THEORY OF FERROELECTRIC PHENOMENA

V. L. Ginzburg

CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

1. Theory of ferroelectric phenomena without allowance for anisotropy and stresses . . . . . . . . . . . . . . . . . . . . 493
2. The case of a phase transition close to the critical Curie point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
3. Anisotropy of dielectric properties and the piezoelectric effect in barium titanate . . . . . . . . . . . . . . . . . . . . 503
4. Properties of Rochelle salt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
5. Pyroelectrics, ferroelectrics, and ferromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

INTRODUCTION

At the present time a fairly large number of crystals are already known that possess anomalous dielectric properties and are called ferroelectrics or ferro-electrics. This latter name, used mainly in the foreign literature, reflects the fact that the substances of the class under consideration are electrical analogues of ferromagnets. However, since none of these substances contains iron, we shall use the term ferroelectric, associated with the first discovered representative of such crystals—Rochelle salt. In addition to Rochelle salt, $\mathrm{NaKC_4O_6\cdot 4H_2O}$, studied in detail by I. V. Kurchatov, P. P. Kobeko, and others,$^{1}$ ferroelectrics include $\mathrm{KH_2PO_4}$, $\mathrm{KH_2AsO_4}$, the same salts with deuterium substituted (for example, $\mathrm{KD_2PO_4}$), certain mixed crystals, and, finally, barium titanate ($\mathrm{BaTiO_3}$), whose ferroelectric properties were discovered quite recently by B. M. Vul and I. M. Goldman.$^{2}$

A distinctive feature of ferroelectrics is the existence of a certain temperature $\theta$, called the Curie temperature, above which the polarization in a weak field is linear in the field, i.e.

\[ P=\frac{\varepsilon-1}{4\pi}E=\chi E, \]
but the susceptibility \(\chi\) obeys the Curie–Weiss law:

\[ \chi=\frac{\mathrm{const}}{T-\Theta}. \]

In the case of Rochelle salt there are two Curie points: an upper one \(\Theta_h\) and a lower one \(\Theta_l\). Above the Curie point (or, in the case of Rochelle salt, at \(T>\Theta_h\) and \(T<\Theta_l\)), when the field \(E\) is equal to zero the polarization in the crystal is zero. Below the Curie point (or, for Rochelle salt, at \(\Theta_l<T<\Theta_h\)), on the contrary, the crystal possesses a spontaneous polarization \(P_0\), i.e. its polarization differs from zero even in a field equal to zero. Thus, at the Curie point (or points) there occurs a phase transition from a non-pyroelectric crystal to a pyroelectric one; moreover, by pyroelectric we mean any crystal with \(P_0\ne0\). The range of temperatures in which, in a ferroelectric, \(P_0\ne0\) is called the ferroelectric region (below, for brevity, we shall usually assume that the ferroelectric region is the entire range of temperatures \(T\) below the Curie temperature \(\Theta\), as is the case in all instances except Rochelle salt). In the ferroelectric region the polarization, even in relatively weak fields, depends nonlinearly on the electric-field strength, and hysteresis is observed. In general, the behavior of ferroelectrics in this region is analogous to the behavior of ferromagnets below the Curie point. In particular, a free (i.e., for example, not placed in a capacitor) ferroelectric crystal, just like a ferromagnet, breaks up into regions (domains) of spontaneous polarization in such a way that the average polarization over the whole crystal is zero.

This splitting is due to the presence of boundaries and is caused by the fact that, in the absence of domain formation, bound electric charges appear on the surface of the crystal; these charges are sources of an electric field inside and outside the crystal. The presence of this field leads to the result that, generally speaking, the absence of splitting into domains is energetically unfavorable. However, the observed splitting into domains may to some extent be regarded as a secondary effect, and under certain conditions it should not occur at all (splitting should not be observed when the ferroelectric is placed in a capacitor; see Section 1). In view of what has been said, in considering a number of important questions (for example, such as the temperature dependence of the spontaneous polarization) one may disregard the division of the ferroelectric into domains. As will be shown below, in order that all phenomena typical of ferroelectrics be observed, it is necessary that the phase transition taking place at the Curie point be a transition of the 2nd order or a transition of the 1st order close to the so-called critical Curie point, which separates, on the \(p,T\)-diagram, the line of Curie points, i.e. points at which a transition of the 2nd order occurs, from the line of phase transitions of the 1st order\(^3\).

Curie temperatures for known ferroelectrics are given in the table.

In the case of barium titanate, the temperature \(\Theta\) is indicated within broad limits, since its value varies rather strongly depending on the presence of impurities and, possibly, under the influence of other factors (stress, inhomogeneity).

Curie temperatures for ferroelectrics

Substance	Curie temperature, in degrees Kelvin	Curie temperature, in degrees Celsius
Rochelle salt \(\mathrm{NaKC_4H_4O_6\cdot 4H_2O}\)	\(\Theta_h=297\) \(\Theta_l=255\)	\(+24\) \(-18\)
\(\mathrm{KH_2PO_4}\)	122	\(-151\)
\(\mathrm{KD_2PO_4}\)	213	\(-60\)
\(\mathrm{KH_2AsO_4}\)	96	\(-177\)
\(\mathrm{BaTiO_3}\)	\(\sim 350 \div 390\)	\(80—120\)

The maximum spontaneous polarization in the case of Rochelle salt is observed at a temperature of \(3^\circ\mathrm{C}\) and is equal to \(P_{0,\max}=735\) CGSE units. For ferroelectrics with one Curie point the polarization is apparently maximal at \(T=0\)*); however, at temperatures several tens of degrees below \(\Theta\), the polarization changes only slightly, and the corresponding value \(P_0\) is close to the maximum. For \(\mathrm{KH_2PO_4}\), \(P_{0,\max}=1.41\cdot 10^4\); for other substances of this type the value of \(P_{0,\max}\) is of the same order; for \(\mathrm{BaTiO_3}\), \(P_{0,\max}\simeq 5\cdot 10^4\).

We shall not present or discuss the experimental data here in any detail, referring to Cady’s book\(^4\) and, with regard to \(\mathrm{BaTiO_3}\), to the review by A. V. Rzhanov\(^5\). The aim of the present article is primarily to set forth a phenomenological (thermodynamic) theory of ferroelectric phenomena, based on works\({}^{6,7}\). In doing so, the main attention is devoted to general questions and to the case of barium titanate. We shall touch with sufficient completeness upon a very important

* We abstract here from the fact that at temperatures around \(+5^\circ\) and \(-70^\circ\mathrm{C}\) in barium titanate there occur phase transformations which, however, are not associated with the disappearance of spontaneous polarization, i.e. take place within the limits of the pyroelectric modification. These transitions are, perhaps, first-order transitions (in favor of this assumption speak the reversibly observed, in places, phenomena of supercooling and superheating, as well as a jump in \(\varepsilon\); see W. Merz, Phys. Rev. 75, 687 (1949)) and are probably connected with changes in the magnitude of the vector of spontaneous polarization and its orientation relative to the axes of the lattice.

of Rochelle salt the author has had no opportunity to discuss, since he has not worked on this question (see, however, Section 4; the properties of Rochelle salt, in contrast to those of BaTiO\(_3\), are discussed in detail in \(^4\)). For the same reason, questions connected with the microscopic (molecular) theory of ferroelectrics are not covered. Thus, the present article does not claim completeness.

In concluding this introduction, we emphasize once again the great importance and guiding role of the works and discoveries of Soviet authors in the field of the study of ferroelectric phenomena (the works of I. V. Kurchatov, B. M. Vul, and a number of other authors; see the cited works of Soviet authors and the references to the literature contained in them).

1. THEORY OF FERROELECTRIC PHENOMENA WITHOUT TAKING ANISOTROPY AND STRESSES INTO ACCOUNT\(^6\)

It was indicated above that the fundamental, determining feature of ferroelectrics is the presence of a Curie point—the point of a second-order phase transition from a non-pyroelectric modification to a pyroelectric one. In order to verify the validity of this statement and, chiefly, to obtain a number of important relations, let us consider the behavior of a crystal near the point of a pyroelectric phase transition of the second order. We shall assume here that mechanical stresses are absent and that the anisotropy of the dielectric properties may be neglected. More precisely, we shall assume that the polarization and the electric field are parallel, which may occur both in the case of isotropy and in the considerably more interesting case when we are interested only in the polarization and field directed along the ferroelectric axis—the axis of spontaneous polarization of the crystal. For all known ferroelectrics, except BaTiO\(_3\), there is only one ferroelectric axis, already distinguished above the Curie point. For such substances the indicated case of polarization along the axis alone is of essential significance.

Under the assumptions made, the state of the body, in addition to the temperature \(T\) and pressure \(p\), is characterized by a single parameter—the polarization \(P\). In the absence of an electric field \(E\), the spontaneous polarization \(P_0\) may be chosen as the parameter \(\xi\) which occurs in the general theory of second-order phase transitions\(^3\), is equal to zero in the disordered (non-pyroelectric) state, and is different from zero in the ordered (pyroelectric) state.

In the case of a second-order transition at the Curie point, by definition, \(P_0 = 0\), and consequently, in the neighborhood of this point the thermodynamic potential \(\Phi\) may, generally speaking, be expanded in a series in powers of \(P_0\) and one may retain only the first significant terms of this series; since on replacing \(P_0\) by \(-P_0\) the value of \(\Phi\) cannot change (because of the complete equivalence of both mutually opposite directions parallel to the ferroelectric axis), the potential must be a function

only \(P_0^2\), and not \(P_0\). We shall write the corresponding expansion, at the same time allowing also for the presence of an electric field \(E\):

\[ \Phi=\Phi_0+\alpha P^2+\frac{\beta}{2}P^4-EP, \tag{1} \]

where \(\Phi_0,\alpha,\beta\) are functions of the pressure \(p\) and the temperature \(T\).

In (1) the independent variables are \(p, T\), and \(E\); if \(p, T\), and \(P\) are chosen as the independent variables, then the role of the potential \(\Phi\) is played by another potential (the free energy)

\[ F=\Phi+EP. \tag{2} \]

For \(E=0\) the potentials \(\Phi\) and \(F\) are equal to one another. Further, as is known,

\[ E=\frac{\partial F}{\partial P},\qquad P=-\frac{\partial\Phi}{\partial E}. \tag{3} \]

Let us first consider the case when the field \(E\) is zero, i.e. \(P=P_0\). In a state of equilibrium the potential \(\Phi\), for given \(E\) and \(T\), must be a minimum, i.e.

\[ \frac{\partial\Phi}{\partial P}=0 \quad\text{and}\quad \frac{\partial^2\Phi}{\partial P^2}>0. \]

The first of these conditions, or the use of one of formulas (3), leads, for \(E=0\), to the equation \(\alpha P_0+\beta P_0^3=0\), which has the solutions

\[ P_0=0, \tag{4a} \]

\[ P_0^2=-\frac{\alpha}{\beta}. \tag{4b} \]

From the condition

\[ \frac{\partial^2\Phi}{\partial P^2}=2\alpha+6\beta P_0^2>0 \]

it follows that solution (4a) corresponds to a state of equilibrium if \(\alpha>0\). Above the Curie point, i.e. for \(T>\Theta\), the spontaneous polarization in a ferroelectric must indeed be equal to zero, and thus it is clear that for \(T>\Theta\), \(\alpha>0\). Below the Curie point, where \(P_0\ne0\), solution (4a) must not correspond to a minimum of \(\Phi\), and consequently, for \(T<\Theta\), \(\alpha<0\). At the Curie point itself, therefore,

\[ \alpha(T=\Theta)\equiv\alpha_\Theta=0. \tag{5} \]

Below the Curie point, where \(\alpha<0\), the minimum of \(\Phi\) corresponds to solution (4b), and since \(P_0^2>0\), the coefficient \(\beta\) must be positive. Near the Curie point one may assume that

\[ \beta(T)=\beta(\Theta)\equiv\beta_\Theta>0 \tag{6} \]

and

\[ \alpha(T)=\left(\frac{\partial\alpha}{\partial T}\right)_{T=\Theta}(T-\Theta)=\alpha'_\Theta(T-\Theta), \tag{7} \]

where \(\alpha'_\Theta>0\).

Expansion (1) is suitable only near the Curie point, since on moving away from this point one must take into account terms proportional to \(P_0^6\), \(P_0^8\), etc. The need to retain in (1) the term of order \(P_0^4\) is evidently connected with the vanishing, at the Curie point, of the coefficient \(a\). Taking (46) and (7) into account, near the Curie point we have

\[ \left. \begin{aligned} P_0^2(T) &= \frac{a'_{\Theta}(\Theta - T)}{\beta_{\Theta}},\\ T &\leqslant \Theta . \end{aligned} \right\} \tag{8} \]

The entropy of the ordered (pyroelectric) phase near the Curie point is equal to

\[ S=-\frac{\partial \Phi}{\partial T} = S_0 - P_0^2 a'_{\Theta} = S_0' + \frac{(a'_{\Theta})^2}{\beta_{\Theta}}(T-\Theta), \tag{9} \]

where \(S_0=\dfrac{\partial \Phi_0}{\partial T}\) is the entropy of the disordered (non-pyroelectric) phase. At \(T=\Theta\), \(S=S_0\), and, as should be the case for a second-order transition, the latent heat of transition is zero. At the Curie point, however, there is a jump in the heat capacity

\[ \Delta c_{\Theta}=\frac{\Theta}{\beta_{\Theta}}(a'_{\Theta})^2, \tag{10} \]

where

\[ \Delta c_{\Theta}=c_p-c_{0p},\qquad c_p=T\left(\frac{\partial S}{\partial T}\right)_p,\qquad c_{0p}=T\left(\frac{\partial S_0}{\partial T}\right)_p . \]

Within the accuracy of the formulas given, which do not take into account terms in (1) of higher order than \(\dfrac{\beta}{2}P_0^4\), the difference of heat capacities \(\Delta c\) near the Curie point does not depend on \(T\) and is equal to expression (10). Taking (8) and (10) into account, one may write

\[ P_0^2(T)=\frac{\Delta c_{\Theta}}{\Theta a'_{\Theta}}(\Theta-T), \tag{11} \]

i.e., eliminate from (8) the quantity \(\beta_{\Theta}\), equal to

\[ \beta_{\Theta}=\frac{\Theta(a'_{\Theta})^2}{\Delta c_{\Theta}}. \tag{12} \]

Formula (11) is important because all the quantities entering it, \(a'_{\Theta}\), \(\Delta c_{\Theta}\), and \(P_0^2\), can be measured independently of one another. Thus \(\Delta c_{\Theta}\) is determined calorimetrically, \(P_0^2\) can be determined from the charge of a capacitor with a ferroelectric (see below), and \(a'_{\Theta}\) is determined from measurements of the dielectric constant, to the calculation of which we now turn.

Putting in (1) the field \(E\) not equal to zero, with the aid of relation (3) we find

\[ 2\alpha P+2\beta P^3=E. \tag{13} \]

The polarization \(P\) in the presence of a field can be represented as the sum of the spontaneous polarization and the induced polarization \(P_i\):

\[ P=P_0+P_i; \tag{14} \]

in essence, formula (14) defines the polarization \(P_i\).

In a sufficiently weak field the polarization \(P_i\), as is justified by the subsequent calculation, is proportional to the field \(E\):

\[ P_i=\frac{\varepsilon-1}{4\pi}E, \tag{15} \]

where the proportionality coefficient is written in the form \(\dfrac{\varepsilon-1}{4\pi}\) in accordance with the usual definition of the dielectric constant \(\varepsilon\). Above the Curie point \(P_0=0\), and in a sufficiently weak field one always has \(2aP=2aP_i=E\); thus, above the Curie point, at any distance from it,

\[ \varepsilon-1=\frac{2\pi}{a(p,T)}\quad (T>\Theta). \tag{16} \]

Near the Curie point, taking (7) into account and neglecting unity in comparison with \(\varepsilon\), we obtain the Curie–Weiss law:

\[ \varepsilon=\frac{2\pi}{a_{\Theta}(T-\Theta)}\quad (T>\Theta). \tag{17} \]

Below the Curie point, from (13) one can obtain the value of \(\varepsilon\) only in the vicinity of this point; using (13), (14), and (15), and assuming

\[ P_i\ll P_0, \tag{18} \]

which in a sufficiently weak field always holds, we find

\[ \varepsilon=-\frac{\pi}{a_{\Theta}(T-\Theta)}\quad (T<\Theta). \tag{19} \]

Since in the initial formula higher-order terms of order \(P^6\), etc., have not been taken into account, expression (19) for \(\varepsilon\) is valid up to a constant \(\varepsilon_0\ll\varepsilon\) [for this reason, in (19) it makes no sense to write \(\varepsilon-1\) instead of \(\varepsilon\)]. From (17) and (19) it is clear that the slope of the straight line \(1/\varepsilon(T)\) above the Curie point is half as large as below this point (“the law of twos”).

The results obtained (the appearance of spontaneous polarization, the Curie–Weiss law for \(\varepsilon\), the jump in heat capacity), corresponding to the principal experimental facts known for ferroelectrics, indicate that the initial interpretation of the ferroelectric transition as a second-order phase transition from a non-pyroelectric modification to a pyroelectric one is correct.

The tendency of \(\varepsilon\) to infinity when approaching the Curie point is physically quite understandable. Near the Curie point the lattice is “loose,” and even with the slightest lowering of temperature below the temperature \(\Theta\), spontaneous polarization \(P_0\) arises. It is clear that in this region a weak external electric field will also cause an enormous polarization, i.e., the value of \(\varepsilon\) in this temperature region must be very large, and as \(T\to\Theta\), \(\varepsilon\to\infty\).

Let us note that in the case of a phase transition of the 2nd kind, not connected with the appearance of pyroelectric polarization and characterized by some parameter \(\xi\) having no relation to the polarization, the quantity \(\varepsilon\) is continuous, while \(\dfrac{d\varepsilon}{dT}\) undergoes a jump. In the case of a transition of the 1st kind, the dielectric constant \(\varepsilon\) itself undergoes a jump.

Among phase transitions of the 1st kind, a special place is occupied by transitions close to transitions of the 2nd kind. It is physically evident that a transition of the 1st kind very close to a transition of the 2nd kind, i.e. lying near the so-called critical Curie point, will differ little from a transition of the 2nd kind. In this connection, from the point of view of ferroelectric phenomena, phase transitions of the 1st kind close to transitions of the 2nd kind and associated with the appearance of pyroelectric polarization acquire interest. We shall dwell on this case, which apparently has a direct relation to ferroelectrics of the type \(KH_2PO_4\), in § 2.

Fig. 1.

Near the Curie point, as is clear from (13) and from what has been said, the polarization depends linearly on the field only in a very weak field. And if in ordinary dielectrics the nonlinearity of the polarization as a function of the field does not manifest itself, since breakdown occurs earlier, then in ferroelectrics this is no longer so. Indeed, for any field \(E\) one can in principle approach the Curie point so closely that the nonlinearity will already be substantial, since in (13) \(\alpha_\Theta=0\) and \(\beta=\beta_\Theta>0\). According to (13),

\[ \frac{dP}{dE}=\frac{\varepsilon_d}{4\pi} =\frac{1}{2\alpha+6\beta P^2} =\frac{1}{2\alpha_\Theta(T-\Theta)+6\beta_\Theta P^2}, \tag{20} \]

where the derivative is taken at constant temperature and pressure, and \(\varepsilon_d\), by definition, is the differential dielectric constant.

As long as the nonlinearity is small, i.e. with accuracy up to terms of order \(E^2\) inclusive, above the Curie point, using (20) and (15)—(17), we have

\[ \varepsilon_d = 4\pi\left(\frac{dP}{dE}\right) = \frac{2\pi}{\alpha'_{\Theta}(T-\Theta)+\dfrac{3\beta_{\Theta}E^2}{4(\alpha'_{\Theta})^2(T-\Theta)^2}} . \tag{21} \]

Of course, as \(E \to 0\), formula (21) goes over into (17).

For \(T < \Theta\) and in the absence of a field, the potential \(\Phi\) as a function of \(P\) has the form schematically represented in Fig. 1, \(a\). The curve \(\Phi(P)\) has two minima, corresponding to the spontaneous polarization \(\pm P_0\), i.e. to a polarization \(|P_0|\) directed in one of two possible mutually opposite directions. Upon application of a field \(E\), the curve \(\Phi(P)\) has only one absolute minimum, corresponding to a polarization directed along the field. At the same time, in not too strong fields, in addition to this absolute minimum there is also a relative minimum, corresponding to a polarization directed against the field (Fig. 1, \(b\)). However, at a certain field \(E_k\) this second minimum disappears, and thus, for \(E > E_k\), the curve has in general only one extremum (a minimum) (Fig. 1, \(v\)). Using formula (13), it is easy to see that the field \(E_k\) is equal to (for \(E=E_k\) equation (13) has a multiple root)

\[ E_k=\frac{4}{3^{3/2}}|\alpha|\,|P_0| = 4\beta_{\Theta} \left[ \frac{\alpha'_{\Theta}(\Theta-T)}{3\beta_{\Theta}} \right]^{3/2}, \tag{22} \]

where, in passing to the second expression, formula (8) has been used.

Up to now we have considered a monocrystalline, uniformly polarized ferroelectric. In fact, however, any isolated ferroelectric specimen in equilibrium and in the absence of a field is, on average, not polarized, but is divided into regions (domains) of spontaneous polarization. The division into regions may be regarded as being due to the finite size of the specimen, and it occurs for the same reason as the division into regions in ferromagnets. The point is that if a finite specimen is polarized, then bound electric charges appear on its surface, leading to the appearance of an electric field in the space surrounding the specimen. The energy of this field, whose density is equal to \(\dfrac{E^2}{8\pi}\), is added to the total free energy of the specimen, increasing it. Therefore it is energetically more advantageous for the specimen to split into domains, within each of which the polarization is homogeneous, but in different domains the directions are different, so that on average the polarization of the specimen is zero. Therefore, if one disregards the stray fields arising at the boundaries of domains, the field outside the specimen is zero. The sizes and configuration of the domains in a state of thermodynamic equilibrium are determined from the condition of minimality of the total free energy of the body, including its volume free energy, the surface energy at the domain boundaries, and the energy of the electric field.

It is very important to emphasize that the division of a ferroelectric into regions can, at least in principle, be eliminated by placing it in a capacitor. Indeed, let us consider a ferroelectric situated in a flat infinite capacitor, with the polarization directed perpendicular to the plates of the capacitor (Fig. 2). At the boundaries of the ferroelectric there are then bound charges with density \(\sigma_{\text{bound}}=\pm P\). If the plates of the capacitor are now charged with free charges of the same magnitude but of the opposite sign, then, evidently, the density of the total charge will be zero, and consequently both inside the capacitor and outside it the field \(E\) will be equal to zero\(^*\). Thus, in the case under consideration \(E=0\), \(P=P_0\), and the electric induction \(D=4\pi P_0\). The above discussion of the properties of ferroelectrics applies directly precisely to a specimen that has not broken up into domains, i.e. one placed in a capacitor and correspondingly “prepared.” If, however, the specimen is divided into domains, then the influence of an external field will affect not only changes in its “true” induced polarization, but also changes in the domain structure itself. This change is connected primarily with the growth of the relative weight of domains with polarization directed along the field at the expense of domains with the opposite direction of polarization (for definiteness we are speaking here of the case where there are only domains of the two indicated types, which, generally speaking, is of course incorrect). The polarization of ferroelectrics in a field, associated with the reorientation of domains and, in general, with a change in the domain structure, corresponds to the process of technical magnetization in ferromagnets. However, whereas in ferromagnets the process of “true” (induced) magnetization (i.e. the change of the magnetization \(M\) within a domain under the influence of an external field) usually, at least in weak fields, plays no role, in ferroelectrics the “true” polarization and susceptibility are very substantial, since the corresponding value of \(\varepsilon\) reaches many thousands. We shall return to this distinction in Section 5.

Fig. 2.

The polarization of ferroelectrics associated with a change in the domain structure is accompanied by usually irreversible processes and leads to hysteresis. Reorientation of individual domains in a number of cases—

\(^*\) To obtain the charge distribution indicated in Fig. 2, one may, for example, connect the capacitor into the circuit of some emf source at \(T>\Theta\). Subsequent cooling will lead to the appearance of polarization, but the field \(E\) will remain equal to zero outside the capacitor. The external emf source can then be disconnected, and the capacitor short-circuited.

occurs very sharply and in its manifestations is analogous to the Barkhausen effect in ferromagnets. The question of the character of the splitting of ferroelectrics into regions and of the influence of the field on the domain structure has been studied experimentally relatively little and has not yet been subjected to theoretical analysis; for this reason we are unable to dwell on it in greater detail.

Let us note that the presence of a large induced polarization near the Curie point makes it impossible in this region to determine the spontaneous polarization directly from hysteresis curves without special analysis. Indeed, for \(P_0 \sim 10^4\) and \(\varepsilon \sim 5 \cdot 10^3\), in a field

\[ E \sim 10 = 3 \cdot 10^3 \ \frac{\text{volts}}{\text{cm}}, \]

\[ P_{\mathrm{i}}=\frac{\varepsilon-1}{4\pi}E \sim 5 \cdot 10^3 \sim P_0; \]

whereas in experiment we measure \(P=P_0+P_{\mathrm{i}}\), and thus \(P_0\) cannot be determined from a single hysteresis curve (one should also not forget that in reality \(P_{\mathrm{i}}\) depends on \(E\), generally speaking, nonlinearly). The value of \(P_0\) can, in principle, be reliably determined by measuring the charge on a capacitor in which there is a ferroelectric not split into domains.

The consideration given above of the true susceptibility of ferroelectrics applied to the case of thermodynamic equilibrium, i.e. in any case to static fields. In an alternating field there will be dispersion, i.e. a dependence of \(\varepsilon\) on the frequency of the electric field \(\omega\). Since ferroelectric properties are connected with the crystal lattice, i.e. the polarization in a ferroelectric is caused by a change in the positions of the ions in the lattice, in the optical region of the spectrum the values \(\varepsilon=n^2\) (\(n\) is the refractive index) for ferroelectrics should not be anomalously large, as is indeed observed experimentally. Thus, over a broad interval of frequencies the dispersion in ferroelectrics is colossal—from \(\varepsilon \sim 10^3 \div 10^4\) in a static field to \(\varepsilon \sim 1\) in the optical part of the spectrum. Finding the dependence \(\varepsilon(\omega)\) is a very complex matter and requires a detailed analysis of lattice vibrations. Some considerations on this question are given in Section 3.

2. CASE OF A PHASE TRANSITION CLOSE TO THE CURIE CRITICAL POINT\(^6\)

As was already mentioned, in the case of ferroelectrics of the type \(KH_2PO_4\) there apparently occurs not an ordinary second-order transition, but either a second-order transition close to a first-order transition, or a first-order transition close to a second-order transition. This is indicated by the very sharp change in heat capacity near the transition point, the enormous value of the experimentally measured heat-capacity jump, equal to \(102 \ \frac{\text{cal.}}{\text{mole}\cdot\text{degree}}\) for \(KH_2PO_4\), and also by a number of other facts. One may therefore think that the experimentally observed continuity of the transition is connected with secondary causes (stresses, splitting into

domains), and in fact the transition is associated with the release of latent heat, as well as with a jump in \(P_0\) and a number of other quantities.

At the same time it must be emphasized that a substance will possess all the typical ferroelectric properties only in the case where the first-order transition occurring in it, associated with the appearance of spontaneous polarization, is close to a second-order transition. This is clear from general considerations (see, in particular, § 1) and follows from the consideration given below of a first-order ferroelectric transition close to a second-order transition. It must also be borne in mind that a first-order phase transition close to a second-order transition is practically very similar to a second-order transition close to a first-order transition. Therefore it is expedient to consider both these cases simultaneously.

The line of Curie points (points of second-order transitions) may, on the \(p,T\)-diagram, pass smoothly into a line of first-order phase transitions; moreover, the point separating transitions of both types is called the critical Curie point. At the critical Curie point, as follows from the general theory\(^3\), not only the coefficient \(\alpha\), but also the coefficient \(\beta\) in expansion (1)\(^*\) vanishes. Therefore expansion (1) near the critical Curie point must be supplemented by one more term, and it takes the form

\[ \Phi=\Phi_0+\alpha P^2+\frac{\beta}{2}P^4+\frac{\gamma}{6}P^6-EP, \tag{23} \]

where \(\gamma>0\).

First-order phase transitions close to second-order transitions, and second-order transitions close to first-order transitions, are evidently situated near the critical Curie point, to which there corresponds a certain temperature \(\Theta_c\), at which \(\beta=\beta_{\Theta_c}=0\). Hence it is clear that, in solving the problem posed above, one must start from expression (23).

Let us begin with the case of second-order transitions close to the critical Curie point. The consideration here is completely analogous to that carried out in § 1, and therefore we shall not repeat the corresponding simple calculations. From the conditions of minimality of \(\Phi\), i.e. the conditions \(\partial \Phi/\partial P=0\) and \(\partial^2\Phi/\partial P^2>0\), it follows that for \(T>\Theta\), where \(\alpha>0\), \(P_0=0\). For \(T<\Theta\), where \(\alpha<0\) and, as in the whole region of second-order transitions, \(\beta>0\), we have

\[ P_0^2=\frac{-\beta+\sqrt{\beta^2-2\alpha\gamma}}{\gamma}. \tag{24} \]

This formula for \(\gamma=0\) passes, of course, into (45) and, in addition, makes it possible to conclude that in (23) one must indeed assume,

\(^*\) For definiteness we shall immediately have in mind the case of transitions of a non-pyroelectric modification into a pyroelectric one.

that \(\gamma>0\). In (24) one may put \(\alpha=\alpha_\Theta(T-\Theta)\) and \(\gamma=\gamma_\Theta\); as regards the quantity \(\beta\), one can only note that in the immediate vicinity of the critical Curie point \(\beta=\beta'_{\Theta c}(T-\Theta_c)\). For the entropy below the Curie point, as in the case of an ordinary second-order transition, the expression \(S=S_0-P^2\alpha'_\Theta\) remains valid [cf. (9)], from which a formula for the heat capacity of the ordered (pyroelectric) phase \(c_p=T\left(\dfrac{\partial S}{\partial T}\right)_p\) may be obtained. On approaching the critical Curie point, as is easily seen from the formulas and is clear from the essence of the matter, the heat capacity \(c_p\) tends to infinity. Neglecting quantities that remain finite as \(\Theta\to\Theta_c\), we have

\[ c_p=\frac{T(\alpha')^2}{\sqrt{\beta^2-2\alpha\gamma}} . \tag{25} \]

Let us introduce the temperature \(\Theta_0\), at which, for the given pressure, \(\beta^2-2\alpha\gamma=0\); at the critical Curie point \(\Theta_0=\Theta_c\), since at this point \(\alpha=\beta=0\). Expanding the radical expression in (25) in powers of \((T-\Theta_0)\) and retaining the most important term, we obtain

\[ c_p=\sqrt{\frac{\Theta_c^2(\alpha'_{\Theta c})^2}{2\gamma_{\Theta c}(\Theta_0-T)}}= \frac{\mathrm{const}}{\sqrt{(\Theta_0-T)}} . \tag{26} \]

Above the Curie point, in weak fields, formulas (16) and (17) remain valid as before. Below the Curie point, instead of formula (19), we have

\[ \varepsilon=-\frac{\pi}{2\alpha+\beta P_0^2}\qquad (T<\Theta). \tag{27} \]

Let us pass to the case of a first-order transition close to a second-order transition. In this case, as in general for a first-order pyroelectric transition, the polarization \(P_0\) changes discontinuously, while the quantities \(\alpha\) and \(\beta\) at the transition point \(\Theta_1\) do not vanish, although they may be very small (the closeness of \(\alpha\) and \(\beta\) to zero is manifested in the closeness of \(\Theta_1\) to the critical Curie point \(\Theta_c\)). At \(T=\Theta_1\) both phases, the ordered (pyroelectric) and the disordered (nonpyroelectric), are in equilibrium and, thus, \(\Phi(P^2=P_c^2)=\Phi(P^2=0)\). Using this condition, formula (23), and the requirement of minimality \(\dfrac{\partial\Phi}{\partial P}=0\), which must hold for \(T\leq\Theta_1\), we find the expression for \(P_0^2\) at the transition point:

\[ P_0^2=-\frac{4\alpha}{\beta}\qquad (T=\Theta_1). \tag{28} \]

In addition, at \(T=\Theta_1\) the relation\(^3\) \(8\alpha\gamma=3\beta^2\) must be satisfied. The latent heat of the transition is equal to

\[ q=\Theta_1(S-S_0)=-\Theta_1P_0^2(\Theta_1)\alpha'_{\Theta_1}. \tag{29} \]

It is practically very difficult to distinguish the case when there is a large jump, and in general an anomaly of the heat capacity, from the case when latent heat of transition is released. It is therefore essential to point out that, if one introduces into consideration the sum \(\int_T^{\Theta_1} \Delta c\,dT - q = Q(T)\), then near the point of a transition of the first kind or the Curie point (in this case \(q=0\) and \(\Theta_1=\Theta\)):

\[ P_0^2=\frac{Q(T)}{\Theta\alpha'_{\Theta}}. \tag{30} \]

Formula (16) remains valid as before, but formula (17) is no longer correct. Near the point \(\Theta_1\) one may put \(\alpha=\alpha_{\Theta_1}+\alpha'_{\Theta_1}(T-\Theta_1)\) and, thus, instead of (17) we have*)

\[ \varepsilon=\frac{2\pi}{\alpha_{\Theta_1}+\alpha'_{\Theta_1}(T-\Theta_1)} \quad (T>\Theta_1). \tag{31} \]

As \(T\) tends to \(\Theta_1\) from the side of higher temperatures (i.e. from the region where \(P_0=0\)), \(\varepsilon\), as is clear from (31), tends to the value \(\varepsilon=\dfrac{2\pi}{\alpha_{\Theta_1}}\). The value of \(\varepsilon\) as \(T\to\Theta_1\) from the side of lower temperatures, where \(P_0\ne0\), as is easily shown using (28), is equal to

\[ \varepsilon=\frac{\pi}{2\alpha_{\Theta_1}} \quad (T=\Theta_1). \tag{32} \]

Thus the jump of \(\varepsilon\) at the transition point is equal to

\[ \Delta\varepsilon=\frac{3\pi}{2\alpha_{\Theta_1}}. \tag{33} \]

From the formulas given it is seen that if the value of \(\alpha_{\Theta_1}\) is small, which precisely indicates the closeness of the transition to the critical Curie point, then the dielectric constant near the transition point is large, and the entire behavior of the substance is close to that occurring in a transition of the second kind considered in Section 1.

3. ANISOTROPY OF DIELECTRIC PROPERTIES AND THE PIEZOEFFECT IN BARIUM TITANATE\(^7\)

The theory of the ferroelectric transition was developed above under the assumption that polarization occurs only along the ferroelectric axis (the axis along which the spontaneous polarization is directed). Such an approach is fully justified in the case of Rochelle salt and substances of the type \(KH_2PO_4\), where the indicated axis is singled out even above

*) Everywhere we assume that \(\varepsilon \gg 1\), and therefore write \(\varepsilon\) instead of \(\varepsilon-1\).

the Curie point, whereas the dielectric properties in directions perpendicular to this axis have no anomalous features. On the contrary, a ferroelectric such as barium titanate, above the Curie point, belongs to the cubic system and, consequently, for \(T>\Theta\) its dielectric properties, at least in weak fields, must be isotropic.

As a result, on approaching the Curie point the dielectric constant \(\varepsilon\) must tend to infinity as \(\dfrac{1}{T-\Theta}\) for any orientation of the crystal axes relative to the external field. From the nature of the ferroelectric transition it is further clear that below the Curie point the properties of \(\mathrm{BaTiO_3}\) must be “anomalous” not only along the ferroelectric axis, but in all directions. What has been said, as well as the desire to take into account the influence of stresses, i.e. the piezoelectric effect, makes it necessary to consider the properties of single crystals of barium titanate and related substances near the Curie point, taking account of anisotropy and elastic stresses.

In the case of an arbitrary crystal, its thermodynamic potential near the Curie point, for an arbitrary mutual orientation of the crystal axes, the polarization, the electric field, and the stresses, may be represented in the form

\[ \Phi=\Phi_0+\alpha_{ik}P_iP_k+\beta_{iklm}P_iP_kP_lP_m+S_{iklm}\sigma_{ik}\sigma_{lm}+ \]
\[ +\delta_{ikl}\sigma_{ik}P_l+\gamma_{iklm}\sigma_{ik}P_lP_m-E_iP_i, \tag{34} \]

where \(\sigma_{ik}\) is the stress tensor, \(P_i\) the polarization vector, \(E_i\) the electric-field strength, and summation is performed over twice-repeated indices. The coefficients \(\alpha,\beta,\gamma\), and \(\delta\) depend on temperature, and at the Curie point some of the coefficients \(\alpha\) are equal to zero.

Expression (34) is made more explicit and, generally speaking, substantially simplified when the symmetry of the crystal under consideration is taken into account. In doing so, in (34) one must take into account the symmetry of the crystal above the Curie point; the lowering of symmetry upon transition into the ferroelectric region, caused by the distortion of the lattice, is then obtained automatically. At a considerable distance from the Curie point toward lower temperatures, an expansion of the type (34), which generalizes expression (1), is insufficient, since the polarization \(P_i\) is large and, generally speaking, one cannot restrict oneself to expanding \(\Phi\) with accuracy up to terms of order \(P^4\).

In application to barium titanate and related substances, the lattice in (34) must be regarded as cubic, with a center of symmetry and axes of the 4th order (a perovskite-type lattice). Therefore, as is known, \(\delta_{ikl}=0\) (i.e. the ordinary piezoelectric effect is impossible), and if the coordinate axes (the axes \(x,y,z\)) are chosen as the axes of the cube, expression (34)

takes the form

\[ \begin{aligned} \Phi ={}& \Phi_0+\alpha\left(P_x^2+P_y^2+P_z^2\right) +\frac{\beta_1}{2}\left(P_x^4+P_y^4+P_z^4\right) \\ &+\beta_2\left(P_x^2P_y^2+P_x^2P_z^2+P_y^2P_z^2\right) +\frac{1}{2}S_{11}\left(\sigma_{xx}^2+\sigma_{yy}^2+\sigma_{zz}^2\right)\\ &+S_{12}\left(\sigma_{xx}\sigma_{yy}+\sigma_{xx}\sigma_{zz}+\sigma_{yy}\sigma_{zz}\right) +\frac{S_{44}}{2}\left(\sigma_{xy}^2+\sigma_{xz}^2+\sigma_{yz}^2\right)\\ &-\gamma_1\left(\sigma_{xx}P_x^2+\sigma_{yy}P_y^2+\sigma_{zz}P_z^2\right)\\ &-\gamma_2\left[\sigma_{xx}\left(P_y^2+P_z^2\right) +\sigma_{yy}\left(P_x^2+P_z^2\right) +\sigma_{zz}\left(P_x^2+P_y^2\right)\right]\\ &-2\gamma_3\left(\sigma_{xy}P_xP_y+\sigma_{xz}P_xP_z +\sigma_{yz}P_yP_z\right) -\left(E_xP_x+E_yP_y+E_zP_z\right). \end{aligned} \tag{35} \]

Here the coefficients \(S\) (elastic moduli) are written in the generally accepted notation with two indices; the coefficient \(\alpha\) in (35) is equal to \(\alpha_{11}=\alpha_{22}=\alpha_{33}\) in (34), etc.; the coefficients \(\gamma\) have been taken in (35) with a minus sign, for a certain convenience in what follows.

If \(\sigma_{ik}=0\) and the polarization is directed along one of the axes of the cube, say the \(z\)-axis, then for \(\Phi\) one obtains expression (1), used in § 1.

From the requirement that the potential \(\Phi\) be minimal (i.e., the equality \(\dfrac{\partial \Phi}{\partial P_i}=0\)), or, according to the formula
\[ E_i=\frac{\partial(\Phi+E_iP_i)}{\partial P_i}, \]
we immediately obtain \((P_1=P_x,\ P_2=P_y,\ P_3=P_z,\ \text{etc.})\):

\[ \begin{aligned} E_x={}&2\left[(\alpha-\gamma_1\sigma_{xx}-\gamma_2\sigma_{yy}-\gamma_2\sigma_{zz}) +\beta_1P_x^2+\beta_2\left(P_y^2+P_z^2\right)\right]P_x\\ &\qquad -2\gamma_3\left(\sigma_{xy}P_y+\sigma_{xz}P_z\right),\\ E_y={}&2\left[(\alpha-\gamma_2\sigma_{xx}-\gamma_1\sigma_{yy}-\gamma_2\sigma_{zz}) +\beta_1P_y^2+\beta_2\left(P_x^2+P_z^2\right)\right]P_y\\ &\qquad -2\gamma_3\left(\sigma_{xy}P_x+\sigma_{yz}P_z\right),\\ E_z={}&2\left[(\alpha-\gamma_2\sigma_{xx}-\gamma_2\sigma_{yy}-\gamma_1\sigma_{zz}) +\beta_1P_z^2+\beta_2\left(P_x^2+P_y^2\right)\right]P_z\\ &\qquad -2\gamma_3\left(\sigma_{xz}P_x+\sigma_{yz}P_y\right). \end{aligned} \tag{36} \]

Let us first assume that \(E_i=\sigma_{ik}=0\). Then equations (36) for \(P_i\) have three essentially different solutions:

\[ P_x=P_y=P_z=0, \tag{37.1} \]

\[ P_x^2=P_y^2=0,\quad P_{z0}^2=P_0^2=-\alpha/\beta_1, \tag{37.2} \]

\[ P_{x0}^2=P_{y0}^2=P_{z0}^2=-\frac{\alpha}{\beta_1+2\beta_2}; \quad P_0^2=-\frac{3\alpha}{\beta_1+2\beta_2}. \tag{37.3} \]

In (37.2) the \(z\)-axis may be replaced by the \(x\)- or \(y\)-axes; moreover, solutions are possible which differ in the signs of the components \(P_{i0}\).

In all, there are six solutions with polarization directed along the axes of the cube [solutions of type (37.2)] and eight solutions with polarization directed along the diagonals of the cube [solutions of type (37.3)]. No-

there are no other solutions, except (37.1–3), apart from the case of degeneracy (i.e. absence of anisotropy), when \(\beta_1=\beta_2=\beta\), \(P_0^2=-\dfrac{\alpha}{\beta}\), and the polarization may be directed arbitrarily. Below, for definiteness, it will be assumed that no degeneracy takes place. The solutions (37.1–3) ensure the extremality of the potential \(\Phi\); finding the solution corresponding to the case of thermodynamic equilibrium can be carried out if one takes into account that in this case the potential must be minimal, i.e. the inequalities \(\dfrac{\partial^2 \Phi_1}{\partial P_i^2}>0\) must be satisfied. If \(\alpha>0\), then for \(\beta_1>0\) the minimum corresponds only to the solution (37.1), i.e. spontaneous polarization is impossible. Therefore, in accordance with the general theory of phase transitions of the second kind, for the existence of a ferroelectric transition it is necessary that at some point (the Curie point) the coefficient \(\alpha\) vanish. Below the Curie point, for \(T<\Theta\), the solution (37.1) corresponds to a maximum, not to a minimum, and one of the solutions (37.2) or (37.3) with spontaneous polarization is realized. Experimentally, barium titanate below the Curie point possesses tetragonal symmetry\(^5\), i.e. the solution (37.2) is realized.

Assuming, for definiteness, that both solutions (37.2) and (37.3) are in general possible, i.e. \(\beta_1>0\) and \(\beta_1+2\beta_2>0\) (if these inequalities are not fulfilled, \(P_0^2<0\), since for \(T<\Theta\) \(\alpha<0\)), it is easy to see that the solution (37.2) corresponds to the absolute minimum of \(\Phi\) if

\[ \beta_2>\beta_1. \tag{38} \]

If, however, \(\beta_2<\beta_1\), then the absolute minimum corresponds to the solution (37.3), corresponding to the rhombohedral structure of the crystal below the Curie point.

In what follows we shall restrict ourselves to the study of the solution (37.2) and of the solutions adjacent to it in the presence of an electric field and stresses. In a weak field the induced part of the polarization \(P_{i\,\mathrm{in}}\) is proportional to the field, i.e. \(P_{i\,\mathrm{in}}=P_i-P_{i0}=\dfrac{\varepsilon_i-1}{4\pi}E_i\), where \(P_i\) is the total polarization, \(P_{i0}\) the spontaneous polarization, and \(\varepsilon_i\) the principal values of the tensor of dielectric constants introduced in this way.

From (36) it is easy to see that above the Curie point

\[ \varepsilon_x=\varepsilon_y=\varepsilon_z=\frac{4\pi}{\alpha} =\frac{4\pi}{\alpha_0(T-\Theta)} \qquad (T>\Theta), \tag{39} \]

where we write \(\varepsilon\) instead of \(\varepsilon-1\), since \(\varepsilon\gg 1\).

We note that below the Curie point the spontaneous polarization is directed along the \(z\)-axis and, according to (37.2), is equal to \(P_{z0}^2=-\dfrac{\alpha}{\beta_1}\).

Then

\[ \left. \begin{aligned} \varepsilon_x=\varepsilon_y&=-\frac{2\pi}{\alpha(\beta_2/\beta_1-1)} =-\frac{2\pi}{\alpha'_\Theta(T-\Theta)(\beta_2/\beta_1-1)},\\ \varepsilon_z&=-\frac{\pi}{\alpha} =-\frac{\pi}{\alpha'_\Theta(T-\Theta)} \qquad (T<\Theta). \end{aligned} \right\} \tag{40} \]

Inequality (38) ensures, of course, the positivity of the quantities \(\varepsilon_x=\varepsilon_y\).

The induced polarization depends on direction, and the crystal is isotropic in this respect only when \(\beta_2=3\beta_1\). In a strong field, which is readily attainable near the Curie point, the polarization depends nonlinearly on the field. Moreover, in a strong field the crystal is dielectrically anisotropic even above the Curie point; the simple relation between \(E\) and \(P\), as is clear from (36), holds only for fields directed along the axes of the cube—in this case \(E=2\alpha P+2\beta_1P^3\).

In the presence of polarization and stresses in the crystal, deformations arise,
\[ u_{ik}=-\frac{\partial\Phi}{\partial\sigma_{ik}}. \]
Thus, for example,

\[ \left. \begin{aligned} u_{xx}&=-\frac{\partial\Phi}{\partial\sigma_{xx}}\\ &=s_{11}\sigma_{xx}-s_{12}(\sigma_{yy}+\sigma_{zz}) +\gamma_1P_x^2+\gamma_2(P_y^2+P_z^2). \end{aligned} \right\} \tag{41} \]

It is important to emphasize that in the ferroelectric region the deformations are not equal to zero even in the absence of stresses and an electric field, since in this case as well \(P^2=P_0^2\ne0\). The presence of such spontaneous deformations leads to a lowering of the symmetry of the lattice below the Curie point.

Let us now consider the piezoelectric effect in barium titanate, i.e. the appearance of induced polarization under the influence of stresses. This effect is fully reflected in equations (36). Suppose, for example, that only the stress component \(\sigma_{zz}\) is nonzero, and that the electric field is zero. Then system (36) takes the form

\[ \left. \begin{aligned} [(\alpha-\gamma_2\sigma_{zz})+\beta_1P_x^2+\beta_2(P_y^2+P_z^2)]P_x&=0,\\ [(\alpha-\gamma_2\sigma_{zz})+\beta_1P_y^2+\beta_2(P_x^2+P_z^2)]P_y&=0,\\ [(\alpha-\gamma_1\sigma_{zz})+\beta_1P_z^2+\beta_2(P_x^2+P_y^2)]P_z&=0. \end{aligned} \right\} \tag{42} \]

We see that, as applied to the solution of interest to us, corresponding to spontaneous polarization directed along an axis of the cube, the stress leads simply to a displacement of the Curie point. This displacement is different for polarization of the crystal along the \(z\)-axis, i.e. along the direction of action of the elastic force, and along the \(x,y\) axes.

In the first case

\[ P_z^2=-\frac{\alpha-\gamma_1\sigma_{zz}}{\beta_1},\qquad P_x=P_y=0; \tag{43} \]

in the second

\[ \left. \begin{gathered} P_x^2=-\frac{\alpha-\gamma_1\sigma_{zz}}{\beta_1},\quad P_y=P_z=0,\\[6pt] \text{or}\\[6pt] P_y^2=-\frac{\alpha-\gamma_1\sigma_{zz}}{\beta_1},\quad P_x=P_z=0. \end{gathered} \right\} \tag{44} \]

We cannot indicate the signs of the coefficients \(\gamma_1\) and \(\gamma_2\). When spontaneous polarization arises, the crystal must evidently expand in the direction of this polarization; if this is so, then \(\gamma_1>0\).

The effect of stresses may be characterized by the difference \(P_{\mathrm{i}}=P-P_0\), where \(P_0\) is the spontaneous polarization in the absence of stresses. Assuming that \(\gamma_1>0\), in the case (43) we have:

\[ \left. \begin{aligned} T>\Theta:\quad & P_{z\mathrm{i}}=P_z=\sqrt{\frac{\gamma_1\sigma_{zz}-\alpha}{\beta_1}},\\[6pt] T<\Theta:\quad & P_{z\mathrm{i}}=-P_{z0}+\sqrt{P_{z0}^{\,2}+\frac{\gamma_1\sigma_{zz}}{\beta_1}}, \end{aligned} \right\} \tag{45} \]

where \(\Theta\) is the Curie-point temperature in the absence of stresses. The polarization \(P_{z\mathrm{i}}\) is maximal at the Curie point, where \(\alpha=0\), \(P_0=0\), and

\[ P_{z\mathrm{i}}=\sqrt{\frac{\gamma_1\sigma_{zz}}{\beta_1}}. \]

At some distance from the Curie point, where \(P_{z\mathrm{i}}^2\ll P_{z0}^{\,2}\),

\[ P_{z\mathrm{i}}=\frac{\gamma_1\sigma_{zz}}{2\beta_1P_{z0}} = \frac{\gamma_1\sigma_{zz}}{2\sqrt{\beta_1\alpha_\Theta(\Theta-T)}}. \tag{46} \]

In the region where formula (46) is valid, the piezopolarization is proportional to the stress \(\sigma_{zz}\), and the piezoeffect has the character of an ordinary linear piezoeffect with an effective piezomodulus

\[ d_{ef}=\frac{\gamma_1}{2\beta_1P_{z0}}. \tag{47} \]

If the stress acts in a direction perpendicular to the polarization \(P_{z0}\), then in (47) \(\gamma_1\) must be replaced by \(\gamma_2\).

As is clear from formula (35), the piezoeffect in barium titanate is associated not with a term proportional to \(P\), as occurs in the ordinary linear piezoeffect, but with terms proportional to \(P^2\) in the expression for \(\Phi\). Therefore the piezoeffect in barium titanate, despite the fact that under certain conditions it is analogous to the linear piezoeffect, may be called a quadratic piezoeffect, or even electrostriction, since this effect is analogous to magnetostriction in ferromagnets and, in particular, in iron.

All that has been said above applies, of course, to single crystals, or, more precisely, even to a single homogeneously polarized region (domain) of a single crystal. In experiment, however, one usually deals with polycrystal-

ceramic BaTiO\(_3\) \({}^{2}\), although single crystals have also already been obtained and have begun to be studied \({}^{5}\).

To construct a fully reliable theory for polycrystals is hardly possible, and here one can hope only for the consideration of limiting cases or for the derivation of approximate formulas. The difficulty of the problem is connected with the fact that the stray fields inevitable in a polycrystal are very large

\[ \left( E \sim P_0 \sim 10^6 \frac{\text{volt}}{\text{cm}} \right) \]

and at the same time, near the Curie point, the constant \(\varepsilon\) is also large; moreover, stresses must arise in the polycrystal, leading to additional complications because of the strong piezoelectric effect. The theory of the piezoelectric effect (electrostriction) in barium titanate ceramics is the subject of work \({}^{9}\). Here, with regard to polycrystals, we shall confine ourselves to only one remark.

In a strong field the ceramic may be polarized to a state in which its total spontaneous polarization no longer changes. If such a specimen is placed in a capacitor, then it must remain—and in experiment does remain—polarized when the external field is switched off (see § 1 and \({}^{5}\)). In a maximally polarized polycrystal, if the influence of internal stray fields is not taken into account, the spontaneous polarization in each grain must be directed along the axis of the cube nearest to the direction of the external field. The largest angle between the field and the nearest cube axis is approximately \(55^\circ\) and corresponds to the case when the field is directed along the diagonal of the cube. If the crystallites are distributed chaotically and are polarized along the axis nearest to the field direction, then the polarization \(P_{\max} \simeq 0.8 P_0\), where \(P_0\) is the spontaneous polarization in a single crystal. Under similar conditions the polarization \(P_{\max}\), as we see, is very close to \(P_0\), and the specimen may, in a dielectric sense, be regarded to a certain approximation as a quasi-single crystal. Observation of the anisotropy of the polarizability, as well as other measurements in such specimens, should be of undoubted interest.

Let us now dwell on the question of the dispersion of the dielectric constant of barium titanate. This problem, as already indicated in § 3, is considerably more complicated than the one considered, since it does not admit a purely thermodynamic treatment. In essence, it can be solved only on the basis of an investigation of the vibrations of the crystal lattice. However, some statements and estimates of the dispersion can be made even without such a consideration. For simplicity we shall restrict ourselves to the case when the field is directed parallel to the spontaneous moment, i.e. along the corresponding axis of the cube. Then, according to (36), in the absence of stresses in the static case,

\[ 2\alpha P + 2\beta_1 P^3 = E, \tag{48} \]

where the index \(z\) is omitted.

Comparing (48) with the equation of motion of an anharmonic oscillator under the influence of an external field, i.e. with the equation

\[ m\ddot{\xi}+r\dot{\xi}+k\xi+s\xi^3=eE, \]

we see that in the static case \(2\alpha=\dfrac{k}{e^2N}\) and \(2\beta_1=\dfrac{s}{e^4N^3}\), since \(P=eN\xi\), where \(e\) is the charge and \(N\) the number of cells per unit volume. By \(\xi\) here is meant, evidently, a generalized coordinate corresponding to the displacement of the Ba atom relative to the Ti and O atoms. From what has been said it is clear that, if the crystal is regarded as an aggregate of the indicated anharmonic oscillators, then for an alternating field \(E=E_0 e^{i\omega t}\), instead of (15) we would have the equation

\[ \mu \ddot{P}+\nu \dot{P}+\alpha P+\beta_1 P^3=\frac{E_0}{2}e^{i\omega t}, \tag{49} \]

where, within the framework of the adopted model,

\[ \mu=\frac{m}{2e^2N}, \qquad \nu=\frac{r}{2e^2N}. \]

In fact, of course, the polarization is determined by various vibrations of the lattice, and formula (49), generally speaking, is not valid. However, at low frequencies this formula may be regarded as the result of expanding \(P\) in a series in powers of the frequency (since \(\ddot{P}=-\omega^2P\), \(\dot{P}=i\omega P\), etc.). In this sense formula (49) is certainly valid so long as the term \(\mu\ddot{P}+\nu\dot{P}\) is a correction term, i.e. substantially smaller than the static terms. From (49), as is easy to see, it follows that

\[ \left. \begin{aligned} T>\Theta:\quad \varepsilon&=\frac{2\pi}{\alpha+i\omega\nu-\omega^2\mu},\\ T<\Theta:\quad \varepsilon&=\frac{\pi}{-\alpha+i\omega\nu-\omega^2\mu}. \end{aligned} \right\} \tag{50} \]

Above, formula (49) was obtained not directly as a result of an expansion in a frequency series, but starting from a crude oscillator model, in order to have the possibility of estimating the magnitude of \(\mu\). Indeed, from this model it follows that \(\mu\sim\dfrac{m}{2e^2N}\), where \(m\sim10^{-22}\) gram is the reduced mass for a system consisting of a Ba atom and a TiO\(_3\) group. Since the lattice constant of BaTiO\(_3\) is \(a\simeq4\text{ Å}\), \(N\simeq2\cdot10^{22}\), and \(\mu\sim10^{-26}\) (\(1/\mu\) is, in order of magnitude, the square of the corresponding Born frequency of the crystal). As for the quantity \(\nu\), if one abstracts from the defect conductivity of the specimen, which makes its own contribution to the damping, one may think that the value of \(\nu\) is small. If for \(T>\Theta\) in a static field \(\varepsilon\sim5000\), then \(\alpha\sim10^{-3}\) and, as is clear from (50), dispersion will begin to manifest itself at \(\omega\sim5\cdot10^{10}\) or

\[ \lambda=\frac{2\pi c}{\omega}\sim 4\ \text{cm} \]
(in this case \(\omega^{2}\mu\sim 3\cdot 10^{-5}\sim 0.03\alpha\)). If \(\mu=10^{-26}\) and \(\alpha=10^{-3}\), then \(\alpha=\omega^{2}\mu\) for \(\omega\simeq 3\cdot 10^{11}\) \(\left(\lambda=\frac{2\pi c}{\omega}\simeq 0.6\ \text{cm}\right)\). In fact, as follows from experiment (see \(^{5}\)), \(\alpha\sim \omega^{2}\mu\) for \(\lambda\simeq 3\ \text{cm}\), and thus \(\mu\sim 10^{-24}\div 10^{-25}\), i.e., the Born frequency of interest to us is lower than according to the rough estimate made above. This circumstance, of course, should not be surprising. The value of formula (50) lies in the fact that it shows in what direction the magnitude of the static dielectric constant affects the dispersion. Thus, before the corresponding consideration, one might have thought that the dispersion is especially large when \(\omega^{2}\mu\sim 1\). On the contrary, as is clear from formulas (50), resonance occurs when \(\omega\mu^{2}\sim \alpha\), and thus the smaller the coefficient \(\alpha\), or, what is the same, the larger the static value \(\varepsilon\), proportional to \(1/\alpha\), the more sharply the dispersion is expressed.

For \(T<\Theta\), the dispersion considered in barium titanate pertains only to the induced polarization, not connected with reorientation of regions of spontaneous polarization. The reorientation of domains in some cases makes a very substantial contribution to the measured value of the polarization and of the differential dielectric constant. The question of the dispersion of this part of the polarization has not yet been considered. One of the possible parameters determining this dispersion may be the time of propagation of sound along a domain, i.e., the time
\[ \tau\sim \frac{l}{c}, \]
where \(l\) is the size of the domain and \(c\) is the speed of sound. For \(l\sim 0.1\ \text{cm}\) and \(c\sim 10^{5}\ \frac{\text{cm}}{\text{sec}}\), \(\tau\sim 10^{-6}\) and \(\omega_{0}=\frac{2\pi}{\tau}\sim 10^{5}\). In any case there is no doubt that the process of reorientation of regions is slower than the process of establishing the induced polarization. Therefore, at sufficiently high frequencies only induced polarization should be observed. The induced polarization must, moreover, appear in arbitrarily weak fields, whereas reorientation of regions in a sufficiently weak field may be strongly slowed down. The remarks made indicate a way by which the induced polarization can be separated from the total polarization in ferroelectrics divided into domains.

4. PROPERTIES OF ROCHELLE SALT

In §§ 1 and 2 the behavior was considered of ferroelectrics possessing, above the Curie point, a certain distinguished axis, which may be called ferroelectric (along this axis the spontaneous polarization is directed in the ferroelectric region). Such an axis is possessed by Rochelle salt, which in the non-ferroelectric region belongs to the rhombic system (class \(V=D_{2}\); ferroelectric axis — the \(a\) or \(X\) axis), and by substances of the type \(KH_{2}PO_{4}\), which above the Curie point belong to the tetragonal system (class \(V_{d}=D_{2d}\); ferroelectric ...).

axis—the \(c\) axis or \(Z^*\)). In the ferroelectric region Rochelle salt belongs to the monoclinic system, and \(\mathrm{KH_2PO_4}\) to the rhombic system. All these substances possess piezoelectric properties already outside the ferroelectric region, and consideration of their piezoeffect is of great importance. The latter is connected primarily with the fact that, at the Curie point, the piezomodulus \(d_{14}\) in Rochelle salt and the piezomodulus \(d_{36}\) in the case of \(\mathrm{KH_2PO_4}\) tend to infinity (the piezomoduli \(d_{25}\) and \(d_{36}\) in Rochelle salt and the piezomoduli \(d_{14}=d_{25}\) in the case of \(\mathrm{KH_2PO_4}\) behave normally). Meanwhile, in §§ 1 and 2 the piezoeffect was explicitly not taken into account, which does not change the results obtained there, but makes them valid only in the absence of stresses. The latter is not entirely obvious, since when the variables \(P_i\) and \(\sigma_{ik}\) are used, certain coefficients, namely the “anomalous” piezomoduli, tend to infinity as \(T=\Theta\). Nevertheless, if \(\sigma_{ik}=0\), expansion (1) is valid, as follows, in particular, from what is given below.

In order to examine in full the properties of a ferroelectric single crystal near the Curie point, it is necessary to expand the potential \(\Phi\) in a series in \(P_i\) and the stresses or strains. In the case of \(\mathrm{BaTiO_3}\) this was done in § 3 in the variables \(P_i\) and \(\sigma_{ik}\). In the case of substances in which some piezomodulus tends to infinity, it is necessary to choose as variables the polarization \(P_i\) and the strains \(u_{ik}\).

Let us dwell from this point of view on the properties of Rochelle salt **). For simplicity we shall take the field to be directed along the ferroelectric axis \(X\) and shall restrict ourselves to consideration of the shear strain \(u_{23}=y_z\), which alone is of interest, since it behaves anomalously owing to the tendency to infinity, as \(T\to\Theta\), of the piezomodulus \(d_{14}\). Taking this into account, the potential \(\Phi\) near each of the Curie points of Rochelle salt may be represented in the form

\[ \Phi=\Phi_0+\alpha_1 P_x^2+\frac{\beta}{2}P_x^4+\frac{1}{2}c_{44}y_z^2+f_{14}y_zP_x-E_xP_x, \tag{51} \]

where it has been taken into account that the crystal lattice is rhombic. By the formulas

\[ E_x=\frac{\partial(\Phi+E_xP_x)}{\partial P_x} \quad\text{and}\quad Y_z=\frac{\partial\Phi}{\partial y_z}, \]

where \(Y_z\equiv\sigma_{23}\) is the corresponding stress, we obtain

\[ E_x=2\alpha_1P_x+2\beta P_x^3+f_{14}y_z, \tag{52} \]

\[ \text{*) A detailed survey of the properties of Rochelle salt, as well as of substances of the type } \mathrm{KH_2PO_4}, \text{ is contained in } {}^4. \]

\[ \text{**) Below in this paragraph some results obtained by A. V. Rzhanov are used.} \]

$$ Y_z=-f_{14}P_x-c_{44}y_z . \tag{53} $$

In the non-ferroelectric region, in a weak field, the term with \(P^3\) in (52) may be neglected, and this equation takes the form

$$ E_x=2a_1P_x+f_{14}y_z \qquad (T>\Theta_h,\; T<\Theta_l). \tag{54} $$

With the aid of (53) and (54) one can express \(P_x\) and \(y_z\) in terms of \(E_x\) and \(Y_z\):

$$ \left. \begin{aligned} P_x&=k_1E_x+d_{14}Y_z,\\ y_z&=-d_{14}E_x-s_{44}Y_z,\\ T&>\Theta_h,\; T<\Theta_l, \end{aligned} \right\} \tag{55} $$

where

$$ \left. \begin{aligned} s_{44}&=\frac{2a_1}{D_{14}}=\frac{1}{\,c_{44}-\dfrac{f_{14}^{2}}{2a_1}\,};\qquad d_{14}=\frac{f_{14}}{D_{14}}=\frac{1}{\,\dfrac{2a_1c_{44}}{f_{14}}-f_{14}\,},\\[6pt] k_1&=\frac{c_{44}}{D_{14}}=\frac{1}{\,2a_1-\dfrac{f_{14}^{2}}{c_{44}}\,};\qquad D_{14}=2a_1c_{44}-f_{14}^{2}=\frac{1}{\,k_1s_{44}-d_{14}^{2}\,}. \end{aligned} \right\} \tag{56} $$

In experiment the quantities \(k_1,d_{14}\), and \(s_{44}\), which have an obvious meaning, are measured directly (for example, \(k_1\) is the dielectric susceptibility of the crystal \(\dfrac{\varepsilon-1}{4\pi}\) in the absence of stresses, since then \(Y_z=0\)).

If the stresses are absent, then equation (53) gives a relation between \(P_x\) and \(y_z\); this relation can be used to eliminate \(y_z\) from (51). As a result we obtain

$$ \Phi=\Phi_0+\left(a_1-\frac{1}{2}\frac{f_{14}^{2}}{c_{44}}\right)P_x^{2}+\frac{\beta P_x^{4}}{2}-E_xP_x . \tag{57} $$

This expression coincides with (1), if one sets

$$ \alpha=a_1-\frac{1}{2}\frac{f_{14}^{2}}{c_{44}} . \tag{58} $$

In other words, as has already been indicated, in the absence of stresses the results of § 1 are fully valid. At the Curie points expression (58) vanishes, as a result of which the coefficients \(s_{44}\), \(d_{14}\), and \(k_1\) tend to infinity at these points [see (56)]. The coefficient \(k_1=\dfrac{\varepsilon-1}{4\pi}\) has, of course, the same value as in the case (16), i.e. \(k_1=\dfrac{1}{2\alpha}\).

The consideration of the ferroelectric region near the Curie points reduces to the investigation of equations (52) and (53) with allowance for the term \(2\beta P_x^3\). We shall confine ourselves here only to giving the expressions for the spontaneous polarization and strain, i.e., the values \(P_x=P_{x0}\) and \(y_z=y_{z0}\) for \(E_x=Y_z=0\):

\[ P_{x0}^2=-\frac{\alpha}{\beta} =-\frac{\alpha_1-\left(\dfrac{1}{2}\dfrac{f_{14}^{\,2}}{c_{44}}\right)}{\beta}, \qquad y_{z0}=-\frac{f_{14}}{c_{44}}\cdot P_{x0}. \tag{59} \]

A characteristic feature of Rochelle salt is that it has two Curie points. This means that the function \(\alpha(T)\) is negative only in a certain temperature interval between the Curie points \(\Theta_h\) and \(\Theta\) (Fig. 3). Formula (51) is strictly applicable only in the vicinity of each of the Curie points. However, in the case of Rochelle salt, taking into account the relative smallness of the quantity \(P_0\) and the narrowness of the ferroelectric region, it is apparently possible everywhere to restrict oneself to the expansion (51) without supplementing it by terms of order \(P^6\), etc. In this case, of course, to put \(\alpha=\alpha_\Theta'(T-\Theta)\), as we did in § 1, remains possible only near the Curie points. Throughout the whole ferroelectric region, however, we do not know the function \(\alpha(T)\), except for its general form shown in Fig. 3.

Fig. 3.

A detailed theoretical consideration of the properties of Rochelle salt was undertaken in due course by Mueller,\(^{10}\) the results of which are also set out in detail in \(^{4}\). In these works, which are phenomenological in character, the expansion (51) was used, sometimes, to be sure, in a somewhat complicated form. However, Mueller does not rely on the general theory of phase transitions of the second kind and introduces a complicated and confused terminology and notation. This leads, in our opinion, to considerable complication and confusion, which are entirely unjustified by the substance of the matter. It therefore seems advisable once again to carry out a complete and consistent analysis of the properties of Rochelle salt on the basis of relation (51), or of an analogous expression taking into account all components of the polarization vector and the strain tensor.

At the same time it is important to determine, using experimental data, to what extent one may restrict oneself to this expansion (i.e., without terms of order \(P^6\), etc.) over the whole ferroelectric interval; and, finally, the whole question must be set out without unnecessary complications, within the framework of the general scheme of a ferroelectric phase transition

of the second kind, similarly to how this was done in § 3 and outlined above in this section. All this work, which it is desirable and even necessary to carry out in inseparable connection with the discussion of the experimental data, has not yet been done.

5. PYROELECTRICS, FERROELECTRICS

AND FERROMAGNETICS

Ferroelectrics are a special case of pyroelectrics, i.e., crystals that possess, in the state of thermodynamic equilibrium, a spontaneous electric polarization. The peculiarities of ferroelectrics are due to the fact that in them there occurs a phase transition of the second kind (or a transition of the first kind close to it) from a non-pyroelectric modification to a pyroelectric one. In other words, a ferroelectric is a pyroelectric only in some temperature range, narrower than the region of existence of the crystalline phase of the substance under consideration. The question of why the presence of a phase transition of the indicated type leads to the appearance of typical ferroelectric properties was clarified in § 1.

It is now essential to understand at least qualitatively why pyroelectrics exist, i.e., why in some cases the state of a crystal with spontaneous polarization proves to be thermodynamically stable. This question is most easily clarified on the basis of simple model considerations. Namely, suppose that the crystal lattice is formed by point electric dipoles with moment \(p\) (the formation and stability of such a lattice may be ensured not only by dipole forces but also by any other forces, which are not of interest to us at present). Within the framework of such a model, the concept of an effective, or acting, electric field \(E_{ef}\) has an exact meaning, i.e., the average field acting on an individual dipole. If the average macroscopic field \(E\) and the polarization \(P\) are parallel (for example, directed along the symmetry axis of the lattice), then

\[ E_{ef}=E+fP, \tag{60} \]

where \(f\) is the so-called internal-field factor (for a cubic lattice \(f=\dfrac{4\pi}{3}\)). The polarization of the “crystal” under consideration in an external electric field may be treated in the same way as the polarization of a gas of dipolar molecules, or as the magnetization of a spin gas in ferromagnetics. Since we pursue here only illustrative aims, we shall not take anisotropy into account and shall apply to the case under discussion, without any changes, the Weiss theory of ferromagnetism, with the obvious replacement of the magnetic field \(H\) by the electric field \(E\), and of the magnetization \(M\) by the polarization \(P\) (see,

for example, \(^{1,11,12}\). As a result we obtain

\[ \frac{P}{P_{\infty}}=L(a),\quad a=\frac{pE_{\mathrm{ef}}}{kT}=\frac{pE+fpP}{kT}, \tag{61} \]

where \(P_{\infty}\) is the polarization at saturation (i.e., at \(T=0\)), \(k=1.38\cdot 10^{-16}\) is Boltzmann’s constant, \(p\) is the electric dipole moment of the dipoles under consideration, and \(L\) is the Langevin function, which for simplicity may be taken in the classical form, i.e., one may put \(L(a)=L_{\infty}=\operatorname{cth}a-\frac{1}{a}\) (if only two mutually opposite orientations of the dipoles are possible, then \(L=\operatorname{th}a\)).

The solution (61) is obtained as the result of a static consideration and corresponds to the case of thermodynamic equilibrium. As is known, it follows from (61) that even in the absence of an external field there is in the “crystal” a certain spontaneous polarization \(P_0\), different from zero in the temperature interval from \(T=0\) to the Curie-point temperature \(\Theta\), equal (for \(L=L_{\infty}\)) to

\[ \Theta=\frac{fpP_{\infty}}{3k}. \tag{62} \]

For \(T>\Theta\), \(P_0=0\); near the Curie point the behavior of our model is completely analogous to the behavior of ferroelectrics (there are the Curie–Weiss law for the susceptibility, a jump in the heat capacity, etc.). Taking \(f\sim 1\), \(p\sim 10^{-18}\), and \(P_{\infty}\sim 10^4 \div 10^5\), i.e., taking quite reasonable values for all quantities,* from (62) we find that \(\Theta\sim 10^2 \div 10^3\) degrees. Thus, we see that the difference between the acting field and the average macroscopic field directly leads to the possibility and, within the framework of the chosen model, even the necessity of the existence of spontaneous polarization, i.e., in the terminology adopted above, to the existence of a pyroelectric phase. An essential point, in connection with which we have dwelt somewhat more fully on the chosen model of the crystal and have not limited ourselves to a simple reference to the theory of ferromagnetism, is that in the electric case, in contrast to ferromagnetism, for the temperature \(\Theta\) one directly obtains a value \(\sim 10^3 \div 10^4\), corresponding to the real Curie temperatures in ferroelectrics. In other words, the ordinary classical electrostatic interaction, leading to values \(f\sim 1\), can ensure the existence of spontaneous polarization up to

\[ \text{* For a cubic lattice of point dipoles } f=\frac{4\pi}{3}. \]
The moment of polar molecules, or the dipole moment referred to one cell of a pyroelectric crystal, is usually of the order \(10^{-18}\). The polarization \(P_{\infty}=pN\), where \(N\) is the number of molecules or the total number of dipoles considered with moment \(p\) per unit volume; if \(N\sim 10^{23}\), \(P_{\infty}\sim 10^5\).

temperatures \(\Theta \sim 10^3\). In the case of ferromagnetism, however, as is known, within Weiss’s theory one must take for \(f\) a value \(\sim 10^3 \div 10^4\), which cannot be due to magnetic interaction. Ferromagnets will be discussed further below.

Real crystals, of course, are very far from the dipole model used. If molecular crystals are not considered, then in a solid one cannot speak of separate dipoles located at the lattice sites. The polarization, for example, of ionic crystals is due to the displacement of ions of different signs from their equilibrium positions, as a result of which each unit cell of the crystal acquires a certain dipole moment. The presence of an internal field \(fP\) makes the existence of polarization energetically favorable even in the absence of a field \(E\), since the electric energy per unit volume in the first approximation is equal to \(-\dfrac{f}{2}P^2\). However, the appearance of polarization is associated with deformation of the lattice and leads to an increase in elastic energy*); in a rough approximation, for small \(P\) the elastic energy is proportional to \(P^2\), since the polarization \(P\) is proportional to certain components of the strain tensor. The question of whether or not a given crystal will be a pyroelectric, i.e., whether it will possess spontaneous polarization, depends on the relative role of the energy associated with the internal field and of the elastic energy.

The number of pyroelectrics is relatively small, since homeopolar crystals, naturally, do not belong to their number. A number of ionic crystals are pyroelectrics, such as tourmaline, lithium sulfate, etc. In a pyroelectric there must be one non-equivalent direction, along which the polarization is precisely directed. Among the possible 32 crystal classes only 10 classes have such a direction and thereby allow the existence of spontaneous polarization. However, strictly speaking, this circumstance does not lead to any restrictions, since for the existence of polarization it is sufficient to have a change of symmetry caused by an arbitrarily small corresponding deviation of the lattice from a more symmetric lattice not permitting the presence of polarization (this is just what happens in ferroelectrics, especially near the Curie point).

Under natural conditions, pyroelectric crystals do not have electric “poles,” i.e., they do not have a resultant electric moment. This is explained by a number of reasons. Suppose that, by some means, a certain homogeneously polarized pyroelectric specimen has been obtained. In this case, on the surface of the specimen there must

*) The fact that elastic energy is ultimately also reducible to electrostatic energy is, here, obviously, immaterial. Moreover, for \(T > 0\) one should speak not only of energy, but also of free energy, or thermodynamic potential.

there are bound electric charges with surface density \(\sigma=-P_n\) (\(P_n\) is the normal component of \(P\)), and if free charges are absent, the electric induction \(D=E+4\pi P=0\) and, consequently, \(E\ne0\). Since the specimen always possesses some nonzero conductivity, the presence of the field will cause the appearance of a current, which will flow until the free charges formed on the surface of the body lead to the disappearance of the field \(E\) in the specimen. Ions settling on the surface of the specimen from the air act in the same direction. In addition to what has been said, it must be borne in mind that in the process of formation of a pyroelectric specimen, for example upon crystallization from a melt under equilibrium conditions and in the absence of an external field, the crystal must break up into domains polarized in different directions. The reasons for such splitting have already been indicated in § 1. To what was said there one can only add that the splitting must also take place for reasons of symmetry. Indeed, in the absence of an external field and, in general, of any distinguished direction in the state of thermodynamic equilibrium, the crystal obtained as a result of crystallization of a liquid phase, which is isotropic, must on the average also remain isotropic. This remark applies to all properties of the crystal, and not only to the question of spontaneous polarization. The well-known twinning in quartz, which leads to the disappearance of the piezoelectric effect, is due to this same reason. (In the case of pyro- and ferroelectrics, domains with different directions of polarization may likewise be called twins.) Of course, upon deviation from the state of equilibrium and, in particular, under nonequilibrium crystallization, as always takes place to one degree or another, the formation of even considerable regions of a crystal with homogeneous polarization and without any twins is possible.

For all the reasons indicated, pyroelectric crystals under ordinary conditions do not have a total electric moment, although the polarization \(P_0\) in them is not equal to zero. Upon heating, the magnitude of the polarization \(P_0\) changes, and this change of polarization \(\Delta P\) with temperature can be observed experimentally (whence, as is known, the term “pyroelectric” derives).* Usually pyroelectrics (not ferroelectrics) remain such up to the melting point of the crystal, or at some temperature pass into a nonpyroelectric crystalline phase as a result of a first-order phase transition. As was indicated in § 1, in such pyroelectrics no anomalies of the dielectric susceptibility should be observed, and if one abstracts from the pyroelectric effect, as well as from the piezoelectric effect present in all pyroelectrics, such “ordinary” pyroelectrics—

* The polarization \(\Delta P\) can be observed by compensating the arising electric field \(\Delta E=-4\pi\Delta P\). With such compensation in the specimen, as before heating, \(E=0\) and no conduction current arises.

crystals do not differ in their behavior from other ionic crystals. This fact is to a considerable extent connected with the fact that in an “ordinary” pyroelectric, broken up, say, into domains, the character of the domain structure practically does not change under the influence of an external field.

This is explained in the following way. In the absence of an external field, each region of the crystal may with equal justification be polarized in either of two mutually opposite directions, since the energy of the crystal is the same for \(P=+P_0\) and \(P=-P_0\) (see Fig. 1, \(a\)). The transition from one state to the other is effected by a definite displacement of the individual ions forming the lattice (a change of polarization \(P\) into \(-P\) takes place, in particular, when the radius-vectors of all ions \(\mathbf r\) are changed into \(-\mathbf r\), i.e., when all lattice sites are inverted with respect to some center). The presence of an electric field \(E\), directed along \(P\), makes the state with polarization \(P\) more favorable than the state with polarization \(-P\). However, both these states are separated by a potential barrier which, generally speaking, is very high. The height of this barrier is of the order
\[ U\sim pE_0\sim 10^{-12}, \]
where \(p\sim 10^{-18}\), and \(E_0\sim 10^6\) is the microscopic electric field in the crystal (\(E_0\sim e/d^2\), where \(d\sim 10^{-8}\) is the interatomic distance and \(e=4.8\cdot 10^{-10}\) is the charge of a monovalent ion).

To the value \(U\sim 10^{-12}\) there corresponds a temperature
\[ \Theta_0=\frac{U}{k}\sim 10^4 \]
degrees. For accessible external fields \(E\sim 10\div 100\) (i.e. \(E\sim 3000\div 30000\ \text{V}/\text{cm}\)) such a high barrier is practically not deformed, and at a low temperature \(T\ll\Theta_0\) reorientation of the regions does not occur, or, more precisely, should occur on the average only over a very long time.

The specific properties of ferroelectrics, as has already been indicated, are due to the presence in them of a second-order phase transition from the non-pyroelectric modification to the pyroelectric one at the temperature \(\Theta\sim 10^2\). Near the Curie point the barrier separating the states with polarizations \(\pm P_0\) is relatively very low (at \(T=\Theta\) this barrier disappears), as a result of which reorientation of the regions in an external field takes place comparatively rapidly. Other features distinguishing ferroelectrics from ordinary pyroelectrics are also due to the presence of a phase transition of the indicated type and have already been discussed above. At a sufficiently low temperature ferroelectrics no longer differ practically in any way from ordinary pyroelectrics (we are speaking, of course, of ferroelectrics for which \(P_0\ne 0\) as \(T\to 0\)).

The theory of ferroelectric phenomena developed in §§ 1—4 is, in essence, phenomenological. The task of a molecular (microscopic) theory is to calculate the electrical properties of a ferroelectric on the basis of a consideration of its structure and its possible changes as functions of temperature and of the strength of the external field. In doing this, of course, as in almost all other questions of the theory of the solid state, it is necessary to use a number of

of simplifying assumptions and the use of empirical data. The simplifying assumptions in question consist in admitting that the ions may be treated as point-like, in neglecting lattice vibrations, in considering the motion only of ions of one type (for example, protons in the case of Rochelle salt), in a given field of other ions, etc. As for the structure of the crystal, the magnitudes of the potential barriers, and the factor of the internal field, all these characteristics are extracted from various experimental data. With this approach the molecular theory proves, in essence, to be very close to the simple Weiss theory, leading to equation (61). A molecular theory of this type has been developed for all the most important ferroelectrics (see^13 for KH$_2$PO$_4$,^14 for Rochelle salt, and^15,16 for BaTiO$_3$). The corresponding constructions, on which we shall not dwell here, have a certain value, but it is far from being as great as might at first sight appear. The point is that all the consequences of the molecular theory which can be obtained on the basis of the phenomenological theory developed above testify only to the fact that the molecular theory does not contradict thermodynamics and a number of simple assumptions, such as the assumption that the thermodynamic potential can be expanded in a series in powers of $P^2$, etc. The real success of the molecular theory can therefore consist only in calculating the functions $\alpha(T)$, $\beta(T)$, and in general all the coefficients entering into the phenomenological relations. Meanwhile, precisely in this direction the achievements of the molecular theory are very modest, since in its present form, in order to determine the coefficients entering the theory, one usually has to use the experimental values of the same functions $\alpha(T)$ and $\beta(T)$ or, what is the same thing, of the function $P_0(T)$ (see, for example,^14). Let us also note that the piezoeffect is usually not taken into account within the framework of the molecular theory and should lead to its substantial complication. Despite all that has been said, molecular theory, even when developed in a relatively primitive form, is, of course, of undoubted interest, since it leads to the construction of a certain model of the ferroelectric under consideration and may have heuristic significance from the standpoint of creating new or changing the properties of known ferroelectrics.

For a better understanding of the properties of ferroelectrics it is useful to compare them with the properties of ferromagnets. For ferromagnets, as is known, the presence of a Curie point is typical, below which there is a spontaneous magnetization $M_0$ in the crystal. In the absence of an external field, a ferromagnet below the Curie point breaks up into domains magnetized in different ways. Further, at the Curie point a jump in the heat capacity is observed, and this point itself is a phase-transition point of the second order. And finally, near the Curie point the magnetic susceptibility obeys the Curie–Weiss law. Thus, ferromagnets are a complete analogue of ferroelectrics or, if one wishes,

on the contrary, ferroelectrics are “ferromagnets” in the domain of electrical phenomena (it is not for nothing that ferroelectrics are often called “ferroelectrics”).

The theory of ferromagnetism is usually developed at once in molecular or, more precisely, quasimolecular form—such is the well-known Weiss theory. However, it is quite possible and useful to consider the behavior of ferromagnets near the Curie point within the framework of a phenomenological (thermodynamic) theory[^1], similar to that which was developed for ferroelectrics in §§ 1–4 of this article.* For this purpose it is sufficient to replace, in the initial equation (1), the polarization \(P\) by the magnetization \(M\), and the electric field \(E\) by the magnetic field \(H\), as a result of which the thermodynamic potential of a ferromagnet is written in the form

\[ \Phi=\Phi_0+\alpha M^2+\frac{\beta}{2}M^4-MH, \tag{63} \]

where, for simplicity, just as in § 1, the vectors \(\mathbf H\) and \(\mathbf M\) are assumed parallel and anisotropy is not taken into account. The relation between the magnetic field and the magnetization is obtained immediately by applying the formula \(M=-\dfrac{d\Phi}{dH}\). In exactly the same way as in § 1, from (63) all the other relations of interest to us can be obtained. Such an approach does not, in essence, lead to any new results in comparison with the Weiss theory, but it makes it possible to separate reliable thermodynamic results from conclusions connected with particular concrete assumptions of the Weiss theory. For example, in the Weiss theory it is assumed that the acting field is equal to

\[ H_{ef}=H+fM, \tag{64} \]

where the internal-field factor \(f\) is constant, i.e. does not depend on temperature. Meanwhile, the formulae obtained under this assumption are not in agreement with experiment, and in order to remove the corresponding discrepancy one must assume that \(f=f(T)\).

Near the Curie point the Weiss theory and the thermodynamic theory lead to identical results if one sets

\[ f=2\Theta\alpha_\Theta', \]

\[ -\frac{fL'''(0)}{12M_\infty^2[L'(0)]^3}=\beta_\Theta, \tag{65} \]

* The author takes this opportunity to point out that after the publication of article[^1] it became known to him that an analogous treatment had been undertaken earlier by S. V. Vonsovskii (see[^11] § 24).

where

\[ a_\Theta=\left(\frac{da}{dT}\right)_\Theta, \]

\(L(a)\) is the Langevin function,

\[ L'(0)=\left(\frac{dL}{da}\right)_{a=0}, \]

\[ L'''(0)=\left(\frac{d^3L}{da^3}\right)_{a=0} \]

and \(M_\infty\) is the magnetization at \(T=0\). At the same time, from the very construction of the thermodynamic theory in question it is clear that it is applicable only near the Curie point and, consequently, only in this region is \(f=2\Theta a'_\Theta=\mathrm{const}\). Away from the Curie point, however, there are no grounds for the constancy of \(f\), and this constancy is not observed experimentally. An analogous remark may be made with respect to the Curie–Weiss law and other relations.

Weiss’s theory for ferromagnets and its quantum-mechanical generalizations are analogous to the molecular theories for ferroelectrics mentioned above. In ferromagnets, however, molecular theory is of much greater importance than in ferroelectrics, for the reason that ferromagnetism is a quantum effect, whereas ferroelectric polarization is entirely explainable within the framework of classical theory. Indeed, as we have seen, allowance for the internal electric field, i.e. ultimately allowance for the classical electrostatic interaction, makes it possible without difficulty to estimate correctly the Curie temperature in ferroelectrics. On the contrary, in order to explain the observed values of the Curie temperature \(\Theta\) in ferromagnets (the factor \(f\) in (64) must take values \(\sim 10^4\), which is quite inexplicable if only magnetic interaction is taken into account*). As is known, the parallel orientation of electron spins in ferromagnets is caused not by magnetic forces, but by the so-called exchange interaction (i.e. an electrostatic interaction that appears only in the quantum-mechanical treatment of electron motion).

Despite the far-reaching analogy between ferroelectrics and ferromagnets, in certain respects these two classes of substances differ substantially from one another. Namely, in ferromagnets, at least in the absence of strong stresses and inhomogene—

*) As follows from Weiss’s theory,

\[ \Theta\sim \frac{\mu f M_\infty}{kT}, \]

where \(\mu\) is the Bohr magneton. Since \(\Theta\sim 10^3\), \(M_\infty\sim 10^3\), and \(\mu\sim 10^{-20}\), the factor \(f\) must have just the value \(\sim 10^4\). In the electric case, according to formula (62), \(f\sim 1\) for \(\Theta\sim 10^3\), since \(p\sim 10^{-18}\sim 100\mu\) and \(P_\infty\sim 10^4 \div 10^5 \sim 10 \div 100\,M_\infty\).

properties, reorientation of domains occurs comparatively easily at all temperatures, with the exception of temperatures immediately adjacent to absolute zero. Therefore, if one does not speak of especially “hard” permanent magnets and, in general, of manifestly nonequilibrium cases, in the region of magnetic phenomena we have no analogue of pyroelectrics (i.e., we have no “pyromagnetics”*). The ease of reorientation of domains in ferromagnets in comparison with the case of pyroelectrics (or ferroelectrics far from the Curie point) is explained as follows. In pyroelectrics (and in ferroelectrics at low temperatures) the change of polarization $\mathbf{P}$ into $-\mathbf{P}$ is connected with the necessity of overcoming a high potential barrier, to which a temperature $\Theta_0 \sim 10^4$ degrees corresponds (see above).

In ferromagnets, however, the direction of the vector $\mathbf{M}$ is determined only by the relatively weak magnetic interaction, whereas the strong exchange interaction does not depend on the orientation of the vector $\mathbf{M}^{11}$. In order of magnitude the magnetic energy in ferromagnets, referred to one electron (magnetic moment), is equal to $\dfrac{\mu^2}{d^3}$, where the magnetic moment $\mu \sim 10^{-20}$ and $d \sim 10^{-8}$ is the distance between the moments; the temperature corresponding to such an interaction is $\Theta_0 = \dfrac{\mu^2}{d^3 k} \sim 1^\circ$. Replacement of the magnetization $\mathbf{M}$ by the magnetization $-\mathbf{M}$, in view of what has been said, requires overcoming only magnetic forces and, for $\Theta \gg \Theta_0 \sim 1^\circ$, occurs comparatively rapidly, even under the influence of a weak external magnetic field**). The process of technical magnetization, as is known, is completely reducible precisely to a change in the domain structure of the ferromagnet. Induced magnetization, i.e. a change of the very value of $M$ within an individual domain under the influence of an external field, at least far from the Curie point, plays no role.

In ferroelectrics, however—and herein lies their second difference from ferromagnets—the induced polarization is of primary importance. The latter is connected both with the relative difficulty of reorientation of domains and with the large value of the dielectric—

*) A crystal possessing spontaneous magnetization, the distribution of which in a specimen (i.e. the domain structure) cannot be changed by accessible or, in any case, not too strong magnetic fields, could be called a pyromagnetic.

**) Replacement of $\mathbf{P}$ by $-\mathbf{P}$ can be achieved as a result of inversion of the lattice with respect to some center, i.e. replacement of all radius vectors of the lattice nodes $\mathbf{r}$ by $-\mathbf{r}$. Such inversion, naturally, is connected with overcoming large electrical forces. Replacement of $\mathbf{M}$ by $-\mathbf{M}$ is achieved as a result of reversal of the spin directions or, in the case of orbital magnetic moments, as a result of changing the velocities of all electrons $\mathbf{v}$ to $-\mathbf{v}$. Reorientation of magnetic moments is opposed only by weak magnetic forces, and it therefore occurs relatively easily.

electric constant in ferroelectrics. For example, the values of \(\varepsilon\) above the Curie point, but near it, may reach many thousands, whereas in ferromagnetics the values of \(\mu\) above the Curie point practically do not exceed \(\mu \simeq 2\). It is not difficult to see the reason for this difference, using the Weiss theory for orientation, both for ferroelectrics and for ferromagnetics. Within this theory, for the dielectric susceptibility \(\chi_\varepsilon = \dfrac{\varepsilon - 1}{4\pi}\) and the magnetic susceptibility \(\chi_\mu = \dfrac{\mu - 1}{4\pi}\) above the Curie point we obtain:

\[ \chi_\varepsilon = \frac{pP_\infty L'(0)}{k(T-\Theta)}, \qquad \chi_\mu = \frac{\mu M_\infty L'(0)}{k(T-\Theta)}, \tag{66} \]

where the derivative of the Langevin function \(L'(0) \sim 1\) (for example, \(L'_\infty(0)=\)

\[ = \frac{1}{3} \, . \]

Putting \(p \sim 10^{-18}\) and \(P_\infty \sim 10^4\), for \(T-\Theta \sim 1^\circ\) we obtain for \(\chi_\varepsilon\) the value \(\chi_\varepsilon \sim 10^2\) and \(\varepsilon \sim 10^3\). In the magnetic case, however, for \(\mu \sim 10^{-20}\), \(M_\infty \sim 10^3\), and \(T-\Theta \sim 1^\circ\), \(\chi_\mu \sim 0.1\) and \(\mu \simeq 2\). Thus, the smallness of \(\mu\) in comparison with \(\varepsilon\) is due to the smallness of the magnetic moment of the electron in comparison with the electric moment of a molecule or of a crystal cell, and also to the smallness of \(M_\infty\) in comparison with \(P_\infty\).

The indicated differences between ferroelectrics and ferromagnetics in no way can alter the conclusion reached earlier concerning the profound analogy existing between substances of both classes. This circumstance, as well as the ever-increasing practical importance of ferroelectrics, makes obsolete the neglect of ferroelectrics observed in all courses and monographs on electrodynamics and electron theory, while ferromagnetics are invariably given much attention. At present there are already all grounds—and it may be thought that in the near future these grounds will become still more weighty—for presenting the theory of ferroelectrics in parallel with the theory of ferromagnetics, or, in any case, for treating both of these cases, and not only one of them. Turning to courses in field theory (see, for example,\(^{12}\), §108, etc.), it is easy to see that the solution of this problem encounters no difficulties whatever and, moreover, gives the entire discussion of the properties of dielectrics and magnetics a certain completeness.

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