{91} \]

\(^1\) Despite the fact that Debye’s last works on this question were published after the appearance of Onsager’s theory.

Combining the Onsager theory with Debye’s local-field theory encounters the following difficulty. The correction for the local field leads, as we already know, to a decrease in the calculated value of the polarization. This decrease improved agreement with experiment in the case of the old theory, which gave an exaggerated value of the polarization. Onsager’s theory, however, in the case of water gives an underestimated value of $\varepsilon$; if Debye’s correction is added to it, the discrepancy between the theoretical results and the experimental data becomes still greater.

In order to obtain better agreement with the experimental data, Piekara$^{13}$ attempted to supplement Debye’s theory in the following way: in addition to the local field $F$, equally probably oriented in space and constant in magnitude, he introduces an additional field of the very nearest neighbor of the molecule under consideration.

The fundamental difference between this local field of the second kind and Debye’s field consists in the fact that it is determined in magnitude and direction by the orientation of the neighboring molecule, which in this respect is regarded as equivalent to the molecule under consideration.

Piekara’s theory is a mathematical formulation of the above-mentioned hypothesis concerning the joining of dipoles in a liquid into pairs. However, if in the case of a gas it might still make sense to take special account of the field of the “neighboring” molecule with which the molecule under consideration is at that moment in the process of collision, then in the case of a liquid, where all molecules are packed closely and all nearest neighbors of the molecule are more or less equivalent, there is no sense in singling out the field of the very nearest of them.

Piekara’s assertion that the field of the nearest molecule plays a greater role than the local field created by the remaining neighbors is without any foundation; in reality the field of the neighboring molecule must be included as part of the local field.

III. KIRKWOOD THEORY

§ 8. Presentation of Kirkwood’s Theory

As was already noted above, the chief shortcomings of Onsager’s theory are the following.

1. Onsager uses an explicitly artificial model of the molecule. Although in some cases artificial models do give correct results, it nevertheless seems desirable to construct a theory in a form that does not depend on any special assumptions about the structure of the molecule.

2. It is assumed that the dielectric constant of the medium retains an unchanged value right up to the molecule itself, i.e. the immediate environment of the molecule is regarded as a continuous medium with a macroscopic dielectric constant.

3. The connection of a molecule with its neighbors is taken into account only by the reactive field $R$, which, being parallel to the direction of the dipole moment of the molecule under consideration, in no way hinders its rotation.

Theory of Polarization of Dielectrics

In 1939 Kirkwood2 published a theory in which the indicated shortcomings of Onsager’s theory are eliminated.

Instead of a single molecule, following Kirkwood, let us single out a certain small specimen of dielectric \(B\), which for simplicity we shall represent as a sphere of radius \(r_0\). If \(r_0\) is sufficiently large in comparison with the distances between neighboring molecules, then one may with full justification regard the medium outside this sphere as continuous and possessing the macroscopic dielectric constant \(\varepsilon\).

Let us take into consideration some one molecule of specimen \(B\), for example the central one, and suppose that the direction of its moment is given. Then the remaining molecules, under the influence of the interaction both with this molecule and with one another, become partially oriented, so that the total moment of the specimen \(\mathfrak{M}\) proves to be different from the dipole moment \(\vec{\mu}\) of the molecule under consideration.

We shall present the mathematical exposition of Kirkwood’s theory in a somewhat simplified form, for the time being adding nothing new to it.

The mean electric moment of a molecule in the presence of a field \(E\) may be expanded in powers of this field. Taking into account both the orientational and the elastic part of the polarization and retaining only the first term, i.e., neglecting saturation, we obtain

\[ p=\bar{\mu}_E+aF, \tag{92} \]

where \(\bar{\mu}_E\) is the mean projection of the intrinsic moment of the molecule on the direction of the field, and \(F\) is the mean effective field in the region occupied by the molecule.

To calculate \(\bar{\mu}_E\), consider a large dielectric sphere \(A\) of volume \(v\) and radius \(R\), consisting of \(N\) molecules and placed in a homogeneous external field \(E_0\). This sphere should not be confused with the sphere \(B\) mentally singled out by us from the dielectric. The latter is cut out somewhere inside the sphere \(A\), which represents the whole dielectric. One of the molecules, say the \(i\)-th, owing to the presence in it of an intrinsic dipole moment \(\vec{\mu}_i\), induces in the sphere \(A\) a certain dipole moment \(M_i\). The total dipole moment of the dielectric is the sum of the dipole moments of all its molecules,

\[ \mathbf{M}=\sum_{k=1}^{N}\vec{\mu}_k . \tag{93} \]

In view of the linearity of the mutual-polarization effect, one may also write

\[ \mathbf{M}=\sum_{k=1}^{N}\mathbf{M}_k . \tag{94} \]

Owing to the presence of this dipole moment, the sphere \(A\) possesses in the external field the potential energy \(-\mathbf{M}\cdot\mathbf{E}_0\). To obtain its total energy, one must add to this the potential energy of the intermolecular forces \(V_N\) (including also the purely dipolar

interaction of the molecules). The mean value of the projection \(\bar{\mu}_{iE}\) of the dipole moment of the \(i\)-th molecule onto the direction \(\mathbf E\) can then be computed by the Gibbs formula:

\[ \bar{\mu}_{iE}= \frac{\displaystyle \int\cdots\int \mu_{iE} e^{-\frac{V_N-ME_0}{kT}}\,d\tau_1\ldots d\tau_N} {\displaystyle \int\cdots\int e^{-\frac{V_N-ME_0}{kT}}\,d\tau_1\ldots d\tau_N}, \tag{95} \]

where \(d\tau_k\) is an element of the configurational space of the \(k\)-th molecule (characterizing both its position and its orientation).

Expanding the integrand in (95) in powers of \(E_0\), we obtain, in the first approximation,

\[ \bar{\mu}_{iE}=\frac{1}{kT}\,\overline{\mu_{iE}M_E}\,E_0+O(E_0^2), \tag{96} \]

where the average on the right-hand side refers to the case of absence of the field \(E_0\). Whereas the mean value of the projection of the dipole moment onto the direction \(\mathbf E\) in the absence of the field is zero, the mean value of the product of it by the projection of the moment of the whole specimen \(A\) is different from zero. Indeed, according to (94), the total moment of the dielectric includes the moment \(M_i\), induced by the \(i\)-th molecule. In an isotropic body \(M_i\) always coincides in direction with \(\mu_i\) and, consequently, forms with the direction \(\mathbf E\) the same angle \(\theta_i\). Hence, in view of the equal probability of all directions of \(\vec{\mu}_i\) for \(E=0\), we have:

\[ \overline{\mu_{iE}M_{iE}}=\mu_i\,\overline{M_i\cos^2\theta_i} =\frac{1}{3}\mu_i\overline{M_i}, \]

where \(\overline{M_i}\) denotes the mean value of the moment \(M_i\), calculated with allowance for the energy of molecular interaction \(V\) by the Gibbs formula:

\[ \overline{M_i}= \frac{\displaystyle \int\cdots\int M_i e^{-\frac{V}{kT}}\,d\tau_1\cdot d\tau_N} {\displaystyle \int\cdots\int e^{-\frac{V}{kT}}\,d\tau_1\ldots d\tau_N}. \]

On the other hand, for \(k\ne i\), \(\overline{\mu_{iE}M_{kE}}=0\). On this basis, omitting the subscript \(i\) (since \(\overline{M_i}\) in a spherical specimen does not depend on the position and orientation of the molecule and, consequently, all \(\overline{M_i}\), as well as \(\mu_i\), are the same for all molecules of the specimen), we may write:

\[ \bar{\mu}_E=\frac{\mu\overline{M}}{3kT}\,E_0. \tag{97} \]

The field \(E\) inside a spherical dielectric is expressed in terms of the external field \(E_0\) by the formula \(E=\dfrac{3}{\varepsilon+2}E_0\), analogous to the Lorentz formula. Thus, the preceding formula can be rewritten in the form

\[ \bar{\mu}_E=\frac{\varepsilon+2}{3}\,\frac{\mu\overline{M}}{3kT}\,E. \tag{98} \]

Substitution of (98) into (92) gives

\[ \frac{\varepsilon-1}{\varepsilon+2} = \frac{4\pi}{3}N\left(\frac{\bar{\mu}M}{3kT} + \frac{3}{\varepsilon+2}\alpha\frac{F}{E}\right). \tag{99} \]

The total moment \(\bar{M}\) (orientational), induced in the spherical dielectric \(A\) of one of its central molecules, is partly due to the charges separated on the surface of the sphere as a result of its polarization. The depolarizing field created by these charges is analogous to the demagnetizing field in the case of ferromagnetic bodies. If the surface of the sphere is moved off to infinity \((R\to\infty)\), then the depolarizing field in any finite (or infinitely low-order) region \(r_0\left(\dfrac{R}{r_0}\to\infty\right)\) tends to zero. Under these conditions the moment \(\bar{M}\) of the sphere \(A\) reduces to the sum of the moment \(\mathfrak{M}(R,r_0)\) of the central spherical region \(B\) of radius \(r_0\), immersed in a continuous medium with dielectric constant \(\varepsilon\), and the moment due to the polarization of this medium:

\[ \bar{M}=\mathfrak{M}(R,r_0)+\int \mathbf{P}\,dv. \tag{100} \]

But the polarization of the dielectric is equal to

\[ \mathbf{P}=\frac{\varepsilon-1}{4\pi}\nabla\psi_i, \tag{101} \]

where \(\psi_i\) is the electric potential inside the sphere \(A\). Substituting (101) into (100) and replacing integration over the volume enclosed between the spheres \(A\) and \(B\) by integration over the surfaces of these spheres (according to Green’s theorem), we obtain

\[ \bar{M} = \mathfrak{M}(R,r_0) - \frac{\varepsilon-1}{4\pi}\int_R \psi_i\,\mathbf{n}\,ds + \frac{\varepsilon-1}{4\pi}\int_{r_0}\psi_i\,\mathbf{n}\,ds, \tag{102} \]

where \(\mathbf{n}\) is the vector of the outward normal.

For the potential we have the following expression:

\[ \psi_i = -\frac{\vec{\mu}^{*}}{\varepsilon}\cdot\nabla\frac{1}{|\mathbf{r}-\mathbf{r}^{1}|} + \chi \quad (\text{for } r>r_0). \tag{103} \]

Here \(\vec{\mu}^{*}\) denotes the “external” moment of the region \(B\) with center at the point \(\mathbf{r}^{1}\), where the fixed molecule is located, while \(\chi\) is the solution of Laplace’s equation without singular points:

\[ \chi=Br\cos\vartheta, \tag{104} \]

corresponding to the homogeneous field \(B\).

Let us note that

\[ -\frac{\mu^{*}}{\varepsilon}\nabla\frac{1}{|\mathbf{r}-\mathbf{r}^{1}|} \simeq \frac{\mu^{*}}{\varepsilon r^{2}}\cos\vartheta. \]

The electric potential outside the sphere \(A\) is equal to

\[ \psi_e=\frac{\bar{M}}{r^{2}}\cos\vartheta. \]

Here \(\vartheta\) is the polar angle with respect to the polarization axis. The boundary conditions of continuity of \(\psi\) and of its normal derivative on the surface of the sphere \(A\), i.e., at \(r=R\),

\[ \psi_i=\psi_e,\qquad \varepsilon \frac{\partial \psi_i}{\partial r}\bigg|_{r=R-0} = \frac{\partial \psi_e}{\partial r}\bigg|_{r=R+0} \]

give:

\[ \frac{\mu^*}{\varepsilon}+BR^3=\overline{M};\qquad -\frac{2\mu^*}{\varepsilon}+BR^3=-\frac{2\overline{M}}{\varepsilon}, \]

whence

\[ \overline{M}=\frac{3}{\varepsilon+2}\mu^*;\qquad BR^3=\frac{2}{\varepsilon}\,\frac{\varepsilon-1}{\varepsilon+2}\,\mu^*. \tag{105} \]

On the other hand, substituting (103) and (104) into (102), we have

\[ \overline{M}=\mathfrak{M}(R,r_0) -\frac{2}{3}\frac{\varepsilon-1}{\varepsilon+2}\mu^* +O\left(\frac{r_0^3}{R^3}\right). \tag{106} \]

Eliminating \(\mu^*\) from these two equations and putting \(\lim R\to\infty\) (or \(R/r_0\to\infty\)), we obtain

\[ \overline{M}=M_\infty = \frac{3}{\varepsilon+2}\cdot \frac{3\varepsilon}{2\varepsilon+1}\,\mathfrak{M}. \tag{107} \]

Let us recall that the vector \(\mathfrak{M}\) represents the total moment of the macroscopic spherical sample \(B\), due to one of its molecules, when the sample is immersed in an unbounded medium \((R=\infty)\) with the same value of \(\varepsilon\) (the macroscopic dielectric constant).

We shall now take elastic polarization into account. Considering it relatively small, Kirkwood identifies the effective field \(F\) in (92) with Onsager’s “cavity” field, i.e., puts

\[ F=\frac{3\varepsilon}{2\varepsilon+1}\,E. \]

Thus, according to (10), (92), (97), and (107), we obtain

\[ \left. \begin{aligned} \frac{\varepsilon-1}{3} &= \frac{3\varepsilon}{2\varepsilon+1}\,P_0;\\ P_0 &= \frac{4\pi}{3}N\left(a+\frac{\mu\mathfrak{M}}{3kT}\right). \end{aligned} \right\} \tag{108} \]

For \(\varepsilon\gg 1\) this formula may be replaced by the approximate one

\[ \varepsilon-1=\frac{9}{2}P_0. \tag{109} \]

As in Onsager’s theory, here the effective dipole moment of a molecule in a liquid, \(\mu\), differs from the intrinsic dipole moment of the molecule, \(\mu_0\) (determined from the polarization of the same substance in the gaseous state), owing to the elastic polarization caused by the reactive field. If the molecule is repre-

THEORY OF POLARIZATION OF DIELECTRICS

viewed, following Onsager, as a complete sphere with a point dipole at the center and dielectric constant \(\varepsilon=1\), then the relation between \(\mu\) and \(\mu_0\) is given by expression (37). The same expression is obtained if the molecule is treated as a sphere with dielectric constant \(\varepsilon_0=n^2\).

It remains now to determine the moment of the macroscopic sample \(\mathfrak{M}\).

If in formula (98) one puts \(M=\vec{\mu}\), i.e., completely neglects the interactions between molecules, then Kirkwood’s theory reduces to the old Debye theory.

If, on the other hand, \(\mathfrak{M}\) is identified with \(\vec{\mu}^{\,*}\), then Kirkwood’s formula (108) passes over into Onsager’s formula. Physically, such an identification means that only the dipole interaction between molecules is taken into account and that the domain of the macroscopic \(\varepsilon\) is assumed to extend up to the molecular surface; consequently, a single molecule is chosen as the sample \(B\).

§ 9. Analysis of the Local Field in Kirkwood’s Theory. Calculation of the Dielectric Constant of Water

In the general case the product \(\mu\mathfrak{M}\) can be determined by the formula

\[ \mu\mathfrak{M} = \mu^2 \left[ 1+ \frac{N}{\mathfrak{v}} \iint_{v_0} \cos\gamma e^{-\frac{W}{kT}}\,d\omega\,dv \right] \tag{110} \]

under the condition that

\[ \iint e^{-\frac{W}{kT}}\,d\omega\,dv=1 \]

and that outside the spherical sample under consideration, with volume \(v\), the dielectric constant is equal to the sought value \(\varepsilon\). Here \(\gamma\) is the angle between the dipole moment of some molecule (contained in the solid angle \(d\omega\)) and the moment of the given molecule; \(W\) is the potential energy of the orientational interaction of these two molecules.

If it is assumed that the volume \(v\) contains only the molecule under consideration and its \(z\) nearest neighbors, then

\[ \left. \begin{aligned} \mu\mathfrak{M} &\simeq \mu^2\left(1+z\overline{\cos\gamma}\right);\\ \overline{\cos\gamma} &= \iint \cos\gamma e^{-\frac{W_0}{kT}}\,d\omega\,dv, \end{aligned} \right\} \tag{111} \]

where \(W_0\) is the potential energy of interaction between two neighboring molecules.

The quantity \(\overline{\cos\gamma}\) measures the correlation between the orientations of neighboring molecules, expressed in hindered rotation or in the replacement of free rotation by rotational oscillations.

Formula (109) then becomes the following:

\[ \varepsilon - 1 = 6\pi N \left[ \alpha + \frac{\mu^2}{3kT}\left(1+z\,\overline{\cos \gamma}\right) \right]. \tag{112} \]

This formula makes it possible to take into account any kind of correlation between the orientations of the molecules of the dielectric, i.e., also nondipole forces. This circumstance is an advantage of Kirkwood’s theory in comparison with preceding theories.

Kirkwood, however, asserts that when only dipole forces are taken into account his theory reduces to Onsager’s theory. The incorrectness of this assertion follows from the latest work of Kirkwood himself[^15]. In this work Kirkwood calculates \(\overline{\cos \gamma}\) taking into account only the dipole interaction between molecules and obtains, for an even number \(z\) of nearest neighbors,

\[ \overline{\cos \gamma} = -s^2, \]

where \(s\) is the solution of the equation \(s = L\left(\dfrac{z\eta s}{2kT}\right)\), and \(\eta\) denotes the height of the potential barrier when two molecules are rotated relative to one another.

If this value of \(\overline{\cos \gamma}\) is substituted into formula (112), it takes the form:

\[ \varepsilon - 1 = 6\pi N \left\{ \alpha + \frac{\mu^2}{3kT}\left[1 - L^2\left(\frac{z\eta}{2kT}s\right)\right]\right\}, \tag{113} \]

which is similar to our formula (88), combining the theories of Onsager and Debye. If, however, one takes into account here that

\[ \varepsilon - 1 = 4\pi P \quad \text{and} \quad \frac{3\varepsilon}{2\varepsilon+1} \simeq \frac{3}{2}, \]

then a discrepancy is found in the term representing elastic polarization. It is explained by the fact that in Kirkwood’s theory this polarization is taken into account incompletely—namely, by identifying the effective field \(F\) with the cavity field \(G\), whereas in fact it is also caused by the reactive field \(R\).

The terms expressing orientational polarization in the two formulas, however, prove to be identical if one sets

\[ 2\mu F = z\eta s, \]

where \(F\) denotes the local Debye field. The assumption of this equality is more or less natural, since the difference between the limiting values of the potential energy of a dipole in the local field, i.e., the energy of its interaction with its neighbors, \(2\mu F\), must, in order of magnitude, coincide with the product of the number of neighbors by the height of the potential barrier in the interaction of two molecules and by the mean value \(\cos \gamma\).

Thus, Kirkwood’s theory, when only the dipole interaction of neighboring molecules is taken into account, reduces to Onsager’s theory with a correction for the local Debye field.

Let us note that Kirkwood’s latest work concerns crystals, but since it deals with the interaction of neighbors, all the forego-

may be applied in the present paragraph with the same justification both to crystals and to liquids.

The combined Onsager–Debye theory may therefore be interpreted as a special case of Kirkwood’s theory. If in one case the local field is introduced, then in the other the height of the potential barrier \(q\) is introduced. Both these quantities can be determined only experimentally. In this respect the two theories are equivalent. However, Kirkwood’s theory is in principle more general and more precise; in particular, it gives a value for the dielectric constant of water that is considerably closer to the experimental data.

In calculating the latter as an example illustrating his theory, Kirkwood, however, does not use the statistical expression for \(\overline{\cos\gamma}\) given by formula (111), but instead assumes that neighboring molecules form a quasi-solid structure, i.e., that the angle \(\gamma\) between the dipoles is fixed once and for all.

Assuming free rotation about neighboring O—H bonds, Kirkwood takes
\[ \cos\gamma=\cos^2 50^\circ\simeq 0.41. \]
The number of nearest neighbors of a water molecule is \(z=4\). Substituting this value in (111), we obtain \(\mu\mathfrak{M}=2.64\,\mu^2\). The four neighbors, under free rotation around OH, create a field equal on the average to
\[ E'=8\cos^2 50^\circ\,\frac{\mu}{a^3} \]
and directed parallel to \(\vec{\mu}_0\). Thus, the total moment is equal to
\[ \vec{\mu}=\vec{\mu}_0+\alpha E', \]
i.e., consequently,
\[ \mu=\frac{\mu_0}{1-3.28\,\dfrac{\alpha}{a^3}}. \]

For \(\alpha=1.5\,\text{\AA}^3\) and molecular radius \(a=3.27\,\text{\AA}\), we obtain \(\mu=1.16\,\mu_0\) and \(\mu\mathfrak{M}=3.55\,\mu_0^2\). For \(\mu_0=1.98\cdot 10^{-18}\) we obtain the following value for the polarization of water:
\[ P_0=3.8+\frac{7.74\cdot 10^4}{T}, \tag{114} \]
which at \(25^\circ\) gives \(\varepsilon=67\). This value is much closer to the experimental one than the result given by the Onsager theory \((\varepsilon=31)\).

If the angle between two O—H bonds in the water molecule is taken to be not 100 but \(90^\circ\), then \(\cos\gamma=0.5\) and \(\varepsilon=82\), which agrees with the experimental data.

Nevertheless, the calculation of \(\varepsilon\) for water given here cannot be considered entirely satisfactory. It is difficult to imagine in what way a molecule rotates while remaining rigidly bound to its neighbors. Each neighbor must in turn be rigidly bound to neighboring molecules, and so on. It would be more correct, instead of a rigid bond, to introduce the potential energy of interaction between neighboring molecules, which is an appropriate function of the angle \(\gamma\).

It is, however, impossible to specify this function until the exact character of the interaction forces is known.

IV. EXTENSION OF THE THEORY TO MORE COMPLEX SYSTEMS1

§ 10. Elastic polarization of a liquid dielectric with anisotropic molecules

The calculation of the polarization of dielectrics set forth above in § 1 was carried out under the assumption that the molecules are isotropic and are characterized by a scalar polarizability \(a\). In this case, in the absence of an intrinsic moment of the molecule, the direction of its induced dipole moment always coincides with the direction of the mean field, and the effective field acting on the molecule may be identified with the Lorentz field. Onsager’s theory gives nothing new here.

A completely different picture is obtained in the case of anisotropic molecules, whose polarizability is characterized not by a scalar \(a\), but by a symmetric tensor with components \(a_{ij}\).

Let us take the system of principal axes \(x_i\) of the polarizability tensor, fixed to the molecule, with versors \(a_i\). The component of the induced moment of the molecule along one of these axes is equal to

\[ p_i = a_i \beta_i E_e, \tag{115} \]

and the moment itself is

\[ \mathbf{p} = \sum_i \mathbf{a}_i a_i \beta_i E_e, \tag{116} \]

where \(a_i\) are the components of the polarizability tensor in the principal system of axes, and \(\beta_i\) are the direction cosines of \(\mathbf{E}_e\) in the same system.

In the general case \(\mathbf{p}\) does not coincide in direction with \(\mathbf{E}_e\); therefore the field exerts on the molecule, in addition to the elastic effect, also an orienting effect. The torque \(\mathbf{M} = [\mathbf{p}, \mathbf{E}_e]\) is equal to zero only in the case when \(\mathbf{E}_e\) coincides in direction with one of the principal axes of polarization of the molecule, and this equilibrium position is stable only when the axis of greatest polarizability is established parallel to the field.

Thus, if no forces other than the electric field act on the molecules, and if the thermal rotation of the molecules may be neglected, they are oriented so that the axis of greatest polarizability \(a_i\) is parallel to the mean field \(\mathbf{E}\). In the presence of thermal motion this orientation is disturbed and, as always, the distribution of molecular orientations is established according to Boltzmann’s law \(f = e^{-W/kT}\), where \(W\) is the orientational potential energy of the molecule.

The same Onsager considerations apply to this case of orientation as to the orientation of molecules with permanent dipoles.

The orienting action is exerted not by the Lorentz field \(\mathbf{E} + \dfrac{4\pi}{3}\mathbf{P}\), but by the cavity field

\[ \mathbf{G} = \frac{3\varepsilon}{2\varepsilon + 1}\mathbf{E} \]

and the reactive field \(\mathbf{R}\), which in the case—

... isotropic bodies (liquids) parallel to the resultant moment \(\mathbf p\). It should be noted that the latter assertion is not quite exact. The presence of the mean field \(\mathbf E\) causes a certain orientation of the molecules; therefore, with respect to the additionally applied field, the dipole + dielectric is not completely isotropic, i.e., the reaction field \(\mathbf R\), generally speaking, does not exactly coincide in direction with \(\mathbf p\). However, in the case of isotropic bodies (liquids) this effect is of higher order with respect to \(E\), and in the first approximation we may neglect it.

The polarizing field is the total effective field

\[ \mathbf E_e=\mathbf G+\mathbf R. \]

In view of the anisotropy of the molecule, the dipole moment produced by the field \(\mathbf R\) does not coincide with it in direction; therefore the influence of the reaction field cannot be reduced to an increase of the orienting moment, as in Onsager’s theory (replacement of \(\mu_0\) by \(\mu\)); rather, the following equation must be solved simultaneously with (115):

\[ \mathbf E_e=\frac{3\varepsilon}{2\varepsilon+1}\mathbf E+ \frac{2(\varepsilon-1)}{(2\varepsilon+1)a^3}\mathbf p \tag{117} \]

for some fixed orientation of the molecule, specified by the direction cosines \(\gamma_i\) of the field \(\mathbf E\) in the system of axes \(x_i\).

Projecting (117) onto the axis \(x_i\), multiplying by \(\alpha_i\), and substituting expression (115), we obtain

\[ p_i=\frac{3\varepsilon}{2\varepsilon+1}E\gamma_i\alpha_i+ \frac{2(\varepsilon-1)}{(2\varepsilon+1)a^3}\alpha_i p_i, \]

whence

\[ p_i=\alpha_i' E\gamma_i, \tag{118} \]

where the notation has been introduced

\[ \alpha_i'=\frac{3\varepsilon \alpha_i a^3} {(2\varepsilon+1)a^3-2(\varepsilon-1)\alpha_i}. \tag{119} \]

We shall call \(\alpha_i'\) the reduced components of the polarizability tensor.

Further,

\[ W=-\mathbf G\cdot\mathbf p =\frac{3\varepsilon}{2\varepsilon+1}E^2\sum_i \alpha_i'\gamma_i^2. \tag{120} \]

The projection of the dipole moment onto the direction of the field \(\mathbf E\) is equal to

\[ p_E=\sum_i p_i\gamma_i =E\sum_i \alpha_i'\gamma_i^2, \tag{121} \]

and the statistical average of this projection according to Boltzmann’s law is

\[ \overline{p}_E= \frac{ \displaystyle \int \sum_i \alpha_i'\gamma_i^2 e^{A\sum_i \alpha_i'\gamma_i^2}\,d\Omega }{ \displaystyle \int e^{A\sum_i \alpha_i'\gamma_i^2}\,d\Omega }E, \tag{122} \]

where

\[ A=\frac{3\varepsilon}{2\varepsilon+1}\frac{E^2}{kT} \tag{123} \]

and \(d\Omega\) is the element of solid angle.

For simplicity we shall consider the case when the polarizability tensor has rotational symmetry, i.e., \(\alpha_2=\alpha_3\). Then, for

taking into account that \(\sum \gamma_i^2=1\), we can rewrite expression (122) as follows:

\[ \bar p_E= \frac{ \displaystyle\int_{-1}^{1} \left[\alpha_1' \gamma_1^2+\alpha_2'(1-\gamma_1^2)\right] e^{A\left[\alpha_1'\gamma_1^2+\alpha_2'(1-\gamma_1^2)\right]}\,d\gamma_1 }{ \displaystyle\int_{-1}^{1} e^{A\left[\alpha_1'\gamma_1^2+\alpha_2'(1-\gamma_1^2)\right]}\,d\gamma_1 }\,E . \tag{124} \]

The exponent of the integrand is very small in comparison with unity; therefore it may be expanded in a series and one may confine oneself to one or two terms.

Let us take the zeroth approximation, replacing the exponential function by unity, i.e. completely neglecting the orientation of the molecules and assuming that the axes \(x_i\) of the different polarizabilities of the molecules are uniformly distributed in all directions. In this case:

\[ \bar p_E=\frac{E}{2} \int_{-1}^{1} \left[\alpha_1'\gamma_1^2+\alpha_2'(1-\gamma_1^2)\right]d\gamma_1 = \frac{E}{3}(\alpha_1'+2\alpha_2'). \tag{125} \]

In the first approximation, accurate up to quantities of order \(A\), we have

\[ \bar p_E= \frac{ \displaystyle\int_{-1}^{1} \left[\alpha_1'\gamma_1^2+\alpha_2'(1-\gamma_1^2)\right] \left\{1+A\left[\alpha_1'\gamma_1^2+\alpha_2'(1-\gamma_1^2)\right]\right\}\,d\gamma_1 }{ \displaystyle\int_{-1}^{1} \left\{1+A\left[\alpha_1'\gamma_1^2+\alpha_2'(1-\gamma_1^2)\right]\right\}\,d\gamma_1 }\,E . \]

After simple calculations we obtain

\[ \bar p_E \simeq \frac{E}{3} \left[ \alpha_1'+2\alpha_2' +\frac{4A}{15}(\alpha_1'-\alpha_2')^2 \right] \]

or, substituting here (119) and (123),

\[ \bar p_E= \frac{E}{3} \left\{ \frac{3\varepsilon a^3\alpha_1}{(2\varepsilon+1)a^3-2(\varepsilon-1)\alpha_1} + \frac{6\varepsilon a^3\alpha_2}{(2\varepsilon+1)a^3-2(\varepsilon-1)\alpha_2} + \frac{ 36E^2\varepsilon^3 a^{12}(2\varepsilon+1)(\alpha_1-\alpha_2)^2 }{ 5kT\left[(2\varepsilon+1)a^3-2(\varepsilon-1)\alpha_1\right]^2 \left[(2\varepsilon+1)a^3-2(\varepsilon-1)\alpha_2\right]^2 } \right\}. \tag{126} \]

The third term on the right-hand side of this equality takes account of the orientation effect. Hence it is seen that orientation increases the elastic polarization.

If we substitute (126) into (10), we obtain an equation of the fifth degree for determining the dielectric constant \(\varepsilon\), which in the present case depends on the temperature and on the field strength \(E\), in contrast to the elastic polarization of isotropic molecules.

Let us note that the simplified Debye theory presented in § 7 for very strong local fields is, in its results, completely equivalent to the theory of elastic polarization of dielectrics with anisotropic molecules (in the zeroth approximation).

Indeed, an external field parallel to the local field \(F\) gives no polarization of the molecule, whereas one perpendicular to it forces the molecule to deviate elastically from the direction of the local field, which is equivalent to a certain elastic polarization. If in formula (125) one sets \(\alpha'_1 = 0\) and \(\alpha'_2 = \dfrac{\mu}{F}\), then it becomes formula (85). Thus, a molecule possessing a permanent dipole moment \(\mu\) and situated in a strong local field \(F\) may be treated as having an anisotropic elastic polarizability with reduced components in the direction of the local field \(\alpha'_1 = 0\) and in the perpendicular directions

\[ \alpha'_2 = \frac{\mu}{F}. \]

The local field is oriented with equal probability in all directions, just as are the axes of different polarizability of molecules in the zeroth approximation.

§ 11. Orientational Melting in Crystals

All the polarization theories set forth above apply chiefly to liquid dielectrics, since they proceed from the assumption that: 1) the medium is isotropic, 2) the molecules are arranged and oriented chaotically, 3) the dipoles may rotate completely freely or are elastically bound by a local field oriented with equal probability in all directions. If, in Kirkwood’s theory, when calculating the dielectric constant of water, a rigid order is assumed in the arrangement of the neighboring molecules of a given molecule, this corresponds only to taking into account the short-range order characteristic of liquids.

For crystalline bodies, by contrast, long-range order in the arrangement and orientation of molecules is primarily characteristic. At absolute zero temperature each dipole in a crystal has a strictly definite orientation, and the orientation of the different dipoles alternates throughout the crystal in a definite order.

As was already noted by Ya. I. Frenkel\(^ {16}\), the order of alternation of dipole orientations may be such that the total moment of the crystal is equal to zero (a nonpolar type of orientation) or is not equal to zero (a polar type of orientation).

Each type of orientation has a minimum of potential energy in comparison with neighboring orientations.

For \(T > 0\), the dipoles perform rotational oscillations about equilibrium orientations of one type or another. In order to judge which type of orientation actually occurs, one must compare the free energies of the different types of orientation. It then turns out that at some temperatures the polar form may be more stable, at others the nonpolar form (depending on the structure of the crystal).

As the temperature is raised, the molecules may pass from rotational oscillations about definite equilibrium orientations to free rotation. This transition does not take place at once,

but in a certain temperature interval. In the work of Frenkel, Izmailov, and Todes^17 it was shown that, proceeding from Pauling’s ideas on the free rotation of dipolar molecules in crystals of the HCl type at high temperatures, one can determine the temperature of transition from rotational oscillations to free rotation if one assumes that the transition under consideration occurs at a quite definite temperature, and not in some temperature interval, as is actually the case.

The transition temperature may be calculated from the condition of equality of the free energies of the two states.

Fowler^18, basing himself on Pauling’s ideas, attempted to give a quantitative theory of the transition, taking into account the circumstance that it occurs in a finite temperature interval. In doing so, Fowler divides the molecules into rotating and oscillating ones and calculates the relative number \(1-s\) of rotating molecules as a function of temperature. The basis of this calculation is the intuitive notion that only those molecules can rotate freely whose kinetic energy of rotation \(W\) is greater than a certain potential barrier \(U\), which in turn is proportional to the relative number \(s\) of non-rotating molecules (since only these molecules create the field that impedes rotation). Starting from these considerations, Fowler arrives at the formula

\[ 1-s=e^{-\beta \frac{W_0 s}{kT}}, \]

where \(\beta\) denotes a certain empirical coefficient which substantially affects the result. By a proper choice of \(\beta\) one can ensure the existence of a “critical temperature” \(T_{\mathrm{cr}}\), above which \(s\) becomes zero.

An analogous result was obtained by Ya. I. Frenkel^19 in a simpler and more rigorous way, free both from Pauling’s ideas on free rotation and from any empirical coefficients. Here it is not the kinetic energy that is considered, but the potential energy of a molecule as a function of the angle \(\theta\) formed by its axis with the equilibrium orientation.

If the thermal oscillations of the molecules could be neglected, their moments would be established along the direction of the internal field, which would then have a quite definite value, \(E_0\). When the thermal oscillations of the molecules are taken into account, however, the orienting field \(E\) acting on a given molecule of the crystal in the equilibrium direction from the other molecules has no constant value, since the orienting field created by each of them is proportional to the mean value of the cosine of the angle of inclination of its dipole moment to the corresponding normal direction.

Taking into account that the mean value of this cosine is the same for all molecules and denoting it by \(\xi\), we arrive at the following expression for \(E\):

\[ E=E_0\xi . \tag{127} \]

On the other hand, by the Langevin formula we have

\[ P=P_0L\left(\frac{pE}{kT}\right). \]

We thus obtain the equation

\[ \xi=L(a_0\xi), \tag{128} \]

which can serve to determine the dependence of \(\xi\) (the degree of orientation) on temperature; here

\[ a_0=\frac{pE_0}{kT}. \tag{129} \]

The critical temperature, at which the quantity \(\xi\) goes to zero, is expressed by the formula

\[ T_{\mathrm{cr}}=\frac{pE_0}{3kT} \tag{130} \]

analogously to the Curie temperature for ferromagnets.

The method set forth for determining \(\xi\) is not flawless, since it is based on the Boltzmann expression for the probability of orientation of a dipole in an external constant field, whereas in our case the effective field \(E=E_0\xi\) is the mean value of a rapidly oscillating variable field. This fact can be taken into account by describing the rotational part of the thermal motion by means of a superposition of a series of “rotational-vibrational” waves. We shall do this for the case of a one-dimensional infinite dipole lattice, which we shall imagine as a chain of equidistant molecules whose axes in the equilibrium state are directed in one and the same direction along the length of the chain. Let the distance between neighboring dipoles be denoted by \(a\). The potential energy of two neighboring molecules as a function of the angles \(\theta_1\) and \(\theta_2\) formed by their axes with the normal direction is expressed by the formula

\[ U_{12}=-\frac{4p^2}{a^3}+\frac{p^2}{a^3}\left(-2\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2\right). \tag{131} \]

For sufficiently small values of \(\theta_1\) and \(\theta_2\) it may be replaced by the quadratic expression

\[ U_{12}=-\frac{4p^2}{a^3}+\frac{p^2}{a^3}\left(\theta_1^2+\theta_2^2+\theta_1\theta_2\right). \tag{132} \]

Hence we obtain the following system of equations for \(\theta_n\), putting, for brevity,

\[ \frac{4p^2}{a^3J}=\mu^2: \]

\[ \frac{d^2\theta_n}{dt^2}=-\mu^2\left(\theta_n+\frac{\theta_{n+1}+\theta_{n-1}}{4}\right). \tag{133} \]

We seek the solution in the form of traveling waves

\[ \theta_n=Ae^{i(\alpha n-\omega t)}. \tag{134} \]

Substituting (134) into (133), we obtain

\[ \omega^2=\mu^2\left(1+\frac{1}{2}\cos\alpha\right). \tag{135} \]

These formulas also remain valid for a finite chain, if it is represented in the form of a closed polygon or cyclic chain. If the number of dipoles is equal to \(g\), then \(\alpha\) may take the values

\[ \alpha=\frac{2\pi r}{g}, \tag{136} \]

where \(r\) is an integer lying between \(\frac{g}{2}\) and \(-\frac{g}{2}\).

In this case

\[ \theta_n=\sum_r A_r e^{i(\alpha_r n-\omega_r t)}, \tag{137} \]

and

\[ U=\frac{p^2}{a^3}\left[\sum_n 2\theta_n^2-\frac{1}{2}\sum_n \theta_n(\theta_{n+1}+\theta_{n-1})\right]. \tag{138} \]

To determine the mean value of \(U\) over time, one must replace \(\theta_n^2\) and \(\theta_n(\theta_{n+1}+\theta_{n-1})\) by \(\theta_n^*\theta_n\) and \(\theta_n^*(\theta_{n+1}+\theta_{n-1})\), and divide in half. After simple calculations we obtain

\[ \overline{U}=\frac{p^2}{a^3}\sum_r A_r^*A_r\left(1-\frac{1}{2}\cos\alpha_r\right). \tag{139} \]

From this it is easy to calculate the mean statistical value of \(\theta_n^2\), namely, from (137) we have

\[ \overline{\theta_n^2}=\frac{a^3}{2p^2}\frac{kT}{g}\sum_r \frac{1}{2-\cos\alpha_r}. \tag{140} \]

Let us compare this formula with the one obtained from the simplified theory for the case of low temperatures. In this case

\[ a\gg 1,\qquad L(a)\simeq 1-\frac{1}{a}; \]

and from (128) \(\xi=1-\frac{kT}{pE_0\xi}\). Putting here \(\xi=\overline{\cos\theta}\simeq 1-\frac{\overline{\theta^2}}{2}\), we obtain

\[ \frac{1}{2}\overline{\theta^2}=-\frac{kT}{pE_0\left(1-\frac{\overline{\theta^2}}{2}\right)},\quad \text{i.e.} \]

\[ \overline{\theta^2}=-\frac{2kT}{pE_0\left(1-\frac{kT}{pE_0}\right)}\simeq \frac{2kT}{pE_0}. \tag{141} \]

If only the action of the nearest neighbors is taken into account, one must set

\[ E_0=-\frac{4p}{a^3}\quad \text{and}\quad \overline{\theta^2}=\frac{a^3}{2p^2}kT. \tag{142} \]

This result differs from (140) by the factor \(\frac{1}{g}\sum_r \frac{1}{2-\cos\alpha_r}\), which is only slightly less than unity. Consequently, the method of orientation of individual dipoles in the effective field \(E=E_0\xi\) is practically equivalent to the rigorous method of orientational waves.

It should be emphasized that the reduction of $\overline{\cos\theta}=\xi$ to zero by no means signifies the transition of the molecules to free rotation, as Fowler thought (following Pauling in this respect). The equality $\xi=0$ means only the disappearance of regularly distributed orientations, i.e., the liquidation of “long-range order” in their distribution over the lattice sites. At the same time, however, generally speaking, there must be preserved “short-range order,” expressed in the presence of irregularly distributed equilibrium orientations about which the molecules perform rotational oscillations and which correspond to Debye’s local field for liquids. This circumstance is evident from the fact that the heat capacity of the crystal above the transition point turns out to be not only no smaller, but even somewhat larger, than before it. The increased value of the heat capacity is explained by the gradual decrease of short-range order in the orientation of the molecules as one moves away from the transition point toward higher temperatures. If the Curie temperature meant a transition from rotational oscillations to free rotation, as Pauling thought, and also Fowler and Frenkel in their original work with Izmailov and Todes, then upon passing through the Curie point the heat capacity should have decreased by $2\ \mathrm{cal}/\mathrm{mol}$ as a consequence of the disappearance of the potential energy corresponding to the rotational oscillations.

In his last work Kirkwood$^{15}$, without mentioning the theory of Ya. I. Frenkel just set forth (with which he was evidently unfamiliar), obtained essentially the same equation (128), but in a much more complicated way. Kirkwood applied the result obtained to determine the heat capacity of a crystal associated with the rotation of dipoles, and to find the temperature dependence of its dielectric constant $\varepsilon$. We shall speak of this below.

The disappearance of the ordered orientation of dipoles at the critical temperature $T_{cr}$ may be interpreted as orientational melting, by analogy with ordinary melting, which is the disappearance of long-range order in the arrangement of the centers of gravity of the molecules.

§ 12. Polarization of crystals

In the preceding paragraph we did not consider the influence on the orientation of dipoles in a crystal of an external electric field, i.e., the phenomenon of polarization of dipolar crystals.

If one neglects the thermal rotational oscillations of the molecules, it is easy to give an approximate theory of this polarization. This can be done in one of two variants. Debye’s simplified theory (§ 7) can be extended to the case of crystals. In doing so, one should assume that not all directions of the local field are equally probable, but that it can have only several different directions. In particular, if all the dipoles are oriented parallel, but in opposite directions (a nonpolar form), then the number of possible directions of the local field is two. Let us calculate the components of the dielectric-constant tensor for crystals of this type, taking into account Onsager’s considerations.

According to formulas (65) and (85), replacing in the latter \(E\) by \(G\), we obtain, if the external field is applied parallel to the axes of the dipoles,

\[ \sin\theta=0 \]

and \(\bar{\mu}_E=0\); consequently, \(\varepsilon_1=0\). If the external field is applied perpendicular to the axes of the dipoles, then

\[ \sin\theta=1,\qquad \bar{\mu}_E=\mu \frac{E_e}{F}\frac{3\varepsilon_2}{2\varepsilon_2+1} \quad\text{and}\quad \varepsilon_2-1=\frac{4\pi}{3}N\frac{\mu}{F}\frac{3\varepsilon_2}{2\varepsilon_2+1} \]

or

\[ \frac{(\varepsilon_2-1)(2\varepsilon_2+1)}{\varepsilon_2} = 4\pi N\frac{\mu}{F}. \tag{143} \]

Let us calculate, analogously to this, the dielectric constant of solid HCl, assuming, following Born and Kornfeld, that this substance crystallizes in the cubic system, with the dipoles oriented along the body diagonals of the cube. In this case, if the field is applied parallel to one of the edges of the cube,

\[ \sin^2\theta=\frac{2}{3}, \]

and for the nonpolar form we obtain:

\[ \bar{\mu}_E=\frac{2}{3}\frac{\mu}{F}\cdot\frac{3\varepsilon}{2\varepsilon+1}\,E \]

and

\[ \frac{(\varepsilon-1)(\varepsilon+2)}{\varepsilon} = \frac{4\pi N}{3}\frac{\mu}{F}. \tag{144} \]

It is also possible to calculate \(\varepsilon\) on the basis of the conception of the elastic character of the polarization of the crystal, taking into account the anisotropy effect, as was shown in § 10. Recalling that the reduced component of the polarizability tensor of an individual molecule in the direction of the local field

\[ a'_1=0, \]

and in the direction perpendicular to it,

\[ a'_2=a'_3=\frac{\mu}{F}, \]

it is not difficult to calculate the components of the polarizability tensor for the whole crystal in the system of principal axes, by averaging over all possible directions of the local field in the crystal. The result of this calculation coincides with the preceding one.

The temperature correction in the first approximation is not difficult to carry out in the same way as was done in § 7. In this case the right-hand sides of formulas (143) and (144) are simply multiplied by

\[ 1-\frac{kT}{\mu F}. \]

With a more exact account of the thermal oscillations of the molecules, one should take into consideration the Onsager reaction field \(R\). In crystals the field \(R\), as was already mentioned above, does not coincide in direction with the dipole moment of the molecule and therefore exerts an orienting action on it. The calculation in this case becomes rather complicated, and therefore we do not give it here.

In the work already cited by us,\(^{15}\) Kirkwood carried out a calculation of the dielectric constant of crystals with a more rigorous account of temperature. He determined the mean degree of orientation of the molecules in a manner similar to that which we used in § 11, calculated the mean value \(\cos\gamma\)—the angle between two neighboring dipoles—and substituted this mean value into formula (112).

Let us note, however, that Kirkwood’s calculation makes sense only for cubic crystals, which are also the ones considered by Kirkwood, since formula (112) was derived under the assumption that the medium is macroscopically isotropic (at least with respect to the dielectric constant). Thus, a general and sufficiently consistent theory of the polarization of dipolar crystals has not yet been created.

Considering the analogous magnetic problem, Van Vleck\(^{20}\) performed a calculation of the magnetic susceptibility of paramagnets with allowance for the interaction of a system of regularly arranged and arbitrarily oriented dipoles. As Van Vleck himself points out, his theory can in principle also be applied to the electric polarization of dielectrics. It seems to us, however, that this application encounters considerable difficulties connected with the fact that the interaction energy of electric dipoles is much greater than the interaction energy of magnetic dipoles. Therefore the admissibility of expanding the distribution function in a series in powers of \(\dfrac{W}{kT}\), which makes the calculation possible, appears highly doubtful; here \(W\) is the energy of orientation of a dipole with respect to the others (for \(W > kT\), for example, this series converges very slowly).

Some crystals, while certainly not dipolar, nevertheless exhibit a number of polarization anomalies similar to those characteristic of dipolar substances (in the liquid state). Their dielectric constant turns out to depend on the frequency of the field; moreover, in the case of a constant field it is much greater than the square of the refractive index for visible rays. The value of \(\varepsilon\) sometimes reaches a very large magnitude, for example, in rutile (129).

These anomalies are explained by the fact that the crystal lattices of the corresponding substances have an ionic structure. Under the action of an electric field, the positive ions are displaced in one direction and the negative ions in the opposite direction, which gives rise to a considerable polarization of an elastic character, differing from ordinary electronic polarization in that it is due to the displacement of heavy charged particles (ions). Under the action of the electric field of light rays, the ions do not have time to be displaced appreciably; therefore light produces only the usual electronic polarization, which is small in comparison with the ionic one. Unlike orientational polarization, elastic ionic polarization does not depend on temperature\(^{21}\). In some crystals the ionic polarization is complicated by the piezoelectric effect. This occurs, for example, in the case of Rochelle salt, which will be discussed in the next paragraph.

§ 13. Ferroelectric Phenomena

The electrical properties of Rochelle salt \(\mathrm{NaKC_4H_4O_6\cdot 4H_2O}\) and of some other compounds of similar type possess a number of specific features.

In a certain temperature range (from 15 to 22.5°), Rochelle salt has, in weak fields, an anomalously high dielectric constant (reaching 200,000), becomes polarized to saturation under an insignificant increase in the field strength, and acquires the ability to retain a constant electric moment after the field is switched off. In an alternating field, Rochelle salt exhibits a broad hysteresis loop. As we already mentioned in § 2, the phenomenon of ferroelectricity is analogous to ferromagnetism.

However, ferroelectrics have one further peculiarity—the existence of a lower Curie point, i.e., a temperature below which the anomalies of polarization disappear. A sample of Rochelle salt polarized at a temperature lying in the interval between the two Curie points loses its permanent moment when cooled to the lower Curie point.

Anomalous polarizability in Rochelle salt is observed only under the action of that component of the electric field which is parallel to one of the crystallographic directions—the so-called electric axis a of the crystal. Closely connected with this anomalous polarizability are the anomalous—in magnitude—piezoelectric properties of crystals of this substance.

When a field is applied parallel to the electric axis, there occurs a slight skewing of the axes b and c perpendicular to it (the angle between them deviates slightly from a right angle). Conversely, when these axes are skewed by a mechanical force, a polarization P arises in the crystal in the direction of the electric axis. These direct and inverse piezoelectric effects are of great importance both in the theory of ferroelectrics and for technology.

Experimental data on the properties of ferroelectrics are given in Kurchatov’s book[^6]. It does not, however, mention a very important fact discovered later and of very substantial significance for the theory of ferroelectric phenomena. Namely, as Sawyer and Tower[^22], David[^23], and Mueller[^24] showed, if a crystal is clamped so as to prevent its deformation, then its dielectric properties change sharply in the sense of a narrowing of the interval between the Curie points, a decrease in the dielectric constant, and a narrowing of the hysteresis loop.

There exist various theories attempting to explain the properties of Rochelle salt. In § 2 we already mentioned Kurchatov’s old theory, based on the Lorentz field.

Another, less trivial, but also apparently incorrect explanation of ferroelectricity, likewise based on the idea of dipole interaction, was given by Fowler in the already cited paper[^18]. He explains the spontaneous character of the internal field orienting the dipoles in Rochelle salt by the anisotropy in the arrangement of the dipoles. Fowler advances the hypothesis that the polar molecules of the water of crystallization are arranged in chains along the electric axis of the crystal a. Under this condition the effective field is equal to \(E_e = E + \frac{4\pi}{3}P + \omega\), where \(\omega\) is a certain addition, due-

THEORY OF POLARIZATION OF DIELECTRICS

conditioned by anisotropy, and at sufficiently large \(\omega\) proves capable of orienting the dipoles independently of the shape of the dielectric and even in the absence of an external field. In calculating \(\omega\), Fowler proceeds from highly hypothetical ideas, hardly corresponding to reality, about the arrangement of water dipoles in Rochelle salt. A shortcoming of Fowler’s theory is also its complete neglect of the difference in the behavior of free and clamped crystals.

An indirect objection to the theory of dipole interaction is the discovery of ferroelectric properties in such substances as \( \mathrm{H}_2\mathrm{KPO}_4 \) and \( \mathrm{H}_2\mathrm{KAsO}_4 \), which contain no polar molecules at all\({}^{25}\). Thus it is quite possible that in the case of Rochelle salt the electric dipoles are of no essential importance and that its electrical anomalies depend on the ions.

Jaffe\({}^{26}\) proposed a theory according to which, at the Curie points, Rochelle salt undergoes a polymorphic transformation, and in the interval between the Curie points its structure is not orthorhombic but monoclinic. It has been established experimentally that, at the Curie points, the right angle between two faces of a Rochelle-salt crystal parallel to its electrical axis \(a\) changes by \(3'\); however, this deformation can also be produced at other temperatures by the action of a mechanical load or an electric field. Therefore it is unclear whether the deformation should be regarded as a primary phenomenon caused by a change in the structure of Rochelle salt, or as the result of an inverse piezoelectric effect caused by spontaneous polarization. This phenomenon can be explained by a change in the elastic properties of the crystal, facilitating, in a certain temperature interval, the piezoelectric effect and spontaneous polarization. According to Jaffe’s theory, at the Curie points the shear modulus of Rochelle salt becomes zero (see below).

The fourth theory, proposed by Cady\({}^{27}\) and independently by Shulvas-Sorokina\({}^{28}\), deserves greater attention. It is based on taking into account the anomalous piezoelectric effect and reduces to the fact that the elastic deformation caused by an electric field (or, more precisely, by the electric polarization directly produced by it) gives rise to an additional polarization of the crystal, which in turn causes a further increase in deformation, as a result of which the crystal proves capable of being polarized spontaneously.

In a recently published work\({}^{29}\), Müller, without going into the mechanism of the phenomenon but proceeding from purely formal considerations, gave a critical assessment of the various theories and at the same time put forward a new theory which is essentially a combination of the preceding ones (dipole interaction, anomaly of elastic properties, and the piezoelectric effect).

Let us consider the change in the free energy of an orthorhombic crystal under a homogeneous deformation with components \(x_x, x_y, y_z\), etc., and polarization in a homogeneous field of arbitrary direction.

This free energy, referred to unit volume, is expressed by the formula

\[ \Phi=\frac{1}{2}(c_{11}x_x^2+c_{22}y_y^2+c_{33}z_z^2)+(c_{23}y_yz_z+c_{31}z_zx_x+c_{12}x_xy_y)+ \]
\[ +\frac{1}{2}(c_{44}y_z^2+c_{55}z_x^2+c_{66}x_y^2)+f_{14}y_zP_x+f_{15}z_xP_y+ \]
\[ +f_{46}x_yP_z+\frac{1}{2}(\chi_1P_x^2+\chi_2P_y^2+\chi_3P_z^2). \tag{145} \]

Here \(P_x, P_y, P_z\) are the components of the polarization vector; \(c_{ik}, f_{ik}, \chi_i\) are the elastic, piezoelectric, and dielectric constants of the crystal, respectively.

Differentiating \(\Phi\) with respect to the components of the strain tensor, we obtain the stresses—\(Y_z=\dfrac{d\Phi}{dy_z}\)—and the components of the electric field \(E_x=\dfrac{d\Phi}{dP_x}\). This gives nine equations of the following form:

\[ -X_x=c_{11}x_x+c_{12}y_y+c_{13}z_z, \tag{146} \]

\[ -Y_z=c_{44}y_z+f_{14}P_x, \tag{147} \]

\[ E_x=f_{14}y_z+\chi_1P_x. \tag{148} \]

Solving this system of equations, we obtain:

\[ -x_x=s_{11}X_x+s_{12}Y_y+s_{13}Z_z, \tag{149} \]

\[ -y_z=s_{44}Y_z+d_{14}E_x, \tag{150} \]

\[ P_x=d_{14}Y_z+k_1E_x. \tag{151} \]

The relations between the first and second systems of constants of the crystal have the following form:

\[ s_{23}=(c_{12}c_{13}-c_{11}c_{23})\cdot\frac{1}{D_{13}};\qquad c_{23}=-(s_{12}s_{13}-s_{11}s_{23})\cdot D_{13}, \tag{152} \]

where

\[ D_{13}= \begin{vmatrix} c_{11} & c_{12} & c_{13}\\ c_{12} & c_{22} & c_{23}\\ c_{13} & c_{23} & c_{33} \end{vmatrix} \tag{153} \]

and

\[ s_{44}=\frac{\chi_1}{D_{14}};\qquad d_{14}=\frac{f_{14}}{D_{14}};\qquad k_1=\frac{c_{44}}{D_{14}}, \tag{154} \]

\[ D_{14}=c_{44}\chi_1-f_{14}^2=-\frac{1}{s_{44}k_1-d_{14}^2}. \tag{155} \]

As has just been noted, the constants characterizing the purely elastic properties of the crystal are \(c_{ik}\), the piezoelectric ones are \(f_{ik}\), and the dielectric ones are \(\chi_i\). The anomalous properties of Seignette salt must be explained by the anomalous temperature dependence of those of these coefficients which correspond to the axis \(x(a)\), i.e., \(c_{44}\), \(f_{14}\), or \(\chi_1\). In order to decide which of the theories of Seignette salt is correct, it is necessary to determine which of the three indicated quantities possesses an anomalous temperature dependence and forms

goes to zero or to infinity at the Curie points (the other constants certainly have no anomalies).

The spontaneous polarization of a free crystal, i.e., one capable of being deformed, can, however, as Müller indicates on the basis of formulas (151), (154), and (155), occur also in the case where all three of the constants mentioned have finite values, but \(D_{14}=0\), i.e., ferroelectricity may be explained by an accidental relation between the values of various constants.

Experimentally, it is not the coefficients \(c_{44}\), \(f_{14}\), and \(\chi_1\) that are determined, but \(s_{44}\), \(d_{14}\), and \(k_1\). Müller calculates the temperature dependences of the first three quantities by means of formulas (152)—(155) from the experimental values of the second group of quantities. It turns out that the elastic constant \(c_{44}\) and the piezoelectric-effect constant \(f_{14}\) have a normal and smooth temperature behavior, so that the theories of Jaffe and Cady, as well as that of Shulvas-Sorokina, do not correspond to reality. The constant \(\chi_1\), however, the reciprocal of the electric susceptibility of the undeformed crystal, vanishes at a certain temperature. In order to obtain correct quantitative results, in particular to determine the positions of the Curie points, and to explain many other properties of ferroelectrics, for example the hysteresis loop, it is necessary, however, to take into account the piezoelectric effect and the elastic deformation of the crystal in an electric field.

V. POLARIZATION IN ALTERNATING FIELDS

§ 14. The Old and New Debye Theory

In 1912 Debye generalized his theory of the polarization of dipolar liquids in constant fields to the case of alternating electric fields of small intensity by applying Einstein’s theory of rotational Brownian motion to the rotational motion of molecules. In this treatment the interaction of the molecules is reduced to a frictional force with a moment proportional to the angular velocity. The coefficient of proportionality is determined by Stokes’ formula

\[ f' = 8\pi a^3\eta, \tag{156} \]

where \(a\) is the radius of the molecule, and \(\eta\) is the coefficient of viscosity of the liquid. The coefficient \(f'\) is related to the coefficient of orientational diffusion \(D'\) of the molecules by Einstein’s relation \(D'f' = kT\). In addition to this kinetic, or “relaxation,” interaction, Debye also took into account the static interaction, determined by the Lorentz formula (6) for the effective external field.

In the case of a weak external field oscillating harmonically according to the law \(E = E_0 e^{i\omega t}\), Debye obtained for the polarization the expression

\[ P = \frac{N\mu^2}{3kT}\cdot \frac{E_0 e^{i\omega t}}{1 + i\frac{\omega f'}{kT}}. \tag{157} \]

This formula, like the formula of the old Debye theory for polarization in a constant field (which is obtained from it for \(\omega = 0\)),

does not agree in all respects with the experimental data. If one takes the macroscopic value \(\eta\) for the corresponding liquid, then the value \(a\), calculated by comparing the results given by the theory with the experimental data, usually proves to be impermissibly small. If, however, one adopts the correct value of the molecular radius, then one has to assume that the coefficient of friction for the molecule is considerably smaller than its macroscopic expression (156).

After revising their old theory of the polarization of a dipolar liquid in a constant field, Debye, together with Ramm\(^{30}\), attempted to apply the idea of the local field also to the case of a variable external field.

The distribution function of the dipoles over angles, \(f\), is determined in the general case by the equation

\[ \frac{\partial f}{\partial t} = \frac{kT}{f'}\nabla^2 f + \frac{1}{f'}\operatorname{div}(f\,\operatorname{grad} U), \tag{158} \]

where \(U\) denotes the potential energy of the dipole, and the symbols of vector differentiation refer to the values of the distribution function \(f\) on the unit sphere (to each molecule there corresponds a point at which its axis intersects this sphere). In the old theory it was assumed that

\[ U=-\mu E_e e^{i\omega t}\cos\vartheta', \tag{159} \]

where \(E_e e^{i\omega t}\) is the effective alternating field in the dielectric \(\left(E+\frac{4\pi}{3}P\right)\), and \(\vartheta'\) is the angle between the dipole axis and this field.

Substituting (159) into equation (158), we obtain its solution in the following approximate form:

\[ f=A\left[1+\frac{\mu E_e e^{i\omega t}}{kT(1+i\omega\tau)}\cos\vartheta'\right], \tag{160} \]

where \(\tau=\frac{f'}{2kT}\) is the “relaxation time.” Hence, for the mean value of the projection of the dipole moments on the direction of the field, the expression obtained is

\[ \bar{\mu} = \frac{\mu^2 E_e e^{i\omega t}}{3kT}\, \frac{1}{1+i\omega\tau}, \tag{161} \]

from which formula (159) follows directly.

The interaction of the dipoles, characterized by the local field \(F\), is taken into account in the theory of Debye and Ramm by replacing \(U_1 e^{i\omega t}\) by

\[ U=U_0+U_1 e^{i\omega t}, \tag{162} \]

where \(U_0=-F\mu\cos\vartheta\); \(U_1=-\mu E_e\cos\vartheta'\); \(F\) is the local field (see § 5); \(\vartheta\) is the angle between the dipole and the local field.

Let us rewrite equation (158) in polar coordinates:

\[ \frac{f'}{kT}\frac{\partial f}{\partial t} = \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta} \sin\vartheta \left[ \frac{\partial f}{\partial\vartheta} + \frac{f}{kT}\frac{\partial U}{\partial\vartheta} \right] + \frac{1}{\sin^2\vartheta}\frac{\partial}{\partial\varphi} \left[ \frac{\partial f}{\partial\varphi} + \frac{f}{kT}\frac{\partial U}{\partial\varphi} \right] \tag{163} \]

and we shall seek its solution in the form

\[ f=f_0+f_1 e^{i\omega t}, \tag{164} \]

where

\[ f_0=ce^{-\frac{U_0}{kT}} \]

is the solution of the equation

\[ 0=\frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta}\sin\vartheta \left[\frac{\partial f_0}{\partial\vartheta}+\frac{f_0}{kT}\frac{\partial U_0}{\partial\vartheta}\right] +\frac{1}{\sin^2\vartheta}\frac{\partial}{\partial\varphi} \left[\frac{\partial f_0}{\partial\varphi}+\frac{f_0}{kT}\frac{\partial U_0}{\partial\varphi}\right], \tag{165} \]

corresponding to the stationary distribution in the absence of an external field.

Substituting (162) and (164) into (163), taking into account (165) and omitting quadratic terms of the form \(f_1\dfrac{\partial U_1}{\partial\vartheta}\), which is permissible so long as the polarization is far from saturation, we obtain the following equation for determining \(f_1\):

\[ i\frac{\omega f'}{kT}f_1= \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta}\sin\vartheta \left[\frac{\partial f_1}{\partial\vartheta}+\frac{f_1}{kT}\frac{\partial U}{\partial\vartheta}\right] +\frac{1}{\sin^2\vartheta}\frac{\partial}{\partial\varphi} \left[\frac{\partial f_1}{\partial\varphi}+\frac{f_1}{kT}\frac{\partial U_0}{\partial\varphi}\right] + \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta}\sin\vartheta \left[\frac{f_0}{kT}\frac{\partial U_1}{\partial\vartheta}\right] +\frac{1}{\sin^2\vartheta}\frac{\partial}{\partial\varphi} \left[\frac{f_0}{kT}\frac{\partial U_1}{\partial\varphi}\right]. \tag{166} \]

To solve this equation, consider the auxiliary equation, similar to it, with unknown function \(X_n\):

\[ \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta}\sin\vartheta \left[\frac{\partial X_n}{\partial\vartheta}+\frac{X_n}{kT}\frac{\partial U_0}{\partial\vartheta}\right] + \]

\[ +\frac{1}{\sin^2\theta} \left[\frac{\partial X_n}{\partial\varphi}+\frac{X_n}{kT}\frac{\partial U_0}{\partial\varphi}\right] +\lambda_n X_n=0. \tag{167} \]

Suppose that we have found such values of the parameter \(\lambda_n\) for which \(X_n\) is a spherical function. For \(U_0=0\) these values coincide with the eigenvalues \(\lambda_n\) for the spherical functions \(\lambda_n=n(n+1)\).

We expand the last two terms of equation (166) and the function \(f_1\) itself in a series in these spherical functions:

\[ \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta}\sin\vartheta \left[\frac{f_0}{kT}\frac{\partial U_1}{\partial\vartheta}\right] +\frac{1}{\sin^2\vartheta}\frac{\partial}{\partial\varphi} \left[\frac{f_0}{kT}\frac{\partial U_1}{\partial\varphi}\right] =\sum c_n X_n, \tag{168} \]

\[ f_1=\sum A_n X_n. \tag{169} \]

Substituting (168) and (169) into (167), we obtain the equation for determining the expansion coefficients of \(f_1\) in the series:

\[ \frac{i\omega f'}{kT}A_n+\lambda_n A_n=c_n, \]

whence

\[ A_n=\frac{c_n}{\lambda_n+\dfrac{i\omega f'}{kT}}. \tag{170} \]

The complete distribution function according to (164) takes the following form:

\[ f=ce^{-\frac{U_0}{kT}}+\sum \frac{c_n X_n}{\lambda_n+\dfrac{i\omega f'}{kT}}e^{i\omega t}. \tag{171} \]

For \(U_0=0\) this formula reduces to formula (160) of the previous theory.

In the case of a weak local field \(\dfrac{F\mu}{kT}=y\ll 1\), one can approximately find \(\lambda_n\) and \(X_n\) by expanding them in powers of \(y\).

To calculate the mean moment \(\bar{\mu}\), as in the static case, it is necessary to average over all possible orientations of \(\mathbf F\) relative to \(\mathbf E\). For a liquid they are all equally probable. As a result we obtain

\[ \frac{\bar{\mu}}{\mu}=\frac{\int f\cos\vartheta' \, d\Omega}{\int f\,d\Omega}. \tag{172} \]

For \(\omega\tau\ll 1\) one may expand \(f\) in a series in powers of \(\omega\tau\). If, at the same time, \(\lambda_n\) and \(X_n\) are expanded in powers of \(y\) and only the first power of \(\omega\tau\) and terms up to \(y^4\) are retained, then, after lengthy calculations, we obtain

\[ \bar{\mu}=\frac{\mu^2 E e^{i\omega t}}{3kT}\left[R(y)-i\omega\tau R'(y)\right], \tag{173} \]

where

\[ R(y)=1-L^2(y)=1-\frac{1}{9}y^2+\frac{2}{135}y^4+\cdots, \tag{174} \]

\[ R'(y)=1-\frac{11}{54}y^2+\frac{113}{3240}y^4+\cdots . \tag{175} \]

In the opposite limiting case of a strong local field, \(F\mu\gg kT\), the expression obtained for \(\bar{\mu}\) is

\[ \bar{\mu}=\frac{\mu^2 E e^{i\omega t}}{\dfrac{3}{2}\mu F\left(1+\dfrac{i\omega f'}{kT}\right)}. \tag{176} \]

Comparing this expression with formula (161), derived without taking into account the local field \(F\), we see that, just as in the static case, the influence of the latter can be accounted for by replacing \(kT\) by the much larger quantity \(\dfrac{\mu F}{2}\). This also means that the relaxation time \(\tau\) decreases by a factor of

\[ \frac{2}{y}, \]

as a result of which the region of dielectric losses shifts toward higher frequencies.

The dielectric losses for \(y\gg 1\) are considerably smaller than for \(y=0\), since they depend on the imaginary part of the mean moment

\[ \frac{\mu^2}{3kT}\frac{\omega\tau}{1+\omega^2\tau^2} \tag{177} \]

and, under the influence of the local field, are multiplied by \(\dfrac{4}{y^2}\).

From the standpoint of the theory presented, it becomes clear why, when determining the relaxation time \(\tau\) according to the old theory from the experimental values of the electrical losses and comparing this value with \(\dfrac{4\pi\eta a^3}{kT}\), excessively small values were obtained for \(\eta\) or for \(a^3\).

Thus, the new Debye theory is in better agreement with experimental data than the old one.

§ 15. Simplified Derivation and Critique of Debye’s New Theory

As was already noted by Ya. I. Frenkel’\(^{31}\), formula (176) can be derived in an elementary way by a direct generalization of formula (85), into which it passes in the case \(\omega = 0\). Let us consider the forced oscillations of an oscillator performing rocking motions about an equilibrium orientation and experiencing friction proportional to the angular velocity \(d\varphi/dt\). Denoting the moment of inertia of the oscillator by \(J\), we have

\[ J\frac{d^2\varphi}{dt^2} = \mu E_e \sin\theta - \mu F \sin\varphi - f'\frac{d\varphi}{dt} \]

or, in view of the smallness of \(\varphi\),

\[ J\frac{d^2\varphi}{dt^2} + \mu F\varphi + f'\frac{d\varphi}{dt} = \mu E_e \sin\theta . \]

Recalling that \(\mu\sin\varphi\sin\theta \simeq \mu\varphi\sin\theta\) is the projection of the molecular moment onto the direction \(E_e\), we obtain, after averaging over all values of \(\theta\):

\[ J\frac{d^2\bar{\mu}}{dt^2} + \mu F\bar{\mu} + f'\frac{d\bar{\mu}}{dt} = \frac{2}{3}\mu^2 E_e . \]

The solution of this equation has the form:

\[ \bar{\mu} = \frac{2}{3}\frac{\mu^2}{J} \frac{E_e}{\omega_0^2-\omega^2+i\frac{\omega f'}{kT}}, \tag{178} \]

where

\[ \omega_0=\sqrt{\frac{\mu F}{J}} \]

is the angular frequency of the free rocking motions of the molecule under the action of the field \(F\).

Since this frequency is much greater than the frequency of the oscillations of the external field \((\omega_0 \simeq 10^{13},\ \omega < 10^9)\), formula (178) reduces to

\[ \bar{\mu} = \frac{2}{3}\frac{\mu^2}{J} \frac{E_e}{\omega_0^2+i\frac{\omega f'}{kT}}, \tag{179} \]

which coincides with Debye’s formula (176).

It is thus clear that the essence of Debye’s new theory consists in replacing the free rotation of molecules (with friction) by rotational rocking motions about their equilibrium orientations, determined by the local field. At the same time, however, Debye proceeds in an obviously inconsistent manner, again introducing the rotational Brownian motion of molecules, but not in connection with a change in the direction of the internal field \(F\), rather as a phenomenon entirely independent of the latter. In his calculations (which we reproduced in the preceding paragraph), Debye in fact regards the magnitude and direction of the field \(F\) as unchanged for each molecule, and then averages the result over all directions of \(F\).

If one assumes, in accordance with Debye’s original ideas, that the equilibrium orientations of molecules rotate discontinuously with time through large angles (in the case of small molecules precisely such a view should be regarded as valid), then Einstein’s theory of rotational Brownian motion proves to be completely inapplicable, since it is based on the assumption of a slow change in the orientations of molecules by successive rotations through very small angles—just as occurs in the case of translational Brownian motion. Only under this condition does the concept of a frictional force have meaning. It is precisely this conception of rotational Brownian motion that underlies Debye’s second theory of the behavior of a dipolar liquid in an alternating field (1912). The magnitude of the local field \(F\) in this theory has significance only as a factor determining the viscosity of the medium, and therefore does not influence the equilibrium value of the polarization in a constant field. In his new (third) theory Debye treats the field \(F\) as the cause not of viscous but of elastic effects, which bring about a decrease of the polarization of the liquid in a constant external field. The basis for such an interpretation, which he himself does not formulate (and apparently does not notice), can only be the representation indicated above of a discontinuous change in the direction of the local field (and of the equilibrium orientation of each molecule determined by it) through large angles. Under this condition the action of the local field reduces to an elastic effect (with the change in its direction with time not noticeably affecting the polarization in a constant external field) and gives no viscous effect whatsoever. These discontinuous rotations of the field \(F\) can affect only the behavior of molecules in an alternating external field, causing friction analogous to the Lorentz damping of electronic oscillations in molecules as a result of their collisions with one another (Stossdämpfung). However, this effect is extremely small and has no practical significance for dielectric losses.

Thus, Debye’s second and third theories, both of which are essentially constructed on the idea of the local field (not explicitly formulated in the first of them), in a certain sense contradict one another. The second theory (1912) corresponds to the idea of a rotation of \(F\) through small angles, which leads to a viscosity effect and does not affect the polarization of the liquid in a constant external field, but only slows the establishment of the equilibrium state; whereas the third theory (1935) corresponds to the idea of a discontinuous rotation of \(F\) through large angles and leads to an elastic effect that reduces the polarization in a constant field.

Debye and Ramm’s fourth theory (1937) attempts to combine the contradictory second and third theories. Here the local field is used twice: on the one hand, as the cause of the frictional force experienced by the molecule, and on the other, as the cause of a static quasi-elastic force. In the opinion of E. V. Kuvshinskii such an interpretation may be justified if the interaction of the molecule with its nearest

with neighbors corresponds to the static effect, while interaction with distant molecules, which determines the slow rotation of the equilibrium orientation, corresponds to the friction effect. Such a notion, however, is hardly valid, since frictional force, characterized by the macroscopic viscosity of a liquid, can hardly depend on the interaction only between distant molecules.

One might think that the frictional force, characterized by the coefficient \(f'\), is due to the anharmonic character of the rotational oscillations, i.e. to the nonlinear character of the dependence of intermolecular forces on the magnitude of the displacement, just as in the case of solids.

However, this notion is not consistent with the extremely sharp increase in the relaxation time, i.e. in the friction, when the temperature is lowered.

Thus, the question of the mechanism of dielectric losses in dipolar liquids should be regarded as completely open.

It should be noted that dipolar substances in the solid state near the crystallization temperature possess an anomalously high polarizability. Thus, for example, in ice a maximum of \(\varepsilon\) is observed (of the order of 120) at a temperature somewhat below zero (about \(-3^\circ\)) in the case of a constant field. In the case of an alternating field, as the frequency is increased this maximum decreases and shifts toward \(0^\circ\). P. P. Kobeko and his school explain these phenomena by “structural relaxation,” i.e. by a change in the structure of large regions of the body, accompanied by a change in the orientation of its molecules. As the temperature is lowered, the rate of this relaxation rapidly decreases, so that there occurs, as it were, a freezing-in of a definite orientation of the molecules, which can no longer change even under the action of a practically constant field. In this case the dielectric constant decreases, reducing to a value dependent on the electronic polarization of the molecules.

§ 16. Formal theory of dielectric losses

Wagner and Gemant\(^{32}\) showed that the dependence of the dielectric constant of a dielectric and of the dielectric losses on frequency can be obtained without resorting to any special mechanism of polarization of the dielectric, but from purely formal considerations, under only one assumption—namely, that the equilibrium state is established in the dielectric according to an exponential law (while that part of the dielectric displacement \(D\) which corresponds to electronic polarization may be established instantaneously). Under these conditions the dielectric constant \(\varepsilon\) must, obviously, be smaller at very high frequencies than in a constant field, and we may put:

\[ \varepsilon_0 = \varepsilon_\infty (1 + k), \tag{180} \]

where \(\varepsilon_0\) is the dielectric constant in a constant field, and \(\varepsilon_\infty\) is that at infinitely high frequency.

If at the instant \(t_0\) a constant field \(E\) is switched on instantaneously, then by the time \(t\) the dielectric displacement can be represented in the form

\[ D=\varepsilon_\infty(1+k)E-\varepsilon_\infty k e^{-\frac{t-t_0}{\tau}}E, \tag{181} \]

where \(\tau\) is the relaxation time. From this formula it is evident that at the very instant of switching on, when \(t=t_0\), \(D=\varepsilon_\infty E\), while for \(t=\infty\), \(D=\varepsilon_0 E\). In the case of switching on an alternating electric field, by the principle of superposition we obtain the following expression:

\[ D=\varepsilon_\infty(1+k)E-\varepsilon_\infty k \int_{-\infty}^{\infty} e^{-\frac{t-t_0}{\tau}} \frac{dE}{dt}\,dt. \tag{182} \]

In particular, in the case of a sinusoidal field \(E=E_0\sin\omega t\), integrating (182) by parts, we obtain

\[ D=\varepsilon_\infty E_0\left[\left(1+\frac{k}{1+\omega^2\tau^2}\right)\sin\omega t -\frac{k\omega\tau}{1+\omega^2\tau^2}\cos\omega t\right] \]

or

\[ D=\varepsilon_\infty E\left(1+\frac{k}{1+\omega^2\tau^2} -i\,\frac{k\omega\tau}{1+\omega^2\tau^2}\right). \tag{183} \]

The real part \(\varepsilon'\) of the complex dielectric constant \(\varepsilon\) represents the dielectric constant in the usual sense, whereas the imaginary part \(\varepsilon''\) characterizes the dielectric losses. From (183) we obtain

\[ \varepsilon'=\varepsilon_\infty\left(1+\frac{k}{1+\omega^2\tau^2}\right). \tag{184} \]

and

\[ \operatorname{tg}\delta=\frac{\varepsilon''}{\varepsilon'}= \frac{k\omega\tau}{1+k+\omega^2\tau^2}, \tag{185} \]

where \(\delta\) is the dielectric loss angle. The last three expressions coincide with those obtained from both the new and the old Debye theory.

Exactly the same formulas are obtained in Wagner’s theory of dielectric losses for two different models of a dielectric, which have nothing in common with the actual mechanism of polarization in dipolar substances.

According to the first model (which has only illustrative significance), the dielectric is represented as consisting of two layers with different complex \(\varepsilon\). According to the second model (which may perhaps correspond to glasses), the dielectric is treated as a homogeneous medium with complex \(\varepsilon_1\), with spheres having complex \(\varepsilon_2\) suspended in it. These results show that the specific features of the structure of the dielectric must be reflected not in the general character of the law determining its dispersion and dielectric losses, but only in the numerical values of the parameters \(k\) and \(\tau\) entering into the expression of this law.

References

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1. This section is original, and the results presented in it are being published for the first time. ↩↩↩

2. Kirkwood. ↩